定义：
衡量光束的横向扩展。

定义平顶分布的激光光束半径是不必要的，但是大多数光束具有其他的横向形状。常得到的是高斯光束，其横向强度分布表示为： 

其中光束半径是指光强降为最大值的 1/e2 (≈ 13.5%)时与光轴之间的距离。这时，电场强度降为最大值的 1/e (≈ 37%)。 

对于任意形状的光束（可能不是高斯光束），存在一些不同的光束半径的定义。可以采用强度变为1/e2时的半径，或者半高全宽（FWHM）对应的半径，还有包含86%光束能量的半径等。采用这些定义主要是由于曲线两翼的强度衰减程度对结果没有影响。如图1，尽管虚线显然比实线更宽，但是两个强度分布具有相同的半高全宽。在有些复杂的强度分布时，FWHM的定义是不合适的。 

图1：FWHM相同的两光束的强度分布曲线。 

推荐的定义是ISO 标准 11146，采用的是强度分布 I(x,y)的二阶矩。例如，在x方向的光束半径为： 

其中坐标x和y是相对于光束的中心而言，即一阶矩为0.这种方法通常称为D4σ方法，因为这里得到的光束直径是强度分布导数的四倍。 

对于高斯光束，D4σ方法与1/e2 方法得到的结果相同，而其它光束形状则会有很大的不同。尤其是在表征非衍射极限光束的光束半径变化时，需要采用 D4σ方法。给定[[M2]]因子的高斯光束采用通常的方法就可以得到正确的光束半径变化情况，而采用其它的光束半径定义就会出现错误。这在设计二极管泵浦激光器时非常重要，因为存在非高斯光束形状的情况。 

二阶矩方法的缺点在于计算光束半径比较复杂（通常需要数值代码），并且结果会由于测量的强度分布的垂直偏移而很容易受到影响（例如，由杂散光或者相机的噪声等）。 
不管光束半径采用何种定义，光束直径都是光束半径的二倍。对于高斯光束，FWHM光束直径是高斯光束半径（(1/e2值）的1.18倍。 
存在许多种类的光束分析仪可以测量光束半径。

Definition: a measure of the transverse extension of a light beam

Formula symbol: w

Units: m

The definition of the radius of a laser beam with a flat-top profile is trivial, but most light beams have other transverse shapes. A frequently obtained shape is the Gaussian one, where the transverse intensity variation is described with the following equation:

where the beam radius w is the distance from the beam axis where the optical intensity drops to 1/e² (≈ 13.5%) of the value on the beam axis. At this radius, the electric field strength drops to 1/e (≈ 37%) of the maximum value.

The usual formula symbol of the beam radius is w. Some authors use ω instead, but that may be confused with an angular frequency.

Definition of Beam Radius for Arbitrary Profile Shapes

For arbitrary (possibly not Gaussian) beam shapes, several different definitions are common. It is possible to still use the 1/e² intensity criterion, or a full width at half-maximum (FWHM), or a radius including 86% of the beam energy, etc. The problem with this type of definitions is essentially that the result does not depend on, e.g., how quickly the intensity decays in the wings of the profile. To illustrate this, Figure 1 shows two intensity profiles which have the same FWHM width, although the dashed curve is clearly wider in a meaningful sense. In the case of complicated intensity patterns, it is even more obvious that an FWHM definition cannot be appropriate.

Figure 1: Intensity profiles of two beams which have the same FWHM.

ISO Standard 11146

For such reasons (and another reason, which is discussed below), the recommended definition is that of ISO Standard 11146, based on the second moment of the intensity distribution I(x,y). For example, the beam radius in the x direction is

where the coordinates x and y must be taken to be relative to the beam center, i.e., such that the first moments vanish. The method is also called the D4σ method, because for the beam diameter, one obtains 4 times the standard deviation of the intensity distribution.

One can define the beam radius w_y in an analogous fashion, just replacing x² with y² in the upper integral.

For Gaussian beams, the D4σ method gives the same result as the 1/e² method, whereas for other beam shapes there can be significant deviations. The D4σ method should be used particularly when trying to predict the evolution of the beam radius for not diffraction-limited beams. It has been shown that the usual rules for Gaussian beam propagation with a certain M² factor then correctly describe this evolution, whereas errors occur when using beam radii defined in some other way. This is important to observe e.g. when designing the pump optics of a diode-pumped laser, because clearly non-Gaussian beam shapes can occur.

Disadvantages of the second-moment method are that the beam radius calculation is somewhat complicated (it usually requires numerical code), and that the result is quite sensitive to the intensity of the outer parts of the profile. For example, it is easily compromised by some vertical offset in the measured intensity distribution (e.g. caused by ambient light) and by noise of the camera. A detector with high dynamic range is therefore required. Often one also uses special smoothing techniques for reducing measurement errors.

Caution: the effective beam radius is not the same as the Gaussian beam radius!

Effective Beam Area

In the context of laser-induced damage, one often uses an effective beam area, which is defined as the optical power divided by the maximum intensity, and is considered to be π times the effective beam radius squared. For a Gaussian beam, that effective beam radius is smaller than the Gaussian beam radius by a factor square root of 2.

The beam diameter is generally defined as twice the beam radius – no matter what the particular definition of beam radius is. For Gaussian beams, the FWHM beam diameter is 1.18 times the Gaussian beam radius (1/e² value).

Measurement of Beam Radius

For the measurement of beam radii, various methods are applied. Obviously, a first step should always be to determine which definition of beam radius is to be used.

Some of the definitions and related measurement methods are applicable only to Gaussian beams. For example, this is mostly true for beam profiles using the knife edge or slit method, i.e., measuring the time-dependent transmitted optical power when a knife edge or a slit is moved through the beam. That method can also be applied with quite simple equipment, e.g. a knife edge mounted on a translation stage and an optical power meter, but the procedure is relatively cumbersome.

For arbitrary beam shapes, one usually uses camera-based beam profilers, with which the beam radius according to ISO 11146 can usually be determined quite quickly. Note, however, that this method often requires the use of an optical attenuator in order to get into the limited dynamic range of the camera, and obviously one must take care that the beam profile is not distorted by the attenuator. Further, not all devices can be applied to laser pulses, or only in certain parameter regions.

Bibliography