= Interface phase matching
{title2=$k_{i,\parallel}=k_{t,\parallel}$}
At a stationary, uniform planar boundary, incident and transmitted <plane waves> have the same variation of <wave phase> along the interface and in <time>. The figure marks places where equal-<wave phase> <wavefronts> meet the boundary. Although their perpendicular spacing differs, their intersections must advance along the boundary together. This is the local mechanism behind <Snell's law>.<Electromagnetism lecture notes by Richard Fitzpatrick/Laws of Geometric Optics>[equations 946–950: common angular frequency and matching tangential wavevector components]{ref}
\Image[/refraction-interface-matched-wavefronts.png]
{title=<Wavefront> spacing changes while interface <wave phase> matches}
{description=Original <plane wave> construction for $n_{\mathrm a}=1$, $n_{\mathrm w}=4/3$ and air incidence $30^\circ$. Both panels have equal horizontal and vertical coordinate scales in units of <vacuum> <wavelength> $\lambda_0$. Coloured lines join at identical interface crossings; arrows show propagation perpendicular to each <wavefront>. The reflected branch is omitted. At <normal incidence>, the <wavefronts> stay parallel to the boundary and only their spacing changes.}
For a short derivation, let $x$ run along the interface and $z$ increase into the transmitted medium. Write each <wave phase> as $\Phi_j=k_{j,x}x+k_{j,z}z-\omega_jt+\phi_j$. The electromagnetic <boundary conditions> must hold at every $x,t$ on $z=0$; their matching requires $\omega_i=\omega_t$ and $k_{i,x}=k_{t,x}$. Constant offsets $\phi_j$ and <wave amplitudes> are determined separately. Since $k_{j,x}=k_j\sin\theta_j$, this gives
$$
\frac{n_i\omega}{c}\sin\theta_i=\frac{n_t\omega}{c}\sin\theta_t.
$$
For nonzero tangential <wavenumber>, the spacing of equal-<wave phase> intersections along the interface is $\lambda_i/\sin\theta_i=\lambda_t/\sin\theta_t$. It is not the perpendicular <wavelength>. At <normal incidence>, the tangential <wavenumber> vanishes and the interface is uniform in <wave phase>.<Electromagnetism lecture notes by Richard Fitzpatrick/Laws of Geometric Optics>[equations 951–955: $k=n\omega/c$ and Snell's law]{ref}
Back to article page
