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Nature of Light

Classical Wave

(1) propagation of oscillations in amedia: need of a medium, non-localization

(2) transfer of energy and momentum

without transporting medium(3) continuous energy

Frequency v, wavelength l

Wave nature: interference and diffraction

Classical Particle

(1) isolated and localized entity

(2) moves without a medium

(3) continuous energy

Mass (m), energy (E) and momentum(p), position

First, lets have a look at:

Classical wave and classical particle.

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Nature of Light

Wave property

(1) light can have interference and

diffraction

(2) it is a part of the electromagneticwave spectra, lfrom 400 to 800 nm

(this is the visible spectra for a humaneye> our definition oflight. But all e.m.

waves obey same physics, no matter l.)

(3) it is described by Maxwell waveequations, requires no medium topropagate

Particle property

(1) light shows particle nature

when interacting with matter, such asabsorption and emission

(2) its energy is quantized, inquanta ofE = hv, called photons

(3) the photon is a massless (the

rest mass = 0, recall relativistic massdefinition) and chargeless particle

(4) photon moves at the speed oflight with momentum: p = h/l

Plancks constant: h = 6.626x10-34 J.s

Speed of light in vacuum: c = 2.998x108 m/s

Light is both a wave and a particle in a non-classical sense.

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Geometrical Optics

Wave nature of light does not play a significant role, no diffraction is considered.

limit {Wave Optics} ----------> {Geometrical Optics}

Both dimension of the optical system and image formed by the system >> l

0

Ray: line drawn in space corresponding to the direction of light propagation

Rectilinear propagation of light:

In an uniform media, light propagates in a straight line. Its speed in themedium is given:

n: index of refraction

c: speed of light in vacuum

e.g. in air and vacuum, n = 1; for most glasses, n 1.5

In geometrical optics, rays are used to trace the light propagation

n

cv

Note: Photons energy & frequency are independent on medium hardness, that is: no matter n, one has E=hn.

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Laws of Reflection and Refraction

(a) When light travels from one medium to another, the speed changes, asdoes the wavelength (but not the frequency!). The index of refraction canalso be stated in terms of wavelength:

n = l/ln l wavelength in vacuum, lnwavelength in the medium

(b) incident, reflection and refraction rays are all in the same plane formed bythe incident ray and surface normal

(c) Angle of incidence equals to the angle of reflection: q1 = qr

(d) Snells Law: n1 sinq1 = n2 sinq2

Angle of refraction

sinq2 = n1 / n2 sinq1

qr

When n1 < n2, q1 > q2 the light bends towards the normal

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Total Internal Reflection

When n1 > n2, q1 < q2 the light bends away from the

normal

Critical angle qc:sinqc = n2 / n1

Light is refracted at 90, in other words, refracted along

the interface

When the q1> qc , there will be no refraction and all thelight are reflected, Total internal reflection

Principle of Reversibility

If the reflected or refracted ray is reversed in direction, it will retrace itsoriginal path.

Note: In the case of total internal reflection, there is still a bit of energy protruding into the n2 medium, right at the

interface. That light is formed by evanescent waves which are decaying exponentially within few l. If an objectis placed within this interval, the evanescent waves are strongly perturbed: near-field microscopy as application.

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Fermats Principle

Fermats Principle: a light ray going from point Sto point P must traverse an optical

path length that is stationary with respect to variations of that path, i.e. it must select theshortest OPL

Optical Path Length (OPL):

n(s): index of refraction as a function of path

In math language, Fermats Principle requires that:

m

i

p

siidssnsnOPL

1

)(

0)( p

s dssnOPL

S

P

Note: example of integration instead of summation: continuously varying n, like nof air above a hotroad or sand (mirages in desserts )

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Fermats Principle

Meaning of OPL: the speed of light in ith medium vi = c / ni

time staying in the ith medium: ti = Si/ vi = ni Si/ c

Therefore:

The actual path between two points taken by a beam of light is the one that istraversed in the least time.

npropagatiooftime1

11

ttsncc

OPL m

i

i

m

i

ii

Example 1: In an uniform medium, n = const. It is obvious that light rays willtravel in a straight line, since the shortest path between any two points is thestraight line connecting these two points.

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Reflection Case

Reflection: The minimum path from source Sto theobservers eye at P must satisfy:

i = r

SAP = SAP; SBP = SBP and SCP = SCP

Second, it is easy to prove that SBP is a straight line: SBO=OBS=PBC.Therefore, since OBC=180, we have SBP= 180

Therefore, SBP is the shortest path.

We can also prove that Snells Law using Fermats Principle (Hecht, P107)

The laws of reflection and refraction will be used to build optical systems.

