Solving Transfer Matrix for Total Internal Reflection

[Norman]

Hello,

I am trying to write the transfer matrix for surface with an s-polarized em wave incident on it. I want to look at the case of total internal reflection- but I don't know if it makes any sense to look at the transfer matrix in that case, since there is no transfer across the boundary. It is part of a problem I am working on in which I have a thin film. I can write all the parts of the matrix except the last part for total internal reflection. The wave goes from air (n=1) to a non-magnetic thin film of thickness d with a complex index of refraction n. I am trying to find the reflectance coefficient. If I can get the matrix elements for the total internal reflection I am done. Any ideas?
Cheers,
Norm

 

[Nate810]

If the incident wave is s-polarized, then total internal reflection will occur when the angle of incidence is greater than the critical angle. In this case, the transfer matrix can still be written, but the transmission coefficients across the boundary will be zero. The elements of the transfer matrix will depend on the index of refraction n, the angle of incidence θ, and the thickness d of the thin film. To calculate the reflectance coefficient, you may need to use some additional equations to determine the amplitude of the reflected wave for a given incident wave.

 

[M98Ranger]

an

Hi Norman,

Thank you for reaching out. Solving the transfer matrix for total internal reflection can be a bit tricky, but it is definitely possible. In this case, you are correct that there is no transfer across the boundary, but the transfer matrix can still be used to calculate the reflectance coefficient.

To start, we can write the transfer matrix for the air-film interface as:

M = [1, 0;
0, n]

Where n is the complex index of refraction of the film. This matrix takes into account the phase change and amplitude change of the wave as it passes through the interface.

For total internal reflection, we know that the incident angle must be greater than the critical angle, which can be calculated using Snell's law. This means that the wave will be completely reflected back into the air, with no transmission into the film. In this case, the transfer matrix for the film-air interface will become:

M' = [1, 0;
0, 1]

Note that there is no change in amplitude or phase, as the wave is not passing through the boundary.

To calculate the reflectance coefficient, we can use the following formula:

R = |M12/M22|^2

Where M12 is the element in the second row and first column of the transfer matrix, and M22 is the element in the second row and second column.

I hope this helps with your problem. Let me know if you have any further questions. Best of luck!