Chapter 9 – Electro-Optics

Gabriel Popescu

University of Illinois at Urbana‐Champaigny p gBeckman Institute

Quantitative Light Imaging Laboratory

Electrical and Computer Engineering, UIUCPrinciples of Optical Imaging

Quantitative Light Imaging Laboratoryhttp://light.ece.uiuc.edu

Electro‐Optics

ECE 460 – Optical Imaging

1st order effect:

Electro Optics

1 ( ) (0)wi ii jj ijk j kP r E E

DC

0

2 2 20 0 ( )i i i i j ijk j kD n E n n r E E DC

11 zO0 0 ( )i i i i j ijk j k

2 4 20 0 0 0 12 3 0 0 133

6 5y Dc e z DCDx n n r E E n n r E E

2 40 0 0 0 213

4

0x DcDy n n r E E 0r kDP

Chapter 9: Electro‐Optics 2

42

0 e zDz n E53 0r

226 ELECTRO-OPTICS

By using the contracted indices (7.1-11), the equation of the index ellipsoid in the presence of an electric field can be written

(:~ + rlkEk ) x2

+ (:; + r2kEk) y2 + (:; + r3kEk) Z2 (7.2-3)

+2yzr4k Ek + 2zxr5kEk + 2xyr6kEk = 0

where Ek (k = 1,2,3) is a component of the applied electric field and summation over repeated indices k is assumed. Here 1,2,3 correspond to the principal dielectric axes x, y, z, and nx ' ny, nz are the principal refrac­tive indices. This new ellipsoid (7.2-3) reduces to the unperturbed ellipsoid (7.1-1) when Ek = O. In general, the principal axes of the ellipsoid (7.2-3) do not coincide with the unperturbed axes (x, y, z).

A new set of principal axes can always be found by a coordinate rotation, which is known as the principal-axis transformation of a quadratic form. The dimensions and orientation of the ellipsoid (7.2-3) are, of course, dependent on the direction of the applied field as well as the 18 matrix elements rlk. We have argued above that in crystals possessing an inversion symmetry (centrosymmetric), rlk = O. The form, but not the magnitude, of the tensor r1k Can be derived from symmetry considerations, which dictate which of the 18 coefficients r1k are zero, as well as the relationships that exist between the remaining coefficients. In Table 7.2 we give the form of the electro-optic tensor for all the noncentrosymmetric crystal classes. The electro-optic coefficients of some crystals are listed in Table 7.3.

7.2.1. Example: 1be Electro-optic Effect in KH 1 P04

Consider the specific example of a crystal of potassium dihydrogen phos­phate (KH2P04 ), also known as KDP. The crystal has a fourfold axis of symmetry, which by strict convention is taken as the z (optic) axis, as well as two mutually orthogonal twofold axes of symmetry that lie in the plane normal to z. These are designated as the x and y axes. The symmetry group of this crystal is 42m. Using Table 7.2, we write the electro-optic tensor in the form

0 0 0 0 0 0 0 0 0

rij = r41 0 0 (7.2-4)

0 r41 0 0 0 r63

'1 I I I I

Table 7.2. Electro-optic: Coefficients in Contracted Notation for All Crystal Symmetry OassesQ

Centrosymmetric (I, 2/m, mmm, 4/m, 4/mmm, 3, 3m 6/m,6/mmm, m3, m3m): 000 000 000 000 000 000

Triclinic:

'Il '12 '13

'21 '22 '23

'31 '32 '33

'41 '42 '43

'51 '52 '53

'61 '62 '63

Monoclinic:

2 (2 II X2) 2 (2" X3) 0 '12 0 0 0 '13

0 '22 0 0 0 '23 0 '32- 0 0 0 '33

'41 0 '43 '41 '42 0 0 '52 0 '51 '52 0

'61 0 '63 0 0 '63

m (m J. X2) m (m J. X3)

'Il 0 '13 'II '12 0

'21 0 '23 '21 '22 0 '31 0 '33 '31 '32 0 0 '42 0 0 0 '43

'51 0 '53 0 0 '53 0 '62 0 '61 '62 0

Orthorhombic:

222 2mm

0 0 0 0 0 '\3 0 0 0 0 0 '23 0 0 0 0 0 '33

'41 0 0 0

0 '42 0 '52 0

'51 0 0 0 0 '63 0 0 0

227

Table 7.1- ( Continued).

