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| align="center" | <math> \begin{pmatrix} M & B \\ 0 & \frac{1}{M} \end{pmatrix} </math>
| ''M'' is the total beam magnification given by <math>M = k_1 k_2 k_3 \cdots k_r</math>, where ''k'' is defined in the previous entry and ''B'' is the total optical propagation distance{{clarify|date=July 2019}} of the multiple prism expander.<ref name=TLO />
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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|last=Nazarathy|first=Moshe|last2=Shamir|first2=Joseph|date=1982-03-01|title=First-order optics—a canonical operator representation: lossless systems|url=http://dx.doi.org/10.1364/josa.72.000356|journal=Journal of the Optical Society of America|volume=72|issue=3|pages=356|doi=10.1364/josa.72.000356|issn=0030-3941}}</ref>.
{| border="1" cellspacing="0" cellpadding="4"
!Element
!Matrix in geometrical optics
!Operator in wave optics
!Remarks
|-
|Scaling
|<math>\begin{pmatrix} b^{-1} & 0\\ 0 & b\end{pmatrix} </math>
|<math>\mathcal{V}[b] u(x)=u(b x)</math>
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|Quadratic phase factor
|<math>\begin{pmatrix} 1 & 0\\ c & 1 \end{pmatrix} </math>
|<math>Q[c]=\exp j \frac{k_{0}}{2} c x^{2}</math>
|<math>k_0</math>: wave number
|-
|Fresnel free-space-propagation operator
|<math>\begin{pmatrix} 1 & d\\ 0 & 1 \end{pmatrix} </math>
|<math>\mathcal{R}[d]\left\{U\left(x_{1}\right)\right\}=\frac{1}{\sqrt{j \lambda d}} \int_{-\infty}^{\infty} U\left(x_{1}\right) e^{j \frac{k}{2 d}\left(x_{2}-x_{1}\right)^{2}} d x_1 </math>
|<math>x_1 </math>: coordinate of the source
<math>x_2 </math>: coordinate of the goal
|-
|Normalized Fourier-transform operator
|<math>\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} </math>
|<math>\mathcal{F}=\left(j \lambda_{0}\right)^{-1 / 2} \int_{-\infty}^{\infty} d x\left[\exp \left(j k_{0} p x\right)\right] \ldots </math>
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