Abstract

The fractional Wigner distribution function, introduced in this paper starting from the fractional Fourier transform, is found to be the appropriate phase-space distribution function for light-beam characterization in the near-field diffraction regime. The properties of the fractional Wigner distribution function and the moment-matrix formalism for beam characterization are studied.

© 1996 Optical Society of America

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References

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  1. V. Namias, “The fractional Fourier transform and its applications in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  4. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I, II,” J. Opt. Soc. Am. A 10, 1875–1881, 2522–2531 (1993).
    [CrossRef]
  5. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [CrossRef] [PubMed]
  6. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  7. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  8. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  9. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  10. T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: continuous-time signals; Part II: discrete-time signals; Part III: relation with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).
  11. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  12. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).
  13. D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
    [CrossRef]
  14. D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
    [CrossRef]
  15. M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
    [CrossRef]
  16. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  17. D. Dragoman, “Wigner distribution function for Gaussian-Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
    [CrossRef] [PubMed]
  18. J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
    [CrossRef]
  19. G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
    [CrossRef]
  20. R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
    [CrossRef]
  21. F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier-transform and Fresnel transform,” Atti Fond. Giorgio Ronchi, 49, 387–390 (1994).
  22. J. C. Wood, D. T. Barry, “Random transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).
  23. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  24. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  25. S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).
  26. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  27. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.

1995

1994

1993

D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I, II,” J. Opt. Soc. Am. A 10, 1875–1881, 2522–2531 (1993).
[CrossRef]

1992

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

J. C. Wood, D. T. Barry, “Random transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

1991

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

1987

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1986

1980

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

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: continuous-time signals; Part II: discrete-time signals; Part III: relation with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

1979

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.

Bagini, V.

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier-transform and Fresnel transform,” Atti Fond. Giorgio Ronchi, 49, 387–390 (1994).

Barnett, S.

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

Barry, D. T.

J. C. Wood, D. T. Barry, “Random transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Barshan, B.

Bastiaans, M. J.

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: continuous-time signals; Part II: discrete-time signals; Part III: relation with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Dragoman, D.

Gori, F.

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier-transform and Fresnel transform,” Atti Fond. Giorgio Ronchi, 49, 387–390 (1994).

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Lohmann, A. W.

Martinez-Herrero, R.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mejias, P. M.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Meklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: continuous-time signals; Part II: discrete-time signals; Part III: relation with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Mendlovic, D.

Namias, V.

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

Niera, J. L. H.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

Onciul, D.

Onural, L.

Ozaktas, H. M.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Piquero, G.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

Sanchez, M.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier-transform and Fresnel transform,” Atti Fond. Giorgio Ronchi, 49, 387–390 (1994).

Serna, J.

Soffer, B. H.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wood, J. C.

J. C. Wood, D. T. Barry, “Random transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Appl. Opt.

Atti Fond. Giorgio Ronchi

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier-transform and Fresnel transform,” Atti Fond. Giorgio Ronchi, 49, 387–390 (1994).

IMA J. Appl. Math.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Niera, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCDoptical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[CrossRef]

Optik

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Philips J. Res.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: continuous-time signals; Part II: discrete-time signals; Part III: relation with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Proc. Int. Conf. Acoust. Speech Signal Process.

J. C. Wood, D. T. Barry, “Random transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Other

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.

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Equations (37)

