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Japanese Journal of Applied Physics

Functional Analytic Formulation of Fresnel Diffraction

Nobuo Aoyagi, Schoichiro Yamaguchi

Published Mar 1, 1973• 5 citations
Publisher

Abstract

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.

DOI
10.1143/jjap.12.366

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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.
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.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
Typically the use of the Rayleigh-Sommerfeld diffraction formula as a photon propagator is widely accepted due to the abundant experimental evidence that suggests that it works. However, a direct link between the propagation of the electromagnetic field in classical optics and the propagation of photons where the square of the probability amplitude describes the transverse probability of the photon detection is still an issue to be clarified. We develop a mathematical formulation for the photon propagation using the formalism of electromagnetic field quantization and the path-integral method, whose main feature is its similarity with a fractional Fourier transform (FRFT). Here we show that because of the close relation existing between the FRFT and the Fresnel diffraction integral, this propagator can be written as a Fresnel diffraction, which brings forward a discussion of the fundamental character of it at the photon level compared to the Huygens-Fresnel principle. Finally, we carry out an experiment of photon counting by a rectangular slit supporting the result that the diffraction phenomenon in the Fresnel approximation behaves as the actual classical limit.
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.
This thesis concerns a general theory of beam propa-gation in optics.Well-known basic ideas are critically reviewed, and the analogy between quantum mechanics and paraxial optics is established through a number of n~w refinements.A parabolic equation for the electrodynamics wave relation is derived using the relativistic front-description of paraxial optics.The •packet-like beam solutions to this equation are shown to be hermite functions of complex argument.1 I The role and applications of canonical transformations in paraxial optics is investigated, attention is focused on some extensions of linear transformations mapping the position and momentum operators "q, '~ to a pair sands of canonically conjugate but not necessarily Hermitian operators.It is shown that for a linear transformation a related space 'jt. of entire functions exists with an inner product over the complex plane, such that the pair s, scan be realized in the Schrbdinger representation.A unitary mapping onto the ordinary Hilbert space of square-integrable functions over :l,t through an integral transform is provided.The well known Fresnel transform is a special case when sands are the harmonic oscillator raising and lowering operators, whose eigen-' functions are the Hermite functions of complex argument.With these functions the use of mathematical distributions in Fourier optics is extended.A generalization of the Cauchy problem is presented, which expresses the field in the plane l = o in terms of the field in any plane \ =~1 > o.The solution is obtained through an extensive use of the Hermite expansion of a distribution.While this expansion is not new, the fact that is not made in the context of the Hilbert space it is.The results obtained establish the applicability of special functions in optics, as well as a precise knowledge of the meaning and accuracy of paraxial wave optics.Applications of the theory to homogeneous and inhomogeneous medium are presented in particular in a selfcontained chapter about relativistic self-focussing in a non-linear medium.This work is primarily concerned with describing 1 1 1 an alternative approach to the study of a pair of canonical operators and it provides new insight over the role of symplectic geometry in quantum mechanics and optics.Chapter I canon i ea 1 tr ans for•ms.Const rue t ion an<1 app l i ea ti ons.< 19) .
Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.