Editing Stokes parameters (section)

== Relationship to Hermitian operators and quantum mixed states ==

From a geometric and algebraic point of view, the Stokes parameters stand in one-to-one correspondence with the closed, convex, 4-real-dimensional cone of nonnegative Hermitian operators on the Hilbert space '''C'''<sup>2</sup>. The parameter ''I'' serves as the trace of the operator, whereas the entries of the matrix of the operator are simple linear functions of the four parameters ''I'', ''Q'', ''U'', ''V'', serving as coefficients in a linear combination of the [[Stokes operators]]. The eigenvalues and eigenvectors of the operator can be calculated from the polarization ellipse  parameters ''I'', ''p'', ''ψ'', ''χ''.

The Stokes parameters with ''I'' set equal to 1 (i.e. the trace 1 operators) are in one-to-one correspondence with the closed unit 3-dimensional ball of [[Quantum state#Mixed states|mixed states]] (or [[density matrix|density operators]]) of the quantum space '''C'''<sup>2</sup>, whose boundary is the [[Bloch sphere]]. The [[Jones vector]]s correspond to the underlying space '''C'''<sup>2</sup>, that is, the (unnormalized) [[pure states]] of the same system. Note that the overall phase (i.e. the common phase factor between the two component waves on the two perpendicular polarization axes) is lost when passing from a pure state |φ⟩ to the corresponding mixed state |φ⟩⟨φ|, just as it is lost when passing from a Jones vector to the corresponding Stokes vector.

In the basis of horizontal polarization state <math>|H\rangle</math> and vertical polarization state <math>|V\rangle</math>, the +45° linear polarization state is <math>|+\rangle =\frac{1}{\sqrt2}(|H\rangle+|V\rangle) </math>, the -45° linear polarization state is <math>|-\rangle =\frac{1}{\sqrt2}(|H\rangle-|V\rangle) </math>, the left hand circular polarization state is <math>|L\rangle =\frac{1}{\sqrt2}(|H\rangle+i|V\rangle) </math>, and the right hand circular polarization state is <math>|R\rangle =\frac{1}{\sqrt2}(|H\rangle-i|V\rangle) </math>. It's easy to see that these states are the [[Pauli matrices#Eigenvectors and eigenvalues|eigenvectors of Pauli matrices]], and that the normalized Stokes parameters (''U/I'', ''V/I'', ''Q/I'') correspond to the coordinates of the [[Bloch sphere#Definition|Bloch vector]] (<math>a_x</math>, <math>a_y</math>, <math>a_z</math>). Equivalently, we have <math>U/I=tr\left(\rho \sigma_x \right)</math>, <math>V/I=tr\left(\rho \sigma_y \right)</math>, <math>Q/I=tr\left(\rho \sigma_z \right)</math>, where <math>\rho</math> is the [[density matrix]] of the mixed state. 

Generally, a linear polarization at angle θ has a pure quantum state <math>|\theta\rangle =\cos{\theta}|H\rangle+\sin{\theta}|V\rangle </math>; therefore, the [[transmittance]] of a [[Linear polarizer|linear polarizer/analyzer]] at angle θ for a mixed state light source with density matrix <math>\rho = \frac{1}{2}\left(I + a_x \sigma_x + a_y \sigma_y + a_z \sigma_z\right)</math> is <math>tr(\rho|\theta\rangle\langle\theta|) 
= \frac{1}{2}\left(1 + a_x \sin{2\theta} + a_z \cos{2\theta}\right)    </math>, with a maximum transmittance of <math> \frac{1}{2} (1+ \sqrt{ a_x^2 + a_z^2 }) </math> at <math>\theta_0 = \frac{1}{2}\arctan{ (a_x/a_z) }    </math> if <math>a_z > 0</math>, or at <math>\theta_0 = \frac{1}{2}\arctan{ (a_x/a_z) }+\frac{\pi}{2}    </math> if <math> a_z < 0</math>; the minimum transmittance of <math> \frac{1}{2} ( 1- \sqrt{ a_x^2 + a_z^2 }) </math> is reached at the perpendicular to the maximum transmittance direction. Here, the ratio of maximum transmittance to minimum transmittance is defined as the [[extinction ratio]] <math>ER = (1 + DOLP) / (1 - DOLP)    </math>, where the [[Degree of polarization|degree of linear polarization]] is <math>DOLP = \sqrt{ a_x^2 + a_z^2 } </math>. Equivalently, the formula for the transmittance can be rewritten as <math>A\cos^2{(\theta- \theta_0)} + B    </math>, which is an extended form of [[Malus's law]]; here, <math> A, B    </math> are both non-negative, and is related to the extinction ratio by <math>ER = (A+B)/B </math>. Two of the normalized Stokes parameters can also be calculated by <math>a_x=DOLP\sin{2\theta_0}, \,  a_z=DOLP\cos{ 2\theta_0}, \, DOLP=(ER-1)/(ER+1)  </math>.

It's also worth noting that a rotation of polarization axis by angle θ corresponds to the [[Bloch sphere#Rotation operators about the Bloch basis|Bloch sphere rotation operator]] <math>R_y (2\theta)   = 
\begin{bmatrix}
        \cos \theta  & -\sin \theta \\
         \sin \theta & \cos \theta 
    \end{bmatrix}</math>. For example, the horizontal polarization state <math>|H\rangle</math> would rotate to <math>|\theta\rangle =\cos{\theta}|H\rangle+\sin{\theta}|V\rangle </math>. The effect of a quarter-wave plate aligned to the horizontal axis is described by <math>R_z (\pi /2)= 
\begin{bmatrix} 
        e^{ -i\pi/4 } & 0 \\
         0 & e^{ +i\pi/4 }
    \end{bmatrix}</math>, or equivalently the [[Quantum logic gate#Phase shift gates|Phase gate S]], and the resulting Bloch vector becomes <math>(-a_y,a_x,a_z)</math>. With this configuration, if we perform the rotating analyzer method to measure the extinction ratio, we will be able to calculate <math>a_y</math> and also verify <math>a_z</math>. For this method to work, the fast axis and the slow axis of the waveplate must be aligned with the reference directions for the basis states.

The effect of a quarter-wave plate rotated by angle θ [[List of quantum logic gates#Rotation operator gates|can be determined]] by [[Rodrigues' rotation formula]] as <math>R_n (\pi/2)=\frac{1}{\sqrt2}I-i\frac{1}{\sqrt2} (\hat{n} \cdot \vec{\sigma} ) </math>, with <math>\hat{n}=\hat{z}\cos{2\theta}+\hat{x} \sin{2\theta}</math>. The transmittance of the resulting light through a linear polarizer (analyzer plate) along the horizontal axis can be calculated using the same Rodrigues' rotation formula and focusing on its components on <math>I</math> and <math>\sigma_z</math>:

:<math>\begin{align}
T&= tr[R_n(\pi/2) \rho R_n (- \pi/2)|H\rangle\langle H|] \\

&= \frac{1}{2}\left[ 1 + a_y \sin{2\theta} + (\hat{n}\cdot \vec{a}) \cos{2\theta}\right] \\
&= \frac{1}{2}\left[ 1 + a_y \sin{2\theta} + (a_x \sin{2\theta} + a_z \cos{2\theta}) \cos{2\theta}\right] \\
&= \frac{1}{2}\left( 1 + a_y \sin{2\theta} +DOLP\times \frac{\cos{(4\theta-2\theta_0) }+\cos{(2\theta_0) }}{2 }\right)
\end{align}   </math>

The above expression is the theory basis of many polarimeters. For [[unpolarized light]], T=1/2 is a constant. For purely circularly polarized light, T has a sinusoidal dependence on angle θ with a period of 180 degrees, and can reach absolute extinction where T=0. For purely linearly polarized light, T has a sinusoidal dependence on angle θ with a period of 90 degrees, and absolute extinction is only reachable when the original light's polarization is at 90 degrees from the polarizer (i.e. <math>a_z =-1</math>). In this configuration, <math>\theta_0=\frac{\pi}{2}</math> and <math>T=\frac{1- 
 \cos{(4\theta)}}{4} </math>, with a maximum of 1/2 at θ=45°, and an extinction point at θ=0°. This result can be used to precisely determine the fast or slow axis of a quarter-wave plate, for example, by using a [[polarizing beam splitter]] to obtain a linearly polarized light aligned to the analyzer plate and rotating the quarter-wave plate in between.

Similarly, the effect of a half-wave plate rotated by angle θ is described by <math>R_n (\pi)=-i(\hat{n} \cdot \vec{\sigma} ) </math>, which transforms the density matrix to:

:<math>\begin{align}
R_n(\pi) \rho R_n (-\pi) &= \frac{1}{2}\left(I+\vec{a}\cdot[-\vec{\sigma}+2\hat{n} (\hat{n}\cdot\vec{\sigma} )]\right) \\
&= \frac{1}{2}\left[I- \vec{a} \cdot \vec{\sigma}+2(\hat{n}\cdot\vec{a} ) (\hat{n}\cdot\vec{\sigma} )\right]
\end{align}   </math>

The above expression demonstrates that if the original light is of pure linear polarization (i.e. <math>a_y= 0 </math>), the resulting light after the half-wave plate is still of pure linear polariztion (i.e. without <math>\sigma_y </math> component) with a rotated major axis. Such rotation of the linear polarization has a sinusoidal dependence on angle θ with a period of 90 degrees.