On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

Abstract

By a quantum mechanical analysis of the additive rule F_α[F_β[f]] = F_α+β[f], which the fractional Fourier transformation (FrFT) F_α[f] should satisfy, we reveal that the position-momentum mutualtransformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.

Keywords

fractional Fourier transform / core operator / IWOP technique / entangled state of continuum variables / Fresnel operator

References

[1] H. Y. Fan and L. Y. Hu, Correpondence between quantumoptical transform and classical-optical transform explored by developing Dirac’s symbolic method, Front. Phys. 7(3), 261 (2012)

[2] V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl.25(3), 241 (1980)

[3] A. C. McBride and F. H. Kerr, On Namias’s fractional Fourier transforms, IMA J. Appl. Math.39(2), 159 (1987)

[4] H. M. Ozaktas and B. Barshan, Convolution, filtering, and mutiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms, J. Opt. Soc. Am. A11(2), 547 (1994)

[5] A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, J. Opt. Soc. Am. A10(10), 2181 (1993)

[6] D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Graded index fibers, Wigner distribution functions and the fractional Fourier transform, Appl. Opt.33(26) 6188 (1994)

[7] L. M. Bernardo and O. D. D. Soares, Fractional Fourier transforms and optical systems, Opt. Commun. 110(5-6), 517 (1994)

[8] H. M. Ozaktas and D. Mendlovic, Fourier transforms of fractional order and their optical implementation, J. Opt. Soc. Am. A10(12), 2522 (1993)

[9] S. Chountasis, A. Vourdas, and C. Bendjaballah, Fractional Fourier operators and generalized Wigner functions, Phys. Rev. A 60(5), 3467 (1999)

[10] D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation (I), J. Opt. Soc. Am. A10(9), 1875 (1993)

[11] H.-Y. Fan, L.-Y. Hu, and J.-S. Wang, Eigenfunctions of complex fractional Fourier transformation obtained in the context of Quantum optics, J. Opt. Soc. Am. A25(4), 974 (2008)

[12] H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt. B5(4), R147 (2003)

[13] A. Wünsche, About integration within ordered products in quantum optics, J. Opt. B2(3), R11 (2000)

[14] H. Y. Fan, H. R. Zaidi, and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D35(6), 1831 (1987)

[15] H. Y. Fan and J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum, Phys. Rev. A49(2), 704 (1994)

[16] H. Y. Fan and X. Ye, Common eigenstates of two particles’ center-of-mass coordinates and mass-weighted relative momentum, Phys. Rev. A51(4), 3343 (1995)

[17] H. Y. Fan and Y. Fan, Representations of two-mode squeezing transformations, Phys. Rev. A54(1), 958 (1996)

[18] H. Y. Fan and J. H. Chen, EPR entangled state and generalized Bargmann transformation, Phys. Lett. A303(5-6), 311 (2002)

[19] A. Einstein, B. Podolsky, and N. Rosen, Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47(3), 777 (1935)

[20] H. Y. Fan, S. Wang, and H. Y. Hu, Evolution of the singlemode squeezed vacuum state in amplitude dissipative channel, Front. Phys. 9(1), 74 (2014)

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg