Before discussing non-linear effects, let us look at a mathematical description
of polarization. The electric field of any polarized beam propagating
along the z-axis may be written as
E = Exi + Eyj ,
where Ex = Axexp(i(kz-wt+fx)), Ey = Ayexp(i(kz-wt+fy)).
We can write the components as a column vector, which is called a Jones vector.
We may factor out and the dependence on z and t and just write
.
The intensity of the beam is proportional to Ax2 + Ay2. The most general Jones vector of a polarized beam propagating along the z-axis is given by the above equation.
The Jones vector for horizontally polarized light is given by
.
Similarly, the Jones vector for vertically polarized light is given by
.
The normalized Jones vector for light polarized at 45o is given by
,
where Ax = Ay = A = 1 and fx = fy = f = p/4.
The normalized Jones vector for right-hand circularly polarized light is given by
.
For the normalized vectors we have
.
If a polarized beam with field vector E is incident on a polarization-changing medium such as a polarizer or a wave plate, and the result is a beam in another polarization state given by E' with E'x = m11Ex + m12Ey, E'y = m21Ex + m22Ey, then we may write
,
where the 2 by 2 transformation matrix is called the Jones matrix. The table below lists the Jones matrices for common optical elements.
| Optical Element | Jones matrix |
| horizontal linear polarizer |
|
| vertical linear polarizer |
|
| linear polarizer at q |
|
| quarter wave plate (fast axis vertical) |
|
| quarter wave plate (fast axis horizontal) |
|
If we require the Jones matrix for an optical element which has been rotated through an angle q with respect to the direction given in the table above, we must multiply the above matrix by the usual matrices for rotation.
M(q) = R(q) M R(-q), where
.
To find the Jones matrix for a sequence of polarization transformations, for example a linear polarizer followed by a quarter wave plate, we simply multiply the individual Jones matrices together in the correct order. If an incident beam of light with field vector E passes through a sequence of four polarizing elements, M1 followed by M2, M3 and M4, then the resultant field vector E' is given by
E' = M4 M3 M2 M1 E.
The eigenvectors of the Jones matrix M of an optical device correspond to the polarization states which propagate through the optical device represented by M unchanged. The beam enters and emerges in the same polarization state. The eigenvalues, which in general are complex numbers, tell us about changes in amplitude and phase produced by the optical device.
Link: The Jones Calculus
Nonlinear Materials
When an electromagnetic wave propagates through a material, the electric field induces a time-varying polarization in the medium. In a linear medium the magnitude of the induced polarization is proportional to the amplitude of the electric field. In nonlinear optical materials this is not the case, especially when the amplitude of the electric field is large. An intense light beam, such as a laser beam, propagating through a nonlinear optical material will produce effects that cannot be observed with weak light beams. For example, it can generate harmonics of the original light frequency. A red beam from a ruby laser can produce an ultraviolet beam as it passes through the nonlinear optical material, while itself still propagating as a red beam.
 | 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 q 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.
|
| Kerr Effect 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 E2. 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. Dn = no - ne = lKE2, 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.
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| Pockels Effect 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 proportional to E. Dn = n0KpE where Kp is the Pockels coefficient, and n0 is the index of refraction of the material with no field applied. 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. |
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 l/2, the device will be transparent. Pockels cells typically require 5 to 10 times lower voltages than equivalent Kerr cells.