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== Eigenvalues of Ray Transfer Matrix ==
A ray transfer matrix can be regarded as a [[linear canonical transformation]]. According to the eigenvalues of the optical system, the system can be classified into several classes.<ref>{{Cite journal|
<math> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}
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== Relation between geometrical ray optics and wave optics ==
The theory of [[Linear canonical transformation]] implies the relation between ray transfermatrix ([[geometrical optics]]) and wave optics.<ref>{{Cite journal|
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== Ray transfer matrices for Gaussian beams ==
The same matrices can also be used to calculate the evolution of [[Gaussian beam]]s.<ref>{{cite journal|last1=Rashidian vaziri|first1=M R|title=New ducting model for analyzing the Gaussian beam propagation in nonlinear Kerr media and its application to spatial self-phase modulations|journal=Journal of Optics|volume=15|issue=3|pages=035202|doi=10.1088/2040-8978/15/3/035202|bibcode=2013JOpt...15c5202R|year=2013|s2cid=123550261 }}</ref> propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength <math>\lambda_0</math>, radius of curvature ''R'' (positive for diverging, negative for converging), beam spot size ''w'' and refractive index ''n'', it is possible to define a [[complex beam parameter]] ''q'' by:<ref name=Lei/>
:<math> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} . </math>
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