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Encyclopedia of Laser Physics and Technology

© Dr. Rüdiger Paschotta

Last update: 2008-04-24

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

Many phase-sensitive nonlinear processes, in particular parametric processes such as frequency doubling, sum and difference frequency generation, parametric amplification and oscillation, as well as four-wave mixing, require phase matching to be efficient. Essentially this means to ensure that a proper phase relationship between the interacting waves (for optimum nonlinear frequency conversion) is maintained along the propagation direction. More precisely, the phase mismatch has to be close to zero; for example, for frequency doubling with collinear beams the phase mismatch is given by

where k₁ and k₂ are the wavenumbers of the fundamental and second-harmonic beam, respectively. Without chromatic dispersion, k₂ = 2 k₁ 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.

Figure 1: Phase mismatch for second-harmonic generation. 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, see below).

Figure 2 illustrates how a phase mismatch keeps the efficiency low: the direction of energy transfer changes periodically according to the change of the phase relation between the interacting waves, so that in a non-phase-matched case the energy oscillates between the waves rather than being transferred in a constant direction. In a frequency doubler, the direction of energy transfer is governed by the complex phase of the term E₁² E₂^*, where E₁ is the complex electric field amplitude of the fundamental wave and E₂ that of the second harmonic. The phase-matching condition k₂ = 2 k₁ means that the phase of the term E₁² E₂^* stays constant along the propagation direction. For a sum frequency generation process, or in a non-degenerate optical parametric oscillator, the corresponding term would be E₁ E₂ E₃^*, and the phase-matching condition would be k₃ = k₁ + k₂.

Figure 2: Growth of second-harmonic power in a crystal along the propagation direction, assuming a constant pump intensity. Solid curve: phase-matched case, with the power growing in proportion to the square of the propagation distance. Dotted curve: non phase-matched case, with the second-harmonic power oscillating between zero and a small value.

For devices like frequency doublers or optical parametric amplifiers, phase matching needs to be achieved, because otherwise the conversion efficiency would be very low. On the other hand, an optical parametric oscillator will 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 varies, as shown in Figure 3. The temperature range in which good conversion efficiency is obtained is inversely proportional to the crystal length. It also depends on the temperature dependence of the involved refractive indices. Similar relations apply to other nonlinear frequency conversion processes.

Figure 3: Second-harmonic power versus temperature deviation from the optimum point, assuming a homogeneous temperature distribution in the crystal and a constant pump intensity (low conversion efficiency).

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 the figure above. 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.

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 of course to processes like degenerate or non-degenerate 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 vectors of all involved beams may have the same direction (→ collinear phase matching) or different directions (→ noncollinear phase matching), where however still the vector sum of the generating beams equals the corresponding sum for the product beams. 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.

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 strongly depend on the application.

An example is sum frequency generation in an LBO crystal, where two inputs are 1064 nm 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 the following:

As an example for the nomenclature, the critical scheme XZ I(ee-o), 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 for a noncritical scheme, take Z / XYX, where the beams propagate in Z direction and are polarized in X (1535 nm), Y (1064 nm), and X (628 nm) direction.

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 involved electric fields with respect to the crystal axes. There are configurations which would e.g. offer 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: compared to the mentioned noncritical scheme, one just slightly rotates the crystal in order to operate at 20°C instead of 13.6°C. The walk-off is correspondingly weak. 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.

Note that the GVM values vary a lot 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 a software which can systematically identify all possible phase-matching configurations for a given nonlinear interaction, based e.g. on Sellmeier equations for the refractive indices of all involved waves. On this basis, the best suitable scheme can be selected, taking into account the concrete requirements for the conversion device to be designed.

Various Remarks

Note that phase matching is in some cases further complicated by the Kerr effect: the resulting intensity-dependent refractive indices also make the phase-matching relations intensity-dependent, and this effect is sometimes quite significant, e.g. for four-wave mixing processes in fibers.

Phase matching is relevant not only for χ⁽²⁾ nonlinearities of nonlinear crystal materials, but also for other nonlinear processes. For example, χ⁽³⁾-based nonlinear interactions in fibers (e.g. parametric amplification or four-wave mixing) can be phase-matched under certain conditions. Such issues are also of great importance in the context of supercontinuum generation. Some nonlinear processes are automatically phase-matched; an example is self-phase modulation, where all involved photons have the same wavelength.

See also: phase-matching bandwidth, birefringent phase matching, critical phase matching, noncritical phase matching, quasi-phase matching, nonlinear frequency conversion, nonlinear polarization, nonlinear crystal materials, Spotlight article 2008-03-10

Categories: methods, nonlinear optics

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics Consulting GmbH. Contact this distinguished expert in laser technology, nonlinear optics and fiber optics, and find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, or staff training) could become very valuable for your business!