atmospheric optics - wiley-vch · 2003-12-12 · 8.1 sun dogs and halos 83 9 clouds 86 9.1 cloud...

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Atmospheric Optics

Craig F. BohrenPennsylvania State University, Department of Meteorology, University Park,Pennsylvania, USAPhone: (814) 466-6264; Fax: (814) 865-3663; e-mail: [email protected]

AbstractColors of the sky and colored displays in the sky are mostly a consequence of selectivescattering by molecules or particles, absorption usually being irrelevant. Molecularscattering selective by wavelength – incident sunlight of some wavelengths beingscattered more than others – but the same in any direction at all wavelengths givesrise to the blue of the sky and the red of sunsets and sunrises. Scattering by particlesselective by direction – different in different directions at a given wavelength – gives riseto rainbows, coronas, iridescent clouds, the glory, sun dogs, halos, and other ice-crystaldisplays. The size distribution of these particles and their shapes determine what isobserved, water droplets and ice crystals, for example, resulting in distinct displays.

To understand the variation and color and brightness of the sky as well as thebrightness of clouds requires coming to grips with multiple scattering: scatterers inan ensemble are illuminated by incident sunlight and by the scattered light fromeach other. The optical properties of an ensemble are not necessarily those of itsindividual members.

Mirages are a consequence of the spatial variation of coherent scattering (refraction)by air molecules, whereas the green flash owes its existence to both coherent scatteringby molecules and incoherent scattering by molecules and particles.

Keywordssky colors; mirages; green flash; coronas; rainbows; the glory; sun dogs; halos; visibility.

1 Introduction 542 Color and Brightness of Molecular Atmosphere 552.1 A Brief History 55

54 Atmospheric Optics

2.2 Molecular Scattering and the Blue of the Sky 572.3 Spectrum and Color of Skylight 582.4 Variation of Sky Color and Brightness 592.5 Sunrise and Sunset 623 Polarization of Light in a Molecular Atmosphere 633.1 The Nature of Polarized Light 633.2 Polarization by Molecular Scattering 644 Scattering by Particles 664.1 The Salient Differences between Particles and Molecules:

Magnitude of Scattering 664.2 The Salient Differences between Particles and Molecules:

Wavelength Dependence of Scattering 674.3 The Salient Differences between Particles and Molecules:

Angular Dependence of Scattering 684.4 The Salient Differences between Particles and Molecules:

Degree of Polarization of Scattered Light 694.5 The Salient Differences between Particles and Molecules:

Vertical Distributions 705 Atmospheric Visibility 716 Atmospheric Refraction 736.1 Physical Origins of Refraction 736.2 Terrestrial Mirages 736.3 Extraterrestrial Mirages 766.4 The Green Flash 777 Scattering by Single Water Droplets 787.1 Coronas and Iridescent Clouds 787.2 Rainbows 807.3 The Glory 828 Scattering by Single Ice Crystals 838.1 Sun Dogs and Halos 839 Clouds 869.1 Cloud Optical Thickness 869.2 Givers and Takers of Light 87

Glossary 89References 90Further Reading 90

1Introduction

Atmospheric optics is nearly synonymouswith light scattering, the only restrictionsbeing that the scatterers inhabit the

atmosphere and the primary source oftheir illumination is the sun. Essentiallyall light we see is scattered light, even thatdirectly from the sun. When we say thatsuch light is unscattered we really meanthat it is scattered in the forward direction;

Atmospheric Optics 55

hence it is as if it were unscattered.Scattered light is radiation from matterexcited by an external source. When thesource vanishes, so does the scattered light,as distinguished from light emitted bymatter, which persists in the absence ofexternal sources.

Atmospheric scatterers are either mole-cules or particles. A particle is an aggrega-tion of sufficiently many molecules thatit can be ascribed macroscopic proper-ties such as temperature and refractiveindex. There is no canonical number ofmolecules that must unite to form abona fide particle. Two molecules clearlydo not a quorum make, but what about10, 100, 1000? The particle size corre-sponding to the largest of these numbersis about 10−3 µm. Particles this smallof water substance would evaporate sorapidly that they could not exist long underconditions normally found in the atmo-sphere. As a practical matter, therefore,we need not worry unduly about scatterersin the shadow region between moleculeand particle.

A property of great relevance to scat-tering problems is coherence, both of thearray of scatterers and of the incident light.At visible wavelengths, air is an array ofincoherent scatterers: the radiant powerscattered by N molecules is N times thatscattered by one (except in the forwarddirection). But when water vapor in aircondenses, an incoherent array is trans-formed into a coherent array: uncorrelatedwater molecules become part of a singleentity. Although a single droplet is a coher-ent array, a cloud of droplets taken togetheris incoherent.

Sunlight is incoherent but not in anabsolute sense. Its lateral coherence lengthis tens of micrometers, which is whywe can observe what are essentiallyinterference patterns (e.g., coronas and

glories) resulting from illumination ofcloud droplets by sunlight.

This article begins with the colorand brightness of a purely molecu-lar atmosphere, including their variationacross the vault of the sky. This nat-urally leads to the state of polarizationof skylight. Because the atmosphere israrely, if ever, entirely free of particles,the general characteristics of scatteringby particles follow, setting the stagefor a discussion of atmospheric visibil-ity.

Atmospheric refraction usually sits byitself, unjustly isolated from all those at-mospheric phenomena embraced by theterm scattering. Yet refraction is anothermanifestation of scattering, coherent scat-tering in the sense that phase differencescannot be ignored.

Scattering by single water droplets andice crystals, each discussed in turn, yieldsfeasts for the eye as well as the mind. Thecurtain closes on the optical propertiesof clouds.

2Color and Brightness of MolecularAtmosphere

2.1A Brief History

Edward Nichols began his 1908 presiden-tial address to the New York meeting of theAmerican Physical Society as follows: ‘‘Inasking your attention to-day, even briefly,to the consideration of the present state ofour knowledge concerning the color of thesky it may be truly said that I am invitingyou to leave the thronged thoroughfaresof our science for some quiet side streetwhere little is going on and you may evensuspect that I am coaxing you into some


blind alley, the inhabitants of which belongto the dead past.’’

Despite this depreciatory statement,hoary with age, correct and completeexplanations of the color of the sky stillare hard to find. Indeed, all the faultyexplanations lead active lives: the bluesky is the reflection of the blue sea;it is caused by water, either vapor ordroplets or both; it is caused by dust.The true cause of the blue sky is notdifficult to understand, requiring only abit of critical thought stimulated by beliefin the inherent fascination of all naturalphenomena, even those made familiar byeveryday occurrence.

Our contemplative prehistoric ancestorsno doubt speculated on the origin of theblue sky, their musings having vanishedinto it. Yet it is curious that Aristotle, themost prolific speculator of early recordedhistory, makes no mention of it in hisMeteorologica even though he deliveredpronouncements on rainbows, halos, andmock suns and realized that ‘‘the sunlooks red when seen through mist orsmoke.’’ Historical discussions of the bluesky sometimes cite Leonardo as the first tocomment intelligently on the blue of thesky, although this reflects a European bias.If history were to be written by a supremelydisinterested observer, Arab philosopherswould likely be given more credit forhaving had profound insights into theworkings of nature many centuries beforetheir European counterparts descendedfrom the trees. Indeed, Moller [1] beginshis brief history of the blue sky withJakub Ibn Ishak Al Kindi (800–870), whoexplained it as ‘‘a mixture of the darknessof the night with the light of the dust andhaze particles in the air illuminated bythe sun.’’

Leonardo was a keen observer of light innature even if his explanations sometimes

fell short of the mark. Yet his hypothesisthat ‘‘the blueness we see in the atmo-sphere is not intrinsic color, but is causedby warm vapor evaporated in minute andinsensible atoms on which the solar raysfall, rendering them luminous against theinfinite darkness of the fiery sphere whichlies beyond and includes it’’ would, withminor changes, stand critical scrutiny to-day. If we set aside Leonardo as sui generis,scientific attempts to unravel the origins ofthe blue sky may be said to have begun withNewton, that towering pioneer of optics,who, in time-honored fashion, reduced itto what he already had considered: inter-ference colors in thin films. Almost twocenturies elapsed before more pieces inthe puzzle were contributed by the exper-imental investigations of von Brucke andTyndall on light scattering by suspensionsof particles. Around the same time Clau-sius added his bit in the form of a theorythat scattering by minute bubbles causesthe blueness of the sky. A better theorywas not long in coming. It is associatedwith a man known to the world as LordRayleigh even though he was born JohnWilliam Strutt.

Rayleigh’s paper of 1871 marks thebeginning of a satisfactory explanationof the blue sky. His scattering law, thekey to the blue sky, is perhaps themost famous result ever obtained bydimensional analysis. Rayleigh argued thatthe field Es scattered by a particle smallcompared with the light illuminating itis proportional to its volume V andto the incident field Ei. Radiant energyconservation requires that the scatteredfield diminish inversely as the distancer from the particle so that the scatteredpower diminishes as the square of r. Tomake this proportionality dimensionallyhomogeneous requires the inverse squareof a quantity with the dimensions of


length. The only plausible physical variableat hand is the wavelength of the incidentlight, which leads to

Es ∝ EiV

rλ2 . (1)

When the field is squared to ob-tain the scattered power, the result isRayleigh’s inverse fourth-power law. Thislaw is really only an often – but notalways – very good approximation. Miss-ing from it are dimensionless proper-ties of the particle such as its refractiveindex, which itself depends on wave-length. Because of this dispersion, there-fore, nothing scatters exactly as the inversefourth power.

Rayleigh’s 1871 paper did not give thecomplete explanation of the color andpolarization of skylight. What he did thatwas not done by his predecessors was togive a law of scattering, which could beused to test quantitatively the hypothesisthat selective scattering by atmosphericparticles could transform white sunlightinto blue skylight. But as far as givingthe agent responsible for the blue sky isconcerned, Rayleigh did not go essentiallybeyond Newton and Tyndall, who invokedparticles. Rayleigh was circumspect aboutthe nature of these particles, settling onsalt as the most likely candidate. It was notuntil 1899 that he published the capstoneto his work on skylight, arguing that airmolecules themselves were the source ofthe blue sky. Tyndall cannot be giventhe credit for this because he consideredair to be optically empty: when purgedof all particles it scatters no light. Thiserroneous conclusion was a result of thesmall scale of his laboratory experiments.On the scale of the atmosphere, sufficientlight is scattered by air molecules to bereadily observable.

2.2Molecular Scattering and the Blue of the Sky

Our illustrious predecessors all gave ex-planations of the blue sky requiring thepresence of water in the atmosphere:Leonardo’s ‘‘evaporated warm vapor,’’Newton’s ‘‘Globules of water,’’ Clausius’sbubbles. Small wonder, then, that waterstill is invoked as the cause of the bluesky. Yet a cause of something is that with-out which it would not occur, and the skywould be no less blue if the atmospherewere free of water.

A possible physical reason for attributingthe blue sky to water vapor is that, becauseof selective absorption, liquid water (andice) is blue upon transmission of whitelight over distances of order meters. Yetif all the water in the atmosphere at anyinstant were to be compressed into a liquid,the result would be a layer about 1 cm thick,which is not sufficient to transform whitelight into blue by selective absorption.

Water vapor does not compensate forits hundredfold lower abundance thannitrogen and oxygen by greater scatteringper molecule. Indeed, scattering of visiblelight by a water molecule is slightly lessthan that by either nitrogen or oxygen.

Scattering by atmospheric moleculesdoes not obey Rayleigh’s inverse fourth-power law exactly. A least-squares fit overthe visible spectrum from 400 to 700 nmof the molecular scattering coefficient of sea-level air tabulated by Penndorf [2] yields aninverse 4.089th-power scattering law.

The molecular scattering coefficient β,which plays important roles in followingsections, may be written

β = Nσs, (2)

where N is the number of moleculesper unit volume and σs, the scatteringcross section (an average because air is


a mixture) per molecule, approximatelyobeys Rayleigh’s law. The form of thisexpression betrays the incoherence ofscattering by atmospheric molecules. Theinverse of β is interpreted as the scatteringmean free path, the average distance aphoton must travel before being scattered.

To say that the sky is blue becauseof Rayleigh scattering, as is sometimesdone, is to confuse an agent with alaw. Moreover, as Young [3] pointed out,the term Rayleigh scattering has manymeanings. Particles small compared withthe wavelength scatter according to thesame law as do molecules. Both canbe said to be Rayleigh scatterers, butonly molecules are necessary for the bluesky. Particles, even small ones, generallydiminish the vividness of the blue sky.

Fluctuations are sometimes trumpetedas the ‘‘real’’ cause of the blue sky. Pre-sumably, this stems from the fluctuationtheory of light scattering by media in whichthe scatterers are separated by distancessmall compared with the wavelength. Inthis theory, which is associated with Ein-stein and Smoluchowski, matter is takento be continuous but characterized by arefractive index that is a random functionof position. Einstein [4] stated that ‘‘it isremarkable that our theory does not makedirect use of the assumption of a discretedistribution of matter.’’ That is, he cir-cumvented a difficulty but realized it couldhave been met head on, as Zimm [5] didyears later.

The blue sky is really caused by scat-tering by molecules – to be more precise,scattering by bound electrons: free elec-trons do not scatter selectively. Because airmolecules are separated by distances smallcompared with the wavelengths of visiblelight, it is not obvious that the power scat-tered by such molecules can be added. Yetif they are completely uncorrelated, as in

an ideal gas (to good approximation theatmosphere is an ideal gas), scattering byN molecules is N times scattering by one.This is the only sense in which the blue skycan be attributed to scattering by fluctua-tions. Perfectly homogeneous matter doesnot exist. As stated pithily by Planck, ‘‘achemically pure substance may be spo-ken of as a vacuum made turbid by thepresence of molecules.’’

2.3Spectrum and Color of Skylight

What is the spectrum of skylight? What isits color? These are two different questions.Answering the first answers the secondbut not the reverse. Knowing the color ofskylight we cannot uniquely determine itsspectrum because of metamerism: A givenperceived color can in general be obtainedin an indefinite number of ways.

Skylight is not blue (itself an impre-cise term) in an absolute sense. Whenthe visible spectrum of sunlight outsidethe earth’s atmosphere is modulated byRayleigh’s scattering law, the result is aspectrum of scattered light that is nei-ther solely blue nor even peaked in theblue (Fig. 1). Although blue does not pre-dominate spectrally, it does predominateperceptually. We perceive the sky to beblue even though skylight contains light ofall wavelengths.

Any source of light may be looked uponas a mixture of white light and light ofa single wavelength called the dominantwavelength. The purity of the source isthe relative amount of the monochromaticcomponent in the mixture. The dominantwavelength of sunlight scattered accordingto Rayleigh’s law is about 475 nm, whichlies solidly in the blue if we take thisto mean light with wavelengths between450 and 490 nm. The purity of this


Fig. 1 Rayleigh’s scattering law (dots), thespectrum of sunlight outside the Earth’satmosphere (dashes), and the product of the two(solid curve). The solar spectrum is taken fromThekaekara, M. P., Drummond, A. J. (1971), Nat.Phys. Sci. 229, 6–9 [6]

scattered light, about 42%, is the upperlimit for skylight. Blues of real skies areless pure.

Another way of conveying the color of asource of light is by its color temperature,the temperature of a blackbody having thesame perceived color as the source. Sinceblackbodies do not span the entire gamutof colors, not all sources of light can beassigned color temperatures. But manynatural sources of light can. The colortemperature of light scattered accordingto Rayleigh’s law is infinite. This followsfrom Planck’s spectral emission functionebλ in the limit of high temperature,

ebλ ≈ 2πckT

λ4 ,hc

λ kT, (3)

where h is Planck’s constant, k is Boltz-mann’s constant, c is the speed of lightin vacuo, and T is absolute temperature.Thus, the emission spectrum of a black-body with an infinite temperature has thesame functional form as Rayleigh’s scat-tering law.

2.4Variation of Sky Color and Brightness

Not only is skylight not pure blue,but its color and brightness vary acrossthe vault of the sky, with the bestblues at zenith. Near the astronomicalhorizon the sky is brighter than overheadbut of considerably lower purity. Thatthis variation can be observed from anairplane flying at 10 km, well abovemost particles, suggests that the skyis inherently nonuniform in color andbrightness (Fig. 2). To understand whyrequires invoking multiple scattering.

Multiple scattering gives rise to observ-able phenomena that cannot be explainedsolely by single-scattering arguments. Thisis easily demonstrated. Fill a blackened pan

Fig. 2 Even at an altitude of 10 km, well abovemost particles, the sky brightness increasesmarkedly from the zenith to theastronomical horizon


with clean water, then add a few drops ofmilk. The resulting dilute suspension il-luminated by sunlight has a bluish cast.But when more milk is added, the suspen-sion turns white. Yet the properties of thescatterers (fat globules) have not changed,only their optical thickness: the blue suspen-sion being optically thin, the white beingoptically thick.

Optical thickness is physical thicknessin units of scattering mean free path, andhence is dimensionless. The optical thick-ness τ between any two points connectedby an arbitrary path in a medium populatedby (incoherent) scatterers is an integralover the path:

τ =∫ 2

1β ds. (4)

The normal optical thickness τn of theatmosphere is that along a radial pathextending from the surface of the Earthto infinity. Figure 3 shows τn over thevisible spectrum for a purely molecularatmosphere. Because τn is generally smallcompared with unity, a photon fromthe sun traversing a radial path in theatmosphere is unlikely to be scatteredmore than once. But along a tangential

Fig. 3 Normal optical thickness of a puremolecular atmosphere

path, the optical thickness is about 35 timesgreater (Fig. 4), which leads to severalobservable phenomena.

Even an intrinsically black object isluminous to an observer because ofairlight, light scattered by all the moleculesand particles along the line of sightfrom observer to object. Provided thatthis is uniformly illuminated by sunlightand that ground reflection is negligi-ble, the airlight radiance L is approxi-mately

L = GL0(1 − e−τ ), (5)

where L0 is the radiance of incidentsunlight along the line of sight with opticalthickness τ . The term G accounts forgeometric reduction of radiance becauseof scattering of nearly monodirectionalsunlight in all directions. If the line ofsight is uniform in composition, τ = βd,where β is the scattering coefficient and dis the physical distance to the black object.

If τ is small (1), L ≈ GL0τ . In apurely molecular atmosphere, τ varieswith wavelength according to Rayleigh’slaw; hence the distant black object insuch an atmosphere is perceived to be

Fig. 4 Optical thickness (relative to the normaloptical thickness) of a molecular atmospherealong various paths with zenith angles between0 (normal) and 90 (tangential)


bluish. As τ increases so does L but notproportionally. Its limit is GL0: The airlightradiance spectrum is that of the source ofillumination. Only in the limit d = 0 isL = 0 and the black object truly black.

Variation of the brightness and color ofdark objects with distance was called aerialperspective by Leonardo. By means of it weestimate distances to objects of unknownsize such as mountains.

Aerial perspective belongs to the samefamily as the variation of color andbrightness of the sky with zenith angle.Although the optical thickness along a pathtangent to the Earth is not infinite, it issufficiently large (Figs. 3 and 4) that GL0

is a good approximation for the radiance ofthe horizon sky. For isotropic scattering (acondition almost satisfied by molecules),G is around 10−5, the ratio of the solidangle subtended by the sun to the solidangle of all directions (4π ). Thus, thehorizon sky is not nearly so bright asdirect sunlight.

Unlike in the milk experiment, whatis observed when looking at the hori-zon sky is not multiply scattered light.Both have their origins in multiple scat-tering but manifested in different ways.Milk is white because it is weakly ab-sorbing and optically thick, and hence allcomponents of incident white light aremultiply scattered to the observer eventhough the blue component traverses ashorter average path in the suspensionthan the red component. White horizonlight has escaped being multiply scat-tered, although multiple scattering is whythis light is white (strictly, has the spec-trum of the source). More light at theshort-wavelength end of the spectrum isscattered toward the observer than at thelong-wavelength end. But long-wavelengthlight has the greater likelihood of being

transmitted to the observer without be-ing scattered out of the line of sight.For a long optical path, these two pro-cesses compensate, resulting in a hori-zon radiance spectrum which is that ofthe source.

Selective scattering by molecules is notsufficient for a blue sky. The atmospherealso must be optically thin, at least formost zenith angles (Fig. 4) (the black-ness of space as a backdrop is takenfor granted but also is necessary, asLeonardo recognized). A corollary of thisis that the blue sky is not inevitable: anatmosphere composed entirely of nonab-sorbing, selectively scattering moleculesoverlying a nonselectively reflecting earthneed not be blue. Figure 5 shows calcu-lated spectra of the zenith sky over blackground for a molecular atmosphere withthe present normal optical thickness aswell as for hypothetical atmospheres 10and 40 times thicker. What we take tobe inevitable is accidental: If our atmo-sphere were much thicker, but identicalin composition, the color of the skywould be quite different from what itis now.

Fig. 5 Spectrum of overhead skylight for thepresent molecular atmosphere (solid curve), aswell as for hypothetical atmospheres 10 (dashes)and 40 (dots) times thicker


2.5Sunrise and Sunset

If short-wavelength light is preferentiallyscattered out of direct sunlight, long-wavelength light is preferentially trans-mitted in the direction of sunlight.Transmission is described by an expo-nential law (if light multiply scatteredback into the direction of the sunlightis negligible):

L = L0e−τ , (6)

where L is the radiance at the observerin the direction of the sun, L0 is theradiance of sunlight outside the atmo-sphere, and τ is the optical thickness alongthis path.

If the wavelength dependence of τ isgiven by Rayleigh’s law, sunlight is red-dened upon transmission: The spectrumof the transmitted light is comparativelyricher than the incident spectrum inlight at the long-wavelength end of thevisible spectrum. But to say that trans-mitted sunlight is reddened is not thesame as saying it is red. The perceivedcolor can be yellow, orange, or red, de-pending on the magnitude of the opticalthickness. In a molecular atmosphere,the optical thickness along a path fromthe sun, even on or below the horizon,is not sufficient to give red light upontransmission. Although selective scatter-ing by molecules yields a blue sky, redsare not possible in a molecular atmo-sphere, only yellows and oranges. Thiscan be observed on clear days, when thehorizon sky at sunset becomes succes-sively tinged with yellow, then orange, butnot red.

Equation (6) applies to the radiance onlyin the direction of the sun. Oranges andreds can be seen in other directionsbecause reddened sunlight illuminates

scatterers not lying along the line ofsight to the sun. A striking exampleof this is a horizon sky tinged withoranges and pinks in the direction oppositethe sun.

The color and brightness of the sunchanges as it arcs across the sky becausethe optical thickness along the line of sightchanges with solar zenith angle . If theEarth were flat (as some still aver), thetransmitted solar radiance would be

L = L0eτn/ cos . (7)

This equation is a good approximationexcept near the horizon. On a flat earth,the optical thickness is infinite for horizonpaths. On a spherical earth, optical thick-nesses are finite although much larger forhorizon than for radial paths.

The normal optical thickness of anatmosphere in which the number densityof scatterers decreases exponentially withheight z above the surface, exp(−z/H), isthe same as that for a uniform atmosphereof finite thickness:

τn =∫ ∞

0β dz = β0H, (8)

where H is the scale height and β0 isthe scattering coefficient at sea level.This equivalence yields a good approx-imation even for the tangential opti-cal thickness. For any zenith angle,the optical thickness is given approxi-mately by

τ

τn=

√R2

e

H2 cos2 + 2Re

H+ 1

− Re

Hcos , (9)

where Re is the radius of the Earth.A flat earth is one for which Re isinfinite, in which instance Eq. (9) yields


the expected relation

limRe→∞

τ

τn= 1

cos . (10)

For Earth’s atmosphere, the molecularscale height is about 8 km. According tothe approximate relation Eq. (9), therefore,the horizon optical thickness is about39 times greater than the normal opticalthickness. Taking the exponential decreaseof molecular number density into accountyields a value about 10% lower.

Variations on the theme of reds andoranges at sunrise and sunset can beseen even when the sun is overhead. Theradiance at an observer an optical distanceτ from a (horizon) cloud is the sumof cloudlight transmitted to the observerand airlight:

L = L0G(1 − e−τ ) + L0Gce−τ , (11)

where Gc is a geometrical factor thataccounts for scattering of nearly monodi-rectional sunlight into a hemisphere ofdirections by the cloud. If the cloudis approximated as an isotropic reflec-tor with reflectance R and illuminatedat an angle , the geometrical factorGc is sR cos/π , where s is thesolid angle subtended by the sun atthe Earth. If Gc > G, the observed ra-diance is redder (i.e., enriched in lightof longer wavelengths) than the incidentradiance. If Gc < G, the observed radi-ance is bluer than the incident radiance.Thus, distant horizon clouds can be red-dish if they are bright or bluish if theyare dark.

Underlying Eq. (11) is the implicit as-sumption that the line of sight is uniformlyilluminated by sunlight. The first termin this equation is airlight; the secondis transmitted cloudlight. Suppose, how-ever, that the line of sight is shadowed

from direct sunlight by clouds (that donot, of course, occlude the distant cloudof interest). This may reduce the firstterm in Eq. (11) so that the second termdominates. Thus, under a partly over-cast sky, distant horizon clouds may bereddish even when the sun is high inthe sky.

The zenith sky at sunset and twilightis the exception to the general rule thatmolecular scattering is sufficient to ac-count for the color of the sky. In theabsence of molecular absorption, the spec-trum of the zenith sky would be essentiallythat of the zenith sun (although greatlyreduced in radiance), hence would notbe the blue that is observed. This waspointed out by Hulburt [7], who showedthat absorption by ozone profoundly af-fects the color of the zenith sky when thesun is near the horizon. The Chappuisband of ozone extends from about 450 to700 nm and peaks at around 600 nm. Pref-erential absorption of sunlight by ozoneover long horizon paths gives the zenithsky its blueness when the sun is nearthe horizon. With the sun more thanabout 10 above the horizon, however,ozone has little effect on the color ofthe sky.

3Polarization of Light in a MolecularAtmosphere

3.1The Nature of Polarized Light

Unlike sound, light is a vector wave, anelectromagnetic field lying in a plane nor-mal to the propagation direction. Thepolarization state of such a wave is deter-mined by the degree of correlation of any


two orthogonal components into whichits electric (or magnetic) field is resolved.Completely polarized light corresponds tocomplete correlation; completely unpolar-ized light corresponds to no correlation;partially polarized light corresponds to par-tial correlation.

If an electromagnetic wave is completelypolarized, the tip of its oscillating electricfield traces out a definite elliptical curve,the vibration ellipse. Lines and circles arespecial ellipses, the light being said tobe linearly or circularly polarized, respec-tively. The general state of polarization iselliptical.

Any beam of light can be consid-ered an incoherent superposition of twocollinear beams, one unpolarized, theother completely polarized. The radianceof the polarized component relative tothe total is defined as the degree of po-larization (often multiplied by 100 andexpressed as a percentage). This can bemeasured for a source of light (e.g., lightfrom different sky directions) by rotat-ing a (linear) polarizing filter and notingthe minimum and maximum radiancestransmitted by it. The degree of (linear)polarization is defined as the differencebetween these two radiances divided bytheir sum.

3.2Polarization by Molecular Scattering

Unpolarized light can be transformed intopartially polarized light upon interactionwith matter because of different changes inamplitude of the two orthogonal field com-ponents. An example of this is the partialpolarization of sunlight upon scatteringby atmospheric molecules, which can bedetected by looking at the sky through a po-larizing filter (e.g., polarizing sunglasses)while rotating it. Waxing and waning of the

observed brightness indicates some degreeof partial polarization.

In the analysis of any scattering prob-lem, a plane of reference is required. Thisis usually the scattering plane, determinedby the directions of the incident and scat-tered waves, the angle between them beingthe scattering angle. Light polarized perpen-dicular (parallel) to the scattering plane issometimes said to be vertically (horizon-tally) polarized. Vertical and horizontal inthis context, however, are arbitrary termsindicating orthogonality and bear no rela-tion, except by accident, to the directionof gravity.

The degree of polarization P of lightscattered by a tiny sphere illuminated byunpolarized light is (Fig. 6)

P = 1 − cos2 θ

1 + cos2 θ, (12)

where the scattering angle θ ranges from0 (forward direction) to 180 (backwarddirection); the scattered light is partiallylinearly polarized perpendicular to thescattering plane. Although this equation

Fig. 6 Degree of polarization of the lightscattered by a small (compared with thewavelength) sphere for incident unpolarizedlight (solid curve). The dashed curve is for asmall spheroid chosen such that the degree ofpolarization at 90 is that for air


is a first step toward understandingpolarization of skylight, more often thannot it also has been a false step, having ledcountless authors to assert that skylight iscompletely polarized at 90 from the sun.Although P = 1 at θ = 90 according toEq. (12), skylight is never 100% polarizedat this or any other angle, and forseveral reasons.

Although air molecules are very smallcompared with the wavelengths of visiblelight, a requirement underlying Eq. (12),the dominant constituents of air are notspherically symmetric.

The simplest model of an asymmetricmolecule is a small spheroid. Althoughit is indeed possible to find a directionin which the light scattered by such aspheroid is 100% polarized, this directiondepends on the spheroid’s orientation.In an ensemble of randomly orientedspheroids, each contributes its mite to thetotal radiance in a given direction, buteach contribution is partially polarized tovarying degrees between 0 and 100%. Itis impossible for beams of light to beincoherently superposed in such a way thatthe degree of polarization of the resultantis greater than the degree of polarizationof the most highly polarized beam.

Because air is an ensemble of randomlyoriented asymmetric molecules, sunlightscattered by air never is 100% polarized.The intrinsic departure from perfection isabout 6%. Figure 6 also includes a curvefor light scattered by randomly orientedspheroids chosen to yield 94% polarizationat 90. This angle is so often singledout that it may deflect attention fromnearby scattering angles. Yet, the degree ofpolarization is greater than 50% for a rangeof scattering angles 70 wide centeredabout 90.

Equation (12) applies to air, not tothe atmosphere, the distinction being

that in the atmosphere, as opposed tothe laboratory, multiple scattering is notnegligible. Also, atmospheric air is almostnever free of particles and is illuminatedby light reflected by the ground. We musttake the atmosphere as it is, whereasin the laboratory we often can eliminateeverything we consider extraneous.

Because of both multiple scatteringand ground reflection, light from anydirection in the sky is not, in general,made up solely of light scattered in asingle direction relative to the incidentsunlight but is a superposition of beamswith different scattering histories, hencedifferent degrees of polarization. As aconsequence, even if air molecules wereperfect spheres and the atmosphere werecompletely free of particles, skylight wouldnot be 100% polarized at 90 to the sun orat any other angle.

Reduction of the maximum degree ofpolarization is not the only consequenceof multiple scattering. According to Fig. 6,there should be two neutral points in thesky, directions in which skylight is unpo-larized: directly toward and away from thesun. Because of multiple scattering, how-ever, there are three such points. Whenthe sun is higher than about 20 above thehorizon there are neutral points within 20of the sun, the Babinet point above it, theBrewster point below. They coincide whenthe sun is directly overhead and moveapart as the sun descends. When the sunis lower than 20, the Arago point is about20 above the antisolar point, the directionopposite the sun.

One consequence of the partial polar-ization of skylight is that the colors ofdistant objects may change when viewedthrough a rotated polarizing filter. If thesun is high in the sky, horizontal airlightwill have a fairly high degree of polariza-tion. According to the previous section,


airlight is bluish. But if it also is partiallypolarized, its radiance can be diminishedwith a polarizing filter. Transmitted cloud-light, however, is unpolarized. Because theradiance of airlight can be reduced morethan that of cloudlight, distant clouds maychange from white to yellow to orangewhen viewed through a rotated polariz-ing filter.

4Scattering by Particles

Up to this point we have considered only anatmosphere free of particles, an idealizedstate rarely achieved in nature. Particlesstill would inhabit the atmosphere evenif the human race were to vanish fromthe Earth. They are not simply by-products of the ‘‘dark satanic mills’’ ofcivilization.

All molecules of the same substance areessentially identical. This is not true ofparticles: They vary in shape and size,and may be composed of one or morehomogeneous regions.

4.1The Salient Differences between Particlesand Molecules: Magnitude of Scattering

The distinction between scattering bymolecules when widely separated andwhen packed together into a droplet isthat between scattering by incoherent andcoherent arrays. Isolated molecules areexcited primarily by incident (external)light, whereas the same molecules forminga droplet are excited by incident light andby each other’s scattered fields. The totalpower scattered by an incoherent array ofmolecules is the sum of their scatteredpowers. The total power scattered by acoherent array is the square of the total

scattered field, which in turn is the sumof all the fields scattered by the individualmolecules. For an incoherent array we mayignore the wave nature of light, whereasfor a coherent array we must take itinto account.

Water vapor is a good example to ponderbecause it is a constituent of air andcan condense to form cloud droplets. Thedifference between a sky containing watervapor and the same sky with the sameamount of water but in the form of a cloudof droplets is dramatic.

According to Rayleigh’s law, scatteringby a particle small compared with thewavelength increases as the sixth powerof its size (volume squared). A droplet ofdiameter 0.03 µm, for example, scattersabout 1012 times more light than does oneof its constituent molecules. Such a dropletcontains about 107 molecules. Thus,scattering per molecule as a consequenceof condensation of water vapor intoa coherent water droplet increases byabout 105.

Cloud droplets are much larger than0.03 µm, a typical diameter being about10 µm. Scattering per molecule in sucha droplet is much greater than scatter-ing by an isolated molecule, but not tothe extent given by Rayleigh’s law. Scat-tering increases as the sixth power ofdroplet diameter only when the moleculesscatter coherently in phase. If a dropletis sufficiently small compared with thewavelength, each of its molecules is ex-cited by essentially the same field andall the waves scattered by them inter-fere constructively. But when a dropletis comparable to or larger than the wave-length, interference can be constructive,destructive, and everything in between,and hence scattering does not increaseas rapidly with droplet size as predicted byRayleigh’s law.


The figure of merit for comparingscatterers of different size is their scatter-ing cross section per unit volume, which,except for a multiplicative factor, is the scat-tering cross section per molecule. A scat-tering cross section may be looked uponas an effective area for removing radiantenergy from a beam: the scattering crosssection times the beam irradiance is theradiant power scattered in all directions.

The scattering cross section per unitvolume for water droplets illuminatedby visible light and varying in sizefrom molecules (10−4 µm) to raindrops(103 µm) is shown in Fig. 7. Scattering bya molecule that belongs to a cloud droplet isabout 109 times greater than scattering byan isolated molecule, a striking exampleof the virtue of cooperation. Yet inmolecular as in human societies there arelimits beyond which cooperation becomesdysfunctional: Scattering by a moleculethat belongs to a raindrop is about 100times less than scattering by a moleculethat belongs to a cloud droplet. Thistremendous variation of scattering bywater molecules depending on their stateof aggregation has profound observationalconsequences. A cloud is optically so much

Fig. 7 Scattering (per molecule) of visible light(arbitrary units) by water droplets varying in sizefrom a single molecule to a raindrop

different from the water vapor out of whichit was born that the offspring bears noresemblance to its parents. We can seethrough tens of kilometers of air ladenwith water vapor, whereas a cloud a fewtens of meters thick is enough to occultthe sun. Yet a rainshaft born out of acloud is considerably more translucentthan its parent.

4.2The Salient Differences between Particlesand Molecules: Wavelength Dependence ofScattering

Regardless of their size and composi-tion, particles scatter approximately asthe inverse fourth power of wavelengthif they are small compared with the wave-length and absorption is negligible, twoimportant caveats. Failure to recognizethem has led to errors, such as that yel-low light penetrates fog better becauseit is not scattered as much as light ofshorter wavelengths. Although there maybe perfectly sound reasons for choosingyellow instead of blue or green as thecolor of fog lights, greater transmissionthrough fog is not one of them: Scat-tering by fog droplets is essentially in-dependent of wavelength over the visiblespectrum.

Small particles are selective scatterers;large particles are not. Particles nei-ther small nor large give the reverse ofwhat we have come to expect as nor-mal. Figure 8 shows scattering of visiblelight by oil droplets with diameters 0.1,0.8, and 10 µm. The smaller dropletsscatter according to Rayleigh’s law; thelarger droplets (typical cloud droplet size)are nonselective. Between these two ex-tremes are droplets (0.8 µm) that scatterlong-wavelength light more than short-wavelength. Sunlight or moonlight seen


Fig. 8 Scattering of visible light by oil dropletsof diameter 0.1 µm (solid curve), 0.8 µm(dashes), and 10 µm (dots)

through a thin cloud of these intermediatedroplets would be bluish or greenish. Thisrequires droplets of just the right size, andhence it is a rare event, so rare that it oc-curs once in a blue moon. Astronomers,for unfathomable reasons, refer to the sec-ond full moon in a month as a blue moon,but if such a moon were blue it would beonly by coincidence. The last reliably re-ported outbreak of blue and green sunsand moons occurred in 1950 and wasattributed to an oily smoke produced inCanadian forest fires.

4.3The Salient Differences between Particlesand Molecules: Angular Dependence ofScattering

The angular distribution of scattered lightchanges dramatically with the size ofthe scatterer. Molecules and particles thatare small compared with the wavelengthare nearly isotropic scatterers of unpo-larized light, the ratio of maximum (at0 and 180) to minimum (at 90) scat-tered radiance being only 2 for spheres,and slightly less for other spheroids. Al-though small particles scatter the same in

the forward and backward hemispheres,scattering becomes markedly asymmetricfor particles comparable to or larger thanthe wavelength. For example, forward scat-tering by a water droplet as small as 0.5 µmis about 100 times greater than backwardscattering, and the ratio of forward to back-ward scattering increases more or lessmonotonically with size (Fig. 9).

The reason for this asymmetry is foundin the singularity of the forward direc-tion. In this direction, waves scatteredby two or more scatterers excited solelyby incident light (ignoring mutual ex-citation) are always in phase regardlessof the wavelength and the separationof the scatterers. If we imagine a par-ticle to be made up of N small sub-units, scattering in the forward direc-tion increases as N2, the only direc-tion for which this is always true. Forother directions, the wavelets scattered bythe subunits will not necessarily all bein phase. As a consequence, scatteringin the forward direction increases withsize (i.e., N) more rapidly than in anyother direction.

Fig. 9 Angular dependence of scattering ofvisible light (0.55 µm) by water droplets smallcompared with the wavelength (dashes),diameter 0.5 µm (solid curve), and diameter10 µm (dots)


Many common observable phenom-ena depend on this forward-backwardasymmetry. Viewed toward the illumi-nating sun, glistening fog droplets on aspider’s web warn us of its presence. Butwhen we view the web with our backsto the sun, the web mysteriously disap-pears. A pattern of dew illuminated bythe rising sun on a cold morning seemsetched on a windowpane. But if we gooutside to look at the window, the patternvanishes. Thin clouds sometimes hoverover warm, moist heaps of dung, but maygo unnoticed unless they lie between usand the source of illumination. These arebut a few examples of the consequencesof strongly asymmetric scattering by sin-gle particles comparable to or larger thanthe wavelength.

