# Spatial Walk-off

Definition: the phenomenon that the intensity distribution of a beam in an anisotropic crystal drifts away from the direction of the wave vector

Opposite term: temporal walk-off

German: räumliches Weglaufen

Categories: nonlinear optics, physical foundations

Units: mrad, °

Formula symbol: <$\rho$>

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/npq

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For a laser beam propagating in an isotropic medium, the transverse intensity distribution propagates along the beam axis as defined by the medium wave vector (= <$k$> vector). In anisotropic (and thus birefringent) crystals, this is not necessarily the case: it can occur that the intensity distribution drifts away from the direction defined by the wave vector, as illustrated in Figure 1, where the gray lines indicate wavefronts and the blue color the region with significant optical intensity. This phenomenon, called *spatial walk-off*, *birefringent walk-off* or *Poynting vector walk-off* (not to be confused with temporal walk-off), is associated with some finite angle <$\rho$> (called *walk-off angle*) between the Poynting vector and the wave vector. The Poynting vector defines the direction of energy transport, whereas the wave vector is normal to the wavefronts.

Spatial walk-off occurs only for a light beam with extraordinary polarization, propagating at some angle <$\theta$> against the optical axes, so that the refractive index <$n_\textrm{e}$> and the phase velocity become dependent on that angle. The walk-off angle can then be calculated from the equation

$$\tan \rho = - \frac{1}{{{n_{\rm{e}}}}}\frac{{\partial {n_{\rm{e}}}}}{{\partial \theta }}$$where the minus sign indicates that the walk-off occurs in the direction where the refractive index would decrease. The extraordinary index <$n_\textrm{e}$> and its derivative are the values occurring for the specific angle <$\theta$>. A beam with ordinary polarization (where the refractive index is not dependent on the propagation angle) does not experience walk-off.

The magnitude of the walk-off angle is exaggerated in Figure 1. In typical cases, it is in the range between a few milliradians and some tens of milliradians. For propagation directions close to one of the axes of the index ellipsoid, the walk-off can even become much smaller.

## An Example Case

As an example, consider a laser beam propagating with a direction in the <$x-z$> plane of a lithium niobate (LiNbO_{3}) crystal. This material is negative uniaxial, meaning that the refractive index is smallest for polarization along the <$z$> axis (which is the optical axis). With some angle <$\theta$> (<90°) between beam axis and <$z$> axis, the refractive index decreases as <$\theta$> increases. Therefore, the walk-off is directed toward larger <$\theta$>, i.e. away from the optical axis. Figure 2 shows the results of a calculation.

## Spatial Walk-off in Nonlinear Interactions

Spatial walk-off is encountered in nonlinear frequency conversion schemes based on critical phase matching in nonlinear crystals. Its consequence is that the waves interacting within a focused beam lose their spatial overlap during propagation because those waves with extraordinary polarization experience the walk-off, whereas this is not the case for those with ordinary polarization. (Note that birefringent phase matching necessarily involves beams with both polarization states.) In effect, the useful interaction length and thus the conversion efficiency can be limited, and the spatial profile of product beams may be broadened and the beam quality reduced.

Unfortunately, it is no solution simply to work with more strongly focused beams, requiring a shorter interaction length because the spatial walk-off becomes more important for smaller beam radii. The problem is reduced, however, for high optical intensities, which allow for good conversion within a short length.

The phenomenon of spatial walk-off is directly related to that of a finite angular phase-matching bandwidth. The equation above shows that a large walk-off angle occurs in situations with a strong angular dependence of the extraordinary refractive index. In such cases, the phase-matching conditions also depend strongly on the propagation angle, and phase matching becomes incomplete when using tightly focused beams, having a large beam divergence.

It is possible to achieve a kind of *walk-off compensation* by using two subsequent nonlinear crystals which are oriented such that the walk-off directions are opposite to each other [3]. There is then still walk-off within these crystals, but its overall effect can be substantially reduced.

Even with a single nonlinear crystal, the impact of the walk-off in sum frequency generation, for example, can be reduced by slightly shifting one of the input beams (the one having walk-off) in the opposite direction.

Spatial walk-off can be avoided altogether by using a noncritical phase matching scheme. This, however, generally requires operation of the crystal at a temperature which is normally not by coincidence close to room temperature.

### Bibliography

[1] | R. Danielius et al., “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses”, Opt. Lett. 21 (13), 973 (1996); https://doi.org/10.1364/OL.21.000973 |

[2] | D. J. Armstrong et al., “Parametric amplification and oscillation with walkoff-compensating crystals”, J. Opt. Soc. Am. B 14 (2), 460 (1997); https://doi.org/10.1364/JOSAB.14.000460 |

[3] | 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 |

[4] | R. J. Gehr et al., “Simultaneous spatial and temporal walk-off compensation in frequency-doubling femtosecond pulses in β-BaB_{2}O_{4}”, Opt. Lett. 23 (16), 1298 (1998); https://doi.org/10.1364/OL.23.001298 |

See also: wave vector, critical phase matching, noncritical phase matching, nonlinear frequency conversion, nonlinear crystal materials, birefringence, temporal walk-off, spotlight 2007-11-19, spotlight 2010-03-15

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!

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