# Critical Phase Matching

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: phase matching of a nonlinear interaction by adjustment of a propagation direction

Alternative term: angle phase matching

More general term: phase matching

Opposite term: noncritical phase matching

Critical phase matching (also called *angle phase matching*) is a technique for obtaining phase matching of some nonlinear process (usually nonlinear frequency conversion) in a birefringent crystal. The interacting beams are aligned at some angle(s) to the axes of the index ellipsoid. In almost all cases, there are one or two waves linearly polarized along one axis of the index ellipsoid (*ordinary beam*), while another one or two waves are polarized at some variable angle with the plane spanned by the other two axes (*extraordinary beams*). Adjustment of the propagation angle affects the refractive index of the extraordinary beam (called *extraordinary refractive index*), whereas the ordinary index stays constant. For some angular position, phase matching may be achieved, i.e., the phase mismatch vanishes. Effectively, this means that the birefringence exactly cancels the phase mismatch which would result from chromatic dispersion. Note that this would not be possible if all involved beams had the same polarization direction, since one would have no effect of the birefringence.

As an example, Figure 1 shows the beam direction and the polarization directions for phase-matched second-harmonic generation (frequency doubling) in LBO based on the type I scheme with polarizations oo-e in the XY plane. This means that the beam propagates within the XY plane, the fundamental polarization is ordinary (o, here in the Z direction) and the second-harmonic polarization is extraordinary (e, with an angle <$\varphi$> to the Y axis). Figure 2 shows the required parameters. For example, for a pump wavelength of 800 nm the phase-matching angle <$\varphi$> would have to be 31.6°, and the effective nonlinearity would be 0.72 pm/V. The walk-off angle of the second-harmonic beam (not shown in the diagram) would be 16.5 mrad.

The angle shown is the angle <$\varphi$> in Figure 1. This phase matching scheme is suitable for frequency doubling from 1064 nm to 532 nm, but less so for shorter wavelengths due to the decreasing nonlinearity and the increasing walk-off angle (not shown).

The attribute “critical” comes from the fact that this technique (in contrast to noncritical phase matching) is relatively sensitive to misalignment of the beams. There is only a finite range of beam angles (the *acceptance angle*, also called angular phase-matching bandwidth) where critical phase matching works; in the example above, it is 0.67 mrad for a 1 cm long crystal. This also implies that the beam divergence must be limited, and that often forces one to use beams with a large beam radius. Efficient nonlinear conversion then requires a higher peak power.

A related problem is the spatial walk-off between ordinary and extraordinary beams, which limits the effective interaction length and may affect both the conversion efficiency and the beam quality. With sufficiently high optical powers, walk-off effects can be minimized by using a short nonlinear crystal and/or large beam diameters.

Nonlinear crystals are often cut such that critical phase matching is possible for close to normal incidence on the input surface. It may be less convenient to use a crystal which needs a substantially non-normal incidence, e.g. because it was designed for use with noncritical phase matching.

A major advantage of critical phase matching is that the crystal temperature can often be close to room temperature, so that a crystal oven is not required. One may only need to stabilize the crystal temperature near room temperature, e.g. using a Peltier element. That also minimizes the required warm-up time.

The above-mentioned limitations of critical phase matching can be severe in some cases, but get less relevant in cases where beams with a high peak power are converted.

Note that critical phase matching (in fact any kind of birefringent phase matching) can work only if the polarization directions of fundamental and second-harmonic light are different. This is not always the case, but there are many configurations where it is so, based on the properties of the nonlinear tensor.

## More to Learn

Encyclopedia articles:

- noncritical phase matching
- phase matching
- phase-matching bandwidth
- spatial walk-off
- nonlinear frequency conversion

## Questions and Comments from Users

2022-12-13

You refer to this diagram as illustrating critical phase matching, even though the phase matching angle theta here is 90 degrees. This is, in fact, what is known as non-critical phase matching.

The author's answer:

No, it is critical phase matching. While <$\theta$> = 90°, we also have <$\varphi$> not being 0 or 90°.

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2021-02-18

In Figure 1, phi appears two times, but not theta. Is that correct?

The author's answer:

Yes, it is. The diagram indicates that <$\varphi$> is the angle between the X axis and the beam direction, and also between the Y axis and the polarization of the second-harmonic wave. In this case, we have <$\theta$> = 90°.