Complex beam parameter

In optics, the complex beam parameter is a complex number that specifies of properties of a Gaussian beam at a particular point in space by its wavelength λ, radius of curvature R and beam radius w (defined at 1/e² intensity). It is usually denoted by q in the equation

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

The parameter is equivalently defined by

$q(z)=z+z_{o}i,$

where z is the location at which q is calculated, relative to the location of the beam waist, and z_o is the Rayleigh length.

Uses of the 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 propagates through 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.

Given the initial beam parameter, q_i, one can use the ray transfer matrix of an optical system, ${\begin{pmatrix}A&B\\C&D\end{pmatrix}}$, to find the resulting beam parameter, q_f, after the beam has traversed the system:

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

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:

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

Solving this equation gives the beam parameter for the chosen starting position in the cavity, and by propagating, the beam parameter for any other location in the cavity can be found.