How can I determine transmission/reflection coefficients for light?

Question

When light rays reflect off a boundary between two materials with different indices of refraction, a lot of the sources I've seen (recently) don't discuss the relation between the amplitude (or equivalently, intensity) of the transmitted/reflected rays and the original ray. Mostly they just discuss the phase difference induced by the reflection, for instance to calculate thin film interference effects.

Is it possible to calculate the transmission coefficient $T$ and reflection coefficient $R$ based on other optical properties of the materials, such as the index of refraction? Or do they need to be looked up from a reference table?

Answer 1

In addition to Fresnel equations, and in response to your question regarding the "... relation between the amplitude of the transmitted/reflected rays and the original ray":

$$T_{\parallel}=\frac{2n_{1}\cos\theta_{i}}{n_{2}\cos\theta_{i}+n_{1}\cos\theta_{t}}A_{\parallel}$$

$$T_{\perp}=\frac{2n_{1}\cos\theta_{i}}{n_{1}\cos\theta_{i}+n_{2}\cos\theta_{t}}A_{\perp}$$

$$R_{\parallel}=\frac{n_{2}\cos\theta_{i}-n_{1}\cos\theta_{t}}{n_{2}\cos\theta_{i}+n_{1}\cos\theta_{t}}A_{\parallel}$$

$$R_{\perp}=\frac{n_{1}\cos\theta_{i}-n_{2}\cos\theta_{t}}{n_{1}\cos\theta_{i}+n_{2}\cos\theta_{t}}A_{\perp}$$

where $A_{\parallel}$ and $A_{\perp}$ is the parallel and perpendicular component of the amplitude of the electric field for the incident wave, respectively. Accordingly for the $T$ (transmitted wave) and $R$ (reflected wave). I think the notation is straightforward to understand. This set of equations are also called Fresnel equations (there are three or four representations).

Answer 2

This was intended as a comment, but for the sake of clarity, I'd better use an answer.

Regarding to the case $\mu \neq 1$, we can start using the following set of equations, which are derived from the Maxwell equations and after applying boundary conditions that demand that across the boundary the tangential components of $E$ and $H$ should be continuous.

$$\cos\theta_{i}(A_{\parallel}-R_{\parallel})=\cos\theta_{t}T_{\parallel}$$ $$A_{\perp}+R_{\perp}=T_{\perp}$$ $$\sqrt{\frac{\epsilon_{1}}{\mu_{1}}}\cos\theta_{i}(A_{\perp}-R_{\perp})=\sqrt{\frac{\epsilon_{2}}{\mu_{2}}}\cos\theta_{t}T_{\perp}$$ $$\sqrt{\frac{\epsilon_{1}}{\mu_{1}}}(A_{\parallel}+R_{\parallel}=\sqrt{\frac{\epsilon_{2}}{\mu_{2}}}T_{\parallel}$$

Then, adding together the first and fourth equation, you obtain

$$T_{\parallel}=\frac{2\cos\theta_{i}\sqrt{\epsilon_{1}\mu_{2}}}{\cos\theta_{t}\sqrt{\mu_{2}\epsilon_{1}}+\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{i}}A_{\parallel}$$

Adding the second and third equation, you have

$$T_{\perp}=\frac{2 \sqrt{\mu_{2}\epsilon_{1}}\cos\theta_{i}}{\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{t}+\cos\theta_{i}\sqrt{\mu_{2}\epsilon_{1}}}A_{\perp}$$

Accordingly for $R_{\parallel}$ and $R_{\perp}$ (in which we have to substitute the value we already found for $T_{\parallel}$ and $T_{\perp}$)

$$R_{\parallel}=\frac{\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{i}-\sqrt{\epsilon_{1}\mu_{2}}\cos\theta_{t}}{\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{i}+\sqrt{\epsilon_{1}\mu_{2}}\cos\theta_{t}}A_{\parallel}$$

$$R_{\perp}=\frac{\sqrt{\epsilon_{1}\mu_{2}}\cos\theta_{i}-\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{t}}{\sqrt{\epsilon_{1}\mu_{2}}\cos\theta_{i}+\sqrt{\epsilon_{2}\mu_{1}}\cos\theta_{t}}A_{\perp}$$

Answer 3

The equations that gives the transmission and reflection coefficients are called Fresnel equations. http://en.wikipedia.org/wiki/Fresnel_equations