In
electrodynamics, elliptical polarization is the
polarization of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
such that the tip of the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
vector describes an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
in any fixed plane intersecting, and
normal to, the direction of propagation. An elliptically polarized wave may be resolved into two
linearly polarized waves in
phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit
chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
.
''
Circular polarization'' and ''
linear polarization'' can be considered to be special cases of ''elliptical polarization''. This terminology was introduced by
Augustin-Jean Fresnel in 1822,
[A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9 December 1822; printed in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol. 1 (1866), pp.731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis", , 2021 (open access); §§9–10.] before the electromagnetic nature of light waves was known.
Mathematical description
The
classical sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
plane wave solution of the
electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous fo ...
for the
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particl ...
fields is (
Gaussian units
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
)
:
:
for the magnetic field, where k is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
,
:
is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
of the wave propagating in the +z direction, and
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
.
Here
is the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the field and
:
is the normalized
Jones vector
In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by ''Jones matrices''. When light crosses an o ...
. This is the most complete representation of polarized electromagnetic radiation and corresponds in general to elliptical polarization.
Polarization ellipse
At a fixed point in space (or for fixed z), the electric vector
traces out an ellipse in the x-y plane. The semi-major and semi-minor axes of the ellipse have lengths A and B, respectively, that are given by
:
and
:
,
where
with the phases
and
.
The orientation of the ellipse is given by the angle
the semi-major axis makes with the x-axis. This angle can be calculated from
:
.
If
, the wave is
linearly polarized. The ellipse collapses to a straight line
) oriented at an angle
. This is the case of superposition of two simple harmonic motions (in phase), one in the x direction with an amplitude
, and the other in the y direction with an amplitude
. When
increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to ''left-handed elliptical polarization''; the semi-major axis is now oriented at an angle
. Similarly, if
becomes negative from zero, the line evolves into an ellipse that is being traced out in the clockwise direction; this corresponds to ''right-handed elliptical polarization''.
If
and
,
, i.e., the wave is
circularly polarized
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to th ...
. When
, the wave is left-circularly polarized, and when
, the wave is right-circularly polarized.
Parameterization
Any fixed polarization can be described in terms of the shape and orientation of the polarization ellipse, which is defined by two parameters: axial ratio AR and tilt angle
. The axial ratio is the ratio of the lengths of the major and minor axes of the ellipse, and is always greater than or equal to one.
Alternatively, polarization can be represented as a point on the surface of the
Poincaré sphere, with
as the
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
and
as the
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
, where
. The sign used in the argument of the
depends on the handedness of the polarization. Positive indicates left hand polarization, while negative indicates right hand polarization, as defined by IEEE.
For the special case of
circular polarization, the axial ratio equals 1 (or 0 dB) and the tilt angle is undefined. For the special case of
linear polarization, the axial ratio is infinite.
In nature
The reflected light from some beetles (e.g. ''
Cetonia aurata'') is elliptical polarized.
See also
*
Ellipsometry
*
Fresnel rhomb
*
Photon polarization
*
References
*
*
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
(1889
Théorie Mathématique de la Lumière, volume 1an
Volume 2(1892) via
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
.
* H. Poincaré (1901
Électricité et Optique : La Lumière et les Théories Électrodynamiques via Internet Archive
External links
Animation of Elliptical Polarization (on YouTube) Comparison of Elliptical Polarization with Linear and Circular Polarizations (YouTube Animation)
{{DEFAULTSORT:Elliptical Polarization
Polarization (waves)
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