Dipole in a Static Electric Field
To understand nonlinearoptics, it is useful to first
consider the case when the electric fields are time independent.
Furthermore, we reduce the problem to one dimension where the fields
and dipole moments point in the same direction. Thus, all quantities
can be treated as scalars.
The figure below shows the potential energy of a charge
in a material as a function of position, r, relative to some
center of force. The sharp increase of the potential at the origin acts
as a wall that keeps the charge from reaching the origin. The bottom of
the well represents the equilibrium position of the charge.
A particle attached to a spring can be represented by
this type of potential. When the spring is fully compressed, it pushes
against the wall. In the other extreme, for large enough extensions,
the spring no longer obeys Hook's law, and the charge becomes free
after the spring has unraveled and breaks.
The figure below show a charged particle attached to a
spring. A force is applied to the charge through the electric field.
The most general model for the dipole
moment, which is proportional to the extension of the spring, is a
series of the electric field,
where p_{0} is the dipole
moment with no field applied, α is called the polarizability, β the
hyperpolarizability, γ the second hyperpolarizability, etc.
Simple Model of the Linear Polarizability
When the displacement of the charge from
its equilibrium position is small, the harmonic approximation holds.
This is equivalent to invoking Hook's law.
In equilibrium, the force applied to the
charge by the electric field is balanced by the force due to the spring,
The dipole moment is defined by p=qx.
Using the above equation, this yields:
As such, the linear spring model predicts
that the polarizability is proportional to the square of the charge and
inversely proportional to the spring constant. This result agrees with
intuition: as the spring constant is made smaller, thus weakening the
restoring force of the spring, the charges moves over a larger distance
in response to an electric field, leading to a larger induced dipole
moment.
Light propagation in a material at low
intensity is governed by the polarizability  leading to
linearoptical phenomena such as refractive bending and light
absorption.
The Nonlinear Spring Model of the
Hyperpolarizability
Small deviations of a spring from Hooks law
can be modeled by adding a nonlinear term of the form,
Solving for the equilibrium position of a
charge in an electric field and in an anharmonic potential, under the
approximation that the electric field is small, we get
,
which yields the dipole moment,
.
This yields a polarizability and
hyperpolarizability,
.
The hyperpolarizability is proportional to
the secondorder spring constant, as expected.

The General Approach
In the above approach, the linear and
nonlinear response of a particular model is determined by expressing
the induced dipole moment as a series in the electric field. As such,
if the dipole moment is given as a function of the electric field, i.e.
p(E), the polarizability and hyperpolarizability can be expressed
as
The higher order terms follow the same
pattern.
When using a thermodynamic model, one first
determines the thermal average of the dipole moment in the presence of
an electric field (see below). Quantum models require the expectation
value of the dipole operator (see the tutorial on the quantum calculation of the
polarizability). Then, the polarizability and higher order terms
are determined through differentiation.
A Thermodynamic Model
As an example of the above approach, we
consider a particle in a onedimensional box, as shown below.
The thermodynamic probability density of
finding the particle at any point in the well is the same, while the
probability density outside the well is zero, as shown by the dashed
line.
In the presence of a uniform and static
electric field, the well tips, as shown below.
The probability density of finding the
charged particle is larger to the right due to the electric field bias.
As a function of temperature, T, applied electric field, E,
and particle's charge q, the probability density is given by,
.
The expectation of the dipole operator is
given by
,
which yields
.
Note that in the limit of infinite
temperature, the polarizability vanishes, as expected on the grounds of
the physical argument that when the electric field is negligible, the
charge is on average at the origin.
Exercises
 Verify the above result for the polarizability of a
particle in a 1D box and calculate the hyperpolarizability.
 Calculate the second hyperpolarizability of the
particle in a 1D box, which is defined by
 Calculate μ_{0} (the static dipole moment),
α, β, and γ for a freely rotating rigid molecule with permanent dipole
moment μ if it is in thermal equilibrium with a bath of temperature T.
 For the dynamic nonlinear spring as described in
Chapter 1 of Boyd's book on Nonlinear Optics, create a
surface plot β(ω_{1},ω_{2}) that clearly shows all of
the resonance's.