S is the mirror image of source S: SO=SO. Therefore,

the OPL from Sto P equals the corresponding OPL fromSto P.

S P

O A B C

Reflec ting

surface

S

shield

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Optical System

Optical system: an imaging system composed of lenses and mirrors

Ideal (stigmatic) system:point source Sis imaged as a geometrical point P(perfect image). S, P are called conjugate points, since a point source placed at Pwould be equally well imaged at S, according to the Principle of Reversibility.

However, in reality, the image quality of a real optical system is always diffraction-limited, i.e., the image is a blurred spot instead of a perfect geometrical point. Thisdiffraction effect limits the spatial resolution, which we will discuss in more detailslater. In geometrical optics, we neglect any diffraction effect.

S P

Object space Image space

Optical system

Image space

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Refraction at spherical surfaces

A point source Sis at distance Sofrom the spherical surface. Lets use FermatsPrinciple to find the image distance:

OPL = n1SA + n2 AP (1)

V: vertex

So: object distance SV

Si: image distance VP

SC: optical axis, AC: radius R

(2)cos22/122

0 RSRRSRSAlSAC oo

(3)cos2 2/122 RSRRSRAPlACP iii

0

d

OPLdSubstitute Eqs: (2) and (3) into eq. (1), use Fermatsprinciple:

A

S PCV

h

qi

qr

qt

Fig 5.6

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Refraction of spherical surfaces

(a) Eq.(5) describes the parameters of rays after refraction at the spherical surface

(b) This relation is strictly held, i.e. no approximations were made

(c) The position P intercepted by different rays (different -> lo,i) is different,indicating that a point source Swill not imaged as point image. That is, a

spherical surface is not an ideal imaging optical system.

We have, (4)

02

sin

2

sin21

i

i

o

o

l

RSRn

l

RSRn

That is: (5)

o

o

i

i

iolSn

lSn

Rln

ln 1221 1

fixed So, R:

Si drops with increasing

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Paraxial ray condition and image formula

Point source S point images P, this one to one correspondence is given by Eq.(7)

Paraxial rays: rays with very small value of , satisfying condition (6).

Gaussian optics: the optics in the paraxial ray region, also called first-order orparaxial optics. All the following discussion are within the paraxial ray region.

However, under the paraxial ray conditions: cos1and sin (6)

The spherical surface can be approximated as an ideal optical system. Underthese conditions, lo = So, li = Si, Eq. (5) can be simplified as:

Rnn

Sn

Sn

io

1221 (7)

fo: first focal (object focal) length: the objectdistance for which Si=

Fo: the corresponding point on the optical axis,first (object) focus

Rnn

nfo

12

1

(8)

Fig. 5.8

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Paraxial ray condition and image formula

fi: second focal (image focal) length, is the imagedistance where So=

Fi: second (image) focus

(9)Rnn

nfi

12

2

Real image: image formed by actual converging rays

Virtual image: image formed by extension of diverging rays

Virtual object: an object is virtual when the rays converge toward it (but notactually crossing with one another)

The nature of images and objects are also reflected by the sign ofSo and Si, seeTable 5.1 in Hecht

SIGN CONVENTION !!

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Lenses

Fo F i

Simple lens: only one element with two refraction surfaces

Compound lens: contain more than one elements

Thin lens: lens thickness is effectively negligible

Thick lens: lens thickness is not negligible

Centered system: all surfaces are rotationally symmetric about the optical axis

Positive (convex, converging) lens: thicker in the

center, tend to focus the rays

Negative (concave, diverging) lens: thinner in thecenter, tend to diverge the rays

F i Fo

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Thin Lens Equations

Simple spherical lens: contains two refraction spherical surfaces

successively using eq. (7) for these two surfaces, after their summation we have

Where all the parameter are defined in Fig. 5.14 in Hecht Book.

For thin lens,d 0, assuming the lens is in the air (nm = 1),

we have the Thin-Lens Equation (Lens maker's Formula):

112121

11

ii

l

ml

i

m

o

m

SdS

dn

RRnnS

n

S

n

21

111

11

RRn

SSl

io

(14)

(15)

For thin lens, the points V1 and V2tend to coalesce, therefore So and Si can be measured

from the either the vertices or the lens center.

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Thin Lens Equations

As in spherical refraction surface, we can define two focal lengths:

(16)

Example: planar-convex lens, nl = 1.5,R1 = ,R2 = -50 mm (LEARN THE TABLE in Fig. 5.12)

limandlim iS

ioS

o SfSfoi

21

1111RR

nf

lTwo focal length are equal, (17)

fSS io

111Gaussian Lens Formula, (18)

100

1

50

1115.1

1

ff = 100 mm

If an object is placed 600 mm from lens (So= 600 mm), the image distance (S

i)

mm

fS

fSS

o

o 120

100600

1006001

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Ray Diagrams

Fo Fi F i Fo

Positive Lens Negative Lens

Fo Fi

Fo Fi F i Fo

F i Fo

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Important Planes in Thin Lens

The Principal Plane: the plane going through the lens center (optical center)and perpendicular to the optical axis: the optical ray bending occurs at this level.

