Electro-optics: Certain materials change their optical properties when subjected to an electric field (caused by forces that distort the positions, orientations, or shapes of the molecules. Electro-optic effect: apply an electric field => refractive index change; two forms (a) n E: linear electro-optic effect or Pockels effect (b) n E2: quadratic electro-optic effect or Kerr effect n is typically very small; the important factor is the phase shift which n and d. The necessary range is 0 < nd < 2.

Applications of the electro-optical effects: A lens with variable refractive index => a lens of controllable focal length; A prism with beam bending ability => optical scanning device. Light transmitted through a transparent plate of controllable refractive index => controllable phase shift => optical phase modulator. An anisotropic crystal with variable refractive index => variable phase retarder => control the polarization properties of light The variable phase retarder placed between two polarizer => optical intensity modulator or optical switch.

Principles of electro-optics: In general, the refractive index of an electro-optic medium is a function n(E) of the applied electric field E. If the function varies slightly with E => Taylor’s expansion

n(E)

E0 0

n n

Pockelsmedium

Kerrmedium

222

11)( EaEanEn

Define two new coefficients: r and s (electro-optic coefficient)

;/ ; / ; )0( where0

22201

EEdEndadEdnann

32

31 / ; /2 nana sr

23213

21)( EnEnnEn sr Neglect higher order

term which is small Expression in terms of electric impermeability

320 /2//1/ ndndn

))(/2()/( 23213

213 EnEnnndnd sr

23213

21)0()( EnEnnEnn sr

Pockels Effect: In many materials the s term is negligible

2)( EEE sr

EE r )( r: Pockels coefficient

Typical value of r 10-12-10-10 m/V (1-100 pm/V); For E=106 V/m => rn3E/2 ~10-6-10-4 (very small).Most common crystals used as Pockels cells:NH4H2PO4 (ADP), KH2PO4 (KDP), LiNbO3,LiTaO3.

Kerr Effect: If the materials is centrosymmetric, as is the case for gases, liquid, and certain crystals, n(E) must be an even function => r = 0

2)( EE s23

21)( EnnEn s

EnnEn 321)( r

s: Kerr coefficient

Typical value of s 10-18-10-14 m2/V2 in crystals; 10-22-10-19 m2/V2 in liquids; for E=106 V/m =>sn3E/2 ~10-6-10-2 in crystals; ~10-10-10-7 in liquids;

Electro-Optic Modulators and Switches: Consider a beam traverses a Pockels cell of length L; an electric field E is applied to the cell => phase shift

00 /)(2)( LEnLkEn 0

300

30 / / /2 ELnELnnL rr

VVdVE // 0 3

0

nLdV r

0

0

V

V: half-wave voltage; depends onmaterials (n and r), wavelength, andgeometry (d/L).

Phasemodulation

V Vd

LVLongitudinal Transverse Traveling wave Transverse

V: ~ several kV for longitudinal; hundreds of V for transverse modulatorsThe operation speed of the device is limited byelectrical capacitive effects and the transit time of thelight through the materials (T). If electric field variesquickly during the time period of T => lights are nottraveling in constant field => transit-time-limitedmodulation bandwidth ~ 1/T. To eliminate thisproblem so that bandwidth could be increased isusing the electrodes as transmission lines forelectrical wave to match the velocity of the opticalwave. => Push operational speed from severalhundred MHz to GHz) EO modulators can also be constructed as integrated-optical (IO) devices. Typically higher

V0V0 d << L

LiNbO3

V: ~ several V

Dynamic Wave Retarders: An anisotropic medium has two linearly polarized normal modes that propagate with different velocities , say c0/n1 and c0/n2. If the medium exhibits the Pockels effect, apply an electric field E, two refractive index are modified:

speed and lower voltage to operate than bulkdevices.

EnnEnEnnEn 3222

122

3112

111 )( ; )( rr

After propagation a distance L, the two modes undergo a phase retardation (relative to each other) given by

LEnEnk )]()([ 210

The medium serves as an electrically controllable dynamic wave retarded.

