# Derivation of the Gaussian Beam Pattern

I am currently reading this book by Mireille Levy. It says (on page 40) that the Gaussian beam pattern with half-power beamwidth $$\beta$$ is gien by:

$$B(\theta)=A\text{exp}\left(-2\text{log}2\frac{\theta^2}{\beta^2}\right)$$

where $$A$$ is the normalization constant, $$\theta$$ is the elevation angle and $$\beta$$ is the half-power beamwidth.

The context is source modelling of the initial vertical field in parabolic equation modelling. I want to understand this equation and understand where it comes from, but I can't find any information online. Could anyone derive it to me or point me to some source, so I can understand it better? Thanks.

The parabolic equation for a beam propagating in the $$z$$ direction comes from inserting $$\varphi(x,y,z,t) = \psi(x,y,z)e^{i(kz-\omega t)}$$ into the wave equation $$\frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2 \varphi}{\partial y^2}+\frac{\partial^2 \varphi}{\partial z^2}- \frac 1{c^2}\frac{\partial^2 \varphi}{\partial t^2}=0$$ to get $$\frac{\partial^2 \psi}{\partial z^2}+2ik \frac{\partial \psi}{\partial z}- k^2 \psi + \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+ \frac{\omega^2}{c^2}\psi.$$ We now drop the $${\partial^2 \psi}/{\partial z^2}$$ term on the grounds that we have included all the fast changes in $$z$$ in the exponential factor $$e^{ikz}$$. The result is the paraxial wave equation $$2ik \frac{\partial \psi}{\partial z}=- \left(\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}\right) +\left(k^2-\frac{\omega^2}{c^2} \right)\psi.$$ This looks like the time-dependent Schroedinger equation, but with time replaced by the distance $$z$$ down the beam and Planck's constant $$\hbar$$ by twice the wavenumber $$2k$$. We can therefore plug in the textbook gaussian solution to Schroedinger to find what the optical beam does.