Phase matching

Phase matching is the condition that the wave vectors of interacting optical fields in a nonlinear-mixing process sum to zero, so that the polarization driving the new field accumulates constructively along the propagation distance. Without phase matching, the new field oscillates in and out of phase with the driving polarization, producing no net energy transfer over distances longer than the coherence length.

Mathematical form for three-wave mixing. For a process where photons at frequencies $\omega_1$ and $\omega_2$ combine to create a photon at $\omega_3 = \omega_1 + \omega_2$ (sum-frequency generation), the phase-matching condition is:

$$\Delta k \;\equiv\; k_3 - k_1 - k_2 \;=\; 0,$$

where each $k_i = n(\omega_i) \omega_i / c$. Substituting:

$$n(\omega_3) \omega_3 \;=\; n(\omega_1) \omega_1 + n(\omega_2) \omega_2.$$

For second-harmonic generation (SHG) with $\omega_1 = \omega_2 = \omega$ and $\omega_3 = 2\omega$:

$$n(2\omega) \;=\; n(\omega).$$

In a normally-dispersive material, $n(2\omega) > n(\omega)$ — phase matching cannot be achieved without special engineering.

Coherence length. Without phase matching, the second-harmonic intensity oscillates with propagation distance with period:

$$L_c \;=\; \frac{\pi}{|\Delta k|} \;=\; \frac{\lambda}{4 [n(2\omega) - n(\omega)]}.$$

For typical materials, $L_c$ is 1 – 50 μm. Beyond this length, the SHG output stops growing and oscillates around its initial value.

Phase-matching techniques.

Technique	Mechanism	Applications
Birefringent phase matching	Use birefringent crystal; ordinary and extraordinary indices match at specific angle and wavelength	LiNbO3, BBO, KDP, KTP — workhorse for early SHG
Quasi-phase-matching (QPM)	Periodically reverse $\chi^{(2)}$ to reset phase mismatch	PPLN (periodically-poled LiNbO3), MgO:PPLN — modern standard
Modal phase matching	Use higher-order waveguide modes whose effective indices match	InP and LiNbO3 waveguides; weak compared to QPM
Counter-propagating phase matching	Use counter-propagating waves; phase mismatch reversed	Slow-light schemes
Cherenkov phase matching	Phase-mismatched but radiates at angle determined by mismatch	Cherenkov SHG in slab waveguides

Quasi-phase-matching (the modern dominant technique). A periodically-poled material has alternating regions of $+\chi^{(2)}$ and $-\chi^{(2)}$, with poling period:

$$\Lambda_\text{QPM} \;=\; \frac{2\pi}{|\Delta k|} \;=\; 2 L_c.$$

The sign reversal of $\chi^{(2)}$ at each domain boundary resets the accumulated phase mismatch, allowing net constructive accumulation over the full crystal length. For 1064 nm pump → 532 nm SHG in lithium niobate: $\Lambda_\text{QPM} \approx 7$ μm. Modern PPLN crystals are poled with periods 5 – 30 μm.

QPM is engineered by the fabrication process: poled domains are formed by applying high electric field through a patterned electrode while heating the crystal. Once poled, the crystal is permanent. Modern QPM crystals are commercially available with custom periods, multi-period sections (for multiple SHG wavelengths in one crystal), and chirped periods (for broadband phase matching).

Bandwidth. Phase matching is exact only at one wavelength. The conversion bandwidth (range of input wavelengths over which significant SHG output is obtained) depends on the crystal length:

$$\Delta \omega \;\sim\; \frac{1}{L \cdot \partial(\Delta k)/\partial\omega}.$$

For a 10 mm PPLN crystal at 1064 nm: full-width at half-maximum SHG bandwidth $\sim 0.2$ nm. For broader bandwidth, shorter crystals (poorer efficiency) or chirped poling periods are used.

Why phase matching matters across photonics.

• Frequency conversion: SHG, sum-frequency, difference-frequency mixing for IR / UV generation
• Optical parametric oscillation (OPO): tunable laser sources via parametric down-conversion
• Four-wave mixing: signal generation in fiber and chip-scale optical processing
• Quantum sources: phase-matched parametric down-conversion produces entangled photon pairs
• Comb generation: phase-matched conversion in microresonators generates optical frequency combs
• Wavelength division multiplexing crosstalk: phase-matched FWM in fiber amplifiers causes intermodulation distortion

Phase matching in third-order nonlinear processes. Four-wave mixing (FWM) and other $\chi^{(3)}$ processes have similar phase-matching requirements but with four photons: $k_1 + k_2 = k_3 + k_4$. For degenerate FWM ($\omega_1 = \omega_2 = \omega_p$, $\omega_3 = \omega_s$, $\omega_4 = \omega_i$):

$$2k_p \;=\; k_s + k_i.$$

In fibers and chip waveguides, dispersion engineering is the standard tool to satisfy this condition, leading to applications like frequency comb generation, wavelength conversion for telecom, and parametric amplification.

References: Boyd, Nonlinear Optics (4th ed., 2020), Ch. 2 and Ch. 3 for the foundational treatment of phase matching; Agrawal, Nonlinear Fiber Optics, Ch. 10 for fiber-FWM phase matching; Hum & Fejer, Quasi-phase matching, C. R. Physique 2007 for the QPM review.