- Linear canonical transformation
Paraxial optical system s implemented entirely withthin lens es and propagation through free space and/or graded index (GRIN) media, areQuadratic Phase System s (QPS). The effect of any arbitrary QPS on an input wavefield can be described using the**linear canonical transform**(LCT), a unitary, additive, four-parameter class of linear integral transform.The former appeared a couple of times before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. A particular case of the latter was developed by Segal (1963) and Bargmann(1961) in order to formalized Fok's boson calculus (1928). [

*K.B. Wolf, "Integral Transforms in Science and Engineering," Ch. 9:Canonical transforms, New York, Plenum Press, 1979.*]The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss-Weierstrass, Bargmann and the Fresnel transforms as particular cases.

**Mathematical Definition**There are maybe several different ways to represent LCT. However, LCT can be viewed as a 2x2 matrix with determinant of the matrix is equal 1.:$X\_\{(a,b,c,d)\}(u)\; =\; sqrt\{-i\}\; cdot\; e^\{-i\; pi\; frac\{d\}\{b\}\; u^\{2\; int\_\{-infty\}^infty\; e^\{-i\; 2\; pi\; frac\{1\}\{b\}\; ut\}e^\{i\; pi\; frac\{a\}\{b\}\; t^2\}\; x(t)\; ,\; dt$ , when $b\; e\; 0\; ,$

:$X\_\{(a,0,c,d)\}(u)\; =\; sqrt\{d\}\; cdot\; e^\{-i\; pi\; cdu^\{2\; x(du)\; ,$ , when $b\; =\; 0\; ,$

:$ad-bc\; =\; 1\; ,$ should be satisfied

**pecial Cases of LCT**Since

**Linear Canonical Transform**is a general term for other transforms, here are some examples of the special case in LCT.**Fourier Transform**Fourier Transform is a special case of LCT.

when $egin\{bmatrix\}\; a\; b\; \backslash \; c\; dend\{bmatrix\}\; =\; egin\{bmatrix\}\; 0\; 1\; \backslash \; -1\; 0end\{bmatrix\}$

**Fractional Fourier Transform**Fractional Fourier Transform is a special case of LCT.

when $egin\{bmatrix\}\; a\; b\; \backslash \; c\; dend\{bmatrix\}\; =\; egin\{bmatrix\}\; cos\; heta\; sin\; heta\; \backslash \; -sin\; heta\; cos\; hetaend\{bmatrix\}$

**Fresnel Transform**Fresnel transform is equivalent towhen $egin\{bmatrix\}\; a\; b\; \backslash \; c\; dend\{bmatrix\}\; =\; egin\{bmatrix\}\; 1\; lambda\; z\; \backslash \; 0\; 1end\{bmatrix\}z:distance\; ;\; lambda:wave\; length$

=Additivity property of the WDF=If we denote the LCT by $O\_F^\{(a,b,c,d)\}\; ,$

i.e., $X\_\{(a,b,c,d)\}(u)\; =\; O\_F^\{(a,b,c,d)\}\; [x(t)]\; ,$

then

:$O\_F^\{(a2,b2,c2,d2)\}\; left\; \{\; O\_F^\{(a1,b1,c1,d1)\}\; [x(t)]\; ight\; \}\; =\; O\_F^\{(a3,b3,c3,d3)\}\; [x(t)]\; ,$

where:$egin\{bmatrix\}\; a3\; b3\; \backslash \; c3\; d3end\{bmatrix\}\; =\; egin\{bmatrix\}\; a2\; b2\; \backslash \; c2\; d2end\{bmatrix\}egin\{bmatrix\}\; a1\; b1\; \backslash \; c1\; d1end\{bmatrix\}$

**Applications**Canonical transforms provide a fine tool for the analysis of a class of differential equations. These include the diffusion, the Schrödinger free-particle, the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the Fokker-Planck equation. Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these. [

*K.B. Wolf, "Integral Transforms in Science and Engineering," Ch. 9&10, New York, Plenum Press, 1979.*]Wave propagation travel through air, lens, and dishes are discussed in here. All of the computations can be reduced to 2x2 matrix algebra. This is the spirit of LCT.

