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The extraordinary wave's Poynting vector can differ in direction from its wavevector. The latter specifies wave phase propagation, while the former specifies energy flux. This angular separation is walk-off.^[ref]

In the lossless, nondispersive model above, take the gradient of $\omega =c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the group velocity, parallel to the Poynting vector. If its angle to the optic axis is $\varphi$, this gives $\tan \varphi =(n_o^2/n_e^2)\tan \theta$. Consequently the extraordinary wave generally carries energy away from its wavevector direction. An internal wavevector normal to a slab face therefore does not guarantee a normal energy path. Birefringent walk-off vanishes at $\theta =0$ and $\theta =\pi /2$; the latter direction still has unequal refractive indices. This is a useful waveplate geometry: overlapping components can acquire phase retardation without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one linear polarization can also hide a double image, for different reasons.

The extraordinary wave's electric displacement field lies in that plane and is perpendicular to the wavevector; its electric field need not be. For angle $\theta$ between the wavevector and optic axis,Thus $n_{\mathrm{ex}}(0)=n_o$ and $n_{\mathrm{ex}}(\pi /2)=n_e$.^[ref]

A short derivation starts from the existing plane wave in an anisotropic dielectric equation. Put $\mathbf{k}=(k_x,0,k_z)$ and $k_0=\omega /c$, with angular frequency $\omega$ and vacuum speed of light $c$. The $y$ component gives the ordinary wave relation $k_x^2+k_z^2=n_o^2{k}_0^2$. The determinant of the $x,z$ system givesSubstituting $k_x=n_{\mathrm{ex}}{k}_0\sin \theta$ and $k_z=n_{\mathrm{ex}}{k}_0\cos \theta$ yields the displayed directional refractive index.

A three-level laser uses a pump transition from ground level1 to a higher level3, rapid relaxation $3\to 2$, and a long-lived upper lasing level2. The laser transition is $2\to 1$, so its lower level is the ground state. Fast feeding and slow upper-state decay allow excitation to accumulate, unlike steady resonant pumping of a simple two-level transition.

Stimulated emission adds photons to the stimulating optical mode, but ground-state photon absorption competes with it. The population inversion condition is $N_2/g_2>N_1/g_1$ when the relevant transition strengths obey the usual Einstein relations. For equal degeneracies and negligible $N_3$, it reduces to $N_2>N_1$, requiring more than half the active population in level2.

For a simple pump rate R per ground-state emitter, fast $3\to 2$ relaxation and negligible laser intensity, the three-level laser rate equation reduces toThis three-level ground-state inversion threshold is only transparency; positive gain must also overcome optical losses. For an illustrative uniformly filled cavity of length $\ell$, equal absorption/emission cross-section $\sigma$, internal intensity-loss coefficient ${\alpha }_i$ and mirror power reflectivities $R_1,R_2$, the threshold condition isHere $n_i$ are number densities, unlike the total populations $N_i$. The optical resonator supplies feedback and selects modes; a partially transmitting mirror couples out light. Spontaneous emission seeds the field, and stimulated transitions deplete the inversion until saturated gain balances loss. A ruby laser is the standard example. Real level degeneracies, branching losses and pump saturation modify the simple equations. ^[ref]

A laser combines pumping, an amplifying medium and optical feedback. Stimulated emission adds photons to an occupied optical mode; net amplification requires population inversion between the relevant levels after accounting for their degeneracies. Pumping through additional levels and rapid lower-level relaxation can sustain inversion; a closed two-level system under steady incoherent resonant pumping cannot sustain it.

An optical resonator selects modes and feeds light back through the gain medium. For intensity gain coefficient $g$, internal intensity loss ${\alpha }_i$, length $L$ and mirror intensity reflectivities $R_1,R_2$, the round-trip laser threshold condition isAbove threshold, saturation limits gain and a partially transmitting mirror provides output. Typical properties are strong directionality, substantial coherence and narrow bandwidth; multiple modes and pulsed operation are possible, so perfect monochromaticity is not universal.^[ref]

Use mode locking to phase-lock many longitudinal cavity modes; their repeated constructive interference produces short optical pulses. A saturable absorber can select this pulsed operation. A $2\mathrm{MHz}$ fundamental repetition rate corresponds to a $500\mathrm{ns}$ round trip, or an optical round-trip length $c/R\simeq 150\mathrm{m}$; a linear air-filled cavity would therefore be about $75\mathrm{m}$ long. A compact alternative is a faster mode-locked oscillator followed by pulse selection and, if required, amplification. Ordinary Q-switching alone generally does not provide $350\mathrm{fs}$ pulses.^[ref]

Type-I phase matching requires $n_{e,2\omega }(\theta )=n_{o,\omega }$, because both pump photons are ordinary and the harmonic is extraordinary. Apply the direction-dependent index equation to the harmonic, not to the pump. Solving givesFor the ruby-laser data this is $0.626210$, whereas the shorter-wavelength data give $0.958873$. ThereforeThe second calculation uses the supplied $257\mathrm{nm}$ indices as rounded harmonic data: the exact half-wavelength of $515\mathrm{nm}$ is $257.5\mathrm{nm}$. The index dispersion between those two ultraviolet values was not supplied.

For an extraordinary wave, its Poynting vector makes an angle ${\theta }_S$ with the optic axis satisfying $\tan {\theta }_S=(n_o^2/n_e^2)\tan \theta$. Thus the spatial walk-off angle relative to the wave vector isBoth angles refer to the harmonic. The ordinary pump does not have this angular walk-off. For beam radius $w_b$, the approximate overlap length is $w_b/\tan \rho$, so the shorter-wavelength geometry has about $2.19$ times the walk-off-limited interaction length.^[ref]

To compare coupling, take propagation azimuth $\phi$ and ordinary unit polarization $(-\sin \phi ,\cos \phi ,0)$. Its quadratic source is $P_z=-{ϵ}_0{d}_{36}{E}_0^2\sin 2\phi$. Projection onto an extraordinary displacement direction with $z$ component proportional to $\sin \theta$ gives $|d_{\mathrm{eff}}|\propto |d_{36}\sin \theta \sin 2\phi |$, up to the small anisotropic normalization correction. The optimal azimuth is $45^{\circ }$. At equal beam area, interaction length, input power and approximately equal nonlinear coefficients, the shorter pump has both stronger angular coupling and the favorable frequency-squared factor:Here $\eta =P_{2\omega }/P_{\omega }$ in the small-conversion, equal-area model, with harmonic index phase-matched to the pump. Its smaller spatial walk-off further favors the $515\mathrm{nm}$ pump in a walk-off-limited geometry. Equal input power alone does not determine an absolute efficiency ranking: ultraviolet absorption, wavelength-dependent $d_{36}$, beam size, crystal length and coatings are unspecified. For example, a sufficiently strongly absorbing $257.5\mathrm{nm}$ output can reverse the ideal ranking. The stated preference is conditional on comparable transparent crystals and comparable focusing, not a prediction independent of those parameters.

A transparent, nonmagnetic linear anisotropic dielectric is uniaxial when two principal permittivities agree. With the optic axis along $z$, its permittivity tensor has the displayed diagonal matrix form. Here $n_o$ and $n_e$ are principal refractive indices at a specified frequency; $n_e$ is not the extraordinary wave's index in every direction.^[ref]