Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Coherency matrix of partially polarized light incomplete?

  1. Aug 22, 2012 #1

    Wox

    User Avatar

    The electric field of quasi-monochromatic, partially polarized light can be expressed by the following random process (Goodman, Statistical optics)
    [tex]\bar{E}(t,\bar{x})=u_{x}(t,\bar{x})\bar{e}_{x}+u_{y}(t,\bar{y})\bar{e}_{y}[/tex]
    [tex]u_{x}(t,\bar{x})=\Psi_{x} e^{i(\bar{k}\cdot\bar{x}-\omega t)}[/tex]
    [tex]u_{y}(t,\bar{x})=\Psi_{y} e^{i(\bar{k}\cdot\bar{x}-\omega t)}[/tex]
    where [itex]\Psi_{x}[/itex] and [itex]\Psi_{y}[/itex] are radom phasor sums which are circular complex Gaussian random variable. The joint statistics of [itex]u_{x}=a+bi[/itex] and [itex]u_{y}=c+di[/itex] describe the polarization state. Knowing that [itex]E(u_{x})=E(u_{y})=0[/itex], the covariance matrix of these two complex is given by
    [tex]C=\begin{bmatrix}
    E(aa)&E(ac)&E(ab)&E(ad)\\
    E(ca)&E(cc)&E(cb)&E(cd)\\
    E(ba)&E(bc)&E(bb)&E(bd)\\
    E(da)&E(dc)&E(db)&E(dd)
    \end{bmatrix}=\begin{bmatrix}
    E(aa)&E(ac)&0&E(ad)\\
    E(ac)&E(cc)&E(bc)&0\\
    0&E(bc)&E(aa)&E(bd)\\
    E(ad)&0&E(bd)&E(cc)
    \end{bmatrix}[/tex]
    This matrix has [itex]6[/itex] free parameters. However, one often states that the polarization is determined by the coherency matrix
    [tex]J=\begin{bmatrix}
    E(u_{x}u_{x}^{\ast})&E(u_{x}u_{y}^{\ast})\\
    E(u_{y}u_{x}^{\ast})&E(u_{y}u_{y}^{\ast})
    \end{bmatrix}=\begin{bmatrix}
    2E(aa)&E(ac)+E(bd)+i(E(bc)-E(ad))\\
    E(ac)+E(bd)-i(E(bc)-E(ad))&2E(cc)
    \end{bmatrix}
    [/tex]
    which has only [itex]4[/itex] free parameters because two pairs of free parameters of [itex]C[/itex] are combined in two free parameters in [itex]J[/itex]. So we lost 2 degrees of freedom. Does this mean that [itex]E(ac)=E(bd)[/itex] and [itex]E(bc)=-E(ad)[/itex] or does this mean that the coherency matrix doesn't contain all information on the polarization state?
     
  2. jcsd
  3. Aug 22, 2012 #2

    Andy Resnick

    User Avatar
    Science Advisor
    Education Advisor
    2016 Award

    Your notion is slightly different than what I am familiar with, but you seem to be (re)discovering the difference between the Stokes vector/Mueller matrix form of optics, which is based on stochastic equations, and the Jones form of optics, which is based on deterministic equations. That is, the Jones calculus is good for highly coherent light (monochromatic, pure polarization state, etc), while the Mueller calculus is valid for partially polarized light.

    You can always transform the Jones calculus to the Mueller calculus, but cannot always do the converse: randomly polarized light cannot be expressed in the Jones calculus.

    Does that help?
     
  4. Aug 22, 2012 #3

    Wox

    User Avatar

    Thanks for your suggestion. However, the coherency matrix definitely treats polarized, unpolarized and partially polarized radiation, just as the Mueller matrix does (and unlike the Jones matrix). For example the coherency matrix of unpolarized radiation is
    [tex]
    J=\begin{bmatrix}
    \frac{I}{2}&0\\
    0&\frac{I}{2}
    \end{bmatrix}
    [/tex]
    I'm not sure whether the Mueller matrix contains the same information as the coherency matrix or as the covariance matrix, as I'm not that familiar with Mueller calculus. Any idea?

    As for stochastic vs. deterministic equations: the expression for the electric field contains random processes, so that makes it stochastic.
     
  5. Aug 22, 2012 #4

    Andy Resnick

    User Avatar
    Science Advisor
    Education Advisor
    2016 Award

    Wolf's 'Introduction to the Theory of Coherence and Polarization of Light' has a chapter on the 2x2 correlation matrix/coherency matrix/polarization matrix. This is indeed equivalent to the Stokes vector and Mueller matrix representation. Wolf doesn't explicitly convert one to the other but says the information is 'treated in many publications'...?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Coherency matrix of partially polarized light incomplete?
  1. Coherent Light Waves (Replies: 3)

  2. Coherent light waves (Replies: 2)

  3. Coherent Length of Light (Replies: 10)

  4. What is coherent light? (Replies: 15)

Loading...