Phase matching lets weak sources build coherently

For sum-frequency generation in a time-independent material, the source frequency is ω₃=ω₁+ω₂. This temporal relation also expresses the photon-energy balance for the corresponding parametric process. Each propagating wave has wavevector magnitude n(ω)ω/c for its particular polarization and direction. Since refractive index depends on frequency, the frequency sum does not generally imply that the output wavevector equals the sum of the inputs. Efficient buildup requires their vector mismatch to be sufficiently small over the interaction length.

For collinear second-harmonic generation, define mismatch Δk=k₂ω−2kω. In a normally dispersive isotropic transparent material this is typically nonzero because the two frequencies see different refractive indices. Birefringent phase matching uses different polarization-dependent indices, with angle or temperature tuning, to reduce mismatch. Material transparency and a nonzero effective tensor coefficient remain necessary. Solving an index equality for a hypothetical material does not by itself prove that an accessible crystal orientation supports the desired interaction.

Take plane waves, negligible absorption, constant pumps, and a uniform effective nonlinearity. In the slowly varying envelope approximation the generated envelope is a constant coupling factor times the integral of exp(−iΔkz) along the crystal. The exponential tracks the phase of each source slice relative to the generated carrier. Evaluate the integral by centering its phase at the midpoint. The result is a length factor multiplied by a sinc and an overall phase that does not affect the output intensity.

At exact matching the amplitude grows linearly with length and the generated intensity grows quadratically, but only until the neglected physics becomes appreciable. Away from matching the first intensity zero occurs at |Δk|L=2π. Define nonlinear coherence length as π/|Δk|, the length over which the source and generated field slip by π. This is unrelated to the temporal coherence length set by laser linewidth. At one nonlinear coherence length the uniform-source intensity is already reduced to 4/π² of its perfectly matched value for that same length.

Quasi-phase matching reverses the sign of the effective nonlinearity periodically. Its reciprocal grating vector compensates the wavevector mismatch; a first-order period is 2π/|Δk| for a simple collinear design. Sign reversals keep successive contributions useful but generally reduce the effective coupling relative to ideal uniform perfect matching. This engineered spatial modulation does not change the frequency sum. Focused beams, finite pulse walk-off, absorption, and pump depletion require more complete coupled propagation models.

The wave lab shows only how two coherent contributions can add or cancel. Reset and set both controls to phase zero degrees and ratio one, predicting amplitude two. Reset and set phase 180 degrees and ratio one for a new run, predicting zero. Regard them as two source-slice contributions brought to one observation point. Their fixed equal frequency permits the phase comparison, but they are not pump and harmonic fields. The lab has no crystal length, wavevector mismatch, poling, or energy-conversion calculation.