**Electromagnetic Wave Propagation**If we assume the system look like this, the wave travel from plane x

_{i}, y_{i}to the plane of x and y.We can use Fresnel Transform to describe the Electromagnetic Wave Propagation in the air.:$U\_0(x,y)\; =\; -\; frac\{i\}\{lambda\}\; frac\{e^\{ikz\{z\}\; int\_\{-infty\}^\{infty\}\; int\_\{-infty\}^\{infty\}\; e^\{j\; frac\{k\}\{2z\}\; [\; (x-x\_i)^2\; -(y-y\_i)^2\; ]\; \}\; U\_i(x\_i,y\_i)\; ,\; dx\_i\; dy\_i$:$k=\; frac\; \{2\; pi\}\{lambda\}:\; ,$ wave number; $lambda\; :\; ,$ wavelength; $z\; :\; ,$ distance of propagation

This is equivalent to LCT (shearing), when:$egin\{bmatrix\}\; a\; b\; \backslash \; c\; d\; end\{bmatrix\}=\; egin\{bmatrix\}\; 1\; lambda\; z\; \backslash \; 0\; 1\; end\{bmatrix\}$

When the travel distance (z) is larger, the shearing effect is larger.

**pherical lens**With the above lens from the image, and refractive index = n, we get::$U\_0(x,y)\; =\; e^\{ikn\; Delta\}\; e^\{-j\; frac\{x\}\{2f\}\; [x^2\; +\; y\; ^2]\; \}\; U\_i(x,y)$:$f:\; ,$ focal lenth $Delta\; :\; ,$ thickness of length

The distortion passing through the lens is similar to LCT, when:$egin\{bmatrix\}\; a\; b\; \backslash \; c\; d\; end\{bmatrix\}=\; egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; f\}\; 1\; end\{bmatrix\}$

This is also a shearing effect, when the focal length is smaller, the shearing effect is larger.

**atellite Dish**Dish is equivalent to LCT, when:$egin\{bmatrix\}\; a\; b\; \backslash \; c\; d\; end\{bmatrix\}=\; egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; R\}\; 1\; end\{bmatrix\}$

This is very similar to lens, except focal length is replaced by the radius of the dish. Therefore, if the radius is larger, the shearing effect is larger.

**Example**If the system is considered like the following image. Two dishes, one is the emitter and another one is the receiver, and the signal travel through a distance of D.

First, for dish A (emitter), the LCT matrix looks like this::$egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; R\_A\}\; 1\; end\{bmatrix\}$

Then, for dish B (receiver), the LCT matrix looks like this::$egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; R\_B\}\; 1\; end\{bmatrix\}$

Last, we need to consider the propagation in air, the LCT matrix looks like this::$egin\{bmatrix\}\; 1\; lambda\; D\; \backslash \; 0\; 1\; end\{bmatrix\}$

If we put all the effects together, the LCT would look like this::$egin\{bmatrix\}\; a\; b\; \backslash \; c\; d\; end\{bmatrix\}=egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; R\_B\}\; 1\; end\{bmatrix\}egin\{bmatrix\}\; 1\; lambda\; D\; \backslash \; 0\; 1\; end\{bmatrix\}egin\{bmatrix\}\; 1\; 0\; \backslash \; frac\{-1\}\{lambda\; R\_A\}\; 1\; end\{bmatrix\}=egin\{bmatrix\}\; 1-frac\{D\}\{R\_A\}\; -\; lambda\; D\; \backslash \; frac\{1\}\{lambda\}\; (R\_A^\{-1\}\; +\; R\_B^\{-1\}\; -\; R\_A^\{-1\}R\_B^\{-1\}D)\; 1\; -\; frac\{D\}\{R\_B\}\; end\{bmatrix\},$

**ee also**Other time-frequency transforms:

*

Fractional Fourier transform

*continuous Fourier transform

*chirplet transform **References**

* J.J. Ding, "time-frequency analysis and wavelet transform course note," the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.

* K.B. Wolf, "Integral Transforms in Science and Engineering," Ch. 9&10, New York, Plenum Press, 1979.

* S.A. Collins, "Lens-system diffraction integral written in terms of matrix optics," "J. Opt. Soc. Amer."**60**, 1168–1177 (1970).

* M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," "J. Math. Phys."**12**, 8, 1772–1783, (1971).

* B.M. Hennelly and J.T. Sheridan, "Fast Numerical Algorithm for the Linear Canonical Transform", "J. Opt. Soc. Am. A"**22**, 5, 928–937 (2005).

* H.M. Ozaktas, A. Koç, I. Sari, and M.A. Kutay, "Efficient computation of quadratic-phase integrals in optics", "Opt. Let."**31**, 35–37, (2006).

* Bing-Zhao Li, Ran Tao, Yue Wang, "New sampling formulae related to the linear canonical transform", "Signal Processing"**'87**', 983–990, (2007).

* A. Koç, H.M. Ozaktas, C. Candan, and M.A. Kutay, "Digital computation of linear canonical transforms", "IEEE Trans. Signal Process.", vol. 56, no. 6, 2383-2394, (2008).

*Wikimedia Foundation.
2010.*