4.4The Salient Differences between Particlesand Molecules: Degree of Polarization ofScattered Light

All the simple rules about polarizationupon scattering are broken when we turnfrom molecules and small particles toparticles comparable to the wavelength.For example, the degree of polarization oflight scattered by small particles is a simplefunction of scattering angle. But simplicitygives way to complexity as particles grow(Fig. 10), the scattered light being partiallypolarized parallel to the scattering planefor some scattering angles, perpendicularfor others.

The degree of polarization of lightscattered by molecules or by small particlesis essentially independent of wavelength.But this is not true for particles comparableto or larger than the wavelength. Scatteringby such particles exhibits dispersion ofpolarization: The degree of polarization at,

Fig. 10 Degree of polarization of light scatteredby water droplets illuminated by unpolarizedvisible light (0.55 µm). The dashed curve is for adroplet small compared with the wavelength; thesolid curve is for a droplet of diameter 0.5 µm;the dotted curve is for a droplet of diameter1.0 µm. Negative degrees of polarizationindicate that the scattered light is partiallypolarized parallel to the scattering plane

Fig. 11 Degree of polarization at a scatteringangle of 90 of light scattered by a water dropletof diameter 0.5 µm illuminated byunpolarized light

say, 90 may vary considerably over thevisible spectrum (Fig. 11).

In general, particles can act as polarizersor retarders or both. A polarizer transformsunpolarized light into partially polarizedlight. A retarder transforms polarized lightof one form into that of another (e.g.,


linear into elliptical). Molecules and smallparticles, however, are restricted to rolesas polarizers. If the atmosphere wereinhabited solely by such scatterers, skylightcould never be other than partially linearlypolarized. Yet particles comparable toor larger than the wavelength oftenare present; hence skylight can acquirea degree of ellipticity upon multiplescattering: Incident unpolarized light ispartially linearly polarized in the firstscattering event, then transformed intopartially elliptically polarized light insubsequent events.

Bees can navigate by polarized sky-light. This statement, intended to evokegreat awe for the photopolimetric pow-ers of bees, is rarely accompanied byan important caveat: The sky must beclear. Figures 10 and 11 show two rea-sons – there are others – why bees, re-markable though they may be, cannot dothe impossible. The simple wavelength-independent relation between the posi-tion of the sun and the direction inwhich skylight is most highly polarized,an underlying necessity for navigatingby means of polarized skylight, is oblit-erated when clouds cover the sky. Thiswas recognized by the decoder of beedances himself von Frisch, [8]: ‘‘Some-times a cloud would pass across the areaof sky visible through the tube; when thishappened the dances became disoriented,and the bees were unable to indicate thedirection to the feeding place. Whateverphenomenon in the blue sky served to ori-ent the dances, this experiment showedthat it was seriously disturbed if theblue sky was covered by a cloud.’’ Butvon Frisch’s words often have been for-gotten by disciples eager to spread thestory about bee magic to those just aseager to believe what is charming eventhough untrue.

4.5The Salient Differences between Particlesand Molecules: Vertical Distributions

Not only are the scattering properties ofparticles quite different, in general, fromthose of molecules; the different verticaldistributions of particles and molecules bythemselves affect what is observed. Thenumber density of molecules decreasesmore or less exponentially with heightz above the surface: exp(−z/Hm), wherethe molecular scale height Hm is around8 km. Although the decrease in numberdensity of particles with height is also ap-proximately exponential, the scale heightfor particles Hp is about 1–2 km. As aconsequence, particles contribute dispro-portionately to optical thicknesses alongnear-horizon paths. Subject to the approxi-mations underlying Eq. (9), the ratio of thetangential (horizon) optical thickness forparticles τtp to that for molecules τtm is

τtp

τtm= τnp

τnm

√Hm

Hp, (13)

where the subscript t indicates a tangentialpath and n indicates a normal (radial) path.Because of the incoherence of scatteringby atmospheric molecules and particles,scattering coefficients are additive, andhence so are optical thicknesses. For equalnormal optical thicknesses, the tangentialoptical thickness for particles is at leasttwice that for molecules. Molecules bythemselves cannot give red sunrises andsunsets; molecules need the help ofparticles. For a fixed τnp, the tangentialoptical thickness for particles is greaterthe more they are concentrated nearthe ground.

At the horizon the relative rate of changeof transmission T of sunlight with zenith


angle is1

T

dT

d= τn

Re

H, (14)

where the scale height and normal opti-cal thickness may be those for moleculesor particles. Not only do particles, be-ing more concentrated near the surface,give disproportionate attenuation of sun-light on the horizon, but they magnifythe angular gradient of attenuation there.A perceptible change in color across thesun’s disk (which subtends about 0.5)on the horizon also requires the helpof particles.

5Atmospheric Visibility

On a clear day can we really see for-ever? If not, how far can we see? Toanswer this question requires qualifyingit by restricting viewing to more or lesshorizontal paths during daylight. Starsat staggering distances can be seen atnight, partly because there is no sky-light to reduce contrast, partly becausestars overhead are seen in directionsfor which attenuation by the atmosphereis least.

The radiance in the direction of a blackobject is not zero, because of light scatteredalong the line of sight (see Sec. 2.4). Atsufficiently large distances, this airlight isindistinguishable from the horizon sky.An example is a phalanx of parallel darkridges, each ridge less distinct than thosein front of it (Fig. 12). The farthest ridgesblend into the horizon sky. Beyond somedistance we cannot see ridges because ofinsufficient contrast.

Equation (5) gives the airlight radi-ance, a radiometric quantity that de-scribes radiant power without taking into

Fig. 12 Because of scattering by molecules andparticles along the line of sight, each successiveridge is brighter than the ones in front of it eventhough all of them are covered with the samedark vegetation

account the portion of it that stimu-lates the human eye or by what relativeamount it does so at each wavelength.Luminance (also sometimes called bright-ness) is the corresponding photometricquantity. Luminance and radiance arerelated by an integral over the visi-ble spectrum:

B =∫

K(λ)L(λ) dλ, (15)

where the luminous efficiency of the hu-man eye K peaks at about 550 nm andvanishes outside the range 385–760 nm.

The contrast C between any object andthe horizon sky is

C = B − B∞B∞

, (16)

where B∞ is the luminance for an infinitehorizon optical thickness. For a uniformlyilluminated line of sight of length d,uniform in its scattering properties, and


with a black backdrop, the contrast is

C = −

∫KGL0 exp(−βd) dλ∫

KGL0 dλ

. (17)

The ratio of integrals in this equationdefines an average optical thickness:

C = − exp(−〈τ 〉). (18)

This expression for contrast reductionwith (optical) distance is mathematically,but not physically, identical to Eq. (6),which perhaps has engendered the mis-conception that atmospheric visibility isreduced because of attenuation. Yet asthere is no light from a black object to beattenuated, its finite visual range cannotbe a consequence of attenuation.

The distance beyond which a darkobject cannot be distinguished from thehorizon sky is determined by the contrastthreshold: the smallest contrast detectableby the human observer. Although thisdepends on the particular observer, theangular size of the object observed,the presence of nearby objects, and theabsolute luminance, a contrast thresholdof 0.02 is often taken as an average. Thisvalue in Eq. (18) gives

− ln |C| = 3.9 = 〈τ 〉 = 〈βd〉. (19)

To convert an optical distance into aphysical distance requires the scatteringcoefficient. Because K is peaked at around550 nm, we can obtain an approximatevalue of d from the scattering coefficientat this wavelength in Eq. (19). At sealevel, the molecular scattering coefficientin the middle of the visible spectrumcorresponds to about 330 km for ‘‘forever’’:the greatest distance at which a blackobject can be seen against the horizon

sky assuming a contrast threshold of 0.02and ignoring the curvature of the earth.

We also observe contrast between ele-ments of the same scene, a hillside mottledwith stands of trees and forest clearings,for example. The extent to which we canresolve details in such a scene depends onsun angle as well as distance.

The airlight radiance for a nonreflectingobject is Eq. (5) with G = p()s, wherep() is the probability (per unit solid angle)that light is scattered in a direction makingan angle with the incident sunlight ands is the solid angle subtended by the sun.When the sun is overhead, = 90; withthe sun at the observer’s back, = 180;for an observer looking directly into thesun, = 0.

The radiance of an object with a finite re-flectance R and illuminated at an angle

is given by Eq. (11). Equations (5) and (11)can be combined to obtain the contrast be-tween reflecting and nonreflecting objects:

C = Fe−τ

1 + (F − 1)e−τ,

F = R cos

πp(). (20)

All else being equal, therefore, contrastdecreases as p() increases. As shown inFig. 9, p() is more sharply peaked in theforward direction the larger the scatterer.Thus, we expect the details of a distantscene to be less distinct when lookingtoward the sun than away from it if theoptical thickness of the line of sight hasan appreciable component contributed byparticles comparable to or larger than thewavelength.

On humid, hazy days, visibility isoften depressingly poor. Haze, however,is not water vapor but rather waterthat has ceased to be vapor. At highrelative humidities, but still well below


100%, small soluble particles in theatmosphere accrete liquid water to becomesolution droplets (haze). Although thesedroplets are much smaller than clouddroplets, they markedly diminish visualrange because of the sharp increase inscattering with particle size (Fig. 7). Thesame number of water molecules whenaggregated in haze scatter vastly more thanwhen apart.

6Atmospheric Refraction

6.1Physical Origins of Refraction

Atmospheric refraction is a consequenceof molecular scattering, which is rarelystated given the historical accident thatbefore light and matter were well un-derstood refraction and scattering werelocked in separate compartments and sub-sequently have been sequestered morerigidly than monks and nuns in neigh-boring cloisters.

Consider a beam of light propagating inan optically homogeneous medium. Lightis scattered (weakly but observably) later-ally to this beam as well as in the directionof the beam (the forward direction). Theobserved beam is a coherent superposi-tion of incident light and forward-scatteredlight, which was excited by the incidentlight. Although refractive indices are of-ten defined by ratios of phase velocities,we may also look upon a refractive indexas a parameter that specifies the phaseshift between an incident beam and theforward-scattered beam that the incidentbeam excites. The connection between(incoherent) scattering and refraction (co-herent scattering) can be divined from theexpressions for the refractive index n of a

gas and the scattering cross section σs of agas molecule:

n = 1 + 12αN, (21)

σs = k4

6π|α|2, (22)

where N is the number density (not massdensity) of gas molecules, k = 2π/λ is thewave number of the incident light, and α

is the polarizability of a molecule (induceddipole moment per unit inducing electricfield). The appearance of the polarizabil-ity in Eq. (21) but its square in Eq. (22) isthe clue that refraction is associated withelectric fields whereas lateral scatteringis associated with electric fields squared(powers). Scattering, without qualification,often means incoherent scattering in alldirections. Refraction, in a nutshell, is co-herent scattering in a particular direction.

Readers whose appetites have beenwhetted by the preceding brief discussionof the physical origins of refraction aredirected to a beautiful paper by Doyle [9]in which he shows how the Fresnelequations can be dissected to reveal thescattering origins of (specular) reflectionand refraction.

6.2Terrestrial Mirages

Mirages are not illusions, any more sothan are reflections in a pond. Reflectionsof plants growing at its edge are notinterpreted as plants growing into thewater. If the water is ruffled by wind,the reflected images may be so distortedthat they are no longer recognizableas those of plants. Yet we still wouldnot call such distorted images illusions.And so is it with mirages. They areimages noticeably different from what theywould be in the absence of atmospheric


refraction, creations of the atmosphere,not of the mind.

Mirages are vastly more common thanis realized. Look and you shall see them.Contrary to popular opinion, they arenot unique to deserts. Mirages can beseen frequently even over ice-coveredlandscapes and highways flanked by deepsnowbanks. Temperature per se is notwhat gives mirages but rather temperaturegradients.

Because air is a mixture of gases, thepolarizability for air in Eq. (21) is anaverage over all its molecular constituents,although their individual polarizabilitiesare about the same (at visible wavelengths).The vertical refractive index gradient canbe written so as to show its dependence onpressure p and (absolute) temperature T :

d

dzln(n − 1) = 1

p

dp

dz− 1

T

dT

dz. (23)

Pressure decreases approximately ex-ponentially with height, where the scaleheight is around 8 km. Thus, the first termon the right-hand side of Eq. (23) is around0.1 km−1. Temperature usually decreaseswith height in the atmosphere. An averagelapse rate of temperature (i.e., its decreasewith height) is around 6 C/km. The aver-age temperature in the troposphere (withinabout 15 km of the surface) is around280 K. Thus, the magnitude of the secondterm in Eq. (23) is around 0.02 km−1. Onaverage, therefore, the refractive-index gra-dient is dominated by the vertical pressuregradient. But within a few meters of thesurface, conditions are far from average.On a sun-baked highway your feet maybe touching asphalt at 50 C while yournose is breathing air at 35 C, which cor-responds to a lapse rate a thousand timesthe average. Moreover, near the surface,temperature can increase with height. In

shallow surface layers, in which the pres-sure is nearly constant, the temperaturegradient determines the refractive indexgradient. It is in such shallow layers thatmirages, which are caused by refractive-index gradients, are seen.

Cartoonists by their fertile imaginationsunfettered by science, and textbook writersby their carelessness, have engenderedthe notion that atmospheric refraction canwork wonders, lifting images of ships, forexample, from the sea high into the sky.A back-of-the-envelope calculation dispelssuch notions. The refractive index of air atsea level is about 1.0003 (Fig. 13). Lightfrom empty space incident at glancingincidence onto a uniform slab with thisrefractive index is displaced in angularposition from where it would have beenin the absence of refraction by

δ = √2(n − 1). (24)

This yields an angular displacement ofabout 1.4, which as we shall see is a roughupper limit.

Trajectories of light rays in nonuniformmedia can be expressed in different ways.According to Fermat’s principle of least

Fig. 13 Sea-level refractive index versuswavelength at −15 C (dashes) and 15 C (solidcurve). Data from Penndorf, R. (1957), J. Opt.Soc. Am. 47, 176–182 [2]


time (which ought to be extreme time),the actual path taken by a ray between twopoints is such that the path integral

∫ 2

1n ds (25)

is an extremum over all possible paths.This principle has inspired piffle about thealleged efficiency of nature, which directslight over routes that minimize travel time,presumably freeing it to tend to importantbusiness at its destination.

The scale of mirages is such that inanalyzing them we may pretend that theEarth is flat. On such an earth, withan atmosphere in which the refractiveindex varies only in the vertical, Fermat’sprinciple yields a generalization

n sin θ = constant (26)

of Snel’s law, where θ is the angle betweenthe ray and the vertical direction. Wecould, of course, have bypassed Fermat’sprinciple to obtain this result.

If we restrict ourselves to nearly hori-zontal rays, Eq. (26) yields the followingdifferential equation satisfied by a ray:

d2z

dy2 = dn

dz, (27)

where y and z are its horizontal and verticalcoordinates, respectively. For a constantrefractive-index gradient, which to goodapproximation occurs for a constant tem-perature gradient, Eq. (27) yields parabolasfor ray trajectories. One such parabola fora constant temperature gradient about 100times the average is shown in Fig. 14.Note the vastly different horizontal andvertical scales. The image is displaceddownward from what it would be in theabsence of atmospheric refraction; hencethe designation inferior mirage. This is the

Fig. 14 Parabolic ray paths in an atmospherewith a constant refractive-index gradient (inferiormirage). Note the vastly different horizontal andvertical scales

familiar highway mirage, seen over high-ways warmer than the air above them. Thedownward angular displacement is

δ = 1

2s

dn

dz, (28)

where s is the horizontal distance betweenobject and observer (image). Even fora temperature gradient 1000 times thetropospheric average, displacements ofmirages are less than a degree at distancesof a few kilometers.

If temperature increases with height,as it does, for example, in air over acold sea, the resulting mirage is calleda superior mirage. Inferior and superior arenot designations of lower and higher castebut rather of displacements downwardand upward.

For a constant temperature gradient,one and only one parabolic ray tra-jectory connects an object point to animage point. Multiple images thereforeare not possible. But temperature gra-dients close to the ground are rarelylinear. The upward transport of energyfrom a hot surface occurs by molecularconduction through a stagnant boundary


layer of air. Somewhat above the surface,however, energy is transported by air inmotion. As a consequence, the tempera-ture gradient steepens toward the groundif the energy flux is constant. This vari-able gradient can lead to two observableconsequences: magnification and multi-ple images.

According to Eq. (28), all image pointsat a given horizontal distance are dis-placed downward by an amount propor-tional to the (constant) refractive indexgradient. A corollary is that the closeran object point is to a surface, wherethe temperature gradient is greatest, thegreater the downward displacement of thecorresponding image point. Thus, non-linear vertical temperature profiles maymagnify images.

Multiple images are seen frequentlyon highways. What often appears tobe water on the highway ahead butevaporates before it is reached is theinverted secondary image of either thehorizon sky or horizon objects lighter thandark asphalt.

6.3Extraterrestrial Mirages

When we turn from mirages of terrestrialobjects to those of extraterrestrial bodies,most notably the sun and moon, wecan no longer pretend that the Earthis flat. But we can pretend that theatmosphere is uniform and bounded.The total phase shift of a vertical rayfrom the surface to infinity is the samein an atmosphere with an exponentiallydecreasing molecular number density as ina hypothetical atmosphere with a uniformnumber density equal to the surface valueup to height H.

A ray refracted along a horizon pathby this hypothetical atmosphere and

originating from outside it had to havebeen incident on it from an angle δ belowthe horizon:

δ =√

2H

R−

√2H

R− 2(n − 1), (29)

where R is the radius of the Earth. Thus,when the sun (or moon) is seen to be onthe horizon it is actually more than halfwaybelow it, δ being about 0.36, whereas theangular width of the sun (or moon) isabout 0.5.

Extraterrestrial bodies seen near thehorizon also are vertically compressed. Thesimplest way to estimate the amount ofcompression is from the rate of change ofangle of refraction θr with angle of inci-dence θi for a uniform slab

dθr

dθi= cos θi√

n2 − sin2 θi

, (30)

where the angle of incidence is that fora curved but uniform atmosphere suchthat the refracted ray is horizontal. Theresult is

dθr

dθi=

√1 − R

H(n − 1), (31)

according to which the sun near thehorizon is distorted into an ellipse withaspect ratio about 0.87. We are unlikelyto notice this distortion, however, be-cause we expect the sun and moon tobe circular, and hence we see themthat way.

The previous conclusions about thedownward displacement and distortion ofthe sun were based on a refractive-indexprofile determined mostly by the verti-cal pressure gradient. Near the ground,however, the temperature gradient is theprime determinant of the refractive-indexgradient, as a consequence of which thesun on the horizon can take on shapes


Fig. 15 A nearly triangular sun on the horizon.The serrations are a consequence of horizontalvariations in refractive index

more striking than a mere ellipse. Forexample, Fig. 15 shows a nearly triangu-lar sun with serrated edges. Assigninga cause to these serrations provides alesson in the perils of jumping to con-clusions. Obviously, the serrations are theresult of sharp changes in the temper-ature gradient – or so one might think.Setting aside how such changes could beproduced and maintained in a real at-mosphere, a theorem of Fraser [10] givespause for thought. According to this the-orem, ‘‘In a horizontally (spherically) ho-mogeneous atmosphere it is impossiblefor more than one image of an extrater-restrial object (sun) to be seen above theastronomical horizon.’’ The serrations onthe sun in Fig. 15 are multiple images.But if the refractive index varies onlyvertically (i.e., along a radius), no mat-ter how sharply, multiple images are notpossible. Thus, the serrations must owetheir existence to horizontal variations ofthe refractive index, a consequence ofgravity waves propagating along a tem-perature inversion.

6.4The Green Flash

Compared to the rainbow, the greenflash is not a rare phenomenon. Beforeyou dismiss this assertion as the ravingsof a lunatic, consider that rainbowsrequire raindrops as well as sunlight toilluminate them, whereas rainclouds oftencompletely obscure the sun. Moreover,the sun must be below about 42. As aconsequence of these conditions, rainbowsare not seen often, but often enough thatthey are taken as the paragon of colorvariation. Yet tinges of green on the upperrim of the sun can be seen every dayat sunrise and sunset given a sufficientlylow horizon and a cloudless sky. Thus,the conditions for seeing a green flashare more easily met than those for seeinga rainbow. Why then is the green flashconsidered to be so rare? The distinctionhere is that between a rarely observedphenomenon (the green flash) and a rarelyobservable one (the rainbow).

The sun may be considered to be acollection of disks, one for each visiblewavelength. When the sun is overhead,all the disks coincide and we see thesun as white. But as it descends in thesky, atmospheric refraction displaces thedisks by slightly different amounts, the redless than the violet (see Fig. 13). Most ofeach disk overlaps all the others exceptfor the disks at the extremes of the visiblespectrum. As a consequence, the upperrim of the sun is violet or blue, its lowerrim red, whereas its interior, the region inwhich all disks overlap, is still white.

This is what would happen in the ab-sence of lateral scattering of sunlight. Butrefraction and lateral scattering go hand inhand, even in an atmosphere free of par-ticles. Selective scattering by atmosphericmolecules and particles causes the color


of the sun to change. In particular, theviolet-bluish upper rim of the low sun canbe transformed to green.

According to Eq. (29) and Fig. 13, theangular width of the green upper rim ofthe low sun is about 0.01, too narrow tobe resolved with the naked eye or even tobe seen against its bright backdrop. Butdepending on the temperature profile, theatmosphere itself can magnify the upperrim and yield a second image of it, therebyenabling it to be seen without the aid of atelescope or binoculars. Green rims, whichrequire artificial magnification, can beseen more frequently than green flashes,which require natural magnification. Yetboth can be seen often by those who knowwhat to look for and are willing to look.

7Scattering by Single Water Droplets

All the colored atmospheric displays thatresult when water droplets (or ice crystals)are illuminated by sunlight have the sameunderlying cause: light is scattered indifferent amounts in different directionsby particles larger than the wavelength,and the directions in which scattering isgreatest depends on wavelength. Thus,when particles are illuminated by whitelight, the result can be angular separationof colors even if scattering integrated overall directions is independent of wavelength(as it essentially is for cloud droplets andice crystals). This description, althoughcorrect, is too general to be completelysatisfying. We need something morespecific, more quantitative, which requirestheories of scattering.

Because superficially different theorieshave been used to describe different op-tical phenomena, the notion has becomewidespread that they are caused by these

theories. For example, coronas are said tobe caused by diffraction and rainbows byrefraction. Yet both the corona and therainbow can be described quantitatively tohigh accuracy with a theory (the Mie the-ory for scattering by a sphere) in whichdiffraction and refraction do not explicitlyappear. No fundamentally impenetrablebarrier separates scattering from (specu-lar) reflection, refraction, and diffraction.Because these terms came into generaluse and were entombed in textbooks be-fore the nature of light and matter was wellunderstood, we are stuck with them. Butif we insist that diffraction, for example, issomehow different from scattering, we doso at the expense of shattering the unityof the seemingly disparate observable phe-nomena that result when light interactswith matter. What is observed dependson the composition and disposition of thematter, not on which approximate theoryin a hierarchy is used for quantitative de-scription.

Atmospheric optical phenomena arebest classified by the direction in whichthey are seen and by the agents respon-sible for them. Accordingly, the followingsections are arranged in order of scatteringdirection, from forward to backward.

When a single water droplet is illumi-nated by white light and the scatteredlight projected onto a screen, the resultis a set of colored rings. But in the atmo-sphere we see a mosaic to which individualdroplets contribute. The scattering patternof a single droplet is the same as themosaic provided that multiple scatteringis negligible.

7.1Coronas and Iridescent Clouds

A cloud of droplets narrowly distributed insize and thinly veiling the sun (or moon)


can yield a spectacular series of coloredconcentric rings around it. This coronais most easily described quantitatively bythe Fraunhofer diffraction theory, a sim-ple approximation valid for particles largecompared with the wavelength and forscattering angles near the forward direc-tion. According to this approximation, thedifferential scattering cross section (crosssection for scattering into a unit solidangle) of a spherical droplet of radius ailluminated by light of wave number k is

|S|2k2 , (32)

where the scattering amplitude is

S = x2 1 + cos θ

2

J1(x sin θ)

x sin θ. (33)

The term J1 is the Bessel function offirst order and the size parameter x =ka. The quantity (1 + cos θ )/2 is usuallyapproximated by 1 since only near-forwardscattering angles θ are of interest.

The differential scattering cross section,which determines the angular distributionof the scattered light, has maxima forx sin θ = 5.137, 8.417, 11.62, . . . Thus,the dispersion in the position of the firstmaximum is

dθ

dλ≈ 0.817

a(34)

and is greater for higher-order maxima.This dispersion determines the upper limiton drop size such that a corona can beobserved. For the total angular dispersionover the visible spectrum to be greaterthan the angular width of the sun (0.5),the droplets cannot be larger than about60 µm in diameter. Drops in rain, evenin drizzle, are appreciably larger thanthis, which is why coronas are not seenthrough rainshafts. Scattering by a dropletof diameter 10 µm (Fig. 16), a typical cloud

Fig. 16 Scattering of light near the forwarddirection (according to Fraunhofer theory) by asphere of diameter 10 µm illuminated by red andgreen light

droplet size, gives sufficient dispersion toyield colored coronas.

Suppose that the first angular maxi-mum for blue light (0.47 µm) occurs for adroplet of radius a. For red light (0.66 µm)a maximum is obtained at the same an-gle for a droplet of radius a + a. Thatis, the two maxima, one for each wave-length, coincide. From this we concludethat coronas require narrow size distri-butions: if cloud droplets are distributedin radius with a relative variance a/agreater than about 0.4, color separation isnot possible.

Because of the stringent requirementsfor the occurrence of coronas, they arenot observed often. Of greater occur-rence are the corona’s cousins, iridescentclouds, which display colors but usuallynot arranged in any obviously regular ge-ometrical pattern. Iridescent patches inclouds can be seen even at the edges ofthick clouds that occult the sun.

Coronas are not the unique signatures ofspherical scatterers. Randomly oriented icecolumns and plates give similar patternsaccording to Fraunhofer theory [11]. As apractical matter, however, most coronas


probably are caused by droplets. Manyclouds at temperatures well below freezingcontain subcooled water droplets. Onlyif a corona were seen in a cloud at atemperature lower than −40 C could oneassert with confidence that it must be anice-crystal corona.

7.2Rainbows

In contrast with coronas, which are seenlooking toward the sun, rainbows areseen looking away from it, and arecaused by water drops much larger thanthose that give coronas. To treat therainbow quantitatively we may pretendthat light incident on a transparent sphereis composed of individual rays, each ofwhich suffers a different fate determinedonly by the laws of specular reflection andrefraction. Theoretical justification for thisis provided by van de Hulst’s ([12], p. 208)localization principle, according to whichterms in the exact solution for scattering bya transparent sphere correspond to moreor less localized rays.

Each incident ray splinters into an infi-nite number of scattered rays: externallyreflected, transmitted without internal re-flection, transmitted after one, two, and soon internal reflections. At any scatteringangle θ , each splinter contributes to thescattered light. Accordingly, the differen-tial scattering cross section is an infiniteseries with terms of the form

b(θ)

sin θ

db

dθ. (35)

The impact parameter b is a sin i, wherei is the angle between an incidentray and the normal to the sphere. Eachterm in the series corresponds to oneof the splinters of an incident ray. Arainbow angle is a singularity (or caustic)

of the differential scattering cross sectionat which the conditions

dθ

db= 0,

b

sin θ= 0 (36)

are satisfied. Missing from Eq. (35) arevarious reflection and transmission coeffi-cients (Fresnel coefficients), which displayno singularities and hence do not deter-mine rainbow angles.

A rainbow is not associated with raysexternally reflected or transmitted withoutinternal reflection. The succession ofrainbow angles associated with one, two,three . . . internal reflections are calledprimary, secondary, tertiary . . . rainbows.Aristotle recognized that ‘‘Three or morerainbows are never seen, because eventhe second is dimmer than the first, andso the third reflection is altogether toofeeble to reach the sun (Aristotle’s viewwas that light streams outward from theeye)’’. Although he intuitively grasped thateach successive ray is associated withever-diminishing energy, his statementabout the nonexistence of tertiary rainbowsin nature is not quite true. Althoughreliable reports of such rainbows are rare(unreliable reports are as common as dirt),at least one observer who can be believedhas seen one [13].

An incident ray undergoes a totalangular deviation as a consequence oftransmission into the drop, one or moreinternal reflections, and transmission outof the drop. Rainbow angles are angles ofminimum deviation.

For a rainbow of any order to exist,

cos i =√

n2 − 1

p(p + 1)(37)

must lie between 0 and 1, where i isthe angle of incidence of a ray that givesa rainbow after p internal reflections and


n is the refractive index of the drop. Aprimary bow therefore requires drops withrefractive index less than 2; a secondarybow requires drops with refractive indexless than 3. If raindrops were composedof titanium dioxide (n ≈ 3), a commonlyused opacifier for paints, primary rainbowswould be absent from the sky and wewould have to be content with onlysecondary bows.

If we take the refractive index of water tobe 1.33, the scattering angle for the primaryrainbow is about 138. This is measuredfrom the forward direction (solar point).Measured from the antisolar point (thedirection toward which one must lookin order to see rainbows in nature), thisscattering angle corresponds to 42, thebasis for a previous assertion that rainbows(strictly, primary rainbows) cannot beseen when the sun is above 42. Thesecondary rainbow is seen at about 51from the antisolar point. Between thesetwo rainbows is Alexander’s dark band, aregion into which no light is scatteredaccording to geometrical optics.

The colors of rainbows are a conse-quence of sufficient dispersion of therefractive index over the visible spectrumto give a spread of rainbow angles thatappreciably exceeds the width of the sun.The width of the primary bow from violetto red is about 1.7; that of the secondarybow is about 3.1.

Because of its band of colors arcingacross the sky, the rainbow has becomethe paragon of color, the standard againstwhich all other colors are compared. Leeand Fraser [14, 15], however, challengedthis status of the rainbow, pointing outthat even the most vivid rainbows arecolorimetrically far from pure.

Rainbows are almost invariably dis-cussed as if they occurred literally in avacuum. But real rainbows, as opposed

to the pencil-and-paper variety, are nec-essarily observed in an atmosphere, themolecules and particles of which scat-ter sunlight that adds to the light fromthe rainbow but subtracts from its purityof color.

Although geometrical optics yields thepositions, widths, and color separationof rainbows, it yields little else. Forexample, geometrical optics is blind tosupernumerary bows, a series of narrowbands sometimes seen below the primarybow. These bows are a consequenceof interference, and hence fall outsidethe province of geometrical optics. Sincesupernumerary bows are an interferencephenomenon, they, unlike primary andsecondary bows (according to geometricaloptics), depend on drop size. This posesthe question of how supernumerary bowscan be seen in rain showers, the dropsin which are widely distributed in size. Ina nice piece of detective work, Fraser [16]answered this question.

Raindrops falling in a vacuum are spher-ical. Those falling in air are distorted byaerodynamic forces, not, despite the de-pictions of countless artists, into teardropsbut rather into nearly oblate spheroids withtheir axes more or less vertical. Fraser ar-gued that supernumerary bows are causedby drops with a diameter of about 0.5 mm,at which diameter the angular positionof the first (and second) supernumerarybow has a minimum: interference causesthe position of the supernumerary bowto increase with decreasing size whereasdrop distortion causes it to increase withincreasing size. Supernumerary patternscontributed by drops on either side of theminimum cancel, leaving only the contri-bution from drops at the minimum. Thiscancellation occurs only near the tops ofrainbows, where supernumerary bows areseen. In the vertical parts of a rainbow, a


horizontal slice through a distorted dropis more or less circular, and hence thesedrops do not exhibit a minimum supernu-merary angle.

According to geometrical optics, allspherical drops, regardless of size, yieldthe same rainbow. But it is not necessaryfor a drop to be spherical for it to yieldrainbows independent of its size. Thismerely requires that the plane defined bythe incident and scattered rays intersectthe drop in a circle. Even distorteddrops satisfy this condition in the verticalpart of a bow. As a consequence, theabsence of supernumerary bows there iscompensated for by more vivid colorsof the primary and secondary bows [17].Smaller drops are more likely to bespherical, but the smaller a drop, theless light it scatters. Thus, the dominantcontribution to the luminance of rainbowsis from the larger drops. At the top of abow, the plane defined by the incident andscattered rays intersects the large, distorteddrops in an ellipse, yielding a range ofrainbow angles varying with the amount ofdistortion, and hence a pastel rainbow. Tothe knowledgeable observer, rainbows areno more uniform in color and brightnessthan is the sky.

Although geometrical optics predictsthat all rainbows are equal (neglectingbackground light), real rainbows do notslavishly follow the dictates of this approx-imate theory. Rainbows in nature rangefrom nearly colorless fog bows (or cloudbows) to the vividly colorful vertical por-tions of rainbows likely to have inspiredmyths about pots of gold.

7.3The Glory

Continuing our sweep of scattering direc-tions, from forward to backward, we arrive

at the end of our journey: the glory. Becauseit is most easily seen from airplanes itsometimes is called the pilot’s bow. Anothername is anticorona, which signals that itis a corona around the antisolar point. Al-though glories and coronas share somecommon characteristics, there are differ-ences between them other than directionof observation. Unlike coronas, which maybe caused by nonspherical ice crystals, glo-ries require spherical cloud droplets. Anda greater number of colored rings may beseen in glories than in coronas becausethe decrease in luminance away from thebackward direction is not as steep as thataway from the forward direction. To seea glory from an airplane, look for coloredrings around its shadow cast on clouds be-low. This shadow is not an essential partof the glory, it merely directs you to theantisolar point.

Like the rainbow, the glory may belooked upon as a singularity in the dif-ferential scattering cross section Eq. (35).Equation (36) gives one set of conditionsfor a singularity; the second set is

sin θ = 0, b(θ) = 0. (38)

That is, the differential scattering crosssection is infinite for nonzero impactparameters (corresponding to incidentrays that do not intersect the center of thesphere) that give forward (0) or backward(180) scattering. The forward directionis excluded because this is the directionof intense scattering accounted for by theFraunhofer theory.

For one internal reflection, Eq. (38) leadsto the condition

sin i = n

2

√4 − n2, (39)

which is satisfied only for refractive indicesbetween 1.414 and 2, the lower refractiveindex corresponding to a grazing-incidence


ray. The refractive index of water liesoutside this range. Although a conditionsimilar to Eq. (39) is satisfied for rays un-dergoing four or more internal reflections,insufficient energy is associated with suchrays. Thus, it seems that we have reachedan impasse: the theoretical condition fora glory cannot be met by water droplets.Not so, says van de Hulst [18] in a sem-inal paper. He argues that 1.414 is closeenough to 1.33 given that geometrical op-tics is, after all, an approximation. Clouddroplets are large compared with the wave-length, but not so large that geometricaloptics is an infallible guide to their opticalbehavior. Support for the van de Hulstianinterpretation of glories was provided byBryant and Cox [19], who showed that thedominant contribution to the glory is fromthe last terms in the exact series for scat-tering by a sphere. Each successive termin this series is associated with ever largerimpact parameters. Thus, the terms thatgive the glory are indeed those correspond-ing to grazing rays. Further unraveling ofthe glory and vindication of van de Hulst’sconjectures about the glory were providedby Nussenzveig [20].

It sometimes is asserted that geometricaloptics is incapable of treating the glory.Yet the same can be said for the rainbow.Geometrical optics explains rainbows onlyin the sense that it predicts singularities forscattering in certain directions (rainbowangles). But it can predict only the anglesof intense scattering, not the amount.Indeed, the error is infinite. Geometricaloptics also predicts a singularity in thebackward direction. Again, this simpletheory is powerless to predict more.Results from geometrical optics for bothrainbows and glories are not the endbut rather the beginning, an invitationto take a closer look with more powerfulmagnifying glasses.

8Scattering by Single Ice Crystals

Scattering by spherical water drops in theatmosphere gives rise to three distinct dis-plays in the sky: coronas, rainbows, andglories. Ice particles (crystals) also can in-habit the atmosphere, and they introducetwo new variables in addition to size: shapeand orientation, the second a consequenceof the first. Given this increase in thenumber of degrees of freedom, it is hardlycause for wonder that ice crystals are thesource of a greater variety of displays thanare water drops. As with rainbows, thegross features of ice-crystal phenomenacan be described simply with geometricaloptics, various phenomena arising fromthe various fates of rays incident on crys-tals. Colorless displays (e.g., sun pillars)are generally associated with reflected rays,colored displays (e.g., sun dogs and halos)with refracted rays. Because of the wealthof ice-crystal displays, it is not possible totreat all of them here, but one exampleshould point the way toward understand-ing many of them.

8.1Sun Dogs and Halos

Because of the hexagonal crystalline struc-ture of ice it can form as hexagonal platesin the atmosphere. The stable position ofa plate falling in air is with the normalto its face more or less vertical, which iseasy to demonstrate with an ordinary busi-ness card. When the card is dropped withits edge facing downward (the supposedlyaerodynamic position that many peopleinstinctively choose), the card somersaultsin a helter-skelter path to the ground. Butwhen the card is dropped with its face par-allel to the ground, it rocks back and forthgently in descent.


A hexagonal ice plate falling throughair and illuminated by a low sun islike a 60 prism illuminated normally toits sides (Fig. 17). Because there is nomechanism for orienting a plate withinthe horizontal plane, all plate orientationsin this plane are equally probable. Statedanother way, all angles of incidence fora fixed plate are equally probable. Yet allscattering angles (deviation angles) of raysrefracted into and out of the plate are notequally probable.

Figure 18 shows the range of scatteringangles corresponding to a range of raysincident on a 60 ice prism that is part of ahexagonal plate. For angles of incidenceless than about 13, the transmittedray is totally internally reflected in theprism. For angles of incidence greaterthan about 70, the transmittance plunges.Thus, the only rays of consequence arethose incident between about 13 and70.

All scattering angles are not equallyprobable. The (uniform) probabilitydistribution p(θi) of incidence angles θi

is related to the probability distribution

Fig. 17 Scattering by a hexagonal ice plateilluminated by light parallel to its basal plane.The particular scattering angle θ shown is anangle of minimum deviation. The scattered lightis that associated with two refractions bythe plate

Fig. 18 Scattering by a hexagonal ice plate (seeFig. 17) in various orientations (angles ofincidence). The solid curve is for red light, thedashed for blue light

P(θ) of scattering angles θ by

P(θ) = p(θi)

dθ/ dθi. (40)

At the incidence angle for whichdθ/ dθi = 0, P(θ) is infinite and scat-tered rays are intensely concentratednear the corresponding angle of mini-mum deviation.

