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Chapter: 2 General physics
    Section: 2.5 Radiation and optics
        SubSection: 2.5.11 Electro-optic materials

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2.5.11 Electro-optic materials

The electro-optic effect

In general, an electric field E applied to a transparent material modifies the refractive index n0 for a particular mode of propagation in accordance with the equation

   

 

1

  =  

1

  + rE + RE2 +

  . . . 

 higher terms

n2

n02


In amorphous materials or centro-symmetric crystals the coefficients of odd powers of E are zero, and the first non-zero term RE2 represents the quadratic Kerr effect, usually characterized for a particular material by the Kerr constant:

 

   

 

B  =  

Δn

  =  

n03

 R

λ0E2

2λ0


where λ0 is the free-space wavelength.

For the case of a birefringent crystal, the field E modifies the index ellipsoid3i = 1 (xi2/ni2) = 1, in the principal axis co-ordinate system, to:

   

where nij = ni, for i = j, and 1/nij = 0 for ij. From a knowledge of the change in orientation and dimensions of the index ellipsoid, as given by this equation, the field-induced birefringence for a ray of given direction and polarization may be calculated.

In practice the electro-optic effect is either predominantly linear or quadratic in E, and is therefore characterized by either rijk or Rijkl, depending on the material. Since rijk is symmetrical in i, j, and Rijkl, symmetrical in i, j and in k, l, pairs of values i, j and k, l are denoted by indices m and n respectively, which run from 1 to 6 according to the scheme:

   1 ↔ 1, 1      2 ↔ 2, 2      3 ↔ 3, 3       4 ↔ 2, 3      5 ↔ 1, 3       6 ↔ 1, 2

When measured at low frequencies (< 104 Hz), the coefficients may include contributions from the elasto-optic effects of piezo-electric and/or electrostrictive strains. The linear electro-optic effect observed at frequencies sufficiently high for these contributions to be negligible is called the Pockels effect. The strain effects may contribute up to 50% of the low-frequency effect.

The quadratic effect is exhibited most markedly by materials in which the permittivity is high and varies rapidly with temperature. Here, since the Rmn vary accordingly, it is more convenient to replace Rmn EkEl in the equation of the index ellipsoid by gmnPkPl (where P = D − ε0 E), and to express the properties of the material in terms of gmn.

Typical parameters for selected electro-optic materials at low frequencies are given in the table at the bottom of this page. Co-ordinate convention: the indicatrix and electro-optic coefficients are referred to the usual crystallographic co-ordinate system as follows. Ox3 ≡ Oz and is the fourfold axis for cubic and tetragonal symmetries, or the threefold axis for trigonal symmetry; Ox1 ≡ Ox; Ox2 ≡ Oy, except for the trigonal case in which Ox1 is perpendicular to the mirror plane. For uniaxial crystals n1 = n1 = n0, n3 = ne.

It should be noted that the half-wave voltage Vπ is used conventionally to characterize the sensitivity of an electro-optic material. It is the voltage required to obtain one half-wavelength of optical path difference between the two vibration components of a wavefront, using a cube of the material of side 1 cm with specified directions of light and applied field. The table overleaf gives the induced birefringence for a number of commonly used configurations:


Material symmetry

Light direction

Field direction

Induced birefringence Δn

 

 

 

 

any transverse Bλ0E2

2

Oz

Oz

n

3

 r63E3

0

3m

Ox

Oz

n

3

 r41E3

0

m3m

Ox

Oz

n

3

(g11g12)P

2

/2

0

3

3m

Ox

Oz

n

3

3

  

r33

  

n1

  

r13

  

E3/2 = 

 n

3

3

 r33E3

n3

4mm

45° to Oz

45° to Oz

(n

3

3

 /4√2)(r13r33)E

 

 

longitudinal

 

 

 

 

 




References

W. R. Cook and H. Jaffe (1979) Piezooptic and electrooptic constants, Landolt–Börnstein New Series, K.-H. Heilwege (ed.), Group III Vol 11, 1979, Springer-Verlag.
J. F. Nye (1964) Physical Properties of Crystals, Oxford University Press.
A. Yariv and P. Yeh (1984) Optical Waves in Crystals, John Wiley & Sons.

O.C.Jones



Properties of selected electro-optic materials at a wavelength of 633 nm

Material

Point-group
Symmetry

Electro-optic coeff.
and value: Low Freq.

High
Freq.
value

Refractive
Index

Relative
permittivity
(LF)

Relative
permittivity
(HF)

r

or

g

10−12 mV−1

m4C−2

 

 

 

 

     

C6H5O2N

B = 4.4 x 10−12mV−2

 

1.55

35.7

 

(nitrobenzene)

 

 

 

 

 

 

Pb0.814La0.124

n

3

e

 r33 − 

n

3

0

r33 = 2320[1]

 

n0 = 2.55   

 

 

    (Ti0.6Zr0.4)O3

 

 

 

 

 

 

    (PLZT)

 

 

 

 

 

 

β-Zns

3m

       r41 = 2.1

−1.6

2.35

16

 12.5

ZnSe

3m

      r41 = 2.0

   2.0

2.60

     9.1

   9.1

ZnTe

3m

        r41 = 4.04

   4.3

2.99

   10.1

 10.1

Bi12SiO20

23

      r41 = 5.0

 

2.54

 

 

KH2PO4

2m

      r41 = 8   

 

n0 = 1.5074

ε1, ε2 = 42

 44   

    (KDP)

 

      r63 = 11

 

ne = 1.4669

     ε3 = 21

21  

KD2PO4

2m

             r41 = 8.8 [1] 

 

n0 = 1.502  

ε1, ε2 = 58

 

     (KD*P)

 

         r63 = 24.1

 

ne = 1.462  

     ε3 = 50

48  

CsH2AsO4

2m

               r41 = 14.8 [2]

 

n0 = 1.572  

 

 

     (CDA)

 

               r63 = 18.2 [2]

 

ne = 1.550  

 

 

BaTiO3

m3m

g11g12 = 0.13

 

2.437

 

4500    

SrTiO3

m3m

g11g12 = 0.14

 

2.38  

 

300   

KTa0.35Nb0.65O3

m3m

            g11 = 0.136

 

2.29  

≈104

 

     (KTN)

 

              g12 = −0.038

 

 

 

 

 

 

            g44 = 0.147

 

 

 

 

Ba0.25Sr0.75 Nb2O6

4mm

r13 = 67 

 

n0 = 2.312

≈ 2 × 104

3400    

 

 

   r33 = 1340

 

ne = 2.299

 

 

 

 

r51 = 42 

 

 

 

 

 

 

rc

1090

 

 

 

LiNbO3

3m

  r13 =   9.6

  8.6

n0 = 2.286

ε1, ε2 = 78

 

 

 

  r22 =   6.8

  3.4

ne = 2.200

 

  43  

 

 

  r33 = 30.9

30.8

 

   ε3 = 32

  28  

 

 

     r51 = 32.6    

28   

 

 

 

 

 

rc = 21.1 

 

 

 

 

LiTaO3

3m

       r13  =   8.4     

  7.5

n0 = 2.176

ε1, ε2 = 51

 

 

 

   r22 = − 0.2

1

ne = 2.180

 

  41  

 

 

    r33 =  30.5

33  

 

     ε3 = 45

  43  

 

 

 rc = 22   

 

 

 

 

 

r51

20

 

 

 

Ag3AsS3

3m

  r22 =       

1.1

n0 = 3.018

 

 

 

 

 rc =      

3.4

ne = 2.739

 

 

KNbO3

2mm

  r13 =  28   

 

n1 = 2.280

 

 

 

 

     r23 =   1.3    

 

n2 = 2.329

 

 

 

 

 r33 =  64  

 

n3 = 2.169

 

 

 

 

 r42 = 380

 

 

 

 

 

 

 r51 = 105

270

 

 

 

             

 

 

 

 

 

 

 

[1] = 546 nm     [2] = 550 nm.




C.Forno

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