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Chinese Physics B

Optical scaled Fresnel–Fourier transform obtained via intermediate coordinate–momentum representation

Chisheng Li, Luo Han-Wen

Published Jan 1, 2010• 0 citations
Publisher

Abstract

Using the intermediate coordinate–momentum representation | x ⟩ s,r , we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F ( r, s ) and the Fourier transform operator F by decomposing U. We also find that the matrix element s,r ⟨| x | U | f ⟩ just corresponds to an optical scaled Fresnel–Fourier transform.

DOI
10.1088/1674-1056/19/1/010308

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Using the intermediate coordinate–momentum representation | x ⟩ s,r , we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F ( r, s ) and the Fourier transform operator F by decomposing U. We also find that the matrix element s,r ⟨| x | U | f ⟩ just corresponds to an optical scaled Fresnel–Fourier 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”.
According to Fan—Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun . 282 3734 ) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinate—momentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator ρ is equal to the marginal integration of the classical Weyl correspondence function of F † ρ F , where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.
Based on the newly-developed coherent-entangled state representation we propose the so-called Fresnel–Hadamard complementary transformation, which is unitary. The new unitary operator plays the role of both Fresnel transformation for and Hadamard transformation for respectively. Physically, and could be a symmetric beamsplitter's two output fields. We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in integration form.
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.