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This technique, as described below, is derived using the ''[[paraxial approximation]]'', which requires that all ray directions (directions normal to the wavefronts) are at small angles ''θ'' relative to the [[optical axis]] of the system, such that the approximation <math>\sin \theta \approx \theta</math> remains valid. A small θ further implies that the transverse extent of the ray bundles (''x'' and ''y'') is small compared to the length of the optical system (thus "paraxial"). Since a decent imaging system where this is ''not'' the case for all rays must still focus the paraxial rays correctly, this matrix method will properly describe the positions of focal planes and magnifications, however [[Optical aberration|aberrations]] still need to be evaluated using full [[Ray tracing (physics)#Optical design|ray-tracing]] techniques.<ref>Extension of matrix methods to tracing (non-paraxial) meridional rays is included [http://spie.org/ETOP/1991/389_1.pdf here].</ref>
== Matrix definition ==
[[File:RayTransferMatrixDefinitions.svg|thumb|300px|In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between <math>(x_1, \theta_1)</math> at the input plane and <math>(x_2, \theta_2)</math> when the ray arrives at the output plane.]]
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Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different [[refractive index|refractive indices]], reflection from [[mirror]]s, etc.
== Eigenvalues
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|last1=Bastiaans|first1=Martin J.|last2=Alieva|first2=Tatiana|date=2007-03-14|title=Classification of lossless first-order optical systems and the linear canonical transformation|url=http://dx.doi.org/10.1364/josaa.24.001053|journal=Journal of the Optical Society of America A|volume=24|issue=4|pages=1053–1062|doi=10.1364/josaa.24.001053|pmid=17361291 |issn=1084-7529}}</ref> Assume the ABCD matrix representing a system relates the output ray to the input according to
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# A pair of two unimodular, complex conjugated eigenvalues <math>e^{i\theta}</math> and <math>e^{-i\theta}</math>. This case is similar to a separable [[Fractional Fourier transform|Fractional Fourier Transform]].
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== Common
There exist infinite ways to decompose a ray transfer matrix <math> \mathbf{T} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} </math> into a concatenation of multiple transfer matrix. For example:
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which represents a periodic function.
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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/>
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