The Focal Planes: the planes going through the twofocal points (F

o

, Fi

) and perpendicular to the optical

axis; in the paraxial ray region, bundles of parallel rayswill focus to a point on the corresponding focal plane.

By knowing the principle and focal planes, plus opticalaxis, one can determine the outgoing directions of anyincoming rays

Fig. 5.18

FoF i

Fi

Fo

(a) Draw a ray parallel to the incoming raythrough the optical center (b) this ray willintersect the 2nd focal plane (c) the outgoingray after lens must pass this point, as shownin the right figure

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Finite Imagery

From now on, we consider the image of an object with finite size (composed ofmany point sources) through an optical system.

Spherical surface case (Fig. 5.19): The object is asegment of a sphere, so, centered at C. Each point on so

has a conjugate point on si, which is also a segment of asphere centered at C. Under the paraxial ray condition,these surfaces (so , si) can be considered planar ( OA).

Fig. 5.19

Thus, a small planar object normal to the optical axis will be imaged into asmall planar region also normal to that optical axis. This is true to any opticalsystem, so long as the paraxial condition is satisfied.

Three most important aspects of an image: location, size and orientation.All ofthem can be determined by a ray diagram, as shown in Fig. 5.22.

Fi

Fo

conver ent

Fo

Fi

diver ent

Ray diagrams!

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Transverse Magnification

From now on, we will follow the sign conventions outlined in Table 5.2.

Since the triangles AOFi P1P2Fi (Fig.5.22), we have

Table. 5.2

Transverse magnification MT: The ratio of the transverse dimensions of the final

image to the corresponding dimension of the object:

fS

f

Y

Y

ii

o

(19)

Similarly triangles S2S1O P2P1O, we have

i

o

i

o

S

S

Y

Y (20)

o

i

o

iT

S

S

Y

YM (24)MT>0, erect (up-right) image; MT

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Newtonian Lens Equation

Using ray diagram, we can also derive Gaussian Lens Equation

From Eqs. (19) & (20), we have:

Table. 5.3Newtonian Lens Equation: Since S2S1F0 BOF0 we have

(22)

(23) Newtonian Equation

xoandximust have the same sign, which indicates that the object and image mustbe on the opposite sides of their respective focal points.

fSSfS

f

S

S

ioii

o 111

fS

f

Y

Y

oo

i

Considering: xo = Sof andxi = Sif, combine Eqs.(19) & (22), we have:

2fxx io

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Image of 3-D object

The Newtonian expression for thetransverse magnification is:

Image of 3-D object: in addition to theMT, we need to consider also the

longitudinal magnification, related to the axial direction, defined as

(27)

Differentiating Eq. (23) leads to: (28)

o

iT

x

f

f

xM (26)

o

iL

dx

dxM

2

2

2

T

o

L Mx

fM

For a thin lens in a single medium, evidently, ML < 0, which implies that a positivedxo corresponding to a negative dxi and vice versa.

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Thin Lens Combinations

Real optical systems usually consists of several lenses, or form a compound lens. Inprinciple, the final image of the system can be determined by using lens equation,MTandML to each lens successively.

Two thin lens system (Fig. 5.28 & 5.30):

ForL1: (29) or (30)111

111

oi SfS

11

11

1

fS

fSS

o

oi

ForL2: or222

111

oi SfS

22

22

2

fS

fSS

o

oi

12 io SdS Substitute (31), we have: (32)

21

21

2

fSd

fSdS

i

ii

Substitute Eq. (30), we have: (33) 11112

1111222

fSfSfd

fSfSfdfS

oo

ooi

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Thin Lens Combinations

Please note that So1 is referenced to O1, while Si2 is referenced to O2

(34)

For the optical system, we usually define:

front focal length (f.f.l.) (35)

Back focal length (b.f.l.) (36)

The total transverse magnification of the system is the product of the individualmagnifications:

111121

21 fSfSd

Sf

MMMoo

i

TTT

21

122

1

lim...ffd

fdfSlfb

iSo

21

211

2

lim...ffd

fdfSlff o

Si

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Thin Lens in Contact

In general,f.f.l. b.f.l., however, ifd 0, that is when the lenses are brought incontact, we have:

(37)

If we have N thin lenses in contact:

2121

21111

or......fffff

fflfblff

Nffff

1111

21

(39)

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Stops

In general, an optical system contains not only lenses, but also stops. They are usedto control :

(1) The amount of light reaching the image

(2) the quality of the image, such as sharpness, distortion

(3) Image size, or the angular width of the object that can be imaged by thesystem (FOV, field of view)

Aperture stop (A.S.): an element determines the amount of light reaching the image

Field stop (F.S.): an element limiting FOV or the image size, see Fig. 5.33

Compared with camera, A.S. is the aperture in the camera, while F.S. is the film edge.