The physical manifestation of this singu-larity (or caustic) at the angle of minimumdeviation for a 60 hexagonal ice plate isa bright spot about 22 from either orboth sides of a sun low in the sky. Thesebright spots are called sun dogs (becausethey accompany the sun) or parhelia ormock suns.

The angle of minimum deviation θm,hence the angular position of sun dogs,depends on the prism angle (60for the plates considered) and refrac-tive index:

θm = 2 sin−1(

n sin

2

)− . (41)

Because ice is dispersive, the separationbetween the angles of minimum deviationfor red and blue light is about 0.7 (Fig. 18),


somewhat greater than the angular widthof the sun. As a consequence, sun dogsmay be tinged with color, most noticeablytoward the sun. Because the refractiveindex of ice is least at the red end of thespectrum, the red component of a sun dogis closest to the sun. Moreover, light of anytwo wavelengths has the same scatteringangle for different angles of incidenceif one of the wavelengths does notcorrespond to red. Thus, red is the purestcolor seen in a sun dog. Away from its redinner edge a sun dog fades into whiteness.

With increasing solar elevation, sundogs move away from the sun. A falling iceplate is roughly equivalent to a prism, theprism angle of which increases with solarelevation. From Eq. (41) it follows that theangle of minimum deviation, hence thesun dog position, also increases.

At this point you may be wondering whyonly the 60 prism portion of a hexagonalplate was singled out for attention. Asevident from Fig. 17, a hexagonal platecould be considered to be made up of 120prisms. For a ray to be refracted twice, itsangle of incidence at the second interfacemust be less than the critical angle. Thisimposes limitations on the prism angle.For a refractive index 1.31, all incident raysare totally internally reflected by prismswith angles greater than about 99.5.

A close relative of the sun dog is the22 halo, a ring of light approximately 22from the sun (Fig. 19). Lunar halos arealso possible and are observed frequently(although less frequently than solar halos);even moon dogs are possible. UntilFraser [21] analyzed halos in detail, theconventional wisdom had been that theyobviously were the result of randomlyoriented crystals, yet another example ofjumping to conclusions. By combiningoptics and aerodynamics, Fraser showedthat if ice crystals are small enough

to be randomly oriented by Brownianmotion, they are too small to yield sharpscattering patterns.

But completely randomly oriented platesare not necessary to give halos, especiallyones of nonuniform brightness. Each partof a halo is contributed to by plates witha different tip angle (angle between thenormal to the plate and the vertical).The transition from oriented plates (zerotip angle) to randomly oriented platesoccurs over a narrow range of sizes. Inthe transition region, plates can be smallenough to be partially oriented yet largeenough to give a distinct contribution tothe halo. Moreover, the mapping betweentip angles and azimuthal angles on thehalo depends on solar elevation. Whenthe sun is near the horizon, plates cangive a distinct halo over much of itsazimuth.

Fig. 19 A 22 solar halo. The hand is not forartistic effect but rather to occlude the bright sun


When the sun is high in the sky,hexagonal plates cannot give a sharp halobut hexagonal columns – another possibleform of atmospheric ice particles – can.The stable position of a falling column iswith its long axis horizontal. When thesun is directly overhead, such columnscan give a uniform halo even if they all liein the horizontal plane. When the sun isnot overhead but well above the horizon,columns also can give halos.

A corollary of Fraser’s analysis is thathalos are caused by crystals with a range ofsizes between about 12 and 40 µm. Largercrystals are oriented; smaller particlesare too small to yield distinct scatter-ing patterns.

More or less uniformly bright halos withthe sun neither high nor low in the skycould be caused by mixtures of hexagonalplates and columns or by clusters of bullets(rosettes). Fraser opines that the latter ismore likely.

One of the by-products of his analysis isan understanding of the relative rarity ofthe 46 halo. As we have seen, the angle ofminimum deviation depends on the prismangle. Light can be incident on a hexagonalcolumn such that the prism angle is 60for rays incident on its side or 90 forrays incident on its end. For n = 1.31,Eq. (41) yields a minimum deviation angleof about 46 for = 90. Yet, although 46halos are possible, they are seen much lessfrequently than 22 halos. Plates cannotgive distinct 46 halos although columnscan. Yet they must be solid and mostcolumns have hollow ends. Moreover, therange of sun elevations is restricted.

Like the green flash, ice-crystal phenom-ena are not intrinsically rare. Halos andsun dogs can be seen frequently – onceyou know what to look for. Neuberger [22]reports that halos were observed in StateCollege, Pennsylvania, an average of 74

days a year over a 16-year period, withextremes of 29 and 152 halos a year. Al-though the 22 halo was by far the mostfrequently seen display, ice-crystal displaysof all kinds were seen, on average, moreoften than once every four days at a loca-tion not especially blessed with clear skies.Although thin clouds are necessary for ice-crystal displays, clouds thick enough toobscure the sun are their bane.

9Clouds

Although scattering by isolated particlescan be studied in the laboratory, parti-cles in the atmosphere occur in crowds(sometimes called clouds). Implicit in theprevious two sections is the assumptionthat each particle is illuminated solelyby incident sunlight; the particles do notilluminate each other to an appreciable de-gree. That is, clouds of water droplets orice grains were assumed to be opticallythin, and hence multiple scattering wasnegligible. Yet the term cloud evokes fluffywhite objects in the sky, or perhaps anovercast sky on a gloomy day. For suchclouds, multiple scattering is not negligi-ble, it is the major determinant of theirappearance. And the quantity that deter-mines the degree of multiple scattering isoptical thickness (see Sec. 2.4).

9.1Cloud Optical Thickness

Despite their sometimes solid appearance,clouds are so flimsy as to be almostnonexistent – except optically. The fractionof the total cloud volume occupied bywater substance (liquid or solid) is about10−6 or less. Yet although the massdensity of clouds is that of air to within


a small fraction of a percent, their opticalthickness (per unit physical thickness) ismuch greater. The number density of airmolecules is vastly greater than that ofwater droplets in clouds, but scattering permolecule of a cloud droplet is also muchgreater than scattering per air molecule(see Fig. 7).

Because a typical cloud droplet is muchlarger than the wavelengths of visible light,its scattering cross section is to goodapproximation proportional to the squareof its diameter. As a consequence, thescattering coefficient [see Eq. (2)] of a cloudhaving a volume fraction f of droplets isapproximately

β = 3f〈d2〉〈d3〉 , (42)

where the brackets indicate an averageover the distribution of droplet diametersd. Unlike molecules, cloud droplets aredistributed in size. Although cloud parti-cles can be ice particles as well as waterdroplets, none of the results in this and thefollowing section hinge on the assumptionof spherical particles.

The optical thickness along a cloudpath of physical thickness h is βh fora cloud with uniform properties. Theratio 〈d3〉/〈d2〉 defines a mean dropletdiameter, a typical value for which is10 µm. For this diameter and f = 10−6, theoptical thickness per unit meter of physicalthickness is about the same as the normaloptical thickness of the atmosphere in themiddle of the visible spectrum (see Fig. 3).Thus, a cloud only 1 m thick is equivalentoptically to the entire gaseous atmosphere.

A cloud with (normal) optical thicknessabout 10 (i.e., a physical thickness of about100 m) is sufficient to obscure the disk ofthe sun. But even the thickest cloud doesnot transform day into night. Clouds are

usually translucent, not transparent, yetnot completely opaque.

The scattering coefficient of clouddroplets, in contrast with that of airmolecules, is more or less independentof wavelength. This is often invoked asthe cause of the colorlessness of clouds.Yet wavelength independence of scatter-ing by a single particle is only sufficient,not necessary, for wavelength indepen-dence of scattering by a cloud of particles(see Sec. 2.4). Any cloud that is opticallythick and composed of particles for whichabsorption is negligible is white uponillumination by white light. Although ab-sorption by water (liquid and solid) is notidentically zero at visible wavelengths, andselective absorption by water can lead toobservable consequences (e.g., colors ofthe sea and glaciers), the appearance of allbut the thickest clouds is not determinedby this selective absorption.

Equation (42) is the key to the vastlydifferent optical characteristics of cloudsand of the rain for which they are theprogenitors. For a fixed amount of water(as specified by the quantity fh), opticalthickness is inversely proportional to meandiameter. Rain drops are about 100 timeslarger on average than cloud droplets, andhence optical thicknesses of rain shafts arecorrespondingly smaller. We often can seethrough many kilometers of intense rainwhereas a small patch of fog on a well-traveled highway can result in carnage.

9.2Givers and Takers of Light

Scattering of visible light by a singlewater droplet is vastly greater in theforward (θ < 90) hemisphere than in thebackward (θ > 90) hemisphere (Fig. 9).But water droplets in a thick cloudilluminated by sunlight collectively scatter


much more in the backward hemisphere(reflected light) than in the forwardhemisphere (transmitted light). In eachscattering event, incident photons aredeviated, on average, only slightly, butin many scattering events most photonsare deviated enough to escape from theupper boundary of the cloud. Here is anexample in which the properties of anensemble are different from those of itsindividual members.

Clouds seen by passengers in an airplanecan be dazzling, but if the airplane wereto descend through the cloud these samepassengers might describe the cloudy skyoverhead as gloomy. Clouds are both giversand takers of light. This dual role isexemplified in Fig. 20, which shows thecalculated diffuse downward irradiancebelow clouds of varying optical thickness.On an airless planet the sky would be blackin all directions (except directly towardthe sun). But if the sky were to be filledfrom horizon to horizon with a thin cloud,the brightness overhead would markedlyincrease. This can be observed in a partlyovercast sky, where gaps between clouds(blue sky) often are noticeably darker than

Fig. 20 Computed diffuse downward irradiancebelow a cloud relative to the incident solarirradiance as a function of cloudoptical thickness

their surroundings. As so often happens,more is not always better. Beyond acertain cloud optical thickness, the diffuseirradiance decreases. For a sufficientlythick cloud, the sky overhead can be darkerthan the clear sky.

Why are clouds bright? Why are theydark? No inclusive one-line answers can begiven to these questions. Better to ask, Whyis that particular cloud bright? Why is thatparticular cloud dark? Each observationmust be treated individually; generaliza-tions are risky. Moreover, we must keepin mind the difference between bright-ness and radiance when addressing thequeries of human observers. Brightness isa sensation that is a property not only ofthe object observed but of its surround-ings as well. If the luminance of an objectis appreciably greater than that of its sur-roundings, we call the object bright. If theluminance is appreciably less, we call theobject dark. But these are relative ratherthan absolute terms.

Two clouds, identical in all respects,including illumination, may still appeardifferent because they are seen againstdifferent backgrounds, a cloud against thehorizon sky appearing darker than whenseen against the zenith sky.

Of two clouds under identical illumi-nation, the smaller (optically) will be lessbright. If an even larger cloud were to ap-pear, the cloud that formerly had been de-scribed as white might be demoted to gray.

With the sun below the horizon, twoidentical clouds at markedly differentelevations might appear quite differentin brightness, the lower cloud beingshadowed from direct illumination bysunlight.

A striking example of dark clouds cansometimes be seen well after the sunhas set. Low-lying clouds that are notilluminated by direct sunlight but are


seen against the faint twilight sky maybe relatively so dark as to seem likeink blotches.

Because dark objects of our everydaylives usually owe their darkness to absorp-tion, nonsense about dark clouds is rife:they are caused by pollution or soot. Yet ofall the reasons that clouds are sometimesseen to be dark or even black, absorptionis not among them.

Glossary

Airlight: Light resulting from scattering byall atmospheric molecules and particlesalong a line of sight.

Antisolar Point: Direction opposite thesun.

Astronomical Horizon: Horizontal direc-tion determined by a bubble level.

Brightness: The attribute of sensation bywhich an observer is aware of differencesof luminance (definition recommendedby the 1922 Optical Society of AmericaCommittee on Colorimetry).

Contrast Threshold: The minimum rela-tive luminance difference that can beperceived by the human observer.

Inferior Mirage: A mirage in which imagesare displaced downward.

Irradiance: Radiant power crossing unitarea in a hemisphere of directions.

Lapse Rate: The rate at which a physicalproperty of the atmosphere (usually tem-perature) decreases with height.

Luminance: Radiance integrated over thevisible spectrum and weighted by the

spectral response of the human ob-server. Also sometimes called photometricbrightness..5.5

Mirage: An image appreciably differentfrom what it would be in the absence ofatmospheric refraction.

Neutral Point: A direction in the sky forwhich the light is unpolarized.

Normal Optical Thickness: Optical thick-ness along a radial path from the surfaceof the earth to infinity.

Optical Thickness: The thickness of a scat-tering medium measured in units ofphoton mean free paths. Optical thick-nesses are dimensionless.

Radiance: Radiant power crossing a unitarea and confined to a unit solid angleabout a particular direction.

Scale Height: The vertical distance overwhich a physical property of the at-mosphere is reduced to 1/e of itsvalue.

Scattering Angle: Angle between incidentand scattered waves.

Scattering Coefficient: The product of scat-tering cross section and number density ofscatterers.

Scattering Cross Section: Effective area of ascatterer for removal of light from a beamby scattering.

Scattering Plane: Plane determined byincident and scattered waves.

Solar Point: The direction toward thesun.


Superior Mirage: A mirage in which im-ages are displaced upward.

Tangential Optical Thickness: Opticalthickness through the atmosphere alonga horizon path.

References

Many of the seminal papers in atmosphericoptics, including those by Lord Rayleigh, arebound together in Bohren, C. F. (Ed.) (1989),Selected Papers on Scattering in the Atmosphere,Bellingham, WA: SPIE Optical EngineeringPress. Papers marked with an asterisk are inthis collection.

[1] Moller, F. (1972), Radiation in the at-mosphere, in D. P. McIntyre (Ed.), Me-teorological Challenges: A History. Ottawa:Information Canada, pp. 43–71.

[2]∗ Penndorf, R. (1957), J. Opt. Soc. Am. 47,176–182.

[3]∗ Young, A. T. (1982), Phys. Today 35(1),2–8.

[4] Einstein, A. (1910), Ann. Phys. (Leipzig)33, 175; English translation in Alexan-der, J. (Ed.) (1926), Colloid Chemistry,Vol. I. New York: The Chemical CatalogCompany, pp. 323–339.

[5] Zimm, B. H. (1945), J. Chem. Phys. 13,141–145.

[6] Thekaekara, M. P., Drummond, A. J.(1971), Nat. Phys. Sci. 229, 6–9.

[7]∗ Hulburt, E. O. (1953), J. Opt. Soc. Am. 43,113–118.

[8] von Frisch, K. (1971), Bees: Their Vision,Chemical Senses, and Language, revisededition, Ithaca, NY: Cornell UniversityPress, p. 116.

[9] Doyle, W. T. (1985), Am. J. Phys. 53,463–468.

[10]∗ Fraser, A. B. (1975), Atmosphere 13, 1–10.[11] Takano, Y., Asano, S. (1983), J. Meteor.

Soc. Jpn. 61, 289–300.[12] van de Hulst, H. C. (1957), Light Scattering

by Small Particles. New York: Wiley-Interscience.

[13] Pledgley, E. (1986), Weather 41, 401.[14] Lee, R., Fraser, A. (1990), New Scientist

127(September), 40–42.

[15] Lee, R. (1991), Appl. Opt. 30, 3401–3407.[16]∗ Fraser, A. B. (1983), J. Opt. Soc. Am. 73,

1626–1628.[17] Fraser, A. B. (1972), J. Atmos. Sci. 29, 211,

212.[18]∗ van de Hulst, H. C. (1947), J. Opt. Soc.

Am. 37, 16–22.[19]∗ Bryant, H. C., Cox, A. J. (1966), J. Opt. Soc.

Am. 56, 1529–1532.[20]∗ Nussenzveig, H. M. (1979), J. Opt. Soc.

Am. 69, 1068–1079.[21]∗ Fraser, A. B. (1979), J. Opt. Soc. Am. 69,

1112–1118.[22] Neuberger, H. (1951), Introduction to Phys-

ical Meteorology. University Park, PA: Col-lege of Mineral Industries, PennsylvaniaState University.

Further Reading

Minnaert, M. (1954), The Nature of Light andColour in the Open Air. New York: DoverPublications, is the bible for those interestedin atmospheric optics. Like accounts of naturalphenomena in the Bible, those in Minnaert’sbook are not always correct, despite which,again like the Bible, it has been and willcontinue to be a source of inspiration.

A book in the spirit of Minnaert’s but with awealth of color plates is by Lynch, D. K., Liv-ingston, W. (1995), Color and Light in Nature.Cambridge, UK: Cambridge University Press.

A history of light scattering, From Leonardo tothe Graser: Light Scattering in Historical Per-spective, was published serially by Hey, J. D.(1983), S. Afr. J. Sci. 79, 11–27, 310–324;Hey, J. D. (1985), S. Afr. J. Sci. 81, 77–91,601–613; Hey, J. D., (1986), S. Afr. J. Sci. 82,356–360. The history of the rainbow is re-counted by Boyer, C. B. (1987), The Rainbow.Princeton, NJ: Princeton University Press.

A unique, beautifully written and illustratedtreatise on rainbows in science and art,both sacred and profane, is by Lee, R. L.,Fraser, A. B. (2001), The Rainbow Bridge,University Park, PA: Penn State UniversityPress.

Special issues of Journal of the Optical Societyof America (August 1979 and December 1983)and Applied Optics (20 August 1991 and 20 July1994) are devoted to atmospheric optics.


Several monographs on light scattering byparticles are relevant to and contain exam-ples drawn from atmospheric optics: vande Hulst, H. C. (1957), Light Scattering bySmall Particles. New York: Wiley-Interscience;reprint (1981), New York: Dover Publica-tions; Deirmendjian, D. (1969), Electromag-netic Scattering on Polydispersions. New York:Elsevier; Kerker, M. (1969), The Scatteringof Light and Other Electromagnetic Radiation.New York: Academic Press; Bohren, C. F.,Huffman, D. R. (1983), Light Scattering bySmall Particles. New York: Wiley-Interscience;Nussenzveig, H. M. (1992), Diffraction Effectsin Semiclassical Scattering. Cambridge, UK:Cambridge University Press.

The following books are devoted to a widerange of topics in atmospheric optics:Tricker, R. A. R. (1970), Introduction to Mete-orological Optics. New York: Elsevier; McCart-ney, E. J. (1976), Optics of the Atmosphere. NewYork: Wiley; Greenler, R. (1980), Rainbows,Halos, and Glories. Cambridge, UK: CambridgeUniversity Press. Monographs of more limitedscope are those by Middleton, W. E. K. (1952),Vision Through the Atmosphere. Toronto: Uni-versity of Toronto Press; O’Connell, D. J. K.(1958), The Green Flash and Other LowSun Phenomena. Amsterdam: North Holland;Rozenberg, G. V. (1966), Twilight: A Study inAtmospheric Optics. New York: Plenum; Hen-derson, S. T. (1977), Daylight and its Spectrum,(2nd ed.), New York: Wiley; Tricker, R. A. R.(1979), Ice Crystal Haloes. Washington, DC:Optical Society of America; Konnen, G. P.(1985), Polarized Light in Nature. Cambridge,UK: Cambridge University Press; Tape, W.(1994), Atmosphere Halos. Washington, DC:American Geophysical Union.

Although not devoted exclusively to atmosphericoptics, Humphreys, W. J. (1964), Physics of theAir. New York: Dover Publications, contains

a few relevant chapters. Two popular sciencebooks on simple experiments in atmosphericphysics are heavily weighted toward atmo-spheric optics: Bohren, C. F. (1987), Clouds ina Glass of Beer. New York: Wiley; Bohren, C. F.(1991), What Light Through Yonder WindowBreaks? New York: Wiley.

For an expository article on colors of the skysee Bohren, C. F., Fraser, A. B. (1985), Phys.Teacher 23, 267–272.

An elementary treatment of the coherenceproperties of light waves was given by For-rester, A. T. (1956), Am. J. Phys. 24, 192–196.This journal also published an expository arti-cle on the observable consequences of multiplescattering of light: Bohren, C. F. (1987), Am. J.Phys. 55, 524–533.

Although a book devoted exclusively to atmo-spheric refraction has yet to be published,an elementary yet thorough treatment of mi-rages was given by Fraser, A. B., Mach, W. H.(1976), Sci. Am. 234(1), 102–111.

Colorimetry, the often (and unjustly) neglectedcomponent of atmospheric optics, is treatedin, for example, Optical Society of Amer-ica Committee on Colorimetry (1963), TheScience of Color. Washington, DC: OpticalSociety of America. Billmeyer, F. W., Saltz-man, M. (1981), Principles of Color Technol-ogy, (2nd ed.), New York: Wiley-Interscience.MacAdam, D. L. (1985), Color Measurement,(2nd ed.), Berlin: Springer.

Understanding atmospheric optical phenom-ena is not possible without acquiring atleast some knowledge of the propertiesof the particles responsible for them. Tothis end, the following are recommended:Pruppacher, H. R., Klett, J. D. (1980), Micro-physics of Clouds and Precipitation. Dor-drecht, Holland: D. Reidel. Twomey, S. A.(1977), Atmospheric Aerosols. New York:Elsevier.

937

Interferometry

Parameswaran HariharanSchool of Physics, University of Sydney, AustraliaPhone: (612) 9413 7159; Fax: (612) 9413 7200; e-mail: hariharan [email protected]

Katherine CreathOptineering, Tucson, Arizona, USAPhone: (520) 882-2950; Fax: (520) 882-6976; e-mail: [email protected]

AbstractThis article reviews the field of interferometry. It begins by outlining the fundamentalsof two-beam and multiple-beam interference. The rest of the article discusses theapplications of interferometry for measurement of length, optical testing, fringe analysis,interference microscopy, interferometric sensors, interference spectroscopy, nonlinearinterferometers, interferometric imaging, space-time and gravitation, holographicinterferometry, Moire techniques, and speckle interferometry. Each section provides areferenced (and cross-referenced) overview of the application area.

Keywordsinterferometry; interference; interferometers; optical testing; optical metrology;nondestructive testing.

1 Introduction 9392 Interference and Coherence 9402.1 Localization of Fringes 9412.2 Coherence 9413 Two-beam Interferometers 9423.1 The Michelson Interferometer 9423.2 The Mach–Zehnder Interferometer 9433.3 The Sagnac Interferometer 943

938 Interferometry

4 Multiple-beam Interference 9434.1 Fringes of Equal Chromatic Order 9445 Measurement of Length 9445.1 Electronic Fringe Counting 9445.2 Heterodyne Interferometry 9445.3 Two-wavelength Interferometry 9455.4 Frequency-modulation Interferometry 9465.5 Laser-feedback Interferometry 9466 Optical Testing 9466.1 Flat Surfaces 9466.2 Homogeneity 9476.3 Concave and Convex Surfaces 9476.4 Prisms 9476.5 Aspheric Surfaces 9486.6 Optically Rough Surfaces 9486.7 Shearing Interferometers 9486.8 The Point-diffraction Interferometer 9497 Fringe Analysis 9497.1 Fringe Tracking and Fourier Analysis 9497.2 Phase-shifting Interferometry 9507.3 Determining Aberrations 9518 Interference Microscopy 9518.1 The Mirau Interferometer 9518.2 The Nomarski Interferometer 9518.3 White-light Interferometry 9529 Interferometric Sensors 9539.1 Laser–Doppler Interferometry 9539.2 Fiber Interferometers 9549.3 Rotation Sensing 95510 Interference Spectroscopy 95510.1 Etendue of an Interferometer 95510.2 The Fabry–Perot Interferometer 95510.3 Wavelength Measurements 95710.4 Laser Linewidth 95810.5 Fourier-transform Spectroscopy 95811 Nonlinear Interferometers 95911.1 Second-harmonic Interferometers 95911.2 Phase-conjugate Interferometers 96011.3 Measurement of Nonlinear Susceptibilities 96112 Interferometric Imaging 96112.1 The Intensity Interferometer 96112.2 Heterodyne Stellar Interferometers 96212.3 Stellar Speckle Interferometry 96312.4 Telescope Arrays 963

Interferometry 939

13 Space-time and Gravitation 96313.1 Gravitational Waves 96313.2 LIGO 96513.3 Limits to Measurement 96514 Holographic Interferometry 96514.1 Strain Analysis 96514.2 Vibration Analysis 96614.3 Contouring 96615 Moire Techniques 96715.1 Grating Interferometry 96716 Speckle Interferometry 96816.1 Electronic Speckle Pattern Interferometry (ESPI) 96816.2 Phase-shifting Speckle Interferometry 96816.3 Vibrating Objects 969


1Introduction

Optical interferometry uses the phe-nomenon of interference between lightwaves to make extremely accurate mea-surements. The interference pattern con-tains, in addition to information on theoptical paths traversed by the waves, infor-mation on the spectral content of the lightand its spatial distribution over the source.

Young was the first to state the principleof interference and demonstrate that thesummation of two rays of light could giverise to darkness, but the father of optical in-terferometry was undoubtedly Michelson.Michelson’s contributions to interferom-etry, from 1880 to 1930, dominated thefield to such an extent that optical inter-ferometry was regarded for many years asa closed chapter. However, the last fourdecades have seen an explosive growth ofinterest in interferometry due to severalnew developments.

The most important of these was thedevelopment of the laser, which madeavailable, for the first time, an intensesource of light with a remarkably highdegree of spatial and temporal coher-ence. Lasers have removed most of thelimitations imposed by conventional lightsources and have made possible many newtechniques, including nonlinear interfer-ometry.

Another development that has revo-lutionized interferometry has been theapplication of electronic techniques. Theuse of photoelectric detector arrays anddigital computers has made possible directmeasurements of the optical path differ-ence at an array of points covering aninterference pattern, with very high accu-racy, in a very short time.

Light scattered from a moving particlehas its frequency shifted by an amountproportional to the component of itsvelocity in a direction determined by thedirections of illumination and viewing.

940 Interferometry

Lasers have made it possible to measurethis frequency shift and, hence, the velocityof the particles, by detecting the beatsproduced by mixing the scattered light andthe original laser beam.

Another major advance has been theuse of single-mode optical fibers tobuild analogs of conventional two-beaminterferometers. Since very long opticalpaths can be accommodated in a smallspace, fiber interferometers are now usedwidely as rotation sensors. In addition,since the length of the optical path insuch a fiber changes with pressure ortemperature, fiber interferometers havefound many applications as sensors fora number of physical quantities.

In the field of stellar interferometry, itis now possible to combine images fromwidely spaced arrays of large telescopesto obtain extremely high resolution. In-terferometry is also being applied to thedetection of gravitational waves from blackholes and supernovae.

Holography (see HOLOGRAPHY) is a com-pletely new method of imaging based onoptical interference. Holographic interfer-ometry has made it possible to map thedisplacements of a rough surface with anaccuracy of a few nanometers, and even tomake interferometric comparisons of twostored wavefronts that existed at differenttimes. Holographic interferometry and arelated technique, speckle interferometry(see SPECKLE AND SPECKLE METROLOGY), arenow used widely in industry for nonde-structive testing and structural analysis.

The applications outlined above providea glimpse of the many areas of optics thatuse interferometry. This article is meantto provide an overview. More detail isavailable in the cross-referenced articlesas well as in the lists of works cited andfurther reading.

2Interference and Coherence

When two light waves are superimposed,the resultant intensity depends on whetherthey reinforce or cancel each other.This is the well-known phenomenon ofinterference (see WAVE OPTICS).

If, at any point, the complex ampli-tudes of two light waves, derived from thesame monochromatic point source andpolarized in the same plane, are A1 =a1 exp(−iφ1) and A2 = a2 exp(−iφ2), theintensity (or the irradiance, units W m−2)at this point is

I = |A1 + A2|2= I1 + I2 + 2(I1I2)

1/2 cos(φ1 − φ2),

(1)

where I1 and I2 are the intensities dueto the two waves acting separately andφ1 − φ2 is the difference in their phases.

The visibility V of the interferencefringes is defined by the relation

V = Imax − Imin

Imax + Imin

= 2(I1I2)1/2

I1 + I2. (2)

Interference effects can be observedquite easily by viewing a transparent plateilluminated by a point source of monochro-matic light. In this case, interference takesplace between the waves reflected from thefront and back surfaces of the plate.

For a ray incident at an angle θ1 on aplane-parallel plate (thickness d, refractiveindex n) and refracted within the plate atan angle θ2, the optical path differencebetween the two reflected rays is

p = 2nd cos θ2 + λ

2, (3)

Interferometry 941

since an additional phase shift of π isintroduced by reflection at one of thesurfaces. The interference fringes arecircles centered on the normal to theplate (fringes of equal inclination, orHaidinger fringes).

With a collimated beam, the interfer-ence fringes are contours of equal opticalthickness (Fizeau fringes). The variationsin the phase difference observed can rep-resent variations in the thickness or therefractive index of the plate. A polished flatsurface can be compared with a referenceflat surface, by placing them in contactand observing the fringes of equal thick-ness formed in the air film between them.Introduction of a small tilt between thetest and reference surfaces produces a setof almost straight and parallel fringes. Anydeviations of the test surface from a planeare seen as a departure of the fringes fromstraight lines. The errors of the test sur-face can then be evaluated, as shown inFig. 1, by measuring the maximum devia-tion (x) of a fringe from a straight lineas well as the spacing between successivefringes (x). Each fringe corresponds to achange in the optical path difference ofhalf a wavelength.

2.1Localization of Fringes

An extended monochromatic source canbe considered as an array of independentpoint sources. Since the light waves fromthese sources take different paths to thepoint where interference is observed, theelementary interference patterns producedby any two of them will not, in general,coincide. Interference fringes are thenobserved with maximum contrast onlyin a particular region (the region oflocalization).

∆x

x

Fig. 1 Evaluation of the errors of a polished flattest surface by interference

With a plane-parallel plate, the inter-ference fringes are localized at infinity.With a wedged thin film, and near-normalincidence, the interference fringes are lo-calized in the wedge.

2.2Coherence

A more detailed analysis [1] shows thatthe interference effects observed dependon the degree of correlation between thewave fields at the point of observation.The intensity in the interference pattern isgiven by the relation

I = I1 + I2 + 2(I1I2)1/2|γ |

× cos(arg γ + 2πντ), (4)

where I1 and I2 are the intensities ofthe two beams, ν is the frequency of theradiation, τ is the mean time delay betweenthe arrival of the two beams and γ is the(complex) degree of coherence between thewave fields.

With two beams of equal intensity, thevisibility of the interference fringes is equalto |γ |, with a maximum value of 1 whenthe correlation between the wave fieldsis complete.

The correlation between the fields at anytwo points, when the difference betweenthe optical paths to the source is small

942 Interferometry

enough for effects due to the spectralbandwidth of the light to be neglected,is a measure of the spatial coherence ofthe light. If the size of the source andthe separation of the two points are verysmall compared to their distance fromthe source, it can be shown that thecomplex degree of coherence is given bythe normalized two-dimensional Fouriertransform of the intensity distributionover the source (see FOURIER AND OTHER

TRANSFORM METHODS).Similarly, the correlation between the

fields at the same point at differenttimes is a measure of the temporalcoherence of the light and is related to itsspectral bandwidth. With a point source(or when interference takes place betweencorresponding elements of the originalwavefront), the visibility of the fringesas a function of the delay is the Fouriertransform of the source spectrum.

To make this analysis complete, we mustalso take into account the polarization ef-fects. In general, for maximum visibility,the beams must start in the same stateof polarization (see POLARIZED LIGHT, BA-

SIC CONCEPTS OF) and interfere in the samestate of polarization. For natural (unpolar-ized) light, the optical path difference mustbe the same for all polarizations. The ef-fects of deviations from these conditions,which can be quite complex, have beendiscussed by [2].

3Two-beam Interferometers

Two methods are used to obtain two beamsfrom a common source.

In wavefront division, two beams areisolated from separate areas of the primarywavefront. This technique was used in

Young’s experiment and in the Rayleighinterferometer.

More commonly, two beams are derivedfrom the same portion of the primary wave-front (amplitude division) using a beamsplitter (a transparent plate coated witha partially reflecting film), a diffractiongrating or a polarizing prism.

3.1The Michelson Interferometer

In the Michelson interferometer, a singlebeam splitter is used, as shown in Fig. 2,to divide and recombine the beams.However, to obtain interference fringeswith white light, the two optical pathsmust contain the same thickness of glass.Accordingly, a compensating plate (of thesame thickness and the same materialas the beam splitter) is introduced inone beam.

The interference pattern observed issimilar to that produced in a plate (n = 1)

bounded by one mirror and the image ofthe other mirror produced by reflectionfrom the beam splitter. With an extendedsource, the interference fringes are circleslocalized at infinity (fringes of equal

BS

Fig. 2 The Michelson interferometer

Interferometry 943

inclination). With collimated light (theTwyman–Green interferometer), straight,parallel fringes of equal thickness (Fizeaufringes) are obtained.

3.2The Mach–Zehnder Interferometer

As shown in Fig. 3, the Mach–Zehnderinterferometer (MZI) uses two beam split-ters to divide and recombine the beams.

The MZI has the advantage that eachoptical path is traversed only once. Inaddition, with an extended source, theregion of localization of the fringes canbe made to coincide with the test section.The MZI has been used widely to maplocal variations of the refractive index inwind tunnels, flames, and plasmas.

A variant, the Jamin interferometer,along with the Rayleigh interferometer, iscommonly used to measure the refractiveindex of gases and mixtures of gases. Accu-rate measurements of the refractive indexof air are a prerequisite for interferometricmeasurements of length (see Sect. 5).

3.3The Sagnac Interferometer

In one form of the Sagnac interferometer,as shown in Fig. 4, the two beams traverse

BS

BS

Fig. 3 The Mach–Zehnder interferometer

BS

Fig. 4 The Sagnac interferometer

exactly the same path in opposite direc-tions. However, with an odd number of re-flections in the path, the wavefronts are lat-erally inverted with respect to each other.

Since the optical paths traversed bythe two beams are always very nearlyequal, fringes can be obtained easily withan extended, white-light source. Modifiedforms of the Sagnac interferometer areused for rotation sensing, since therotation of the interferometer with anangular velocity about an axis makingan angle θ with the normal to theplane of the beams introduces an opticalpath difference

p =(

4A

c

)cos θ, (5)

between the two beams, where A is thearea enclosed by the beams and c is thespeed of light.

4Multiple-beam Interference

With two highly reflecting surfaces, wehave to take into account the effects

944 Interferometry

of multiply reflected beams (see WAVE

OPTICS). The intensity in the interferencepattern formed by the transmission is

IT(φ) = T2

1 + R2 − 2R cos φ, (6)

where R and T are, respectively, thereflectance and transmittance of the sur-faces and φ = (4π/λ)nd cos θ2. As thereflectance R increases, the intensity at theminima decreases, and the bright fringesbecome sharper.

The finesse, defined as the ratio of theseparation of adjacent fringes to the fullwidth at half maximum (FWHM) of thefringes (the separation of points at whichthe intensity is equal to half its maximumvalue), is

F = πR1/2

1 − R. (7)

The interference fringes formed by thereflected beams are complementary tothose obtained by transmission.

4.1Fringes of Equal Chromatic Order

With a white-light source, interferencefringes cannot be seen for optical path dif-ferences greater than a few micrometers.However, if the reflected light is exam-ined with a spectroscope, the spectrumwill be crossed by dark bands correspond-ing to interference minima. With a thinfilm (thickness d, refractive index n), if λ1

and λ2 are the wavelengths correspondingto adjacent dark bands, we have

d = λ1λ2

2n|λ2 − λ1| . (8)

With two highly reflecting surfaces en-closing a thin film, very sharp fringes ofequal chromatic order (FECO fringes) canbe obtained, using a white-light source

and a spectrograph. FECO fringes permitmeasurements with a precision of λ/500.

A major application of FECO fringes hasbeen to study the microstructure of sur-faces [3, 4]. However, the test surface mustbe coated with a highly reflective coating.

5Measurement of Length

One of the earliest applications of interfer-ometry was in measurements of lengths.Because of the limited distance over whichinterference fringes could be observedwith conventional light sources, Michel-son had to perform a laborious seriesof comparisons to measure the numberof wavelengths of a spectral line in thestandard meter. The extremely narrowspectral bandwidth of light from a laserhas led to the development of a number ofinterferometric techniques for direct mea-surements of large distances. The valuesobtained for the optical path length are di-vided by the value of the refractive index ofair, under the conditions of measurement,to obtain the true length.

5.1Electronic Fringe Counting

If an additional phase difference of π/2is introduced between the beams inone half of the field, two detectors canprovide signals in quadrature to drive abidirectional counter. These signals canalso be processed to obtain an estimate ofthe fractional interference order [5].

5.2Heterodyne Interferometry

In the Hewlett–Packard interferome-ter [6], a He-Ne laser is forced to oscillate

Interferometry 945

SolenoidBeam

expander

Laser l/4 plate

Detectors

DR DS

2 − 1± ∆

2 − 1

Polarizers

Counter

Counter

Subtractor Display

C1

2± ∆

21 C2

Referencesignal

Fig. 5 Fringe-counting interferometer using a two-frequency laser (after Dukes, J. N., Gordon, G. B.(1970), Hewlett-Packard J. 21, 2–8 [6]. Hewlett–Packard Company. Reproduced with permission)

simultaneously at two frequencies sep-arated by about 2 MHz by applying alongitudinal magnetic field. As shown inFig. 5, these two frequencies that have op-posite circular polarizations pass through aλ/4 plate that converts them to orthogonallinear polarizations.

A polarizing beam splitter reflects onefrequency to a fixed cube-corner, while theother is transmitted to a movable cube-corner. Both frequencies return along acommon axis and, after passing througha polarizer set at 45, are incident on aphotodetector. The beat frequencies fromthis detector and a reference detector go toa differential counter. If one of the cube-corners is moved, the net count gives thechange in the optical path in wavelengths.

Very small changes in length can bemeasured by heterodyne interferometry.In one technique [7], a small frequencyshift is introduced between the two beams,typically by means of a pair of acousto-opticmodulators operated at slightly differentfrequencies. The output from a detector

viewing the interference pattern containsa component at the difference frequency,and the phase of this heterodyne signal cor-responds to the phase difference betweenthe interfering beams.

In another technique, the two mirrors ofa Fabry–Perot interferometer are attachedto the two ends of the sample, and thewavelength of a laser is locked to atransmission peak [8]. A change in theseparation of the mirrors results in achange in the wavelength of the laser and,hence, in its frequency. These changes canbe measured with high precision by mixingthe beam from the laser with the beamfrom a reference laser, and measuring thebeat frequency.

5.3Two-wavelength Interferometry

If an interferometer is illuminated simul-taneously with two wavelengths λ1 and λ2,the envelope of the fringes yields the inter-ference pattern that can be obtained with

946 Interferometry

a synthetic wavelength

λs = λ1λ2

|λ1 − λ2| . (9)

One way to implement this technique iswith a CO2 laser, which is switched rapidlybetween two wavelengths, as one of themirrors of an interferometer is movedover the distance to be measured. Theoutput signal from a photodetector is thenprocessed to obtain the phase difference atany point [9].

5.4Frequency-modulation Interferometry

Absolute measurements of distance can bemade with a semiconductor laser by sweep-ing its frequency linearly with time [10]. Ifthe optical path difference between the twobeams in the interferometer is L, one beamreaches the detector with a time delay L/c,and they interfere to yield a beat signalwith a frequency

f =(

L

c

)(df

dt

), (10)

where df /dt is the rate at which the laserfrequency varies with time.

5.5Laser-feedback Interferometry

If, as shown in Fig. 6, a small fraction ofthe output of a laser is fed back to it byan external mirror, the output of the laservaries cyclically with the position of themirror [11]. A displacement of the mirror

by half a wavelength corresponds to onecycle of modulation.

A very simple laser-feedback interfer-ometer can be set up with a single-modesemiconductor laser. An increased mea-surement range and higher accuracy canbe obtained by mounting the mirror on apiezoelectric translator, and using an ac-tive feedback loop to hold the optical pathconstant [12].

6Optical Testing

A major application of interferometry isin testing optical components and opticalsystems (see OPTICAL METROLOGY).

6.1Flat Surfaces

The Fizeau interferometer (see Fig. 7) isused widely to compare a polished flatsurface with a standard flat surface withoutplacing them in contact and riskingdamage to the surfaces. Measurementsare made on the fringes of equal thicknessformed with collimated light in the airspace separating the two surfaces.

To determine absolute flatness, it ispossible to use a liquid surface as areference [13]; however, a more often-usedmethod is to test a set of three nominallyflat surfaces in pairs. Errors of each of thethree surfaces can be evaluated using thistechnique without the need for a knownstandard flat surface [14–16].

Detector Laser

M

Fig. 6 Laser-feedback interferometer

Interferometry 947

Laser

Test surface

Reference flat

Fig. 7 Fizeau interferometer used to test flat surfaces

6.2Homogeneity

The homogeneity of a material can bechecked by preparing a plane-parallel sam-ple and placing it in the test path ofthe interferometer. The effects of sur-face imperfections and systematic er-rors can be minimized by submergingthe sample in a refractive-index match-ing oil and making measurements withand without the sample in the testpath [17].

6.3Concave and Convex Surfaces

The Fizeau and Twyman–Green interfer-ometers can be used to test concave andconvex surfaces [18, 19]. Typical test con-figurations for curved optical surfaces areshown in Fig. 8.

6.4Prisms

Figure 9 shows a test configuration for a60 prism. With a prism having a roof

Reference flat Converging lens

Concave surface test

Convex surface test

Fig. 8 Fizeau interferometer used to test concave and convex surfaces

948 Interferometry

Prism

Fig. 9 Twyman–Green interferometer used totest a prism

angle of 90, the beam is retro-reflectedback through the system.

6.5Aspheric Surfaces

Problems can arise in testing an asphericsurface against a spherical reference wave-front because the fringes in some partsof the resulting interferogram may be tooclosely spaced to be resolved. One way tosolve this problem is to use a compen-sating null-lens [20]; another is to use acomputer-generated hologram to producea reference wavefront matching the de-sired aspheric wavefront [21, 22]. Shearinginterferometry is yet another way to reducethe number of fringes in the interfero-gram [23, 24].

6.6Optically Rough Surfaces

One way to test fine ground surfaces,before they are polished, is by infraredinterferometry with a CO2 laser at awavelength of 10.6 µm [25]. A simpler al-ternative, with nominally flat surfaces,

is to use oblique incidence [26]. An-other means of measuring these sur-faces uses scanning white-light tech-niques similar to those described inSect. 8.3.

6.7Shearing Interferometers

Shearing interferometers, in which inter-ference takes place between two imagesof the test wavefront, have the advan-tage that they eliminate the need for areference surface of the same dimen-sions as the test surface [23, 24]. Witha lateral shear, as shown in Fig. 10(a),the two images undergo a mutual lateraldisplacement. If the shear is small, thewavefront aberrations can be obtained byintegrating the phase data from two in-terferograms with orthogonal directionsof shear. With a radial shear, as shownin Fig. 10(b), one of the images is con-tracted or expanded with respect to the

s

x

y

(a)

(b)

d1 d2

Fig. 10 Images of the test wavefront in(a) lateral and (b) radial shearing interferometers

Interferometry 949

other. If the diameter of one image isless than (say) 0.3 of the other, the inter-ferogram obtained is very similar to thatobtained with a Fizeau or Twyman–Greeninterferometer.

Other forms of shear, such as rotational,inverting or folding shears, can also beused for specific applications.

6.8The Point-diffraction Interferometer

As shown in Fig. 11, the point-diffractioninterferometer [27] consists of a pinhole ina partially transmitting film placed at thefocus of the test wavefront.

The interference pattern formed bythe test wavefront, which is transmittedby the film, and a spherical referencewave produced by diffraction at the pin-hole corresponds to a contour map ofthe wavefront aberrations. Both wave-fronts traverse the same path, makingthis compact interferometer insensitive tovibrations.

7Fringe Analysis

High-accuracy information can be ex-tracted from the fringe pattern, includingthe calculation of aberration coefficients(see OPTICAL ABERRATIONS; [28]), by usingan electronic camera interfaced with acomputer to measure and process theintensity distribution in the interferencepattern. Several methods are available forthis purpose (see [29]).

7.1Fringe Tracking and Fourier Analysis

Early approaches to fringe analysis werebased on fringe tracking [30]. In orderto analyze a single fringe pattern, it isdesirable to introduce a tilt between theinterfering wavefronts so that a largenumber of nominally straight fringes areobtained. The shape of the fringe willbe modified by the errors of the testwavefront. Fourier analysis of the fringes

Image ofpoint source

Transmittedwave

Pinhole

Test wavefrontPartially

transmittingfilm

Diffracted sphericalreference wave

Fig. 11 The point-diffraction interferometer [27]

950 Interferometry

can then determine the test wavefrontdeviation [31–33].

7.2Phase-shifting Interferometry

Direct measurement of the phase differ-ence between the beams at a uniformlyspaced array of points offers many advan-tages. In order to determine the phaseof the wavefront at each data point, atleast three interferograms are required.The phase difference between the inter-fering beams is usually varied linearlywith time, and the intensity signal isintegrated at each point over a num-ber of equal phase segments coveringone period of the sinusoidal output sig-nal. This technique is often simplified byadjusting the phase difference in equalsteps.

The most common way to accomplishthe phase shift between the object andreference beams is by changing the op-tical path difference between the beamsthrough a shift of the reference mirroralong the optical axis. Other ways include

tilting a glass plate, moving a grating,frequency-shifting, or rotating a half-waveplate or analyzer. Typically, intensity infor-mation from four or five interferogramsare used to calculate the original phasedifference between the wavefronts on apoint-by-point basis [34, 35]. The repeata-bility of measurements is around λ/1000.

Because the phase-calculation algorithmutilizes an arctangent function, which doesnot yield any information on the integralinterference order, it is necessary to usea phase unwrapping procedure to detectchanges in the integral interference orderand remove discontinuities in the retrievedphase [29, 36].

Figure 12 shows a three-dimensionalplot of the errors of a flat surface pro-duced by an interferometer using a digitalphase-measurement system.

Normally, to implement such a phaseunwrapping procedure, it is necessary tohave at least two measurements per fringespacing; this constraint, limits the phasegradients that can be measured. However,techniques are available that can be usedin special situations, with some a priori

109.9 mm

105.7

Fig. 12 Three-dimensional plot of the errors of a flat surface (326 nm Peak-to-Valley)obtained with a phase-measurement interferometer (Courtesy of VeecoInstruments Inc.)

Interferometry 951

knowledge about the test surface, to workaround this limitation [37].

7.3Determining Aberrations

With most optical systems, it is thenconvenient to express the deviations ofthe test wavefront as a linear combina-tion of Zernike circular polynomials inthe form

W(ρ, θ) =n∑

k=0

k∑l=0

ρk

× (Akl cos lθ + Bkl sin lθ), (11)

where ρ and θ are polar coordinatesover the pupil, and (k − l) is an evennumber. If the optical path differencesat a suitably chosen array of pointsare known, the coefficients Akl and Bklcan be calculated from a set of linearequations [38, 39].

8Interference Microscopy

Interference microscopy provides a non-contact method for studies of sur-faces as well as a method for study-ing living cells without the need tostain them.

Two-beam interference microscopeshave been described using optical systemssimilar to the Fizeau and Michelsoninterferometers. For high magnifications,a suitable configuration is that describedby Linnik [40] in which a beam splitterdirects the light onto two identicalobjectives; one beam is incident on thetest surface, while the other is directed tothe reference mirror.

8.1The Mirau Interferometer

The Mirau interferometer permits a verycompact optical arrangement. As shownschematically in Fig. 13, light from anilluminator is incident through the mi-croscope objective on a beam splitter. Thetransmitted beam falls on the test surface,while the reflected beam falls on an alu-minized spot on a reference surface. Thetwo reflected beams are recombined at thesame beam splitter and return throughthe objective.

As shown in Fig. 14, very accuratemeasurements of surface profiles canbe made using phase shifting. With arough surface, the data can be processedto obtain the rms surface roughnessand the autocovariance function of thesurface [41].

8.2The Nomarski Interferometer

Common-path interference microscopesuse polarizing elements to split and re-combine the beams [42]. In the Nomarskiinterferometer (see Fig. 15), two polarizing

Microscope objective

Reference surface

Beam splitter

Sample

Fig. 13 The Mirau interferometer

952 Interferometry

Fig. 14 Pits (approximately 90 nm deep) on the surface of a mass-replicatedCD-ROM. 11 µm × 13 µm field of view (Courtesy of Veeco Instruments Inc.)

Objective

Object

Condenser

Fig. 15 The Nomarski interferometer

prisms (see MICROSCOPY) introduce a lat-eral shear between the two beams.

With small isolated objects, two im-ages are seen covered with fringes thatmap the phase changes introduced bythe object. With larger objects, the in-terference pattern is a measure of thephase gradients, revealing edges and localdefects.

The use of phase-shifting techniques toextract quantitative information from theinterference pattern has been describedby [43].

8.3White-light Interferometry

With monochromatic light, ambiguitiescan arise at discontinuities and stepsproducing a change in the optical pathdifference greater than a wavelength.This problem can be overcome by usinga broadband (white-light) source. Whenthe surface is scanned in height andthe corresponding variations in intensityat each point are recorded, the heightposition corresponding to equal opticalpaths at which the visibility of the fringes

Interferometry 953

Laser Flow

Detector

2q

Fig. 16 Laser–Doppler interferometer for measurements of flow velocities

is a maximum yields the height of thesurface at that point.

The method most commonly used to re-cover the fringe visibility function fromthe fringe intensity is by digital filter-ing [44]. More recent techniques combinephase-shifting techniques with signal de-modulation [45, 46]. More detail on thesetechniques is available in OPTICAL METROL-

OGY.Another technique that can be used

with white light is spectrally resolvedinterferometry [47, 48]. A spectroscope isused to analyze the light from each pointon the interferogram. The optical pathdifference between the beams at this pointcan then be obtained from the intensitydistribution in the resulting channeledspectrum (see Sect. 4.1).

White-light interferometry techniquesused for biomedical applications are gen-erally referred to as optical coherencetomography or coherence radar (BIOMEDI-

CAL IMAGING TECHNIQUES; [49, 50]).

9Interferometric Sensors

Interferometers can be used as sensors forseveral physical quantities.

9.1Laser–Doppler Interferometry

Laser–Doppler interferometry [51] makesuse of the fact that light scattered by

a moving particle has its frequencyshifted.

In the arrangement shown in Fig. 16,two intersecting laser beams makingangles ±θ with the direction of observationare used to illuminate the test field.

The frequency of the beat signal ob-served is

ν = 2v sin θ

λ, (12)

where v is the component of the velocityof the particle in the plane of the beams atright angles to the direction of observation.

Simultaneous measurements of the ve-locity components along orthogonal direc-tions can be made by using two pairs ofilluminating beams (with different wave-lengths) in orthogonal planes.

It is also possible to use a self-mixingconfiguration for velocimetry, in which thelight reflected from the moving object ismixed with the light in the laser cavity. Avery compact system has been describedby [52] using a laser diode operated nearthreshold with an external cavity to ensuresingle-frequency operation.

Very small vibration amplitudes can bemeasured by attaching one of the mirrorsin an interferometer to the vibrating object.If the reflected beam is made to interferewith a reference beam with a fixed-frequency offset, the time-varying outputcontains, in addition to a component atthe offset frequency (the carrier), twosidebands [53]. The vibration amplitude is

954 Interferometry

given by the relation

a =(

λ

2π

)(Is

Ic

), (13)

where Ic and Is are, respectively, the powerin the carrier and the sidebands.

9.2Fiber Interferometers

Since the length of the optical path in anoptical fiber changes when it is stretched,or when its temperature changes, inter-ferometers in which the beams propagatein single-mode optical fibers (see FIBER OP-

TICS and SENSORS, OPTICAL) can be used assensors for a number of physical quanti-ties [54]. High sensitivity can be obtained,because it is possible to have very long,noise-free paths in a very small space.

In the interferometer shown in Fig. 17,light from a laser diode is focused on theend of a single-mode fiber and opticalfiber couplers are used to divide andrecombine the beams. Fiber stretchers are

used to shift and modulate the phase ofthe reference beam. The output goes to apair of photodetectors, and measurementsare made with a heterodyne system or aphase-tracking system [55].

It is also possible, as shown in Fig. 18, touse a length of a birefringent single-modefiber, in a configuration similar to a Fizeauinterferometer, as a temperature-sensingelement [56].

The outputs from the two detectors areprocessed to give the phase retardationbetween the waves reflected from the frontand rear ends of the fiber. Changes intemperature of 0.0005 C can be detectedwith a 1-cm-long sensing element.

Measurements of electric and magneticfields can also be made with fiber interfer-ometers by bonding the fiber sensor to apiezoelectric or magnetostrictive element.In addition, it is possible to multiplex sev-eral optical fiber sensors in a single systemto make measurements of various quanti-ties at a single location or even at differentlocations (see [57, 58]).

Splitter Combiner

Oscillator

Detectors

Detectorsystem

Phaseshifter

Laser

Phase modulator

Optical fiberreference path

Optical fibersensing element

Fig. 17 Interferometer using a single-mode fiber as a sensing element.(Giallorenzi, T. G., Bucaro, J. A., Dandridge, A., Sigel, Jr G. H., Cole, J. H.,Rashleigh, S. C., Priest, R. G. (1982), IEEE J. Quantum Electron. QE-18, 626–665 [55] IEEE, 1982. Reproduced with permission.)

Interferometry 955

Laser

DetectorSignal

processing

Detector

Polarizingbeam splitter

l/2 plate

Monomode highbirefringence fiber

Beamsplitter

Fig. 18 Interferometer using a birefringent single-mode fiber as asensing element [56]

9.3Rotation Sensing

Another application of fiber interferom-eters is in rotation sensing [59, 60]. Theconfiguration used, in which the two wavestraverse a closed multiturn loop in oppo-site directions, is the equivalent of a Sagnacinterferometer (see Sect. 3.3). A typical sys-tem is shown in Fig. 19 [61].

10Interference Spectroscopy

Interferometric techniques are now usedwidely in high-resolution spectroscopy(see SPECTROSCOPY, LASER) because theyoffer, in addition to higher resolution, ahigher throughput.

10.1Etendue of an Interferometer

The throughput of an optical system isproportional to a quantity known as its

etendue (see OPTICAL RADIATION SOURCES

AND STANDARDS).In the optical system shown in Fig. 20,

the effective areas AS and AD of thesource and detector are images of eachother.

The etendue of the system is

E = ASS = ADD, (14)

where S is the solid angle subtended bythe lens LS at the source and D is thesolid angle subtended by the lens LD atthe detector.

Since the etendue of a conventional spec-troscope is limited by the entrance slit, amuch higher etendue can be obtained withan interferometer.

10.2The Fabry–Perot Interferometer

The Fabry–Perot interferometer (FPI) [62]uses multiple-beam interference betweentwo flat, parallel surfaces coated with

956 Interferometry

Polarizationcontroller

PolarizationcontrollerPolarizer

Dead ends

Coupler Coupler

Fiberloop

Phasemodulator

Detector

Modulated signal

Rotation signal

Chart recorder Lock-in amplifier AC generator

Reference signal

Semiconductorlaser

Fig. 19 Fiber interferometer for rotation sensing [61]

Source

AS

ΩS LS LDΩD

Detector

AD

Interferometer

Fig. 20 Etendue of an interferometer

semitransparent, highly reflecting coat-ings. With a fixed spacing d, any singlewavelength produces a system of sharp,bright rings (fringes of equal inclination),defined by Eq. (6), centered on the normalto the surfaces.

For any angle of incidence, with abroadband source, it also follows fromEq. (6) that the separation of succes-sive intensity maxima corresponds to a

wavelength difference λ2/2nd. This wave-length difference, known as the freespectral range (FSR), is the range ofwavelengths that can be handled by theFPI without successive interference ordersoverlapping.

The resolving power of the FPI isobtained by dividing the FSR by thefinesse (see Eq. 7) and is given by therelation

Interferometry 957

R = λ

λ

=(

2nd

λ

)[πR1/2

1 − R

], (15)

where λ is the half-width of the peaks,and R is the reflectance of the surfaces.

One way to overcome the limited FSR ofthe FPI is by imaging the fringes onto theslit of a spectrometer, but this procedurelimits the etendue of the system. A betterway is to use two or more FPIs, withdifferent values of d, in series.

Another important characteristic of anFPI is the contrast factor, defined by theratio of the intensities of the maxima andminima, which is

C =[

1 + R

1 − R

]2

. (16)

For applications such as Brillouin spec-troscopy, in which a weak spectrum linemay be masked by the background due toa neighboring strong spectral line, a muchhigher contrast factor can be obtained bypassing the light several times through thesame FPI [63].

A much higher throughput can beobtained with the confocal Fabry–Perotinterferometer shown in Fig. 21 that uses

M M

Fig. 21 Confocal Fabry–Perot interferometer

two spherical mirrors separated by adistance equal to their radius of curvature,so that their foci coincide. Since the opticalpath difference is independent of the angleof incidence, a uniform field is obtained.An extended source can be used, and thetransmitted intensity is recorded as theseparation of the plates is varied [64].

10.3Wavelength Measurements

Accurate measurements of the wavelengthof the output from a tunable laser, suchas a dye laser, can be made with aninterferometric wavelength meter.

In the dynamic wavelength meter shownin Fig. 22, a beam from the dye laser aswell as a beam from a reference laser,whose wavelength is known, traverse thesame two paths. The wavelength of the dyelaser is determined by counting fringessimultaneously at both wavelengths, as theend reflector is moved [65].

Reference laser (l2)

Dye laser (l1)

Detector D1 (l1)

Detector D2 (l2)

Movablecube-corner

Fig. 22 Dynamic wavelength meter [65]

958 Interferometry

In a simpler arrangement [66], thefringes of equal thickness formed in awedged air film are imaged on a linear de-tector array. The spacing of the fringes isused to evaluate the integral interferenceorder, and their position to determine thefractional interference order.

10.4Laser Linewidth

The extremely small spectral bandwidth ofthe output from a laser can be measuredby mixing, as shown in Fig. 23, light fromthe laser with a reference beam from thesame laser that has undergone a frequencyshift and a delay [67].

10.5Fourier-transform Spectroscopy

Major applications of Fourier-transformspectroscopy include measurements ofinfrared absorption spectra as well asemission spectra from faint astronomicalobjects (see SPECTROMETERS, ULTRAVIOLET

AND VISIBLE LIGHT and SPECTROMETERS, IN-

FRARED).With a scanning spectrometer, the total

time of observation T is divided between,say, m elements of the spectrum. Sincein the infrared, the main source of noiseis the detector, the signal-to-noise (S/N)

ratio is reduced by a factor m1/2. Thisreduction in the S/N ratio can be avoidedby varying the optical path difference inan interferometer linearly with time, inwhich case each element of the spectrumgenerates an output modulated at afrequency that is inversely proportionalto its wavelength. It is then possible torecord all these signals simultaneously(or, in other words, to multiplex them)and then, by taking the Fourier transformof the recording (see FOURIER AND OTHER

TRANSFORM METHODS), to recover thespectrum [68–70].

As shown in Fig. 24, a Fourier-transform spectrometer is basically aMichelson interferometer illuminatedwith an approximately collimated beamfrom the source whose spectrum is to be

Laser Opticalisolator

Acousto-optic

modulatorB1

B2

Single-modeoptical fiber

Detector

Detector

SubtractorSpectrumanalyzer

Fig. 23 Measurement of laser linewidth by heterodyne interferometry [67]

Interferometry 959

Detector

Movable reflector

Source

Fig. 24 Fourier-transform spectrometer

recorded. The mirrors are often replacedby ‘‘cat’s eye’’ reflectors to minimizeproblems due to tilting as they are moved.

The interferogram is then sampled ata number of equally spaced points. Toavoid ambiguities (aliasing) the changein the optical path difference betweensamples must be less than half the short-est wavelength in the spectrum. Finally,the spectrum is computed using the fastFourier-transform algorithm [71]. Errorsin the computed spectrum can be reducedby a process called apodization, wherethe interference signal is multiplied bya symmetrical weighting function whosevalue decreases gradually with the opticalpath difference.

11Nonlinear Interferometers

The high light intensity available withpulsed lasers has opened up completely

new areas of interferometry based onthe use of nonlinear optical materials(see NONLINEAR OPTICS).

11.1Second-harmonic Interferometers

Second-harmonic interferometers producea fringe pattern corresponding to thephase difference between two second-harmonic waves generated at differ-ent points in the optical path fromthe original wave at the fundamentalfrequency.

Figure 25 is a schematic of an inter-ferometer using two frequency-doublingcrystals that can be considered as ananalog of the Mach–Zehnder interferom-eter [72].

In this interferometer, the infraredbeam from a Q-switched Nd:YAG laser(λ1 = 1.06 µm) is incident on a frequency-doubling crystal, and the green (λ2 =

960 Interferometry

IR laser

Doubler DoublerTest piece

Green

Fig. 25 Second-harmonic interferometer using two frequency-doubling crystals

0.53 µm) and infrared beams emergingfrom this crystal pass through the testpiece. At the second crystal, the infraredbeam undergoes frequency doubling toproduce a second green beam that in-terferes with the one produced at thefirst crystal. The interference order at anypoint is

N(x, y) = (n2 − n1) d(x, y)

λ2, (17)

where d(x, y) is the thickness of the testspecimen at any point (x, y) and n1 and n2

are its refractive indices for infrared andgreen light, respectively.

Other types of interferometers that areanalogs of the Fizeau, Twyman, and point-diffraction interferometers have also beendescribed [73, 74].

11.2Phase-conjugate Interferometers

In a phase-conjugate interferometer, thetest wavefront is made to interfere with itsconjugate, eliminating the need for a ref-erence wave and doubling the sensitivity.

In the phase-conjugate interferometershown in Fig. 26, which can be regardedas an analog of the Fizeau interferometer,a partially reflecting mirror is placed infront of a single crystal of barium titanate,which functions as a self-pumped phase-conjugate mirror [75, 76].

An interferometer in which both mirrorshave been replaced by a single-phaseconjugator has the unique property thatthe field of view is normally completelydark and is unaffected by misalignmentor by air currents. However, because ofthe delay in the response of the phase

Input wavefront

Beam splitter

Outputwavefronts

Referencesurface

Phase-conjugatemirror

Fig. 26 Phase-conjugate interferometer [76]

Interferometry 961

conjugator, any sudden local change in theoptical path produces a bright spot thatslowly fades away [77].

11.3Measurement of Nonlinear Susceptibilities

A modified Twyman–Green interferom-eter can be used for measurements ofnonlinear susceptibilities. A system thatcan be used to measure the relative phaseshift between two-phase conjugators, aswell as the ratio of their susceptibilities,and yields high sensitivity, even with weaksignals, has been described by [78].

12Interferometric Imaging

Interferometric imaging started with thedevelopment of techniques to measurethe diameters of stars that could notbe resolved with conventional telescopes(see ASTRONOMICAL TELESCOPES AND IN-

STRUMENTATION).Michelson’s stellar interferometer [79]

used the fact that the angular diameter of astar can be calculated from observationsof the visibility of the fringes in aninterferometer using light from the starreaching the surface of the earth at twopoints separated by a known distance.

If we assume the star to be a uniformcircular source with an angular diameter2α, and D is the separation of two mirrorsreceiving the light from the star andfeeding it to a telescope, as shown inFig. 27, the visibility of the fringes is

V = 2J1(u)

u, (18)

where u = 2παD/λ and J1 is a first-orderBessel function. The fringe visibility drops

Telescope

Fig. 27 Michelson’s stellar interferometer

to zero when

D = 1.22λ

2α. (19)

Measurements with Michelson’s stellar in-terferometer over baselines longer than6 m presented serious difficulties becauseof the difficulty of maintaining the opticalpath difference between the beams stableand small enough not to affect the visibilityof the fringes. However, modern detection,control, and data handling techniques havemade possible a new version of Michel-son’s stellar interferometer [80] designedto make measurements over baselines upto 640 m.

12.1The Intensity Interferometer

The problem of maintaining the equalityof the two paths was minimized inthe intensity interferometer, which usedmeasurements of the degree of correlationbetween the fluctuations in the outputsof two photodetectors at the foci of two

962 Interferometry

large light collectors separated by a variabledistance [81, 82].

The actual instrument used light collec-tors operated with separations up to 188 m.With a bandwidth of 100 MHz it was onlynecessary to equalize the two optical pathsto within 30 cm, but, because of the narrowbandwidth, measurements could only bemade on 32 of the brightest stars.

12.2Heterodyne Stellar Interferometers

In these instruments, as shown in Fig. 28,light from the star is received by twotelescopes and mixed with light from alaser at two photodiodes. The resultingheterodyne signals are multiplied in acorrelator. The output signal is a measure

Telescope

5 MHz offset

Heterodyne signals (< 1500 MHz)

Delay lines

Multiplier

Processor

5 MHz signal

Computer

Fringe amplitude

Laser

Detector Detector

Laser

Telescope

Fig. 28 Schematic of a heterodyne stellar interferometer [83]

Interferometry 963

of the degree of coherence of the wavefields at the two photo detectors [83].

As with the intensity interferometer,this technique only requires the two op-tical paths to be equalized to within afew centimeters; however, the sensitivityis higher, since it is proportional to theproduct of the intensities of the laserand the star. An infrared interferome-ter comprising two telescopes with anaperture of 1.65 m has been constructed,capable of yielding an angular resolu-tion of 0.001 second of arc [84]. Largertelescopes are planned (see ASTRONOMICAL

TELESCOPES AND INSTRUMENTATION).

12.3Stellar Speckle Interferometry

Stellar speckle interferometry makes useof the fact that, due to local inhomo-geneities in the earth’s atmosphere, theimage of a star produced by a largetelescope, when observed under high mag-nification, has a speckle structure [85](see also ASTRONOMICAL TELESCOPES AND

INSTRUMENTATION). However, individualspeckles have dimensions correspond-ing to a diffraction-limited image of thestar. Reference [86] showed that a high-resolution image could be extracted from anumber of such speckled images recordedwith sufficiently short exposures to freezethe speckles.

While the angular resolution that canbe obtained by speckle interferometry islimited by the aperture of the telescope, ithas been applied successfully to a numberof problems, including the study of closedouble stars.

12.4Telescope Arrays

The ultimate objective would be the abil-ity to produce high-resolution images of

stars. Unfortunately, with a two-elementinterferometer, it is only possible to ob-tain information on the fringe amplitudebecause the value of the phase is affectedby instrumental and atmospheric effects.However, with a triangular array, the clo-sure phase is determined only by thecoherence function. As the number of ele-ments increases, the image becomes betterconstrained [87].

Some images have already been ob-tained from a large, multielement inter-ferometer (the Cambridge Optical Aper-ture Synthesis Telescope) [88] and severalother telescope arrays are nearing comple-tion (see ASTRONOMICAL TELESCOPES AND

INSTRUMENTATION).

13Space-time and Gravitation

Michelson’s classical experiment to testthe hypothesis of a stationary ether showedan effect that was less than one-tenth ofthat expected. This experiment was re-peated by [89], with a much higher degreeof accuracy, by locking the frequency ofa He-Ne laser to a resonance of a ther-mally isolated Fabry–Perot interferometermounted along with it on a rotating hor-izontal granite slab. When the frequencyof this laser was compared to that of astationary, frequency-stabilized laser, thefrequency shifts were found to be lessthan 1 part in 106 of those expected with astationary ether.

13.1Gravitational Waves

It follows from the general theory ofrelativity that binary systems of neutronstars, collapsing supernovae, and blackholes should be the sources of gravita-tional waves.

964 Interferometry

Because of the transverse quadrupolarnature of a gravitational wave, the localdistortion of space-time due to it stretchesspace in a direction normal to the directionof propagation of the wave, and shrinksit along the orthogonal direction. Thislocal strain could, therefore, be measuredby a Michelson interferometer in whichthe beam splitter and the end mirrorsare attached to separate, freely suspendedmasses [90, 91].

Theoretical estimates of the intensityof gravitational radiation due to variouspossible events, suggest that a sensitivityto strain of the order of 10−21 over abandwidth of a kilohertz would be needed.This would require an interferometer withunrealistically long arms.

One way to obtain a substantial increasein sensitivity is, as shown in Fig. 29, byusing two identical Fabry–Perot cavities,with their mirrors mounted on freelysuspended test masses, as the arms of

the interferometer. The frequency of thelaser is locked to a transmission peak ofone interferometer, while the optical pathlength in the other is continually adjustedso that its peak transmittance is also at thisfrequency [92].

A further increase in sensitivity canbe obtained by recycling the availablelight. Since the interferometer is normallyadjusted so that observations are madeon a dark fringe, to avoid overloadingthe detector, most of the light is returnedtoward the source and is lost. If this lightis reflected back into the interferometer inthe right phase, by an extra mirror placedin the input beam, the amount of lighttraversing the arms of the interferometercan be increased substantially [93].

Two other techniques that can be com-bined with these techniques for obtainingeven higher sensitivity are signal recy-cling [94] and resonant side-band extrac-tion [95].

End mirrors

T = 0%

T = 3%

T = 3%

T = 3%

T = 50%

T = 0%

Cornermirrors

Todetector

Recyclingmirror

Fromlaser

L1 L 2

Fig. 29 Schematic of an interferometric gravitational wave detector [91]

Interferometry 965

13.2LIGO

The laser interferometer gravitational ob-servatory (LIGO) project [96, 97] involvesthe construction of three laser interferom-eters with arms up to 4-km long, two atone site and the third at another site sep-arated from the first by almost 3000 km.The test masses and the optical paths inthese interferometers are housed in a vac-uum. Correlating the outputs of the threeinterferometers should make it possible todistinguish the signals due to gravitationalwaves from the bursts of instrumental andenvironmental noise.

13.3Limits to Measurement

The limit to measurements of such smalldisplacements is ultimately related to thenumber of photons n that pass throughthe interferometer in the measurementtime. The resulting uncertainty in mea-surements of the phase difference betweenthe beams has been shown to be [98]

φ ≥ 1

2√

n, (20)

and is known as the standard quantumlimit (SQL).

14Holographic Interferometry

Holography (see HOLOGRAPHY andOPTICAL TECHNIQUES FOR MECHANICAL

MEASUREMENT) makes it possible tostore and reconstruct a perfect three-dimensional image of an object. Thereconstructed wave can then be made tointerfere with the wave generated by theobject to produce fringes that contour, in

real time, any changes in the shape of theobject. Alternatively, two holograms can berecorded with the object in two differentstates and the wavefronts reconstructedby these two holograms can be madeto interfere.

Since holographic interferometry makesit possible to measure, with very high pre-cision, changes in the shape of objects withrough surfaces, it has found many applica-tions including nondestructive testing andvibration analysis [99–101].

14.1Strain Analysis

The phase difference at any point (x, y) inthe interferogram is given by the relation(see Fig. 30)

φ = L(x, y) · (k1 − k2) = L(x, y) · K,

(21)

where L(x, y) is the vector displacement ofthe corresponding point on the surface ofthe object, k1 and k2 are the propagationvectors of the incident and scattered light,and K = k1 − k2 is known as the sensitivityvector [102].

To evaluate the vector displacements(out-of-plane and in-plane), it is convenientto use a single direction of observation and

Deformedobject

Toobserver

Image Incidentlight

k1

k2

P′

K P

L

Fig. 30 Phase difference produced by adisplacement of the object

966 Interferometry

record four holograms with the object illu-minated from two different angles in thevertical plane and two different angles inthe horizontal plane. Phase shifting is usedto obtain the phase differences at a networkof points [103]. These values can then beused, along with information on the shapeof the object, to obtain the strains.

14.2Vibration Analysis

One way to study the vibrating objects is torecord a hologram of the vibrating objectwith an exposure time that is much longerthan the period of the vibration [104]. Theintensity at any point (x, y) in the image isthen given by the relation

I(x, y) = I0(x, y)J0[K · L(x, y)], (22)

where I0(x, y) is the intensity with thestationary object, J0 is a zero-order Besselfunction and L(x, y) is the amplitude ofvibration of the object. The fringes ob-served (time-average fringes) are contoursof equal vibration amplitude, with the dark

fringes corresponding to the zeros of theBessel function.

Alternatively, a hologram can be recor-ded of the stationary object, and the real-time interference pattern obtained with thevibrating object can be viewed using stro-boscopic illumination. A brighter imagecan be obtained by recording the hologramwith stroboscopic illumination, synchro-nized with the vibration cycle, and viewingthe interference fringes formed with thestationary object, using continuous illu-mination. Phase-shifting techniques canthen be used to map the instantaneousdisplacement of the vibrating object (seeFig. 31) [105].

14.3Contouring

The simplest method of contouring anobject is by recording two hologramswith the object illuminated from slightlydifferent angles. More commonly usedtechniques are two-wavelength contouringand two-refractive-index contouring.

In two-wavelength contouring [106], atelecentric lens system is used to image

1.0

0.5

25 50(mm)

75

(µm

)

Fig. 31 Three-dimensional plot of the instantaneous displacement of a metal platevibrating at 231 Hz obtained by stroboscopic holographic interferometry usingphase shifting[105]

Interferometry 967

the object on the hologram plane andexposures are made with two differentwavelengths, λ1 and λ2. When the holo-gram is illuminated with one of thewavelengths (say λ2), fringes are seen con-touring the reconstructed image, separatedby an increment of height

|δz| = λ1λ2

2(λ1 − λ2)(23)

In two-refractive-index contouring [107],the object is placed in a cell with aglass window and imaged by a telecen-tric system.

Two holograms are recorded on a plateplaced near the stop of the telecentricsystem with the cell filled with liquidshaving refractive indices n1 and n2, re-spectively. Contours are obtained with aspacing

|δz| = λ

2(n1 − n2). (24)

Digital phase-shifting techniques canbe used with both these methods ofholographic contouring. Figure 32 showsa three-dimensional plot of a wear markon a flat surface obtained by phase

shifting, using the two-refractive-indextechnique [108].

15Moire Techniques

Moire techniques complement hologra-phic interferometry and can be used wherea contour interval greater than 10 µm is re-quired [109] (see also OPTICAL TECHNIQUES

FOR MECHANICAL MEASUREMENT).A simple way to obtain Moire fringes

is to project interference fringes (or agrating) onto the object and view itthrough a grating of approximately thesame spacing. The contour interval isdetermined by the fringe (grating) spacingand the angle between the illuminationand viewing directions. Phase shifting ispossible by shifting one grating or theprojected fringes.

15.1Grating Interferometry

The in-plane displacements of nearlyflat objects can be measured with

200

100

0

(µm

)

Fig. 32 Three-dimensional plot of a wear mark on a flat surfaceobtained by phase shifting, using the two-refractive-indextechnique [108]

968 Interferometry

submicrometer sensitivity by gratinginterferometry [109, 110].

A reflection grating is attached to theobject under test with a suitable adhesiveand illuminated by two coherent beamssymmetrical to the grating normal. Theinterference pattern produced by the twodiffracted beams reflected from the gratingyields a map of the in-plane displacements.Polarization techniques can be used forphase shifting.

16Speckle Interferometry

The image of an object illuminated by alaser is covered with a stationary gran-ular pattern known as a speckle pat-tern (see SPECKLE AND SPECKLE METROL-

OGY). Speckle interferometry [111] utilizesinterference between the speckled imageof an object illuminated by a laser and areference beam derived from the samelaser. Any change in the shape of theobject results in local changes in the in-tensity distribution in the speckle pattern.If two photographs of the speckled imageare superimposed, fringes are obtained,corresponding to the degree of correlationof the two speckle patterns that contourthe changes in shape of the surface [112].

Speckle interferometry can be a verysimple way of measuring the in-plane

displacements using an optical system inwhich the surface is illuminated by twobeams making equal but opposite anglesto the normal.

16.1Electronic Speckle Pattern Interferometry(ESPI)

Measurements can be made at video ratesusing a TV camera interfaced to a com-puter [111, 113]. As shown in Fig. 33, theobject is imaged on a CCD array along witha coaxial reference beam. This techniquewas originally known as electronic specklepattern interferometry (ESPI).

If an image of the object in its originalstate is subtracted from an image ofthe object at a later stage, regions inwhich the speckle pattern has not changed,corresponding to the condition

K · L(x, y) = 2mπ, (25)

where m is an integer, appear dark, whileregions where the pattern has changed arecovered with bright speckles [114, 115].

This technique is also known asTV holography.

16.2Phase-shifting Speckle Interferometry

Each speckle can be regarded as anindividual interference pattern and the

ObjectTV

camera

Reference beam

Single-modefiber

Fig. 33 System for ESPI

Interferometry 969

phase difference between the two beamsat this point can be measured by phaseshifting, with the object before and afteran applied stress. The result of subtractingthe second set of values from the first isthen a contour map of the deformation ofthe object [116, 117].

