Chapter 6. Optics of Solids Part 4 Propagation of light in ... (9)_0.pdf · Chapter 6. Optics of...

Changhee Lee, SNU, Korea

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2016. 11. 17.

Changhee Lee

School of Electrical and Computer Engineering

Seoul National Univ.

[email protected]

Chapter 6. Optics of Solids

Part 4 – Propagation of light in crystals


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https://en.wikipedia.org/wiki/Plasmon

Propagation of light in crystals

Birefringence is the optical property of a material having a refractive index that depends on the

polarization and propagation direction of light. These optically anisotropic materials are said to be

birefringent (or birefractive). Crystals with non-cubic crystal structures are often birefringent, as

are plastics under mechanical stress.

A calcite crystal laid upon a

graph paper with blue lines

showing the double

refraction

https://en.wikipedia.org/wiki

/Birefringence

A model to illustrate the anisotropic polarizability of a crystal


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333231

232221

131211

333231

232221

131211

tensordielectric

,

χ

χ)(1

Eχ)E(1D χEP

o

oo

z

y

x

o

z

y

x

E

E

E

P

P

P

ε

ε


For ordinary nonabsorbing crystals the tensor is symmetric so there always exists a set of coordinate

axes, called principal axes, such that the tensor assumes the diagonal form

33

22

11

00

00

00

χ

The tensor assumes the diagonal form where three ’s are known as the principal susceptibilies.

constants dielectric principal ,1 ,1 ,1 333322221111 KKK


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2

2

2

2

2

2

2

2 ~~

)(tc

N

tc

N

Eχ

EE


2

2

2

2

2

1)(

ttco

PEE

zzyxyyzxxz

yzzyyzxxxy

xzzxyyxxzy

Ec

Ec

kkEkkEkk

Ec

EkkEc

kkEkk

Ec

EkkEkkEc

kk

33

22

11

2

2

2

222

2

2

2

222

2

2

2

222

)(

)(

)(

For monochromatic plane wave of the usual form )( tie rk

χEEEEkk

χEEEkk

2

2

2

22

2

2

2

2

-)(

)(

cck

cc


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axis in the points if velocity phase ,1 0, If

axis in the points if velocity phase ,1 0, If

)(

)(

axis) the toe transversis field (

33

33

22

22

2

2

2

22

2

2

2

22

2

2

2

2

33

22

33

22

11

zK

c

kK

cckE

yK

c

kK

cckE

Ec

Ec

k

Ec

Ec

k

xEEc

Ec

x

y

zz

yy

xxx

E

E

E

Consider a wave propagating in the direction of one of the principal axes, say the x axis. In this case

of the usual form 0 , zyx kkkk


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In the principal coordinate systems,

33

22

11

00

00

00

χ

33333

22222

11111

1

1

1

Kn

Kn

Kn

Introduce the three principal indices of refraction

0

)(

)(

)(

222

3

222

2

222

1

y

y

x

yxyzxz

zyzxxy

zxyxzy

E

E

E

kknc

kkkk

kkkknc

kk

kkkkkknc

0)1(-)(2

22 EχEEkk

ck

can be written as


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ellipse)an of(equation 1)/()/(

circle) a of(equation )(

0)()()(

2

1

2

2

2

2

2

3

22

2222

2

22

1

222

3

cn

k

cn

k

nc

kk

kkknc

knc

kknc

yx

yx

yxxyyx

Consider any one of the coordinate planes, say the xy plane. In this plane 0zk

Similar equations are obtained for the xz and the yz planes. The intercept of the k surface with each

coordinate plane consists of one circle and one ellipse.

In order for a nontrivial solution 0) det(

0

)(

)(

)(

222

3

222

2

222

1

yxyzxz

zyzxxy

zxyxzy

kknc

kkkk

kkkknc

kk

kkkkkknc


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For any given direction of the wave vector k,

there are two possible values for k, i.e., two

values of the phase velocity corresponding to two

mutually orthogonal directions of polarization.

A light wave of arbitrary polarization can always

be resolved into two orthogonally polarized

waves, Thus, when unpolarized light, or light of

arbitrary polarization propagates through a

crystal, it can be considered to consist of two

independent waves that are polarized

orthogonally with respect to each other and

travelling with different phase velocities.

For propagation in the direction of an optic axis,

there is only one value of k. Thus, two

orthogonally polarized waves propagate in this

direction with the same phase velocity.

