The complex beam parameter, $q(z)$ describes the geometry of a [[Gaussian beam]] as it propagates in $z.$ This parameter depends on the $z$ coordinate along the beam axis and in simplest terms it is expressed as
$q(z)=z+q_0=z+iz_R$
where $z_R$ is the [Rayleigh range](Rayleigh%20range.md) where we defined $q_0=iz_R$ where $q_0$ is the complex parameter at the [beam waist](Gaussian%20beam.md#beam%20radius) Thus we may rewrite it as
$q(z)=z+q_0=z+i\frac{i\pi n w_0^2}{\lambda_0}$
where $w_0$ is the [beam radius](Gaussian%20beam.md#beam%20radius) at its narrowest point (the _beam waist_.)
Finally, here $n$ is the [[Index of refraction]].
# Relationship with other Gaussian beam parameters
The complex beam parameter has embedded in it the following key parameters that characterize [Gaussian beam](Gaussian%20beam.md)s.
## Confocal parameter
Alternatively we may also define the [complex beam parameter](Complex%20beam%20parameter.md) in terms of the [confocal parameter](Confocal%20parameter.md) such that
$q(z)=z+q_0=z+i\frac{1}{2}b$
## Wavefront curvature and beam radius
It is generally convenient to express the complex beam parameter as $\frac{1}{q(z)}$ where we find that
$\frac{1}{q(z)}=\frac{1}{R(z)}-i\frac{\lambda}{\pi n w^2(z)}$ ^a3e349
where $R(z)$ is the [wave front curvature](Gaussian%20beam.md#wave%20front%20curvature) and $w(z)$ is the [beam radius](Gaussian%20beam.md#beam%20radius).
Thus we may conclude that
$R(z)=\mbox{Re}\bigg(\frac{1}{q(z)}\bigg)$
and
$w(z)=\mbox{Im}\bigg(\frac{1}{q(z)}\bigg)$
# Lens transformations
Given an incident complex beam parameter $q_1$ on an optical element, its transformation to $q_2$ is expressed in terms of [ray transfer matrix elements](Ray%20transfer%20matrix.md#The%20matrix%20elements) as
$q_2=\frac{Aq_1+B}{Cq_1+D}$
The case where $q=q_1=q_2$ describes a ray transfer matrix that models a round trip in a [stable optical cavity](Cavity%20stability.md) since, in an ideal stable cavity the beam remains unchanged with each round trip. In this case we can solve for $q$ such that the beam is stable with the following quadratic equation
$Cq^2+(D-A)q-B=0$
## Derivation of lens transformation
# Derivation of $q(z)$
The complex beam parameter appears in the ansatz for solutions to the [cylindrically symmetric paraxial wave equation](Paraxial%20Wave%20Equation.md#In%20cylindrical%20coordinates) given as 
#Electromagnetism/Optics/waveOptics/GaussianBeamOptics