Further developments in speckleinterferometry with phase-measurementtechniques, high-resolution CCD arrays,and real-time processing have createdmany techniques encompassing what isnow most commonly known as digitalholography (see OPTICAL TECHNIQUES

FOR MECHANICAL MEASUREMENT andHOLOGRAPHY).

16.3Vibrating Objects

If the period of the vibration is smallcompared to the exposure time or thescan time of the camera, the contrastof the speckles at any point is given bythe expression

C = 1 + 2αJ20[K · L(x, y)]1/2

1 + α, (26)

where α is the ratio of the intensities ofthe reference beam and the object beam,K is the sensitivity vector and L(x, y) is thevibration amplitude at that point. Regionscorresponding to the zeros of the J0 Besselfunction appear as dark fringes.

Phase-shifting techniques can also beapplied to the analysis of vibrations [118].

Glossary

Beam splitter: An optical element thatdivides a single beam of light into twobeams of the same wave form.

Coherence: A complex quantity whosemagnitude denotes the correlation be-tween two wave fields; its phase de-notes the effective phase difference be-tween them.

Degree of Coherence: The value of thecoherence expressed as a fraction ofthat for complete correlation between thewave fields.

Fringes of Equal Inclination: Interferencefringes created from two collinear in-terfering beams having wavefronts withdifferent radii of curvature.

Fringes of Equal Thickness: Interferencefringes created with collimated beamswhen the optical path difference dependsonly on the thickness and refractive index.

Interference Order: The number of wave-lengths in the optical path differencebetween two interfering beams.

Interferogram: The varying part of aninterference pattern, after subtracting anyuniform background.

Moire Fringes: Relatively coarse fringesproduced by the superposition of twofine fringe patterns with slightly differentspacings or orientations.

Optical Path Difference: The differencein the optical path length between twointerfering beams.

Optical Path Length: The product of therefractive index of the medium traversedby a beam and the length of the path inthe medium.

Optical Phase: The resultant phase of alight beam after allowing for changes dueto the optical path traversed and reflectionat any surfaces.

970 Interferometry

Phase Shifting: A technique that shifts thephase of one interfering beam relative tothe other in order to determine opticalpath difference from the intensity in aninterference fringe pattern.

Region of Localization: The region in whichinterference fringes are observed withmaximum contrast.

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Further Reading

Adrian, R. J. (Ed.) (1993), Laser Doppler Velocime-try, Vol. MS78. Bellingham: SPIE.

Brown, G. M. (Ed.) (2000), Modern Interferometry,Selected SPIE Papers on CD-ROM, Vol. 15.Bellingham: SPIE.

Hariharan, P. (1985), Optical Interferometry. Syd-ney: Academic Press.

Hariharan, P. (Ed.) (1991), Interferometry, Vol.MS28. Bellingham: SPIE.

Hariharan, P. (1992), Basics of Interferometry. SanDiego: Academic Press.

Hariharan, P. (1995), Optical Holography. Cam-bridge: Cambridge University Press.

Hariharan, P., Malacara-Hernandez, D. (Eds.)(1995), Interference, Interferometry and

Interferometric Metrology, Vol. MS110. Belling-ham: SPIE.

Lawson, P. R. (Ed.) (1997), Long Baseline Stel-lar Interferometry, Vol. MS139. Bellingham:SPIE.

Meinischmidt, P., Hinsch, K. D., von Ossiet-zky, C., Sirohi, R. S. (Eds.) (1996), Elec-tronic Speckle Pattern Interferometry: Princi-ples and Practice, Vol. MS132. Bellingham:SPIE.

Sirohi, R. S., Hinsch, K. D., von Ossietzky, C.(Eds.) (1998), Holographic Interferometry: Prin-ciples and Techniques, Vol. MS144. Bellingham:SPIE.

Steel, W. H. (1983), Interferometry. Cambridge:Cambridge University Press.

Udd, E. (Ed.) (1991), Fiber Optic Sensors: AnIntroduction for Engineers and Scientists. NewYork: John Wiley & Sons.

Udd, E., Tatam, R. P. (Eds.) (1994), Interferometry’94: Interferometric Fiber Sensing: 16–20 May1994, Warsaw, Poland, SPIE Proceedings No.2341, Bellingham: SPIE.

975

Laser Cooling and Trapping of Neutral Atoms

Harold J. MetcalfDepartment of Physics, State University of New York, Stony Brook, N.Y. 11794-3800, USAPhone: (631) 632-8185; Fax: (631) 632-8176; e-mail: [email protected]

Peter van der StratenDebye Institute, Department of Atomic and Interface Physics, Utrecht University, 3508 TAUtrecht, The NetherlandsPhone: 31-30-2532846; Fax: 31-30-2537468; e-mail: [email protected]

AbstractThis article presents a review of some of the principal techniques of laser cooling andtrapping that have been developed during the past 20 years. Its approach is primarilyexperimental, but its quantitative descriptions are consistent in notation with most ofthe theoretical literature.

Keywordslaser cooling; atom trapping; optical lattice; Bose–Einstein condensation.

1 Introduction 9771.1 Temperature and Entropy 9771.2 Phase Space Density 9782 Optical Forces on Neutral Atoms 9782.1 Radiative Optical Forces 9792.2 Dipole Optical Forces 9792.3 Density Matrix Description of Optical Forces 9802.3.1 Introduction 9802.3.2 Open Systems and the Dissipative Force 9802.3.3 Solution of the OBEs in Steady State 981

976 Laser Cooling and Trapping of Neutral Atoms

2.3.4 Radiative and Dipole Forces 9822.3.5 Force on Moving Atoms 9823 Laser Cooling 9823.1 Slowing Atomic Beams 9823.2 Optical Molasses 9843.2.1 Doppler Cooling 9843.2.2 Doppler Cooling Limit 9853.2.3 Atomic Beam Collimation – One-dimensional Optical

Molasses – Beam Brightening 9863.2.4 Experiments in Three-dimensional Optical Molasses 9873.3 Cooling Below the Doppler Limit 9883.3.1 Introduction 9883.3.2 Linear ⊥ Linear Polarization Gradient Cooling 9883.3.3 Origin of the Damping Force 9903.3.4 The Limits of Sisyphus Laser Cooling 9914 Traps for Neutral Atoms 9914.1 Dipole Force Optical Traps 9924.1.1 Single-beam Optical Traps for Two-level Atoms 9924.1.2 Blue-detuned Optical Traps 9934.2 Magnetic Traps 9944.2.1 Introduction 9944.2.2 Magnetic Confinement 9944.2.3 Classical Motion of Atoms in a Quadrupole Trap 9964.2.4 Quantum Motion in a Trap 9974.3 Magneto-optical Traps 9974.3.1 Introduction 9974.3.2 Cooling and Compressing Atoms in an MOT 9994.3.3 Capturing Atoms in an MOT 9994.3.4 Variations on the MOT Technique 10005 Optical Lattices 10005.1 Quantum States of Motion 10005.2 Properties of 3-D Lattices 10025.3 Spectroscopy in 3-D Lattices 10035.4 Quantum Transport in Optical Lattices 10046 Bose–Einstein Condensation 10056.1 Introduction 10056.2 Evaporative Cooling 10056.2.1 Simple Model 10066.2.2 Application of the Simple Model 10076.2.3 Speed of Evaporation 10086.2.4 Limiting Temperature 10086.3 Forced Evaporative Cooling 10097 Conclusion 1010

Glossary 1010References 1012

Laser Cooling and Trapping of Neutral Atoms 977

1Introduction

The combination of laser cooling andatom trapping has produced astoundingnew tools for atomic physicists [1]. Theseexperiments require the exchange of mo-mentum between atoms and an opticalfield, usually at a nearly resonant fre-quency. The energy of light hω changesthe internal energy of the atom, and theangular momentum h changes the orbitalangular momentum of the atom, as de-scribed by the well-known selection rule = ±1. By contrast, the linear momen-tum of light p = hω/c = hk cannot changethe internal atomic degrees of freedom,and therefore must change the momen-tum of the atoms in the laboratory frame.The force resulting from this momentumexchange between the light field and theatoms can be used in many ways to controlatomic motion, and is the subject of thisarticle.

1.1Temperature and Entropy

The idea of ‘‘temperature’’ in laser cool-ing requires some careful discussion anddisclaimers. In thermodynamics, temper-ature is carefully defined as a parameterof the state of a closed system in thermalequilibrium with its surroundings. This,of course, requires that there be thermalcontact, that is, heat exchange, with the en-vironment. In laser cooling, this is clearlynot the case because a sample of atomsis always absorbing and scattering light.Furthermore, there is essentially no heatexchange (the light cannot be consideredas heat even though it is indeed a formof energy). Thus, the system may verywell be in a steady state situation, but cer-tainly not in thermal equilibrium, so the

assignment of a thermodynamic ‘‘temper-ature’’ is completely inappropriate.

Nevertheless, it is convenient to use thelabel of temperature to describe an atomicsample whose average kinetic energy 〈Ek〉has been reduced by the laser light, andthis is written simply as kBT/2 = 〈Ek〉,where kB is the Boltzmann’s constant (forthe case of one dimension, 1-D). It mustbe remembered that this temperatureassignment is absolutely inadequate foratomic samples that do not have aMaxwell–Boltzmann velocity distribution,whether or not they are in thermal contactwith the environment; there are infinitelymany velocity distributions that have thesame value of 〈Ek〉 but are so differentfrom one another that characterizing themby the same ‘‘temperature’’ is a severeerror. (In the special case where thereis a true damping force, F ∝ −v, andwhere the diffusion in momentum spaceis a constant independent of momentum,solutions of the Fokker–Planck equationcan be found analytically and can leadto a Maxwell–Boltzmann distribution thatdoes indeed have a temperature.)

Since laser cooling decreases the tem-perature of a sample of atoms, there isless disorder and therefore less entropy.This seems to conflict with the second lawof thermodynamics, which requires theentropy of a closed system to always in-crease with time. The explanation lies inthe consideration of the fact that in lasercooling, the atoms do not form a closedsystem. Instead, there is always a flow oflaser light with low entropy into the sys-tem and fluorescence with high entropyout of it. The decrease of entropy of theatoms is accompanied by a much largerincrease in entropy of the light field. En-tropy considerations for a laser beam arefar from trivial, but recently it has beenshown that the entropy lost by the atoms


is many orders of magnitude smaller thanthe entropy gained by the light field.

1.2Phase Space Density

The phase space density ρ(r, p, t) isdefined as the probability that a singleparticle is in a region dr around r andhas momentum dp around p at time t.In classical mechanics, ρ(r, p, t) is just thesum of the ρ(r, p, t) values of each of the Nparticles in the system divided by N. Sincethe phase space density is a probability, itis always positive and can be normalizedover the six-dimensional volume spannedby position r and momentum p. For a gasof cold atoms, it is convenient to choose theelementary volume for ρ(r, p, t) to be h3,so it becomes the dimensionless quantity

ρφ = nλ3deB, (1)

where λdeB is the deBroglie wavelength ofthe atoms in the sample and n is theirspatial density.

The Liouville theorem requires that ρφ

cannot be increased by using conservativeforces. For instance, in light optics onecan focus a parallel beam of light with alens to a small spot. However, that simplyproduces a high density of light rays in thefocus in exchange for the momentum partof ρφ because the beam entering the lensis parallel but the light rays are divergentat the focus.

For classical particles, the same principleapplies. By increasing the strength of thetrapping potential of particles in a trap, onecan increase the density of the atoms in thetrap but at the same time, the compressionof the sample results in a temperatureincrease, leaving the phase space densityunchanged.

In order to increase the phase spacedensity of an atomic sample, it is necessary

to use a force that is not conservative,such as a velocity-dependent force. In lasercooling, the force on the atoms can bea damping force, that is, always directedopposite to the atomic velocity, so thatthe momentum part of ρφ increases. Thisprocess arises from the irreversible natureof spontaneous emission.

2Optical Forces on Neutral Atoms

The usual form of electromagnetic forces isgiven by F = q(E + v × B), but for neutralatoms, q = 0. The next order of force is thedipole term, but this also vanishes becauseneutral atoms have no inherent dipolemoment. However, a dipole moment canbe induced by a field, and this is mostefficient if the field is alternating near theatomic resonance frequency. Since thesefrequencies are typically in the opticalrange, dipole moments are efficientlyinduced by shining nearly resonant lighton the atoms.

If the light is absorbed, the atommakes a transition to the excited state,and the subsequent return to the groundstate can be either by spontaneous or bystimulated emission. The nature of theoptical force that arises from these twodifferent processes is quite different andwill be described separately.

The spontaneous emission case isdifferent from the familiar quantum-mechanical calculations using state vec-tors to describe the system. Spontaneousemission causes the state of the sys-tem to evolve from a pure state intoa mixed state and so the density ma-trix is needed to describe it. Sponta-neous emission is an essential ingredientfor the dissipative nature of the opticalforces.


2.1Radiative Optical Forces

In the simplest case–the absorption ofwell-directed light from a laser beam–themomentum exchange between the lightfield and the atoms results in a force

F = dpdt

= hkγp, (2)

where γp is the excitation rate of the atoms.The absorption leaves the atoms in theirexcited state, and if the light intensityis low enough that they are much morelikely to return to the ground state byspontaneous emission than by stimulatedemission, the resulting fluorescent lightcarries off momentum hk in a randomdirection. The momentum exchange fromthe fluorescence averages zero, so the nettotal force is given by Eq. (2).

The excitation rate γp depends on thelaser detuning from atomic resonance δ ≡ωl − ωa, where ωl is the laser frequencyand ωa is the atomic resonance frequency.This detuning is measured in the atomicrest frame, and it is necessary that theDoppler-shifted laser frequency in the restframe of the moving atoms be used tocalculate γp. In Sect. 2.3.3, we find thatγp for a two-level atom is given by theLorentzian

γp = s0γ /2

1 + s0 + [2(δ + ωD)/γ ]2, (3)

where γ ≡ 1/τ is an angular frequencycorresponding to the natural decay rate ofthe excited state. Here, s0 = I/Is is the ratioof the light intensity I to the saturationintensity Is ≡ πhc/3λ3τ , which is a fewmW cm−2 for typical atomic transitions.The Doppler shift seen by the movingatoms is ωD = −k · v (note that k oppositeto v produces a positive Doppler shift for

the atoms). The force is thus velocity-dependent and the experimenters’ task isto exploit this dependence to the desiredgoal, for example, optical friction for lasercooling.

The maximum attainable decelerationis obtained for high intensities of light.High-intensity light can produce faster ab-sorption, but it also causes equally faststimulated emission; the combination pro-duces neither deceleration nor cooling.The momentum transfer to the atomsby stimulated emission is in the oppo-site direction to what it was in absorption,resulting in a net transfer of zero mo-mentum. At high intensity, Eq. (3) showssaturation of γp at γ /2, and since the forceis given by Eq. (2), the deceleration satu-rates at a value amax = hkγ /2M.

2.2Dipole Optical Forces

While detuning |δ| γ , spontaneousemission may be much less frequent thanstimulated emission, unlike the case of thedissipative radiative force that is necessaryfor laser cooling, given by Eqs. (2) and (3).In this case, absorption is most often fol-lowed by stimulated emission, and seemsto produce zero momentum transfer be-cause the stimulated light has the samemomentum as the exciting light. How-ever, if the optical field has beams with atleast two different k-vectors present, suchas in counterpropagating beams, absorp-tion from one beam followed by stimulatedemission into the other indeed produces anonzero momentum exchange. The resultis called the dipole force, and is reversibleand hence conservative, so it cannot beused for laser cooling.

The dipole force is more easily calculatedfrom an energy picture than from amomentum picture. The force then derives


from the gradient of the potential of anatom in an inhomogeneous light field,which is appropriate because the force isconservative. The potential arises from theshift of the atomic energy levels in thelight field, appropriately called the ‘‘lightshift”, and is found by direct solution of theSchrodinger equation for a two-level atomin a monochromatic plane wave. Aftermaking both the dipole and rotating waveapproximations, the Hamiltonian can bewritten as

H = h

2

[−2δ

∗ 0

](4)

where the Rabi frequency is || = γ√

s0/2for a single traveling laser beam. Solutionof Eq. (4) for its eigenvalues providesthe dressed state energies that are light-shifted by

ωls =[√||2 + δ2 − δ

]2

. (5)

For sufficiently large detuning |δ| ||,approximation of Eq. (5) leads to ωls ≈||2/4δ = γ 2s0/8δ.

In a standing wave in 1-D with |δ| ||,the light shift ωls varies sinusoidally fromnode to antinode. When δ is sufficientlylarge, the spontaneous emission ratemay be negligible compared with that ofstimulated emission, so that hωls may betreated as a potential U. The resultingdipole force is

F = −∇U = − hγ 2

8δIs∇I, (6)

where I is the total intensity distributionof the standing-wave light field of periodλ/2. For such a standing wave, the opticalelectric field (and the Rabi frequency) at theantinodes is double that of each travelingwave that composes it, and so the total

intensity Imax at the antinodes is four timesthat of the single traveling wave.

2.3Density Matrix Description of OpticalForces

2.3.1 IntroductionUse of the density matrix ρ for pure statesprovides an alternative description to themore familiar one that uses wave functionsand operators but adds nothing new. Itsequation of motion is ih(dρ/dt) = [H, ρ],and can be derived directly from theSchrodinger equation. Moreover, it is astraightforward exercise to show that theexpectation value of any operator A thatrepresents an observable is 〈A〉 = tr (ρA).

Application of the Ehrenfest theoremgives the expectation value of the force as〈F〉 = −tr (ρ∇H). Beginning with the two-level atom Hamiltonian of Eq. (4), we findthe force in 1-D to be

〈F〉 = h

(∂

∂zρ∗

eg + ∂∗∂z

ρeg

). (7)

Thus, 〈F〉 depends only on the off-diagonalelements ρeg = ρ∗

ge, terms that are calledthe optical coherences.

2.3.2 Open Systems and the DissipativeForceThe real value of the density matrixformalism for atom-light interactions isits ability to deal with open systems. Bynot including the fluorescent light that islost from an atom-laser system undergoingcooling, a serious omission is being madein the discussion above. That is, the closedsystem of atom plus laser light that can bedescribed by Schrodinger wave functionsand is thus in a pure state, undergoesevolution to a ‘‘mixed’’ state by virtue ofthe spontaneous emission. This omissioncan be rectified by simple ad hoc additions


to the equation of motion, and the result iscalled the optical Bloch equations (OBE).These are written explicitly as

dρgg

dt= +γρee + i

2(∗ρeg − ρge)

dρee

dt= −γρee + i

2(ρge − ∗ρeg)

dρge

dt= −

(γ

2+ iδ

)ρge + i

2∗(ρee − ρgg)

dρeg

dt= −

(γ

2− iδ

)ρeg + i

2(ρgg − ρee),

(8)

where ρeg ≡ ρege−iδt for the coherences.In these equations, the terms propor-

tional to the spontaneous decay rate γ havebeen put in ‘‘by hand”, that is, they havebeen introduced into the OBEs to accountfor the effects of spontaneous emission.The spontaneous emission is irreversibleand accounts for the dissipation of thecooling process. For the ground state, thedecay of the excited state leads to an in-crease of its population ρgg proportional toγρee, whereas for the excited state, it leadsto a decrease of ρee, also proportional to

γρee. These equations have to be solved inorder to evaluate the optical force on theatoms.

2.3.3 Solution of the OBEs in Steady StateIn most cases, the laser light is appliedfor a period long compared to the typicalevolution times of atom-light interaction,that is, the lifetime of the excited state τ =1/γ . Thus, only the steady state solution ofthe OBEs have to be considered, and theseare found by setting the time derivatives inEq. (8) to zero. Then the probability ρee tobe in the excited state is found to be

ρee = γp

γ= s0/2

1 + s0 + (2δ/γ )2 = s/2

1 + s,

(9)

where s ≡ s0/[1 + (2δ/γ )2] is the off-resonance saturation parameter. Theexcited-state population ρee increases lin-early with the saturation parameter s forsmall values of s, but for s of the order ofunity, the probability starts to saturate to avalue of 1/2. The detuning dependence ofγp (see Eq. 3) showing this saturation forvarious values of s0 is depicted in Fig. 1.

–10 –5 0 5 100.00

0.10

0.20

0.30

0.40

0.50

Detuning d (g)

Sca

tterin

g ra

te g

p (g

)

s0 = 100.0s0 = 10.0s0 = 1.0s0 = 0.1

Fig. 1 Excitation rate γp as a function of the detuning δ for severalvalues of the saturation parameter s0. Note that for s0 > 1, the lineprofiles start to broaden substantially from power broadening


2.3.4 Radiative and Dipole ForcesSome insight into these forces emergesby expressing the gradient of the Rabifrequency of Eq. (7) in terms of a realand imaginary part so that (∂/∂z) =(qr + iqi). Then Eq. (7) becomes

F = hqr(ρ∗eg + ∗ρeg)

+ ihqi(ρ∗eg − ∗ρeg) (10)

Thus, the first term of the force is relatedto the dispersive part of the atom-lightinteraction, whereas the second term isrelated to the absorptive part of the atom-light interaction.

To appreciate the utility of the separationof ∇ into real and imaginary parts,consider the interaction of atoms with atraveling plane wave E(z) = E0(ei(kz−ωt) +c.c.)/2. In this case, qr = 0 and qi = k, andso the force is caused only by absorption.The force is given by Fsp = hkγρee and isthe radiative force of Eqs. (2) and (3).

For the case of counterpropagating planewaves, there is a standing wave whoseelectric field is E(z) = E0 cos(kz)(e−iωt +c.c.). Thus, qr = −k tan(kz) and qi = 0, sothere is only the dispersive part of theforce, given by

Fdip = 2hkδs0 sin 2kz

1 + 4s0 cos2 kz + (2δ/γ )2 . (11)

This replaces Eq. (6) for the dipole forceand removes the restriction |δ| ||,thereby including saturation effects. Eventhough the average of this force over awavelength vanishes, it can be used totrap atoms in a region smaller than thewavelength of the light.

2.3.5 Force on Moving AtomsIn order to show how these forces can beused to cool atoms, one has to considerthe force on moving atoms. For the case of

atomic velocities that are small comparedwith γ /k, the motion can be treated as asmall perturbation in the atomic evolutionthat occurs on the time scale 1/γ . Then thefirst-order result is given by

d

dt= ∂

∂t+ v

∂

∂z= ∂

∂t+ v(qr + iqi).

(12)

For the case of atoms moving in a standingwave, this results in the same dampingforce as Eq. (13) below.

3Laser Cooling

3.1Slowing Atomic Beams

Among the earliest laser cooling exper-iments was the deceleration of atomsin a beam [2]. The authors exploited theDoppler shift to make the momentumexchange (hence the force) velocity de-pendent. It worked by directing a laserbeam opposite an atomic beam so that theatoms could absorb light, and hence mo-mentum hk, very many times along theirpaths through the apparatus as shown inFig. 2 [2, 3]. Of course, excited-state atomscannot absorb light efficiently from thelaser that excited them, so between ab-sorptions they must return to the groundstate by spontaneous decay, accompaniedby the emission of fluorescent light. Thespatial symmetry of the emitted fluores-cence results in an average of zero netmomentum transfer from many such fluo-rescence events. Thus, the net force on theatoms is in the direction of the laser beam,and the maximum deceleration is limitedby the spontaneous emission rate γ .

Since the maximum deceleration amax =hkγ /2M is fixed by atomic parameters, it isstraightforward to calculate the minimum


C.M.

PMT

Laser 1

Laser 2

80-MHzAOM

80-MHzAOM

Pumpbeam

40 cm 40 cm

4 cm

125 cm 25 cm

Profilemagnet

BiasmagnetSodium

source

Extractionmagnets

l/4-plate

Probebeam

zs zc zp

Fig. 2 Schematic diagram of apparatus for beam slowing. The tapered magnetic field isproduced by layers of varying length on the solenoid

stopping length Lmin and time tmin forthe rms velocity of atoms v = 2

√kBT/M

at the source temperature. The result isLmin = v2/2amax and tmin = v/amax. InTable 1 are some of the parameters forslowing a few atomic species of interestfrom the peak of the thermal velocitydistribution.

Maximizing the scattering rate γp re-quires δ = −ωD in Eq. (3). If δ is chosenfor a particular atomic velocity in thebeam, then as the atoms slow down, theirchanging Doppler shift will take themout of resonance. They will eventuallycease deceleration after their Doppler shifthas been decreased by a few times thepower-broadened width γ ′ = γ

√1 + s0,

corresponding to v of a few timesvc = γ ′/k. Although this v of a fewm s−1 is considerably larger than the typ-ical atomic recoil velocity vr = hk/M of afew cm s−1, it is still only a small frac-tion of the average thermal velocity v ofthe atoms, such that significant furthercooling or deceleration cannot be accom-plished.

Tab. 1 Parameters of interest for slowingvarious atoms. The stopping length Lmin andtime tmin are minimum values. The oventemperature Toven that determines the peakvelocity is chosen to give a vapor pressure of1 Torr. Special cases are H at 1000 K fordissociation of H2 into atoms, and He in themetastable triplet state, for which two rows areshown: one for a 4-K source and another for thetypical discharge temperature

Atom Toven[K]

v[m s−1]

Lmin[m]

tmin[ms]

H 1000 5000 0.012 0.005He* 4 158 0.03 0.34He* 650 2013 4.4 4.4Li 1017 2051 1.15 1.12Na 712 876 0.42 0.96K 617 626 0.77 2.45Rb 568 402 0.75 3.72Cs 544 319 0.93 5.82

In order to achieve deceleration thatchanges the atomic speeds by hundredsof m s−1, it is necessary to maintain(δ + ωD) γ by compensating such largechanges of the Doppler shift. This can be


done by changing ωD through the angulardependence of k · v, or changing δ eithervia ωl or ωa. The two most commonmethods for maintaining this resonanceare sweeping the laser frequency ωlalong with the changing ωD of thedecelerating atoms [4–6], or by spatiallyvarying the atomic resonance frequencywith an inhomogeneous d.c magneticfield to keep the decelerating atomsin resonance with the fixed frequencylaser [2, 3, 7].

The use of a spatially varying magneticfield to tune the atomic levels along thebeam path was the first method to succeedin slowing atoms [2, 3]. It works as longas the Zeeman shifts of the ground andexcited states are different so that theresonant frequency is shifted. The fieldcan be tailored to provide the appropriateDoppler shift along the moving atom’spath. A solenoid that can produce sucha spatially varying field has layers ofdecreasing lengths. The technical problemof extracting the beam of slow atoms fromthe end of the solenoid can be simplified byreversing the field gradient and choosing atransition whose frequency decreases withincreasing field [9].

For alkali atoms such as sodium, atime-of-flight (TOF) method can be usedto measure the velocity distribution ofatoms in the beam [8]. It employs twoadditional beams labeled pump and probefrom Laser 1 as shown in Fig. 2. Becausethese beams cross the atomic beam at 90,ωD = −k · v = 0, and they excite atoms atall velocities. The pump beam is tunedto excite and empty a selected groundhyperfine state (hfs), and it transfersmore than 98% of the population as theatoms pass through its 0.5 mm width.To measure the velocity distribution ofatoms in the selected hfs, this pumplaser beam is interrupted for a period of

t = 10 − 50 µs with an acoustic opticalmodulator (AOM). A pulse of atoms inthe selected hfs passes the pump regionand travels to the probe beam. The timedependence of the fluorescence inducedby the probe laser, tuned to excite theselected hfs, gives the time of arrival, andthis signal is readily converted to a velocitydistribution. Figure 3 shows the measuredvelocity distribution of the atoms slowedby Laser 2.

3.2Optical Molasses

3.2.1 Doppler CoolingA different kind of radiative force arisesin low intensity, counterpropagating lightbeams that form a weak standing wave. Itis straightforward to calculate the radiativeforce on atoms moving in such a standingwave using Eq. (3). In the low intensitycase where stimulated emission is notimportant, the forces from the two lightbeams are simply added to give FOM =

125 130 135 140 145 150 155

Velocity (m s–1)

0

2

4

6

Ato

m fl

ux (

arb.

uni

ts)

Instrumental resolution

2.97 m s–1

(FWHM)

Fig. 3 The velocity distribution measured withthe TOF method. The experimental width ofapproximately 1

6 (γ /k) is shown by the dashedvertical lines between the arrows. The Gaussianfit through the data yields an FWHM (full widthat half maximum) of 2.97 m s−1 (figure takenfrom Molenaar, P. A., vander Straten, P.,Heideman, H. G. M., Metcalf, H. (1997), Phys.Rev. A 55, 605–614)


F+ + F−, where F± are found from Eqs. (2)and (3). Then the sum of the two forces is

FOM ∼= 8hk2δs0vγ (1 + s0 + (2δ/γ )2)2 ≡ −βv,

(13)

where terms of order (kv/γ )4 and higherhave been neglected. The force is pro-portional to velocity for small enoughvelocities, resulting in viscous dampingfor δ < 0 [10, 11] that gives this techniquethe name ‘‘optical molasses’’ (OM).

These forces are plotted in Fig. 4. Forδ < 0, this force opposes the velocityand therefore viscously damps the atomicmotion. The force FOM has maxima nearv ≈ ±γ

√s0 + 1/2k and decreases rapidly

for larger velocities.

3.2.2 Doppler Cooling LimitIf there were no other influence on theatomic motion, all atoms would quicklydecelerate to v = 0 and the sample wouldreach T = 0, a clearly unphysical result. In

laser cooling and related aspects of opticalcontrol of atomic motion, the forces arisebecause of the exchange of momentumbetween the atoms and the laser field.These necessarily discrete steps of size hkconstitute a heating mechanism that mustbe considered.

Since the atomic momentum changesby hk, their kinetic energy changes onan average by at least the recoil energyEr = h2k2/2M = hωr. This means that theaverage frequency of each absorption isat least ωabs = ωa + ωr. Similarly, the en-ergy hωa available from each spontaneousdecay must be shared between the out-going light and the kinetic energy ofthe atom recoiling with momentum hk.Thus, the average frequency of each emis-sion is ωemit = ωa − ωr. Therefore, thelight field loses an average energy ofh(ωabs − ωemit) = 2hωr for each scatter-ing event. This loss occurs at a rate of 2γp

(two beams), and the energy is convertedto atomic kinetic energy because the atoms

–4 –2 0 2 4– 0.4

– 0.2

0.0

0.2

0.4

Velocity (g /k)

For

ce (

hkg)

Fig. 4 Velocity dependence of the optical damping forces for 1-Doptical molasses. The two dotted traces show the force from eachbeam, and the solid curve is their sum. The straight line showshow this force mimics a pure damping force over a restrictedvelocity range. These are calculated for s0 = 2 and δ = −γ , sothere is some power broadening evident


recoil from each event. The atomic sampleis thereby heated because these recoils arein random directions.

The competition between this heatingand the damping force of Eq. (13) resultsin a nonzero kinetic energy in steady state,where the rates of heating and cooling areequal. Equating the cooling rate, FOM · v, tothe heating rate, 4hωrγp, we find the steadystate kinetic energy to be (hγ /8)(2|δ|/γ +γ /2|δ|). This result is dependent on|δ|, and has a minimum at 2|δ|/γ = 1,whence δ = −γ /2. The temperature foundfrom the kinetic energy is then TD =hγ /2kB, where TD is called the Dopplertemperature or the Doppler cooling limit.For ordinary atomic transitions, TD istypically below 1 mK.

Another instructive way to determineTD is to note that the average momentumtransfer of many spontaneous emissionsis zero, but the rms scatter of theseabout zero is finite. One can imaginethese decays as causing a random walkin momentum space, similar to Brownianmotion in real space, with step size hk

and step frequency 2γp, where the factorof 2 arises because of the two beams. Therandom walk results in an evolution ofthe momentum distribution as describedby the Fokker–Planck equation, and canbe used for a more formal treatment oflaser cooling. It results in diffusion inmomentum space with diffusion coeffi-cient D0 ≡ 2(p)2/t = 4γp(hk)2. Thenthe steady state temperature is given bykBT = D0/β. This turns out to be hγ /2 asabove for the case s0 1 when δ = −γ /2.This remarkable result predicts that thefinal temperature of atoms in OM is inde-pendent of the optical wavelength, atomicmass, and laser intensity (as long as it isnot too large).

3.2.3 Atomic BeamCollimation – One-dimensional OpticalMolasses – Beam BrighteningWhen an atomic beam crosses a 1-D OMas shown in Fig. 5, the transverse motionof the atoms is quickly damped whilethe longitudinal component is essentiallyunchanged. This transverse cooling of

Opticalmolasses

Opticalmolasses

Laser ormagnetic lens

Very brightatomic beam

Oven

Fig. 5 Scheme for optical brightening of an atomic beam. First,the transverse velocity components of the atoms are damped outby an optical molasses, then the atoms are focused to a spot,and finally the atoms are recollimated in a second opticalmolasses (figure taken from Sheehy, B., Shang, S. Q., van derStraten, P., Metcalf, H. (1990), Chem. Phys. 145, 317–325)


an atomic beam is an example of amethod that can actually increase itsbrightness (atoms/s-sr-cm2) because suchactive collimation uses dissipative forcesto compress the phase space volumeoccupied by the atoms. By contrast, theusual focusing or collimation techniquesfor light beams and most particle beamsis restricted to selection by apertures orconservative forces that preserve the phasespace density of atoms in the beam.

This velocity compression at low in-tensity in one dimension can be easilyestimated for two-level atoms to be aboutvc/vD = √

γ /ωr ≡ √1/ε. Here vD is the

velocity associated with the Doppler limitfor laser cooling discussed above: vD =√

hγ /2M. For Rb, vD = 12 cm s−1, vc =γ /k 4.6 m s−1, ωr 2π × 3.8 kHz, and1/ε 1600. (The parameter ε character-izes optical forces on atoms.) Includ-ing two transverse directions along withthe longitudinal slowing and coolingdiscussed above, the decrease in three-dimensional 3-D phase space volume forlaser cooling of an Rb atomic beam fromthe momentum contribution alone canexceed 106. Clearly optical techniquescan create atomic beams enormouslymore times intense than ordinary thermalbeams and also many orders of magnitudebrighter.

3.2.4 Experiments in Three-dimensionalOptical MolassesBy using three intersecting orthogonalpairs of oppositely directed beams, themovement of atoms in the intersectionregion can be severely restricted in all 3-D,and many atoms can thereby be collectedand cooled in a small volume.

Even though atoms can be collectedand cooled in the intersection region, itis important to stress that this is not atrap (see Sect. 4 below), that is, atoms that

wander away from the center experienceno force directing them back. They areallowed to diffuse freely and even escape,as long as there is enough time fortheir very slow diffusive movement toallow them to reach the edge of theregion of intersection of the laser beams.Since the atomic velocities are randomizedduring the damping time M/β = 2/ωr,atoms execute a random walk in positionspace with a step size of 2vD/ωr =λ/(π

√2ε) ∼= few µm. To diffuse a distance

of 1 cm requires about 107 steps or about30 s [13, 14].

In 1985, the group at Bell Labs was thefirst to observe 3-D OM [11]. Preliminarymeasurements of the average kinetic en-ergy of the atoms were done by blinkingoff the laser beams for a fixed interval.Comparison of the brightness of the fluo-rescence before and after the turnoff wasused to calculate the fraction of atoms thatleft the region while it was in the dark.The dependence of this fraction on theduration of the dark interval was used to es-timate the velocity distribution and hencethe temperature. This method, which isusually referred to as release and recapture,is specifically designed to measure the tem-perature of the atoms, since the usual wayof measuring temperatures cannot be ap-plied to an atomic cloud of a few millionatoms. The result was consistent with TD

as calculated from the Doppler theory, asdescribed in Sect. 3.2.2.

Later a more sensitive ballistic techniquewas devised at NIST that showed theastounding result that the temperature ofthe atoms in OM was very much lower thanTD [15]. These experiments also found thatOM was less sensitive to perturbationsand more tolerant of alignment errorsthan was predicted by Doppler theory.For example, if the intensities of the twocounterpropagating laser beams forming


an OM were unequal, then the forceon the atoms at rest would not vanish,but the force on the atoms with somenonzero drift velocity would vanish. Thisdrift velocity can be easily calculated byusing unequal intensities s0+ and s0−,to derive an analog of Eq. (13). Thus,atoms would drift out of an OM, and thecalculated rate would be much faster thanobserved by deliberately unbalancing thebeams in the experiments [16].

3.3Cooling Below the Doppler Limit

3.3.1 IntroductionIt was an enormous surprise to observethat the ballistically measured temperatureof the Na atoms was as much as 10times lower than TD = 240 µK [15], thetemperature minimum calculated fromtheory. This breaching of the Doppler limitforced the development of an entirely newpicture of OM that accounts for the factthat in 3-D, a two-level picture of atomicstructure is inadequate. The multilevelstructure of atomic states, and opticalpumping among these sublevels, must beconsidered in the description of 3-D OM.

In response to these surprising mea-surements of temperatures below TD, twogroups developed a model of laser coolingthat could explain the lower tempera-tures [17, 18]. The key feature of this modelthat distinguishes it from the earlier pic-ture is the inclusion of the multiplicity ofsublevels that make up an atomic state(e.g., Zeeman and hfs). The dynamicsof optically pumping the moving atomsamong these sublevels provides the newmechanism for producing ultralow tem-peratures [19].

The dominant feature of these modelsis the nonadiabatic response of movingatoms to the light field. Atoms at rest in a

steady state have ground-state orientationscaused by optical pumping processes thatdistribute the populations over the differ-ent ground-state sublevels. In the presenceof polarization gradients, these orienta-tions reflect the local light field. In thelow-light-intensity regime, the orientationof stationary atoms is completely deter-mined by the ground-state distribution;the optical coherences and the excited-state population follow the ground-statedistribution adiabatically.

For atoms moving in a light fieldthat varies in space, optical pumpingacts to adjust the atomic orientationto the changing conditions of the lightfield. In a weak pumping process, theorientation of moving atoms always lagsbehind the orientation that would exist forstationary atoms. It is this phenomenon ofnonadiabatic following that is the essentialfeature of the new cooling process.

Production of spatially dependent opticalpumping processes can be achieved inseveral different ways. As an example,consider two counterpropagating laserbeams that have orthogonal polarizations,as discussed below. The superposition ofthe two beams results in a light fieldhaving a polarization that varies on thewavelength scale along the direction of thelaser beams. Laser cooling by such a lightfield is called polarization gradient cooling.In a 3-D OM, the transverse wave characterof light requires that the light field alwayshas polarization gradients.