)()( 322

31102

1210 ELnnkLnnk rr

VVdVE // 0 311

311

0

nnLdV rr

where

0

0

V

Phaseretardation

Intensity Modulators: Use of a phase modulator in an interferometer;

0

V

A

B

C

V

T(V) E.g. the Mach-Zehnder interferometer

50%

50%

22

21

21

0 coscos iii IIII (=1-2): the phase difference of the two branches

T: transmittance (=I0/Ii)

1

0.5

0

1

2

Can be operated as a linear intensity modulator by adjusting the optical path difference so that 0 =/2 => operating in the nearly linear region around T = 0.5. Can be operated as an optical switch by T(0) = 1 and T(V) = 0. Mach-Zehnder intensity may also be constructed in the form of an integrated-optical device.

1: modulated by VV /011 VVVV // 020121

VVV

22cos)( 02T

V

0

Commercial available integrated-opticalModulator generally operate at speeds ofa few GHz, but speed of 25 GHz havebeen achieved!

Intensity Modulators: Use of a Retadrer between crossed Polarizers;

0

V

A

B

C

V

T(V)1

0.5

0

t

t

)2/(sin)( 2 VT

VV /0

VVV

22sin)( 02T

: retardation

Linear modulator if T(V) = 0.5 and V << V; let 0=/2

VV

VVV

221

24sin)( 2

T

0 could be adjusted either optically (by an additional phase retarder or a compensator) or electrically (by adding a constant bias to V).

Scanners: An optical beam can be deflected dynamically by using a prism with an electrically controlled refractive index.

+V

-V

Ld

D

An electro-optic prismAn electro-optic

double prism

i

)1( nθ

For small

dVnEnnθ / 3213

21 rr

Changing V => changing => scanning! Several prisms can be cascaded by alternating the direction of the electric field. An important parameter: resolution! An optical beam of width D and wavelength 0, has an angular divergence 0/D. (ref. Eq. 3.2-15 of textbook)

To minimize the divergence angle, the beam should be as wide as possible (cover the entire prism). For a given maximum voltage V corresponding to a scanned angle , the number of independent spots is given by

)//()/ (/ 03

21 DdVnθθN r

)/)(/( ; / 30 nLdVDL r VVN 2/

V => to make N point scan => require 2NV

voltage very high => not a popular method! The process of double refraction can be used to shift an incident beam by changing its polarization

or Position switch

Birefringent crystal

Directional Couplers: Control the coupling between two parallel waveguides in an integrated-optical device. Coupling of light between two parallel single-mode planar waveguides

P1(0) P1(0)

P1(z)

P2(z)

P2(L0)

L0

2/1 202

2

20

022

1

02 121csin

2)(sin

2)0(P)(P

L

LLLT

T : power transfer ratio 021 /2 n

= 0 => transfer distance L0

can be controlled by an applied voltage For a waveguide of length L0 and 0, T is a function of the phase mismatch

T1

0 31/2

L0

T has a maximum at L0 = 0; at L0 = 31/2, the optical power is not transferred to waveguide 2. Electrical controlled is achieved by using the electro-optical effect => phase shift L0 :

ELnLEnnLL 03

0003

21

000 )/(/) (2/2 rr

VdLnL )/( )/(2 03

00 r

d

VG

L0

E = V/d for one waveguide and -V/d for another waveguide =>

The necessary switching voltage (V0) is L0 = 31/2,

3)/( )/(2 003

00 VdLnL r

- sign is not important!Only the relative shiftcounts

rrr 30

30

30

00

322/

32

3n

dn

dnL

dV

CC

The coupling length L0 is inversely proportional to the coupling coefficient C, and its switching voltage (V0) is directly proportional to C. The key parameter is therefore C, which is governed by the geometry and the refractive indices. E.g. Diffusion Ti into LiNbO3 substrate: V0 ~ 10V; operation speed can exceed 10GHz.

