 # Dragon Notes UNDER CONSTRUCTION
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## Evanescent Waves

Suppose an EM-wave incident from [#1] onto a less dense medium 2 $$[\epsilon_1>\epsilon_2 ]$$. Then, $$θ_t>θ_i$$. At a certain angle $$θ_i=θ_c$$, $$θ_t=\pi/2$$ (critical angle) – and the incident wave is said to be totally reflected. When $$\theta_i>\theta_c$$, both $${\overrightarrow{\boldsymbol{E}}}_t$$ and $${\overrightarrow{\boldsymbol{E}}}_t$$ vary spatially in accordance with the following factor $e^{-{\alpha }_2z}e^{-j{\beta }_{2x}x},\ \left[1^*\right]$ 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)}.\$$[𝕯]

Thus, 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]. 