Linear canonical transformation

It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain).

The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group.

The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered.

Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates.

The LCT can be represented in several ways; most easily,[1] it can be parameterized by a 2×2 matrix with determinant 1, i.e., an element of the special linear group SL2(C).

with ad − bc = 1, the corresponding integral transform from a function

The Fourier transform corresponds to a clockwise rotation by 90° in the time–frequency plane, represented by the matrix

The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL2(R), represented by the matrices

The Fresnel transform corresponds to shearing, and are a family of parabolic elements, represented by the matrices

The Laplace transform corresponds to rotation by 90° into the complex domain and can be represented by the matrix

The fractional Laplace transform corresponds to rotation by an arbitrary angle into the complex domain and can be represented by the matrix[2]

Occasionally the product of transforms can pick up a sign factor due to picking a different branch of the square root in the definition of the LCT.

Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded-index (GRIN) media, are quadratic-phase systems (QPS); these were known before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics.

The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform, a particular case of which was developed by Segal (1963) and Bargmann (1961) in order to formalize Fock's (1928) boson calculus.

[3] In quantum mechanics, linear canonical transformations can be identified with the linear transformations which mix the momentum operator with the position operator and leave invariant the canonical commutation relations.

Canonical transforms are used to analyze differential equations.

These include diffusion, the Schrödinger free particle, the linear potential (free-fall), and the attractive and repulsive oscillator equations.

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.

[4] Wave propagation through air, a lens, and between satellite dishes are discussed here.

The Fresnel transform is used to describe electromagnetic wave propagation in free space:

With the lens as depicted in the figure, and the refractive index denoted as n, the result is[5]

where f is the focal length, and Δ is the thickness of the lens.

The distortion passing through the lens is similar to LCT, when

This is very similar to lens, except focal length is replaced by the radius R of the dish.

Therefore, if the radius is smaller, the shearing effect is larger.

The relation between the input and output we can use LCT to represent

In this part, we show the basic properties of LCT Given a two-dimensional column vector

we show some basic properties (result) for the specific input below: The system considered is depicted in the figure to the right: two dishes – one being the emitter and the other one the receiver – and a signal travelling between them over a distance D. First, for dish A (emitter), the LCT matrix looks like this:

Then, for dish B (receiver), the LCT matrix similarly becomes:

Last, for the propagation of the signal in air, the LCT matrix is: