The linear and quadratic electro-optic effects are the results of  three and four wave mixing processes, respectively.  The linear electro-optic effect (LEO) or Pockels effect is a 2nd-order nonlinear effect.  It does not occur in isotropic media or media with inversion symmetry.

Polarization components for an incident field containing two frequency components w₁ and w₂ produced by the c⁽²⁾ tensor are

P_i⁽²⁾(w')/e₀ = c_ijk⁽²⁾E_j'(w_n)E_k'(w_m)d(w_n + w_m - w').

where  w_n and w_m can take on the values ±w₁ and ±w₂ and E'(w_n) is the field strength associated with w_n.

Let the incident field contain two frequencies.  Let w₁= w be associated with an electromagnetic wave E₀ and w₂ = 0 be associated with a static field E.  Then the c⁽²⁾ tensor produces  polarization components P_i⁽²⁾(w) which are proportional to E₀E.  These polarization components vary linearly with E.

The linear electro-optic effect modifies the dielectric tensor e.

e_ij ® e_ij + e₀ c_ijk⁽²⁾ E_k

A anisotropic, uniaxial crystal is traditionally characterized by its electro-optic coefficients r_ijk defined through

e₀k_ij ® e₀k_ij + r_ijk E_k

D(e₀k_ij) = r_ijk E_k, 

where k = e^-1, i.e. the inverse of the dielectric tensor.  D = eE, E = kD.  The r_ijk are related to the c_ijk⁽²⁾ and have magnitudes on the order of 10^-12 C/N or 10^-12 m/V.

The tensor r_ijk is symmetric in the indices i and j.  It is customary to combine these two subscripts into one subscript I that assumes 6 values from 1 to 6.  So r_ijk ® r_Ik, where the following convention is used.

In general, the linear electro-optic effect changes the shape of the index ellipsoid of the anisotropic, uniaxial crystal  and rotates the ellipsoid in space.

Let

x²/n_x² + y²/n_y² + z²/n_z² = x²/n₀² + y²/n₀² + z²/n_e² = 1 

be the equation defining the index ellipsoid of the crystal when the external field is zero.  

Then 

x²/n₀² + y²/n₀² + z²/n_e² + r_ijkE_kx_ix_j = 1

is the equation of the ellipsoid when an external field is present.  (The summation convention is implied.)

For an anisotropic, uniaxial crystal like potassium dihydrogen phosphate (KDP) or lithium niobate (LiNbO₃) an external field applied along the z-direction will introduce new indices n_x' and n_y' along the x'-axis and the y'-axis, respectively.  The external field will turn the uniaxial crystal into a biaxial crystal.  Let us choose our coordinate system so that the x-axis and y-axis are rotated about the z-direction by 45^o with respect to the x'-axis and y'-axis.

For a crystal with the symmetry of KDP we find

n_x' @ n₀ + n₀³(r₆₃/2)E_z,    n_y' @ n₀ - n₀³(r₆₃/2)E_z.

An electromagnetic wave polarized along the x-direction propagating through the crystal can be decomposed into a superposition of waves polarized along x' and y' respectively.   Both component waves propagate undeflected along the z-axis.  When entering the crystal E_x' and E_y' oscillate in phase.  After passage through the crystal of length L one component will be retarded with respect to the other.

E_x' = 2^-1/2E₀ exp(i(wn_x'/c)L),   E_y' = 2^-1/2E₀ exp(i(wn_y'/c)L),   Df = (w(n_x' - n_y')/c)L.

For a crystal with the symmetry of KDP we have  Df = (w/c)n₀³r₆₃E_zL.
E_zL = V, so Df  is proportional to the voltage applied across the crystal.

Df is approximately proportional to the applied field E and depends on the electro-optic coefficients of the material.  To construct an electro-optic amplitude modulator we place an analyzer behind the crystal.  The analyzer only passes the y-component of the field and measures the intensity of the transmitted beam.

E_y = 2^-1/2(E_y' - E_x') = (1/2)E₀ exp(i(wn_x'/c)L)[exp(-i(w(n_x' - n_y')/c)L) -1] 

= (1/2)E₀ exp(i(wn_x'/c)L)[exp(-iDf) -1].

I/I₀ = sin²(Df/2).

When the argument of sin²(Df/2) is approximately p/4, then the transmitted intensity varies approximately linearly with Df.   Since n_x' and n_y' vary approximately linearly with E, this happens for a particular value if EL = V, where V is the applied voltage.  For common uniaxial crystal V has a magnitude of several kV.  V increases with wavelength.

A device based on the  process described above is a longitudinal amplitude modulator. 

We can construct a transverse amplitude modulator by letting an electromagnetic wave polarized along a direction making an angle of 45^o with the z-axis propagate through the crystal along the x' axis and measuring the intensity of the beam transmitted through the crystal with polarization rotated by 90^o with respect to the incident polarization.

The emerging beam is elliptically polarized even in the absence of an external field because of the natural birefringence.  The external field introduces an additional phase difference, which is proportional to Ed.  Let d denote the extend of the crystal along the x' direction and L the extend along the z-direction.  Then Df = (w(n_e - n_y')/c)d.  But n_e - n_y' µ E_z = V/L, so Df µ wVd/L.  By making d larger, we can reduce the applied voltage V and obtain the same phase shift.  Using integrated optics and making L very small (~mm) and d large compared to L (mm) one can obtain phase shifts of p with voltages of a few V.

Systems similar to amplitude modulators can be used as phase modulators.  If the external field is modulated sinusoidally with frequency w_m << w, then the output beam is phase modulated.  If the field is along the z-axis and the beam passes through the crystal along the z-axis as in the longitudinal  amplitude modulator we have Df µ sin(w_mt).  Interference techniques can then be used to monitor the modulation.

Longitudinal amplitude modulators are particularly useful for light beams with large cross-sectional areas.  Transverse amplitude modulators avoid the use of transparent electrodes in the light path.  In addition, the voltage required for a given retardation can be reduced by increasing the ratio of the light path length to the electrode spacing.  For longitudinal amplitude modulators the required voltage is independent of the dimensions of the crystal.

Table:  Optical Properties of KDP

Link:

Crystals which belong to twenty symmetry classes which lack a center of symmetry can show a linear electro-optic effect.  The symmetry conditions for the occurrence of this effect are exactly the same as for the occurrence of the piezoelectric effect.  The linear electro-optic effect  produces a linear change in the refractive index as a function of electric field and the piezoelectric effect  produces a linear geometric deformation as a function of electric field.  Practical electric fields (< ~20 kV) at room temperature produce only very small changes in the refractive index (on the order of 10^-4).  This index change is too small to change refraction angles noticeably, but it is large enough  to produce retardations of the order of one wavelength and therefore observable interference phenomena.

The quadratic electro-optic effect (QEO) is a third order nonlinear optical process.  Unlike the LEO effect it does not require a material without inversion symmetry but can be generated in materials with any molecular orientation. 

Polarization components for an incident field containing two frequency components w₁ and w₂ produced by the c⁽³⁾ tensor are

P_i⁽³⁾(w')/e₀ = c_ijkl⁽³⁾(E_j'(w_n)E_k'(w_m)E_l'(w_p)d(w_n + w_m + w_p - w')).

where  w_n, w_m and w_p can take on the values ±w₁ and ±w₂ and E'(w_n) is the field strength associated with w_n.

Let the incident field contain two frequencies.  Let w₁= w be associated with an electromagnetic wave E₀ and w₂ = 0 be associated with a static field E.  Then the c⁽³⁾ tensor produces  polarization components 

P_i⁽³⁾(w)/e₀ = c_ijko⁽³⁾(E_j'(w)E_k'(0)E_l'(0)d(w + 0 + 0 - w')),

which are proportional to E_oE².  These polarization components vary quadratically with E.  This is called the DC Kerr effect.  Liquids such as nitrobenzene and carbon disulfide consist of anisotropic molecules and possess particularly large Kerr coefficients, because the molecules tend to align under the influence of the electric field.

A quadratic electro-optic effect can also be produced with an optical drive.  Let w₂ = w''.  Then the c⁽³⁾ tensor produces  polarization components 

P_i⁽³⁾(w)/e₀ = c_ijko⁽³⁾(E_j'(w)E_k'(w'')E_l'(-w'')d(w + w'' - w'' - w')),

which are proportional to E₀E².  This is called the optical Kerr effect.

The QEO effect is much smaller than the LEO effect, but for isotropic materials, it is the first non-zero nonlinear optical effect, and the largest source of nonlinearity.