# Phase Matching

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a group of techniques for achieving efficient nonlinear interactions in a medium

More specific terms: critical phase matching, noncritical phase matching, birefringent phase matching, quasi-phase matching, type-I or type-II phase matching, collinear and noncollinear phase matching

Many phase-sensitive nonlinear processes, in particular parametric processes such as frequency doubling, sum and difference frequency generation, parametric amplification and oscillation, and also four-wave mixing, require *phase matching* to be efficient. Essentially, this means ensuring that a proper phase relationship between the interacting waves (for optimum nonlinear frequency conversion) is maintained along the propagation direction. Only if that condition is fulfilled, amplitude contributions from different locations to the product wave are all in phase at the end of the nonlinear crystal. In other words, some *phase mismatch* should be close to zero in order to obtain an effective nonlinear interaction. For example, for type I phase matching of frequency doubling with collinear beams the phase mismatch is given by

where <$k_1$> and <$k_2$> are the wavenumbers of the fundamental and second-harmonic beam, respectively. Without chromatic dispersion, <$k_2 = 2 k_1$> would hold, so that the phase mismatch vanishes. However, dispersion generally causes a non-zero phase mismatch (Figure 1) if no special measures (as discussed below) are taken to avoid this.

Due to chromatic dispersion, the wavenumber of the second harmonic is more than twice as large as that for the fundamental wave. This can be avoided, e.g., by choosing a different polarization in a birefringent crystal (→ *birefringent phase matching*).

Figure 2 illustrates how a phase mismatch keeps the efficiency low. Here, the arrows illustrate the phasors corresponding to the complex amplitude contributions from different parts of the nonlinear crystal to the harmonic wave. Only when phase matching is achieved, these contributions add up constructively, and a high power conversion efficiency is achieved. Otherwise, the direction of energy transfer changes periodically (possibly thousands of times during the passage through the crystal) according to the change in the phase relation between the interacting waves. The energy then oscillates between the waves rather than being transferred in a constant direction. The effect on the power conversion is illustrated in Figure 3.

Solid curve: phase-matched case, with the power growing in proportion to the square of the propagation distance. Dashed curve: non phase-matched case, with the second-harmonic power oscillating between zero and a small value.

In a frequency doubler, the direction of energy transfer is governed by the complex phase of the term <$E_1^2 E_2^*$>, where <$E_1$> is the complex electric field amplitude of the fundamental wave and <$E_2$> that of the second harmonic. The asterisk indicates complex conjugation. The phase-matching condition <$k_2 = 2 k_1$> implies that the phase of the term <$E_1^2 E_2^*$> remains constant along the propagation direction. For a sum frequency generation process, or in a nondegenerate optical parametric oscillator, the corresponding term would be <$E_1 E_2 E_3^*$>, and the phase-matching condition would be <$k_3 = k_1 + k_2$>.

For devices such as frequency doublers or optical parametric amplifiers, phase matching needs to be actively arranged. On the other hand, an optical parametric oscillator may automatically choose its signal wavelength so that phase matching is achieved. Wavelength tuning can thus be achieved by influencing the phase-matching conditions e.g. via temperature changes or angular adjustments.

## Phase-matching Curves

When the crystal temperature is varied around the optimum point, the phase mismatch and thus the conversion efficiency also vary, as shown in Figure 4. The temperature range in which a high conversion efficiency is obtained is inversely proportional to the crystal length. It also depends on the temperature dependence of the refractive indices involved. Similar relations apply to other nonlinear frequency conversion processes.

Similar curves are obtained e.g. for critical phase matching when the angular orientation of the crystal is varied.

The phase-matching curve actually does not need to be as symmetric as shown in Figure 4. For example, it becomes asymmetric if the crystal temperature is lower at the crystal end faces, as compared with the middle of the crystal. It is possible to quantify the temperature homogeneity in a crystal oven on the basis of the measured phase-matching curve. Based on such data, it can be estimated whether the conversion efficiency could be increased e.g. by using a longer crystal.

## Influence of Optical Intensities

A high optical intensity can cause substantial nonlinear phase changes (→ Kerr effect), which in some cases can significantly affect phase-matching conditions. For example, this is often relevant in the context of parametric processes in optical fibers, which are based on a <$\chi^{(3)}$> nonlinearity. That leads to phase-matching conditions which also involve optical intensities and not only wavenumbers. On the other hand, nonlinear interactions in nonlinear crystal materials with a <$\chi^{(2)}$> nonlinearity normally take place at low enough optical intensities to make such nonlinear influences on the phase matching negligible.

## Phase-matching Techniques

The usual technique for achieving phase matching in nonlinear crystals is birefringent phase matching, where one exploits birefringence to cancel the phase mismatch. This technique comes in many variations:

*Type I phase matching*means that, e.g., in sum frequency generation the two fundamental beams have the same polarization, perpendicular to that of the sum frequency wave. Conversely, in*type II phase matching*, the two fundamental beams have different polarization directions; this can be appropriate when the birefringence is relatively strong (overcompensating the dispersion in a type I scheme) and/or the phase velocity mismatch is small. The distinction between type I and type II similarly applies to frequency doubling, and to processes such as degenerate or nondegenerate parametric amplification. The different polarization arrangements can have various practical implications, for example for the combination of several nonlinear conversion stages, or for intracavity frequency doubling.*Critical phase matching*means that an angular adjustment of the crystal (or the beam) is used to find a phase-matching configuration, whereas in*noncritical phase matching*all polarization directions are along the crystal axes, and the angular position is then not a sensitive parameter.- The wave vector of all involved beams may have the same direction (
*collinear phase matching*) or different directions (*noncollinear phase matching*), where, however, the vector sum of the generating beams equals the wave vector of the product beam. A special case is*achromatic phase matching*where at least one of the interacting beams is angularly dispersed so that each frequency component of the signal is properly phase-matched.

A special technique of significant importance is *quasi-phase matching*, where real phase matching does not occur, but high conversion efficiencies are nevertheless obtained in a crystal where the sign (or strength) of the nonlinearity varies periodically. Such a periodic variation of nonlinearity can be achieved e.g. by periodic poling. Normally, one then has equal polarization states of all involved waves (as the birefringence is not needed), and this configuration is sometimes called *type-0 phase matching*.

## Phase-matching Bandwidth and Group Velocity Mismatch

When phase matching is achieved, the group velocities of the interacting waves are in general still not matched; there is a certain group velocity mismatch, which limits the interaction length for pulses and (for a given interaction length) the spectral range (called phase-matching bandwidth) in which phase matching is achieved. Also, there is only a finite range of beam angles where phase matching works – particularly for critical phase matching. This range of angles is usually called the *angular phase-matching bandwidth*. See the article on phase-matching bandwidth for more details.

## Choice of Phase-matching Configurations: an Example

For any given crystal and nonlinear interaction (characterized by the participating wavelengths), there may be multiple phase-matching configurations, the comparison of which involves a number of important properties. Which configuration is most suitable, can depend strongly on the application.

An example is sum frequency generation in an LBO crystal, where two inputs are 1064 and 1535 nm generate a red output at 628 nm. For this interaction, there exist three critical and two noncritical phase-matching schemes, considering only collinear phase matching. Also, the cases with critical phase matching are restricted to a crystal temperature of 20 °C. These schemes are listed in Table 1.

Scheme | nonlinearity | walk-off | GVM |
---|---|---|---|

XZ I(ee-o), <$\theta$> = 88.0° | 0.85 pm/V | <$\rho_1$> = −1.73 mrad <$\rho_2$> = −1.82 mrad |
GVM_{13} = −0.30 fs/mm GVM _{23} = +6.0 fs/mm |

XZ II(eo-e), <$\theta$> = 19.4° | 0.50 pm/V | <$\rho_1$> = −15.1 mrad <$\rho_2$> = −16.3 mrad |
GVM_{13} = +50.4 fs/mm GVM _{23} = −30.3 fs/mm |

YZ II(eo-o), <$\theta$> = 41.1° | 0.51 pm/V | <$\rho_1$> = −9.27 mrad | GVM_{13} = −74.6 fs/mm GVM _{23} = +51.9 fs/mm |

X / ZZY at 13.6 °C | 0.85 pm/V | – | GVM_{13} = −0.29 fs/mm GVM _{23} = +6.0 fs/mm |

Z / XYX at 250 °C | 0.67 pm/V | – | GVM_{13} = +50.4 fs/mm GVM _{23} = −29.5 fs/mm |

As an example of the nomenclature, the critical scheme XZ I(ee-o), <$\theta$> = 88.0° means that the beams propagate in the <$XZ$> plane with an angle of 88.0° to the <$Z$> axis. Both inputs have extraordinary (e) polarization (*type I*), whereas the product wave has ordinary (o) polarization. As an example of a noncritical scheme of type II, take Z / XYX, where the beams propagate in the <$Z$> direction and are polarized in the <$X$> (1535 nm), <$Y$> (1064 nm), and <$X$> (628 nm) directions.

It is important to realize that the choice of phase-matching configuration also influences the effective strength of the nonlinearity because it determines the directions of the electric fields involved with respect to the crystal axes. There are configurations which would offer, e.g., a large gain bandwidth but are hardly usable since the effective nonlinearity would be very weak.

In principle, the noncritical scheme X / ZZY should offer the best performance, having the highest nonlinearity and no spatial walk-off. However, the critical scheme XZ I(ee-o) is actually very similar: starting from the noncritical scheme X / ZZY, one just rotates the crystal by 2° in order to get phase matching at 20 °C instead of 13.6 °C. The walk-off is very weak, as the beam direction is close to the X axis direction.

In the case that a type II scheme is required (e.g. when the input beams are already collinear but have orthogonal polarization directions), the scheme YZ II(eo-o) might be considered, even though the lower group velocity mismatch of scheme XZ II(eo-e) could be slightly more advantageous for ultrashort pulses because the dispersive pulse broadening is reduced.

The group velocity mismatch (GVM) e.g. between some pump and signal waves can be important for the conversion of ultrashort pulses, as is related to the phase-matching bandwidth. Note that the GVM values vary considerably between different schemes, demonstrating that the choice of the material alone does not determine how important this issue is. For each particular phase-matching scheme, the optimum crystal length (particularly for ultrashort pulses) and beam waist need to be found.

Obviously, it is very useful to have software which can systematically identify all possible phase-matching configurations for a given nonlinear interaction, based on, e.g., Sellmeier equations for the refractive indices of all involved waves. On this basis, the most suitable scheme can be selected, taking into account the specific requirements for the conversion device to be designed.

## Phase Matching in Optical Fibers

Fibers also exhibit optical nonlinearities, and the impact of some of those is also subject to constraints of phase matching. For example, this can be the case for cross-phase modulation, four-wave mixing and parametric amplification, but not e.g. self-phase modulation, where all photons involved have the same wavelength, and for Raman scattering. As one is dealing with a <$\chi^{(3)}$> nonlinearity, the relevant physical processes and equations somewhat differ from those for interactions in nonlinear crystals. An additional complication is that phase matching is often significantly influenced by the Kerr effect: the resulting intensity-dependent refractive indices also make the phase-matching relations intensity-dependent.

Most fibers do not have a substantial amount of birefringence. Therefore, phase matching is largely controlled by the chromatic dispersion properties. These can be modified to a substantial extent by the used fiber design – by the refractive index profile of the fiber core, and in the case of photonic crystal fibers through a proper placement of the air holes on the fiber cross-section.

Another possible complication is related to the fluctuations of chromatic dispersion along the fiber. Such fluctuations may be caused by fluctuations of the core diameter, for example, as can result from fluctuations in the fiber fabrication process. Of course, what matters is not e.g. only the *total* group delay dispersion along the fiber, but that the group velocity dispersion is approximately constant along the fiber. Therefore, the results for different fibers, having different amounts of dispersion fluctuations, can substantially differ. Unfortunately, it is not easy to measure chromatic dispersion with spatial resolution.

The explained issues are of particularly great importance in the context of supercontinuum generation, where high optical intensities are applied and a substantial part of the employed nonlinear processes is subject to phase matching.

## More to Learn

Encyclopedia articles:

- phase-matching bandwidth
- birefringent phase matching
- critical phase matching
- noncritical phase matching
- quasi-phase matching

Blog articles:

- The Photonics Spotlight 2007-11-19: “Walk-Off and Phase-Matching Bandwidth in Nonlinear Crystals”
- The Photonics Spotlight 2008-03-10: “Automatic Phase Matching”

### Bibliography

[1] | P. D. Maker et al., “Effects of dispersion and focusing on the production of optical harmonics”, Phys. Rev. Lett. 8 (1), 21 (1962); https://doi.org/10.1103/PhysRevLett.8.21 |

[2] | M. V. Hobden, “Phase-matched second harmonic generation in biaxial crystals”, J. Appl. Phys. 38 (11), 4365 (1967); https://doi.org/10.1063/1.1709130 |

[3] | R. Eckhardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation”, IEEE J. Quantum Electron. 20 (10), 1178 (1984); https://doi.org/10.1109/JQE.1984.1072294 |

[4] | A. V. Smith et al., “Increased acceptance bandwidths in optical frequency conversion by use of multiple walk-off-compensating nonlinear crystals”, J. Opt. Soc. Am. B 15 (1), 122 (1998); https://doi.org/10.1364/JOSAB.15.000122; see also references therein |

[5] | A. M. Schober et al., “Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion”, J. Opt. Soc. Am. B 22 (8), 1699 (2005); https://doi.org/10.1364/JOSAB.22.001699 |

[6] | O. Isaienko and E. Borguet, “Generation of ultra-broadband pulses in the near-IR by non-collinear optical parametric amplification in potassium titanyl phosphate”, Opt. Express 16 (6), 3949 (2008); https://doi.org/10.1364/OE.16.003949 |

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics AG. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

## Questions and Comments from Users

2023-06-19

For an optical modulator, I got some explanations which I don't understand: “To avoid this, our devices have an index mismatch between the microwave phase and optical group index of the TE00 mode of Δn ~ 0.17. For an impedance matched, 1 cm long lossless modulator, this index mismatch corresponds to a theoretical bandwidth of ~80 GHz.” Could you explain?

The author's answer:

This is a different kind of phase matching, occurring in a travelling-wave optical modulator where a microwave drive signals propagates together with the modulated optical signal. Due to the mentioned mismatch between group indices, a short signal pulse, for example, travels with a velocity which somewhat deviates from that of the drive signal. Therefore, the drive signal in one moment of time will affect not just precisely the optical signal at the same time, but in the finite temporal range. Of course, that effect becomes stronger as the modulator is made longer. As a result, you obtain a limitation of the modulator's bandwidth which gets more severe with increasing device length.

2024-07-30

When I have a very broadband fundamental beam, does the SHG only involve the center frequency/wavelength?

The author's answer:

That depends on the phase-matching bandwidth. Phase matching may work for some limited part of the spectrum, or possibly for the whole spectrum.

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2022-12-30

Why do we limit the crystal length to achieve phase matching?

The author's answer:

You can in principle achieve phase matching for any crystal length, but with a long length the detrimental effects of any deviations from ideal phase matching are more severe.