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The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse.

Definitions

The Poincaré sphere is the parametrisation of the last three Stokes' parameters in spherical coordinates

The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations below and the figure at right.

$\begin{array}{cc}{S}_0& =I\\ S_1& =pI\cos 2\psi \cos 2\chi \\ S_2& =pI\sin 2\psi \cos 2\chi \\ S_3& =pI\sin 2\chi \end{array}$

Here $p$, $I$, $2\psi$ and $2\chi$ are the spherical coordinates of the three-dimensional vector of cartesian coordinates $(S_1,S_2,S_3)$. $I$ is the total intensity of the beam, and $p$ is the degree of polarization. The factor of two before $\psi$ represents the fact that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before $\chi$ 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.

If given the Stokes parameters one can solve for the spherical coordinates with the following equations:

$\begin{array}{cc}I& =S_0\\ p& =\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}\\ 2\psi & =\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\frac{S_2}{S_1}\\ 2\chi & =\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\frac{S_3}{\sqrt{S_1^2+S_2^2}}\\ \end{array}$

Stokes vectors

The Stokes parameters are often combined into a vector, known as the Stokes vector:

$\overrightarrow{S}=\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}I\\ Q\\ U\\ V\end{array}\right)$

The Stokes vector spans the 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 coherent light. The four Stokes parameters do not form a preferred basis of the space, but rather were chosen because they can be easily measured or calculated.

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.

Examples

Below are shown some Stokes vectors for common states of polarization of light.

$\left(\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right)$	Linearly polarized (horizontal)
$\left(\begin{array}{c}1\\ -1\\ 0\\ 0\end{array}\right)$	Linearly polarized (vertical)
$\left(\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right)$	Linearly polarized (+45°)
$\left(\begin{array}{c}1\\ 0\\ -1\\ 0\end{array}\right)$	Linearly polarized (−45°)
$\left(\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right)$	Right-hand circularly polarized
$\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right)$	Left-hand circularly polarized
$\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right)$	Unpolarized

Alternate explanation

A monochromatic plane wave is specified by its propagation vector, $\overrightarrow{k}$, and the complex amplitudes of the electric field, $E_1$ and $E_2$, in a basis $({\widehat{ϵ}}_1,{\widehat{ϵ}}_2)$. Alternatively, one may specify the propagation vector, the phase, $\varphi$, and the polarization state, $\Psi$, where $\Psi$ is the curve traced out by the electric field in a fixed plane. The most familiar polarization states are linear and circular, which are degenerate cases of the most general state, an ellipse.

One way to describe polarization is by giving the semi-major and semi-minor axes of the polarization ellipse, its orientation, and the sense of rotation (See the above figure). The Stokes parameters $I$, $Q$, $U$, and $V$, 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.

Definitions

The Stokes parameters are defined by

$\begin{array}{ccc}I& \equiv & ⟨E_x^2⟩+⟨E_y^2⟩\\ & =& ⟨E_a^2⟩+⟨E_b^2⟩\\ & =& ⟨E_l^2⟩+⟨E_r^2⟩,\\ Q& \equiv & ⟨E_x^2⟩-⟨E_y^2⟩,\\ U& \equiv & ⟨E_a^2⟩-⟨E_b^2⟩,\\ V& \equiv & ⟨E_l^2⟩-⟨E_r^2⟩.\end{array}$

where the subscripts refer to three bases: the standard Cartesian basis ($\hat{x},\hat{y}$), a Cartesian basis rotated by 45° ($\hat{a},\hat{b}$), and a circular basis ($\hat{l},\hat{r}$). The circular basis is defined so that $\hat{l}=(\hat{x}+i\hat{y})/\sqrt{2}$. 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.

Representations in fixed bases

In a fixed ($\hat{x},\hat{y}$) basis, the Stokes parameters are

$\begin{array}{ccc}I& =& |E_x{|}^2+|E_y{|}^2,\\ Q& =& |E_x{|}^2-|E_y{|}^2,\\ U& =& 2\text{Re}(E_x{E}_y^{\ast }),\\ V& =& -2\text{Im}(E_x{E}_y^{\ast }),\end{array}$

while for $(\hat{a},\hat{b})$, they are

$\begin{array}{ccc}I& =& |E_a{|}^2+|E_b{|}^2,\\ Q& =& -2\text{Re}(E_a^{\ast }{E}_b),\\ U& =& |E_a{|}^2-|E_b{|}^2,\\ V& =& 2\text{Im}(E_a^{\ast }{E}_b).\end{array}$

and for $(\hat{l},\hat{r})$, they are

$\begin{array}{ccc}I& =& |E_l{|}^2+|E_r{|}^2,\\ Q& =& 2\text{Re}(E_l^{\ast }{E}_r),\\ U& =& -2\text{Im}(E_l^{\ast }{E}_r),\\ V& =& |E_l{|}^2-|E_r{|}^2.\end{array}$

Properties

For purely monochromatic coherent radiation, one can show that

$\begin{array}{c}{Q}^2+U^2+V^2=I^2,\end{array}$

whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality:

$\begin{array}{c}{Q}^2+U^2+V^2\le I^2.\end{array}$

However, we can define a total polarization intensity $I_p$, so that

$\begin{array}{c}{Q}^2+U^2+V^2=I_p^2,\end{array}$

where $I_p/I$ is the total polarization fraction.

Let us define the complex intensity of linear polarization to be

$\begin{array}{ccc}L& \equiv & |L|e^{i2\theta }\\ & \equiv & Q+iU.\end{array}$

Under a rotation $\theta \to \theta +{\theta }^{\prime }$ of the polarization ellipse, it can be shown that $I$ and $V$ are invariant, but

$\begin{array}{ccc}L& \to & e^{i2{\theta }^{\prime }}L,\\ Q& \to & \text{Re}\left(e^{i2{\theta }^{\prime }}L\right),\\ U& \to & \text{Im}\left(e^{i2{\theta }^{\prime }}L\right).\end{array}$

With these properties, the Stokes parameters may be thought of as constituting three generalized intensities:

$\begin{array}{ccc}I& \ge & 0,\\ V& \in & ℝ,\\ L& \in & ℂ,\end{array}$

where $I$ is the total intensity, $|V|$ is the intensity of circular polarization, and $|L|$ is the intensity of linear polarization. The total intensity of polarization is $I_p=\sqrt{|L{|}^2+|V{|}^2}$, and the orientation and sense of rotation are given by

$\begin{array}{ccc}\theta & =& \frac{1}{2}\arg (L),\\ h& =& \mathrm{sgn}(V).\end{array}$

Since $Q=\text{Re}(L)$ and $U=\text{Im}(L)$, we have

$\begin{array}{ccc}|L|& =& \sqrt{Q^2+U^2},\\ \theta & =& \frac{1}{2}{\tan }^{-1}(U/Q).\end{array}$

Relation to the polarization ellipse

In terms of the parameters of the polarization ellipse, the Stokes parameters are

$\begin{array}{ccc}{I}_p& =& A^2+B^2,\\ Q& =& (A^2-B^2)\cos (2\theta ),\\ U& =& (A^2-B^2)\sin (2\theta ),\\ V& =& 2ABh.\end{array}$

Inverting the previous equation gives

$\begin{array}{ccc}A& =& \sqrt{\frac{1}{2}(I_p+|L|)}\\ B& =& \sqrt{\frac{1}{2}(I_p-|L|)}\\ \theta & =& \frac{1}{2}\arg (L)\\ h& =& \mathrm{sgn}(V).\end{array}$

See also

• Mueller calculus
• Jones calculus
• Polarization (waves)
• Rayleigh Sky Model
• Stokes operators
• Polarization mixing

References

• E. Collett, Field Guide to Polarization, SPIE Field Guides vol. FG05, SPIE (2005). ISBN 0-8194-5868-6.
• E. Hecht, Optics, 2nd ed., Addison-Wesley (1987). ISBN 0-201-11609-X.
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
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  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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External links

• Stokes parameters and polarisation