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Right-handed/Clockwise circularly polarized Light
(as seen from the receiver)

In electrodynamics, circular polarization (also circular polarisation) of electromagnetic radiation is a polarization such that the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. The electric vector, at one point in time, describes a helix along the direction of wave propagation (see the polarization article for pictures). The magnitude of the electric field vector is constant as it rotates. Circular polarization is a limiting case of the more general condition of elliptical polarization. The other special case is the easier-to-understand linear polarization.

Circular (and elliptical) polarization is possible because the propagating electric (and magnetic) fields can have two orthogonal components with independent amplitudes and phases (and the same frequency).

A circularly polarized wave may be resolved into two linearly polarized waves, of equal amplitude, in phase quadrature (90 degrees apart) and with their planes of polarization at right angles to each other.

1 Left/Right Handedness Conventions
2 FM radio
3 Circular dichroism
4 Circularly polarized luminescence
5 Mathematical description
6 See also
7 References

Left/Right Handedness Conventions

Circular polarization may be referred to as right handed or left handed, depending on the direction in which the electric field vector rotates. Unfortunately, two opposing historical conventions exist. In physics, astronomy, and optics, polarization is defined as seen from the receiver, such as a telescope or radio telescope.^[1] Using this convention, left or right handedness is determined by pointing ones left or right thumb towards the source and matching the curling of ones fingers to the temporal rotation of the field as it passes a given point in space.
When using this convention one can alternatively consider the sense of rotation of the field in space. If one freezes the wave in time, the handedness of the screw type nature of the field, matches the defined handedness of polarization.

When determining if the wave is clockwise or counter-clockwise circularly polarized, one again takes the point of view of the receiver and, looking towards the source, one observes the direction of the temporal rotation of the field.

As a specific example, refer to the circularly polarized wave in the illustration. It is defined to be right-handed and the helix forms a right handed screw. Note that it is the nature of all screws and helices that it doesn't matter in which direction you point your thumb when determining its handedness. It is also considered clockwise circularly polarized because for a receiver observing the wave approaching, the field rotates in the clockwise direction as it passes a given point in space. This rotation is being referred to as its "temporal" rotation.<sp> This convention matches the U.S., Federal Standard 1037C convention.^[2]

In electrical engineering however, it is more common to define polarization as seen from the source, such as from a transmitting antenna. When using this convention, left or right handedness is determined by pointing ones left or right thumb away from the source and matching the curling of ones fingers to the temporal rotation of the field.^[3] In this case the handedness of the helix does not match the defined handedness of the wave.
When determining if the wave is clockwise or counter-clockwise circularly polarized, one takes the point of view of the transmitter, and while looking in the same direction which the wave is propagating, observes the direction of the temporal rotation of the field.
This convention is in conformity with the IEEE standard.^[4]

To avoid confusion, it is good practice to specify "as seen from the receiver" (or transmitter) when discussing polarization matters. In each instance the thumb points away from the viewer.

FM radio

The term "circular polarization" is often used erroneously to describe mixed polarity signals used mostly in FM radio (87.5 to 108.0 MHz), where a vertical and a horizontal component are propagated simultaneously by a single or a combined array. This has the effect of producing greater penetration into buildings and difficult reception areas than a signal with just one plane of polarization. This would be an instance where the polarization would more appropriately be called random polarization (or simply unpolarized). See Stokes parameters.

Circular dichroism

Main article: Circular dichroism

Circular dichroism (CD), is the differential absorption of left- and right-handed circularly polarized light. Circular dichroism is the basis of a form of spectroscopy that can be used to determine the optical isomerism and secondary structure of molecules.

In general, this phenomenon will be exhibited in absorption bands of any optically active molecule. As a consequence, circular dichroism is exhibited by most biological molecules, because of the dextrorotary (e.g. some sugars) and levorotary (e.g. some amino acids) molecules they contain. Noteworthy as well is that a secondary structure will also impart a distinct CD to its respective molecules. Therefore, the alpha helix, beta sheet and random coil regions of proteins and the double helix of nucleic acids have CD spectral signatures representative of their structures.

Circularly polarized luminescence

Circularly polarized luminescence (CPL) can occur when either a luminophore or an ensemble of luminophores is chiral. The extent to which emissions are polarized is quantified in the same way it is for circular dichroism, in terms of the dissymmetry factor[1], also sometimes referred to as the anisotropy factor. This value is given by

$g_{em} \ =\ 2\left ( {\theta_\mathrm{left} - \theta_\mathrm{right} \over \theta_\mathrm{left} + \theta_\mathrm{right} } \right )$

where $θ left$ corresponds to the quantum yield of left-handed circularly polarized light, and $θ right$ to that of right-handed light. The maximum absolute value of g_em, corresponding to purely left- or right-handed circular polarization, is therefore 2. Meanwhile the smallest absolute value that g_em can achieve, corresponding to linearly polarized or unpolarized light, is zero.

Mathematical description

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is

$\mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \}$

$\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )$

where k is the wavenumber,

$\omega_{ }^{ } = c k$

is the angular frequency of the wave, and $c$ is the speed of light.

Here

$\mid \mathbf{E} \mid$

is the amplitude of the field and

$|\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}$

is the Jones vector in the x-y plane.

If $α y$ is rotated by $π / 2$ radians with respect to $α x$ and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector is

$|\psi\rangle = {1\over \sqrt{2}} \begin{pmatrix} 1 \\ \pm i \end{pmatrix} \exp \left ( i \alpha_x \right )$

where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If basis vectors are defined such that

$|R\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 1 \\ i \end{pmatrix}$

and

$|L\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 1 \\ -i \end{pmatrix}$

then the polarization state can be written in the "R-L basis" as

$|\psi\rangle = \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) |R\rangle + \left ( {\cos\theta + i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) |L\rangle = \psi_R |R\rangle + \psi_L |L\rangle$

where

$\psi_R \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right )$

and

$\psi_L \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta +i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right )$ .

References

Jackson, John D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
Born, M. and Wolf, E. (1999). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.). Cambridge University Press. ISBN 0521642221.

^ HANDBOOK OPTICS Volume I,Devices , Measurements and Properties,Michael Bass Page 272 Footnote: "Right-circularly polarized light is defined as a clockwise rotation of the electric vector when the observer is looking against the direction the wave is traveling."
^ Federal Standard 1037C Circular Polarization http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm ”Circular polarization may be referred to as "right-hand" or "left-hand," depending on whether the helix describes the thread of a right-hand or left-hand screw, respectively. “
^ Electromagnetic Waves & Antennas – S. J. Orfanidis Pg 44 "Curl the fingers of your left and right hands into a fist and point both thumbs towards the direction of propagation"
^ Electromagnetic Waves & Antennas – S. J. Orfanidis: Footnote pg45: "Most engineering texts use the IEEE convention and most physics texts, the opposite convention."

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