# Reflection and transmission coefficients

Suppose we have a plane wave with s-polarization travelling through a medium with refractive index $n_1$ in direction $\vec{k}$ perpendicular to a surface of a dieletric with refractive index $n_2$; at a distance $L$ fram the surface of separation between $n_1$ and $n_2$ there is a third dielectric $n_3$ (all the separation surfaces are parallel to each other). Suppose $n_3<n_1,n_2$, but we don't know any relation between $n_1$ and $n_2$. Suppose every medium is lossless, so all refractive indeces are real quantities. (see figure below)

At each of the separation surfaces, the electric field is in part reflected and in part transmitted, so we define the quantities $r_i,t_i$ ($i=1,2$) by $$\begin{cases} E_{r_i}=r_iE_{in}\\ E_{t_i}=t_iE_{in} \end{cases}$$ In our case, these coefficients are related to the refractive indeces by \begin{aligned} & r_1=\frac{n_1-n_2}{n_1+n_2}\\ & t_1=\frac{2n_1}{n_1+n_2}=1+r_1\\ & r_2=\frac{n_2-n_3}{n_2+n_3}\\ & t_2=\frac{2n_1}{n_1+n_2}=1+r_2\\ \end{aligned}

In the case $n_1>n_2$, we get that $r_1>0$, so $t_1>1$, so by definition $E_{t,1}>E_{in}$. Can this happen and why?

By considering all possible internal reflections, how can you evaluate the total reflection and transmission coefficients $R$ and $T$?

• Short answer: yes, it's possible, it's the transmitted power that should be less than the incident one . Hint: consider the Poynting vector. To evaluate the total transmission and reflection coefficients, the quickest approach is to use the so-called transmission line analog of the stratified media. Unfortunately I don't have the time to write a complete answer. – Massimo Ortolano Feb 14 '18 at 14:59