Abstract

The fractional Fourier transform (FRT) is known to be optically implementable with use of a medium with a perfect radial quadratic-index profile. Using the quantum-mechanical operator formalism, we examine the effects on the FRT action of such a medium that are due to small random inhomogeneities in the longitudinal direction, the direction of propagation, and we formulate the random fractional Fourier transform (RFRT). Applying the RFRT to a self-fractional Fourier function, a Gaussian function, we discuss both the total power and the variance. The random Fourier transform is examined as a special limiting case.

© 1999 Optical Society of America

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References

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  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  4. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  6. M. Imai, S. Kikuchi, T. Matsumoto, Y. Kinoshita, “Mode conversion due to fluctuations in a lens-like medium,” J. Opt. Soc. Am. 59, 904–913 (1969).
  7. T. Asakura, Y. Kinoshita, M. Suzuki, “Further correlation studies of Gaussian-beam fluctuations caused by a random medium,” J. Opt. Soc. Am. 59, 913–920 (1969).
  8. G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
    [CrossRef]
  9. M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre wave guide,” Opt. Quantum Electron. 8, “Part I,” 503–508; “Part II,” 509–512 (1976).
  10. A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  12. S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A Math. Gen. 27, 4179–4187 (1994);Corrigenda 27, 7937 (1994).
    [CrossRef]
  13. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef] [PubMed]
  14. M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).
  15. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  16. Information is available from Sumiyoshi Abe, College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi Chiba 274-8501, Japan, or John T. Sheridan, Physics Department, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland.
  17. L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
    [CrossRef]

1996

L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
[CrossRef]

1995

1994

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A Math. Gen. 27, 4179–4187 (1994);Corrigenda 27, 7937 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

1993

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1979

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

1976

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre wave guide,” Opt. Quantum Electron. 8, “Part I,” 503–508; “Part II,” 509–512 (1976).

1973

G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
[CrossRef]

1969

Abe, S.

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A Math. Gen. 27, 4179–4187 (1994);Corrigenda 27, 7937 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

Asakura, T.

Bansal, N. K.

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Burridge, R.

G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
[CrossRef]

Charnotskii, M. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.

Chen, L.-Y.

L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
[CrossRef]

Eve, M.

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre wave guide,” Opt. Quantum Electron. 8, “Part I,” 503–508; “Part II,” 509–512 (1976).

Ghatak, A. K.

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

Goldenfeld, N.

L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Goyal, I. C.

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.

Hannay, J. H.

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre wave guide,” Opt. Quantum Electron. 8, “Part I,” 503–508; “Part II,” 509–512 (1976).

Imai, M.

Kikuchi, S.

Kinoshita, Y.

Lohmann, A. W.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Matsumoto, T.

McLaughlin, D.

G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
[CrossRef]

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Oono, Y.

L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
[CrossRef]

Ozaktas, H. M.

Papanicolaou, G. C.

G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
[CrossRef]

Scully, M. O.

M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).

Sharma, A.

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A Math. Gen. 27, 4179–4187 (1994);Corrigenda 27, 7937 (1994).
[CrossRef]

Suzuki, M.

Tatarskii, V. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.

Zubairy, M. S.

M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).

Fiber Integr. Opt.

A. Sharma, I. C. Goyal, N. K. Bansal, A. K. Ghatak, “Propagation of Gaussian beams through parabolic index optical waveguides with random dielectric constant gradient,” Fiber Integr. Opt. 2, 299–314 (1979).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Math. Phys.

G. C. Papanicolaou, D. McLaughlin, R. Burridge, “A stochastic Gaussian beam,” J. Math. Phys. 14, 84–89 (1973).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A Math. Gen.

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A Math. Gen. 27, 4179–4187 (1994);Corrigenda 27, 7937 (1994).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

M. Eve, J. H. Hannay, “Ray theory and random mode coupling in an optical fibre wave guide,” Opt. Quantum Electron. 8, “Part I,” 503–508; “Part II,” 509–512 (1976).

Phys. Rev. E

L.-Y. Chen, N. Goldenfeld, Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376–394 (1996).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Information is available from Sumiyoshi Abe, College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi Chiba 274-8501, Japan, or John T. Sheridan, Physics Department, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics XXXII, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1993), Chap. IV, pp. 203–266.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

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Equations (60)

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H^0=12 p^2+12 x^2.
i z |ψ0(z)=H^0|ψ0(z),
|ψ0(z)=exp[-iH^0(z-z0)]|ψ(z0).
i ψ0(x, z)z=-12 2ψ0(x, z)x2+12 x2ψ0(x, z).
Hˆ(z)=12 p^2+12 ω2(z)x^2.
ω2(z)=1+2 f(z).
f(z)=0,(fz)f(z)=Dδ(z-z),
i z |ψ(z)=Hˆ(z)|ψ(z).
i ψ(x, z)z=-12 2ψ(x, z)x2+12 ω2(z)x2ψ(x, z).
|ψ(z)=Wˆ(z, z0)|ψ(z0).
i z Wˆ(z, z0)=Hˆ(z)Wˆ(z, z0),
Wˆ(z, z0)=T exp-iz0zdτHˆ(τ).
Wˆ(z, z0)=exp[-iH^0(z-z0)]Uˆ(z, z0),
Uˆ(z, z0)=T exp-iz0zdτf(τ)exp[iH^0(z-z0)]x^2 ×exp[-iH^0(z-z0)].
Wˆ(z, z0)=Vˆ(z, z0)exp[-iH^0(z-z0)],
Vˆ(z, z0)=exp[-iH^0(z-z0)]Uˆ(z, z0)×exp[H^0(z-z0)].
Vˆ(z, z0)=1-iz0zdτf(τ)X^2(τ-z)+(-i)22! z0zdτz0zdτf(τ)f(τ)×T[X^2(τ-z)X^2(τ-z)]+(termsofhigherorderinf ),
Xˆ(τ-z)=exp[iH^0(τ-z)]xˆ exp[-iH^0(τ-z)]=xˆ cos(τ-z)+pˆ sin(τ-z).
ρ(x, x; z)=x|ψ(z)ψ(z)|x=ψ(x, z)ψ*(x, z).
ρ(x, x; z)=x|Vˆ(z, z0)exp[-iH^0(z-z0)]|ψ(z0)×ψ(z0)|exp[iH^0(z-z0)]V^(z, z0)|x,
ρR(x, x; z)=ρ(x, x; z),
ρR(x, x; z)
=x|ψ0(z)ψ0(z)|x+Dz0zdτx|X^2(τ-z)|ψ0(z)ψ0(z)|X^2(τ-z|x-D2 z0zdτ[x|ψ0(z)ψ0(z)|X^4(τ-z)|x
+x|X^4(τ-z)|ψ0(z)ψ0(z)|x],
[Fθ(n)ψ](x)
=12π|sin θ| exp-iϕ+i2 x2 cot θ×-dx0x0n expi2 x02 cot θ-ixx0 cosec θψ(x0),
θ=z-z0.
ρR(x, x; z)ρθ(x, x)=1-D2 [3 cot θ-θ(1+3 cot2 θ)]-iD2 (x2-x2)(2+3 cot2 θ-3θ cot θ cosec2 θ)+D16 (x2-x2)2(5 cot θ+3 cot3 θ-3θ cosec4 θ)[Fθψ](x)[Fθψ]*(x)+-iDx[3 cot θ-θ(1+3 cot2 θ)]cosec θ+D4 x(x2-x2)(2+3 cot2 θ-3θ cot θ cosec2 θ)cosec θ[Fθψ](x)[Fθ(1)ψ]*(x)+i  Dx[3 cot θ-θ(1+3 cot2 θ)]cosec θ-D4 x(x2-x2)(2+3 cot2 θ-3θ cot θ cosec2 θ)cosec θ[Fθ(1)ψ](x)[Fθψ]*(x)
