Hecht; 10/12/2010; 8-1

Chapter 8. Polarization 8.1 The Nature of Polarized Light A. Linear Polarization Sum of two waves ( ) ( ) ( )ω ω ε= − + − +ˆ ˆ, cos cosox oyE z t xE kz t yE kz t For 0, 2 ...ε π= ± , ( ) ( )ˆ ˆ cosox oyE xE yE kz tω= + −

For ...ε π= ± , ( ) ( )ˆ ˆ cosox oyE xE yE kz tω= − −

B. Circular Polarization Right-circularly polarized

For o ox oyE E E= = and /2 2 mε π π= − +

( ) ( )ˆ ˆcos sinoE E x kz t y kz tω ω= − + −⎡ ⎤⎣ ⎦ E is constant but direction of E changes in space and in time.

At oz z= and t=0 → ( )ˆ cosx o oE xE kz= , ( )ˆ siny o oE yE kz=

At oz z= and /ot kz ω= → ˆx oE xE= , 0yE = E is rotating clockwise in time Left-circularly polarized

For o ox oyE E E= = and /2 2 mε π π= +

( ) ( )ˆ ˆcos sinoE E x kz t y kz tω ω= − − −⎡ ⎤⎣ ⎦

C. Elliptic Combin xE

yE

→

It is an

tan

Various

States o mI=== b -state

Right-c

cal Polarizane two wave

(cosoxE kz=

(cosoyE k=2

x

ox

EE E

⎛⎛ ⎞+ ⎜⎜ ⎟ ⎜⎝ ⎠ ⎝

ellipse mak

2

2n2 ox

ox

E EE

α =

s polarizatio

of polarizatioI===iI===b -st

e is given by

circularly po

ation es

)kz tω−

)kz tω ε− + 2

2y

oy

E EE E

⎞ ⎛−⎟ ⎜⎟ ⎝⎠

king an ang

2

cosoy

oy

EE

ε−

on configura

ion tates

y a sum of

olarization

coyx

ox oy

EEE E

⎛ ⎞⎞⎜ ⎟⎟ ⎜ ⎟⎠ ⎝ ⎠

gle α with x

ations

o -state an

2os sinε ε=

x-axis

nd i -state

e.

Hechht; 10/12/22010; 8-2

8.2 Pola Natural (Random Principl Dic

A linear The elec TE The tra (I θ

arizers l light mly polarize

les of polarichroism(Sele

r polarizer h

ctric field cocosoE θ=

↑ ↑ T The input

ansmitted i) ( )0 cosIθ =

ed)

izer ective absor

has transmi

omponent t

The angle bt electric fie

intensity 2s θ

→ ↑ Polariz

rption), Ref

ission axis a

that is paral

between oE eld.

Pol

zer

flection, Bir

and extinct

llel to the tr

and the tra

: Malus’s L

larized light

refringence(D

tion axis.

ransmission

ansmission

Law

t

Double refr

n axis can o

axis.

Hech

raction)

only be tran

ht; 10/12/2

nsmitted.

2010; 8-3

8.3 Dich Dichroic A. The Wi y-polari → Cur → Rer → For Bac x-polari B. Dichro Absorpt → Ani Tourma

Electric → Tra Some ab (Tourma C. Polaro J-sheet H-sheet

hroism c material a

ire-Grid Po

ized light (Vrrent in y-dradiation inrward radiackward rad

ized light pr

oic Crystalstion and conisotropic cry

aline crystalc field perpeansmission

bsorption aalin is usua

id : Herapathi ↑ Dichro

t: Long mole

absorbs one

olarizer

Vertical diredirection (Son forward anation canceliation form

ropagates th

s nductivity dystals

ls endicular toalong optic

along the opally green co

ite needles a

ic crystal

ecules, not

e of the two

ction) ome energy nd backwars the incides a reflected

hrough the

depend on d

o the optic acs axis.

ptic axis depolored)

aligned in o

dichroic, fo

orthogonal

loss by jou

rd directionsent wave. d wave.

grid with n

direction

axis is stron

pends on wa

one directio

orm the wire

l m -states

ule heat). s.

no change

ngly absorb

avelength.

n

e grid.

bed.

