Circular polarization

Handedness conventions

Circular polarization may be referred to as right-handed or left-handed, and clockwise or anti-clockwise, depending on the direction in which the electric field vector rotates. Unfortunately, two opposing historical conventions exist.

From the point of view of the source

Using this convention, polarization is defined from the point of view of the source. When using this convention, left- or right-handedness is determined by pointing one's left or right thumb away from the source, in the same direction that the wave is propagating, and matching the curling of one's fingers to the direction of the temporal rotation of the field at a given point in space. When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the source, and while looking away from the source and in the same direction of the wave's propagation, one observes the direction of the field's temporal rotation.

Using this convention, the electric field vector of a left-handed circularly polarized wave is as follows: ${\displaystyle \left(E_x,E_y,E_z\right)\propto \left(\cos \frac{2\pi }{\lambda }\left(ct-z\right),-\sin \frac{2\pi }{\lambda }\left(ct-z\right),0\right).}$

As a specific example, refer to the circularly polarized wave in the first animation. Using this convention, that wave is defined as right-handed because when one points one's right thumb in the same direction of the wave's propagation, the fingers of that hand curl in the same direction of the field's temporal rotation. It is considered clockwise circularly polarized because, from the point of view of the source, looking in the same direction of the wave's propagation, the field rotates in the clockwise direction. The second animation is that of left-handed or anti-clockwise light, using this same convention.

This convention is in conformity with the Institute of Electrical and Electronics Engineers (IEEE) standard and, as a result, it is generally used in the engineering community.

Quantum physicists also use this convention of handedness because it is consistent with their convention of handedness for a particle's spin.

Radio astronomers also use this convention in accordance with an International Astronomical Union (IAU) resolution made in 1973.

From the point of view of the receiver

In this alternative convention, polarization is defined from the point of view of the receiver. Using this convention, left- or right-handedness is determined by pointing one's left or right thumb toward the source, against the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field.

When using this convention, in contrast to the other convention, the defined handedness of the wave matches the handedness of the screw type nature of the field in space. Specifically, if one freezes a right-handed wave in time, when one curls the fingers of one's right hand around the helix, the thumb will point in the direction of progression for the helix, given the sense of rotation. Note that, in the context of the nature of all screws and helices, it does not matter in which direction you point your thumb when determining its handedness.

When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the receiver and, while looking toward the source, against the direction of propagation, one observes the direction of the field's temporal rotation.

Just as in the other convention, right-handedness corresponds to a clockwise rotation, and left-handedness corresponds to an anti-clockwise rotation.

Many optics textbooks use this second convention. It is also used by SPIE as well as the International Union of Pure and Applied Chemistry (IUPAC).

Uses of the two conventions

As stated earlier, there is significant confusion with regards to these two conventions. As a general rule, the engineering, quantum physics, and radio astronomy communities use the first convention, in which the wave is observed from the point of view of the source. In contrast, many textbooks dealing with optics and chemistry, the second convention is used, in which the light is observed from the point of view of the receiver.

To avoid confusion, it is good practice to specify "as defined from the point of view of the source" or "as defined from the point of view of the receiver" when discussing polarization matters.

The archive of the US Federal Standard 1037C proposes two contradictory conventions of handedness.

Note that the IEEE defines RHCP and LHCP the opposite as those used in optics. The IEEE 1979 Antenna Standard will show RHCP on the South Pole of the Poincaré sphere (Poincaré sphere article shows handedness as per optics convention). The IEEE defines RHCP using the right hand with thumb pointing in the direction of transmit (wave is observed from the point of view of the source), and the fingers showing the direction of rotation of the E field with time. The rationale for the opposite conventions used for optics and chemistry is that observations are usually done from the point of view of the receiver with the incoming wave traveling toward the observer, whereas for most engineers, they are assumed to be standing behind the transmitter watching the wave traveling away from them. This article is not using the IEEE 1979 Antenna Standard and is not using the +t convention typically used in IEEE work.

However, as per more recent standards publication, the IEEE defines the sense of polarisation as:

"the sense of polarization, or handedness ... is called right handed (or left handed) if the direction of rotation is clockwise (or anti-clockwise) for an observer looking in the direction of propagation"

J.D. Kraus (the inventor of the helical antenna) states "The left-handed helix responds to left-circular polarisation, and the right handed helix to right-circular polarisation (IEEE definition)". Thus a right-handed helix radiates a wave which is right-handed, the electric field vector rotating clockwise looking in the direction of propagation.

FM radio

FM broadcast radio stations sometimes employ circular polarization to improve signal penetration into buildings and vehicles. It is one example of what the International Telecommunication Union refers to as "mixed polarization", i.e. radio emissions that include both horizontally- and vertically-polarized components. In the United States, Federal Communications Commission regulations state that horizontal polarization is the standard for FM broadcasting, but that "circular or elliptical polarization may be employed if desired".

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.

Also, under the right conditions, even non-chiral molecules will exhibit magnetic circular dichroism — that is, circular dichroism induced by a magnetic field.

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, also sometimes referred to as the anisotropy factor. This value is given by:

${\displaystyle g_{em}=2\left(\frac{{\theta }_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}-{\theta }_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}{{\theta }_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}+{\theta }_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}\right)}$

where ${\displaystyle {\theta }_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}}$ corresponds to the quantum yield of left-handed circularly polarized light, and ${\displaystyle {\theta }_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}$ 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:

${\displaystyle \begin{array}{cc}\mathbf{E}(\mathbf{r},t)& =\left|\mathbf{E}\right|\mathrm{R}\mathrm{e}\left\{\mathbf{Q}|\psi ⟩\exp \left[i\left(kz-\omega t\right)\right]\right\}\\ \mathbf{B}(\mathbf{r},t)& ={\displaystyle \frac{1}{c}}\hat{\mathbf{z}}\times \mathbf{E}(\mathbf{r},t)\end{array}}$

where k is the wavenumber;

${\displaystyle \omega =ck}$

is the angular frequency of the wave; ${\displaystyle \mathbf{Q}=\left[\hat{\mathbf{x}},\hat{\mathbf{y}}\right]}$ is an orthogonal ${\displaystyle 2\times 2}$ matrix whose columns span the transverse x-y plane; and ${\displaystyle c}$ is the speed of light.

