Quasi-phase-matching is a technique in nonlinear optics which allows a positive net flow of energy from the pump frequency to the signal and idler frequencies by creating a periodic structure in the nonlinear medium. Momentum is conserved, as is necessary for phase-matching, through an additional momentum contribution corresponding to the wavevector of the periodic structure. Consequently, in principle any three-wave mixing process that satisfies energy conservation can be phase-matched. For example, all the optical frequencies involved can be collinear, can have the same polarization, and travel through the medium in arbitrary directions. This allows one to use the largest nonlinear coefficient of the material in the nonlinear interaction. Quasi-phase-matching ensures that there is positive energy flow from the pump frequency to signal and idler frequencies even though all the frequencies involved are not phase locked with each other. Energy will always flow from pump to signal as long as the phase between the two optical waves is less than 180 degrees. Beyond 180 degrees, energy flows back from the signal to the pump frequencies. The coherence length is the length of the medium in which the phase of pump and the sum of idler and signal frequencies are 180 degrees from each other. At each coherence length the crystal axes are flipped which allows the energy to continue to positively flow from the pump to the signal and idler frequencies. The most commonly used technique for creating quasi-phase-matched crystals has been periodic poling. A popular material choice for this is lithium niobate. More recently, continuous phase control over the local nonlinearity was achieved using nonlinear metasurfaces with homogeneous linear optical properties but spatially varying effective nonlinear polarizability. Optical fields are strongly confined within or surround the nanostructures, nonlinear interactions can therefore be realized with an ultra-small area down to 10 nm to 100 nm and can be scattered in all directions to produce more frequencies. Thus, relaxed phase matching can be achieved at the nanoscale dimension.
Mathematical description In nonlinear optics, the generation of new frequencies is the result of the nonlinear polarization response of the crystal due to a typically monochromatic high-intensity pump frequency. When the crystal axis is flipped, the polarization wave is shifted by 180°, thus ensuring that there is a positive energy flow to the signal and idler beam. In the case of sum-frequency generation, where waves at frequencies ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} are mixed to produce ω 3 = ω 1 + ω 2 {\displaystyle \omega _{3}=\omega _{1}+\omega _{2}} , the polarization equation can be expressed by
P 3 = 4 d A 1 A 2 e i ( k 1 + k 2 ) z , {\displaystyle P_{3}=4dA_{1}A_{2}e^{i(k_{1}+k_{2})z},}
where d {\displaystyle d} is the nonlinear susceptibility coefficient, i {\displaystyle i} represents the imaginary unit, A {\displaystyle A} are the complex-valued amplitudes, and k = ω / c {\displaystyle k=\omega /c} is the wavenumber. In this frequency domain vector representation, the sign of the d {\displaystyle d} coefficient is flipped when the nonlinear (anisotropic) crystal axis is flipped,
P 3 = − 4 d A 1 A 2 e i ( k 1 + k 2 ) z = 4 d A 1 A 2 e i ( ( k 1 + k 2 ) z e i π . {\displaystyle P_{3}=-4dA_{1}A_{2}e^{i(k_{1}+k_{2})z}=4dA_{1}A_{2}e^{i((k_{1}+k_{2})z}e^{i\pi }.}
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