Hechht; 10/12/22010; 8-4

8.4 Bire Birefrin Electron → z-ax A. Calcite 3CaCO Calcite

[Fig 8.16

A single → Two The Principa

Principa The ordina The sec

efringencngent crysta

n binding foxis become

e has 3-fold

in a cleavag6] [Fig 8.18

e beam into o beams aftey are linea

al plane al section

ary ray condary wav

ce al has two d

orces are ids optic axis

symmetry a

ge form form8]

Calcite ter the crys

arly polarize

: A plane c: A princip

velets have

different ind

entical in xs

about the o

ms rhomboh

tal: Ordinaed and ortho

containing tpal plane tha

the speed o

dices of refra

x- and y-axi

ptic axis

hedron

ary ray, Extogonal to ea

the optic axat is norma

of v⊥

action.

s

raordinary ach other

xis al to two sur

ray.

rfaces of th

o-ray ⊥ k-e-ray ⊥ k-

Hech

e cleavage f

-vector and-vector and

ht; 10/12/2

form

d optic axis.d o-ray.

2010; 8-5

The extrao The elec

Ray dire From It’s p It is n B. Birefrin Cubic c Optic Singl Hexagon Symm Unia Orthorh Biax Uniaxia

Birefrin

ordinary rayctric field =

ection : m the origin parallel to thnot normal

ngent Crystcrystal cally isotrople index of r

nal, Tetragometric in xy

axial crysta

hombic, Molxial crystals

al crystal hangence is de

y perpendicu ↑ =v⊥

of a wavelehe energy fl to the wave

tals

pic. refraction.

onal, Trigony-plane. als with opti

loclinic, Tris with two o

as two indicefined as nΔ

ular + paral ↑=

et to the tanlow directioefront.

nal

ic axis alon

iclinic optic axes

ces of refrac

e on n n≡ −

Neg

llel to optic v . (Two di

ngent point n (Poynting

g z-axis.

ction: on c=

gative and p

axis ifferent spee

of the waveg vector).

/ and ec v n⊥

positive unia

eds)

elet with the

/e c v=

axial crysta

Hech

e wavefront

als

ht; 10/12/2

t.

2010; 8-6

C. BirefrinNicol prism Split ca ↑ on = The inci o-ray is

Glan-Fouc Same p o-ray is Field o Max in Cemen Field o Max in

Wollaston o- and The dev

ngent Polarm alcite rhomb

1.6584, en

ident beam totally inte

cault polarizpropagations totally intf view = 10

ntensity = 10

nted calcite f view = 30

ntensity = 1W

prism e-rays sepaviation ang

rizers

bohedron is

1.4864e =

m refracts internally refle

zer (or Glann direction oernally refle degree,

200 /W cm

prisms is ca degree,

2/W cm

arate at the

gle ranges fr

cemented b

to o- and e-cted at the

n-Air) of o- and e-ected at the

alled Glan-

e interface. rom 15 to 4o

by Canada ↑ n=1.55

-rays in the interface.

rays in the e interface.

-Thompson

45o .

balsam

e first prism

first prism.

n polarizer

m.

.

Hech

ht; 10/12/22010; 8-7

8.6 Pola When θ Using S tan

At pθ , n Plate pola If iθ = θ

Polarizing Multipl

Degree of

V =

max, I I

→ pI

→ V =

arization 90o

i tθ θ+ =

Snell’s law n /p t in nθ =

no reflection

arizer pθ , a single

g Cube le layers at

Polarizatiop

p n

II I

=+

minI are mea

max minI I= −

max min

max min

I II I

−=

+

by Refle, iθ is call

sini pn n=θ

n of parallel

plate is a p

the diagona

on

asured afte

n , nI

n

n

ction ed Brewste

sint t tn n⇒θ

l polarizatio

perfect pola

al surface (

, p nI I

r the analyz

min2I=

er angle (or

sin(90op−θ

on beam.

arizer.

Similar as p

: Intensitie

zer

polarization

)p

pile-of-plate

es of polariz

n angle, pθ

es)

zed and unp

Hech

)

polarized lig

ht; 10/12/2

ght.