Fig. 5.33, 5.34, 5.35Chief ray and Marginal ray

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Stops

Pupil: used to determining whether or not a given ray will transverse the entireoptical system

Entrance pupil: image of A.S. as seen from an axial point in the object space

Exit pupil: image of A.S. as seen from an axial point in the image space

As shown in the Figs. 5.34 &5.35, the cone of light entering the optical system isdetermined by the entrance pupil, whereas the cone of light leaving it is controlled bythe exit pupil. No rays from the source point proceeding outside either cone will makeit to the image plane.

To determine which stop is of A.S. : (1) using ray diagram to find the images all thestops in the optical system in object space (2) the image subtends the smallest coneto a point on the optical axis is entrance pupil (3) the corresponding stop is A.S. (4)its image in the image space is exit pupil. Note that which stop is A.S. also dependson the position of the point on the optical axis.

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Relative Aperture and f-Number

The total light energy gathered by the lens (or optical system) is proportional to thearea of the entrance pupil:

pupil)entranceofdiameter:(collectedenergylight 2 DD

The image area is proportional to the square of its lateral dimension, i.e.:

o

Tx

fMf areaimage 2

Therefore, the light intensity (energy/area) or flux density at the image plane variesas (D/f)2. We introduce the f-number (f/#, or focal ratio):

D

ff # (40)

Therefore, a smaller f-number lens permits more light to reach the image plane. Its afaster lens, since the exposure time is proportional to the square of f-number.

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Prisms

The prism is one of the most important optical elements, it can serve as beam splitter,polarizer, dispersion tool and interferometer.

Dispersing Prism (used in optical spectrum analyzer)

Dispersion refers to the frequency dependence of index of refraction, i.e. n(w) orn(l) . This implies that the speed of light in a dispersive medium is also a function offrequency (or wavelength), v(w) = v(l) = c/n(w) = c/n(l).

A dispersing prism is an optical element using the dispersive nature of the medium toseparate spatially rays of light with different frequencies.

A ray entering a dispersing prism, as shown in Fig. 5.56, will be deflected from itsoriginal direction by an angle , known as angular deviation. By using Snells Law andgeometry, we have:

qqq cossinsinsinsin 121

1

221

1 iii n

Fig. 5.56

(53)

Fig. 5.57

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Dispersing Prisms

Where is the prism apex angle, n is its index of refraction.

Since n is a function ofl(or w), is also a function ofl(or w),

This dependence of angular deviation on incoming wavelength enables prism to

separate the light of different frequency spatially.Minimum deviation angle

mand its applications

:

For given l, is only a function of incident angle qi1, as shown in Fig. 5.57. Thesmallest value of is know as the minimum deviation angle .

By setting : we can prove that form

,

qi1= qt2, also qt1= qi2, which indicates for an isosceles prism, the deviation isminimum when rays transverse the prism symmetrically, that is, parallel to its base.Under such conditions, we have:

2sin2sin

mn (54)This equation provides one of the most

accurate way to determine n for a transparentmedium.

01

1

2

1

21

1

i

t

i

ti

i d

d

d

d

d

d

q

q

q

qq

q

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Reflecting Prisms

Under the minimum deviation conditions, the outgoingray direction is stationary with respect to the prismposition, i.e., when the prism is translated along thedirection perpendicular to its base, the outgoing raypath will not be affected.

Reflecting prisms:

Prisms used to change either the direction of light propagation or orientation of theimage, or both. Here, the prisms are used for reflection, and no refraction is desiredsince it introduces dispersion.

At glass-air interface, the critical angle for total internal reflection is ~42.Reflecting prism are shaped to let the angle of incidence larger than this 42.

Some examples:

Right angle prism (Fig. 5.61):it deviates rays normal to the incident face by 90.

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Reflecting Prisms

Some examples:

Porro prism (Fig. 5.62):it deviates incoming rays normal to the incident face by 180.Image right-handed in will leave right-handed out.

Penta prism (Fig. 5.66):it deviates the beam by 90 with out changing theorientation of the image.

Double Porro prism (Fig. 5.68): invert image, right-handed in and right-handed out.

Corner-cube prism : it has three mutually perpendicular faces, can be made by cuttinga corner of a cube. It will reflect all incoming rays back along their original directions.