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DivGradCurl

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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.

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

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

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

[tex]\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} [/tex]

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

[tex]\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} [/tex]

where f is a random variable.

A) Reflectance

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

[tex]R = r \, r^{\ast}[/tex]

Therefore, for the diffuse case, we have:

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

B) Transmittance

B.1) TE

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

[tex]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 [/tex]

Therefore, for the diffuse case, we have:

[tex]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 [/tex]

B.1) TM

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

[tex]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 [/tex]

Therefore, for the diffuse case, we have:

[tex]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 [/tex]

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

Please let me know what you think. Thanks.

__Definitions:__1. The expected value of the amplitude reflectance is [tex]\mbox{E} \left[ r(f) \right][/tex]

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

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

[tex]\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} [/tex]

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

[tex]\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} [/tex]

where f is a random variable.

__My thoughts:__A) Reflectance

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

[tex]R = r \, r^{\ast}[/tex]

Therefore, for the diffuse case, we have:

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

B) Transmittance

B.1) TE

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

[tex]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 [/tex]

Therefore, for the diffuse case, we have:

[tex]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 [/tex]

B.1) TM

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

[tex]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 [/tex]

Therefore, for the diffuse case, we have:

[tex]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 [/tex]

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

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