The beam radius along the length of a [[Gaussian beam]] is expressed as $w(z)=w_0\bigg[1+\bigg(\frac{z}{z_R}\bigg)^2\bigg]$ ^58a26f or $w(z)=w_0\sqrt{1+\bigg(\frac{2z}{b}\bigg)^2}$ ^55a54d in terms of the [confocal parameter,](Confocal%20parameter.md) $b.$ Here $z$ is the axis along which the beam propagates and $z_R$ is the _[[Rayleigh range]],_ which is one-half the [confocal parameter](Gaussian%20beam.md#confocal%20parameter). The _minimum beam radius_ is denoted by $w_0.$ This the width of a Gaussian beam at its _beam waist._ ^a9abfb Shown below is the Gaussian beam radius, $w$ and its minimum $w_0$ in relation to the overall geometry of the Gaussian beam.  # Gaussian beam radius at the far-field limit The beam approaches the [far-field propagation](Far-field%20propagation.md) limit as $z\rightarrow \infty.$ Thus the beam radius,  is rewritten as $w(z\rightarrow \infty)=w_0\frac{2z}{b}=\frac{2z}{kw_0}=\frac{\lambda z}{\pi w_0}.$ ^4caed6 Here, the envelope is linearized such that we find its slope to be $\frac{\lambda}{\pi w_0}=\tan{\theta}$ and $\theta$ is the angle from the $z$ axis derived from the slope. In practice, this angle is typically in the milliradian scale. Thus we take $\tan{\theta}=\theta$ and we may define the _[divergence angle half angle](Divergence%20angle.md)_ as [$\theta=\frac{\lambda}{\pi w_0}$](Divergence%20angle.md#^11f1c0) ^760ae0 One way in which this quantity is useful is in finding how tightly the Gaussian beam may be focused using the [beam parameter product](Beam%20parameter%20product.md). #Electromagnetism/Optics/waveOptics/GaussianBeamOptics