Editing Gaussian beam (section)

===Beam divergence===
{{Further|Beam divergence}}
Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where {{math|1=''r'' = ''w''(''z'')}}. That is where the intensity has dropped to {{math|1/''e''<sup>2</sup>}} of its on-axis value. Now, for {{math|''z'' ≫ ''z''<sub>R</sub>}} the parameter {{math|''w''(''z'')}} increases linearly with {{mvar|z}}. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose {{math|1=''r'' = ''w''(''z'')}}) and the beam axis ({{math|''r'' {{=}} 0}}) defines the ''divergence'' of the beam:
<math display="block">\theta = \lim_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right).</math>

In the paraxial case, as we have been considering, {{mvar|θ}} (in radians) is then approximately<ref name="svelto153" />
<math display="block">\theta = \frac{\lambda}{\pi n w_0}</math>

where {{mvar|n}} is the refractive index of the medium the beam propagates through, and {{mvar|λ}} is the free-space wavelength. The total angular spread of the diverging beam, or ''apex angle'' of the above-described cone, is then given by
<math display="block">\Theta = 2 \theta\, .</math>

That cone then contains 86% of the Gaussian beam's total power.

Because the divergence is inversely proportional to the spot size, for a given wavelength {{mvar|λ}}, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to ''minimize'' the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section ({{math|''w''<sub>0</sub>}}) at the waist (and thus a large diameter where it is launched, since {{math|''w''(''z'')}} is never less than {{math|''w''<sub>0</sub>}}). This relationship between beam width and divergence is a fundamental characteristic of [[diffraction]], and of the [[Fourier transform]] which describes [[Fraunhofer diffraction]]. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam.<ref>Siegman (1986) p. 630.</ref> From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about {{math|2''λ''/''π''}}.

[[Laser beam quality]] is quantified by the [[beam parameter product]] (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size {{math|''w''<sub>0</sub>}}. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as {{math|''M''<sup>2</sup>}} ("[[M squared]]"). The {{math|''M''<sup>2</sup>}} for a Gaussian beam is one. All real laser beams have {{math|''M''<sup>2</sup>}} values greater than one, although very high quality beams can have values very close to one.

The [[numerical aperture#Laser physics|numerical aperture]] of a Gaussian beam is defined to be {{math|1=NA = ''n'' sin ''θ''}}, where {{mvar|n}} is the [[index of refraction]] of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by 
<math display="block">z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} .</math>