Complex beam parameter ($q$)

The complex beam parameter combines a Gaussian beam's wavefront curvature $R_b$ and spot radius $w$: $1/q=1/R_b-i\lambda /(\pi w^2)$. With $q=z+i{z}_R$, the beam waist is $q=i{z}_R$. A real paraxial ray transfer matrix maps it by $q^{\prime }=(Aq+B)/(Cq+D)$. Positive imaginary part selects the physical finite-width branch in this convention.

A real paraxial ray matrix transforms a Gaussian beam parameter by the displayed fractional-linear map. For a lossless round trip with determinant one, a fixed point with positive imaginary part generically requires $|A+D|<2$. Scalar matrices $\pm I$ are degenerate exceptions: their fractional-linear map fixes every physical beam parameter even at the boundary. Finite apertures and resonance phases impose additional physical constraints.