17 Jun 2025
Reflection and Refraction at a Dielectric Interface
Phase matching, reflection and refraction laws, Fresnel formulae, Brewster's law, and power coefficients.
Let a plane interface at $z=0$ separate two lossless, isotropic, nonmagnetic dielectrics with refractive indices $n_1$ and $n_2$. The plane containing the incident wave vector and the interface normal is the plane of incidence.
Laws from phase matching
At every point of the stationary interface and at every time, the tangential fields must match. Their phase factors must therefore agree:
\[\omega_i=\omega_r=\omega_t, \qquad (\mathbf k_i)_\parallel=(\mathbf k_r)_\parallel=(\mathbf k_t)_\parallel.\]Since $k_j=n_j\omega/c$,
\[k_1\sin\theta_i=k_1\sin\theta_r=k_2\sin\theta_t.\]The first equality gives the law of reflection,
\[\boxed{\theta_r=\theta_i},\]and the second gives Snell’s law,
\[\boxed{n_1\sin\theta_i=n_2\sin\theta_t}.\]The frequency is unchanged; the wavelength changes because $k$ changes.
Boundary equations for amplitudes
With no free surface charge or current,
\[\hat{\mathbf n}\times(\mathbf E_2-\mathbf E_1)=\mathbf0, \qquad \hat{\mathbf n}\times(\mathbf H_2-\mathbf H_1)=\mathbf0.\]Write $r=E_{0r}/E_{0i}$ and $t=E_{0t}/E_{0i}$ using polarization unit vectors tied to each propagation direction. For nonmagnetic dielectrics, $\eta_j=\eta_0/n_j$.
Perpendicular or s polarization
For s polarization, $\mathbf E$ is perpendicular to the plane of incidence. Tangential-$E$ continuity gives
\[E_{0i}+E_{0r}=E_{0t}.\]The tangential magnetic components are $E\cos\theta/\eta$, with the reflected contribution carrying the opposite propagation sign:
\[\frac{E_{0i}-E_{0r}}{\eta_1}\cos\theta_i =\frac{E_{0t}}{\eta_2}\cos\theta_t.\]Solving the two simultaneous equations,
\[\boxed{r_s=\frac{n_1\cos\theta_i-n_2\cos\theta_t} {n_1\cos\theta_i+n_2\cos\theta_t}},\] \[\boxed{t_s=\frac{2n_1\cos\theta_i} {n_1\cos\theta_i+n_2\cos\theta_t}}.\]Parallel or p polarization
For p polarization, $\mathbf E$ lies in the plane of incidence. Applying the same two tangential conditions gives
\[\boxed{r_p=\frac{n_2\cos\theta_i-n_1\cos\theta_t} {n_2\cos\theta_i+n_1\cos\theta_t}},\] \[\boxed{t_p=\frac{2n_1\cos\theta_i} {n_2\cos\theta_i+n_1\cos\theta_t}}.\]With the conventional p-polarization unit vectors, $r_p$ and $r_s$ have opposite signs at normal incidence. This is a basis-orientation sign; the measurable reflected fraction depends on $\lvert r\rvert^2$.
Reflection and transmission coefficients
The normal component of average Poynting flux is
\[\langle S_n\rangle=\frac{\lvert E_0\rvert^2}{2\eta}\cos\theta.\]Therefore the power reflection and transmission coefficients are
\[\boxed{R_s=\lvert r_s\rvert^2,\qquad R_p=\lvert r_p\rvert^2},\] \[\boxed{T_s=\frac{n_2\cos\theta_t}{n_1\cos\theta_i}\lvert t_s\rvert^2, \qquad T_p=\frac{n_2\cos\theta_t}{n_1\cos\theta_i}\lvert t_p\rvert^2}.\]For lossless media,
\[\boxed{R_s+T_s=1,\qquad R_p+T_p=1}.\]At normal incidence,
\[R=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2, \qquad T=\frac{4n_1n_2}{(n_1+n_2)^2}.\]These are power fractions and hence dimensionless.
Brewster’s law
The p-polarized reflection vanishes when the numerator of $r_p$ is zero:
\[n_2\cos\theta_B=n_1\cos\theta_t.\]Combine this with Snell’s law $n_1\sin\theta_B=n_2\sin\theta_t$. Division gives
\[\frac{\sin\theta_B}{\sin\theta_t} =\frac{\cos\theta_t}{\cos\theta_B}.\]Hence $\sin(2\theta_B)=\sin(2\theta_t)$. For unequal media the physical solution is $\theta_B+\theta_t=90^\circ$. Snell’s law then becomes
\[n_1\sin\theta_B=n_2\cos\theta_B,\]and hence
\[\boxed{\tan\theta_B=\frac{n_2}{n_1}}.\]At Brewster incidence the reflected and refracted rays are perpendicular, and the reflected beam contains only s polarization.
The s and p energy balances and the Brewster zero are checked symbolically in the Unit II Maxima worksheet; every printed residual is zero.
Discussion