| Atomic, Molecular, and Optical Physics |
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On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations |
Hong-Yi Fan1,2( ),Jun-Hua Chen2,*() |
1. Department of Physics, Ningbo University, Ningbo 315211, China
2. Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China |
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Abstract By a quantum mechanical analysis of the additive rule Fα[Fβ[f]] = Fα+β[f], which the fractional Fourier transformation (FrFT) Fα[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.
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| Keywords
fractional Fourier transform
core operator
IWOP technique
entangled state of continuum variables
Fresnel operator
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Corresponding Author(s):
Jun-Hua Chen
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Issue Date: 10 February 2015
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| 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)
https://doi.org/10.1007/s11467-011-0206-z
|
| 2 |
V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl. 25(3), 241 (1980)
https://doi.org/10.1093/imamat/25.3.241
|
| 3 |
A. C. McBride and F. H. Kerr, On Namias’s fractional Fourier transforms, IMA J. Appl. Math. 39(2), 159 (1987)
https://doi.org/10.1093/imamat/39.2.159
|
| 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. A 11(2), 547 (1994)
https://doi.org/10.1364/JOSAA.11.000547
|
| 5 |
A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, J. Opt. Soc. Am. A 10(10), 2181 (1993)
https://doi.org/10.1364/JOSAA.10.002181
|
| 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)
https://doi.org/10.1364/AO.33.006188
|
| 7 |
L. M. Bernardo and O. D. D. Soares, Fractional Fourier transforms and optical systems, Opt. Commun. 110(5-6), 517 (1994)
https://doi.org/10.1016/0030-4018(94)90242-9
|
| 8 |
H. M. Ozaktas and D. Mendlovic, Fourier transforms of fractional order and their optical implementation, J. Opt. Soc. Am. A 10(12), 2522 (1993)
https://doi.org/10.1364/JOSAA.10.002522
|
| 9 |
S. Chountasis, A. Vourdas, and C. Bendjaballah, Fractional Fourier operators and generalized Wigner functions, Phys. Rev. A 60(5), 3467 (1999)
https://doi.org/10.1103/PhysRevA.60.3467
|
| 10 |
D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation (I), J. Opt. Soc. Am. A 10(9), 1875 (1993)
https://doi.org/10.1364/JOSAA.10.001875
|
| 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. A 25(4), 974 (2008)
https://doi.org/10.1364/JOSAA.25.000974
|
| 12 |
H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt. B 5(4), R147 (2003)
https://doi.org/10.1088/1464-4266/5/4/201
|
| 13 |
A. Wünsche, About integration within ordered products in quantum optics, J. Opt. B 2(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. D 35(6), 1831 (1987)
https://doi.org/10.1103/PhysRevD.35.1831
|
| 15 |
H. Y. Fan and J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum, Phys. Rev. A 49(2), 704 (1994)
https://doi.org/10.1103/PhysRevA.49.704
|
| 16 |
H. Y. Fan and X. Ye, Common eigenstates of two particles’ center-of-mass coordinates and mass-weighted relative momentum, Phys. Rev. A 51(4), 3343 (1995)
https://doi.org/10.1103/PhysRevA.51.3343
|
| 17 |
H. Y. Fan and Y. Fan, Representations of two-mode squeezing transformations, Phys. Rev. A 54(1), 958 (1996)
https://doi.org/10.1103/PhysRevA.54.958
|
| 18 |
H. Y. Fan and J. H. Chen, EPR entangled state and generalized Bargmann transformation, Phys. Lett. A 303(5-6), 311 (2002)
https://doi.org/10.1016/S0375-9601(02)01312-9
|
| 19 |
A. Einstein, B. Podolsky, and N. Rosen, Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47(3), 777 (1935)
https://doi.org/10.1103/PhysRev.47.777
|
| 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)
https://doi.org/10.1007/s11467-013-0367-z
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