### Re: how to obtain circular polarization in oblique incidence phase retarder

Date: Mon Dec 25 04:21:46 2006
Posted By: Zehra Sarac, PhD, Department of Electric and Electronic Engineering
Area of science: Physics
ID: 1165838281.Ph
Message:
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Firstly, we should explain how wave plates (oblique incidence phase
retarder) works to convert light of one polarization state to another.
The first step is the understanding polarization. The light is accepted
as an electromagnetic wave, composed of an electric field(E) and a
magnetic field (H) that travel together at the same velocity and in the
same direction, k E and H are vector quantities. Thus we usually deal
only with E and define a wave’s polarization as the orientation of its E-
field.
There are basically three polarization states: linear, circular, and
elliptical. These terms describe the path traced out by the tip of the
electric-field vector as it propagates in space. The output light from a
laser is typically highly polarized, that is, it consists almost entirely
of one linear polarization. On the other hand, un-polarized light, such
as light from a light bulb, an LED, or the sun, is a random superposition
of all possible polarization states. A linearly polarized wave has two
components of electric field(E), which are in phase with each other. In
circularly polarized wave E field vector tip forms a helix or corkscrew
shape. It can be either left-handed or right handed, depending on the
clockwise or counter-clockwise nature of the rotation. For a circular
wave, two linear e-field components must be of the same magnitude. If
they are not equal in magnitude, the result is an elliptically polarized
wave. Indeed elliptical polarization is the most general case of
polarization, For instance, you can think of a circle as being a special
ellipse with equal major and minor axes. In general, an elliptically
polarized wave will be of the form:

E=Ex.(cos(wt))+Ey.cos(wt+T)

Notice that the ellipse degenerates into a linear wave when

T=-….360,-180, 0, 180, 360… (0 and 180 degrees)

The polarization is circular when

T=…-270, -90, 90, 270…. (90 degrees) and Ex and Ey is equal in magnitude.

In here we should review the properties of uniaxial crystals. When light
travels through a transparent material such as a crystal, it interacts
with the atoms in the lattice. Consequently, the speed of light inside
the crystal is slower than that in a vacuum or air, typically by a factor
between 1 and 3. The speed v varies inversely with the crystal’s
refractive index n. That is, v=c/n. The larger the refractive index, the
more the light is retarded. The amount of phase retardation (or delay) T
that a monochromatic wave acquires from traveling through the crystal is
related to its speed (n, refractive index), wavelength, and the path
length L inside the crystal.

The simplest class of crystals is those with cubic symmetry. In a cubic
crystal, all 3 crystallographic directions or axes are equivalent.
nx=ny=nz, and the crystal is optically isotropic. Regardless of how the
light is polarized with respect to the crystal, it will experience the
same refractive index and phase delay. Therefore, any polarized light,
aside from accumulating a constant phase delay, remains unchanged after
traveling through a defect-free, isotropic crystal. The Phase Retarder is
extremely useful for applications where you want to synthesize and
analyze light of different polarization states. For example, using a
quarter-wave plate, you can convert an input beam from linear
polarization to circular (or elliptical) polarization and vice versa.
Using a half-wave plate, you can continuously adjust the polarization
angle of a linearly-polarized beam.
An input beam that is normally incident on the wave plate will be
resolved into ordinary and extraordinary axis components, each with a
different refractive index. The beam that emerges has a phase-delay
difference or retardation between the axes of

If the wave plate thickness L is chosen, the retardation corresponds to
90 radians (or 90o) then it is called a quarter-wave plate. A phase shift
of T=90 will convert linearly polarized light to circular and vice versa.
Half-wave plates have 180 radians of retardation. A retardation of T=180o
will flip linearly polarized light. If the incoming beam is at angle
theta with respect to the fast axis, the light will be flipped 2.theta
around fast axis. This is especially convenient since your laser or
apparatus is often too large to rotate. Quarter and half wave are not
measures of physical thickness, rather they are in reference to a
specific wavelength. Therefore, all fixed-thickness wave plates should be
properly labeled with the wavelength of light they were designed for.
In table 1, the most common applications of phase retarders is given. In
order to follow the prescriptions in Table , you need to find the fast
and slow axes of your wave plate and then rotate the retarder so that the
input or output polarization is at the correct angle.

Table 1: Common polarization conversions using phase retarder

Input	Output
Quarter-wave
Linear, theta=45o	Right circular
Linear, theta=-45o	Left circular
Right circular	Linear, theta=-45o
Left circular	Linear, theta=45o
Linear, any theta but not 45o	Elliptical
Half-wave
Linear, angle theta	Linear, angle theta
Left circular	Right circular
Right circular	Left circular
Any wave plate
Linear, angle=0o or 90o	Unchanged

References

Born, M.; Wolf, E. 1980, Principles of Optics. Oxford: Pergammon Press

Yariv, A.; Yeh P. 1984, Optical Waves in Crystals, New York: John Wiley
and Sons.

Best Wishes

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