An interaction among three waves mediated by a second-order nonlinear optical susceptibility. In sum-frequency generation, two input angular frequencies satisfy ${\omega }_3={\omega }_1+{\omega }_2$. The same coupling permits reverse conversion. Phase matching determines whether successive generated contributions reinforce; the Manley-Rowe relations express the associated lossless exchange constraints.

Loss of spatial overlap between interacting finite beams because their energy flux directions differ. In an anisotropic crystal, the direction of the Poynting vector need not coincide with the wave vector. For beam radius $w$ and relative walk-off angle $\rho$, an overlap length is of order $w/|\tan \rho |$. A full conversion calculation also includes beam profiles, diffraction and phase mismatch.

Self-defocusing is the spreading tendency caused when a beam lowers the refractive index most strongly where its own intensity is largest. In the local optical Kerr effect model $n=n_0+n_{2}I$, a beam brightest on its axis produces this behavior for $n_2<0$: its transverse phase profile resembles that of a diverging optical lens.

Compare the beam width after the same sample at several input powers while holding the input profile fixed. A power-dependent change can reveal nonlinear spreading, but ordinary diffraction, optical absorption and heating must also be considered. Self-defocusing modifies a beam's propagation rather than removing the ordinary diffraction it already experiences.

Cross-phase modulation occurs when one beam changes the wave phase of another through an intensity-dependent optical response. In an optical Kerr effect medium, a pump changes the refractive index sampled by a probe; propagation over length $L$ gives the probe an additional phase approximately $(2\pi /{\lambda }_0)L\Delta n$.

A delayed weak probe can therefore acquire a time-dependent phase from a strong pump optical pulse. Interfering the probe with an unaffected reference makes the phase change observable as an intensity change. Changing the pump intensity or blocking the pump tests the coupling. The coefficient depends on the beams' relative polarization and the material response; a phase change need not imply appreciable transfer of energy to the probe.

A beam with a transverse profile sustained by a balance between diffraction and a nonlinear change of refractive index. A one-transverse-dimensional focusing cubic model supports a stationary bright profile after its propagation wave phase is removed. Existence in that model does not establish stability in two transverse dimensions or in a saturating, absorbing material.

A third-order nonlinear optical susceptibility produces a polarization term proportional to $E_1{E}_2{E}_3^{\ast }$. Its temporal and spatial phases give the displayed output frequency and the source wave vector ${\mathbf{k}}_1+{\mathbf{k}}_2-{\mathbf{k}}_3$. Efficient emission also requires that this source match a propagating optical mode. The pump waves supply the generated energy.

Treat the strong input pump amplitudes as prescribed while solving for weaker fields. It is self-consistent only when conversion removes a small fraction of pump energy and the assumed pump phases remain valid. An apparent divergent gain is evidence of breakdown of this approximation, not unlimited energy generation.

An optical wave envelope whose spreading is balanced by nonlinear refraction. A temporal optical soliton balances dispersion, while a spatial optical soliton balances diffraction. Persistence of a profile in an ideal propagation model does not by itself establish robustness in a lossy, higher-dimensional or delayed-response medium.

A resonance with no resolved scalar splitting under the acquisition conditions. This does not prove that every coupling is exactly zero.

A pair of lines with the first-order splitting expected for one spin-one-half neighbour. Additional unresolved or coincident interactions can complicate the pattern.

A three-line first-order pattern, often from two equivalent spin-one-half neighbours, with intensities 1:2:1.

A four-line first-order pattern, often from three equivalent spin-one-half neighbours, with intensities 1:3:3:1. Equal couplings to different groups can mimic it.

A seven-line First-order NMR multiplet. Coupling to six equivalent spin-one-half neighbours gives the intensity ratio $1:6:15:20:15:6:1$.

A first-order NMR multiplet split successively by two distinct spin-one-half couplings and by two equivalent spin-one-half nuclei. Ideally it has twelve resolved lines before coincidences or further unresolved couplings.

A first-order NMR multiplet split into a doublet by one spin-one-half nucleus and into quartets by three equivalent spin-one-half nuclei. Ideally it has eight lines; overlap and second-order behavior can change its appearance.

A first-order NMR multiplet produced by coupling to two distinct sets of three equivalent spin-one-half nuclei. Sixteen lines are possible; equal couplings can collapse the pattern to a septet.

An approximately six-line scalar-coupling pattern. Similar coupling to five spin-one-half partners can mimic a sextet even when those partners are not all chemically equivalent.

Vicinal coupling is spin-spin coupling between nuclei separated by three bonds, often hydrogens on neighboring carbons, and is denoted ${}^{3}J$. Its dependence on the intervening dihedral angle is described approximately by a Karplus relation.

This makes a measured coupling useful for estimating molecular conformations. The relationship also depends on substituents and bonding, and different angles can give similar coupling magnitudes. A single coupling therefore rarely determines a conformation uniquely; several measurements and stereochemical constraints are normally combined.