Editing Ray transfer matrix analysis (section)

== Relation between geometrical ray optics and wave optics ==
The theory of [[Linear canonical transformation]] implies the relation between ray transfer matrix ([[geometrical optics]]) and wave optics.{{sfnp|Nazarathy|Shamir|1982}}

{| class="wikitable plainrowheaders"
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! scope="col" style="max-width: 10em;" | Element
! scope="col" style="max-width: 8em;" | Matrix in geometrical optics
! scope="col" | Operator in wave optics
! scope="col" | Remarks
|-
! scope="row" | Scaling
| style="text-align: center;" | <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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|-
! scope="row" | Quadratic phase factor
| style="text-align: center;" | <math>\begin{pmatrix} 1 & 0\\ c & 1 \end{pmatrix} </math>
|<math>Q[c]=\exp i \frac{k_{0}}{2} c x^{2}</math>
|<math>k_0</math>: wave number
|-
! scope="row" | Fresnel free-space-propagation operator
| style="text-align: center;" | <math>\begin{pmatrix} 1 & d\\ 0 & 1 \end{pmatrix} </math>
|<math>\mathcal{R}[d]\left\{U\left(x_{1}\right)\right\}=\frac{1}{\sqrt{i \lambda d}} \int_{-\infty}^{\infty} U\left(x_{1}\right) e^{i \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
|-
! scope="row" | Normalized Fourier-transform operator
| style="text-align: center;" | <math>\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} </math>
|<math>\mathcal{F}=\left(i \lambda_{0}\right)^{-1 / 2} \int_{-\infty}^{\infty} d x\left[\exp \left(i k_{0} p x\right)\right] \ldots </math>
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