2010; 8-8

8.7 Reta Two m

→ Chan A. Wave P After th

( ,E z

The pha

ϕΔ =

Fast axi The Half-W ϕ πΔ = It rotate It conve The Quart

/ϕ πΔ = It conve Retarders Single-o Relat Thin Multiple Relat Less Compou Fast Less B. Compe A comp Babinet Optic Soleil C

arders -states expenge of the p

Plates and he plate

) ˆ, co= oxt xE

ase differen

(2πλ

= −eo

d n

is has smal

Wave Plate

es the polarerts o stat

ter-Wave Pl

/2 erts m stat

in generalorder retardtive phase o and difficu

e-order retative phase os expensive

und zero-or axis of one sensitive to

ensators anensator can

t Compensac axes of we

Compensato

erience diffepolarization

Rhombs

s ω⎛

− +⎜⎝kz t

ce

)on

ller refractiv

rization direte to i sta

late

te to o /i

l der : of 2Δ <ϕ π . ult to make

arder : of 2 mϕ πΔ +, but narrow

rder retardee plate is oveo temperatu

nd Variablen control th

ator edge calcite

r

ferent phase state.

2 ˆπλ

⎞+⎟

⎠e

o

dn y

ve index tha

ection by 2θate and vice

i state and

but large fi

m w FOV, sen

er : erlapped wiure but still

e Retardershe amount o

es are shown

e changes in

ˆ cos⎛⎜⎝

oyyE kz

an that of s

θ . e versa.

d vice versa

eld-of-view

nsitive to inc

ith slow axil narrow fiel

s of the retard

n by dots

n the retard

2πωλ

− +o

t dn

low axis

.

cident angle

is of the othld-of-view.

dance.

der.

⎞⎟⎠

on

e, λ and T.

her.

Hech

ht; 10/12/22010; 8-9

8.8 Circ Linear Left-cir 8.9 PolaA. Bandw Monoch Polychr Quasim B. Interfe

Phase d Blue ma 8.10 Op Opticall Looking Dext Levor A. A Usefu A wave → Elec → Rota B. Optica Synthes But one Natu Prote Antib Five am They co

cular Pola polarizer rcular polar

arization width and Chromatic wa

omatic wav

monochroma

erence Colo

difference, Δakes constr

ptical Actly active: A

g in the diretrorotatory (rotatory (l-r

ful Model into helical

ctric and maated polariz

ally Active sized organi

e is dominanural sugar iseins consistbiotics cont

mino acids inontain an eq

arizers + 90o retrizer allows

of PolychCoherence Tave

ve

atic wave

ors

↑ Polarizer .

ϕΔ , after thructive inter

tivity linear plan

ection of the(d-rotatory) rotatory)

l molecules agnetic dipozation

Biological ic molecule

nt in naturas always d-rts of amino tain amino a

n a meteoritqual numbe

tarder. only the lef

hromaticTime of a P

: Single

: A ran Const

: A nar

↑ Wave plat

he wave platrference wh

e wave app

e source : Clock

: Coun

ole moment

Substances have an e

al organic mrotatory. acids(compacids that a

te in Austraer of l- and

ft-circular l

c Light Polychromae fequency.

nge of frequetant polariz

rrow bandw

↑ te. Analyz

te can depehile yellow d

ears to rota

kwise nter-clockwi

ts

s equal numb

molecules.

pounds of Care d-rotato

alia. d-rotatories

light.

atic Wave

encies. zation state

width .

zer.

nds of λ. destructive i

ate.

ise

ber of l- and

C, H, O, N), ory.

s

only during

interference

d d-isomers.

generally l-

Hecht

g a short tim

e at the out

.

-rotatory

t; 10/12/20

me.

tput.

010; 8-10

Hecht; 10/12/2010; 8-11 8.11 Induced Optical Effects-Optical Modulators A. Photoelasticity Isotropic substances can be made optically anisotropic by mechanical stress. B. The Faraday Effect Linear polarization rotates in a magnetic field applied along the beam direction. Bdβ = s : Rotation angle in minutes Verdet constant, magnetic flux density, length. Other magneto-optic effects Voigt and Cotton-Mouton effects occur for a magnetic field perpendicular to the beam direction. ↑ ↑ Vapors Liquids C. The Kerr and Pockels Effects Some isotropic substances become birefringent in an applied electric field. The birefringence in this case. 2

on KEλΔ = K is Kerr constant E is electric field in statvolts( ≈300V) Pockels Effect It is called linear electro-optic effect ~n EΔ . 20 out of 32 crystal symmetry groups with noncentrosymmetry

