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In total internal reflection light inside a dense medium reflects from the boundary to a less dense medium. Since by Snell's law there is no allowed refracted ray, all energy continues along the reflected ray. In the wave picture there is an evanescent wave decaying exponentially in the thinner medium but not transmitting any energy outward.

As noted in the answer to the previous question "Why is light energy 100% reflected in total internal reflection?" this is not exactly 100% efficient. However, there was no referenced exact answer of how efficient the reflection can be. My question is basically: given a perfect crystal slab of some material with an infinite perfect vacuum above, what is the efficiency of reflection in terms of material properties?

The most obvious limit would presumably be due to material impedance: the surface wave is moving at a certain velocity, pulling local charges back and forth in such a way that some energy is lost. Unavoidable surface imperfections may also interfere. There could also be some photon tunnelling across the interface (although that moves the analysis into the quantum realm). But are there any actual measurements or theoretical calculations of how much energy loss there is to a vacuum from the slab?

Optical fibres give some hints, although the geometry is more complex and the beam is usually parallel to the fibre rather than slanted. A length constant of $\sim 10$ km is fairly normal.

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  • $\begingroup$ What do you mean by "a perfect crystal slab of some material"? Is the material dispersive and/or absorbent? If so, what kinds of absorbance are you allowing for? $\endgroup$ Feb 15, 2023 at 13:58
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    $\begingroup$ Moreover, regarding tunnelling: no, this does not "move the analysis into the quantum realm" -- it is simply a feature if the evanescent wave extends long enough to couple into some other material where propagation is allowed (called evanescent-wave coupling). But in the setting you mentioned, with an infinite vacuum above, this is ruled out. $\endgroup$ Feb 15, 2023 at 14:01
  • $\begingroup$ I am assuming an infinite slab of matter, let's say diamond for specificity. But what I really would like to know is how much loss there would be as a function of material properties. $\endgroup$ Feb 17, 2023 at 13:28
  • $\begingroup$ That doesn't answer either of the comments above, and actually contradicts the premise of the question. You presumably mean semi-infinite slab? And what is on the other side of the boundary? $\endgroup$ Feb 17, 2023 at 16:13
  • $\begingroup$ In any case, how do you quantify "how much loss" there is? If the material absorbs, there will always be losses, even in uniform propagation. So what are you comparing TIR against? Reflection from a metal mirror? (And if so, how do you account for losses there, given that those mirrors might be more lossy than TIR?) And how do you separate any losses from the TIR itself from the inevitable losses from transmission in an absorbing medium? $\endgroup$ Feb 17, 2023 at 16:15

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The evanescent wave propagates parallel to the surface with an intensity I that decays exponentially with perpendicular distance z from the surface:

$$I = I_0\exp(-z/d)$$

which from that you can calculate the efficiency.

References:

https://doi.org/10.1083/jcb.89.1.141

https://www.nature.com/articles/s41467-020-20863-0

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  • $\begingroup$ No, this is incorrect. The evanescent wave holds energy but it does not transmit any of it. $\endgroup$ Feb 15, 2023 at 13:55

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