### 2.1 Laguerre–Gauss beams

Because we are developing models based on axisymmetry, we pay special attention to cylindrical
coordinates, and consequently to the Laguerre–Gauss family of modes. If is the coordinate along the
optical axis, the radial coordinate and the azimuthal, a Laguerre–Gauss mode () of
parameter has the following complex amplitude at (the wavelength is and ):
where is the Gouy phase,
is the Rayleigh parameter, and is the beam width at . Parameter
represents the minimum width (waist) of the beam, localized at abscissa . The normalized
radial function is given by
are the generalized Laguerre polynomials. Figure 1 shows the intensity pattern of a
nonaxisymmetric Laguerre–Gauss mode of cosine angular parity. Obviously the origin of angles is arbitrary,
so that we can replace the cosine by a sine, and even combine a cosine mode with a sine mode to obtain a
mode having an axisymmetric intensity. For such an axisymmetric (in intensity) mode, the normalization is
slightly different, so that
See Figure 2 for the intensity pattern of such a readout beam.