# Optics - Scattered Fields in Stratified Media

In summary, scattered fields in stratified media refer to electromagnetic fields that are generated when an incident wave interacts with a layered medium. The behavior of these fields differs from that in homogeneous media due to the presence of multiple layers and interfaces. Factors such as incident angle and polarization, refractive index, and material properties affect the intensity and direction of scattered fields. Various methods, such as the transfer matrix method, can be used to calculate and analyze the properties of these fields. The study of scattered fields in stratified media has applications in optics, including designing optical devices and understanding light behavior at the nanoscale.
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:
The quantities \hat{n}_i, \hat{n}_s, \hat{\theta}_i and \hat{\theta}_s are the refractive indices and angles of incidence and reflection, respectively.

In summary, to obtain the diffuse reflectance and transmittance, we need to calculate the expected values and variances of the amplitude reflectance and transmittance, and then use these values in the corresponding equations for the diffuse case. It is important to note that the expressions for diffuse reflectance and transmittance may differ depending on the type of polarization (TE or TM) and the specific conditions of the stratified media.

In summary, the diffuse reflectance and transmittance can be obtained by taking the expected value and variance of the amplitude reflectance and transmittance, and combining them with the appropriate coefficients for the TE and TM cases. These expressions allow us to better understand the behavior of scattered fields in stratified media, and can be useful in various applications such as remote sensing and optical coatings. Further analysis and experimentation may also be necessary to fully understand the behavior of diffuse reflectance and transmittance in different scenarios.

## 1. What is the definition of scattered fields in stratified media?

Scattered fields in stratified media refer to the electromagnetic fields that are generated when an incident electromagnetic wave interacts with a layered medium, such as a dielectric or a metal film. These scattered fields propagate in all directions and can be observed both inside and outside of the medium.

## 2. How is the behavior of scattered fields in stratified media different from that in homogeneous media?

The behavior of scattered fields in stratified media differs from that in homogeneous media because the presence of multiple layers introduces new boundary conditions and interfaces that affect the propagation and reflection of the fields. In homogeneous media, the fields propagate in a straight line without any changes in direction or intensity.

## 3. What factors affect the intensity and direction of scattered fields in stratified media?

The intensity and direction of scattered fields in stratified media are affected by several factors, including the incident angle and polarization of the incoming wave, the refractive index and thickness of each layer in the medium, and the properties of the materials used in the layers, such as their dielectric constant and conductivity.

## 4. How are the properties of scattered fields in stratified media calculated and analyzed?

The properties of scattered fields in stratified media can be calculated and analyzed using various techniques, such as the transfer matrix method, the boundary element method, or the finite-difference time-domain method. These methods involve solving Maxwell's equations to determine the electric and magnetic fields at different points in the medium and analyzing their behavior and interactions with the layers.

## 5. What are some applications of studying scattered fields in stratified media?

The study of scattered fields in stratified media has various applications in optics, including designing optical devices such as waveguides, filters, and sensors, understanding the behavior of light in natural and artificial photonic structures, and developing techniques for controlling and manipulating light at the nanoscale.

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