Home | About | Table of Contents | Advanced Search | Copyright | Feedback | Privacy

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 index ellipsoid ∑³_{i = 1} (x_i²/n_i²) = 1, in the principal axis co-ordinate system, to:

where n_ij = n_i, for i = j, and 1/n_ij = 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 r_ijk or R_ijkl, depending on the material. Since r_ijk is symmetrical in i, j, and R_ijkl, 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 R_mn vary accordingly, it is more convenient to replace R_mn E_kE_l in the equation of the index ellipsoid by g_mnP_kP_l (where P = D − ε₀ E), and to express the properties of the material in terms of g_mn.

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 = n_e.

It should be noted that the half-wave voltage V_π 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 (g11 − g12)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)(r13 − r33)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

[1] = 546 nm     [2] = 550 nm.

C.Forno

Home | About | Table of Contents | Advanced Search | Copyright | Feedback | Privacy | ^ Top of Page ^

This site is hosted and maintained by the National Physical Laboratory © 2008.