When an electromagnetic wave propagates through a material, the particles of
the medium are displaced from their equilibrium positions. Positively charged
particles move in the direction of the field, while negatively charged particles
move in the direction opposite to the direction of the electric field. This
relative displacement of positively and negatively charged particles creates
dipole moments. The dipole moment per unit volume describes the induced
polarization **P** of the medium.

In a linear material the relative displacement of positively and negatively charged particles is proportional to the instantaneous magnitude of the electric field. But all real materials, when exposed to a high enough light intensity show a nonlinear response. In return the medium modifies the electromagnetic field in a nonlinear way.

When the magnitude of the external field is small, the magnitude of the
electric polarization is approximately linearly proportional to the electric
field **E**. In an isotropic material we write

**P **= ε_{0}χ_{e}**E**.

In an anisotropic material **P** and **E** can have a different direction.

When the magnitude of external electric field is large, the induced polarization has a nonlinear dependence on the electric field and can be expressed as a power series with respect to the electric field. In an isotropic material we have

**P **= ∑_{n}ε_{0}χ^{(n)}**E**^{n}.

In an anisotropic material **P** and **E**^{n} can have a different direction.

χ^{(1) }is the linear susceptibility linear absorption
and refraction. χ^{(2) }is the first non-linear term. It only is
nonzero in anisotropic materials. χ^{(3) }is the second non-linear
term, which can be nonzero in all materials.

Electric fields associated with light from ordinary non-coherent sources are
usually too small to trigger an observable nonlinear response. But the magnitude
of electric fields from pulsed laser beams focused onto a small spot can
approach the magnitude of typical internal fields in crystals. In general it
takes fields greater than of 10^{5} V/m to observe nonlinear optical
phenomena. These optical fields are easily generated by lasers. Equally
important, the coherence of the laser light makes it possible to observe many
nonlinear phenomena. When the molecules in the material respond coherently to
the laser light, even the weakest nonlinear effects can be detected.

Assume E is proportional to cos(ωt). Then P is is proportional to cos(ωt)
in a linear material. All charges oscillate with angular frequency ω and produce electromagnetic wave with angular frequency ω. In nonlinear
materials P also contains non-negligible terms that are proportional to E^{2}
and therefore to cos^{2}(ωt) = (1/2)(1 + cos(2ωt)), and maybe
terms proportional to E^{3} and therefore to cos^{3}(ωt)
= (1/4)(cos(3ωt) + 3cos(ωt)). The displacement of the charges is
a superposition of a constant displacement and oscillations with angular
frequencies ω, 2ω, and maybe 3ω. The material generates EM wave with twice or tree times the frequency of the
incident light and also a constant electric field.

**What are possible consequences?**

The **nonlinear index of refraction** is defined as a variation of the
index of refraction of the material proportional to the intensity of the light
propagating through the medium. Let us consider an optical wave
propagating through an isotropic material with inversion symmetry (χ^{(2)}
= 0). The linear susceptibility χ^{(1)} is related to the ordinary
index of refraction in the linear regime.

n_{0} = (1 + χ^{(1)})^{½} .

In the nonlinear regime the magnitude of the polarization can be expressed as:

P = ε_{0} (χ^{(1)}E + χ^{(3)} E^{3}).

This yields

D = ε_{0}E + P = ε_{0}(1 + χ^{(1)} + χ^{(3)}
E^{2})E = εE.

The index of refraction is

n = (ε/ε_{0})^{½} = (1 + χ^{(1)} + χ^{(3)}
E^{2})^{½} ~ n_{0}(1 + (χ^{(3)} E^{2})/2n_{0}^{2})
= n_{0 }+ n_{2}I,

where I is the intensity of the optical wave, I
E^{2},
and n_{2} represents the nonlinear index of refraction, which depends on
the intensity of the applied field.

The nonlinear index of refraction is responsible for self-focusing and soliton propagation in fibers.

**The electro-optic effect **is a nonlinear effect that mixes
a static field with the field of and electromagnetic wave.
If an isotropic dielectric is placed in an electric field **E** and a
beam of light is passed through the sample perpendicular to the field then
the material displays induced birefringence proportional to E^{2}.
The ordinary and the extraordinary ray propagate in the same direction
perpendicular to **E**, but the index of refraction is different for the
ordinary and extraordinary ray. Δn = n_{o}
- n_{e} = λKE^{2}, where K is the
Kerr coefficient. The **Kerr effect** is a third order effect, and it is
sometimes referred to as the quadratic electro-optic effect. It can be
generated in materials with any molecular orientation. It can be
used to construct wave plates.