3.3.2 Linear ⊥ Linear PolarizationGradient CoolingOne of the most instructive models fordiscussion of sub-Doppler laser coolingwas introduced in Ref. [17] and very welldescribed in Ref. [19]. If the polarizationsof two counterpropagating laser beamsare identical, the two beams interfere


and produce a standing wave. Whenthe two beams have orthogonal linearpolarizations (same frequency ωl) withtheir E vectors perpendicular (e.g., x andy), the configuration is called lin ⊥ linor lin-perp-lin. Then the total field is thesum of the two counterpropagating beamsgiven by

E = E0 x cos(ωlt − kz) + E0y cos(ωlt + kz)

= E0[(x + y) cos ωlt cos kz

+ (x − y) sin ωlt sin kz]. (14)

At the origin, where z = 0, this becomes

E = E0(x + y) cos ωlt, (15)

which corresponds to linearly polarizedlight at an angle +π/4 to the x-axis. Theamplitude of this field is

√2E0. Similarly,

for z = λ/4, where kz = π/2, the field isalso linearly polarized but at an angle −π/4to the x-axis.

Between these two points, at z = λ/8,where kz = π/4, the total field is

E = E0

[x sin

(ωlt + π

4

)

− y cos(ωlt + π

4

)]. (16)

Since the x and y components havesine and cosine temporal dependence,they are π/2 out of phase, and soEq. (16) represents circularly polarizedlight rotating about the z-axis in thenegative sense. Similarly, at z = 3λ/8where kz = 3π/4, the polarization iscircular but in the positive sense. Thus,in this lin ⊥ lin scheme the polarizationcycles from linear to circular to orthogonallinear to opposite circular in the space ofonly half a wavelength of light, as shown inFig. 6. It truly has a very strong polarizationgradient.

l/2

py

px

l/2

0

0 l/4

s−

s +k1→

k2→

Fig. 6 Polarization gradient field for the lin ⊥ linconfiguration

Since the coupling of the different statesof multilevel atoms to the light field de-pends on its polarization, atoms movingin a polarization gradient will be coupleddifferently at different positions, and thiswill have important consequences for lasercooling. For the Jg = 1/2 → Je = 3/2 tran-sition (one of the simplest transitions thatshow sub-Doppler cooling [20]), the opti-cal pumping process in purely σ+ lightdrives the ground-state population to theMg = +1/2 sublevel. This optical pump-ing occurs because absorption alwaysproduces M = +1 transitions, whereasthe subsequent spontaneous emission pro-duces M = ±1, 0. Thus, the averageM ≥ 0 for each scattering event. For σ−-light, the population is pumped toward theMg = −1/2 sublevel. Thus, atoms trav-eling through only a half wavelength inthe light field, need to readjust their pop-ulation completely from Mg = +1/2 toMg = −1/2 and back again.

The interaction between nearly resonantlight and atoms not only drives transitionsbetween atomic energy levels but alsoshifts their energies. This light shift ofthe atomic energy levels, discussed inSect. 2.2, plays a crucial role in this schemeof sub-Doppler cooling, and the changingpolarization has a strong influence on thelight shifts. In the low-intensity limit oftwo laser beams, each of intensity s0Is, the


light shifts Eg of the ground magneticsubstates are given by a slight variation ofthe approximation to Eq. (5) that accountsfor the multilevel structure of the atoms.We write

Eg = hs0C2geγ

2

8δ, (17)

where Cge is the Clebsch–Gordan coeffi-cient that describes the coupling betweenthe particular levels of the atom and thelight field.

In the present case of orthogonal linearpolarizations and J = 1/2 → 3/2, the lightshift for the magnetic substate Mg = 1/2is three times larger than that of theMg = −1/2 substate when the light fieldis completely σ+. On the other hand,when an atom moves to a place wherethe light field is σ−, the shift of Mg =−1/2 is three times larger. So, in thiscase, the optical pumping discussed abovecauses a larger population to be therein the state with the larger light shift.This is generally true for any transitionJg → Je = Jg + 1. A schematic diagramshowing the populations and light shiftsfor this particular case of negative detuningis illustrated in Fig. 7.

3.3.3 Origin of the Damping ForceTo discuss the origin of the cooling processin this polarization gradient scheme,consider atoms with a velocity v at aposition where the light is σ+-polarized, asshown at the lower left of Fig. 7. The lightoptically pumps such atoms to the stronglynegative light-shifted Mg = +1/2 state. Inmoving through the light field, atoms mustincrease their potential energy (climb ahill) because the polarization of the light ischanging and the state Mg = 1/2 becomesless strongly coupled to the light field. Aftertraveling a distance λ/4, atoms arrive at

0 λ /4 λ /2 3λ /4

Position (z)

0

Ene

rgy

M = +1/2

M = −1/2

Fig. 7 The spatial dependence of the light shiftsof the ground-state sublevels of theJ = 1/2 ⇔ 3/2 transition for the case of lin ⊥ linpolarization configuration. The arrows show thepath followed by atoms being cooled in thisarrangement. Atoms starting at z = 0 in theMg = +1/2 sublevel must climb the potentialhill as they approach the z = λ/4 point where thelight becomes σ− polarized, and they areoptically pumped to the Mg = −1/2 sublevel.Then they must begin climbing another hilltoward the z = λ/2 point where the light is σ+polarized and they are optically pumped back tothe Mg = +1/2 sublevel. The process repeatsuntil the atomic kinetic energy is too small toclimb the next hill. Each optical pumping eventresults in the absorption of light at a frequencylower than emission, thus dissipating energy tothe radiation field

a position where the light field is σ−-polarized, and are optically pumped toMg = −1/2, which is now lower than theMg = 1/2 state. Again, the moving atomsare at the bottom of a hill and start toclimb. In climbing the hills, the kineticenergy is converted to potential energy,and in the optical pumping process, thepotential energy is radiated away becausethe spontaneous emission is at a higherfrequency than the absorption (see Fig. 7).Thus, atoms seem to be always climbinghills and losing energy in the process.This process brings to mind a Greekmyth, and is thus called ‘‘Sisyphus lasercooling’’.


The cooling process described above iseffective over a limited range of atomicvelocities. The force is maximum foratoms that undergo one optical pumpingprocess while traveling over a distance λ/4.Slower atoms will not reach the hilltopbefore the pumping process occurs andfaster atoms will travel a longer distancebefore being pumped toward the othersublevel, so E/z is smaller. In bothcases, the energy loss is smaller andtherefore the cooling process less efficient.Nevertheless, the damping constant β

for this process is much larger than forDoppler cooling, and therefore the finalsteady state temperature is lower [17, 19].

In the experiments of Ref. [21], thetemperature was measured in a 3-Dmolasses under various configurationsof the polarization. Temperatures weremeasured by a ballistic technique, in whichthe flight time of the released atoms wasmeasured as they fell through a probea few centimeters below the molassesregion. The lowest temperature obtainedwas 3 µK, which is a factor 40 below theDoppler temperature and a factor 15 abovethe recoil temperature of Cs.

3.3.4 The Limits of Sisyphus Laser CoolingThe extension of the kind of thinkingabout cooling limits in the case of Dopplercooling to the case of the sub-Dopplerprocesses must be done with some care,because a naive application of similarideas would lead to an arbitrarily lowfinal temperature. In the derivation inSect. 3.2.2, it is explicitly assumed thateach scattering event changes the atomicmomentum p by an amount that is a smallfraction of p and this clearly fails whenthe velocity is reduced to the region ofvr ≡ hk/M.

This limitation of the minimum steadystate value of the average kinetic energy to

a few times Er ≡ kBTr/2 = Mv2r /2 is intu-

itively comforting for two reasons. First,one might expect that the last spontaneousemission in a cooling process would leaveatoms with a residual momentum of theorder of hk, since there is no control overits direction. Thus, the randomness asso-ciated with this would put a lower limitof vmin ∼ vr on such cooling. Second, thepolarization gradient cooling mechanismdescribed above requires that atoms belocalizable within the scale of ∼ λ/2π inorder to be subject to only a single polariza-tion in the spatially inhomogeneous lightfield. The uncertainty principle then re-quires that these atoms have a momentumspread of at least hk.

The recoil limit discussed here hasbeen surpassed by evaporative coolingof trapped atoms [22] and two differentoptical cooling methods, neither of whichcan be described in simple notions. One ofthese uses optical pumping into a velocity-selective dark state [23]. The other oneuses carefully chosen, counterpropagating,laser pulses to induce velocity-selectiveRaman transitions, and is called Ramancooling [24].

4Traps for Neutral Atoms

In order to confine any object, it isnecessary to exchange kinetic for poten-tial energy in the trapping field, and inneutral atom traps, the potential energymust be stored as internal atomic energy.Thus, practical traps for ground-state neu-tral atoms are necessarily very shallowcompared with thermal energy becausethe energy-level shifts that result fromconvenient size fields are typically con-siderably smaller than kBT for T = 1 K.Neutral atom trapping therefore depends


on substantial cooling of a thermal atomicsample, and is often connected with thecooling process. In most practical cases,atoms are loaded from magneto-opticaltraps (MOTs) in which they have beenefficiently accumulated and cooled to mKtemperatures (see Sect. 4.3), or from op-tical molasses, in which they have beenoptically cooled to µK temperatures (seeSect. 3.2).

The small depth of typical neutralatom traps dictates stringent vacuumrequirements because an atom cannotremain trapped after a collision with athermal energy background gas molecule.Since these atoms are vulnerable targetsfor thermal energy background gas, themean free time between collisions mustexceed the desired trapping time. Thecross section for destructive collisions isquite large because even a gentle collision(i.e., large impact parameter) can impartenough energy to eject an atom from atrap. At pressure P sufficiently low to beof practical interest, the trapping time is ∼(10−8/P) s, where P is in Torr.

4.1Dipole Force Optical Traps

4.1.1 Single-beam Optical Traps forTwo-level AtomsThe simplest imaginable optical trap con-sists of a single, strongly focused Gaussianlaser beam (see Fig. 8) [25, 26] whose in-tensity at the focus varies transversely withr as

I(r) = I0e−2r2/w20 , (18)

Laser

2w0

Fig. 8 A single focused laser beam producesthe simplest type of optical trap

where w0 is the beam waist size. Sucha trap has a well-studied and importantmacroscopic classical analog in a phe-nomenon called optical tweezers [27–29].

With the laser light tuned below reso-nance (δ < 0), the ground-state light shiftis negative everywhere, but largest atthe center of the Gaussian beam waist.Ground-state atoms, therefore, experiencea force attracting them toward this cen-ter, given by the gradient of the lightshift, which is found from Eq. (5), and forδ/γ s0 is given by Eq. (6). For the Gaus-sian beam, this transverse force at the waistis harmonic for small r and is given by

F hγ 2

4δ

I0

Is

r

w20

e−2r2/w20 . (19)

In the longitudinal direction, there isalso an attractive force but it is morecomplicated and depends on the detailsof the focusing. Thus, this trap producesan attractive force on the atoms in threedimensions.

Although it may appear that the trap doesnot confine atoms longitudinally becauseof the radiation pressure along the laserbeam direction, careful choice of the laserparameters can indeed produce trappingin 3-D. This can be accomplished becausethe radiation pressure force decreases as1/δ2 (see Eqs. 2 and 3), but by contrast,the light shift and hence the dipole forcedecreases only as 1/δ for δ (seeEq. 5). If |δ| is chosen to be sufficientlylarge, atoms spend very little time inthe untrapped (actually repelled), excitedstate because its population is proportionalto 1/δ2. Thus, a sufficiently large valueof |δ| produces longitudinal confinementand also maintains the atomic populationprimarily in the trapped ground state.

The first optical trap was demonstratedin Na with light detuned below the D-lines [26]. With 220 mW of dye laser light


tuned about 650 GHz below the atomictransition and focused to an ∼10 µmwaist, the trap depth was about 15hγ

corresponding to 7 mK.Single-beam dipole force traps can

be made with the light detuned by asignificant fraction of its frequency fromthe atomic transition. Such a far-off-resonance trap (FORT) has been developedfor Rb atoms using light detuned by nearly10% to the red of the D1 transition atλ = 795 nm [30]. Between 0.5 and 1 W ofpower was focused to a spot about 10 µm insize, resulting in a trap 6 mK deep wherethe light-scattering rate was only a fewhundreds per second. The trap lifetimewas more than half a second.

There is a qualitative difference when thetrapping light is detuned by a large frac-tion of the optical frequency. In one suchcase, Nd : YAG light at λ = 1064 nm wasused to trap Na whose nearest transitionis at λ = 596 nm [31]. In a more extremecase, a trap using λ = 10.6 µm light froma CO2 laser has been used to trap Cswhose optical transition is at a frequency∼12 times higher (λ = 852 nm) [32]. Forsuch large values of |δ|, calculations ofthe trapping force cannot exploit the rotat-ing wave approximations as was done forEqs. (4) and (5), and the atomic behavior issimilar to that in a DC field. It is impor-tant to remember that for an electrostatictrap Earnshaw’s theorem precludes a fieldmaximum, but that in this case there is in-deed a local 3-D intensity maximum of thefocused light because it is not a static field.

4.1.2 Blue-detuned Optical TrapsOne of the principal disadvantages of theoptical traps discussed above is that thenegative detuning attracts atoms to theregion of highest light intensity. This mayresult in significant spontaneous emissionunless the detuning is a large fraction of

the optical frequency such as the Nd : YAGlaser trap [31] or the CO2 laser trap [32].More important in some cases is that thetrap relies on Stark shifting of the atomicenergy levels by an amount equal to thetrap depth, and this severely compromisesthe capabilities for precision spectroscopyin a trap [33].

Attracting atoms to the region of low-est intensity would ameliorate both theseconcerns, but such a trap requires positivedetuning (blue), and an optical configu-ration having a dark central region. Oneof the first experimental efforts at a bluedetuned trap used the repulsive dipoleforce to support Na atoms that were oth-erwise confined by gravity in an optical‘‘cup’’ [34]. Two rather flat, parallel beamsdetuned by 25% of the atomic resonancefrequency were directed horizontally andoriented to form a V-shaped trough. TheirGaussian beam waists formed a region1 mm long where the potential was deep-est, and hence provided confinement alongtheir propagation direction as shown inFig. 9. The beams were the λ = 514 nmand λ = 488 nm from an argon laser, and

0.6

0.4

0.2

0

0

0

y

x

1000

– 1000

– 5050

Inte

nsity

Fig. 9 The light intensity experienced by anatom located in a plane 30 µm above the beamwaists of two quasi-focused sheets of lighttraveling parallel and arranged to form aV-shaped trough. The x and y dimensions are inµm (figure taken from Davidson, N., Lee, H. J.,Adams, C. S., Kasevich, M., Chu, S. (1995), Phys.Rev. Lett. 74, 1311–1314)


the choice of the two frequencies was notsimply to exploit the full power of the mul-tiline Ar laser, but also to avoid the spatialinterference that would result from the useof a single frequency.

Obviously, a hollow laser beam wouldalso satisfy the requirement for a blue-detuned trap, but conventional textbookwisdom shows that such a beam is notan eigenmode of a laser resonator [35].Some lasers can make hollow beams,but these are illusions because theyconsist of rapid oscillations between theTEM01 and TEM10 modes of the cavity.Nevertheless, Maxwell’s equations permitthe propagation of such beams, andin the recent past there have beenstudies of the LaGuerre–Gaussian modesthat constitute them [36–38]. The severalways of generating such hollow beamshave been tried by many experimentalgroups and include phase and amplitudeholograms, hollow waveguides, axicons orrelated cylindrical prisms, stressing fibers,and simply mixing the TEM01 and TEM10

modes with appropriate cylindrical lenses.An interesting experiment has been

performed using the ideas of Sisyphuscooling (see Sect. 3.3) with evanescentwaves combined with a hollow beamformed with an axicon [39]. In the pre-viously reported experiments with atomsbouncing under gravity from an evanes-cent wave field [40, 41], they were usuallylost to horizontal motion for several rea-sons, including slight tilting of the surface,surface roughness, horizontal motion as-sociated with their residual motion, andhorizontal ejection by the Gaussian pro-file of the evanescent wave laser beam.The authors of Ref. [39] simply confinedtheir atoms in the horizontal direction bysurrounding them with a wall of blue-detuned light in the form of a verticalhollow beam. Their gravito-optical surface

trap cooled Cs atoms to 3 µK at a densityof 3 × 1010/cm3 in a sample whose 1/eheight in the gravitational field was only19 µm. Simple ballistics gives a frequencyof 450 bounces per second, and the 6-slifetime (limited only by background gascollisions) corresponds to several thousandbounces. However, at such low energies,the deBroglie wavelength of the atoms is1/4 µm, and the atomic motion is nolonger accurately described classically, butrequires deBroglie wave methods.

4.2Magnetic Traps

4.2.1 IntroductionMagnetic trapping of neutral atoms iswell suited for use in very many areas,including high-resolution spectroscopy,collision studies, Bose–Einstein conden-sation (BEC), and atom optics. Althoughion trapping, laser cooling of trapped ions,and trapped ion spectroscopy were knownfor many years [42], it was only in 1985that neutral atoms were first trapped [43].Such experiments offer the capability ofthe spectroscopic ideal of an isolated atomat rest, in the dark, available for interactionwith electromagnetic field probes.

Because trapping requires the exchangeof kinetic energy for potential energy, theatomic energy levels will necessarily shiftas the atoms move in the trap. Theseshifts can severely affect the precision ofspectroscopic measurements. Since oneof the potential applications of trappedatoms is in high-resolution spectroscopy,such inevitable shifts must be carefullyconsidered.

4.2.2 Magnetic ConfinementThe Stern–Gerlach experiment in 1924first demonstrated the mechanical actionof inhomogeneous magnetic fields on


neutral atoms having magnetic moments,and the basic phenomenon was subse-quently developed and refined. An atomwith a magnetic moment µ can be con-fined by an inhomogeneous magnetic fieldbecause of an interaction between the mo-ment and the field. This produces a forcegiven by

F = ∇(µ · B) (20)

since E = −µ · B. Several different mag-netic traps with varying geometries thatexploit the force of Eq. (20) have beenstudied in some detail in the literature.The general features of the magnetic fieldsof a large class of possible traps has beenpresented [44].

W. Paul originally suggested a quadru-pole trap composed of two identical coilscarrying opposite currents (see Fig. 10).This trap clearly has a single center inwhich the field is zero, and is the simplestof all possible magnetic traps. When thecoils are separated by 1.25 times theirradius, such a trap has equal depth in theradial (x-y plane) and longitudinal (z-axis)

I

I

Fig. 10 Schematic diagram of the coilconfiguration used in the quadrupole trap andthe resultant magnetic field lines. Because thecurrents in the two coils are in oppositedirections, there is a |B| = 0 point at the center

directions [44]. Its experimental simplicitymakes it most attractive, both because ofease of construction and of optical accessto the interior. Such a trap was used in thefirst neutral atom trapping experiments atNIST on laser-cooled Na atoms for timesexceeding 1 s, and that time was limitedonly by background gas pressure [43].

The magnitude of the field is zero atthe center of this trap, and increases in alldirections as

B = ∇B√

ρ2 + 4z2, (21)

where ρ2 ≡ x2 + y2 and the field gradientis constant (see Ref. [44]). The field gradi-ent is constant along any line through theorigin, but has different values in differ-ent polar directions because of the ‘4’ inEq. (21). Therefore, the force of Eq. (20)that confines the atoms in the trap isneither harmonic nor central, and orbitalangular momentum is not conserved.

The requisite field for the quadrupoletrap can also be provided in two dimen-sions by four straight currents as indicatedin Fig. 11. The field is translationally in-variant along the direction parallel to thecurrents, so a trap cannot be made thisway without additional fields. These areprovided by end coils that close the trap, asshown.

Although there are very many differentkinds of magnetic traps for neutral parti-cles, this particular one has played a special

II+

−

−

+

Fig. 11 The Ioffe trap has four straight currentelements that form a linear quadrupole field. Theaxial confinement is accomplished with end coilsas shown. These fields can be achieved withmany different current configurations as long asthe geometry is preserved


role. There are certain conditions requiredfor trapped atoms not to be ejected in aregion of zero field such as occurs at thecenter of a quadrupole trap (see Sects. 4.2.3and 4.2.4). This problem is not easily cured;the Ioffe trap has been used in many of theBEC experiments because it has |B| = 0everywhere.

4.2.3 Classical Motion of Atoms in aQuadrupole TrapBecause of the dependence of the trappingforce on the angle between the fieldand the atomic moment (see Eq. 20), theorientation of the magnetic moment withrespect to the field must be preservedas the atoms move about in the trap.Otherwise, the atoms may be ejectedinstead of being confined by the fieldsof the trap. This requires velocities lowenough to ensure that the interactionbetween the atomic moment µ and thefield B is adiabatic especially when theatom’s path passes through a regionwhere the field magnitude is small andtherefore the energy separation betweenthe trapping and nontrapping states issmall. This is especially critical at thelow temperatures of the BEC experiments.Therefore energy considerations that focusonly on the trap depth are not sufficientto determine the stability of a neutralatom trap; orbit and/or quantum statecalculations and their consequences mustalso be considered.

For the two-coil quadrupole magnetictrap of Fig. 10, stable circular orbits ofradius ρ1 in the z = 0 plane can be foundclassically by setting µ∇B = Mv2/ρ1, sothat v = √

ρ1a, where a ≡ µ∇B/M isthe centripetal acceleration supplied bythe field gradient (cylindrical coordinatesare appropriate). Such orbits have anangular frequency of ωT = √

a/ρ1. Fortraps of a few centimeter size and a few

hundred Gauss depth, a ∼ 250 m s−2, andthe fastest trappable atoms in circularorbits have vmax ∼ 1 m s−1 so ωT/2π ∼20 Hz. Because of the anharmonicityof the potential, the orbital frequenciesdepend on the orbit size, but in general,atoms in lower-energy orbits have higherfrequencies.

For the quadrupole trap to work, theatomic magnetic moments must be ori-ented with µ · B < 0 so that they arerepelled from regions of strong fields.This orientation must be preserved whilethe atoms move around in the trap eventhough the trap fields change directionsin a very complicated way. The condi-tion for adiabatic motion can be writtenas ωZ |dB/dt|/B, where ωZ = µB/h isthe Larmor precession rate in the field.

Since v/ρ1 = v∇B/B = |dB/dt|/B for auniform field gradient, the adiabaticitycondition is

ωZ ωT . (22)

More general orbits must satisfy a simi-lar condition. For the two-coil quadrupoletrap, the adiabaticity condition can be eas-ily calculated. Using v = √

ρ1a for circularorbits in the z = 0 plane, the adiabatic con-dition for a practical trap (∇B ∼ 1 T/m) re-quires ρ1 (h2/M2a)1/3 ∼ 1 µm as wellas v (ha/M)1/3 ∼ 1 cm s−1. Note thatviolation of these conditions (i.e., v ∼1 cm s−1 in a trap with ∇B ∼ 1 T/m) re-sults in the onset of quantum dynamics forthe motion (deBroglie wavelength ≈ orbitsize).

Since the nonadiabatic region of the trapis so small (less than 10−18 m3 comparedwith typical sizes of ∼2 cm, correspondingto 10−5 m3), nearly all the orbits of mostatoms are restricted to regions wherethey are adiabatic. Therefore, most ofsuch laser-cooled atoms stay trapped for


many thousands of orbits correspondingto several minutes. At laboratory vacuumchamber pressures of typically 10−10 torr,the mean free time between collisions thatcan eject trapped atoms is ∼2 min, so thetransitions caused by nonadiabatic motionare not likely to be observable in atomsthat are optically cooled.

4.2.4 Quantum Motion in a TrapSince laser and evaporative cooling havethe capability to cool atoms to energieswhere their deBroglie wavelengths are onthe scale of the orbit size, the motionaldynamics must be described in termsof quantum mechanical variables andsuitable wave functions. Quantization ofthe motion leads to discrete bound stateswithin the trap having µ · B < 0, and also acontinuum of unbound states having µ · Bwith opposite sign.

Studying the behavior of extremely slow(cold) atoms in the two-coil quadrupoletrap begins with a heuristic quantizationof the orbital angular momentum usingMr2ωT = nh for circular orbits. The en-ergy levels are then given by

En = 3

2E1n2/3, where

E1 = (Ma2 h2)1/3 ∼ h × 5 kHz, (23)

For velocities of optically cooled atoms ofa few cm s−1, n ∼ 10 − 100. It is readilyfound that ωZ = nωT , so that the adiabaticcondition of Eq. (22) is satisfied only forn 1.

These large-n bound states have smallmatrix elements coupling them to theunbound continuum states [45]. This canbe understood classically since they spendmost of their time in a stronger field, andthus satisfy the condition of adiabaticityof the orbital motion relative to theLarmor precession. In this case, the

separation of the rapid precession fromthe slower orbital motion is reminiscentof the Born–Oppenheimer approximationfor molecules.

On the other hand, the small-n states,whose orbits are confined to a region nearthe origin where the field is small, havemuch larger coupling to the continuumstates. Thus, they are rapidly ejected fromthe trap. The transitions to unbound statesresulting from the coupling of the motionwith the trapping fields are called Majoranaspin flips, and effectively constitute a ‘‘hole’’at the bottom of the trap. The evaporativecooling process used to produce very cold,dense samples reduces the average totalenergy of the trapped atoms sufficientlythat the orbits are confined to regionsnear the origin and so, such lossesdominate [44, 45].

There have been different solutionsto this problem of Majorana losses forconfinement of ultracold atoms for theBEC experiments. In the JILA-experiment,the hole was rotated by rotating themagnetic field, and thus, the atoms donot spend sufficient time in the hole tomake a spin flip. In the MIT experiment,the hole was plugged by using a focusedlaser beam tuned to the blue side of atomicresonance, which expelled the atoms fromthe center of the magnetic trap. In theRice experiment, the atoms were trappedin an Ioffe trap, which has a nonzerofield minimum. Most BEC experimentsare now using the Ioffe trap solution.

4.3Magneto-optical Traps

4.3.1 IntroductionThe most widely used trap for neutralatoms is a hybrid employing both opticaland magnetic fields to make a magneto-optical trap (MOT), first demonstrated in


1987 [46]. The operation of an MOT de-pends on both inhomogeneous magneticfields and radiative selection rules to ex-ploit both optical pumping and the strongradiative force [46, 47]. The radiative in-teraction provides cooling that helps inloading the trap and enables very easy op-eration. MOT is a very robust trap thatdoes not depend on precise balancing ofthe counterpropagating laser beams or ona very high degree of polarization.

The magnetic field gradients are modestand have the convenient feature thatthe configuration is the same as thequadrupole magnetic traps discussed inSect. 4.2.2. Appropriate fields can readilybe achieved with simple, air-cooled coils.The trap is easy to construct because it canbe operated with a room-temperature cellin which alkali atoms are captured fromthe vapor. Furthermore, low-cost diodelasers can be used to produce the lightappropriate for many atoms, so the MOThas become one of the least expensive waysto make atomic samples with temperaturesbelow 1 mK.

Trapping in an MOT works by opti-cal pumping of slowly moving atoms ina linearly inhomogeneous magnetic fieldB = B(z) (see Eq. 21), such as that formedby a magnetic quadrupole field. Atomictransitions with the simple scheme ofJg = 0 → Je = 1 have three Zeeman com-ponents in a magnetic field, excited by eachof three polarizations, whose frequenciestune with the field (and therefore with po-sition) as shown in Fig. 12 for 1-D. Twocounterpropagating laser beams of oppo-site circular polarization, each detunedbelow the zero-field atomic resonance byδ, are incident as shown.

Because of the Zeeman shift, the excitedstate Me = +1 is shifted up for B >

0, whereas the state with Me = −1 isshifted down. At position z′ in Fig. 12,

z ′

Me = +1

Me = 0

Me = −1

Mg = 0Position

Energy

s− beam

d+d

wl

d−

s+ beam

Fig. 12 Arrangement for an MOT in 1-D. Thehorizontal dashed line represents the laserfrequency seen by an atom at rest in the center ofthe trap. Because of the Zeeman shifts of theatomic transition frequencies in theinhomogeneous magnetic field, atoms at z = z′are closer to resonance with the σ− laser beamthan with the σ+ beam, and are therefore driventoward the center of the trap

the magnetic field, therefore, tunes theM = −1 transition closer to resonanceand the M = +1 transition further out ofresonance. If the polarization of the laserbeam incident from the right is chosento be σ− and correspondingly σ+ for theother beam, then more light is scatteredfrom the σ− beam than from the σ+ beam.Thus, the atoms are driven toward thecenter of the trap where the magnetic fieldis zero. On the other side of the center ofthe trap, the roles of the Me = ±1 states arereversed and now more light is scatteredfrom the σ+ beam, again driving the atomstoward the center.

So far, the discussion has been limitedto the motion of atoms in 1-D. However,the MOT scheme can easily be extendedto 3-D by using six instead of two laserbeams. Furthermore, even though veryfew atomic species have transitions assimple as Jg = 0 → Je = 1, the schemeworks for any Jg → Je = Jg + 1 transition.Atoms that scatter mainly from the σ+


laser beam will be optically pumped towardthe Mg = +Jg substate, which forms aclosed system with the Me = +Je substate.

4.3.2 Cooling and Compressing Atoms inan MOTFor a description of the motion of atomsin an MOT, consider the radiative force inthe low intensity limit (see Eqs. 2 and 3).The total force on the atoms is given byF = F+ + F−, where F± can be found fromEqs. (2) and (3), and the detuning δ± foreach laser beam is given by δ± = δ ∓ k ·v ± µ′B/h. Here, µ′ ≡ (geMe − ggMg)µB

is the effective magnetic moment for thetransition used. Note that the Dopplershift ωD ≡ −k · v and the Zeeman shiftωZ = µ′B/h both have opposite signs foropposite beams.

The situation is analogous to the velocitydamping in an OM from the Doppler effectas discussed in Sec. 3.2, but here the effectalso operates in position space, whereasfor molasses it operates only in velocityspace. Since the laser light is detunedbelow the atomic resonance in both cases,compression and cooling of the atoms isobtained simultaneously in an MOT.

When both the Doppler and Zeemanshifts are small compared to the detuningδ, the denominator of the force can beexpanded as for Eq. (13) and the resultbecomes

F = −βv − κr, (24)

where the damping coefficient β is definedin Eq. (13). The spring constant κ arisesfrom the similar dependence of F on theDoppler and Zeeman shifts, and is givenby κ = µ′β∇B/hk

The force of Eq. (24) leads to dampedharmonic motion of the atoms, where thedamping rate is given by MOT = β/Mand the oscillation frequency ωMOT =

√κ/M. For magnetic field gradients ∇B ≈

0.1 T m−1, the oscillation frequency is typ-ically a few kHz, and this is much smallerthan the damping rate that is typically afew hundred kHz. Thus, the motion isoverdamped, with a characteristic restor-ing time to the center of the trap of2MOT/ω2

MOT ≈ several milliseconds fortypical values of detuning and intensity ofthe lasers.

4.3.3 Capturing Atoms in an MOTAlthough the approximations that lead toEq. (24) for force hold for slow atoms nearthe origin, they do not apply for the captureof fast atoms far from the origin. In thecapture process, the Doppler and Zeemanshifts are no longer small compared tothe detuning, so the effects of the positionand velocity can no longer be disentangled.However, the full expression for the forcestill applies and the trajectories of theatoms can be calculated by numericalintegration of the equation of motion [48].

The capture velocity of an MOT isserendipitously enhanced because atomstraveling across it experience a decreasingmagnetic field just as in beam decelerationdescribed in Sect. 3.1. This enables reso-nance over an extended distance and ve-locity range because the changing Dopplershift of decelerating atoms can be com-pensated by the changing Zeeman shift asatoms move in the inhomogeneous mag-netic field. Of course, it will work thisway only if the field gradient ∇B does notdemand an acceleration larger than themaximum acceleration amax. Thus, atomsare subject to the optical force over a dis-tance that can be as long as the trap size,and can therefore be slowed considerably.

The very large velocity capture rangevcap of an MOT can be estimated by usingFmax = hkγ /2 and choosing a maximumsize of a few centimeters for the beam


diameters. Thus, the energy change canbe as large as a few K, correspondingto vcap ∼ 100 m s−1 [47]. The number ofatoms in a vapor with velocities below vcapin the Boltzmann distribution scales asv4

cap, and there are enough slow atoms tofall within the large MOT capture rangeeven at room temperature, because a fewK includes 10−4 of the atoms.

4.3.4 Variations on the MOT TechniqueBecause of the wide range of applicationsof this most versatile kind of atom trap, anumber of careful studies of its propertieshave been made [47, 49–56], and severalvariations have been developed. One ofthese is designed to overcome the densitylimits achievable in an MOT. In thesimplest picture, loading additional atomsinto an MOT produces a higher atomicdensity because the size of the trappedsample is fixed. However, the densitycannot increase without limit as moreatoms are added. The atomic densityis limited to ∼1011 cm−3 because thefluorescent light emitted by some trappedatoms is absorbed by others.

One way to overcome this limit is tohave much less light in the center of theMOT than at the sides. Simply loweringthe laser power is not effective in reducingthe fluorescence because it will also reducethe capture rate and trap depth. But thoseadvantageous properties can be preservedwhile reducing fluorescence from atoms atthe center if the light intensity is low onlyin the center.

The repumping process for the alkaliatoms provides an ideal way of imple-menting this idea [57]. If the repumpinglight is tailored to have zero intensity atthe center, then atoms trapped near thecenter of the MOT are optically pumpedinto the ‘‘wrong’’ hfs state and stop fluo-rescing. They drift freely in the ‘‘dark’’ at

low speed through the center of the MOTuntil they emerge on the other side, intothe region where light of both frequen-cies is present and begin absorbing again.Then they feel the trapping force and aredriven back into the ‘‘dark’’ center of thetrap. Such an MOT has been operated atMIT [57] with densities close to 1012/cm3,and the limitations are now from colli-sions in the ground state rather than frommultiple light scattering and excited-statecollisions.

5Optical Lattices

5.1Quantum States of Motion

As the techniques of laser cooling ad-vanced from a laboratory curiosity to a toolfor new problems, the emphasis shiftedfrom attaining the lowest possible steadystate temperatures to the study of ele-mentary processes, especially the quantummechanical description of the atomic mo-tion. In the completely classical descriptionof laser cooling, atoms were assumed tohave a well-defined position and momen-tum that could be known simultaneouslywith arbitrary precision. However, whenatoms are moving sufficiently slowly thattheir deBroglie wavelength precludes theirlocalization to less than λ/2π , these de-scriptions fail and a quantum mechanicaldescription is required. Such exotic be-havior for the motion of whole atoms, asopposed to electrons in the atoms, had notbeen considered before the advent of lasercooling simply because it was too far out ofthe range of ordinary experiments. A seriesof experiments in the early 1990s provideddramatic evidence for these new quantumstates of motion of neutral atoms, and


led to the debut of deBroglie wave atomoptics.

The quantum description of atomicmotion requires that the energy of suchmotion be included in the Hamiltonian.The total Hamiltonian for atoms movingin a light field would then be given by

H = Hatom + Hrad + Hint + Hkin, (25)

where Hatom describes the motion of theatomic electrons and gives the internalatomic energy levels, Hrad is the energy ofthe radiation field and is of no concern herebecause the field is not quantized, Hintdescribes the excitation of atoms by thelight field and the concomitant light shifts,and Hkin is the kinetic energy operator ofthe motion of the center of mass of theatoms. This Hamiltonian has eigenstatesof not only the internal energy levels andthe atom-laser interaction that connectsthem, but also that of the kinetic energyoperatorHkin ≡ P2/2M. These eigenstateswill therefore be labeled by quantumnumbers of the atomic states as well asthe center of mass momentum p. An atomin the ground state, |g; p〉, has an energyEg + p2/2M, that can take on a range ofvalues.

In 1968, V.S. Letokhov [58] suggestedthat it is possible to confine atoms in thewavelength-size regions of a standing waveby means of the dipole force that arisesfrom the light shift. This was first accom-plished in 1987 in 1-D with an atomic beamtraversing an intense standing wave [59].Since then, the study of atoms confinedto wavelength-size potential wells has be-come an important topic in optical controlof atomic motion because it opens up con-figurations previously accessible only incondensed matter physics using crystals.

The limits of laser cooling discussed inSect. 3.3.4 suggest that atomic momentacan be reduced to a ‘‘few’’ times hk. This

means that their deBroglie wavelengths areequal to the optical wavelengths divided bya ‘‘few’’. If the depth of the optical potentialwells is high enough to contain such veryslow atoms, then their motion in potentialwells of size λ/2 must be describedquantum mechanically, since they areconfined to a space of size comparableto their deBroglie wavelengths. Thus, theydo not oscillate in the sinusoidal wells asclassical localizable particles, but insteadoccupy discrete, quantum mechanicalbound states [60], as shown in the lowerpart of Fig. 13.

The basic ideas of the quantum me-chanical motion of particles in a periodicpotential were laid out in the 1930s with theKronig–Penney model and Bloch’s theo-rem, and optical lattices offer importantopportunities for their study. For example,

0 l/4 l/2

Position

3l/4 l

Ene

rgy

Fig. 13 Energy levels of atoms moving in theperiodic potential of the light shift in a standingwave. There are discrete bound states deep inthe wells that broaden at higher energy, andbecome bands separated by forbidden energiesabove the tops of the wells. Under conditionsappropriate to laser cooling, optical pumpingamong these states favors populating the lowestones as indicated schematically by the arrows


these lattices can be made essentially freeof defects with only moderate care inspatially filtering the laser beams to as-sure a single transverse mode structure.Furthermore, the shape of the potentialis exactly known and does not dependon the effect of the crystal field or theionic energy level scheme. Finally, thelaser parameters can be varied to mod-ify the depth of the potential wells withoutchanging the lattice vectors, and the lat-tice vectors can be changed independentlyby redirecting the laser beams. The sim-plest optical lattice to consider is a 1-Dpair of counterpropagating beams of thesame polarization, as was used in the firstexperiment [59].

Of course, such tiny traps are usuallyvery shallow, so loading them requirescooling to the µK regime. Even atomswhose energy exceeds the trap depthmust be described as quantum mechanicalparticles moving in a periodic potentialthat display energy band structure [60].Such effects have been observed in verycareful experiments.

Because of the transverse nature of light,any mixture of beams with different k-vectors necessarily produces a spatiallyperiodic, inhomogeneous light field. Theimportance of the ‘‘egg-crate’’ array ofpotential wells arises because the asso-ciated atomic light shifts can easily becomparable to the very low average atomickinetic energy of laser-cooled atoms. Atypical example projected against two di-mensions is shown in Fig. 14.

Atoms trapped in wavelength-sizedspaces occupy vibrational levels similarto those of molecules. The optical spec-trum can show Raman-like sidebands thatresult from transitions among the quan-tized vibrational levels [61, 62] as shown inFig. 15. These quantum states of atomic

Fig. 14 The ‘‘egg-crate’’ potential of an opticallattice shown in two dimensions. The potentialwells are separated by λ/2

motion can also be observed by stimu-lated emission [62, 63] and by direct RFspectroscopy [64, 65].

5.2Properties of 3-D Lattices

The name ‘‘optical lattice’’ is used ratherthan optical crystal because the fillingfraction of the lattice sites is typicallyonly a few percent (as of 1999). The limitarises because the loading of atoms intothe lattice is typically done from a sampleof trapped and cooled atoms, such as anMOT for atom collection, followed by anOM for laser cooling. The atomic density insuch experiments is limited by collisionsand multiple light scattering to a few times1011 cm−3. Since the density of lattice sitesof size λ/2 is a few times 1013 cm−3, thefilling fraction is necessarily small. Withthe advent of experiments that load atomsdirectly into a lattice from a BEC, the fillingfactor can be increased to 100%, and insome cases it may be possible to load morethan one atom per lattice site [66, 67].

In 1993 a very clever scheme wasdescribed [68]. It was realized that an n-dimensional lattice could be created byonly n + 1 traveling waves rather than


0

0.2

0.4

0.6

−200 −100 0 100 200 −200 −100 0 100 200

Flu

ores

cenc

e (a

rb. u

nits

)

Flu

ores

cenc

e (a

rb. u

nits

)

×20×10

0

0.1

0.2

0.3

0.4

0.5

Frequency (kHz)Frequency (kHz)(a) (b)

Fig. 15 (a) Fluorescence spectrum in a 1-D lin ⊥ lin optical molasses. Atoms are firstcaptured and cooled in an MOT, and then the MOT light beams are switched off leaving a pairof lin ⊥ lin because of spontaneous emission of the atoms to the same vibrational state fromwhich they are excited, whereas the sideband on the left (right) is due to spontaneousemission to a vibrational state with one vibrational quantum number lower (higher) (seeFig. 13). The presence of these sidebands is a direct proof of the existence of the bandstructure. (b) Same as (a) except that the 1-D molasses is σ+ − σ−, which has no spatiallydependent light shift and hence no vibrational states (figure taken from Jessen, P. S., Gerz, C.,Lett, P. D., Phillips, W. D., Rolston, S. L., Spreeuw, R. J. C., Westbrook, C. I. (1992), Phys. Rev.Lett. 69, 49–52)

2n. The real benefit of this scheme isthat in case of phase instabilities in thelaser beams, the interference pattern isonly shifted in space, but the interferencepattern itself is not changed. Insteadof producing optical wells in 2-D withfour beams (two standing waves), theseauthors used only three. The k-vectorsof the coplanar beams were separated by2π/3, and they were all linearly polarizedin their common plane (not parallelto one another). The same immunityto vibrations was established for a 3-D optical lattice by using only fourbeams arranged in a quasi-tetrahedralconfiguration. The three linearly polarizedbeams of the 2-D arrangement describedabove were directed out of the planetoward a common vertex, and a fourthcircularly polarized beam was added.All four beams were polarized in thesame plane [68]. The authors showed thatthis configuration produced the desiredpotential wells in 3-D.

5.3Spectroscopy in 3-D Lattices

The group at NIST developed a newmethod that superposed a weak probebeam of light directly from the laser uponsome of the fluorescent light from theatoms in a 3-D OM, and directed thelight from these combined sources ontoon a fast photodetector [70]. The resultingbeat signal carried information about theDoppler shifts of the atoms in the opticallattices [61]. These Doppler shifts wereexpected to be in the sub-MHz range foratoms with the previously measured 50-µK temperatures. The observed featuresconfirmed the quantum nature of themotion of atoms in the wavelength-sizepotential wells (see Fig. 15) [15].

The NIST group also studied atomsloaded into an optical lattice using oflaser light from the spatially orderedarray [71]. They cut off the laser beamsthat formed the lattice, and before the


atoms had time to move away from theirpositions, they pulsed on a probe laserbeam at the Bragg angle appropriate forone of the sets of lattice planes. TheBragg diffraction not only enhanced thereflection of the probe beam by a factorof 105, but by varying the time betweenthe shut-off of the lattice and turn-on ofthe probe, they could also measure the‘‘temperature’’ of the atoms in the lattice.The reduction of the amplitude of theBragg-scattered beam with time providedsome measure of the diffusion of the atomsaway from the lattice sites, much like theDebye–Waller factorDebye–Waller factorin X-ray diffraction.

5.4Quantum Transport in Optical Lattices

In the 1930s, Bloch realized that applyinga uniform force to a particle in a periodicpotential would not accelerate it beyond acertain speed, but instead would resultin Bragg reflection when its deBrogliewavelength became equal to the latticeperiod. Thus, an electric field applied toa conductor could not accelerate electrons

to a speed faster than that correspondingto the edge of a Brillouin zone, andthat at longer times the particles wouldexecute oscillatory motion. Ever since then,experimentalists have tried to observethese Bloch oscillations in increasinglypure and/or defect-free crystals.

Atoms moving in optical lattices are ide-ally suited for such an experiment, as wasbeautifully demonstrated in 1996 [69]. Theauthors loaded a 1-D lattice with atomsfrom a 3-D molasses, further narrowed thevelocity distribution, and then instead ofapplying a constant force, simply changedthe frequency of one of the beams of the1-D lattice with respect to the other ina controlled way, thereby creating an ac-celerating lattice. Seen from the atomicreference frame, this was the equivalent ofa constant force trying to accelerate them.After a variable time ta, the 1-D latticebeams were shut off and the measuredatomic velocity distribution showed beau-tiful Bloch oscillations as a function of ta.The centroid of the very narrow velocitydistribution was seen to shift in velocityspace at a constant rate until it reached

Ato

m n

umbe

r (a

rb. u

nits

)

Atomic momentum (hk)

ta = t

ta = 3t/4

ta = t/2

ta = t/4

ta = 0

Quasi-momentum (k)

Mea

n ve

loci

ty

−2 −1 0 1 2

−0.5

0

0.5

−1 0 1

(a) (b)

Fig. 16 Plot of the measured velocity distribution verses. time in the accelerated 1-D lattice.(a) Atoms in a 1-D lattice are accelerated for a fixed potential depth for a certain time ta andthe momentum of the atoms after the acceleration is measured. The atoms accelerate onlyto the edge of the Brillouin zone where the velocity is +vr, and then the velocity distributionappears at −vr. (b) Mean velocity of the atoms as a function of the quasi-momentum, thatis, the force times the acceleration time (figure taken from Ben Dahan, M., Peik, E.,Reichel, J., Castin, Y., Salomon, C. (1996), Phys. Rev. Lett. 76, 4508–4511)


vr = hk/M, and then it vanished and reap-peared at −vr as shown in Fig. 16. Theshape of the ‘‘dispersion curve’’ allowedmeasurement of the ‘‘effective mass’’ ofthe atoms bound in the lattice.

6Bose–Einstein Condensation

6.1Introduction

In the 1920s, Bose and Einstein predictedthat for sufficiently high phase spacedensity, ρφ ∼ 1 (see Sect. 1.2), a gas ofatoms undergoes a phase transition that isnow called Bose–Einstein condensation.It took 70 years before BEC could beunambiguously observed in a dilute gas.From the advent of laser cooling andtrapping, it became clear that this methodcould be instrumental in achieving BEC.

BEC is another manifestation of quanti-zation of atomic motion. It occurs in theabsence of resonant light, and its onset ischaracterized by cooling to the point wherethe atomic deBroglie wavelengths are com-parable to the interatomic spacing. This isin contrast with the topics discussed inSect. 5 where the atoms were in an opticalfield and their deBroglie wavelengths werecomparable to the optical wavelength λ.

Laser cooling alone is inherently inca-pable of achieving ρφ ∼ 1. This is easilyseen from the recoil limit of Sect. 3.3.4that limits λdeB to λ/‘‘few’’. Since the crosssection for optical absorption near reso-nance is σ ∼ λ2 near ρφ ∼ 1, this limit ofλdeB ∼ λ/‘‘few’’ results in the penetrationdepth of the cooling light into the samplebeing smaller than λ. Thus, the samplewould have to be extremely small and con-tain only a few atoms, hardly a systemsuitable for investigation.

Temperatures lower than the recoil limitare readily achieved by evaporative cooling,and so all BEC experiments employ it intheir final phase. Evaporative cooling isinherently different from the other coolingprocesses discussed in Sect. 3, and hencediscussed here separately.

Since the first observations in 1995, BEChas been the subject of intense investiga-tion, both theoretical and experimental. Noattempt is made in this article to even ad-dress, much less cover, the very rich rangeof physical phenomena that have been un-veiled by these studies. Instead, we focuson the methods to achieve ρφ ∼ 1 andBEC.

6.2Evaporative Cooling

Evaporative cooling is based on thepreferential removal of those atoms froma confined sample with an energy higherthan the average energy followed by arethermalization of the remaining gas byelastic collisions. Although evaporation isa process that occurs in nature, it wasapplied to atom cooling for the first timein 1988 [72].

One way to think about evaporativecooling is to consider cooling of a con-tainer of hot liquid. Since the mostenergetic molecules evaporate from theliquid and leave the container, the remain-ing molecules obtain a lower temperatureand are cooled. Furthermore, it requiresthe evaporation of only a small fractionof the liquid to cool it by a considerableamount.

Evaporative cooling works by remov-ing the higher-energy atoms as sug-gested schematically in Fig. 17. Thosethat remain have much lower average en-ergy (temperature) and so they occupy asmaller volume near the bottom of the


Fig. 17 Principle of the evaporation technique.Once the trap depth is lowered, atoms withenergy above the trap depth can escape and theremaining atoms reach a lower temperature

trap, thereby increasing their density. Fortrapped atoms, it can be achieved by lower-ing the depth of the trap, thereby allowingthe atoms with energies higher than thetrap depth to escape, as discussed first byHess [73]. Elastic collisions in the trap thenlead to a rethermalization of the gas. Thistechnique was first employed for evapora-tive cooling of hydrogen [72, 74–76]. Sinceboth the temperature and the volume de-crease, ρφ increases.

Recently, more refined techniques havebeen developed. For example, to sustainthe cooling process the trap depth canbe lowered continuously, achieving acontinuous decrease in temperature. Sucha process is called forced evaporation and isdiscussed in Sect. 6.3 below.

6.2.1 Simple ModelThis section describes a simple model ofevaporative cooling. Since such cooling isnot achieved for single atoms but for thewhole ensemble, an atomic description ofthe cooling process must be replaced bythermodynamic methods. These methodsare completely different from the rest of thematerial in this article, and will thereforeremain rather elementary.

Several models have been developedto describe this process, but we presenthere the simplest one [77] because of itspedagogical value [22]. In this model, thetrap depth is lowered in one single stepand the effect on the thermodynamicquantities, such as temperature, density,and volume, is calculated. The process canbe repeated and the effects of multiplesteps added up cumulatively.

In such models of evaporative cooling,the following assumptions are made:

1. The gas behaves sufficiently ergodically,that is, the distribution of atomsin phase space (both position andmomentum) depends only on theenergy of the atoms and the natureof the trap.

2. The gas is assumed to begin theprocess with ρφ 1 (far from the BECtransition point), and so it is describedby classical statistics.

3. Even though ρφ 1, the gas is coldenough that the atomic scattering ispure (s-wave) quantum mechanical,that is, the temperature is sufficientlylow that all higher partial waves donot contribute to the cross section.Furthermore, the cross section forelastic scattering is energy-independentand is given by σ = 8πa2, where a is thescattering length. It is also assumed thatthe ratio of elastic to inelastic collisionrates is sufficiently large that the elasticcollisions dominate.

4. Evaporation preserves the thermal na-ture of the distribution, that is, thethermalization is much faster than therate of cooling.

5. Atoms that escape from the trap neithercollide with the remaining atoms norexchange energy with them. This iscalled full evaporation.


6.2.2 Application of the Simple ModelThe first step in applying this simple modelis to characterize the trap by calculatinghow the volume of a trapped sampleof atoms changes with temperature T .Consider a trapping potential that can beexpressed as a power law given by

U(x, y, z) = ε1

∣∣∣∣ x

a1

∣∣∣∣s1

+ ε2

∣∣∣∣ y

a2

∣∣∣∣s2

+ ε3

∣∣∣∣ z

a3

∣∣∣∣s3

, (26)

where aj is a characteristic length and sj thepower, for a certain direction j. Then thevolume occupied by trapped atoms scalesas V ∝ Tξ [78], where

ξ ≡ 1

s1+ 1

s2+ 1

s3. (27)

Thus, the effect of the potential on thevolume of the trapped sample for agiven temperature can be reduced to asingle parameter ξ . This parameter isindependent of how the occupied volumeis defined, since many different definitionslead to the same scaling. When a gas is heldin a 3-D box with infinitely high walls, thens1 = s2 = s3 = ∞ and ξ = 0, which meansthat V is independent of T , as expected.For a harmonic potential in 3-D, ξ = 3/2;for a linear potential in 2-D, ξ = 2; and fora linear potential in 3-D, ξ = 3.

The evaporative cooling model itself [77]starts with a sample of N atoms in volumeV having a temperature T held in an in-finitely deep trap. The strategy for usingthe model is to choose a finite quantity η,and then (1) lower the trap depth to a valueηkBT , (2) allow for a thermalization of thesample by collisions, and (3) determine thechange in ρφ .

Only two parameters are needed to com-pletely determine all the thermodynamic

quantities for this process (the values af-ter the process are denoted by a prime).One of these is ν ≡ N′/N, the fraction ofatoms remaining in the trap after the cool-ing. The other is γ (This γ is not to beconfused with the natural width of the ex-cited state.), a measure of the decrease intemperature caused by the release of hotatoms and subsequent cooling, modifiedby ν, and defined as

γ ≡ log(T ′/T)

log(N′/N)= log(T ′/T)

log ν. (28)

This yields a power-law dependence forthe decrease in temperature caused bythe loss of the evaporated particles, thatis, T ′ = Tνγ . The dependence of theother thermodynamic quantities on theparameters ν and γ can then be calculated.

The scaling of N′ = Nν, T ′ = Tνγ , andV ′ = Vνγ ξ can provide the scaling ofall the other thermodynamic quantitiesof interest by using the definitions forthe density n = N/V , the phase spacedensity ρφ = nλ3

deB ∝ nT−3/2, and theelastic collision rate kel ≡ nσ v ∝ nT1/2.The results are given in Table 2. For agiven value of η, the scaling of all quantitiesdepends only on γ . Note that for successivesteps j, ν has to be replaced with ν j.

Tab. 2 Exponent q for the scaling of thethermodynamic quantities X ′ = Xνq with thereduction ν of the number of atoms in the trap

Thermodynamicvariable

Symbol Exponent q

Number ofatoms

N 1

Temperature T γ

Volume V γ ξ

Density n 1 − γ ξ

Phase spacedensity

ρ 1 − γ (ξ + 3/2)

Collision rate k 1 − γ (ξ − 1/2)


The fraction of atoms remaining is fullydetermined by the final trap depth η fora given potential characterized by the trapparameter ξ . In order to determine thechange of the temperature in the coolingprocess, it is necessary to consider indetail the distribution of the atoms in thetrap, and this is discussed more fully inRefs. [1, 22, 77].

6.2.3 Speed of EvaporationSo far, the speed of the evaporative coolingprocess has not been considered. If thetrap depth is ramped down too quickly, thethermalization process does not have timeto run its course and the process becomesless efficient. On the other hand, if thetrap depth is ramped down too slowly,the loss of particles by inelastic collisionsbecomes important, thereby making theevaporation inefficient.

The speed of evaporation can be foundfrom the principle of detailed balance [22].Its application shows that the ratio of theevaporation time and the elastic collisiontime is

τev

τel=

√2eη

η. (29)

Note that this ratio increases exponentiallywith η.

Experimental results show that ∼2.7elastic collisions are necessary to rether-malize the gas [79]. In order to model therethermalization process, Luiten et al. [80]have discussed a model based on the Boltz-mann equation where the evolution of thephase space density ρ(r, p, t) is calculated.This evolution is not only caused by thetrapping potential, but also by collisionsbetween the particles. Only elastic col-lisions, whose cross section is given byσ = 8πa2 with a as the scattering length,are considered. This leads to the Boltz-mann equation [81].

6.2.4 Limiting TemperatureIn the models discussed so far, onlyelastic collisions have been considered,that is, collisions where kinetic energyis redistributed between the partners.However, if part of the internal energyof the colliding partners is exchanged withtheir kinetic energy in the collision, thenit is inelastic. Inelastic collisions can causeproblems for two reasons: (1) the internalenergy released can cause the atoms toheat up and (2) the atoms can change theirinternal states, and the new states mayno longer be trapped. In each case, suchcollisions can lead to trap loss and aretherefore not desirable.

Apart from collisions with the back-ground gas and three-body recombination,there are two inelastic processes that areimportant for evaporative cooling of alkaliatoms: dipolar relaxation and spin relax-ation. The collision rate nkdip for themat low energies is independent of veloc-ity [82]. Since the elastic collision rate isgiven by kel = nσ vrel, the ratio of good(= elastic) to bad (= relaxation) collisionsgoes down when the temperature does.This limits the temperature to a value Te

near which the ratio between good andbad collisions becomes unity, and Te isgiven by

kBTe =πMk2

dip

16σ 2 . (30)

The limiting temperature for the alkalisis of the order of 1 nK, depending on thevalues of σ and kdip.

In practice, however, this ratio has tobe considerably larger than unity, andso the practical limit for evaporativecooling occurs when the ratio is ∼103 [22].In the model of Ref. [80], the authorsdiscuss different strategies for evaporativecooling. Even for the strategy of the


lowest temperature, the final temperatureis higher than Te.

The collision rate between atoms withone in the excited state (S + P collisions) isalso much larger at low temperatures thanthe rate for such collisions with both atomsin the ground state (S + S collisions). SinceS + P collisions are generally inelastic andthe inelastic energy exchange generallyleads to a heating of the atoms, increasingthe density increases the loss of coldatoms. To achieve BEC, resonant lightshould therefore be avoided, and thus lasercooling is not suitable for achieving BEC.

6.3Forced Evaporative Cooling

In all the earliest experiments that achievedBEC, the evaporative cooling was ‘‘forced’’by inducing rf transitions to magneticsublevels that are not bound in themagnetic trap. Atoms with the highestenergies can access regions of the trapwhere the magnetic field is stronger, andthus their Zeeman shifts would be larger.A correspondingly high-frequency rf fieldwould cause only these most energeticatoms to undergo transitions to statesthat are not trapped, and in doing so, thedeparting atoms carry away more than theaverage energy. Thus, a slow sweep ofthe rf frequency from high to low wouldcontinuously shave off the high-energy tailof the energy distribution, and therebycontinuously drive the temperature lowerand the phase space density higher. Theresults of evaporative cooling from the firstthree groups that have obtained BEC haveshown that using this rf shaving technique,it is much easier to select high-energyatoms and waste them than it is to coolthem.

For the evaporation of the atoms, it isimportant that atoms with an energy above

the cutoff are expelled from the trap. Byusing RF-evaporation, one can expel theatoms in all three dimensions equally andthus obtain a true 3-D evaporation. In thecase of the time orbiting potential (TOP)trap, the atoms are evaporated along theouter side of the cloud that is exposed tothe highest magnetic field on the average.This is a cylinder along the direction ofrotation axis of the magnetic field and thusthe evaporation takes place in 2-D.

Once the energy of the atoms becomesvery small, the atoms sag because of gravityand the outer shell of the cloud is no longerat a constant magnetic field. Atoms at thebottom of the trap have the highest energyand thus the evaporation becomes 1-D. Incase of harmonic confinement, Utrap =U′′z2/2, the equipotential surface is atz ≈ √

2ηkBT/U′′. Now, the gravitationalenergy is given by Ugrav = mgz andthus the limiting temperature for 1-Devaporation to take place is given by [22]

kBT <2η(mg)2

µB′′ (31)

For a curvature of B′′ = 500 T m−2, thelimiting temperature becomes 1 µK for 7Li,10 µK for 23Na, and 150 µK for 87Rb. Belowthis temperature, evaporation becomesless efficient.

In the three experiments that obtainedBEC for the first time in 1995, the problemof this ‘‘gravitational sag’’ was not known,but it did not prevent the experimentalistfrom observing BEC. The solution usedin those experiments was because of thelight mass (7Li), tight confinement (23Na),and TOP trap (87Rb). In the last case, theaxis of rotation is in the z-direction andthus the evaporation always remains 2-D. Table 3 shows typical values of ρφ forvarious situations.


Tab. 3 Typical numbers for the phase space density as obtained in theexperiments aimed at achieving BEC. The different stages of cooling andtrapping the atoms are discussed in the text

Stages T λdeB n ρφ

Oven 300 C 0.02 nm 1010 cm−3 10−16

Slowing 30 mK 2 nm 108 cm−3 10−12

Pre-cooling 1 mK 10 nm 109 cm−3 10−9

Trapping 1 mK 10 nm 1012 cm−3 10−6

Cooling 1 µK 0.3 µm 1011 cm−3 3 × 10−3

Evaporation 70 nK 1 µm 1012 cm−3 2.612

7Conclusion

In this article, we have reviewed someof the fundamentals of optical control ofatomic motion. The reader is cautionedthat this is by no means an exhaustivereview of the field, and that many impor-tant and current topics have been omitted.Much of the material here was taken fromour recent textbook [1], and the reader isencouraged to consult that source for theorigin of many of the formulas presentedin the present text, as well as for furtherreading and more detailed references tothe literature.

Glossary

Atomic Beam Slowing: Using laser light toslow down the velocities of atoms in abeam.

Bad Collisions: Inelastic scattering ofatoms leading to loss of atoms from thetrap during evaporation.

Beam Brightening: Increasing the bright-ness of an atomic beam by using lasercooling.

Bloch Oscillations: The oscillatory motionof particles moving through a periodic

potential, predicted in the 1930s by Blochfor electrons in solid state.

Bose–Einstein Condensation (BEC): Newstate of matter theoretically described inthe 1920s by Bose and Einstein andexperimentally first observed in dilutegases in 1996, where the interatomicspacing is smaller than the deBrogliewavelength of the atoms.

Brightness: The number of atoms emittedfrom a source or in a beam per unit oftime, per unit of solid angle, and per unitof area of the source.

Capture Range: The range in velocity,where the optical forces are effective.

Damping Force: The optical force on atomsthat leads to damping of their velocity.

Dark-spot Magneto-optical Trap: Magneto-optical trap, where in the center of thetrap the atoms are not kept in the cyclingtransition, which reduces the loss of atomsdue to inelastic collisions.

Dipole Force Trap: Trap using the dipoleoptical force to trap atoms.

Dipole Forces: The optical force on theatom caused by the gradient of the lightshift of the atom.


Doppler Cooling Limit: Limit of coolingatoms given by the Doppler theory.

Doppler Cooling Techniques: Using theDoppler shift and the detuning of the lightfrom atomic resonance in order to coolatoms.

Doppler Theory: Theory describing thecooling of two-level atoms by exploiting theDoppler shift of the atoms in combinationwith the detuning of the laser light fromresonance.

Ehrenfest Theorem: Theorem by Ehrenfestmaking the correspondence between clas-sical relations, like F = −gradV , and theirquantum-mechanical counterpart, in thiscase 〈F〉 = −grad〈H〉.Entropy: Thermodynamic measure of thedisorder in the system.

Evaporative Cooling: The preferential re-moval of high-energy atoms from a gassample, thereby reducing the temperatureof the remaining atoms.

Far-off-resonance Traps: Dipole force trap,where the light is far detuned from reso-nance, which minimizes the scattering oflight due to spontaneous emission.

Fokker-Planck Equation: Equation describ-ing the evolution of the atomic momentumdistribution under the combined influenceof force and diffusion.

Good Collisions: Elastic scattering of atomsleading to the thermalization of the atomsduring evaluation.

Laser Cooling: Using the interaction be-tween laser light and atoms to reduce theaverage kinetic energy of the atoms.

lin ⊥ lin Cooling: Polarization gradientcooling, where the two beams haveperpendicular, linear polarization.

Magnetic Traps: Traps where atoms aretrapped due to their magnetic moment byan inhomogeneous magnetic field.

Magneto-optical Trap (MOT): Combinationof light forces and inhomogeneous mag-netic field, leading to cooling and trappingof atoms.

Majorana Losses: Loss of atoms frommagnetic traps, where the atoms in thecenter of the trap undergo a motion-induced transition to an untrapped state.

Multilevel Atoms: Atoms that have degen-erate magnetic substates in ground andexcited state, which are coupled by laserlight.

Optical Bloch Equations (OBE): Relationsdescribing the evolution of the state vectorof the internal state of the atom due to theinteraction with laser light.

Optical Forces: The force induced on theatom by laser light.

Optical Lattice: The periodic trapping po-tential for atoms created by the interfer-ence of laser beams.

Optical Molasses: Viscous damping ofthe atomic velocities by the use ofcounterpropagating laser beams.

Phase Space Density: Probability of find-ing a particle in a certain region of space,with a certain range of velocities.

Polarization Gradient Cooling: The use ofpolarization gradients to cool atoms belowthe Doppler limit.

Polarization Gradient: The interferencepattern that occurs when two counter-propagating laser beams have differentpolarization. The resulting polarization ofthe light field is not constant in space, thatis, shows a gradient.


Quadrupole Trap: The simplest magnetictrap, where the magnetic field is generatedby two separated identical coils carryingopposite currents.

Radiative Forces: The optical force on theatom caused by the scattering of light.

Recoil Limit: Limit in laser cooling deter-mined by the recoil by one photon on theatom.

Saturation Intensity: Intensity of the lightwhere the rate for spontaneous emissionand the rate for stimulated emission areequal.

Saturation Parameter: Ratio between theintensity of the laser light and thesaturation intensity.

σ+-σ− Cooling: Polarization gradient cool-ing, where the two beams have opposite,circular polarization.

Sisyphus Cooling: The continuous trans-fer of kinetic energy to potential en-ergy in laser cooling, where the poten-tial energy is radiated away by sponta-neously emitted photons. A special caseof Sisyphus cooling is polarization gra-dient cooling to sub-Doppler tempera-ture.

Spontaneous Emission: The return of theatom from the excited state to the groundstate by the emission of one photon in arandom direction.

Stimulated Emission: The return of theatom from the excited state to the groundstate by the emission of one photon in thedirection of the laser light.

Sub-Doppler Cooling: Using laser cool-ing techniques beyond the Doppler tech-niques to cool atoms below the Dopplerlimit.

s-wave Scattering: Scattering of atoms atlow temperatures, where only the lowest-order partial wave is effective.

Temperature: A measure of the averagekinetic energy of the atoms, that is,kBT/2 = 〈Ek〉.Time-of-flight (TOF) Method: Technique todetermine the flight time of atoms over awell-defined flight path in order to detectthe atomic velocity.

Time-orbiting Potential (TOP) trap: Mag-netic quadrupole trap, where the zero ofthe magnetic field oscillates in space inorder to reduce trap loss due to Majoranaflips.

Two-level Atoms: Hypothetical atoms thatonly have one nondegenerate groundstate and one nondegenerate excited state,which are resonant with the laser light.

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1251

Lasers, Semiconductor

Nagaatsu OgasawaraUniversity of Electro-Communications, Department of Electronics Engineering, Tokyo, Japan

Ryoichi ItoMeiji University, Department of Physics, Kawasaki, JapanPhone/Fax: +81-44-934-7425; e-mail: [email protected]

AbstractFollowing a brief history of semiconductor lasers, this article describes their opera-tion principles in relation to their device structures and materials. Their fundamentaloperation characteristics, including the static, spectral, and dynamic aspects, are thenpresented. Finally, the current state of the art in semiconductor laser technology is de-scribed by reviewing up-to-date devices incorporating new structures and new functions.

Keywordssemiconductor laser; laser diode; photonics; compact light source; heterostructure;fiber-optic telecommunications; optical disk.

1 Introduction 12522 History 12533 Structures, Materials, and Operation Principles 12543.1 Double Heterostructure 12543.2 Fabry–Perot Cavity 12573.3 Lasing Threshold Condition 12573.4 Stripe Geometry 12583.5 Materials and Emission Wavelengths 12593.5.1 III–V Compounds 12593.5.2 IV–VI Compounds 12603.5.3 II–VI Compounds 1260

1252 Lasers, Semiconductor

4 Fundamental Characteristics 12604.1 Light–Current Characteristics 12604.2 Beam Profile and Polarization 12624.3 Spectral Characteristics 12634.4 Dynamic Characteristics 12655 New Structures and Functions 12675.1 Quantum-well Lasers 12675.2 Distributed-feedback and Distributed Bragg-reflector Lasers 12685.3 Semiconductor-laser Arrays 12695.4 Surface-emitting Lasers 12706 Summary 1270


1Introduction

Semiconductor lasers have now grown tobe key components in modern photonicstechnology, most notably as light sourcesin fiber-optic telecommunications systemsand optical disk systems. They havealso found an increasing number ofapplications ranging from instrumentssuch as bar code scanners and computerprinters to solid-state laser pumps andmeasuring and sensoring equipment forengineering use.

Semiconductor lasers have the followingfeatures that distinguish them from otherlasers.

1. Compactness: The typical size of a laserchip is 300 × 200 × 100 µm3. The smallchip contains all the ingredients of a laserstructure: a resonator, a waveguide, anactive medium, and a p-n junction to pumpthe active medium.2. High efficiency: Semiconductor laserscan be driven by low electrical power[(several tens of mA) × (1–2 V)]. The

efficiency of converting electrical powerinto optical power is several tens ofpercent, which should be compared withthat of 0.1% or less for gas and otherlasers.3. Capability for high-speed direct modula-tion: Light output can be modulated atfrequencies of 10 GHz or more simply bymodulating the pumping current.

4. Wide emission spectrum: The emissionwavelength is determined by the band-gap energy of the active-layer material.The blue-near-infrared region is coveredby III–V compounds. The infrared re-gion (3–30 µm) is covered by IV–VIcompounds.

5. High reliability: Semiconductor lasershave become highly reliable devices be-cause of the remarkable improvement incrystal quality, although rapid degradationwas a great problem at the early stage oftheir development.

6. Sensitivity to temperature: The thresholdcurrent and the lasing wavelength stronglydepend on temperature. This is a disadvan-tage for most applications. On the other

Lasers, Semiconductor 1253

hand, wavelength tunability is useful inspectroscopic research.

This article is intended to serve as auseful reference for those who wish tograsp the basic concepts of semiconductorlasers and utilize them in engineering andresearch activities. Following this intro-duction is a brief history of semiconductorlasers. Next, the operation principles ofsemiconductor lasers are described in re-lation to their structures and materials.The fourth section presents a general viewof fundamental operation characteristicscovering the static, spectral, and dynamicaspects. The fifth section gives the cur-rent state of the art in semiconductorlaser technology by reviewing up-to-datedevices incorporating new structures andnew functions. Because of the limitedspace, citation of published papers hasbeen kept to a minimum. Several bookslisted in the (Further Reading) section canbe consulted both for further study and foraccess to the relevant published papers.

2History

The advent of semiconductor injectionlasers dates back to 1962, only two yearslater than the first achievement of laseraction in ruby, when stimulated emis-sion from forward-biased GaAs diodeswas demonstrated [1–3]. The stimulatedemission resulted from the radiative re-combination of electrons and holes inthe direct-gap semiconductor GaAs. Theelectrons and holes were injected by for-ward bias into the depletion region inthe vicinity of the p-n junction. Opticalfeedback was provided by polished facetsperpendicular to the junction plane. Thedemonstration of GaAs lasers prompted

the exploration of many other III–Vand IV–VI compound semi-conductors.Unfortunately, these early homojunctiondevices, consisting of a single semicon-ductor, were not considered for seriousapplications since continuous operationat room temperature was not feasible be-cause of their high threshold current den-sities for lasing (several tens of kA cm−2).

A breakthrough was brought about in1970 by employing a double heterostruc-ture grown by liquid-phase epitaxy [4, 5].In the double heterostructure, stimulatedemission occurred only within a thin ac-tive layer of GaAs sandwiched betweenp- and n-doped AlGaAs layers that havea wider band gap. The threshold currentdensity was dramatically reduced to severalkA cm−2 or less, and continuous opera-tion at room temperature was eventuallyaccomplished. The invention of the dou-ble heterostructure was obviously the mostimportant step in the history of the devel-opment of semiconductor lasers towardpractical utility. For the invention of het-erostructure, Alferov and Kroemer werehonored with a Nobel Prize in physics in2000, while Alferov, Hayashi and Panishshared the Kyoto Prize in 2001.

There still remained, however, a trou-blesome problem of degradation; the reli-ability of the early double-heterostructuredevices was very poor and many of themstopped their continuous operation withina few or several tens of minutes. For-tunately, intensive investigations of thedegradation mechanism led to the identi-fication of the causes of major failure: therecombination-enhanced growth of dis-locations, and facet erosion [6 Part B,Chapter 8]. These studies have revealedthat a long operating life is obtained byeliminating dislocation generation duringmaterial growth and device processing and


by coating the mirror facets with dielectricthin films.

Another task undertaken in parallelwith the improvement of reliability wasthe stabilization of the optical mode inthe lateral transverse direction (parallelto the junction plane). It had beenrecognized by the late 1970s that thelateral-mode instability often observed athigher output powers was detrimentalto most applications and that such aninstability could be effectively suppressedby incorporating a built-in refractive-indexprofile in the lateral direction [7]. A largenumber of laser structures have beenproposed and demonstrated to introducelateral variations of the refractive index.

As a result of these research and develop-ment efforts during the first two decades,the maturity of AlGaAs lasers reached thestage of practical use in printers and diskplayers early in the 1980s. Meanwhile, therapid reduction of transmission loss inoptical fibers achieved during the 1970sstimulated intensive development effortson GaInPAs lasers, which have emissionwavelengths (1.3–1.55 µm) in the low-lossand dispersion-free region of silica fibers.During the first half of the 1980s, GaIn-PAs lasers emerged as indispensable lightsources in telecommunications systemsusing silica fibers (see OPTICAL COMMUNI-

CATIONS).Progress has continued to date in both

device structures and materials. In thefollowing, let us look at some major topics.

The idea of utilizing periodic gratingsincorporated in a laser medium as ameans of optical feedback was proposedby Kogelnik and Shank in [8]. Distributed-feedback and distributed-Bragg-reflectorsemiconductor lasers utilizing this prin-ciple have since become the major devicestructures used in long-haul, high bit-ratetelecommunications systems because of

the stable single–longitudinal-mode oscil-lation they exhibit even under high-speedmodulation.

Recent advances in growth technol-ogy of semiconductor ultrathin layers bymolecular-beam epitaxy and metal-organicvapor-phase epitaxy have also created anew class of laser structures. Quantum-well lasers incorporating such ultrathinactive layers (∼20 nm or less) have beenshown to display a variety of superior lasercharacteristics, such as low-threshold cur-rents and wide modulation bandwidths,which are attributed to the size quanti-zation of electrons in the ultrathin activelayers [9]. Quantum-well structures as wellas multiple quantum-well structures arenow commonly employed in most semi-conductor lasers.

In the visible spectrum, GaInP laserswere first commercialized as red-emittinglasers (0.63–0.69 µm) in 1988 by Sony,Toshiba, and NEC. They are now exten-sively used in DVD systems. Blue–greenlaser emission was achieved first inZnCdSe lasers [10] and then in InGaNlasers [11, 12]. InGaN blue lasers are be-ing employed in second-generation DVDplayers (see DATA STORAGE, OPTICAL).

3Structures, Materials, and OperationPrinciples

3.1Double Heterostructure

A semiconductor laser is a diode, asshown schematically in Fig. 1. When aforward current is passed through the p-njunction, electrons and holes injected intothe active region from the n and p regionsrespectively, recombine to emit radiation.The wavelength λ of the radiation is


x

z

y

ElectrodeCurrent Hole

Laseremission

Electron

Cleaved facet

p-type

n -type

Active layer100 µm

300 µm

200 µm

Fig. 1 Schematic illustration of a double-heterostructure semiconductor laser

basically determined by the relation λ ∼hc/Eg, where h is the Planck constant, c isthe light velocity, and Eg is the band-gapenergy of the active-region material.

For low injection currents, light is emit-ted through spontaneous emission. Inorder for lasing to take place, a suf-ficiently high concentration of carriersmust be accumulated within the active re-gion to induce population inversion. Thisis effectively accomplished by adoptinga double-heterojunction (DH) structure,where a thin active layer, typically ∼0.1 µmthick, but as thin as 10 nm in quantum-well lasers, is sandwiched between n- andp-type cladding layers, which have widerband gaps than the active layer. Electronsand holes injected into the active layerthrough the heterojunctions are confinedwithin the thin active layer by the poten-tial barriers at the heteroboundaries, asillustrated in Fig. 2(a). The DH structureforms an efficient optical waveguide aswell, because of the refractive-index dif-ference between the active and claddinglayers shown in Figs. 2(b) and 2(c). Thus,

the DH structure facilitates the interactionneeded for laser action between the opticalfield and the injected carriers.

Let us use a GaAs(active)/AlxGa1−xAs(cladding) DH structure as an example.The dependence of the band-gap energy ofAlxGa1−xAs on the AlAs mole fraction xcan be approximated for x < 0.42 (direct-gap region) by [6 Part A, p. 193].

Eg = 1.424 + 1.247x eV. (1)

The energy-gap difference between theactive and cladding layers Eg is dividedinto the band discontinuities Ec in theconduction band and Ev in the valenceband according to [13].

Ec ∼ 0.62Eg (2)

andEv ∼ 0.38Eg. (3)

The efficiency of the carrier confinementdepends strongly on the magnitude ofthe band discontinuity. The AlAs molefraction x of the cladding layers is usuallychosen to be greater than that of the active


Ev

Ec

p -claddinglayer

n -claddinglayer

Electron

Hole

eV

Activelayer

(a)

dEle

ctro

n en

ergy

(b)

Ref

ract

ive

inde

x

y

Γ

(c)

Ligh

tin

tens

ity

Fig. 2 Diagram illustrating carrier confinement andwaveguiding in a double heterostructure (a) Energy-banddiagram at high forward bias; (b) refractive-index distribution;(c) light intensity distribution. Ec and Ev: The edges of theconduction and valence bands. V: Applied voltage. d:Active-layer thickness (∼0.1 µm). : Confinement factor

layer by ∼0.3. This gives rise to a Ecapproximately 10 times as large as theroom-temperature thermal energy, whichis sufficient to suppress electron diffusionover the heterobarriers. The hole leakagecurrent is less important because of thesmaller diffusion constant for holes.

