Complex beam parameter

In optics, the complex beam parameter is a complex number that specifies of properties of a Gaussian beam such as its 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 in ray transfer matrix analysis, which allows the calculation of the beam properties at any given point as it traverses an optical system, 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

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