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Phase Matching/Quasi-Phase Matching

Phase matching

        To achieve efficient conversion of nonlinear wavelengths, both photon energy and photon momentum conservation must be satisfied simultaneously in nonlinear optical processes, as shown in Figure 2. In the three-wave nonlinear coupled-wave equation, the phase mismatch factor Δβ = k3 - k1 - k2 plays a crucial role. If Δβ ​​= 0, the nonlinear interaction is enhanced; if Δβ ≠ 0, the three-wave interactions interfere with each other, thus reducing the number of output photons. To obtain a strong nonlinear optical process, it is generally desirable for Δβ = 0, which is a constraint on photon momentum conservation, also known as the "phase matching" condition. However, the refractive index of optical materials depends on the wavelength of light propagating through the material. Therefore, due to the relative phase delay between mixed photons, the phase matching condition may not be satisfied, as shown in Figure 3.

         

Figure 2-a Conservation of photon energy;             Figure 2-b Conservation of photon kinetic energy

 

                                      

Figure 3. Curves showing the variation of photon phase matching and phase mismatch with distance.

 

Generally, the nonlinear interaction of the three waves occurs in the transparent region of the medium, where the medium and the light field exchange energy. In this case, the three waves should satisfy the laws of conservation of energy and momentum (Δβ=0). Methods to achieve phase matching include:

① Angular phase matching using crystal birefringence;

② Temperature phase matching, where the crystal's refractive index is sensitive to temperature;

③ Quasi-phase matching by periodically reversing the polarization directions of the ferroelectric domains in the crystal.

 

Birefringence phase matching

Generally, the frequencies of the three light waves involved in the interaction are constant. Phase matching is achieved by utilizing the birefringence and dispersion properties of nonlinear optical crystals to change the relative magnitudes of the refractive indices of the three waves, thus satisfying the phase matching condition. This is the traditional birefringence phase matching (BPM) technique, which orients the crystal axis to a specific angle or adjusts the crystal temperature to achieve the phase matching condition between the three waves.

For uniaxial crystals, light waves are divided into o-rays (light waves) and e-rays (light waves). The refractive index of the e-ray varies with the angle θ between the wave vector and the optical axis. Changing the angle θ of the light wave causes a change in the refractive index of the e-ray. When the angle θ changes to a certain point, the refractive index of the e-ray exactly meets the phase matching condition. This method of achieving phase matching by changing the incident angle of the crystal is called angular phase matching or critical phase matching, as shown in Figure 4.

    

Figure 4 shows that the refractive index of the fundamental wave must be equal to that of the SHG wave to achieve birefringence phase matching. Therefore, the incident angle of the frequency-harmonic wave must be obliquely incident along the optical axis (c-axis) of the nonlinear optical crystal at an angle θ or Φ.

However, during the propagation of the e-ray in the crystal, the wave vector and energy flow direction are inconsistent, with an angle α between them, called the dispersion angle α. This causes the frequency-harmonic e-ray and the fundamental o-ray to separate in space. Therefore, to eliminate the influence of the dispersion effect, phase matching is usually achieved after angle matching in phase matching, and then phase matching is achieved by utilizing the temperature sensitivity of the crystal's refractive index; this is called temperature phase matching or non-critical phase matching.

However, this technique cannot effectively utilize the fully transparent region of the material under optimal conversion efficiency. This is because the dispersion angle of the BPM technique causes special waves to undergo birefringence and escape, while ordinary rays do not. This separation problem (i.e., the frequency-harmonic wave deviating from the fundamental wave) distorts the beam quality, limits the effective interaction length, and restricts the overall conversion efficiency.

 

Quasi-phase matching   

Another method to achieve phase matching is quasi-phase matching (QPM). Momentum conservation is required in nonlinear frequency conversion, which is difficult to achieve in ordinary nonlinear crystals due to dispersion, especially in cases involving multiple simultaneous nonlinear interactions. However, the reciprocal lattice vectors provided by nonlinear periodic structures can more easily achieve phase matching. By constructing periodic structures in nonlinear media (nonlinear photonic crystals), nonlinear frequency conversion can be effectively achieved. Compared to the usual perfect phase matching (temperature matching, angle matching), this method, called quasi-phase matching (QPM), can more easily utilize larger nonlinear coefficients. Therefore, this technique is now widely used in the field of nonlinear optics and has enabled the realization of phenomena that are difficult to achieve in ordinary crystals.

 

             

Figure 5: a. Photon energy conservation; b. Photon momentum conservation; c. Photon momentum conservation is used to establish wave vectors to compensate for wave vector mismatch between mixed waves.

At equal distances, the phase shift between the mixed waves is compensated using wave vectors. By selecting the correct reversal period, newly generated photons effectively interfere with previously generated photons, thereby increasing the number of generated photons as light propagates through the QPM material. Figure 5 illustrates the principle and effect of quasi-phase matching in the nonlinear mixing process, where wave vectors related to the periodic polarization modulation of the nonlinear material are used to compensate for wave vector mismatch between the mixed waves, thus effectively achieving the phase matching condition.

 

Comparison of Quasi-Phase Matching and Non-Phase Matching

Figure 6 shows a comparison of quasi-phase-matched and non-phase-matched phase waves, and the phase-matching condition for generating SHG. In a nonlinear medium, the distance by which the accumulated phase difference of the mixing wave reaches π is the "coherence length" Lc.

        

Figure 6: Output power increases with distance: (i) birefringence phase matching, (ii) non-phase matching, (iii) quasi-phase matching. The + & - symbols represent the symbols for second-order nonlinear polarization magnetization.

 

 

Comparison of Birefringence Phase Matching (BPM) and Quasi-Phase Matching (QPM)

Figure 7 summarizes the requirements for efficient nonlinear mixing processes and provides a comparison between BPM and QPM.

 

        

        

Figure 7. Summary and comparison of nonlinear mixing wave BPM and QPM techniques.