Editing Polarization (waves) (section)

== Polarization state{{anchor|State}} ==
{{further|Linear polarization|Circular polarization|Elliptical polarization}}
[[File:Polarisation rectiligne.gif|class=skin-invert-image|thumb|upright=0.25|Electric field oscillation]]
Polarization can be defined in terms of pure polarization states with only a coherent sinusoidal wave at one optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser (whose oscillation frequency would be typically {{val|e=15}} times faster). The field oscillates in the {{mvar|xy}}-plane, along the page, with the wave propagating in the {{mvar|z}} direction, perpendicular to the page.
The first two diagrams below trace the electric field vector over a complete cycle for linear polarization at two different orientations; these are each considered a distinct ''state of polarization'' (SOP). The linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave (as in the leftmost figure) and a vertically polarized wave of the same amplitude {{em|in the same phase}}.

<div style="float:left;width:100px">
[[File:Polarisation state - Linear polarization parallel to x axis.svg|class=skin-invert-image|center|100px]]
</div>
<div style="float:left;width:100px">
[[File:Polarisation state - Linear polarization oriented at +45deg.svg|class=skin-invert-image|center|100px]]
</div>
<div style="float:left;width:100px">
[[File:Polarisation state - Right-elliptical polarization A.svg|class=skin-invert-image|100px]]
</div>
<div style="float:left;width:100px">
[[File:Polarisation state - Right-circular polarization.svg|class=skin-invert-image|100px]]
</div>
<div style="float:left;width:100px">
[[File:Polarisation state - Left-circular polarization.svg|class=skin-invert-image|100px]]
</div>
{{Clear}}
[[File:Wave Polarisation.gif|thumb|upright=0.7|Animation showing four different polarization states and three orthogonal projections.]]
[[File:Rising circular.gif|thumb|upright=0.7|A circularly polarized wave as a sum of two linearly polarized components 90° out of phase]]
Now if one were to introduce a [[Phase (waves)#phase shift|phase shift]] in between those horizontal and vertical polarization components, one would generally obtain elliptical polarization<ref>{{cite book | last=Chandrasekhar | first=Subrahmanyan | title=Radiative Transfer | url=https://archive.org/details/radiativetransfe0000chan | url-access=registration | publisher=Dover | year=1960 | isbn=0-486-60590-6 | oclc=924844798 | page=[https://archive.org/details/radiativetransfe0000chan/page/27 27]}}</ref>  as is shown in the third figure. When the phase shift is exactly ±90°, and the amplitudes are the same, then '''circular polarization''' is produced (fourth and fifth figures). Circular polarization can be created by sending linearly polarized light through a [[quarter-wave plate]] oriented at 45° to the linear polarization to create two components of the same amplitude with the required phase shift. The superposition of the original and phase-shifted components causes a rotating electric field vector, which is depicted in the animation on the right. Note that circular or elliptical polarization can involve either a clockwise or counterclockwise rotation of the field, depending on the relative phases of the components. These correspond to distinct polarization states, such as the two circular polarizations shown above.

The orientation of the {{mvar|x}} and {{mvar|y}} axes used in this description is arbitrary. The choice of such a coordinate system and viewing the polarization ellipse in terms of the {{mvar|x}} and {{mvar|y}} polarization components, corresponds to the definition of the Jones vector (below) in terms of those [[basis functions|basis]] polarizations. Axes are selected to suit a particular problem, such as {{mvar|x}} being in the plane of incidence. Since there are separate reflection coefficients for the linear polarizations in and orthogonal to the plane of incidence (''p'' and ''s'' polarizations, see below), that choice greatly simplifies the calculation of a wave's reflection from a surface.

Any pair of [[Orthogonality|orthogonal]] polarization states may be used as basis functions, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence (optical activity) or circular dichroism.

<!-- Following sections (had been) linked from [[coherence (physics)]] and from [[decibel#examples]] and from [[axial ratio]] -->
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{{anchor|Parameterization}}

=== Polarization ellipse ===
{{Main|Polarization ellipse}}
[[File:Polarisation ellipse2.svg|class=skin-invert-image|left|200px]]
For a purely polarized monochromatic wave the electric field vector over one cycle of oscillation traces out an ellipse.
A polarization state can then be described in relation to the geometrical parameters of the ellipse, and its "handedness", that is, whether the rotation around the ellipse is clockwise or counter clockwise. One parameterization of the elliptical figure specifies the '''orientation angle''' {{mvar|ψ}}, defined as the angle between the major axis of the ellipse and the {{mvar|x}}-axis<ref name=Stetten>{{cite book | last1=Sletten | first1=Mark A. | last2=Mc Laughlin | first2=David J. | title=Encyclopedia of RF and Microwave Engineering | chapter=Radar Polarimetry | publisher=John Wiley & Sons, Inc. |editor-last=Chang|editor-first=Kai | date=2005-04-15 | isbn=978-0-471-65450-6 | doi=10.1002/0471654507.eme343 }}</ref> along with the '''ellipticity''' {{math|1=''ε'' = ''a/b''}}, the ratio of the ellipse's major to minor axis.<ref>{{cite book | chapter=6 Reflector Antennas | chapter-url=http://ww.helitavia.com/skolnik/Skolnik_chapter_6.pdf | editor-first=Merrill Ivan | editor-last=Skolnik | first1=Helmut E. | first2=Gary E. | first3=Daniel | last1=Schrank | url=http://www.geo.uzh.ch/microsite/rsl-documents/research/SARlab/GMTILiterature/PDF/Skolnik90.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.geo.uzh.ch/microsite/rsl-documents/research/SARlab/GMTILiterature/PDF/Skolnik90.pdf |archive-date=2022-10-09 |url-status=live | title=Radar Handbook | publisher=McGraw-Hill | last2=Evans | last3=Davis | year=1990 | pages=6.30, Fig 6.25 | isbn=978-0-07-057913-2}}</ref><ref>{{cite book|editor-last=Ishii|editor-first=T. Koryu|title=Handbook of Microwave Technology|url=https://books.google.com/books?id=c3fNCgAAQBAJ&pg=PA177|volume=2: Applications|year=1995|publisher=Elsevier|isbn=978-0-08-053410-7|page=177}}</ref><ref>{{cite book|last=Volakis|first=John|title=Antenna Engineering Handbook, Fourth Edition|url=https://books.google.com/books?id=bmdFAAAAYAAJ|year=2007|publisher=McGraw-Hill|isbn=9780071475747|at=Sec. 26.1 | postscript=: ''Note'': in contrast with other authors, this source initially defines ellipticity reciprocally, as the minor-to-major-axis ratio, but then goes on to say that "Although [it] is less than unity, when expressing ellipticity in decibels, the minus sign is frequently omitted for convenience", which essentially reverts to the definition adopted by other authors.}}</ref> (also known as the [[axial ratio]]). The ellipticity parameter is an alternative parameterization of an ellipse's [[Eccentricity (mathematics)#Ellipses|eccentricity]] <math display=inline>e=\sqrt{1 - b^2/a^2},</math> or the '''ellipticity angle''', <math display=inline>\chi=\arctan b/a</math> <math display=inline>=\arctan 1/\varepsilon</math> as is shown in the figure.<ref name=Stetten /> The angle {{mvar|χ}} is also significant in that the latitude (angle from the equator) of the polarization state as represented on the Poincaré sphere (see below) is equal to {{math|±2''χ''}}. The special cases of linear and circular polarization correspond to an ellipticity {{mvar|ε}} of infinity and unity (or {{mvar|χ}} of zero and 45°) respectively.

=== Jones vector ===
{{Main|Jones vector}}
Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional [[complex number|complex]] vector (the [[Jones calculus|Jones vector]]):

<math display="block"> \mathbf{e} = \begin{bmatrix}
  a_1 e^{i\theta_1} \\
  a_2 e^{i\theta_2}
\end{bmatrix}.</math>

Here {{math|''a''{{sub|1}}}} and {{math|''a''{{sub|2}}}} denote the amplitude of the wave in the two components of the electric field vector, while {{math|''θ''{{sub|1}}}} and {{math|''θ''{{sub|2}}}} represent the phases. The product of a Jones vector with a complex number of unit [[Absolute value|modulus]] gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of [[absolute phase]]. The [[Basis (linear algebra)|basis]] vectors used to represent the Jones vector need not represent linear polarization states (i.e. be [[real numbers|real]]). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero [[inner product]]. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization.

=== Coordinate frame ===
Regardless of whether polarization state is represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about the propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in [[astronomy]] the [[equatorial coordinate system]] is generally used instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the [[horizontal coordinate system]]) corresponding to due north.

==== ''s'' and ''p'' designations{{anchor|s and p}} ====
{{see also|Fresnel equations#S and P polarizations}}

[[File:E xy deformation.webm|thumb|upright=1.2|Electromagnetic vectors for {{math|'''E'''}}, {{math|'''B'''}}, and {{math|'''k'''}} with {{math|1='''E''' = '''E'''(''x'',''y'')}} along with 3 planar projections and a deformation surface of total electric field. The light is always ''s''-polarized in the {{mvar|xy}}-plane. {{mvar|θ}} is the polar angle of {{math|'''k'''}} and {{math|''φ''{{sub|'''E'''}}}} is the azimuthal angle of {{math|'''E'''}}.]]

Another coordinate system frequently used relates to the ''[[plane of incidence]]''. This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. <!--The rays in this plane are illustrated in the diagram to the right.--> The component of the electric field parallel to this plane is termed ''p-like'' (parallel) and the component perpendicular to this plane is termed ''s-like'' (from {{lang|de|senkrecht}}, German for 'perpendicular'). Polarized light with its electric field along the plane of incidence is thus denoted ''{{dfn|p-polarized}}'', while light whose electric field is normal to the plane of incidence is called ''{{dfn|s-polarized}}''. ''P''-polarization is commonly referred to as ''transverse-magnetic'' (TM), and has also been termed ''pi-polarized'' or ''{{pi}}-polarized'', or ''tangential plane polarized''. ''S''-polarization is also called ''transverse-electric'' (TE), as well as ''sigma-polarized'' or ''σ-polarized'', or ''sagittal plane polarized''.

=== Degree of polarization {{anchor|Degree}} ===
'''Degree of polarization''' ('''DOP''') is a quantity used to describe the portion of an [[electromagnetic wave]] which is polarized. {{abbr|DOP|degree of polarization}} can be calculated from the [[Stokes parameters]]. A perfectly polarized wave has a {{abbr|DOP|degree of polarization}} of 100%, whereas an unpolarized wave has a {{abbr|DOP|degree of polarization}} of 0%.  A wave which is partially polarized, and therefore can be represented by a superposition of a polarized and unpolarized component, will have a {{abbr|DOP|degree of polarization}} somewhere in between 0 and 100%. {{abbr|DOP|degree of polarization}} is calculated as the fraction of the total power that is carried by the polarized component of the wave.

{{abbr|DOP|degree of polarization}} can be used to map the [[Strain (materials science)|strain]] field in materials when considering the {{abbr|DOP|degree of polarization}} of the [[photoluminescence]]. The polarization of the photoluminescence is related to the strain in a material by way of the given material's [[photoelasticity tensor]].

{{abbr|DOP|degree of polarization}} is also visualized using the [[Poincaré sphere (optics)|Poincaré sphere]] representation of a polarized beam.  In this representation, {{abbr|DOP|degree of polarization}} is equal to the length of the [[Vector (geometric)|vector]] measured from the center of the sphere.

=== Unpolarized and partially polarized light ===
{{excerpt|Unpolarized light}}