Here,

${\displaystyle \left|\mathbf{E}\right|}$

is the amplitude of the field, and

${\displaystyle |\psi ⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(\begin{array}{c}{\psi }_x\\ {\psi }_y\end{array}\right)=\left(\begin{array}{c}\cos \theta \exp \left(i{\alpha }_x\right)\\ \sin \theta \exp \left(i{\alpha }_y\right)\end{array}\right)}$

is the normalized Jones vector in the x-y plane.

If ${\displaystyle {\alpha }_y}$ is rotated by ${\displaystyle \pi /2}$ radians with respect to ${\displaystyle {\alpha }_x}$ and the x amplitude equals the y amplitude, the wave is circularly polarized. The Jones vector is:

${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ \pm i\end{array}\right)\exp \left(i{\alpha }_x\right)}$

where the plus sign indicates left circular polarization, and the minus sign indicates right 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:

${\displaystyle |\mathrm{R}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)}$

and:

${\displaystyle |\mathrm{L}⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)}$

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

${\displaystyle |\psi ⟩={\psi }_{\mathrm{R}}|\mathrm{R}⟩+{\psi }_{\mathrm{L}}|\mathrm{L}⟩}$

where:

${\displaystyle \begin{array}{cc}{\psi }_{\mathrm{R}}& \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\cos \theta +i\sin \theta \exp \left(i\delta \right)\right)\exp \left(i{\alpha }_x\right)\\ {\psi }_{\mathrm{L}}& \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{2}}\left(\cos \theta -i\sin \theta \exp \left(i\delta \right)\right)\exp \left(i{\alpha }_x\right)\end{array}}$

and:

${\displaystyle \delta ={\alpha }_y-{\alpha }_x.}$

Antennas

A number of different types of antenna elements can be used to produce circularly polarized (or nearly so) radiation; following Balanis, one can use dipole elements:

"... two crossed dipoles provide the two orthogonal field components.... If the two dipoles are identical, the field intensity of each along zenith ... would be of the same intensity. Also, if the two dipoles were fed with a 90° degree time-phase difference (phase quadrature), the polarization along zenith would be circular.... One way to obtain the 90° time-phase difference between the two orthogonal field components, radiated respectively by the two dipoles, is by feeding one of the two dipoles with a transmission line which is 1/4 wavelength longer or shorter than that of the other," p.80;

or helical elements:

"To achieve circular polarization [in axial or end-fire mode] ... the circumference C of the helix must be ... with C/wavelength = 1 near optimum, and the spacing about S = wavelength/4," p.571;

or patch elements:

"... circular and elliptical polarizations can be obtained using various feed arrangements or slight modifications made to the elements.... Circular polarization can be obtained if two orthogonal modes are excited with a 90° time-phase difference between them. This can be accomplished by adjusting the physical dimensions of the patch.... For a square patch element, the easiest way to excite ideally circular polarization is to feed the element at two adjacent edges.... The quadrature phase difference is obtained by feeding the element with a 90° power divider," p.859.

In quantum mechanics

In the quantum mechanical view, light is composed of photons. Polarization is a manifestation of the spin angular momentum of light. More specifically, in quantum mechanics, the direction of spin of a photon is tied to the handedness of the circularly polarized light, and the spin of a beam of photons is similar to the spin of a beam of particles, such as electrons. In the physics convention (from the point of view of the source), a right-handed circular polarization corresponds to a positive spin (denoted ${\displaystyle {\sigma }^{+}}$), whereas a left-handed circular polarization corresponds to a negative spin (denoted ${\displaystyle {\sigma }^{-}}$).

In nature

Only a few mechanisms in nature are known to systematically produce circularly polarized light. In 1911, Albert Abraham Michelson discovered that light reflected from the golden scarab beetle Chrysina resplendens is preferentially left-polarized. Since then, circular polarization has been measured in several other scarab beetles such as Chrysina gloriosa, as well as some crustaceans such as the mantis shrimp. In these cases, the underlying mechanism is the molecular-level helicity of the chitinous cuticle. Two species of mantis shrimp have been reported to be able to detect circular polarized light.

The bioluminescence of the larvae of fireflies is also circularly polarized, as reported in 1980 for the species Photuris lucicrescens and Photuris versicolor. For fireflies, it is more difficult to find a microscopic explanation for the polarization, because the left and right lanterns of the larvae were found to emit polarized light of opposite senses. The authors suggest that the light begins with a linear polarization due to inhomogeneities inside aligned photocytes, and it picks up circular polarization while passing through linearly birefringent tissue.

Circular polarization has been detected in light reflected from leaves and photosynthetic microbes.

Water-air interfaces provide another source of circular polarization. Sunlight that gets scattered back up towards the surface is linearly polarized. If this light is then totally internally reflected back down, its vertical component undergoes a phase shift. To an underwater observer looking up, the faint light outside Snell's window therefore is (partially) circularly polarized.

Weaker sources of circular polarization in nature include multiple scattering by linear polarizers, as in the circular polarization of starlight, and selective absorption by circularly dichroic media.

Radio emission from pulsars can be strongly circularly polarized.