000

0 /3||3

|| VVLVVL

2/1 2

0

22

3121csin

2 VVT Coupling

efficiency

Spatial light modulators: A device that modulates the intensity of the light at different positions by prescribed factors.

incidentlight

modulatedlight

TransmittanceT(x, y)

y

x

+

+

+

---

A planar optical element of controllable intensity transmittance T(x, y); the transmitted light intensity I0(x, y) is related to the incident light intensity Ii(x, y)

),(),(),(0 yxyxIyxI i T T(x, y) is like a image stored in the medium and the incident beam is like a read beam! If T(x, y) is controllable, such as an array of E-O materials on crossed electrodes! (need addressing!)

Optically addressed electro-optic spatial light modulators: One method to addressing the E-O spatial light modulator is based on the use of a thin layer of photoconductive materials to create the electric field required to operate the modulator. The conductivity of the photoconductive materials the intensity of light it exposed

Transparentelectrodes

Mirror

EOPhotoconductive

materials

Writebeam

Incidentbeam

Modulatedbeam

Two electrodes to charge up the photoconductivity materials => the write beam IW(x, y) illuminate on PCM => local conductance G(x, y) IW(x, y) => higher charge leakage at regions exposed to

stronger light intensity => local voltage 1/G(x, y) i.e. E(x, y)1/G(x, y)1/IW(x, y)!

Pockels Readout Optical Modulator (PROM): The device uses a crystal of bismuth silicon oxide Bi12SiO20 (BSO). BSO exhibits the Pockels effect, is photoconductive for blue light (but not for red light), and is a good insulator in the dark.

Modulatedlight

Incidentread light

Polarizingbeamsplitter

Transparent electrodeDichroic reflector pf red light

BSO

Priming: a large potential (~ 4kV) is applied to charge up the capacitor (BSO) Writing: a blue light IW(x, y) illuminate BSO => the spatial pattern of refractive index change n(x, y) is stored. Reading: uniform red light is

used to read n(x, y) with the polarizing beamsplitter playing the role of the crossed polarizers. Erasing: uniform flash of blue light

Incoherent-to-Coherent Optical Converters: In an optically addressed spatial light modulator (such as PROM), the light used to write a spatial pattern could be incoherent. One could use coherent light as the read light => conversion of a spatial distribution of natural incoherent light into a proportional spatial distribution of coherent light! * Useful in many optical data and image processing applications.

Electro-optics of Anisotropic Media: Pockels and Kerr Effects: When a steady electric field E, (E1, E2, E3), is applied to a crystal, elements of the tensor are altered => ij = ij(E)

3,2,1,,, ; )( lkjiEEEEkl

lkijklk

kijkijij sr

lkijijklkijijkijij EEE / ; / ; )0( where 221 sr

27 coefficients for rijk (Pockels coefficients); 81 coefficients for sijkl (Kerr coefficients); Symmetry: is symmetric => ij = ji => r and s are invariant under permutations of i and j => rijk = rjik and sijkl= sjikl. s is also invariant to permutations of k and l => sijkl= sijlk; => rijk: 18 coefficient; sijk: 36 coefficient

For simplicity, replace (i, j) with a single index I. (1, 1)1, (2, 2)2, (3, 3)3, (2,3) = (3,2)4, (1,3) = (3,1)5, (1,2) = (2,1)6. Similarly, replace (k, l) with a single index K. => rijk = rIk; sijkl = sIK; e.g. r112 = r12; r12k = r6k; s1112 = s16; s1231 = s65; Crystal Symmetry: add more constrains to the entries of the r and s matrices. For example, centrosymmetric crystals r vanishes. Ref: F. Nye, Physical Properties of Crystals: Their representation by Tensors and Matrices, Oxford University Press, New York, 1984.

