Complex beam parameter

Defining complex beam parameter

The complex beam parameter is a complex number that consist of properties of a beam such as wavelength λ, radius of curvature R and beam radius w. It is usually denoted by q in the equation:

${\frac {1}{q}}={\frac {1}{R}}-{\frac {i\lambda }{\pi w^{2}}}$.

Uses of complex beam parameter

The complex beam parameter is usually used with the Ray transfer matrix, which allows the calculation of the beam properties at any given point if the ray matrix and the initial complex beam parameter is known. This same method can also be used to find the fundamental mode size of a stable Optical resonator.

Ray matrix and complex beam parameter

Given the initial beam parameter, $q_{i}$, one can use the Ray transfer matrix, ${\begin{pmatrix}A&B\\C&D\end{pmatrix}}$, to find the resulting beam parameter, $q_{f}$, with the equation:

$q_{f}={\frac {Aq_{i}+B}{Cq_{i}+D}}$.

Complex beam parameter of stable resonator

To find the complex beam parameter of a stable optical resonator, one need to find the ray matrix of the cavity. This is done by tracing the path of beam in the cavity. Assuming a starting point, find the matrix that goes through the cavity and return until the beam is in the same position and direction as the starting point. With this matrix and by making $q_{i}=q_{f}$, a quadratic is formed as

$Cq_{f}^{2}+(D-A)q_{f}-B=0$