Three decades ago I began my PhD with the great Allister Ferguson. My goal would become to design and build the world’s first **single-frequency Ti:sapphire laser**. In January 1989 I began by designing a Nd:YAG ring laser. Below is a photograph of the first page of my PhD notebook.

To design a laser cavity, you start with the equations that describe the propagation of laser beams, and on the left (in the image) you can see a matrix equation that describes how the **complex beam parameter** *q* of a laser beam changes when it passes through a lens with focal length *f* (or is reflected by a curved mirror). I had learnt this equation from Geoff Brooker, except that his equation (see here) included an odd factor called “cancel me”. At the time I did not understand why I needed to “cancel me” but as long the equations gave the right answers (which they do) I could start building lasers.

The Nd:YAG laser worked and we published a paper, then I started work on the Ti:sapphire laser. Using the same equations this is the desgin I came up with.

Using an early version of the laser we published a paper on Doppler-free spectroscopy of rubidium atoms. Later this laser was developed into a commercial product by Microlase who subsequently became Coherent Scotland. Hundreds of these lasers (perhaps one thousand by now costing around 100k each) have been sold worldwide. I mention this because it helps us to appreciate the utility of both maths and physics.

Thirty years on, while writing a textbook on optics, I found the motivation to revisit those old equations on the complex beam parameter and rewrite them in a more elegant form. As I suspected there is no need for “cancel me” or even to write the equation in the form of a matrix times a vector. The point is that a gaussian beam is simply a paraxial spherical wave with a complex argument (see p. 178 in Optics *f*2*f*) and an ideal lens or curved mirror is simply an element which modifies the curvature of a spherical wave (see p. 27-8 of Optics *f*2*f*).

This beautiful piece of mathematics (p. 181 in Optics *f*2*f*), with real applications, has been putting `bread on the table’ for many, over many years.

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