Abstract

The quantum optical complex fractional Fourier transform (FRFT) has been related to the classical FRFT using both classical and quantum formalisms. In particular, it was shown that the kernel of the complex FRFT can be classically produced with rotated astigmatic optical systems that mimic the quantum entanglement property.

© 2009 Optical Society of America

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References

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  1. H.-Y. Fan, L. Hu, and J. Wang, “Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics,” J. Opt. Soc. Am. A 25, 974-978 (2008).
    [CrossRef]
  2. H. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
    [CrossRef]
  3. R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368-2381 (2000).
    [CrossRef]
  4. A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531-596 (2002).
    [CrossRef]
  5. D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).
  6. S. Chávez-Cerda, J. R. Moya-Cessa, and H. M. Moya-Cessa, “Quantumlike systems in classical optics: Applications of quantum optical methods,” J. Opt. Soc. Am. B 24, 404-407 (2007).
    [CrossRef]
  7. K. Wódkiewicz and G. H. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815-821 (1998).
    [CrossRef]
  8. D. Dragoman, “Classical optical analogs of quantum Fock states,” Optik (Jena) 111, 393-396 (2000).
  9. D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 42, 433-496 (2002).
    [CrossRef]
  10. D. Dragoman and M. Dragoman, “Quantum coherent versus classical coherent light,” Opt. Quantum Electron. 33, 239-252 (2001).
    [CrossRef]
  11. H. Fan and A. Wuensche, “Wavefunctions of two-mode states in entangled-state representation,” J. Opt. B: Quantum Semiclassical Opt. 7, R88-R102 (2005).
    [CrossRef]
  12. C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express 8, 76-85 (2001).
    [CrossRef] [PubMed]
  13. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  14. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543-559 (1992).
    [CrossRef]

2008 (1)

2007 (1)

2005 (1)

H. Fan and A. Wuensche, “Wavefunctions of two-mode states in entangled-state representation,” J. Opt. B: Quantum Semiclassical Opt. 7, R88-R102 (2005).
[CrossRef]

2002 (3)

H. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531-596 (2002).
[CrossRef]

D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 42, 433-496 (2002).
[CrossRef]

2001 (2)

D. Dragoman and M. Dragoman, “Quantum coherent versus classical coherent light,” Opt. Quantum Electron. 33, 239-252 (2001).
[CrossRef]

C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express 8, 76-85 (2001).
[CrossRef] [PubMed]

2000 (2)

D. Dragoman, “Classical optical analogs of quantum Fock states,” Optik (Jena) 111, 393-396 (2000).

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368-2381 (2000).
[CrossRef]

1998 (1)

K. Wódkiewicz and G. H. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815-821 (1998).
[CrossRef]

1992 (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543-559 (1992).
[CrossRef]

1970 (1)

Chávez-Cerda, S.

Collins, S. A.

Dragoman, D.

D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 42, 433-496 (2002).
[CrossRef]

D. Dragoman and M. Dragoman, “Quantum coherent versus classical coherent light,” Opt. Quantum Electron. 33, 239-252 (2001).
[CrossRef]

D. Dragoman, “Classical optical analogs of quantum Fock states,” Optik (Jena) 111, 393-396 (2000).

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

Dragoman, M.

D. Dragoman and M. Dragoman, “Quantum coherent versus classical coherent light,” Opt. Quantum Electron. 33, 239-252 (2001).
[CrossRef]

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

Fan, H.

H. Fan and A. Wuensche, “Wavefunctions of two-mode states in entangled-state representation,” J. Opt. B: Quantum Semiclassical Opt. 7, R88-R102 (2005).
[CrossRef]

H. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

Fan, H.-Y.

Fan, Y.

H. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

Gerry, C. C.

Herling, G. H.

K. Wódkiewicz and G. H. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815-821 (1998).
[CrossRef]

Hu, L.

Moya-Cessa, H. M.

Moya-Cessa, J. R.

Simon, R.

Torre, A.

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531-596 (2002).
[CrossRef]

Wang, J.

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543-559 (1992).
[CrossRef]

Wódkiewicz, K.

K. Wódkiewicz and G. H. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815-821 (1998).
[CrossRef]

Wolf, K. B.

Wuensche, A.

H. Fan and A. Wuensche, “Wavefunctions of two-mode states in entangled-state representation,” J. Opt. B: Quantum Semiclassical Opt. 7, R88-R102 (2005).
[CrossRef]

Eur. Phys. J. D (1)

H. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543-559 (1992).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

H. Fan and A. Wuensche, “Wavefunctions of two-mode states in entangled-state representation,” J. Opt. B: Quantum Semiclassical Opt. 7, R88-R102 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Quantum Electron. (1)

D. Dragoman and M. Dragoman, “Quantum coherent versus classical coherent light,” Opt. Quantum Electron. 33, 239-252 (2001).
[CrossRef]

Optik (Jena) (1)

D. Dragoman, “Classical optical analogs of quantum Fock states,” Optik (Jena) 111, 393-396 (2000).

Phys. Rev. A (1)

K. Wódkiewicz and G. H. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815-821 (1998).
[CrossRef]

Prog. Opt. (2)

D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 42, 433-496 (2002).
[CrossRef]

A. Torre, “The fractional Fourier transform and some of its applications to optics,” Prog. Opt. 43, 531-596 (2002).
[CrossRef]

Other (1)

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

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Figures (1)

Fig. 1
Fig. 1

Classical optical system that produces an integral transform of the input field with a kernel identical to that of the complex FRFT.