Certain crystals exhibit a linear electro-optic effect. Birefringence
occurs when the material is placed in an electric field **E** and a beam
of light passes through the material parallel to the field. The
ordinary and the extraordinary ray propagate in the same direction parallel
to **E**, but the index of refraction is different for the ordinary and
extraordinary ray. The induced birefringence is proportional to E. Δn
= n_{0}K_{p}E where K_{p} is the Pockels
coefficient, and n_{0} is the index of refraction of the material
with no field applied. The
linear electro-optic effect (LEO) or **Pockels effec**t is a 2nd-order
nonlinear effect. It does not occur in isotropic media or media with
inversion symmetry.

Both the Pockels and the Kerr effects can be used to construct very fast
optical shutters (10^{-10 }s) by placing the material between crossed
polarizers. When the retardation is λ/2, the device will be
transparent. Pockels cells typically require 5 to 10 times lower voltages
than equivalent Kerr cells.

**Second harmonic generation** is a second
order nonlinear process (ω + ω --> 2ω).**
Parametric up-conversion** in a crystal can be used to convert a signal from a
low frequency ω_{1} to a high frequency ω_{3} by mixing it with a strong
laser beam of frequency ω_{2} (ω_{1} + ω_{2}
--> ω_{3}). **Parametric amplification**
involves an input signal at at ω_{1} together with an intense pump beam
at ω_{2}, ω_{2 }> ω_{1}. A photon of
the pump beam with frequency ω_{2}
interacts with a photon with frequency ω_{1} and splits into two
photons, one with frequency (ω_{2} - ω_{1}) and one
with frequency ω_{1}. The coupling is provided by the non-linearity
of the crystal. The wave at ω_{1 }is amplified. This is
accompanied by the generation of an idler beam at ω_{3} = ω_{2}
- ω_{1}. The idler beam then interacts with the pump beam to
produce additional amounts of signal and idler light. A photon of the pump
beam with frequency ω_{2}
interacts with a photon of frequency ω_{3} = ω_{2} - ω_{1} and splits into two photons, one with
frequency ω_{2} - ω_{3} = ω_{1} and one with
frequency ω_{3}.

In most materials the index of refraction depends on the wavelength of the EM
wave, EM waves with different angular frequencies ω propagate with
slightly different speeds. Consider, for example, second harmonic
generation. Light with frequency
2ω generated at some point P_{1} in the medium will not be
in phase with light with frequency 2ω generated at some other point P_{2.
}As a result, the two waves from P_{1}and P_{2 }will
interfere destructively at some point P_{3}. The problem of
destructive interference can be overcome by letting the fundamental be the
ordinary ray and the second harmonic be the extraordinary ray in an anisotropic crystal. When the direction of propagation of the
fundamental is chosen so that the index of refraction of the fundamental and
second harmonic are the same, both waves will stay in phase. The crystal then is
said to be phase-matched or index-matched. Phase matching can also be
achieved in some crystals by varying their temperature. The extraordinary index
is in general more temperature-dependent than the ordinary index. By varying the
temperature one can adjust the birefringence of the crystal until phase matching
is obtained.

**Photo-elasticity**

An isotropic material can become birefringent when placed under stress.
Under compression it becomes a negative uniaxial crystal and under tension
it becomes a positive uniaxial crystal. The direction of the stress
defines the direction of the optical axis. The induced birefringence
is proportional to the stress. Photo-elasticity can be used to study
stress patterns in complex objects, for example bridges, by building a
transparent scale model of the object.

**Faraday Rotation in solids**

If an isotropic dielectric is placed in a magnetic field
**B** and a beam
of linearly polarized light is passed through the sample in the direction of
the field then a rotation of the plane of polarization will occur. The angle
θ through which the direction of polarization is
rotated is proportional to B and the thickness of the material. The
proportionality constant is called the Verdet constant. The direction
of the rotation depends on whether light is traveling parallel or
anti-parallel to **B**.