Gaussian beam

Peak Intensity

From the definitions (1) and (2): \[ I(r,z) = I_0 \left(\frac{w_0}{w(z)}\right)^2 \exp \left(\frac{-2r^2}{w(z)^2}\right) \tag{1}\] \[w(z) = w_0\sqrt{1 + \left(\frac{z}{z_R}\right)^2} \tag{2}\] we can rewrite: \[ \left(\frac{w_0}{w(z)}\right)^2 = \frac{1}{1 + \left(\frac{z}{z_R}\right)^2} \tag{3}\] and therefore: \[ I(r,z) = I_0 \frac{1}{1 + \left(\frac{z}{z_R}\right)^2} \frac{1}{\exp \left(\frac{2r^2}{w(z)^2}\right)} \tag{4}\] Clearly \( I(0,0) = I_0\), and \(I(r,z) \leq I_0\) for any real value of \(r\) or \(z\). Therefore \(I_0\) is the peak intensity.

It is common to normalize to 'unity' (or 1) using equation (1) in the form: \[ \frac{I(r,z)}{I_0} = \left(\frac{w_0}{w(z)}\right)^2 \exp \left(\frac{-2r^2}{w(z)^2}\right) \tag{5}\]