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Radiative Transfer Equation: Point Source and a Thin Lens

  1. Aug 18, 2015 #1
    1. The problem statement, all variables and given/known data

    Write the Radiative Transfer Equation for an isotropic incoherent point source a distance p away from a thin lens. Assume that scattering in air can be ignored but absorption cannot be ignored.

    2. Relevant equations

    1. Radiative Transfer Equation (RTE):
    [tex]\frac{dw}{dt} = \left[ \frac{dw}{dt} \right] _{\mbox{absorption}} +\left[ \frac{dw}{dt} \right] _{\mbox{emission}} + \left[ \frac{dw}{dt} \right] _{\mbox{propagation}} + \left[ \frac{dw}{dt} \right] _{\mbox{scattering}} [/tex]
    which can be expressed as
    [tex]\frac{dw}{dt} = \left[ -\frac{c}{n} \mu w \right] _{\mbox{absorption}} +\left[ \Xi _{p,\mathcal{E}} \right] _{\mbox{emission}} + \left[ -\frac{c}{n} \, \hat{s} \cdot \nabla w \right] _{\mbox{propagation}} + \left[ Kw \right] _{\mbox{scattering}} [/tex]
    where K is an integral operator and w is the phase-space distribution function.

    2. The source term can be expressed as
    [tex]\Xi (x,z,\mathcal{E}) = \delta (x) \delta (z+p) \Xi (\mathcal{E}) [/tex]
    for a a source at z = -p relative to a lens at z = 0.

    3. The ABCD matrix of a the system is
    [tex]M_{\mbox{system}} =
    \begin{bmatrix}
    1& q \\
    0& 1 \end{bmatrix}
    \begin{bmatrix}
    1& 0 \\

    -1/f & 1 \end{bmatrix}
    \begin{bmatrix}
    1& p \\

    0& 1 \end{bmatrix} [/tex]
    where
    [tex]q = \frac{fp}{f+p}[/tex]
    via the imaging equation
    [tex] \frac{1}{p} + \frac{1}{q} = \frac{1}{f} [/tex]

    3. The attempt at a solution

    Let me go term by term:

    As the problem says, scattering is ignored, so
    [tex] \left[ Kw \right] _{\mbox{scattering}} = 0 [/tex]
    Absorption should have the same generic form:
    [tex] \left[ -\frac{c}{n} \mu w \right] _{\mbox{absorption}} [/tex]
    Emission should have the form
    [tex] \left[ \Xi _{p,\mathcal{E}} \right] _{\mbox{emission}} = \frac{1}{\mathcal{E}} \delta (x) \delta (z+p) \Xi (\mathcal{E})[/tex]
    I'm not sure about propagation. Specifically how to relate the term
    [tex] \left[ -\frac{c}{n} \, \hat{s} \cdot \nabla w \right] _{\mbox{propagation}}[/tex]
    to the radiance L. I know the radiance is conserved and that I can write
    [tex] L_{\mbox{out}} \left(
    \begin{bmatrix}
    \vec{r} \\
    \hat{s}_{\perp} \end{bmatrix} \right) = L_{\mbox{in}} \left(
    M_{\mbox{system}} \begin{bmatrix}
    \vec{r} \\
    \hat{s}_{\perp} \end{bmatrix} \right)= L_{\mbox{in}} \left(
    \begin{bmatrix}
    1& q \\

    0& 1 \end{bmatrix}
    \begin{bmatrix}
    1& 0 \\

    -1/f & 1 \end{bmatrix}
    \begin{bmatrix}
    1& p \\

    0& 1 \end{bmatrix} \begin{bmatrix}
    \vec{r} \\
    \hat{s}_{\perp} \end{bmatrix} \right)[/tex]
    So, can I write
    [tex] \left[ -\frac{c}{n} \, \hat{s} \cdot \nabla w \right] _{\mbox{propagation}} = -\frac{c}{n} L_{\mbox{in}} \left(
    M_{\mbox{system}} \begin{bmatrix}
    \vec{r} \\
    \hat{s}_{\perp} \end{bmatrix} \right)= -\frac{c}{n} L_{\mbox{in}} \left(
    \begin{bmatrix}
    1& q \\

    0& 1 \end{bmatrix}
    \begin{bmatrix}
    1& 0 \\

    -1/f & 1 \end{bmatrix}
    \begin{bmatrix}
    1& p \\

    0& 1 \end{bmatrix} \begin{bmatrix}
    \vec{r} \\
    \hat{s}_{\perp} \end{bmatrix} \right)[/tex]
    the above?

    I would appreciate any comments or suggestions that can lead me to the answer or the answer, if you just happen to know. :smile:
     
    Last edited: Aug 18, 2015
  2. jcsd
  3. Aug 23, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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