= ABCD law for Gaussian beams
{c}
{title2=$q'=(Aq+B)/(Cq+D)$}
= ABCD law
{c}
{synonym}
A real <paraxial approximation>[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.
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