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Chinese Physics B

Relation between Fresnel transform of input light field and the two-parameter Radon transform of Wigner function of the field

Fan Hong‐Yi, Liyun Hu

Published Feb 1, 2009• 15 citations
Publisher

Abstract

This paper proves a new theorem on the relationship between optical field Wigner function's two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field Ψ( x ') propagates through an optical [ D (− B ) (− C ) A ] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D , B . It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A , C .

DOI
10.1088/1674-1056/18/2/037

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This paper proves a new theorem on the relationship between optical field Wigner function's two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field Ψ( x ') propagates through an optical [ D (− B ) (− C ) A ] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D , B . It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A , C .
Similar in spirit to the preceding work 9 Fan, H-Y and Hu, L-Y. 2009. Opt. Commun., 282: 3734–3736. [Crossref], [Web of Science ®] , [Google Scholar], where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e. where F 2 is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find where is the conjugated state to .
Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.
Based on the correspondence between the Collins diffraction formula (optical Fresnel transform) and the transform matrix element of a three-parameter two-mode squeezing operator in the entangled state representation 1 Fan, H-Y and Lu, H-L. 2006. Opt. Lett., 31: 2622–2624. [Crossref], [PubMed] , [Google Scholar] we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip, both in the spatial domain and the frequency domain. The entangled Wigner function is introduced for recapitulating the result.
Based on the correspondence between Collins diffraction formula (optical Fresnel transform) and the transformation matrix element of a three-parameters two-mode squeezing operator in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip both in spacial domain and frequency domain. The entangled Wigner function is introduced for recapitulating the result.
Abstract The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.