Quasi-phase matching (QPM)

Quasi-phase matching (QPM) compensates the phase mismatch between interacting waves in a nonlinear crystal by flipping the sign of the nonlinear coefficient periodically along the propagation direction. The flip is placed every coherence length, just where the generated wave would otherwise begin to convert back, so the output keeps growing along the whole crystal. In practice the flip is made by reversing ferroelectric domains, a technique called periodic poling, in lithium niobate (PPLN), lithium tantalate (PPLT) or KTP (PPKTP). Periods range from a few micrometers for visible and blue generation to around 30 µm for mid-infrared parametric oscillators.

The poling period

In a dispersive crystal, second-harmonic light drifts out of phase with its driving polarization over a coherence length $L_c = \pi/\Delta k$. Ordinary phase matching removes $\Delta k$ with birefringence; QPM instead adds a grating vector $2\pi m/\Lambda$ that cancels it:

$$\Delta k - \frac{2\pi m}{\Lambda} = 0, \qquad \Lambda = 2 m L_c,$$

where $m$ is the odd QPM order. For second-harmonic generation of a fundamental at wavelength $\lambda$,

$$\Lambda = \frac{m\,\lambda}{2\,(n_{2\omega} - n_\omega)}.$$

Worked example: for 1064 nm doubled to 532 nm in lithium niobate with both waves polarized along the crystal $z$ axis, approximate extraordinary indices are $n_\omega \approx 2.156$ and $n_{2\omega} \approx 2.234$. The index difference 0.078 gives $L_c = 3.4$ µm and a first-order period of 6.8 µm. This is an estimate: design periods come from a temperature-dependent Sellmeier equation for the specific material (congruent or MgO-doped) and can differ from it noticeably. Bulk PPLN for doubling 1550 nm uses periods of roughly 19–20 µm, and thin-film lithium niobate waveguides need shorter periods, around 4 µm, because the waveguide dispersion adds to the material dispersion.

Effective nonlinear coefficient

A square-wave modulation of the sign of $d$ has a Fourier component at the grating period of amplitude $2/(m\pi)$, so

$$d_\text{eff} = \frac{2}{m\pi}\, d_{33}.$$

For first order this is 0.64 $d_{33}$, and conversion efficiency, proportional to $d_\text{eff}^2$, is 41% of what perfect phase matching with $d_{33}$ would give. The advantage lies in the choice of coefficient. With QPM all three waves can share the same polarization along $z$ and use $d_{33}$, the largest nonlinear coefficient of lithium niobate at roughly 25 pm/V, so $d_\text{eff} \approx 16$ pm/V. Birefringent phase matching in lithium niobate relies mainly on $d_{31}$, roughly 4–5 pm/V, so QPM gives about 10–16 times higher efficiency for the same length and intensity. Third-order QPM, used when the first-order period is too short to fabricate, has $d_\text{eff}$ reduced by a further factor of 3.

Fabrication and measurement

Periodic poling is done by patterning metal electrodes with the grating period on a $z$-cut wafer and applying high-voltage pulses above the coercive field, about 21 kV/mm for congruent lithium niobate, so that domains under the electrodes invert. The domain pattern is inspected after a selective hydrofluoric-acid etch, which attacks the two domain orientations at different rates, or non-destructively by SHG microscopy. The practical test is a phase-matching curve: harmonic power versus crystal temperature or wavelength should follow a $\mathrm{sinc}^2$ shape, and side lobes or a broadened peak indicate period errors or a nonuniform crystal.

Where QPM is used

• Frequency doublers for green and visible lasers, and in integrated form in thin-film lithium niobate waveguides.
• Mid-infrared optical parametric oscillators and difference-frequency sources, often with several gratings or a fan-out grating on one chip for tuning.
• Photon-pair sources based on PPKTP and PPLN for quantum optics, where type-0 or type-II QPM sets the polarization of the pairs.

Because propagation can be along a principal axis, QPM interactions are noncritically phase-matched and free of spatial walk-off, which allows long crystals and tight focusing.

Pitfalls

Duty-cycle errors multiply the effective coefficient by $\sin(\pi D)$ for duty cycle $D$, and the efficiency by its square: a 40:60 grating keeps 90% of the ideal efficiency. Undoped PPLN suffers photorefractive damage with visible light and is usually run hot; MgO doping reduces this and lowers the coercive field. At high green power, absorption induced by the green light heats the crystal and detunes the phase matching. Long gratings also give narrow temperature and wavelength acceptance bandwidths.

Common questions

Who proposed quasi-phase matching?

Armstrong, Bloembergen, Ducuing and Pershan described it in 1962. Practical devices had to wait for reliable electric-field poling of lithium niobate in the early 1990s; Fejer and coworkers set out the tuning and tolerance theory in 1992.

Why is the QPM efficiency penalty worth paying?

The $(2/\pi)^2$ penalty is more than recovered by access to $d_{33}$, by the absence of walk-off and by being able to design any wavelength within the transparency window by choosing the period.

References: J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918 (1962); M. M. Fejer, G. A. Magel, D. H. Jundt, R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992); L. E. Myers et al., J. Opt. Soc. Am. B 12, 2102 (1995); R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020).