Saleh and Teich, pp.214-215


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Biaxial crystal Uniaxial positive crystal

no < ne

Uniaxial negative crystal

no > ne

Isotropic crystal, n=n1=n2=n3 example, cubic crystal


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• Uniaxial crystal: There is a single direction governing the optical anisotropy whereas all

directions perpendicular to it are optically equivalent. Thus rotating the material around this axis

does not change its optical behavior. This special direction is known as the optic axis of the

material.

• A ray with a linear polarization direction perpendicular to the optic axis is called an ordinary ray.

• A ray with a linear polarization in the direction of the optic axis is called an extraordinary ray.

• The ordinary ray will always experience a refractive index of no, whereas the refractive index of

the extraordinary ray will be in between no and ne, depending on the ray direction as described by

the index ellipsoid. The magnitude of the difference is quantified by the birefringence: Dn=ne – no

• The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as

if there were no birefringence involved. However the extraordinary ray, as its name suggests,

propagates unlike any wave in a homogenous optical material. Its refraction (and reflection) at a

surface can be understood using the effective refractive index (a value in between no and ne).

• Biaxial crystals are characterized by three refractive indices corresponding to three principal axes

of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays

but with different effective refractive indices.


https://en.wikipedia.org/wiki/Birefringence


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https://en.wikipedia.org/wiki/Bravais_lattice

a = b = c

α = β = γ = 90°

a ≠ b ≠ c

α ≠ β ≠ γ ≠ 90°

a ≠ c

α = γ = 90°, β ≠ 90°

a ≠ b ≠ c

α = β = γ = 90°

a = b ≠ c

α = β = γ = 90°

a = b = c

α = β = γ ≠ 90°

a = b

α = β = 90°, γ = 120°

Crystals in three dimensional space


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0)()(

)()()(

2222

2

222222

3

22

22222

3

222

2

222

1

zyxzxzyxyx

zyyxzxzy

kkknc

kkkkknc

kk

kkkknc

kknc

kknc

In general, we can expand the determinant as follows:

0

)(

)()(

2222222222222222

222

3

222

2

222

1

22222

3

22222

2

22222

12

2

222

1

2

3

222

3

2

2

222

2

2

1

2

2

22

3

2

2

2

1

3

2

2

zyxzyxzyzyyxxzzy

yxxzzyyzxzzyxyzxyx

xzzyyx

kkkkkkkkkkkkkkkk

kknkknkknkkkknkkkknkkkknc

kknnkknnkknnc

nnnc

222222

22222222

zyxzyx

zyyxxzzy

kkkkkk

kkkkkkkk

][][)(][][][)(

)(

2

3

2

2

22222

12

22

3

2

2

2222222

12

2

222

3

222

2

222

1

22222

3

22222

2

22222

12

2

nnkkkknc

nnkkkkkknc

kknkknkknkkkknkkkknkkkknc

zyxxzyzxyx

yxxzzyyzxzzyxyzxyx


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(1) Isotropic crystal, n1=n2=n3=n

0)()( 222

2

2

2

1

2

2

3

2

1

2

2

3

2

2

2

2

2

22

2

1

22

2

3

22

2

22

2

2

zyx

zyxxzzyyxkkk

nn

k

nn

k

nn

k

n

kk

n

kk

n

kk

cc

2

3

2

2

2

12

2

)()equation Previous( nnnc

)degenerate(doubly 0)(

0)())((2)(

2

2

2

2

2

2

22

2

22

2

2

nc

kn

k

c

n

k

n

k

cc

(2) Uniaxial crystal, n1=n2=no, n3=ne

)( ),( ,0)()(

0)()(

2

222

2

2

2

2

2

22

2

2

2

2

2

2

2

2

2

22

2

2

2

22

2

2

2

22

2

22

2

22

2

22

2

2

cnk

cn

k

n

kk

cn

k

cn

k

n

kk

n

k

n

kk

n

k

n

kk

n

kk

n

kk

cc

o

o

z

e

yx

oo

z

e

yx

o

z

e

yx

oo

zy

o

zx

e

yx


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Alternative form

0)()(

)()()(

2222

2

222222

3

22

22222

3

222

2

222

1

zyxzxzyxyx

zyyxzxzy

kkknc

kkkkknc

kk

kkkknc

kknc

kknc

We define effective refractive index:

0)()()(

)()()(

22222

2

2222

3

2222

1

22

222

3

222

2

222

1

zyxzxyxzy

zyx

kkkknc

kkknc

kkknc

kk

kknc

kknc

kknc

0)()()()(

)()()()()(

22

2

22

1

222

1

22

3

2

22

3

22

2

222

3

22

2

22

1

knc

knc

kknc

knc

k

knc

knc

kknc

knc

knc

zy

x

1

)()()( 2

3

2

2

2

2

2

2

2

1

2

2

n

ck

k

nc

k

k

nc

k

k zyx

22

3

2

2

2

2

2

2

2

1

2

21

nnn

s

nn

s

nn

s zyx

kcn

),,(),,( zyxzyx ssskkkk


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Propagation of light in uniaxial crystals

Saleh and Teich, p.218

2

2

2

2

2

sincos

)(

1

eo nnn


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The Poynting vector and the ray velocity

SkES

Dk

PED 0,D

toparallenot is

o

cos

vty ray veloci

u

o

HES

DHk

HEk

2

2

2

2

22

2

22

2

2

vcos

-)(

)(

Dc

ED

ck

ck

c

o

o

o

o

DE

DDDE

DEEkk

DEkk


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Propagation of light in uniaxial crystals



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Phase-velocity surface

0

/

/

/

22242

3

22242

2

22242

1

2

yxyzxz

zyzxxy

zxyxzy

vvcvnvvkv

vvvvcvnvv

vvvvvvcvn

v

vk

2

4

2

1

2

2

2

2

2

3

2222

vvv

vvv

plane, For the

cnn

n

c

xy

yx

yx


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The Poynting vector and the ray velocity

If the coordinate axes are principal axes of the crystal,

, ,2

333

2

222

2

111 n

DDE

n

DDE

n

DDE zz

zo

yy

yoxx

xo

0)(

0)(

0)(

22

2

3

2

22

2

2

2

22

2

1

2

yxzyzyxzx

zyzzxyxyx

zxzyxyzyx

uun

cDuuDuuD

uuDuun

cDuuD

uuDuuDuun

cD

0

/

/

/

222

3

2

222

2

2

222

1

2

yxyzxz

zyzxxy

zxyxzy

uuncuuuu

uuuuncuu

uuuuuunc

ellipse)(an circle) (a 222

1

22

22

3

222 cunun

n

cuu yxyx

The equations of the intercepts in the xy plane are obtained by setting 0zu


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6.8 Double refraction (birefringence)

boundary)(at rkrk o

21

11

sinsin

sinsin

kk

kk

o

o


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eee

ooo

o

o

n

n

nknk

kk

sin)(sinfor which at (TM)on polarizati parallel of ary waveextraordinAn

sinsinfor which at (TE)on polarizati orthogonal of aveordinary wAn

crystal, uniaxial afor wavesrefracted Two

sin)(sin ,)( medium, canisotropian In

sinsin condition matching-phase

1

1

1

1



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Polarizing prisms

Suppose we have a negative uniaxial crystal, such

as calcite, and the internal angle of incidence is such

that

In this case we have total internal reflection for the

ordinary wave but not for the extraordinary wave.

Thus, the refracted wave is completely polarized.

oe nn sin

1


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Polarizing prisms


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6.9 Optical activity

Optical activity = The ability to rotate the plane of polarization of light passing through the medium.

When a beam of linearly polarized light is passed through an optically active medium, the light

emerges with its plane of polarization turned through an angle that is proportional to the length of the

path of the light through the medium.

Specific rotatory power = the amount of rotation per unit length of travel.

If the sense of rotation of the plane of polarization is to the right, as a right-handed screw pointing in

the direction of propagation, the substance is called dextrorotatory or right-handed. If the rotation is to

the left, the substance is called levorotatory or left-handed.


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6.9 Optical activity

)(power rotatory specific

)(2

)(

RR

RRRR

nn

lnn

c

lnn


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6.10 Faraday rotation in solids

constant)Verdet ( VVBl

The presence of the field causes the dielectric to become optically active. This phenomenon was

discovered in 1845 by Michael Faraday.


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6.11 Other magneto-optic and electro-optic effects

Kerr electro-optic effect: When an optically

isotropic substance is placed in a strong

electric field, it becomes doubly refracting,

discovered in 1875 by J. Kerr.

- 2

|| oλKEnn


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6.11 Other magneto-optic and electro-optic effects

Cotton-Mouton effect: Magnetic analogue of the Kerr electro-optic effect.

Pockels effect: When certain kinds of birefringent crystals are placed in an electric field, their

indices of refraction are altered by the presence of the field.


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6.12 Nonlinear Optics (NLO)

Chapter 6. Optics of Solids Part 4 Propagation of light in ... (9)_0.pdf · Chapter 6. Optics of...

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