Phase matching

A relation between wave phases that lets contributions from successive interaction regions add coherently. In Cherenkov radiation, the condition along the particle trajectory is $\omega =\mathbf{k}\cdot \mathbf{v}$, relating angular frequency, wave vector and particle velocity.

The daughter photons occupy orthogonal ordinary and extraordinary polarization classes. Alternative assignments to two collected paths can interfere if timing, spectrum and spatial mode do not distinguish them. spatial walk-off and unequal group delays often require compensation.

In a common birefringent down-conversion arrangement, the two daughter photons have the same polarization class, ordinarily both ordinary or both extraordinary relative to the crystal. The pump has a suitable different index for phase matching. This classification alone does not establish polarization entanglement.

Compensation of phase mismatch by a spatial modulation of the coupling coefficient. Write $d(z)={\sum }_m{d}_m{e}^{imKz}$. The driven amplitude contains $e^{i(\Delta k+mK)z}$, so the selected Fourier component accumulates coherently when $\Delta k+mK=0$. For first-order compensation the full spatial period is $2\pi /|\Delta k|$, twice the nonlinear optical coherence length.

An ideal equal-duty pattern alternating $+d$ and $-d$ has first Fourier coefficient of magnitude $2d/\pi$. Hence its conversion intensity is $4/{\pi }^2$ of a uniformly phase-matched crystal using that same coefficient, under the same undepleted plane-wave assumptions. The modulation can nevertheless allow a larger tensor coefficient or an otherwise inaccessible wavelength. Periodically poled lithium niobate implements such sign reversals using ferroelectric domains.

Periodically poled lithium niobate is lithium niobate whose ferroelectric domain orientation is reversed periodically along an optical interaction path. Reversal changes the sign of the relevant second-order nonlinear optical susceptibility, supplying the spatial modulation needed for quasi-phase-matching.

For a uniform collinear conversion mismatch $\Delta k$, a first-order poling period $\Lambda$ can compensate it through $\Delta k-2\pi /\Lambda \simeq 0$, with signs chosen consistently. The generated contributions then reinforce over many domains instead of repeatedly canceling. The poling period, temperature and input wavelengths determine the useful conversion band; domain defects and finite length affect efficiency. ^[ref]

The residual wave vector difference in a wave-conversion process. For collinear sum-frequency generation, choose $\Delta k=k_1+k_2-k_3$. Contributions separated by $z$ then differ by wave phase $\Delta kz$. At phase matching this residual is zero. The sign convention changes some amplitude phases but not the unseeded conversion factor ${\mathrm{sinc}}^2(\Delta kL/2)$.

The distance over which a constant phase mismatch accumulates phase difference $\pi$. In unseeded, undepleted sum-frequency generation, the generated intensity first reaches a maximum at this length and returns to zero at $2{\ell }_c$. This length concerns relative phases of interacting waves, rather than the temporal optical coherence length of one source.