# Pockels Effect

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: the phenomenon that the refractive index of a medium exhibits a modification which is proportional to the strength of an applied electric field (*linear electro-optic effect*)

The Pockels effect (first described in 1906 by the German physicist Friedrich Pockels) is the linear electro-optic effect, where the refractive index of a medium is modified in proportion to the applied electric field strength. This effect can occur only in non-centrosymmetric materials. The most important materials of this type are crystal materials such as lithium niobate (LiNbO_{3}), lithium tantalate (LiTaO_{3}), potassium di-deuterium phosphate (KD^{*}P), β-barium borate (BBO), potassium titanium oxide phosphate (KTP), and compound semiconductors such as gallium arsenide (GaAs) and indium phosphide (InP). A relatively new development is that of poled polymers, containing specifically designed organic molecules. Some of these polymers exhibit a huge nonlinearity, with nonlinear coefficients which are an order of magnitude larger than those of highly nonlinear crystals.

Mathematically, the Pockels effect is best described via the induced deformation of the index ellipsoid, which is defined by

$${\left( {\frac{1}{{{n^2}}}} \right)_1}{x^2} + {\left( {\frac{1}{{{n^2}}}} \right)_2}{y^2} + {\left( {\frac{1}{{{n^2}}}} \right)_3}{z^2} + 2{\left( {\frac{1}{{{n^2}}}} \right)_4}yz + 2{\left( {\frac{1}{{{n^2}}}} \right)_5}xz + 2{\left( {\frac{1}{{{n^2}}}} \right)_6}xy = 1$$in a Cartesian coordinate system. An electric field can now change the coefficients according to

$$\Delta {\left( {\frac{1}{{{n^2}}}} \right)_i} = \sum\limits_{j = 1}^3 {{r_{ij}}{E_j}} $$with the electro-optic tensor components <$r_{ij}$>. Note that the first index (<$i$>) runs from 1 to 6 in this contracted notation, where e.g. <$i = 4$> corresponds to the *y-z* component.

Usually, only some of the coefficients <$r_{ij}$> are nonzero, depending on the crystal symmetry and the orientation of the coordinate system with respect to the crystal axes. For example, for lithium niobate (LiNbO_{3}) or lithium tantalate (LiTaO_{3}), which belong to the symmetry group 3m, the non-zero coefficients for the commonly used coordinate system are <$r_{12} = -r_{22} = r_{61}$>, <$r_{13} = r_{23}$>, <$r_{33}$>, <$r_{42}$> = <$r_{51}$>. For application of these materials e.g. in Pockels cells for electro-optic modulators, the largest tensor element (<$r_{33}$>) is often used. Its magnitude is of the order of 30 pm/V for LiNbO_{3}, with some wavelength dependence. Most other nonlinear crystal materials (e.g. BBO) have significantly lower electro-optic coefficients of a few pm/V, whereas some electrically poled polymers exhibit substantially higher values than LiNbO_{3}.

The equation describes the change of <$n^{-2}$>, rather than directly the change of the refractive index. As the index changes are usually small, the approximation

$$\Delta \left( {\frac{1}{{{n^2}}}} \right) \approx - 2{n^{ - 3}}\Delta n\quad \Rightarrow \quad \Delta n \approx - \frac{{{n^3}}}{2}\Delta \left( {\frac{1}{{{n^2}}}} \right)$$(based on a first-order Taylor expansion) is often used.

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2024-05-10

What in the crystalline structure determines how strong the Pockels effect is?

The author's answer:

The symmetry of the crystal structure is crucial: with a too high symmetry, there is no Pockels effect at all. Further, the crystal orientation matters a lot, and the bonding structure. However, I don't know a simple criterion to judge the strength of the Pockels effect in a material.