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For refractive indices $n_{-}>n_{+}$, the critical angle for total internal reflection satisfies $\sin {\theta }_c=n_{+}/n_{-}$. Above it, the transmitted normal wavenumber is imaginary and the transmitted field is an evanescent wave.

The critical angle for total internal reflection is the boundary between outgoing refraction and an evanescent wave, rather than an angle at which the external electric field abruptly disappears. Write $n_1=n_{-}$ and $n_2=n_{+}$. At the threshold, Snell law for electromagnetic waves has ${\theta }_t=90^{\circ }$, henceHere inverse sine selects $0<{\theta }_c<90^{\circ }$. For $n_1=1.5$ and $n_2=1$, $\sin {\theta }_c=2/3$ and ${\theta }_c\approx 41.8103149^{\circ }$. The threshold is independent of polarization of an electromagnetic wave under the stated assumptions.

A short proof using phase matching at a planar wave interface reveals all three cases at once. Choose a Cartesian coordinate system with boundary $z=0$, incident medium $z<0$, and transmitted medium $z>0$. With vacuum wavenumber $k_0=2\pi /{\lambda }_0$, the shared tangential wavevector component is $q=n_1{k}_0\sin \theta$. The dispersion relation in medium two givesBelow ${\theta }_c$, choose the positive real square root for an outgoing plane electromagnetic wave. At ${\theta }_c$, $k_{2z}=0$: there is a grazing wave, with zero normal average power but no exponential confinement in $z$. Above ${\theta }_c$, choose $k_{2z}=i\kappa$ with $\kappa >0$, so the wave amplitude decays into the second half-space.

If $n_1\le n_2$, no nongrazing angle of incidence produces this threshold. As $n_2/n_1\to 1$ from below, ${\theta }_c\to 90^{\circ }$; as that ratio tends to zero, ${\theta }_c\to 0$. These limits explain why a larger refractive index contrast permits confinement over a wider range of internal angles.

The source-free Maxwell equations in a linear medium areHere the constitutive relations are $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

Substitution of the sinusoidal plane waves into the two divergence equations gives the transversality of an electromagnetic plane wave,Faraday's law and the Ampère-Maxwell equation give, respectively,Taking $\mathbf{k}\times$ the first equation and using transversality yieldsThus a nonzero plane electromagnetic wave in a linear medium must satisfyIn particular, ${\mathbf{E}}_0$, ${\mathbf{B}}_0$, and $\mathbf{k}$ are mutually orthogonal, with $|{\mathbf{B}}_0|=\sqrt{\mu \epsilon }|{\mathbf{E}}_0|$.

Take the interface to be $x=0$ and the plane of incidence to be the $xy$-plane. The incident, reflected, and transmitted wavevectors may be writtenThe phase matching at a planar wave interface requires one common angular frequency and one common tangential wavevector:Since $k_{\pm }=\omega \sqrt{{\epsilon }_{\pm }\mu }=n_{\pm }\omega /c$, this gives the Snell law for electromagnetic waves,

For the final polarization, take the signed electric amplitudes along $\hat{\mathbf{z}}$. The dielectric-interface boundary conditions require continuity of tangential $\mathbf{E}$ and $\mathbf{H}$. Sincetheir $z$- and $y$-components giveandEliminating $E_I$ yieldsUsing ${\theta }_R={\theta }_I$ and Snell's law,Therefore the reflected-to-transmitted transverse-electric amplitude ratio iswith the polarization directions chosen so that the displayed signed factor is nonnegative; otherwise its absolute value gives the magnitude ratio.

The condition ${\mathbf{E}}_I\cdot \hat{\mathbf{x}}=0$, together with transversality ${\mathbf{k}}_I\cdot {\mathbf{E}}_I=0$, selects polarization along $\hat{\mathbf{y}}$ away from the normal-incidence degeneracy. Phase matching at a planar wave interface givesThe two dispersion relations give $|{\mathbf{k}}_T{|}^2=({\epsilon }_{+}/{\epsilon }_{-})k_I^2$. Choosing propagation into $x>0$ therefore yieldsEquivalently, this is Snell law for electromagnetic waves with refractive index proportional to $\sqrt{\epsilon }$ because the two media have equal magnetic permeability.

For each $X\in \{I,R,T\}$, substitution of a complex plane electromagnetic wave in a linear medium into the source-free Maxwell equations in a linear medium requireswhere ${\epsilon }_I={\epsilon }_R={\epsilon }_{-}$ and ${\epsilon }_T={\epsilon }_{+}$. The accompanying magnetic field isThe incident and transmitted waves have $k_{Ix}>0$ and $k_{Tx}>0$, while the reflected wave has $k_{Rx}<0$. Matching the fields at the stationary interface additionally imposes the phase matching at a planar wave interface conditions ${\omega }_I={\omega }_R={\omega }_T$ and equality of all tangential wavevector components.

Choose the interface as $x=0$ and write the incident, reflected, and transmitted fields as plane waves with wavevectors $k_I,k_R,k_T$. The boundary conditions must hold for every tangential coordinate and every time. The phase matching at a planar wave interface therefore requiresand equality of tangential wavevectors:

The incident and reflected waves are in the same medium, so
$|k_I|=|k_R|$. Equality of their tangential components giveshence

At fixed frequency, $|k_{\pm }|=\omega /v_{\pm }$ is proportional to the refractive index $n_{\pm }$. Tangential phase matching between incident and transmitted waves gives the Snell law for electromagnetic waves:orHere $n_{\pm }=c/v_{\pm }=c\sqrt{{ϵ}_{\pm }\mu }$ if the vacuum speed $c$ is used to normalize the index.

For a linear stationary interface, phase matching at a planar wave interface preserves frequency: the incident and transmitted electric fields must oscillate together at each boundary point. Their phase velocities can differ, so their wavelengths differ. Writing $\omega =2\pi f$, the definitions giveThis explains the compressed wavefront spacing in the figure without changing the time between crests at the interface. Moving interfaces or time-dependent material responses need not preserve frequency. ^[ref]

The notation $n({\lambda }_0)$ labels refractive index by vacuum wavelength; it does not mean the wavelength inside the material. Dependence on ${\lambda }_0$, equivalently on frequency, is optical dispersion. The field driving the material induces oscillating electric charges; their emitted electromagnetic fields combine with the applied field to change the resulting wave phase. This collective response motivates refractive index, rather than a picture of independently delayed little pieces of light. ^[ref]

To distinguish three speeds, track a wavefront for phase velocity, a narrow wave envelope for group velocity, and the earliest newly introduced disturbance for front velocity. For a transparent differentiable propagation branch,The denominator is the group refractive index. It describes envelope transit when loss and reshaping are small. It need not equal $n$, and neither phase velocity nor a reshaped pulse peak generally measures signal arrival. The front velocity article explains the separate relativistic causality constraint and why a detector threshold is not an exact causal-front measurement.