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W ( x , p ) = φ ( x + x 2 ) φ * ( x - x 2 ) exp ( i p x ) d x ,
F ( p ) = 1 2 π φ ( x ) exp ( i x p ) d x ,
W ( x , p ) = F ( p + p 2 ) F * ( p - p 2 ) exp ( - i x p ) d p .
F α ( p ) = exp [ i ( π α ^ / 4 - α / 2 ) ] 2 π sin α exp ( - i 2 p 2 cot α ) × exp ( i x p sin α - i 2 x 2 cot α ) φ ( x ) d x ,             0 < α < π ,
W α ( x , p ) = F α ( p + p 2 ) F α * ( p - p 2 ) exp ( - i x p ) d p .
W α ( x , p ) = φ ( p cos α + x sin α + p 2 ) × φ * ( p cos α + x sin α - p 2 ) × exp [ i ( p sin α - x cos α ) p ] d p = F ( p sin α - x cos α + x 2 ) × F * ( p sin α - x cos α - x 2 ) × exp [ - i ( p cos α + x sin α ) x ] d x .
W α ( x , p ) = W ( p cos α + x sin α , p sin α - x cos α ) ;
F α ( x + x 2 ) F α * ( x - x 2 ) exp ( i p x ) d x = W α ( - p , x ) = W ( x cos α - p sin α , x sin α + p cos α ) ,
φ ( x ) W ( x , p ) W ( x , p ) W ( x cos α - p sin α , p cos α + x sin α ) = W α ( - p , x ) W α ( - p , x ) W - 1 [ W α ( - p , x ) ] = F α ( x ) .
F α ( 2 x ) F α * ( 0 ) = 1 2 π φ ( x cos α - p sin α + x 2 ) × φ * ( x cos α - p sin α - x 2 ) × exp [ i x ( p cos α + x sin α ) - i 2 p x ] d p d x .
W α + β = W α W β .
W α ( x , p ) d x d p = W ( x , p ) d x d p = E total ,
W α ( x , p ) d x = 2 π F α ( p ) 2 ,
W α ( x , p ) d p = | F α ( p ) exp ( - i x p ) d p | 2 ,
φ ( x 1 ) φ * ( x 2 ) = 1 2 π W α ( x 1 + x 2 2 sin α - p cos α , x 1 + x 2 2 cos α + p sin α ) exp [ - i p ( x 1 - x 2 ) ] d p , F ( p 1 ) F * ( p 2 ) = 1 2 π W α ( x sin α - p 1 + p 2 2 cos α , x cos α + p 1 + p 2 2 sin α ) exp [ i x ( p 1 - p 2 ) ] d x .
x i p j ¯ α = ( x - x ¯ α ) i ( p - p ¯ α ) j W α ( x , p ) d x d p W α ( x , p ) d x d p ,
ξ ¯ α = ξ W α ( x , p ) d x d p / W α ( x , p ) d x d p ,
W o ( x , p ) = W i ( D x - B p , - C x + A p ) ,
W α o ( x , p ) = W α i ( D x - B p , - C x + A p ) ,
S = [ A B C D ] = [ sin α - cos α cos α sin α ] [ A B C D ] × [ sin α cos α - cos α sin α ] .
[ x p ] o = [ A B C D ] [ x p ] i .
M j α = ( x p ) [ x p ] ¯ α j times ,
M j o α = ( S S ) [ j / 2 ] times M j i α ( S S ) ( j - [ j / 2 ] ) times T ,
F α ( p ) exp ( i p 2 2 cot α ) exp ( i α 2 - i π 4 ) = 1 cos α E tan α ( p sin α ) ,
E σ ( p ) = 1 σ 2 π φ ( x ) exp ( - i x 2 2 σ 2 + i x p ) d x .
S T JS = J ,
J = [ 0 - 1 1 0 ] .
( S S ) T j times ( J J ) j times ( S S ) j times = ( J J ) j times .
K o = TK i T - 1 ,
K = M 2 j α ( J J ) j times ,             T = ( S S ) j times ,
K = M 2 j + 1 α ( J J ) j + 1 times M 2 j + 1 α T ( J J ) j times , T = ( S S ) j times ,             j 0 ,
K = M 2 j + 1 α T ( J J ) j times M 2 j + 1 α ( J J ) j + 1 times , T = ( S S ) j + 1 times ,             j > 0.
Λ o = Λ i ,             P o = T P i ,
K = P Λ P - 1 .
I 2 j α = ω 2 j W α ( x , p ) W α ( x , p ) d x d p d x d p W α ( x , p ) W α ( x , p ) d x d p d x d p ,
ω ( x , p , x , p ) = ( x p ) J [ x p ] .
Q α = x 2 ¯ α p 2 ¯ α - ( x p ¯ α ) 2 = det M 2 α = I 2 α / 2 ,

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