Optics - Scattered Fields in Stratified Media

1. Jan 29, 2009

In the case of scattered fields in stratified media, one can obtain expressions for the expected value and variance of the amplitude reflectance and transmittance. My question is: from this information, how do you get the diffuse reflectance and transmittance?

Please let me know what you think. Thanks.

Definitions:

1. The expected value of the amplitude reflectance is $$\mbox{E} \left[ r(f) \right]$$

2. The expected value of the amplitude transmittance is $$\mbox{E} \left[ t(f) \right]$$

3. The variance of the amplitude reflectance is $$\mbox{Var} \left[ r(f) \right]$$

$$\mbox{Var} \left[ r(f) \right] = \mbox{E} \left[ r(f) \, r(f) ^{\ast} \right] - \mbox{E} \left[ r(f) \right] \mbox{E} \left[ r(f) \right] ^{\ast}$$

4. The variance of the amplitude transmittance is $$\mbox{Var} \left[ t(f) \right]$$

$$\mbox{Var} \left[ t(f) \right] = \mbox{E} \left[ t(f) \, t(f) ^{\ast} \right] - \mbox{E} \left[ t(f) \right] \mbox{E} \left[ t(f) \right] ^{\ast}$$

where f is a random variable.

My thoughts:

A) Reflectance

In the specular case, the reflectance is obtained as follows:

$$R = r \, r^{\ast}$$

Therefore, for the diffuse case, we have:

$$R = \mbox{E} \left[ r(f) \right] \, \mbox{E} \left[ r(f) \right] ^{\ast} \pm \mbox{Var} \left[ r(f) \right]$$

B) Transmittance

B.1) TE

In the specular case, the transmittance is obtained as follows:

$$T = \Re \left\{ \frac{ \hat{n}_s \cos \hat{\theta}_s }{ \hat{n}_i \cos \hat{\theta}_i } \right\} \left| t_{\mbox{TE}} \right| ^2$$

Therefore, for the diffuse case, we have:

$$T = \Re \left\{ \frac{ \hat{n}_s \cos \hat{\theta}_s }{ \hat{n}_i \cos \hat{\theta}_i } \right\} \left| \mbox{E} \left[ t_{\mbox{TE}} (f) \right] \pm \sqrt{\mbox{Var} \left[ t_{\mbox{TE}} (f) \right]} \right| ^2$$

B.1) TM

In the specular case, the transmittance is obtained as follows:

$$T = \Re \left\{ \frac{ \hat{n}_s ^{\ast} \cos \hat{\theta}_s }{ \hat{n}_i ^{\ast} \cos \hat{\theta}_i } \right\} \left| t _{\mbox{TM}} \right| ^2$$

Therefore, for the diffuse case, we have:

$$T = \Re \left\{ \frac{ \hat{n}_s ^{\ast} \cos \hat{\theta}_s }{ \hat{n}_i ^{\ast} \cos \hat{\theta}_i } \right\} \left| \mbox{E} \left[ t_{\mbox{TM}} (f) \right] \pm \sqrt{\mbox{Var} \left[ t_{\mbox{TM}} (f) \right]} \right| ^2$$

where the subscript "i" stands for incidence medium, and "s" corresponds to the substrate.

Last edited: Jan 29, 2009