# Evanescent wave in absorbing medium

Classic setup : Reflection and refraction with Snell-Descartes' law.

Having 2 medium (1 and 2) with the normal n being oriented from 1 to 2 and placing ourselves in the incidence plane, we assume n in the z axis, and the interface in the x axis.

We can derive the transmitted wave vector in the z direction

\begin{align*} k_z=\omega/c*\sqrt{n_1^2sin\theta_1^2-n_2^2} \end{align*}

having the case $$n_2 < n_1sin\theta$$, $$k_z$$ becomes pure imaginary, leading to evanescent wave of the form :

\begin{align*} \vec E_t = \vec E_{0t}*exp(ik_xx-i\omega t)*exp(-|k_z| z) \end{align*} And the Poynting vector is $$0$$

Yet this is assuming that $$n_1$$ and $$n_2$$ are real.

Now let's suppose that $$n_2$$ is complex. $$n=c\sqrt{\epsilon \mu}$$ with $$\epsilon = \epsilon_r + i\epsilon_i$$ and after supposing that $$\epsilon_i << \epsilon_r$$ we can obtain $$n_2=c\sqrt{\mu \epsilon_r}+i \frac{c}{2} \sqrt{\frac{\mu}{\epsilon_r} }\epsilon_i$$ After trying to get anything from $$k_z$$ with these equations, I cannot figure out what will actually change.

Will the critical angle (total internal reflection) change ? In non-absorbing conditions, we have total internal reflection producing the evanescent field, will it be coupled to the loss in this material ?

What would be the energy lost in medium 2 when in situation of total internal reflection ?

It looks like the wave will be evanescent and will also transmit power (so maybe evanescent is not the right word anymore ?)

Any hints to light my lantern would be greatly appreciated !

EDIT : After applying Fresnel equation : $$R_s = \left| \frac{n_1 \cos{\theta_i} - n_2 \cos{\theta_t}}{n_1 \cos{\theta_i} + n_2 \cos{\theta_t}} \right|^2$$

If I want to compute $$\theta_t$$ I have $$\theta_t = \arcsin{(\frac{n_1}{n_2} \sin{\theta_i})}$$

but the quantity n1/n2 becomes complex, I must be mistaken somewhere...

Are we sure that Fresnel equations are compatible with complex valued index ?

• You can apply the Fresnel equations to find reflectivity. – my2cts Jan 7 at 20:25