A. La Rosa Lecture Notes APPLIED OPTICSLecture-15 PROPAGATION of LIGHT______________________________________________________________________ Here we explore different approaches used to describe the propagation of light.I. Huygens-Fresnel principle

Wavefront, Huygens principle, Huygens-Fresnel principle

II. Rayleigh scattering (Scattering from small particles)II.1. Self-sustained propagation of electromagnetic fieldsII.2. Elastic scattering from particles of size smaller than the wavelength

II.2.1 Propagation of light in tenuous mediaCase: Lateral scatteringCase: Forward propagation

II.2.2 Propagation of light in dense media

III.Scattering from large particlesIV. Transmission and the Index of RefractionV. Diffraction

Contrasting geometrical optics and diffraction. Difference between interference and diffraction. Far- and near-field diffractionV. 1 Fraunhofer DiffractionV.2 Fresnel Diffraction

I. Huygens-Fresnel principle I.1 Wavefront: It is the leading surface of a wave

disturbance

Wavefront ∑ will be distorted after passing the piece of glass.

How to determine the new wavefront ∑ '?

I.2 Huygens Principle Ref: Hecht, “Optics”, Section 4.4.2

I.3 Huygens-Fresnel Principle Ref: Hecht, “Optics”, Section 10.1

II. RAYLEIGH SCATTERINGReference:Eugene Hecht, "Optics," 4th Edition, Addison Wesley: Sections 4.1, 4.2

II.1 Self-sustained propagation of electromagnetic fieldsIn the empty space, light propagates forever.

Fig.1 The time-dependent electric and magnetic fields self-sustain each other, resulting in an electromagnetic wave that propagates forever.

∂2 Ez

∂ y2 −1c2

∂2 E z

∂ t2 =0 where

c≡ 1√ μo εo

E z ( y , t )=f ( y−ct ) propagating wave

II.2 Elastic scattering from particles of size smaller than the wavelength

\When light propagate in the Earth atmosphere, it interacts with molecules of nitrogen, oxygen,

These molecules have resonance in the uv-region; therefore the incident visible light cannot cause resonance absorption. The medium is therefore transparent.

Still, the electron clouds of the molecules can be driven by the incident radiation. A photon is absorbed and re-emitted with the same frequency; i. e. light is elastically scattered, randomly everywhere. This happens particularly in a tenuous region of the atmosphere.

Fig.2 The presence of small particles (size smaller than lambda) gives rise to elastic scattering

Scattering from particles whose size is smaller than the (1)wavelength (let’s say size < /15) is referred as Rayleigh Scattering

Analytical description. One classical approach considers each molecule as a little oscillator of resonance frequency 0 (where 0 is one of the resonance absorption frequencies of the molecules.)

Fig. 3 Incident light of frequency interacts with the molecules, and light of the same frequency is (re-emitted) scattered in all directions.

Typical of resonant phenomena:The closer is to 0 the higher the amplitude A(ω ) , (2)

the stronger the scattering

This implies: Blue light scatters while red light scatters (3)

strongly comparatively weaker

Equivalently, the forward propagating light is richer in redthe scattered out light will be richer in blue

Thus, Rayleigh scattering contributes to the blue color of the sky.

Fig. 4 The forward beam and the light scattered laterally present different coloration

Quantitatively, how much stronger is blue light scattered compared to red light?To answer this question, let’s use the argument that an accelerated charge produces an electromagnetic field.From the figure above:

x=[ Ae j ϕ ] e jω t, v= jω [ Ae j ϕ ] e jωt

, a=−ω2[ Ae j ϕ ] e jωt

Electric field ~ acceleration E ~

2

Light intensity I ~ E2 ~

4 (4)

EARTH ATMOSPHERE

Scattered light stronger in blue

‘White’ light

Forward beam stronger in red

~ (1/)4

Ired / Iblue = ( blue / red)4 = (450 / )4 = ( 0.7 )4 = 0.22

II.2.1 Propagation of light in tenuous mediaLateral scattering and forward propagation

We want to understand why the denser the material the less the lateral scattering

II.2.1.A Case: Lateral scatteringMolecules randomly distributed is spaceSeparation d between molecules greater than the wavelength

Fig. 5 Molecules in a tenuous media (average distance between molecules is greater than the participating wavelengths) behave as independent oscillators, hence producing a net intensity at P.

Let’s analyze the interference between the waves re-emitted by two arbitrary molecules. For simplicity, let use two plane waves as a model (see also Fig.6).

E1(at P)=E10COS (k r1−ω t+φ1 ) (5)

E2 (at P)=E20COS (k r2−ω t+φ2 )

Let’s assume also that both E10 and E20 point in the direction out the plane of the figure.

