### 7.3 Properties of Gaussian beams

The basic or ‘lowest-order’ Hermite–Gauss mode is equivalent to what is usually called a Gaussian beam and is given by
The parameters of this equation are explained in detail below. The shape of a Gaussian beam is quite simple: the beam has a circular cross section, and the radial intensity profile of a beam with total power is given by
with the spot size, defined as the radius at which the intensity is times the maximum intensity . This is a Gaussian distribution, see Figure 40, hence the name Gaussian beam.

Figure 41 shows a different cross section through a Gaussian beam: it plots the beam size as a function of the position on the optical axis.

Such a beam profile (for a beam with a given wavelength ) can be completely determined by two parameters: the size of the minimum spot size (called beam waist) and the position of the beam waist along the z-axis.

To characterise a Gaussian beam, some useful parameters can be derived from and . A Gaussian beam can be divided into two different sections along the z-axis: a near field – a region around the beam waist, and a far field – far away from the waist. The length of the near-field region is approximately given by the Rayleigh range . The Rayleigh range and the spot size are related by

With the Rayleigh range and the location of the beam waist, we can usefully write
This equation gives the size of the beam along the z-axis. In the far-field regime (), it can be approximated by a linear equation, when

The angle between the z-axis and in the far field is called the diffraction angle and is defined by

Another useful parameter is the radius of curvature of the wavefront at a given point z. The radius of curvature describes the curvature of the ‘phase front’ of the electromagnetic wave – a surface across the beam with equal phase – intersecting the optical axis at the position z. We obtain the radius of curvature as a function of z:

We also find: