Radiative Transfer Equation: Point Source and a Thin Lens

[DivGradCurl]

Homework Statement

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.

Homework Equations

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

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

3. The ABCD matrix of a the system is
$$M_{\mbox{system}} = \begin{bmatrix} 1& q \\ 0& 1 \end{bmatrix} \begin{bmatrix} 1& 0 \\ -1/f & 1 \end{bmatrix} \begin{bmatrix} 1& p \\ 0& 1 \end{bmatrix}$$
where
$$q = \frac{fp}{f+p}$$
via the imaging equation
$$\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$$

The Attempt at a Solution

Let me go term by term:

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

 

[mark726]

Your approach seems correct. The propagation term can be related to the radiance by using the radiative transfer equation and the imaging equation. You can write it as:
\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}\frac{1}{\mathcal{E}}\delta(x)\delta(z+p)\int L_{\mbox{in}} \left(M_{\mbox{system}} \begin{bmatrix} \vec{r} \\ \hat{s}_{\perp} \end{bmatrix} \right) d\mathcal{E}
Using the ABCD matrix and the imaging equation, you can simplify this to:
\left[ -\frac{c}{n} \, \hat{s} \cdot \nabla w \right] _{\mbox{propagation}} = -\frac{c}{n}\frac{1}{\mathcal{E}}\delta(x)\delta(z+p)L_{\mbox{in}} \left(M_{\mbox{system}} \begin{bmatrix} \vec{r} \\ \hat{s}_{\perp} \end{bmatrix} \right) = -\frac{c}{n}\frac{1}{\mathcal{E}}\delta(x)\delta(z+p)L_{\mbox{in}} \left(\begin{bmatrix} 1& q \\ 0& 1 \end{bmatrix} \begin{bmatrix} \vec{r} \\ \hat{s}_{\perp} \end{bmatrix} \right) = -\frac{c}{n}\frac{1}{\mathcal{E}}\delta(x)\delta(z+p)L_{\mbox{in}} \left(\begin{bmatrix} \vec{r} + q\hat{s} \\ \hat{s}_{\perp} \end{bmatrix} \right)
This can be further simplified using the imaging equation