Fig. 6 Light emitted by the two sources modeled as two plane waves

The intensity at P will be given by,

I (at P)=ε0 c ¿¿ (6)

where the symbol ⟨ ⟩ stands for a time average evaluation

⟨ f ⟩=1τ∫0

τ

f ( t )dt (7)

The integration time varies in different situations, depending on the detector integration time or the dimensions of the device used to detect the interference.

I (at P )=ε0 c¿¿

=ε0 c ¿¿

=I 1( at P) + I2 (at P ) + 2 ε0 c¿¿¿

(8) The interference term

2 E1¿ E2=¿2 E10¿ E20 COS (k r1−ω t+φ1)COS(k r2−ω t+φ2 ) ¿

Using Cos(A+B) + Cos(A-B) = 2Cos(A) Cos(B)

2 E1¿ E2= E10 ¿ E20 COS [ k (r1+r 2) − 2ω t + φ1+φ2 ] +

+¿ E10 ¿E20 COS [ k ( r1−r2) +φ1−φ2 ] ¿

The first terms changes so rapidly at optical frequencies , which should lead to an average equal to zero.

¿¿¿ (9)

Because the two molecules under analysis,not only move randomly, but also their scattering emission (as well as excitation) occurs randomly.Hence, their phase difference (φ1−φ2 ) changes randomly as a function of time, which lead to a zero interference term.

¿¿¿ Expression (8) becomes,

I (at P )=I 1 (at P ) + I 2( at P ) ¿ 0 (10)That is, randomly and widely spread molecules behave as independent scattering centers of light.

The generalization is straightforward for N independent scattering centers,

I (at P )=∑n=1

N

I n (at P ) ¿ 0 (11)

Phasor analysisIn terms of phasors addition, finding the total electric field at P, at a given instantaneous time t, consists of adding N phasors whose phases have no correlation among them. In fact, the phase of the individual phasors are expected to differ greatly, because scatterers are far apart in term of .

E( at P )=∑n=1

N

En0 ej (krn−ωt+φn)=SUM

(12)I (at P )=ε0 c ⟨|SUM|2 ⟩ ≠0 lateral scattering in tenuous media

Despite the random phase value of the different individual phasors, the total phasor SUM cannot be zero all the time. (Otherwise the average intensity would be zero, which would contradict expression (11).)

That is, the phasor SUM changes with time randomly (basically due to the random

value of (φn( t ) ) but in such a way that its time-averaged amplitude-square is equal to the value given in (11)

Fig.7 Lateral scattering described in terms of phasors. The picture shows schematically the different phasors at a given instantaneous time. The phase of the individual phasors differ greatly (because scatterers are far apart in term of ).

In short, in lateral scattering,The different phasors adding at the point P have uncorrelated phases. Their phases differ greatly among them (because the scatterers are farapart between them, compared to .) (13)Hence, the scatterers act independently, and the intensities from these individual scatterers add up. In conclusion, unimpeded by interference, light stream out of the forward direction

II.2.1.B Case: Forward propagationFor the forward direction, the calculation of the light intensity reaching a point

like Q should in principle be similar to the case of lateral scattering (i.e. adding phasors from randomly located scatterers).

However, when the incident light and the scattered light occupy the same space, there exists a peculiar behavior:

the re-emitted light from the scatterers (despite their random location) tend to cooperate, marching in phase in the forward (14)direction.

This is illustrated in the figure below.

First, consider an incident lane wave, as sketched in the figure below. The profile at the bottom of the diagram shows the amplitude of the electric filed.

The following diagram shows the sequential excitation of two scatterers A and B. The description is facilitated if the lateral separation between A and B is l/2, but this specific separation is not necessary. (What matter is the assumption that the phase separation between the excited filed and the scattered field is constant.)

Fig. 8 The incident light is a plane wave ( Despite their random location of the molecules, the light scattered in the forward direction march in phase on a planar wavefront.

II.2.2 Propagation of light in dense media

In contrast to a tenuous media, as depicted in Fig. 5, consider a denser medium where there are ~ 1 million of molecules in a cube of side . The average separation between molecules is d ~ (500 nm)3 /106 = 5 nm. If the incident light is in the visible region (~500 nm) then we are in the regime d << At a given time, the scattering from all the molecules in that hypothetical cube may have some correlation. In the language of phasors, the phase of the different phasors contributing to the field at point P may not be random; they will display some interference since phasors from contiguous molecules will have similar phases, as illustrated at the bottom in figure 9.

Fig. 9 Scattering by molecules in a dense medium. Since the average separation d between molecules is much smaller than , the phase of phasors originated from contiguous molecules do not differ greatly. On average the contribution from the molecules in a cube of side would be close to zero. .

E( at P )=∑n=1

N

En0 ej (krn−ωt+φn)=SUM

Summation carried over all the molecules

inside the cube of side .

Further, if we considered that the incident beam has a plane wave wavefront profile, for a given phasor associated to the A source, given the compact arrangement of the molecules there will be another source B located /2 apart so that these two phasors will interfere destructively. Given the larger than lateral extension of the bean, one can find pairs of molecules that interfere destructively. Hence, we expect then that the net interference at P will be very small, if not zero.

Fig. 10 Due to the dens medium, for a given scaterer A, there will be another scatter B such that their electric field at P will differ p radians in phase, hence given a net zero field.

The argument above would be valid also for analyzing wave scatered in the backward direction.

It is plausible to conclude then that, different than the case of lateral scattering in tenuous media,

Almost no light is scattered laterally or backwards (15)when light propagates in a dense homogeneous medium

“The more dense, uniform, and ordered the medium, is(the more nearly homogeneous), the more complete will be the lateral destructive interference and the smaller amount of non forward scattering. Thus, most of the energy will go into the forward direction, and the beam will advance essentially undiminished.” E. Hecht. Optics 4th edition, p. 91

III. Scattering from large particlesWhen the size of the particle is smaller than the wavelength

the scattered light from the different compacted atoms interfere almost constructively, thus the scattering is strong.

As the particle increases in size approaching the wavelength ()The molecules located on the extremes scatter radiation that is not any more in phase; hence the overall coherence decreases; this happens first (as the particle size increases) at the short wavelengths (blue).Accordingly, for incident white light, the particle scatters more coherently in the red than in the blue

Mie scattering Gustav Mie in 1908 published a theoretical analysis of scattering from spherical particles of any size.

Why do clouds look white (sometimes)?Although water is essentially transparent, water vapor appears white. The reason:

If the grain size (drop of water) is small but larger than the wavelength , light will enter each transparent particle and (after reflection and refraction) emerge. This will happen regardless of the wavelength, so the emerging light will appear “white”.

This is the mechanism why few things appear white, like, for example, sugar, salt, paper, clouds, snow, paint.

Busted concept: Even though we tend to consider paper, talcum powder, and sugar as

“opaque” white substances, that is not correct. (Spread some sugar grains on a sheet of paper, and you will see through them without difficulty.

White paint results from suspending colorless transparent particles (zinc oxide, titanium, or lead) in an equally transparent medium (linseed oil, acrylics)

Indeed, if the index of refraction of the particles np is different than the index of refraction of the medium nm ( np ≠ nm ), then there will be reflections (at all wavelengths) and the paint will appear white and opaque. To color paint, one needs only to dye the particles so they absorb all wavelength radiation, except the desired color.

But if np ~ nm, then there will be no reflection from the grain boundaries. The transparent medium will remain transparent, whereby decreasing the whitness of the object.Ref: E. Hecht, 4th Ed., p.132.

IV. Transmission and the Index of RefractionPhotons exists only at speed c, still the transmission of light through a

homogeneous medium appears to occur at a lower speed v = c/n, where n is the identified as the index of refraction.

This apparent contradiction was explicitly addressed in Lecture-3. We found that there is no need to postulate that light slows down when traveling through a medium. Rather, the introduction of the index of refraction (i. e. the concept that light slows down) simply offers an alternative convenient and compact way to express the forward net electric field that results from adding the field of the light source Es and the field E’ generated from mediu’s molecules excited by the light source. In fact the net electric field at a point located in the forward direction was calculated without invoking a slowdown of the speed of light.

Fig. 11 Primary wave Es propagate though the medium and drive the electric dipole of the molecules in the medium. The accelerated molecules radiate a secondary wave Es. Primary and secondary waves interfere at P.

We found that the electric field E’ (produced by the oscillating charges in the material) is given by,

E'(z, t )= - Ndq2 ε o c

iω xo eiω ( t -z/c)

=¿Nd q2 εo

ωc

xo e-i π /2 eiω (t -z/c )¿

Here xo is the amplitude of oscillations of the individual charges q, which is

obtained through the solution of the equation, m( d2

dtx +γ d

dtx+ω

o2 x )=qEo eiω t

That gives xo= q

m1

(ωo2 - ω2 ) +i γ ω

Eo

.

