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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
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:
where λ0 is the free-space wavelength.
For the case of a birefringent crystal, the field E modifies the
where nij = ni, for i = j, and 1/nij = 0 for i ≠ j. 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:
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.
 = 546 nm  = 550 nm.
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