+-iD2 (1-3 cosec2 θ+3θ cot θ cosec2 θ)
-D8 (x2-3x2)[3 cot θ-θ(1+3 cot2 θ)]cosec2θ[Fθψ](x)[Fθ(2)ψ]*(x)
+iD2 (1-3 cosec2 θ+3θ cot θ cosec2 θ)
-D8 (x2-3x2)[3 cot θ-θ(1+3 cot2 θ)]cosec2 θ[Fθ(2)ψ](x)[Fθψ]*(x)
-D2 xx[3 cot θ-θ(1+3 cot2 θ)]cosec2 θ[Fθ(1)ψ](x)[Fθ(1)ψ]*(x)
+D4 x(1-3 cosec2 θ+3θ cot θ cosec2 θ)cosec θ[Fθψ](x)[Fθ(3)]*(x)
+D4 x(1-3 cosec2 θ+3θ cot θ cosec2 θ)cosec θ[Fθ(3)ψ](x)[Fθψ]*(x)
-D4 x(1-3 cosec2 θ+3θ cot θ cosec2 θ)cosec θ[Fθ(1)ψ](x)[Fθ(2)ψ]*(x)
-D4 x(1-3 cosec2 θ+3θ cot θ cosec2 θ)cosec θ[Fθ(2)ψ](x)[Fθ(1)ψ]*(x)
-D8 (2 cot θ-3 cot θ cosec2 θ+3θ cosec4 θ)[Fθ(2)ψ](x)[Fθ(2)ψ]*(x)
+D16 (2 cot θ-3 cot θ cosec2 θ+3θ cosec4 θ)[Fθψ](x)[Fθ(4)ψ]*(x)
+D16 (2 cot θ-3 cot θ cosec2 θ+3θ cosec4 θ)[Fθ(4)ψ](x)[Fθψ]*(x).
ρθ(x, x)=1-D2 [3 cot θ-θ(1+3 cot2 θ)]|[Fθψ](x)|2+2Dx[3 cot θ-θ(1+3 cot2 θ)]cosec θ Im{[Fθψ](x)[Fθ(1)ψ]*(x)}+D[1-3 cosec2 θ+3θ cot θ cosec2 θ]Im{[Fθψ](x)[Fθ(2)ψ]*(x)}+D2 x2[3 cot θ-θ(1+3 cot2 θ)]cosec2 θ Re{[Fθψ](x)[Fθ(2)ψ]*(x)}-D2 x2[3 cot θ-θ(1+3 cot2 θ)]cosec2 θ|[Fθ(1)ψ](x)|2+D2 x(1-3 cosec2 θ+3θ cot θ cosec2 θ)cosec θ(Re{[Fθψ](x)[Fθ(3)ψ]*(x)}-Re{[Fθ(1)ψ](x)[Fθ(2)ψ]*(x)})-D8 (2 cot θ-3 cot θ cosec2 θ+3θ cosec4 θ)(|[Fθ(2)ψ](x)|2-Re{[Fθψ](x)[Fθ(4)ψ]*(x)}).
ψG(x0)=A exp(-x02/2),
[Fθ(0)ψG](x)=[FθψG](x)=B2πα,
[Fθ(1)ψG](x)=-β[FθψG](x),
[Fθ(2)ψG](x)=1α+β2[FθψG](x),
[Fθ(3)ψG](x)=-β3α+β2[FθψG](x),
[Fθ(4)ψG](x)=3α2+6β2α+β4[FθψG](x),
B=A2π|sin θ| exp-iϕ-x22,
α=-i exp(iθ)cosec θ,
β=-x exp(-iθ).
ρG,θ(x, x)/|[FθψG](x)|2
=1+D4 [12 cot θ+6 sin3 θ cos θ-17 sin θ cos θ+θ(11-6 cot2 θ-6 cosec2 θ)]+Dx2[-15 cot θ-6 sin3 θ+19 sin θ cos θ+θ2 (-23+15 cot2 θ+15 cosec2 θ)]+Dx4[(3 cot θ-2 sin θ cos θ)(1+cos2 θ)
+θ(1-6 cot2 θ)].
0θπ/2,
ρG,θ(x, x)[1-Dθ(x4-10x2+2)]|[FθψG](x)|2.
ρG,π/2(x, x)=1+πD2 x4-41+3πx2+54×|[Fπ/2ψG](x)|2.
Pout=-dxρG,θ(x, x)=πA2[1-3D sin3 θ(1-cos θ)].
(Δx)2=-dxx2ρG,θ(x, x)/-dxρG,θ(x, x).
(Δx)2=12 {1+D[cot θ+2 cot θ cos2 θ+9 sin3 θ(cos θ-1)]-Dθ(1+3 cot2 θ)}/
[1-3D sin3 θ(1-cos θ)].

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