41

41

41

000000000000000

rr

r

3m4 cubice.g. GaAs,CdTe, InAs

63

41

41

000000000000000

rr

r

2m4 Tetragonale.g. KDP,ADP

000000

0000

22

51

51

33

1322

1322

rr

rrrrrr

-

--3m Trigonal

e.g. LiNbO3,LiTaO3;

44

44

44

111212

121112

121211

000000000000000000000000

ss

ssssssssss

mediumIsotropic

2/)( 121144 sss Sequence to determine the optical properties of an anisotropic materials exhibitinf the Pockels effect in the presence of E: (1) Find the principal axes and principal refractive indices n1, n2, and n3 in the absence of E. (2) Find the coefficients (rijk )by using the appropriate matrix for rIk. (3) Determine the elements of the impermeability tensor using ij(0) is a diagonal matrix with elements 1/n1

2, 1/n2

2, and 1/n32.

k

kijkijij EE r)0()(

(4) Write the equation for the modified index ellipsoid;

(5) Determine the principal axes of the modified index ellipsoid by diagonalizing the matrix ij(E) and find the corresponding principal refractive indices n1(E), n2(E), and n3(E). (6) Given the direction of light propagation, find the normal modes and their associated refractive indices by using the index ellipsoid.

1)( jiij

ij xxE

Examples 1: Trigonal 3m crystal; uniaxial (n1 = n2

= no, n3 = ne ); assume E = (0,0,E), i.e. the electricfield points along the optical axis

z(optical

axis)E

x y

2

2

2

/1000/1000/1

e

o

o

nn

n

11(0) =1(0) = 1/no2;

22(0) =2(0) = 1/no2;

33(0) =3(0) = 1/ne2;

23 = 32=4 = 0; 13 = 31=5 = 0; 12 = 21=6 = 0

E00

000000

0000

)0()0()0()0()0()0(

)()()()()()(

22

51

51

33

1322

1322

6

5

4

3

2

1

6

5

4

3

2

1

rr

rrrrrr

EEEEEE

1)(2

23

2

22

21

eo nx

nxx

EnbEna

e

o

333

21

133

21

rr

a a

b

EnEn oo 132

2132

1 /1)( ; /1)( rr EE 0)()()( ; /1)( 65433

23 EEEE Ene r

EnEn

En

e

o

o

332

132

132

/1000/1000/1

rr

rE

The modified index ellipsoid is

1)(1)(1 23332

22

21132

xE

nxxE

n eo

rr

Enn

Enn eeoo

33221322

1)(

1 ; 1)(

1 rr EE

The terms r13E and r33E are usually small!EnnEnEnnEn eeeooo 33

321

133

21 )( ; )( rr

Examples 2: Tetragonal 42m crystal; uniaxial(n1 = n2 = no, n3 = ne ); assume E = (0,0,E), i.e.the electric field points along the optical axis

z(optical

axis)E

x y

a a

ne

E00

000000000000000

)0()0()0()0()0()0(

)()()()()()(

63

41

41

6

5

4

3

2

1

6

5

4

3

2

1

rr

r

EEEEEE

2

2

2

/1000/1000/1

e

o

o

nn

n

23

22

21 /1)( ; /1)( ; /1)( eoo nnn EEE

E63654 )( ; 0)()( r EEE

Ena o 633

21 r

The modified index ellipsoid is

12 21632

23

2

22

21 xEx

nx

nxx

eo

r

Rotating the coordination system 45o about the z axis. 33212211 ; 2/)( ; 2/)(let xuxxuxxu

1)()()( 2

3

23

22

22

21

21

Enu

Enu

Enu

eoo

nEnEnEn

EnEn

)( ; 1)(

1 ; 1)(

1 where 363222

63221

rr

The terms r63E is usually small!

ee

oo

oo

nEnEnnEnEnnEn

)()()(

633

21

2

633

21

1

rr

The optical properties of a Kerr medium can be determined by using similar sequence!

