 # Dragon Notes UNDER CONSTRUCTION
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## [Der] Evanescent Waves

Assume an $$EM$$-wave incident from [#1] onto a less dense medium $$\left[{\epsilon }_1>{\epsilon }_2\right]$$. Then, $${\theta }_t>{\theta }_i$$. At a certain angle $${\theta }_i={\theta }_c$$, $${\theta }_t=\pi /2$$ (critical angle) – and the incident wave is said to be totally reflected. Now consider the case when $${\theta }_i>{\theta }_c:$$ ${\mathrm{sin} \left({\theta }_i\right)\ }>{\mathrm{sin} \left({\theta }_c\right)\ }=\sqrt{{\epsilon }_1/{\epsilon }_2}$ $\Rightarrow {\mathrm{si}\mathrm{n} \left({\theta }_t\right)\ }=\sqrt{{\epsilon }_1/{\epsilon }_2}\ {\mathrm{sin} \left({\theta }_i\right)\ }>1$ – which does not yield a real solution for $${\theta }_t$$. Although $${\mathrm{sin} \left({\theta }_t\right)\ }$$ is still real, $${\mathrm{cos} \left({\theta }_t\right)\ }$$ becomes imaginary: ${\mathrm{cos} \left({\theta }_t\right)\ }=\sqrt{1-{{\mathrm{sin}}^2 \left({\theta }_t\right)\ }}=\pm j\sqrt{\left[{\epsilon }_1/{\epsilon }_2\right]{{\mathrm{sin}}^2 \left({\theta }_i\right)\ }-1}.\ \left[1^*\right]$ In [#2], the unit vector $${\widehat{\boldsymbol{n}}}_t$$ in the direction of propagation of a typical transmitted wave is ${\widehat{\boldsymbol{n}}}_t=\widehat{\boldsymbol{x}}{\mathrm{sin} \left({\theta }_t\right)\ }+\widehat{\boldsymbol{z}}{\mathrm{cos} \left({\theta }_t\right)\ }.$ Both $${\overrightarrow{\boldsymbol{E}}}_t$$ and $${\overrightarrow{\boldsymbol{H}}}_t$$ vary spatially in accordance with the following factor $e^{-j{\beta }_2{\widehat{\boldsymbol{n}}}_t\boldsymbol{\cdot }\overrightarrow{\boldsymbol{R}}}=e^{-j{\beta }_2\left(x{\mathrm{sin} \left({\theta }_t\right)\ }+z{\mathrm{cos} \left({\theta }_t\right)\ }\right)},$ which, for $${\theta }_i>{\theta }_c$$, becomes $e^{-{\alpha }_2z}e^{-j{\beta }_{2x}x}\ \left[2^*\right],$ where ${\alpha }_2={\beta }_2\sqrt{\left({\epsilon }_1/{\epsilon }_2\right){{\mathrm{sin}}^{\mathrm{2}} \left({\theta }_i\right)\ }-1};\ {\beta }_{2x}={\beta }_2\sqrt{{\epsilon }_1/{\epsilon }_2}{\mathrm{sin} \left({\theta }_i\right)\ }.$ The + in $$[1^{*}]$$ has been abandoned as it'd lead to the impossible result of an increasing field as $$z$$ increases. We can conclude from $$[2^{*}]$$ that for $${\theta }_i>{\theta }_c$$ an evanescent wave exists along the interface (in the $$x$$-direction), which is attenuated exponentially in [#2] in the normal direction ($$z$$-direction). This wave is tightly bound to the interface and is called a surface wave. It is a non-uniform plane wave, and transmits no power into [#2]. 