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We saw that when the index of refraction of the incident material is greater than the transmitting material we can get total internal reflection at thecritical angle. An interesting question is "what happens at larger angles of incidence?" This actually is somewhat subtle. From simple trigonometry we knowthat $${\mathrm{cos}}{\theta}_{t}=\sqrt{1-{{\mathrm{sin}}}^{2}{\theta}_{t}}\text{.}$$ We also know from Snell's law that $${\mathrm{sin}}{\theta}_{t}=\frac{{n}_{i}}{{n}_{t}}{\mathrm{sin}}{\theta}_{i}$$ so we have $${\mathrm{cos}}{\theta}_{t}=\sqrt{1-\frac{{n}_{i}^{2}}{{n}_{t}^{2}}{{\mathrm{sin}}}^{2}{\theta}_{i}}\text{.}$$ So we see that if ${n}_{i}>{n}_{t}$ ${\mathrm{cos}}{\theta}_{t}$ can become an imaginary number! For convenience we will write this as $${\mathrm{cos}}{\theta}_{t}=i\sqrt{\frac{{n}_{i}^{2}}{{n}_{t}^{2}}{{\mathrm{sin}}}^{2}{\theta}_{i}-1}$$ Now lets write down the expression for the transmitted wave: $${E}_{t}={E}_{0t}{e}^{i(\stackrel{\u20d7}{{K}_{t}}\cdot \stackrel{\u20d7}{r}-\omega t)}$$ For simplicity we will assume that the interface lies in the $y=0$ plane and thus the $y$ direction is normal to the interface. Also, we assume the $z=0$ plane defines then plane of incidence. Then we can write $$\stackrel{\u20d7}{{K}_{t}}=({K}_{tx},{K}_{ty},0)$$ or $$\stackrel{\u20d7}{{K}_{t}}=({K}_{t}{\mathrm{sin}}{\theta}_{t},{K}_{t}{\mathrm{cos}}{\theta}_{t},0)\text{.}$$ Also $$\stackrel{\u20d7}{r}=(x,y,0)\text{.}$$
So now we can write that the wave as $${E}_{t}={E}_{0t}{e}^{i(\stackrel{\u20d7}{{K}_{t}}\cdot \stackrel{\u20d7}{r}-\omega t)}$$ $${E}_{t}={E}_{0t}{e}^{i{K}_{t}{\mathrm{sin}}{\theta}_{t}x}{e}^{i{K}_{t}{\mathrm{cos}}{\theta}_{t}y}{e}^{-i\omega t}$$ or $${E}_{t}={E}_{0t}{e}^{i{K}_{t}{\mathrm{sin}}{\theta}_{t}x}{e}^{-\sqrt{\frac{{n}_{i}^{2}}{{n}_{t}^{2}}{{\mathrm{sin}}}^{2}{\theta}_{i}-1}y}{e}^{-i\omega t}\text{.}$$ It is interesting to note the effect of the term $${e}^{-\sqrt{\frac{{n}_{i}^{2}}{{n}_{t}^{2}}{{\mathrm{sin}}}^{2}{\theta}_{i}-1}y}$$ in that expression. This is an exponential decay. The amplitude of the wave drops rapidly to zero.
So there is a transmitted wave but its amplitude drops precipitously. This is referred to as the evanescent wave.
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