Properties of some electro-optic materials at 633 nm

Material Crystal symmetry r coefficient (pm/V) n Inorganic

KDP (KH2PO4)

KD*P (KD2PO4)

LiNbO3

BSN

GaAs

OrganicMNADCNP

42m

42m

3m

4mm

43m

mm

r41 = 8;r63 = 11r63 = 24

r13 = 9.6;r41 = 30.9r13 = 67;

r33 = 1340r41 = 0.97

r11 = 65r33 = 82

n0 = 1.5074ne = 1.4669

ne = 1.462n0 = 2.286ne = 2.200n0 = 2.312ne = 2.299n = 3.32

nx = 1.9; ny = 2.8

1 = 442 = 213 = 48

1 = 783 = 32

10

Modulators: Phase modulators 3

0

nLdV r

When an electric field is directed along the optical axis of 3m crystal (previous example) Longitudinal Modulator: the linearly polarized light travels along the optical axis (z) => n = no

, r = r13, and d = L. Transverse Modulator: the linearly polarized light travels along the x axis => n = ne and r = r33. Intensity Modulator

322

311

0

nnLdV rr

When an electric field is directed along the optical axis of 42m crystal; assume a longitudinal modulator (d = L)=> n1 = n2 = no, r1 = r63, r2 = -r63 3

63

0

2 onV r

Photorefractive Materials: Photorefractive materials exhibit photoconductive and electro-optic behavior => photoinduced charges => space charge distribution => internal electric field => alter the refractive index (local) => detect and store the spatial distributions of optical intensity Light shines on photorefractive material => free charge carriers (electrons or holes) generated from impurity level (Rate Power) => carriers diffuse away leaving charges of opposite sign (ions) => inhomogeneous space charge (remain in place for a period of time after the light is removed ) => localized internal field => local refractive index difference by virtue of Pockels effect!

Important photorefractive material: BaTiO3 (barium titanate), BSO, LiNbO3 (lithium niobate), KNbO3 (potassium niobate), GaAs, SBN (strontium barium niobate)

Valenceband

Conductionband

Fe2+ Fe3+a

b

c+++ ------ a: photoionization

b: diffusionc: recombination

Fe2+ Fe3++e–

Fe3+

light

ee

Simplified Theory of Photorefractivity: A photorefractive material is illuminated by light of intensity I(x) that varies in the x direction => n(x) Photogeneration: rate of photoionization G(x) I(x) and available ionization sites

)()()( xINNsxG DD

ND: number density of donor; ND+: number density of ionized donor;

s: probability of ionization; ionization cross section

Diffusion: I(x) is nonuniform => number density of electron n(x) is also nonuniform => electron diffused Recombination: the electron recombination rate R(x) n(x) and number density of ionized donors ND

+ (traps). DR NxnxR )()( R: a constant; probability of recombination;

In equilibrium, R(x)=G(x))())(()()( xGNNxsINxnxR DDDR

)()( xIN

NNsxnD

DD

R

Space charge: electron generated leaves behind a positive ionic charge; when the electron is trapped, its negative charge is deposited at a different site.

Electric field: space charge => electric field; determined by observing that in steady state the drift current and diffusion current must be balanced

dxdnTkxExneJ eBe /)()( TkD Be /e: electron mobility; D: diffusivity Einstein relation

dxdn

xneTkxE B

)(1)(

Refractive index: )()( 321 xEnxn r

Assume that (ND/ND+-1) ~ constant, independent of

x => n(x) I(x)

dxdI

xIeTkxE B

)(1)(

dxdI

xIeTknxn B

)(1

21)( 3r

Example: an intensity distribution in the form of a sinusoidal grating of period , contrast m, and mean intensity I0.

)2cos1()( 0

xmIxI

The electric field and the refractive index distributions:

)/2cos(1)/2sin()(

)/2cos(1)/2sin()(

max

max

xmxnxn

xmxExE

------

------

------

------

------

------

------

++++++

++++++

++++++

I

n

E

nRecombinationat traps

max3

21

max

max )/(2Enn

eTkmE B

r