定义：一束激光在各向同性介质中传播，横向光强分布会用沿梁轴线的介质中的波矢量定义（= kvector）。在各向异性（从而折射）晶体，这是不一定的情况下：可出现的光强分布偏离的波矢量定义的方向，如图1所示，其中灰色线表示波前和显着的光强度的区域的蓝色。这种现象，称为空间走离，双折射走离或坡印廷矢量走离（不要与时域走离混淆），是一些有限的角度ρ相关（所谓的走离角）坡印廷矢量与波矢之间。坡印廷矢量定义了能量传输的方向，而波矢量是垂直于波前。

图1：空间走离：各向异性晶体中光束的强度分布与波矢稍有不同的方向传播


空间走离只对异常偏振光束发生，在某些角度θ对光轴传播，使折射率ne和相速度变得依赖那个角度。可以从方程中计算出走离角：

负号表示在折射率下降的方向上会发生偏离。具有普通偏振（折射率不依赖于传播角）的光束不会发生偏离。
图1夸大了走行角的大小。在典型情况下，它是在几毫弧度，弧度数之间的距离。对于接近指数椭球轴之一的传播方向，走离甚至可以变得更小。
一个例子

作为一个例子，考虑一个激光光束在X−Z平面上的铌酸锂（LiNbO3）晶体的方向。这种材料是负单轴，这意味着折射率是沿z轴（这是光轴）最小的极化。以一定角度θ（＜90°）梁轴和Z轴之间，折射率随θ增加。因此，走离是为了更大的θ，即远离光轴。图2显示了计算结果。

图2：在室温下铌酸锂晶体中635 nm激光束的折射率和走离角与z轴的传播角有关

非线性交互作用下的空间走离
空间走离现象与有限角相位匹配带宽直接相关。上面的方程表明，在异常折射率有很强的角度依赖性的情况下，出现了一个大的走离角。在这种情况下，相位匹配条件也强烈地依赖于传播角，当聚焦光束具有较大的光束发散时，相位匹配变得不完整。
利用两个后续的非线性晶体，从而使偏离方向彼此相对，就可以实现一种偏离补偿。然后在这些晶体中仍然走，但其整体效果可以大大减少。即使有一个非线性晶体，例如在频率和频移中的走频的影响，也可以通过稍微改变相反方向的输入光束之一（一个已经离开的）来减小。
空间走离可以通过使用一个非临界相位匹配方案完全避免。然而，这通常要求晶体在不接近室温的温度下操作晶体。

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

Formula symbol: ρ

Units: mrad, °

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 ρ (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.

Figure 1: Spatial walk-off: the intensity distribution of a beam in an anisotropic crystal propagates in a direction which is slightly different to that of the wave vector.

Spatial walk-off occurs only for a light beam with extraordinary polarization, propagating at some angle θ against the optical axes, so that the refractive index n_e and the phase velocity become dependent on that angle. The walk-off angle can then be calculated from the equation

where the minus sign indicates that the walk-off occurs in the direction where the refractive index would decrease. The extraordinary index n_e and its derivative are the values occurring for the specific angle θ. 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 (LiNbO₃) 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 θ (<90°) between beam axis and z axis, the refractive index decreases as θ increases. Therefore, the walk-off is directed toward larger θ, i.e. away from the optical axis. Figure 2 shows the results of a calculation.

Figure 2: Refractive indices and walk-off angle for a 635-nm laser beam in LiNbO₃ at room temperature as functions of the propagation angle with respect to the z axis.

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