Tetragonal: 4 4

0 0 '13 0 0 '13 0 0 0 '13 0 0 -'13 0

0 0 '33 0 0 0 0

0 '41 -'51 0 '41 '41 '51

0 '51 '41 0 0 '51 -'41

0 0 0 0 0 '63 0

4mm 42m (211 XI)

0 0 '\3 0 0 0 0 0 '\3 0 0 0 0 0 '33

0 0 0

0 '51 0 '41 0 0

0 0 0 '41 0 '51

0 0 0 0 0 '63

Trigonal: 3 32

'II -'22 '13 '11 0 0 -'II '22 '\3 -'11 0 0

0 0 '33 0 0 0 '41 '51 0 '41 0 0

'51 -'41 0 0 -'41 0

-'22 -'II 0 0 -'11 0

3m (m .1 XI) 3m (m .1 X2)

0 -'22 '13 '11 0 '13 0 '22 '13 -'II 0 '13 0 0 '33 0 0 '33

0 '51 0 0 '51 0

'51 0 0 '51 0 0

-'22 0 0 0 -'11 0

228

"r

422

0 0 0 0 0 0 0 0

-'41 0

0 0

THE LINEAR ELECTRO-OPTIC EFFECT 229

Table 7.2. ( Continued).

Hexagonal: 6 6mm 622

0 0 '\3 0 0 '13 0 0 0 0 0 '13 0 0 '\3 0 0 0 0 0 '33 0 0 '33

0 0 0

'41 '51 0 0 '51 0 '41 0 0

'51 -'41 0 '51 0 0 0 0 0 0 0 0

0 -'41 0

0 0 0

6 6m2 (m .L XI) 6m2 (m.l X2)

'11 -'22 0 0 -'22 0 'II 0 0 -'11 '22 0 0 '22 0 -'11 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-'22 -'II 0 -'22 0 0 0 -'11 0

Cubic:

43m,23 432 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

'41 0 0 0 0 0 0 '41 0 0 0 0 0 0 '41

0 0 0

°The symbol over each matrix is the conventional symmetry-group designation.

so that the only nonvanishing elements are '41 = '52 and '63' Using Eqs. (7.2-3) and (7.2-4), we obtain the equation of the index ellipsoid in the presence of a field E(Ex' Ey ' Ez) as

where the constants involved in the first three terms do not depend on the field and, since the crystal is uniaxial, are taken as nx = ny = no' nz = ne' We thus find that the application of an electric field causes the appearance of "mixed" terms in the equation of the index ellipsoid. These are the terms with xv. xz. vz. This means that the major axes of the ellipsoid, with a field

Electro‐Optics

ECE 460 – Optical Imaging

Electro Optics2 4

0 0 63 0Dcn n r E 4 2

0 0 63 02

0

0 0ij Dc

e

n r E n

n

' ( ) ( )ij W W 2

0cos sin cos sinn 2

0sin cos sin cosn

2 0 0n ib0

20 0

2

0 0

0 0 ;4 ij

n

n

i

40 63 ( )zn r E DC

3

20 0 en

Chapter 9: Electro‐Optics

biaxial crystal

Modulators

ECE 460 – Optical Imaging

Modulators

Eg           (tetra          )  only three nonzero elementsKDP 42m 41 52 63, ,r r rg ( ) yKDP 41 52 63

2 20 ;xn n

20 0 0 0 0 02 2

0 0

1 112 2xn n n n n n n

n n

1

012yn n

30 0 63

1 ( )2 zn n r E DC

012y r

4Chapter 9: Electro‐Optics

Modulators

ECE 460 – Optical Imaging

Modulators

32 2( ') ( )n n d n r E DC d 0 63

0 0( ') ( )x y zn n d n r E DC d

n

V

03

0 632V

n r

linearize

2sinT 2sin VQW T

linearize

Adds2 sin

4 2QW T

V

Add

5Chapter 9: Electro‐Optics

Modulators

ECE 460 – Optical Imaging

Modulators

Let  sinm mV V t m m

2sin sin4 2 m mT t 4 2

1 11 cos sin 1 sin sint t 1 cos sin 1 sin sin2 2 2m m m mt t

linear

11 1 sin2m m m mT t

6Chapter 9: Electro‐Optics

Quadratic (Kerr)

ECE 460 – Optical Imaging

Quadratic (Kerr)

( ) 1 ( ) ( ) ( )P E E DC E DC( )

0

1 ( ) ( ) ( )i ii jj ijk j k eP s E E DC E DC

7Chapter 9: Electro‐Optics

Applications of EO

ECE 460 – Optical Imaging

Applications of EO

longitudinalg transverse For LiNiO3 : 

modulators

30 0 13

1n n n r E 0 0 13

30 0 13

212

x

y

n n n r E

n n n r E

333

212z e en n n r E

30 0 13

0

2 n L n r Vv

Ed3

0 13

vVn r

Phase mod(indep. of polariz.)

