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| The '''Stokes parameters''' are a set of values that describe the [[Polarization (waves)|polarization]] state of [[electromagnetic radiation]]. They were defined by [[George Gabriel Stokes]] in 1852,<ref>S. Chandrasekhar 'Radiative Transfer'', Dover Publications, New York, 1960, ISBN 0-486-60590-6, page 25</ref> as a mathematically convenient alternative to the more common description of [[coherence (physics)|incoherent]] or partially polarized radiation in terms of its total [[field strength|intensity]] (''I''), (fractional) [[degree of polarization]] (''p''), and the shape parameters of the [[polarization ellipse]].
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| ==Definitions==
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| [[File:Poincaré sphere.svg|right|thumb|The [[Poincaré sphere]] is the parametrisation of the last three Stokes' parameters in [[spherical coordinates]]]]
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| The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations below and the figure at right.
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| :<math> \begin{align}
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| S_0 &= I \\
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| S_1 &= p I \cos 2\psi \cos 2\chi\\
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| S_2 &= p I \sin 2\psi \cos 2\chi\\
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| S_3 &= p I \sin 2\chi
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| \end{align} </math>
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| Here <math>p</math>, <math>I</math>, <math>2\psi</math> and <math>2\chi</math> are the [[spherical coordinates]] of the three-dimensional vector of [[cartesian coordinates]] <math>(S_1, S_2, S_3)</math>. <math>I</math> is the total intensity of the beam, and <math>p</math> is the degree of polarization. The factor of two before <math>\psi</math> represents the fact that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before <math>\chi</math> indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped accompanied by a 90° rotation. The four Stokes parameters are sometimes denoted ''I'', ''Q'', ''U'' and ''V'', respectively.
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| If given the Stokes parameters one can solve for the [[spherical coordinates]] with the following equations:
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| :<math> \begin{align}
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| I &= S_0 \\
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| p &= \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0} \\
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| 2\psi &= \mathrm{atan} \frac{S_2}{S_1}\\
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| 2\chi &= \mathrm{atan} \frac{S_3}{\sqrt{S_1^2+S_2^2}}\\
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| \end{align} </math>
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| | |
| ===Stokes vectors===
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| The Stokes parameters are often combined into a vector, known as the '''Stokes vector''':
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| :<math>
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| \vec S \ =
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| \begin{pmatrix} S_0 \\ S_1 \\ S_2 \\ S_3\end{pmatrix}
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| =
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| \begin{pmatrix} I \\ Q \\ U \\ V\end{pmatrix}
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| </math>
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| The Stokes vector spans the [[vector space|space]] of unpolarized, partially polarized, and fully polarized light. For comparison, the [[Jones vector]] only spans the space of fully polarized light, but is more useful for problems involving [[coherence (physics)|coherent]] light. The four Stokes parameters do not form a preferred [[Basis (linear algebra)|basis]] of the space, but rather were chosen because they can be easily measured or calculated.
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| The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying [[Mueller calculus]], to obtain the Stokes vector of the light leaving the system. | |
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| ====Examples====
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| Below are shown some Stokes vectors for common states of polarization of light.
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| :{|
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| |-
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| | <math>\begin{pmatrix} 1 \\ 1 \\ 0 \\ 0\end{pmatrix}</math> || Linearly polarized (horizontal)
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| |-
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| | <math>\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0\end{pmatrix}</math> || Linearly polarized (vertical)
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| |-
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| | <math>\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0\end{pmatrix} </math> || Linearly polarized (+45°)
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| |-
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| | <math>\begin{pmatrix} 1 \\ 0 \\ -1 \\ 0\end{pmatrix}</math> || Linearly polarized (−45°)
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| |-
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| | <math>\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1\end{pmatrix} </math> || Right-hand circularly polarized
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| |-
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| | <math>\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1\end{pmatrix}</math> || Left-hand circularly polarized
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| |-
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| | <math>\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix}</math> || Unpolarized
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| |}
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| ==Alternate explanation==
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| [[File:Polarisation ellipse.svg|250px|right]]
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| A [[monochromatic]] [[plane wave]] is specified by its propagation vector, <math>\vec{k}</math>, and the complex amplitudes of the [[electric field]], <math>E_1</math> and <math>E_2</math>, in a [[Basis (linear algebra)|basis]] <math>(\hat{\epsilon}_1,\hat{\epsilon}_2)</math>. Alternatively, one may specify the propagation vector, the [[Phase (waves)|phase]], <math>\phi</math>, and the polarization state, <math>\Psi</math>, where <math>\Psi</math> is the curve traced out by the electric field in a fixed plane. The most familiar polarization states are linear and circular, which are [[Degeneracy (mathematics)|degenerate]] cases of the most general state, an [[ellipse]].
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| One way to describe polarization is by giving the [[Semi-major axis|semi-major]] and [[Semi-minor axis|semi-minor]] axes of the polarization ellipse, its orientation, and the sense of rotation (See the above figure). The Stokes parameters <math>I</math>, <math>Q</math>, <math>U</math>, and <math>V</math>, provide an alternative description of the polarization state which is experimentally convenient because each parameter corresponds to a sum or difference of measurable intensities. The next figure shows examples of the Stokes parameters in degenerate states.
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| [[File:StokesParameters.png|center]]
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| ===Definitions===
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| The Stokes parameters are defined by
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| :<math> \begin{matrix}
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| I & \equiv & \langle E_x^{2} \rangle + \langle E_y^{2} \rangle \\
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| ~ & = & \langle E_a^{2} \rangle + \langle E_b^{2} \rangle \\
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| ~ & = & \langle E_l^{2} \rangle + \langle E_r^{2} \rangle, \\
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| Q & \equiv & \langle E_x^{2} \rangle - \langle E_y^{2} \rangle, \\
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| U & \equiv & \langle E_a^{2} \rangle - \langle E_b^{2} \rangle, \\
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| V & \equiv & \langle E_l^{2} \rangle - \langle E_r^{2} \rangle.
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| \end{matrix} </math>
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| where the subscripts refer to three bases: the standard [[Cartesian coordinate system|Cartesian basis]] (<math>\hat{x},\hat{y}</math>), a Cartesian basis rotated by 45° (<math>\hat{a},\hat{b}</math>), and a circular basis (<math>\hat{l},\hat{r}</math>). The circular basis is defined so that <math>\hat{l} = (\hat{x}+i\hat{y})/\sqrt{2}</math>. The next figure shows how the signs of the Stokes parameters are determined by the helicity and the orientation of the semi-major axis of the polarization ellipse.
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| [[File:StokesParamSign1.png|center]]
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| ===Representations in fixed bases===
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| In a fixed (<math>\hat{x},\hat{y}</math>) basis, the Stokes parameters are
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| :<math> \begin{matrix} | |
| I&=&|E_x|^2+|E_y|^2, \\
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| Q&=&|E_x|^2-|E_y|^2, \\
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| U&=&2\mbox{Re}(E_xE_y^*), \\
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| V&=&-2\mbox{Im}(E_xE_y^*), \\
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| \end{matrix}
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| </math>
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| while for <math>(\hat{a},\hat{b})</math>, they are
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| :<math> \begin{matrix}
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| I&=&|E_a|^2+|E_b|^2, \\
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| Q&=&-2\mbox{Re}(E_a^{*}E_b), \\
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| U&=&|E_a|^{2}-|E_b|^{2}, \\
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| V&=&2\mbox{Im}(E_a^{*}E_b). \\
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| \end{matrix}
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| </math>
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| and for <math>(\hat{l},\hat{r})</math>, they are
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| :<math> \begin{matrix}
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| I &=&|E_l|^2+|E_r|^2, \\
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| Q&=&2\mbox{Re}(E_l^*E_r), \\
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| U & = &-2\mbox{Im}(E_l^*E_r), \\
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| V & =&|E_l|^2-|E_r|^2. \\
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| \end{matrix} </math>
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| ==Properties==
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| For purely [[monochromatic]] [[Coherence (physics)|coherent]] radiation, one can show that
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| :<math>
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| \begin{matrix}
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| Q^2+U^2+V^2 = I^2,
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| \end{matrix}
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| </math>
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| whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality:<ref>H. C. van de Hulst ''Light scattering by small particles'', Dover Publications, New York, 1981, ISBN 0-486-64228-3, page 42</ref>
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| :<math>
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| \begin{matrix}
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| Q^2+U^2+V^2 \le I^2.
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| \end{matrix}
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| </math>
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| However, we can define a total polarization intensity <math>I_p</math>, so that
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| :<math> | |
| \begin{matrix}
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| Q^{2} + U^2 +V^2 = I_p^2,
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| \end{matrix}
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| </math>
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| where <math>I_p/I</math> is the total polarization fraction.
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| Let us define the complex intensity of linear polarization to be
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| :<math>
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| \begin{matrix}
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| L & \equiv & |L|e^{i2\theta} \\
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| & \equiv & Q +iU. \\
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| \end{matrix}
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| </math>
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| Under a rotation <math>\theta \rightarrow \theta+\theta'</math> of the polarization ellipse, it can be shown that <math>I</math> and <math>V</math> are invariant, but
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| :<math>
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| \begin{matrix}
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| L & \rightarrow & e^{i2\theta'}L, \\
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| Q & \rightarrow & \mbox{Re}\left(e^{i2\theta'}L\right), \\
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| U & \rightarrow & \mbox{Im}\left(e^{i2\theta'}L\right).\\
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| \end{matrix}
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| </math>
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| With these properties, the Stokes parameters may be thought of as constituting three generalized intensities:
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| :<math>
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| \begin{matrix}
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| I & \ge & 0, \\
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| V & \in & \mathbb{R}, \\
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| L & \in & \mathbb{C}, \\
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| \end{matrix}
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| </math>
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| where <math>I</math> is the total intensity, <math>|V|</math> is the intensity of circular polarization, and <math>|L|</math> is the intensity of linear polarization. The total intensity of polarization is <math>I_p=\sqrt{|L|^2+|V|^2}</math>, and the orientation and sense of rotation are given by
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| :<math>
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| \begin{matrix}
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| \theta &=& \frac{1}{2}\arg(L), \\
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| h &=& \sgn(V). \\
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| \end{matrix}
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| </math>
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| Since <math>Q=\mbox{Re}(L)</math> and <math>U=\mbox{Im}(L)</math>, we have
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| :<math>
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| \begin{matrix}
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| |L| &=& \sqrt{Q^2+U^2}, \\
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| \theta &=& \frac{1}{2}\tan^{-1}(U/Q). \\
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| \end{matrix}
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| </math> | |
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| ==Relation to the polarization ellipse==
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| In terms of the parameters of the polarization ellipse, the Stokes parameters are
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| :<math>
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| \begin{matrix}
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| I_p & = & A^2 + B^2, \\
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| Q & = & (A^2-B^2)\cos(2\theta), \\
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| U & = & (A^2-B^2)\sin(2\theta), \\
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| V & = & 2ABh. \\
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| \end{matrix}
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| </math>
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| Inverting the previous equation gives
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| :<math>
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| \begin{matrix}
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| A & = & \sqrt{\frac{1}{2}(I_p+|L|)} \\
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| B & = & \sqrt{\frac{1}{2}(I_p-|L|)} \\
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| \theta & = & \frac{1}{2}\arg(L)\\
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| h & = & \sgn(V). \\
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| \end{matrix}
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| </math>
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| ==See also==
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| * [[Mueller calculus]]
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| * [[Jones calculus]]
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| * [[Polarization (waves)]]
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| * [[Rayleigh Sky Model]]
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| * [[Stokes operators]]
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| * [[Polarization mixing]]
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| ==Notes==
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| <references />
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| ==References==
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| * E. Collett, ''Field Guide to Polarization'', SPIE Field Guides vol. '''FG05''', SPIE (2005). ISBN 0-8194-5868-6.
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| * E. Hecht, ''Optics'', 2nd ed., Addison-Wesley (1987). ISBN 0-201-11609-X.
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| * {{cite journal | author = William H. McMaster
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| | title = Polarization and the Stokes Parameters
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| | journal = Am. J. Phys.
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| | page = 351
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| | volume = 22
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| | year = 1954
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| | doi = 10.1119/1.1933744|bibcode = 1954AmJPh..22..351M }}
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| * {{cite journal | author = William H. McMaster
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| | title = Matrix representation of polarization
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| | journal = Rev. Mod. Phys.
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| | page = 33
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| | volume = 8
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| | year = 1961
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| | doi = 10.1103/RevModPhys.33.8|bibcode = 1961RvMP...33....8M }}
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| ==External links==
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| *[http://www.atnf.csiro.au/computing/software/atca_aips/node11.html Stokes parameters and polarisation]
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| {{DEFAULTSORT:Stokes Parameters}}
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| [[Category:Polarization (waves)]]
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| [[Category:Radiometry]]
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