8.12 Liq Long cig Positive Alignme No posit ↑ lik Parallel Nema Ra Two Rubb

The Liqu The b

The r

(1) In

(2) (3) LC

Twiste

Two The i Mole Incid Appl → R Voltage LC ce

quid Crysgar-shaped e uniaxial bient directiontional order ke liquid

l nematic catic crystalandom in p

opposite gla

bed ITO sur

uid Crystal birefringenc

retardance

ncident E p“ n

C cell btwn

ed nematic glass platesinput direct

ecules are grdent E paral

ied electric Reduced bir

controlled ell between

stals molecules.irefringent wn of LC molr, but large-

cell s:

position but

ass plates a↑ ~ 10 mμ

rface →

Variable Rce : Δ

: Δ

parallel to dinot parallel 45o± cross

cell s are relativtor is horizoradually rotllel to the in

field tilts mrefringence.

switch crossed po

with optic alecules is ca-scale orien

nearly para

are coated w

ParalleMolecu

Retarder ( )en n V= −

2

o

d n= Δπϕλ

irector “ s polarizers

vely rotated ontal and thtated betwenput directo

molecules pe.

olarizers

axis in the ealled directontational ord

↑ like crysta

allel to each

with indium ↑ Con Tra

el microgrooules parallel

on

→ phase m→ Retarde

→ Irradian

by 90o. he output deen plates. or is rotated

erpendicula

elongated dor. der.

al

h other.

m tin oxide ↑ nducting mansparent fr

oves. l to the glas

modulator er. nce modula

director is ve

d by 90o .

ar to the ele

irection.

metallic film.

rom 450nm

ss and to ea

ator.

ertical.

ectrode.

Hecht

. m to 1800nm

ach other

t; 10/12/20

m.

010; 8-12

8.13 A MA. The St B. The Jo They ar The elec

E⎡

= ⎢⎣

m -stat

→ E =

→ E E=

o -stat

E =o

i -stat

E =i

The sum

Elliptica

15⎡⎢⎣

Two ligh if their J

E •o

hE •

Note E •o

E •o

Mathematokes Param

ones Vectorre used only

ctric field ve( )( )

ox

y o

EE tE t E

⎡⎡ ⎤= ⎢⎥⎢⎣ ⎦ ⎣

Comple

e at 45o

( )( )

x

y

E t EE t E

⎡⎡ ⎤= ⎢⎢ ⎥

⎣ ⎦ ⎣

11

oioE e ⎡ ⎤

⇒⎢ ⎥⎣ ⎦

ϕ

e light

( /2)

o

o

i

o i

eEe −

⎡= ⎢

⎣

ϕ

ϕ π

te light 11

2 i⎡ ⎤

= ⎢ ⎥⎣ ⎦

m E E+ =o i

al polarizati2i⎤⎥− ⎦

ht polarizatJones vecto

( )1 12

E ⎡= ⎣Gi

( ) ( )v 1 0E ⎡= ⎣G

E E• = •G

o i

E E• = •Gi i

atical Desmeters

rs y for polariz

ector x

y

iox o

ioy

e Ee

⎤ ⎡⇒⎥ ⎢

⎥ ⎣⎦

ϕ

ϕ

↑ ex form H

o

o

io

io

E eE e

⎤⎥⎦

ϕ

ϕ

1112⎡ ⎤⎢ ⎥⎣ ⎦

)

112 i

⎤ ⎡ ⎤⇒⎥ ⎢ ⎥−⎣ ⎦⎦

1202⎡ ⎤

= ⎢ ⎥⎣ ⎦

ion for exam

ions are ortors are orth

( ) ( ) ( )*1 i i+ −

) ( ) ( )* *0 1 ⎤+ =⎦

1E =Gi

0E =Go

scription

zed lights.

xiox

oy

oeEo

⎡⎤+ ⎢⎥⎢⎦ ⎣

ϕ

↑ Horizontal m

Ve

: Norm

⎤

⎦

mple

thogonal ogonal.

)* 0⎤ =⎦

0=

of Polari

yi

o

e⎤⎥⎥⎦

ϕ

↑ m -state ertical m -s

malized to ha

ization

tate

ave unity irrradiance

Hechtt; 10/12/20010; 8-13

Hecht; 10/12/2010; 8-14 C. The Jones and Mueller Matrices The input and output Jones vectors are related by a transmission matrix A t iE E= A

m -state at 45o is incident into a quarter-wave plate with vertical fast axis

1 0 1 10 1

tx

ty

EE i i⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⇒⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ : o -state

A wave passing through a series of optical elements 1, 2, 3…n 3 2 1..t n iE E= A A A A