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Acta Physica Sinica

Mutual-induced fractional Fourier transform in strongly nonlocal nonlinear medium

Zhao Bao-Ping, Zhenjun Yang, Daquan Lu, Wei Hu

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The effect of mutual-induced fractional Fourier transform (FRFT) between a weak signal beam and a strong pump beam in a strongly nonlocal nonlinear medium is described. The signal beam is an FRFT during propagation and the order of FRFT is dependent on the propagation distance and the power of the pump beam. When the propagation distance is fixed, the order of FRFT of the signal beam is proportional to the square root of the power of the pump beam. Mutual-induced FRFT is another method of light controlling light, and its properties contribute to the development of new type FRFT devices, optical information processing and optical imaging.

DOI
10.7498/aps.60.084214

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The effect of mutual-induced fractional Fourier transform (FRFT) between a weak signal beam and a strong pump beam in a strongly nonlocal nonlinear medium is described. The signal beam is an FRFT during propagation and the order of FRFT is dependent on the propagation distance and the power of the pump beam. When the propagation distance is fixed, the order of FRFT of the signal beam is proportional to the square root of the power of the pump beam. Mutual-induced FRFT is another method of light controlling light, and its properties contribute to the development of new type FRFT devices, optical information processing and optical imaging.
It is well known that the fractional Fourier transform (FrFT) is a non-trivial generalization of the classical Fourier transform. This transform has been extensively utilized in signal analysis and image processing. The uncertainty principle is one of the fundamental results related to the FrFT. In the present paper, we first propose the different method in proving a variant of Heisenberg uncertainty principles associated with the FrFT. This method combines the Heisenberg-type uncertainty principle pertaining to the FrFT and the additive property of the FrFT. We then provide an alternative proof of the logarithmic inequality for the FRFT and find that this result differs slightly that in the literature. Finally, we explicitly expand the variant of Heisenberg uncertainty principle to a more general case.
A new optical security system for image encryption based on a fractional joint transform correlator and nonlinear filtering is proposed.The position of the lens in the proposed optical encryption setup can be chosen, so that an additional key is introduced in the security system.The distributions at the input and output planes of the encryption system are related by a fractional Fourier transform (FrFT) at a given fractional order that is defined, among other parameters, by the focal length and the position of the lens in the setup; this fractional order acts as an additional key of the security system.The optical intensity of the complex distribution in the fractional Fourier domain (output plane) is captured by a CCD camera.The nonlinearity introduced in the last step of the encryption process, maintains the encrypted function as a real-valued function.The security system proposed in this work is an generalization of the encryption system based on a conventional joint transform correlator (JTC), from the Fourier domain to the fractional Fourier domain.Additional advantages of the proposed system are: new degrees of freedom for the optical setup, alleviated alignment requirement, and the introduction of an additional key (the fractional order of the FrFT) that improves security.Numerical simulations verify the validity of this new optical security system.
Light processing plays a central role in quantum communication, since optical quantum networks that will allow for secure quantum communication [1] and linking quantum computers [2] are of significant scientific and engineering interest [3], [4]. In this work, we present the experimental implementation of optical Fractional Fourier Transform. The protocol is realised in gradient echo memory. Hermite-Gaussian modes of light are encoded in rubidium atoms with the coupling impulse linearly chirped in frequency - this corresponds to the temporal lens [5]. Then the ac-Stark beam is applied to the atoms which leads to them acquiring the spectral phase corresponding to the propagation through a spectral lens. This operation results in the rotation of the Wigner function of the input state [6] and is exactly the Fractional Fourier Transform (FrFT) of the input function. The Hermite-Gaussian modes of light are the eigenfunctions of FrFT.
Fractional Fourier transform (FrFT) is an important tool used in signal processing. It is usually utilized to filter the noise in acoustic signals and it is generalization of Fourier transform. In recent years it was also presented that it has a crucial role in optics, especially mode sorting [2]. One of the important interpretations of FrFT is rotation of the Wigner distribution in time-frequency domain called chronocyclic Wigner function. We have shown that it is possible to perform rotation of arbitrary angleφ\varphiφcoresponding to angle of FrFT. Despite its usefulness, it is still very difficult to perform FrFT in the time-frequency domain. Our experiment is based on the gradient echo memory built around cloud of ultracold rubidium 87 atoms trapped in the magneto-optical trap. The memory protocol uses two-photon Raman transition to store signal pulses in atomic coherence. By applying spatial lens with ac-Stark modulation and imposing two temporal lenses in specific order [1]. We have experimentally performed FrFT in spectro-temporal domain for the cat state.
The fractional Fourier transform (FrFT), a fundamental operation in physics that corresponds to a rotation of phase space by any angle, is also an indispensable tool employed in digital signal processing for noise reduction. Processing of optical signals in their time-frequency degree of freedom bypasses the digitization step and presents an opportunity to enhance many protocols in quantum and classical communication, sensing, and computing. In this Letter, we present the experimental realization of the fractional Fourier transform in the time-frequency domain using an atomic quantum-optical memory system with processing capabilities. Our scheme performs the operation by imposing programmable interleaved spectral and temporal phases. We have verified the FrFT by analyses of chroncyclic Wigner functions measured via a shot-noise limited homodyne detector. Our results hold prospects for achieving temporal-mode sorting, processing, and superresolved parameter estimation.
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.