Wednesday, May 28, 2014

optics - How can I determine transmission/reflection coefficients for light?


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.



reflection/refraction diagram


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



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).


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