8

0Ed

Chapter 9: Electro‐Optics

Chapter 9 – Acousto-optics

Gabriel Popescu

University of Illinois at Urbana‐Champaigny p gBeckman Institute

Quantitative Light Imaging Laboratory

Electrical and Computer Engineering, UIUCPrinciples of Optical Imaging

Quantitative Light Imaging Laboratoryhttp://light.ece.uiuc.edu

Acousto‐optics

ECE 460 – Optical Imaging

Acousto optics

2 2P S E 2 20i j i ijkl kl j

jklP n n p S E

12 6 • ac wave:x

S S12 613 523 4

• ac wave: z 13 5S S

( ) cos( )U z t xA t kz ( , ) cos( )U z t xA t kz

Chapter 9: Acousto‐optics 10

Chapter 1: Introduction 11

Acousto‐optics

2kK

Acousto optics

k1 k2

K

1k

02 sinnk K

0 2 /2 /

kK

sin ;B Bragg angle

Acousto‐optics

ECE 460 – Optical Imaging

Acousto optics

( )ll k k 0 0( )small k k

Δk ΔvDopplerShift sKv

Δk Δv

Quantum mechanics

'k k '

conservation of momentumconservation of energymechanics ' conservation of energy

Chapter 9: Acousto‐optics 13

Anisotropic media

ECE 460 – Optical Imaging

Anisotropic media

'k k 'k

k

'n-different - negative

k

'sin ' sin ; 'k k

2 ' 2 2 20 0

2 ' 2 2sin ' sin ;n n

2

0sin ' sin' '

nn n

wavelength of sound

Chapter 9: Acousto‐optics 14

Anisotropic

ECE 460 – Optical Imaging

Anisotropic

C

Ex:

k'

Ck - (e)

'k - scattered in prop. Plane – (o)

'k (o)

0

0sin ' sin

'en

n n

0 '2 2

' ,

0

00

0

en n

0 0

0 0e en n n n

Chapter 9: Acousto‐optics 15

,2 2

0

0 en n

0 0e e

Small angle Scattering

ECE 460 – Optical Imaging

Small angle Scattering

2sin ( )scattI L sin ( )inc

LI

3/2( )k 3/20 1 2

1 2

( )4 cos cos

i ke jijkek n n e p S e

Kin. Energy/ V = ½ Wtotal

1 2 1 1Usv

12ac sI v 2 2 3 21 1| | [ | |]

2 2s sU v u v Ut

US U 231I S

Chapter 9: Acousto‐optics 16

z 312ac sI v S

Small angle Scattering

ECE 460 – Optical Imaging

Small angle Scattering

2IS3/2

0 1 2( ) 2k n n IP 32 ac

s

ISv

0 1 23

1 2

( ) 24 cos cos

ac

s

k n n IPv

Small cos 1

26

3s

n pMv

ps

table

3/2 30 1 2( ) 2s ack n n v M I MI0 1 2

6 3( ) 2

4s ac

s

k n n v M In v

02 acMI

Chapter 9: Acousto‐optics 17

Small angle Scattering

ECE 460 – Optical Imaging

Small angle Scattering

Detuning: sin B g sin2B k

2 1 2sin sin ; Bk

1 1 2 2sin sin( ) 2 sin

( ) cos cos

k kBragg k

k k

1 1 1 2 2 2

1 2

1 2

( ) cos cos

( ) (cos cos )

k k

k

1 2( ) ( )

2 sin Bk 22

22 sin 1 ;

2scatI LI

22 2 1

2s

Chapter 9: Acousto‐optics 18

22 21

2incI

2

Finite Beams

ECE 460 – Optical Imaging

Finite Beams

A’

B’

0

2 ;nw L

B

size of acoustic beam

; 1 ;2 2L

2 2L

0

0 0 0

2 cos 2 4 coss ss

nv vfnw w

- Full12sv

0 0 0 2

0

1 ;fW

0

2 cos( ) snvfL

or

19Chapter 9: Acousto‐optics

0

! Not overlap with undiffracted order

N spots

ECE 460 – Optical Imaging

N spots

0 0

2 cos 2f nWN

0

2W f N

02 cos 2snv 2 sv

f 2B

sf

nv ; cond B

0 0

2 sf

L nv

2

0 0

2f nf L

20Chapter 9: Acousto‐optics