Equations (17)

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K α cl ( x , x ) = exp ( i α i π 2 ) 2 π sin α exp ( i ( x 2 + x 2 ) 2 tan α i x x sin α ) ,
K α q ( ξ , η ) = exp ( i π 2 i α ) 2 π sin α exp [ i ( η 2 + ξ 2 ) 2 tan α + ξ * η ξ η * 2 sin α ] ,
ξ = 1 π exp ( ξ 2 2 + ξ a 1 + + ξ * a 2 + a 1 + a 2 + ) 0 , 0 = 1 π m , n = 0 1 m ! n ! H m , n ( ξ , ξ * ) exp ( ξ 2 2 ) m , n ,
η = 1 π exp ( η 2 2 + η a 1 + η * a 2 + + a 1 + a 2 + ) 0 , 0 = 1 π m , n = 0 i m + n m ! n ! H m , n ( i η , i η * ) exp ( η 2 2 ) m , n ,
H m , n ( λ , λ * ) = l = 0 min ( m , n ) ( 1 ) l m ! n ! l ! ( m l ) ! ( n l ) ! λ m l λ * n l ,
m , n = 0 H m , n ( λ , σ ) exp ( λ σ 2 ) H m , n ( ρ , k ) exp ( ρ k 2 ) t n s m m ! n ! = exp [ ( σ λ ρ k ) 2 ] 1 t s exp ( λ σ + t k σ + s λ ρ t s k ρ 1 t s ) ,
K α q ( η , ξ ) = η exp [ i ( α + π 2 ) ( a 1 + a 1 + a 2 + a 2 ) ] ξ .
m , n = 0 H m , n ( ξ , ξ * ) exp ( ξ 2 2 ) H m , n ( ξ * , ξ ) exp ( ξ 2 2 ) ( 1 ) n + m exp [ i α ( n + m ) ] m ! n ! = exp [ ( ξ 2 ξ 2 ) 2 ] 1 exp ( 2 i α ) exp ( ξ 2 + exp ( i α ) ( ξ ξ * + ξ ξ * ) exp ( 2 i α ) ξ 2 1 exp ( 2 i α ) ) ,
m , n = 0 H m , n ( ξ , ξ * ) exp ( ξ 2 2 ) H m , n ( ξ * , ξ ) exp ( ξ 2 2 ) exp [ i α ( n + m ) ] m ! n ! = exp ( i α i π 2 ) 2 sin α exp [ i ( ξ 2 + ξ 2 ) 2 tan α i ( ξ ξ * + ξ ξ * ) 2 sin α ] ,
K α cl ( x , x ) = m , n , m , n = 0 ξ m n m n exp [ i α ( a 1 + a 1 + a 2 + a 2 ) ] m , n m , n ξ = ξ exp [ i α ( a 1 + a 1 + a 2 + a 2 ) ] ξ
m , n = 0 H m , n ( i η , i η * ) exp ( η 2 2 ) H m , n ( i η * , i η ) exp ( η 2 2 ) ( 1 ) m + n exp [ i α ( n + m ) ] m ! n ! = exp ( i α i π 2 ) 2 sin α exp [ i ( η 2 + η 2 ) 2 tan α i ( η η * + η η * ) 2 sin α ] ,
K α cl ( x , x ) = m , n , m , n = 0 η m n m n exp [ i α ( a 1 + a 1 + a 2 + a 2 ) ] m , n m , n η = η exp [ i α ( a 1 + a 1 + a 2 + a 2 ) ] η .
x 1 ( x 1 x 2 ) 2 , x 2 ( x 1 + x 2 ) 2 ,
ζ 1 x 1 = ( ζ 1 ζ 2 ) 2 , ζ 2 x 2 = ( ζ 1 + ζ 2 ) 2 .
exp ( i α + i π 2 ) 2 π sin α exp { i [ ( x 1 x 2 ) 2 + ( x 1 + x 2 ) 2 + ( x 1 + x 2 ) 2 + ( x 1 + x 2 ) 2 ] 4 tan α + i [ ( x 1 x 2 ) ( x 1 + x 2 ) + ( x 1 + x 2 ) ( x 1 + x 2 ) ] 2 sin α } = K α q ( x 1 , x 2 , x 1 , x 2 ) .
M = 1 2 ( 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 ) ( cos α 0 sin α 0 0 cos α 0 sin α sin α 0 cos α 0 0 sin α 0 cos α ) ( 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 ) = ( A B C D ) ,
K opt ( x , x ) = exp [ i 2 ( x T B 1 A x + x T D B 1 x 2 x T B 1 x ) ] .

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