E'( z, t ) = Ndq2

2m εo

ωc

1(ω

o2 - ω2 ) +i γ ωe -i π /2

Eo eiω (t -z/c )

(16)

We notice that the lagging of the field E’ with respect to the field of the sourceEs(z,t )¿Eo ei(ωt−kz )

has two origins:a) Within the elementary treatment that models the atom as an oscillator, the factor

1(ω

o2 - ω2 ) + i γ ω is the typical frequency dependent lagging caused by the fact

that the oscillator cannot respond instantaneously to the driving force.

b) The factor e-i π /2

accounts for the accumulated contribution from the different field produced by the individual oscillator in the medium

The electric field given in (16) can also be obtained if we postulate a slowing down of the light speed by a factor of n when travelling in a medium, provided that n is given by,

n =¿1+N q2

2m εo

1(ω

o2 - ω2 ) +i γ ω¿

(17)

In passing, notice the phase of the complex index of refraction is identical to the phase difference between the dipole moment p=−ex of in the individual atoms in the medium and the applied external field Es(as discussed in Lecture-5).

p= e2

m1

(ωo2 - ω2

+i γ ω )Es

(18)

It is very illustrative to compare i) the lagging phase of E’ with respect to Es

(expression (16), and ii) the phase displayed by the complex index of refraction or the lagging phase of the electric dipole relative also to Es.

Fig. 12 The three phasors propagate counter-clockwise at angular frequency .

Notice that at resonance, when = o, E’ and Es are 180 degrees out of phase;

that is, they are just opposite to each other. The net electric field is just weaker and, equivalently, there would be no need to invoke a time delay due to the slowdown of the wave while travelling to the medium. This is compatible with the fact that at = o the index of refraction is 1.

Fig. 13 Top: Dipole amplitude. Bottom: Real part of the index of refraction.

V. DIFFRACTIONContrasting diffraction and geometrical optics

Diffraction is any deviation from geometrical optics that results from the obstruction of a wavefront of light.

Fig. 14 Edges of an optical image are blurred due to diffraction. It is a consequence of the wave character of light. (More often, however, the sharpness of an optical image is more seriously degraded by the optical lens’ aberrations. Diffraction limited is actually acceptable optics in a variety of applications.)

Contrasting diffraction and interference phenomenaActually, they describe the same on fundamental principle; both phenomena result from the addition of waves. Still is customary to find the following classification:Interference: The addition of waves from a discrete number sourcesDiffraction: The addition of waves from a continuum sources of light

Fig. 15 Interference and diffraction are both based on the addition of waves.

Classification of diffraction according to the far-field or near-field excitation/detection distances

Fraunhofer, or far-field, diffraction:Both the source of light and observation screen arefar enough from the diffraction aperture,so that the wavefronts arriving at the aperture and observation screen may be considered planes.

Fig. 16 Fraunhofer diffraction. The source of light “P” is so far away such that the wavefronts arriving to the aperture can be considered to be plane waves.

Fig. 17 In Franhoufer diffraction the viewing screen is far away from the aperture, such that as the viewing screen is moved relative to the aperture, the size of the diffraction pattern scales uniformly, but the shape of the diffraction pattern does not change.

Fresnel, or near-field, diffractionThe source of light and/or the observation screen areare so close that the curvature of the wavefront must be taken into account.

In this approximation, both the shape and size of the diffraction pattern depend on the distance between the aperture and the screen.

Criterion for evaluating the far- and near- field conditionIn this section we establish a criterion to evaluate whether or not we have

Fraunhofer or Fresnel diffractionWhat is the required separation distance between the source and the aperture, as well as between the aperture and the detector, in order to have Fraunhofer or Fresnel diffraction?

For simplicity, let’s consider the case in which the source P and the detector Q are located on the axis that passes through the center of the aperture.

Notice in the figure below that the field at Q will depend on whether we use optical paths like PAQ (A on the plane wavefront) or PBQ (B on the spherical wavefront). The key parameter to consider in our analysis is , for, if “p” were sufficiently large such that becomes much smaller than the

wavelength , then the optical path lengths PAQ and PBQ cannot produce too much difference in phase.Hence no major difference in the calculation of the field at Q will result when using either optical path. This condition would lead to Fraunhufer diffraction.

Otherwise, if P were too close to the aperture such that makes (or ), then whether we use SAP or SBP will make a difference (the phases of the corresponding phasors eiOPL would differ significantly).This condition would correspond to Fesnel diffraction.

More straightforward will the be to find the condition under which the Fraunhofer diffraction occurs. With that in mind, let’s find out a relationship involving and the other parameters involved in the diffraction.

Δ = r '−p = r '− √(r ' )2−h2

= r '− r ' √1−(h/r ' )2

Consider the case where (h/r ' )<< 1

Δ ≈ r '− r '(1- 1

2h2

r '2)= 1

2h2

r '

Hence, in addition to (h/r ' )<<1 , Fraunhofer diffraction occurs when h2

r '<< λ

, or,

h2

λ<<r '

. Since h2 scales with the area of the aperture, the last result can be put in more general terms:

Fraunhofer diffraction: r ' >> aperture area

λ

Fresnel diffraction: r '< aperture area

λAlternative derivation: