Since the Pockels coefficients are so small, in order to get a noticeable
effect, what is often done is to apply a strong bias electric field. If 
denotes the bias field (on the order of 100V or 1kV or higher) and E
the field of a laser beam at frequency 
, then the nonlinear
component of the polarization density involves an 
term, an 
term, and an term. The first is constant, and may be lumped with
the bias field; the third is several orders of magnitude smaller than the
second ( 
). Therefore, we retain only
the factor , which is at the frequency .
Thus, the total polarization density at frequency is:
Hence, the material responds to the laser beam as if it were linear with effective dielectric constant:
For this reason, the electro-optic effect is sometimes considered a linear
effect. Its application is that we can control the dielectric constant
(equivalently, the index of refraction 
, the ratio of
the speed of light in a vacuum to the speed of light in the material) with a
bias electric field. The disadvantage is that the bias field must be large
to have a noticeable effect.
This can be used to construct an optical switch controlled by an external electric field, called a Mach-Zender interferometer, illustrated in Figure 4.1. A laser beam propagates through an optical waveguide. The optical waveguide is basically a channel of dielectric material surrounded by a substrate material of lower index of refraction n. The mechanism which confines the field can be approximated as total internal reflection (light incident from a material with larger n to lower n will be entirely reflected if the angle of incidence is sufficiently high).
The light beam is split into two components, and these components travel through waveguides of equal length before being combined at the output. Normally, both paths have equal index of refraction, so the beams undergo equal phase shifts as they propagate, and are combined constructively. Thus, the full power of the beam passes through the interferometer. However, high voltage electrodes are placed around one of the two paths. When a strong bias is applied across the electrodes, the index of refraction in that path is changed, and hence the two beams emerge from the paths with unequal phases. The bias voltage is precisely what is needed to cause perfect destructive interference, and no output beam appears at all!
