The Evanescent Wave

Consider the following scenario

Evanescent Wave Diagram

The input wave can be written as

$LaTeX: \overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i \left(\overset{\rightharpoonup }{k}_i\cdot \overset{\rightharpoonup }{r}-\omega t\right)}$ $\overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i \left(\overset{\rightharpoonup }{k}_i\cdot \overset{\rightharpoonup }{r}-\omega t\right)}$

in this case we have

$LaTeX: \overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i \left(k_xx+k_zz-\omega t\right)}$ $\overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i \left(k_xx+k_zz-\omega t\right)}$

from the diagram we see that

$LaTeX: k_z=k_0n_t\cos \left(\theta _t\right)$ $k_z=k_0n_t\cos \left(\theta _t\right)$

from Snell's Law we have

$LaTeX: \theta _t=\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)$ $\theta _t=\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)$

inserting this we have

$LaTeX: k_z=k_0n_t\cos \left(\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)\right)=k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}$ $k_z=k_0n_t\cos \left(\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)\right)=k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}$

and similarly

$LaTeX: k_x=k_0n_t\sin \left(\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)\right)=k_0n_i\sin \left(\theta _i\right)$ $k_x=k_0n_t\sin \left(\arcsin \left(\frac{n_i}{n_t}\sin \left(\theta _i\right)\right)\right)=k_0n_i\sin \left(\theta _i\right)$

we can now insert these into our representation for the transmitted wave

$LaTeX: \overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i k_zz}e^{i k_xx}e^{-i \omega t}=\overset{\rightharpoonup }{E}_{0,t}e^{i k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}z}e^{i k_0n_i\sin \left(\theta _i\right)x}e^{-i \omega t}$ $\overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}e^{i k_zz}e^{i k_xx}e^{-i \omega t}=\overset{\rightharpoonup }{E}_{0,t}e^{i k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}z}e^{i k_0n_i\sin \left(\theta _i\right)x}e^{-i \omega t}$

rearranging this for some clarity

$LaTeX: \overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}\left[e^{i k_0n_i\sin \left(\theta _i\right)x}e^{-i \omega t}\right]e^{i k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}z}$ $\overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t}\left[e^{i k_0n_i\sin \left(\theta _i\right)x}e^{-i \omega t}\right]e^{i k_0n_t\sqrt{1-\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2}z}$

and focusing on the last term noting that in the case of total internal reflection LaTeX: n_i>n_t $n_i>n_t$ and $LaTeX: \left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2>1$ $\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2>1$ , we can write this as

$LaTeX: e^{-k_0n_t\sqrt{\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2-1}z}=e^{-\alpha z}$ $e^{-k_0n_t\sqrt{\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2-1}z}=e^{-\alpha z}$

where I've defined

$LaTeX: \alpha \equiv k_0n_t\sqrt{\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2-1}$ $\alpha \equiv k_0n_t\sqrt{\left(\frac{n_i}{n_t}\right){}^2\sin \left(\theta _i\right){}^2-1}$

as a real, positive number. Putting this all together we have

$LaTeX: \overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t} e^{i \left(k_0n_i\sin \left(\theta _i\right)x- \omega t\right)}e^{-\alpha z}$ $\overset{\rightharpoonup }{E}_t\left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{E}_{0,t} e^{i \left(k_0n_i\sin \left(\theta _i\right)x- \omega t\right)}e^{-\alpha z}$

This is a wave disturbance that moves along the interface between the two materials and decays very quickly away from the interface due to the decaying exponential in the z-direction. This is called an evanescent wave. It can be shown that the z-component of the evanescent field carries no power away from the interface although there is power carried along the interface.