The characteristics of a three-layer slabwaveguide are conveniently described interms of the normalized waveguide thick-ness D, defined as

D =(

2π

λ

)d√

η2a − η2

c , (4)

where ηa and ηc are the refractive indices ofthe active and cladding layers respectivelyand d is the active-layer thickness. Forexample, the condition that a waveguide

supports only the lowest-order fundamen-tal mode is expressed as

D < π. (5)

Meanwhile, the refractive index ofAlxGa1−xAs, η(x), can be approximatedfor a light wavelength of ∼0.9 µm by [6,Part A, p. 45]

η(x) = 3.590 − 0.710x + 0.091x2. (6)

Thus, for a waveguide with a GaAsactive layer surrounded by Al0.3Ga0.7Aslayers and a wavelength λ of 0.87 µm, thecondition of Eq. (5) simply becomes

d < 0.36 µm. (7)


Since the thickness d is typically 0.1to 0.2 µm, the single, fundamental-modecondition is almost always satisfied inpractical devices.

The confinement factor , defined as thefraction of the electromagnetic energy ofthe guided mode that exists within theactive layer, is an important parameterrepresenting the extent to which thewaveguide mode is confined to the activelayer. for a fundamental mode isapproximately given by [14]

∼ D2

2 + D2 . (8)

For a GaAs/Al0.3Ga0.7As waveguidewith d = 0.1 µm, ∼ 0.27.

3.2Fabry–Perot Cavity

In addition to the optical gain, opticalfeedback is the other ingredient for laseroscillation. This is provided by a pairof mirror facets at both ends of thedevices. These facets are normally formedsimply by cleaving the crystal. Since therefractive index of major semiconductorlaser materials is ∼3.6, the reflectivityof the mirror is ∼0.3. This is very lowcompared to other types of lasers but isstill sufficient to provide optical feedbackin semiconductor lasers.

3.3Lasing Threshold Condition

When a sufficient number of electronsand holes is accumulated to form aninverted population, the active regionexhibits optical gain and can amplifylight passing through it since stimulatedemission overcomes interband absorption.The condition for self-sustained laseroscillation to occur is that the light makes

a full round trip in the cavity withoutattenuation; that is, the optical gain shouldequal the losses both inside the cavity andthrough the partially reflecting end facets.Thus, the gain coefficient at threshold gthis given by the relation

gth = αa + (1 − )αc + αs︸︷︷︸αi

+1

Lln

1

R.

(9)

Here, αa and αc denote the losses inthe active and cladding layers respectively,due to free-carrier absorption. αs accountsfor scattering loss due to heterointerfacialimperfections. The first three loss termson the right-hand side combined aretermed internal loss αi and add up to 10to 20 cm−1 [6, Part A, p. 174–176]. Thereflection loss L−1 ln R−1 due to outputcoupling (∼40 cm−1 for L ∼ 300 µm, R ∼0.3) is normally the largest among theloss terms. Despite the fact that boththe internal and the reflection losses areexceptionally large as compared to otherlasers, lasing is brought about by virtue ofthe high optical gain in semiconductors.

It is not an easy task to analyze preciselythe optical gain spectrum in semiconduc-tor lasers. Fortunately, however, we canutilize a phenomenological linear relation-ship between the maximum gain g (thepeak value of the gain spectrum for a givencarrier density) and the injected carrierdensity n,

g(n) = ∂g

∂n(n − nt), (10)

to a good approximation [15–17]. Here,∂g/∂n is termed differential gain, and nt

denotes the carrier density required toachieve transparency where stimulatedemission balances against interband ab-sorption corresponding to the onset of


population inversion. For GaAs lasers,

∂g

∂n∼ 3.5 × 10−16 cm2 (11)

and

nt ∼ 1.5 × 1018 cm−3. (12)

Substituting Eqs. (10) to (12) intoEq. (9) and assuming that = 0.27, αi =10 cm−1, and L−1 ln R−1 = 40 cm−1, weget a threshold carrier density nth of∼2 × 1018 cm−3.

The threshold current density Jth isexpressed as

Jth = ednth

τs, (13)

where τs is the carrier lifetime due tospontaneous emission. Assuming thatτs = 2–4 ns and d = 0.1 µm, we ob-tain a threshold current density Jth or∼1 kA cm−2.

3.4Stripe Geometry

Most modern semiconductor lasers adopta stripe geometry, where current is injectedonly within a narrow region beneath astripe contact several µm wide, in orderto keep the threshold current low and tocontrol the optical field distribution in thelateral direction. As compared with broad-area lasers, where the entire laser chipis excited, the threshold current of thestripe-geometry lasers is reduced roughlyproportional to the area of the contact.

Figure 3(a) shows the simplest form ofthe stripe geometry [18]. Lasing occurs ina limited region of the active layer beneaththe stripe contact where a high density ofcurrent flows. Such lasers are termed gain-guided lasers because the optical intensitydistribution in the lateral direction (x

ElectrodeInsulator

p -cladding layer

p -burying layer

Active layern -burying layer

n -cladding layer

n -substrate

Electrode

Electrode

Insulator

p -cladding layer

Active layer

n -cladding layer

n -substrate

Electrode

x

y

z

(a)

(b)

w

w

Fig. 3 Two representative examples ofstripe-geometry lasers (front view of the endfacet) (a) Stripe-contact laser; (b) Buriedheterostructure (BH) laser. W: Stripe width(2–10 µm)

direction) is determined by the gain profileproduced by carrier density distribution.

Devices incorporating a built-in refr-active-index variation in the lateral di-rection are termed index-guided lasers.Figure 3(b) shows the structure of a buriedheterostructure (BH) laser [19] as a repre-sentative example of index-guided lasers.The active region is surrounded by materi-als with lower refractive indices in both thevertical (y) and lateral (x) transverse direc-tions, thus forming a waveguide structurein both directions. In the BH laser, the


width of current flow is delineated by thep-n junction formed by the burying layers,which is reverse biased when the activeregion is forward biased.

3.5Materials and Emission Wavelengths

Semiconductor-laser materials that havebeen studied range over III–V, IV–VI,and II–VI compounds as listed in Table 1.

3.5.1 III–V CompoundsIII–V compounds are the most impor-tant and popular laser materials [6, PartB, Chapter 5]. In particular, AlGaAs (ac-tive layer)/AlGaAs (cladding layer)/GaAs(substrate) lasers emitting at 0.7 to0.9 µm and GaInP/AlGaInP/GaAs lasersemitting at 0.63 to 0.69 µm are exten-sively used in optical disk systems and

laser printers. GaInPAs/InP/InP lasersemitting at 1.2 to 1.6 µm are usedin fiber-optic communications systems.GaInN/AlGaN/sapphire lasers emittingat 0.38 to 0.45 µm are starting to beused in second-generation DVD players.All of them exhibit low-threshold, room-temperature continuous wave (cw) opera-tion with high reliability.

The common features among them areas follows:

1. The active layer consists of direct-gapmaterials.

2. Binary compounds (GaAs, InP) areused as the substrate except forGaInN/AlGaN/sapphire lasers, forwhich GaN substrate will be used inthe future.

3. The active and cladding layers havenearly the same lattice constant as thesubstrate.

Tab. 1 Major semiconductor-laser materials and their emissionwavelengths

Materials (active/cladding/substrate) Emissionwavelengths [µm]

III–V compoundsAlGaAs/AlGaAs/GaAs 0.7–0.9GaInPAs/InP/InP 1.2–1.6GaInP/AlGaInP/GaAs 0.66–0.69GaInPAs/GaInP/GaAs or GaPAs 0.65–0.9GaInPAs/AlGaAs/GaAs 0.62–0.9AlGaAsSb/AlGaAsSb/GaSb 1.1–1.7GaInAsSb/AlGaAsSb/GaSb or InAs 2–4InPAsSb/InPAsSb/GaSb or InAs 2–4GaInAs(strained)/AlGaAs/GaAs 0.9–1.1GaInPAs(strained)/GaInPAs/InP ∼1.55GaInN/AlGaN/sapphire 0.38–0.45

IV–VI compoundsPbSnTe/PbSnSeTe/PbTe 6–30PbSSe/PbS/PbS 4–7PbEuTe/PbEuTe/PbTe 3–6

II–VI compoundsZnCdSe/ZnSSe/GaAs ∼0.5


The use of direct-gap semiconductors,featuring efficient radiative recombina-tion, as the active-region material (1) isessential in achieving laser action. Fea-tures 2 and 3 are related to the crystalquality of the hetero-interfaces. To min-imize the generation of lattice defects thatmay impair the device reliability, the lat-tice parameter of the active and claddinglayers should be matched to that of thehigh-quality binary substrate (the lattice-matching condition). The lattice parameterof AlxGa1−xAs is essentially independentof alloy concentration x. This is the onlyexception in ternary alloys, arising fromthe fortuity that GaAs and AlAs havevirtually the same lattice parameter, thedifference being as small as ∼0.1%. Inquaternary alloys like GaxIn1−xPyAs1−y

and AlxGayIn1−x−yP, the band-gap en-ergy can be tuned in a certain rangefor a fixed lattice parameter by choos-ing appropriate pairs of x and y values.The design of lattice-matched DH struc-tures is facilitated by taking advantageof this degree of freedom in quater-nary alloys. In some quantum-well struc-tures, a certain degree of strain in theactive material caused by inevitably ordeliberately introduced slight lattice mis-matching is used to provide better laserperformance.

3.5.2 IV–VI CompoundsPbSnTe and PbSSe lasers emitting in theinfrared region (3–30 µm) [20] are used inhigh-resolution gas spectroscopy and in airpollution monitoring. These lasers operateonly at cryogenic temperatures. However,an appreciable spectral tuning is feasibleby simply changing temperature and injec-tion current, which is extremely desirablefor spectroscopy.

3.5.3 II–VI CompoundsZnCdSe/ZnSSe/ZnMgSSe/GaAs quan-tum-well lasers emitting in the blue–greenspectral region emerged around 1991 asa result of success in overcoming thedifficulty of p-type doping into these ma-terials. [10, 21] Despite extensive efforts,however, these lasers have never attaineddevice life times exceeding several hun-dred hours and hence have not been putinto practical use.

4Fundamental Characteristics

In this section, several aspects of fun-damental characteristics are described.Numerical examples are given based onAlGaAs (λ ∼ 0.8 µm) and GaInPAs (λ ∼1.3 µm) lasers.

4.1Light–Current Characteristics

Figure 4 shows an example of outputpower (P) versus DC injection current (I)characteristics. The ordinate is the lightpower emitted from one end facet, andessentially the same power is emitted fromthe other end facet.

The threshold current at room temper-ature is typically several tens of mA. Thedependence of the threshold current Ith ondevice temperature T is phenomenologi-cally expressed as

Ith ∝ exp(

T

T0

), (14)

where T0 is a constant referred to asthe characteristic temperature and takesa value of 100 to 150 and 50 to80 K for AlGaAs and GaInPAs lasers,respectively.


20 40 6000

5

10

15

Current (mA)

Ligh

t out

put P

(m

W)

Thresholdcurrent

∆I

∆P

Fig. 4 Light output versus injection currentcurve in an AlGaAs laser

The light power emitted by injectionabove threshold is expressed as

P = hωlI − Ith

eηi

12 L−1 ln(1/R)

αi + L−1 ln(1/R). (15)

Here, (I − Ith)/e is the rate of excesscarrier injection beyond the threshold. ηiis the internal quantum efficiency repre-senting the fraction of injected carriersthat recombine radiatively and gener-ate photons of energy hωl. The fac-tor 1/2L−1 ln R−1/(αi + L−1 ln R−1) repre-sents the fraction of the generated photonsthat are coupled out of the cavity throughone of the end facets. From Eq. (15), thedifferential efficiency ηD, defined as theslope of output power versus current curveabove threshold, is written as

ηD = P

I= hωl

eηi

12 L−1 ln(1/R)

αi + L−1 ln(1/R).

(16)

The differential external quantum effi-ciency ηext, defined as the ratio of thenumber of photons coupled out to thenumber of carriers injected, is given by

ηext = P/hωl

I/e= ηi

12 L−1 ln(1/R)

αi + L−1 ln(1/R).

(17)

Above threshold, the stimulated emis-sion predominates over the nonradia-tive processes and ηi ∼ 1. Typical val-ues of L = 300 µm, R = 0.3, and αi =20 cm−1 give ηext ∼ 0.3. Since hωl ∼ Eg,ηD ∼ (Eg/e)ηext. For GaAs lasers (Eg =1.424 eV), we get ηD ∼ 0.4 W/A.

The maximum rating for output poweris typically 5 to 30 mW. Figure 5 showsschematically the three phenomena thatdetermine the maximum power rating.

1. Kink: The nonlinearity in the curveof light versus current caused by lateral-mode instability is referred to as a kink.Since a kink is often associated with abeam-profile shift and the generation ofintensity noise, lasers cannot be used

Catastrophicfailure

Kink

Thermallyinducedsaturation

Current

Ligh

t out

put

Fig. 5 Schematic illustration of the threephenomena occurring at high injection levels


in the kink region for most applica-tions. The lateral-mode instability arisesfrom the optical power dependence ofthe refractive index. By incorporating arefractive-index profile in the lateral di-rection to form a waveguide, the powerlevel for kink generation can be apprecia-bly increased.

2. Catastrophic optical damage: For opti-cal power densities higher than severalMW cm−2, the end facets of semiconduc-tor lasers may be melted. This results ina sudden decrease of output power andfailure of the device. The catastrophic dam-age is considered to be brought about bythe enhanced light absorption associatedwith surface states at the end facets. Theimportance of this process seems to de-pend on materials and photon energies.In GaInPAs lasers (λ ∼ 1.3 µm), the catas-trophic optical damage practically does notoccur.3. Thermally induced saturation: Heatingof the junction under high current in-jections raises the threshold current.This may lead to saturation of outputpower with an increase in injection cur-rent, especially in lasers with lower T0values.

4.2Beam Profile and Polarization

Typical radiation intensity patterns of laserdiodes are shown in Figs. 6 and 7. Shownin Fig. 6 are the near-field patterns, thatis, the spatial distributions of opticalintensity on the end facet in the direc-tions (a) perpendicular and (b) parallel tothe junction plane. The far-field pattern(Fig. 7) is the angular distribution of radi-ant intensity measured at distances severalmm or more away from the facet, whichis mathematically a Fourier transform ofthe near-field pattern. The single-lobedradiation patterns show that the trans-verse mode of the device is single andfundamental. Transverse-mode-stabilizedlasers incorporating appropriate waveg-uide structures exhibit such beam profilesstably up to reasonable output powerlevels.

Normally, the angular width or beamdivergence (the full width at half maxi-mum) perpendicular to the junction, θ⊥(20 –60), is larger than that parallel tothe junction θ⊥ (10 –30) since the near-field spot is an ellipsoid because of thelarge difference between the stripe width

−2 0 +2

Inte

nsity

(arb

. uni

t)

−2 0 +2

Inte

nsity

(arb

. uni

t)

Distance (µm) Distance (µm)(a) (b)

Fig. 6 Near-field optical intensity patterns (a) perpendicular; (b) parallel to thejunction plane


−20 0 +20

Rad

iant

inte

nsity

(arb

. uni

t)

Angle (deg)(a)

−20 0 +20

Rad

iant

inte

nsity

(arb

. uni

t)

Angle (deg)(b)

Fig. 7 Far-field radiant intensity patterns (a) perpendicular; (b) parallel to the junction plane

(several µm) and the active-layer thickness(∼0.1 µm). θ⊥ can be reduced by adoptingthinner active layers; then, most of the op-tical energy penetrates into the claddinglayers, increasing the spot size perpendic-ular to the junction.

The output beam from gain-guidedlasers can be considerably astigmatic;the beam waist perpendicular to thejunction plane is located essentially onthe end facet, while the virtual beamwaist along the junction plane is formedin the cavity at a position several tensof µm behind the end facet. This isbrought about by the difference of thewaveguiding mechanism between the twodirections. In contrast, in index-guidedlasers, the beam waist, both parallel andperpendicular to the junction plane, islocated within several µm of the facet.The astigmatism should be taken intoconsideration when the beam is coupledinto lenses.

The laser beam is linearly polarizedalong the junction plane. This is because,in a slab waveguide, the facet reflectivity forthe TE mode is higher than that for the TMmode [22]. Since the gain needed to reachthreshold depends on the facet reflectivity(see Eq. (9)), the TE mode is selected for

oscillation in DH lasers because of thehigher reflectivity.

4.3Spectral Characteristics

Figure 8 shows the variations of lasingspectrum with output power in an index-guided AlGaAs laser. At lower powers,the laser oscillates in several longitudinalmodes. Here, the longitudinal-mode wave-length λN is determined by the relation

(1

2

λN

η

)N = L (N : positive integers),

(18)

where η is the effective refractive index ofthe waveguide and L is the cavity length.The mode spacing λ is given by

λ = λ2

2ηgL, (19)

where ηg = η − (∂η/∂λ)λ is the group in-dex. λ is typically 3 A (λ = 0.8 µm)–8 A(λ = 1.3 µm) for L = 300 µm.

As injection is increased, the laserpower tends to concentrate in a single-longitudinal mode, while the power in the


30

835

1 5 10

834

833

40 50 60

l

l

Current I (mA)

Light output P (mW)

Wav

elen

gth

l (

nm)

Fig. 8 Lasing wavelength versus injectioncurrent characteristics in an AlGaAs laser

remaining modes saturates. This is com-mon among index-guided lasers. Gain-guided lasers, in contrast, tend to exhibitmultiple–longitudinal-mode spectra evenat higher powers.

It can be seen in Fig. 8 that the lasingwavelength shifts toward longer wave-lengths as the output power is increased.This is induced by the temperature risein the active region with the increasein injection current. Shown in Fig. 9 isan example of the wavelength shift dueto a heat sink temperature change at afixed injection current. Each longitudinal-mode shifts at a rate of 0.5 A K−1 (λ ∼0.8 µm)–0.8 A K−1 (λ ∼ 1.3 µm) becauseof the temperature dependence of therefractive index. In addition, the lasingwavelength jumps toward longer wave-lengths, as temperature is raised, ata rate of 2 A K−1 (λ ∼ 0.8 µm)–5 A K−1

(λ ∼ 1.3 µm). This is caused by the tem-perature dependence of the band-gapenergy.

20

837

836

83525 30

Temperature T (°C)

Wav

elen

gth

λ (µ

m)

Fig. 9 Temperature dependence of the lasingwavelength

At ranges of temperature and bias cur-rent where the oscillation mode jumps toa neighboring mode, it is often observedthat the lasing is randomly switched be-tween the two longitudinal modes. Therandom switching is associated with alarge-intensity noise since the outputpower is different between modes by(0.1–1%). Some devices exhibit hysteresisin the lasing wavelength versus tem-perature and/or injection-current charac-teristics. These phenomena have beeninterpreted in terms of nonlinear modecoupling among longitudinal modes [17].

The spectral linewidth of a single-longitudinal mode is inversely propor-tional to the output power to a good ap-proximation. The product of the linewidthf and the output power P is typically inthe range of 1 to 100 MHz mW. This valueis 10 to 50 times larger than fST obtainedfrom the well-known Schawlow–Townesformula. An analysis shows that the en-hancement is expressed as [23]

f = fST(1 + α2), (20)

where α is the linewidth enhancementfactor defined as

α = −4π

λ

(∂η/dn)

(∂g/dn)(21)


by the derivatives of the refractive indexη and the gain g with respect to the car-rier density n. α represents the magnitudeof the amplitude-phase coupling, inher-ent in semiconductor lasers, originatingfrom the strong dependence of the re-fractive index on the carrier density. Thelinewidth is determined not only by thedirect phase fluctuation caused by spon-taneous emission but there also exists anadditional contribution from the ampli-tude fluctuation since it is coupled intothe phase fluctuation through the carrierdensity fluctuation.

α takes a value of 2 to 6 dependingon the active-region material, the injectedcarrier density, and the lasing photonenergy [24], while in most gas and solid-state lasers, α can be virtually taken tobe zero. This is because the gain spec-trum in semiconductor lasers based ona band-to-band transition is asymmetricwith respect to the gain-peak frequency(corresponding to the laser frequency) and,as a consequence, the associated refractiveindex in the vicinity of the lasing fre-quency varies appreciably with the injectedcarrier density. In contrast, in ordinarylasers where the lasing transition takesplace between two discrete levels, the gainspectrum is symmetric and the associatedrefractive-index dispersion crosses zero atthe lasing frequency. The nonzero value ofthis parameter affects a number of semi-conductor laser characteristics includinglinewidth broadening, lateral-mode insta-bility, and frequency chirping.

4.4Dynamic Characteristics

The capability of direct-current modula-tion is one of the important advantages ofsemiconductor lasers. The laser responseto a stepwise current pulse is schematically

Time

1/fr

Ith

td

Ligh

t out

put

Cur

rent

0

0

Fig. 10 Laser response to a staircase injectioncurrent. τd: Turn-on delay time. fr: Relaxationoscillation frequency

shown in Fig. 10. There is a delay time τdfor the turn-on of lasing because a finitetime is required before the injection car-riers are accumulated to form populationinversion. τd is approximated by [25]

τd = τs

√Ith

Itanh−1

√Ith

I, (22)

where τs is the carrier lifetime at threshold(∼2 ns), Ith is the threshold current, andI is the current pulse height. τd can bereduced by prebiasing the laser. Then τdis expressed by the DC bias current I0

(<Ith) as

τd = τs

√Ith

I

(tanh−1

√Ith

I

− tanh−1

√I0

I

). (23)

After the onset of laser oscillation, re-laxation oscillations are generated for a


time interval of several nanoseconds be-fore the steady state is attained. Therelaxation oscillation occurs because ofthe resonant interaction between pho-tons and carriers; the energy stored inthe cavity is transferred back and forthbetween the photon and carrier subsys-tems. Therefore, the periodic oscillationof the laser output is accompanied by amodulation of carrier density. The tem-poral variation of gain spectrum causedby the carrier density modulation leadsto a multiple–longitudinal-mode oscilla-tion. Furthermore, the frequency of eachindividual mode is modulated becauseof the carrier-density–dependent refrac-tive index, leading to a broadening ofthe time-averaged spectral linewidth. Thisphenomenon is referred to as frequencychirping. The magnitude of the chirping isgoverned by the linewidth enhancementfactor α.

When a laser is biased above threshold,the bandwidth of the dynamic responseof the laser is essentially determined bythe relaxation oscillation frequency fr. LetI(ω)eiωt be a small-amplitude sinusoidalcurrent superimposed on a DC bias I0

(>Ith). Then the photon density insidethe active region is made up of a DCcomponent S0 and an AC componentS(ψ)eiωt. The magnitude of the mod-ulated component S(ω) is analyzed tobe [26]

S(ω) = τpω2r

−ω2 + iω + ω2r

I(ω)

eVa. (24)

Here, the angular relaxation oscillationfrequency ωr is given by

ωr = 2π fr =√

S0

τp

c

ηg

∂g

∂n. (25)

represents the damping of the relax-ation oscillation and is given by

= 1

τs+ S0

c

ng

∂g

∂n. (26)

τp is the photon lifetime of the cavity(∼2 ps) expressed as

τ−1p = c

ηg

[αi + 1

Lln

(1

R

)]. (27)

Va denotes the active-region volume. Themagnitude of the modulated componentof output power P(ω) is proportional toS(ω); that is,

P(ω) = 1

2

c

ηghωl

1

Lln

(1

R

)S(ω)

Va

.

(28)

Figure 11 shows an example of relativemodulation response |P(ω)/P(0)| =ω2

r /[(ω2 − ω2r )

2 − ω22]1/2. A flat re-sponse at modulation frequencies lessthan fr is followed by a peak at fr and asharp drop at modulation frequencies ex-ceeding fr. Therefore fr is the uppermostuseful modulation frequency. fr is propor-tional to the square root of output power

Mod

ulat

ion

resp

onse

(dB

)

10

0

–10

108 109 1010

fr

Modulation frequency (Hz)

Fig. 11 Calculated small-signal modulationresponse. fr is chosen to be 1.3 GHz


P and typically takes a value of 3 to 5 GHzat P = 10 mW. In practice, however, themodulation bandwidth may be limited bythe electrical parasitics associated with aspecific device structure that leads to arolloff in the modulation response. Thecutoff frequency due to the parasitic rolloffvaries 3 to 30 GHz depending on the devicestructure.

5New Structures and Functions

5.1Quantum-well Lasers

Quantum-well (QW) lasers have an ac-tive region composed of ultrathin lay-ers (∼20 nm thick or less) formingnarrow potential wells for injected car-riers. Illustrated in Fig. 12 are thetwo representative examples of theactive-region structures in AlGaAs QWlasers. Lasers comprising multiple wells(Fig. 12a) are termed multiple-quantum-well (MQW) lasers [27, 28]. Shown inFig. 12(b) is the active region of a

GRIN (graded index)–SCH (separate con-finement heterostructure)–SQW (single-quantum-well) laser [27, 28]. In the latterstructure, the injected carriers are confinedin the SQW, while the laser light is guidedby the GRIN waveguide, thus ensuringan effective interaction between carriersand light without resort to multiple-wellstructure.

The quantization of electronic states inthe well gives rise to a variety of superiorlaser characteristics over the conventionallasers with bulk active region. A majorbenefit is the reduction of the thresholdcurrent. Figure 13 helps us understand thelow-threshold characteristics of QW lasers,where the energy distributions of thedensity of states and the injected carriersare compared between a bulk material anda QW structure. As the parabolic densityof states in the bulk material changes intothe staircase density of states in the QWstructure, the energy distribution of theinjected carriers narrows. The narrowerenergy distribution of the injected carriersleads to a narrower gain spectrum, with ahigher peak gain value for a given carrier

n-claddinglayer

p-claddinglayer

Activelayer

n-claddinglayer

p-claddinglayer

Activelayer

Al m

ole

frac

tion

Al m

ole

frac

tion

Alx Ga1-x As wells GRIN waveguide

Alx Ga1-x As well

Aly Ga1-y As barriers

Aly Ga1-y AsAlz Ga1-z As

cladding layer

Alz Ga1-z Ascladding layer

y y(a) (b)

Fig. 12 Two representative examples of active-region structures in AlGaAs quantum-well (QW)lasers (a) Multiple-quantum well (MQW) laser; (b) Graded index (GRIN)–separate confinementheterostructure (SCH)–single-quantum-well (SQW) laser


Electron energy

Ele

ctro

n po

pula

tion

Bulk

Bulk

Electron population

QW

QWDensityof states

Fig. 13 Comparison of energy distribution ofinjected carriers between bulk and quantum-wellactive regions

density. Because of the staircase densityof states, however, optical gain in QWlasers saturates for higher injection levels.Since the saturated gain value dependson the number of QW layers, SQW ispreferable for high-Q cavities while MQWstructure should be employed for lossycavities. A threshold current as low as0.55 mA has been demonstrated in ahigh-reflectivity coated (R ∼ 0.8) GRIN-SCH-SQW laser [29]. The narrowing of thegain spectrum in QW lasers as comparedto bulk lasers is accompanied by anenhancement of ∂g/∂n (nearly doubled)and a reduction of α (see Eq. (21)). Thisleads to an enhancement of modulationbandwidths (see Eq. (25)) and a reductionof linewidths (see Eqs. (20) and (21)) underboth DC and pulsed excitation [30].

The DH structure is generally formedby successive epitaxial growth of materi-als lattice-matched to the substrate. Thelattice-matching condition imposes severelimitations on the combination of DHstructure materials. However, the limita-tion is alleviated to a certain extent if thegrown layer is thin enough to be elasti-cally strained without any generation ofdislocations. Applying this principle to theactive region, we can obtain lasers whoseemission wavelengths cannot be realizedby lattice-matched systems. In QW lasers

comprising a strained GaxIn1−xAs welland AlGaAs cladding layers grown ona GaAs substrate, laser emission at 0.9to 1.1µm can be realized by varying thecomposition x and the well width. Thiswavelength region corresponds to the gapbetween AlGaAs and GaInPAs laser wave-lengths and is fit for such applicationsas the excitation of Er-doped fibers andblue–green light generation by the second-harmonic generation technique.

Furthermore, it is predicted theoret-ically that the strain is beneficial inobtaining such good laser properties aslow-threshold currents, high-relaxation os-cillation frequencies, and low linewidth-enhancement factors [31, 32]. This is re-lated to the strain-induced alleviation ofthe asymmetry in the density of states be-tween the conduction and valence bands.The strain effects on high-speed modula-tion characteristics in GaInPAs lasers arenow being studied experimentally.

5.2Distributed-feedback and DistributedBragg-reflector Lasers

In place of mirror facets in Fabry–Perotcavities, periodic gratings incorporatedwithin laser waveguides can be uti-lized as a means of optical feedback.Integrated optical feedback from theperiodic grating provides strong wave-length selectivity. Devices incorporatingthe grating in the pumped region aretermed distributed-feedback (DFB) lasers(Fig. 14a), while those incorporating thegrating in the passive region are termeddistributed Bragg-reflector (DBR) lasers(Fig. 14b) [33]. By virtue of the strong wave-length selectivity, DFB and DBR lasersoscillate in a single-longitudinal modeeven under high-speed modulation, incontrast to Fabry–Perot lasers, which


Current

Current

(a)

(b)

Active layer

Active layer

Waveguide

p-type

n-type

p-type

n-typeDBR DBR

Fig. 14 (a) Distributed-feedback (DFB);(b) distributed Bragg-reflector (DBR) lasers

exhibit multiple–longitudinal-mode oscil-lation when pulsed rapidly. This is anadvantageous feature for optical data trans-mission using fibers because the spec-tral width determines the maximum bitrate transmitted in the presence of fiber-chromatic dispersion.

The carrier-density dependence of therefractive index can be exploited to providewavelength tunability to the periodic grat-ings [34]. Illustrated in Fig. 15 is a tunable

Active region Tuning region

I1 I2 I3

Active layer Waveguide

Fig. 15 Wavelength-tunable distributedBragg-reflector (DBR) laser

laser utilizing this principle, which canbe tuned continuously over a wavelengthrange of several nanometers. Here, theBragg wavelength of the DBR structure,at which the reflection loss is minimized,is tuned by the injection current I3. Thecurrent I2 is adjusted so that one of the res-onant wavelengths of the cavity is broughtinto coincidence to the Bragg wavelength.Thus, by choosing appropriate pairs of I2

and I3, the oscillation wavelength can bevaried continuously without mode jump-ing. Tunable lasers are expected to be usedin future wavelength-division multiplexingcommunications systems.

5.3Semiconductor-laser Arrays

The major drawbacks of semiconductorlasers, the relatively low output powers,and the conspicuous beam divergencecan be alleviated by integrating multiplelaser stripes into an array as shown inFig. 16. The stripes are closely spaced sothat the radiation from neighboring stripesis coupled to form coherent modes ofthe entire array. The array modes, oftenreferred to as supermodes, are phase-locked combinations of the individualstripe modes and are characterized bythe phase relationship between the opticalfields supported by adjacent stripes. Ifan array is properly designed so that

Fig. 16 Semiconductor-laser array


one particular supermode, which is auniphase superposition of the individualstripe modes (so called 0 shift mode),is excited, a single-lobed, low-divergencebeam is attained; the divergence angle θ‖is approximated by the relation θ‖ ∼ λ/Ns,where s is the center-to-center separationof adjacent stripes and N is the number ofstripes.

A serious problem in the performance oflaser arrays is their liability to multisuper-mode oscillation; the difference among themodal gains of the individual supermodesis so small that the lateral-mode behavior ofan array is susceptible to spatial hole burn-ing and temperature distribution in thelateral direction induced at high-excitationlevels. The multisupermode oscillation isgenerally accompanied by a multilobedoutput beam, which greatly reduces theutility of arrays for most potential applica-tions. The maximum output power for thesingle-lobed beam operation is typically300 to 500 mW.

5.4Surface-emitting Lasers

Conventional semiconductor lasers utiliz-ing cleaved end facets as a means of opticalfeedback are not fit for monolithic inte-gration. For monolithic integration, it isdesirable to have laser output normal to thewafer surface. Illustrated in Fig. 17 is thebasic configuration of a surface-emittinglaser incorporating two mirrors parallel tothe surface to form a vertical cavity typically5 to 10 µm long [35]. It is crucially impor-tant to increase the reflectivity of mirrors,since the reflection loss otherwise becomesvery high because of the very short gainregion. Bragg reflectors composed of mul-tilayered semiconductors or dielectrics canbe exploited to provide reflectivities exceed-ing 95%. Carrier injection is performed

Light output

Electrode

Electrode

Active region

DBR

DBR

Fig. 17 Vertical-cavity Surface-emitting Laser

through a pair of electrodes in the upperand lower surfaces to excite a cylindricalactive region typically of 10-µm diameter,1 to 3 µm thick.

The surface-emitting laser has sev-eral advantages over the conventionaledge-emitting laser, such as the capabil-ity of being integrated on a wafer toform a two-dimensional densely packedlaser array, the feasibility of stablesingle–longitudinal-mode operation be-cause of the large mode spacing dueto the short cavity length, and the at-tainability of a narrow circular outputbeam.

6Summary

We have reviewed the history and the stateof the art of semiconductor lasers. We note


here that their successful development isthe result of contributions of scientistsand engineers from a number of fields,including semiconductor physics, quan-tum electronics, crystal growth, processingtechnologies, and system applications, asis most notably represented by the appli-cation to the fiber-optic communicationssystems.

Ongoing research and development ac-tivities indicate that the same will be truein the future. There is growing inter-est in improving laser characteristics byminiaturizing device structures. The nat-ural extension of the QW lasers will bequantum wire and box lasers, where wecan take advantage of the two- and three-dimensional quantization of electronicsystems. Researchers in quantum opticsare interested in the physics of microcavitysemiconductor lasers, which have cavitylengths comparable to light wavelength.Such a microcavity will drastically alterthe nature of spontaneous emission andcan lead to ultralow-threshold (∼1 µA orless), low-noise, and high-speed lasers. Inorder to convert the microcavity lasers aswell as the quantum wire and box lasersfrom a laboratory curiosity to a practicallight source, further progress in growthand processing technology of semicon-ductor microstructures is indispensable.Integration of these superior devices willfind new application fields such as opticalinterconnection in large-scale-integrationcircuits and two-dimensional informationprocessing.

Glossary

Active Layer: A layer, typically ∼0.1 µmthick in which stimulated emission occurs.

Cladding Layers: Layers that sandwich anactive layer made of materials that havewider band gaps than the active layer, pand n doped to facilitate carrier injectioninto the active layer.

Distributed Bragg Reflector Laser: A laserincorporating a periodic grating at bothends of the cavity as a means of opti-cal feedback.

Distributed-feedback Laser: A laser incor-porating a periodic grating in the vicinityof the active layer as a means of opti-cal feedback.

Double Heterojunction Structure: Anactive-region structure in which an ac-tive layer of one material is sandwichedbetween two cladding layers of an-other material.

Gain-guided Laser: A laser in which theoptical field distribution in the transversedirection is determined by the gain profileproduced by carrier density distribution.

Index-guided Laser: A laser in which theoptical field distribution in the transversedirection is determined by a built-inrefractive-index profile.

Linewidth-enhancement Factor: The ratiobetween the carrier-induced variations ofthe real and imaginary parts of susceptibil-ity χ (n), i.e.,

α = ∂ [Reχ(n)]/∂n

∂ [Imχ(n)]/∂n, (29)

where n is the carrier density. An equiva-lent expression is given in Eq. (21).

Phase-locked Laser Array: An array ofstripe lasers spaced closely so that a


phase-locked oscillation among the laserstripes is attained.

Quantum-well Laser: A laser that has ultra-thin active layers forming narrow potentialwells for injected carriers and giving riseto a size quantization of the electronicstates.

Stripe-geometry Laser: A laser in which thecurrent is injected only within a narrowregion beneath a stripe contact severalµm wide.

Surface-emitting Laser: A laser that emitslight normal to the wafer surface.

References

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[2] Nathan, M. I., Dumke, W. P., Burns, G.,Dill Jr, F. H., Lasher, G. (1962), Appl. Phys.Lett. 1, 62–64.

[3] Quist, T. M., Rediker, R. H., Keyes, R. J.,Krag, W. E., Lax, B., McWhorter, A. L., Zei-gler, H. J. (1962), Appl. Phys. Lett. 1, 91–92.

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[20] Horikoshi, Y. (1985), in W. T. Tsang (Ed.),Semiconductors and Semimetals, Vol. 22.Part C, Orlando, FL: Academic Press,Chap. 3.

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[25] Chinone, N., Ito, R., Nakada, O. (1974),IEEE J. Quantum Electron. QE-10, 81–84.

[26] Yariv, A. (1989), Quantum Electronics. NewYork: Wiley, Chap. 11.

[27] Tsang, W. T. (1981a), Appl. Phys. Lett. 39,786–788.

[28] Tsang, W. T. (1981b), Appl. Phys. Lett. 39,134–137.

[29] Lau, K. Y., Derry, P. L., Yariv, A. (1988),Appl. Phys. Lett. 52, 88–90.

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[33] Mroziewicz, B., Bugajski, M., Nakwaski, W.(1991), Physics of Semiconductor Lasers. Am-sterdam: North Holland, Chap. 6.


[34] Yoshikuni, Y. (1991), in Y. Yamamoto (Ed.),Coherence, Amplification, and QuantumEffects in Semiconductor Lasers. New York:Wiley, Chap. 4.

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Further Reading

Agrawal, G. P., Dutta, N. K. (1986), Long-Wave-length Semiconductor Lasers. New York: VanNostrand Reinhold.

Casey Jr, H. C., Panish, M. B. (1978), Heterostruc-ture Lasers, Parts A and B, New York: AcademicPress.

Kressel, H., Butler, J. K. (1977), SemiconductorLasers and Heterojunction LEDs. New York:Academic Press.

Mroziewicz, B., Bugajski, M., Nakwaski, W.(1991), Physics of Semiconductor Lasers.Amsterdam: North Holland.

Nakamura, S., Chichibu, S. F. (2000), NitrideSemiconductor Blue Lasers and Light EmittingDiodes. London: Taylor & Francis.

Nakamura, S., Fasol, G., Pearton, S. J. (2000),The Blue Laser Diode: The Complete Story. NewYork: Springer.

Peterman, K. (1988), Laser Diode Modulationand Noise. Dordrecht: Kluwer AcademicPublishers.

Thompson, G. H. B. (1980), Physics of Semicon-ductor Laser Devices. Chichester: Wiley.

Yamamoto, Y. (Ed.) (1991), Coherence, Amplifi-cation, and Quantum Effects in SemiconductorLasers. New York: Wiley.

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