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Japanese Journal of Applied Physics

Generalized Fresnel Transformations and Their Properties

Nobuo Aoyagi, Shoichiro Yamaguchi

Published Sep 1, 1973• 6 citations
Publisher

Abstract

A theory of generalized Fresnel transformations, which is an important part of transformation of scalar-wave propagating between two quadratic surfaces, is presented. The generalized Fresnel transformation is formulated in terms of generalized Fresnel transform operator T( Z ). A set of { T( Z )} has group properties. The differential formulation of the generalized Fresnel transform is obtained by means of functional analytic method. In a special case of the generalized Fresnel transform, one may obtain the differential formula of Fresnel diffraction and Fourier transform.

DOI
10.1143/jjap.12.1343

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A theory of generalized Fresnel transformations, which is an important part of transformation of scalar-wave propagating between two quadratic surfaces, is presented. The generalized Fresnel transformation is formulated in terms of generalized Fresnel transform operator T( Z ). A set of { T( Z )} has group properties. The differential formulation of the generalized Fresnel transform is obtained by means of functional analytic method. In a special case of the generalized Fresnel transform, one may obtain the differential formula of Fresnel diffraction and Fourier transform.
The theory of Fresnel diffraction is developed strictly by means of functional analytic method. Fresnel transforms and inverse Fresnel transforms, which give a basis for Fresnel diffraction, are formulated systematically in terms of Fresnel diffraction operator T( z ). Then, it is shown that the set { T( z )} is a one-parameter group of unitary and factor-type operators from the algebraic and topological properties of T( z ). Furthermore, from the representation with respect to the infinitesimal operator of T( z ), the differential formulation of Fresnel diffraction is obtained. This infinitesimal operator, which is -i k times the paraxial approximate expression of the quantized Hamiltonian operator of the variational problem which is given by Fermat's principle in optics, is closely related to the light ray slope.
Via the route of developing Dirac's symbolic method and following Dirac's assertion: "⋯ for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory", we find the generalized Fresnel operator (GFO) corresponding to the generalized Fresnel transform (GFT) in classical optics. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in quantum optics. With the aid of entangled state representation the entangled Fresnel transform is proposed; new eigenfunctions of the complex fractional Fourier transform and fractional Hankel transform are obtained; the two-variable Hermite eigenmodes of light propagation are used in studying the Talbot effect in quadratic-index media; the complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Moreover, quantum optical version of classical z-transforms is obtained on the basis of the eigenvector of creation operator. Throughout our discussions, the coherent state, squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used.
In this paper it is shown the performing of an optical transform to state the scalar diffraction in the formulation of the wavelet transform and the 'wave equations'. From there, a bridge is build between equations of spherical waves presented in 1678 by Huygens and the continuous wavelet transform. For such a purpose, wavelets are introduced that meet the principles of waves and the properties of wavelets. The following equations are applied in solution to show a correspondence between the Huygens-Fresnel diffraction and the wavelet transform.
By virtue of the new technique of performing integration over Dirac’s ket-bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.
Based on the displacement-squeezing related squeezed coherent state representation | z⟩ g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens–Fresnel integration transformation describing optical diffraction). The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz - rz * in the | z⟩ g representation, while | z⟩ g in phase space is graphically denoted by an ellipse.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.