Linear canonical transformation

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Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are Quadratic Phase Systems (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). ^[1]

The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss-Weierstrass, Bargmann and the Fresnel transforms as particular cases.

1 Mathematical Definition
2 Special Cases of LCT
3 Additivity property of the WDF
4 Applications
5 Example
6 See also
7 References

[edit] 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.

$X_{(a,b,c,d)}(u) = \sqrt{-j} \cdot e^{-j \pi \frac{d}{b} u^{2}} \int_{-\infty}^\infty e^{-j2 \pi \frac{1}{b} ut}e^{j \pi \frac{a}{b} t^2} x(t) \, dt$ , when $b \ne 0 \,$

$X_{(a,0,c,d)}(u) = \sqrt{d} \cdot e^{-j \pi cdu^{2}} x(du) \,$ , when $b = 0 \,$

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

[edit] Special Cases of LCT

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

[edit] Fourier Transform

Fourier Transform is a special case of LCT.

when $\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$

[edit] Fractional Fourier Transform

Fractional Fourier Transform is a special case of LCT.

when $\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \end{bmatrix}$

[edit] Fresnel Transform

Fresnel transform is equivalent to

when $\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & \lambda z \\ 0 & 1 \end{bmatrix} z:distance ; \lambda:wave length$

[edit] 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

$O_F^{(a2,b2,c2,d2)} \left \{ O_F^{(a1,b1,c1,d1)}[x(t)] \right \} = O_F^{(a3,b3,c3,d3)}[x(t)] \,$

where

$\begin{bmatrix} a3 & b3 \\ c3 & d3 \end{bmatrix} = \begin{bmatrix} a2 & b2 \\ c2 & d2 \end{bmatrix} \begin{bmatrix} a1 & b1 \\ c1 & d1 \end{bmatrix}$

[edit] 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.^[2]

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.

[edit] 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

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & \lambda z \\ 0 & 1 \end{bmatrix}$

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

[edit] Spherical 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

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda f} & 1 \end{bmatrix}$

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

[edit] Satellite Dish

Dish is equivalent to LCT, when

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda R} & 1 \end{bmatrix}$

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.

[edit] 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:

$\begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda R_A} & 1 \end{bmatrix}$

Then, for dish B (receiver), the LCT matrix looks like this:

$\begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda R_B} & 1 \end{bmatrix}$

Last, we need to consider the propagation in air, the LCT matrix looks like this:

$\begin{bmatrix} 1 & \lambda D \\ 0 & 1 \end{bmatrix}$

If we put all the effects together, the LCT would look like this:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda R_B} & 1 \end{bmatrix} \begin{bmatrix} 1 & \lambda D \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda R_A} & 1 \end{bmatrix} = \begin{bmatrix} 1-\frac{D}{R_A} & - \lambda D \\ \frac{1}{\lambda} (R_A^{-1} + R_B^{-1} - R_A^{-1}R_B^{-1}D) & 1 - \frac{D}{R_B} \end{bmatrix} \,$

[edit] See also

Other time-frequency transforms:

[edit] References

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

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).

Categories: Integral transforms | Fourier analysis