Abstract

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input–output relationship is then broken down into a number of rotation-type input–output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

© 2002 Optical Society of America

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  3. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  4. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  5. W. Mecklenbräuker, F. Hlawatsch, eds., The Wigner Distribution—Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997).
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light beams,” in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of ICOL’98, the International Conference on Optics and Optoelectronics, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 101–115.
  7. International Organization for Standardization, Technical Committee/Subcommittee 172/SC9, “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle and beam propagation factor,” (International Organization for Standardization, Geneva, 1999).
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  9. G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
    [CrossRef]
  10. B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
    [CrossRef]
  11. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  12. C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
    [CrossRef]
  13. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  14. J. Serna, F. Encinas-Sanz, G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001).
    [CrossRef]
  15. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953), Chap. 7.
  16. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  17. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 284–295.
  18. L. Cohen, Time-Frequency Signal Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  19. F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.
  20. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  21. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  22. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.
  23. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.
  24. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).
  25. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).
  26. T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
    [CrossRef]
  27. N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

2001 (1)

2000 (1)

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

1998 (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

1997 (1)

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

1994 (2)

1993 (1)

1992 (1)

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.

1991 (1)

1988 (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1986 (1)

1979 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1974 (1)

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

1932 (1)

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

Alieva, T.

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light beams,” in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of ICOL’98, the International Conference on Optics and Optoelectronics, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 101–115.

Boudreaux-Bartels, G. F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.

Cohen, L.

L. Cohen, Time-Frequency Signal Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

de Bruijn, N. G.

N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

Encinas-Sanz, F.

J. Serna, F. Encinas-Sanz, G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Eppich, B.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Gao, C.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Hlawatsch, F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Lohmann, A. W.

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

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

Marti´nez, C.

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Marti´nez-Herrero, R.

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Meji´as, P. M.

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Nemes, G.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Papoulis, A.

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 284–295.

Serna, J.

Siegman, A. E.

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Weber, H.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Wigner, E.

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

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953), Chap. 7.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.

IEEE Signal Process. Lett. (1)

T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 320–323 (2000).
[CrossRef]

IEEE Signal Process. Mag. (1)

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag., April1992, pp. 21–67.

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Opt. Laser Technol. (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Phys. Rev. (1)

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

Other (11)

W. Mecklenbräuker, F. Hlawatsch, eds., The Wigner Distribution—Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997).

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light beams,” in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of ICOL’98, the International Conference on Optics and Optoelectronics, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 101–115.

International Organization for Standardization, Technical Committee/Subcommittee 172/SC9, “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle and beam propagation factor,” (International Organization for Standardization, Geneva, 1999).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953), Chap. 7.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 284–295.

L. Cohen, Time-Frequency Signal Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Vol. 38, pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform—with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001).

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Figures (1)

Fig. 1
Fig. 1

Two simple coherent-optical fractional FT systems,20 whose point-spread functions take the form of Eq. (14), (a) using one thin (cylindrical) lens with focal length f, preceded and followed by two identical distances d of free space, and (b) using two identical thin (cylindrical) lenses with focal lengths f, separated by a distance d. The relation between d, f, and the fractional angle α reads d=2 f sin2 12α in both setups; the real space coordinates in the input and output planes are proportional to x and u, respectively, by the proportionality factor (λf sin α)1/2 in setup (a) and (λf tan 12α)1/2 in setup (b), where λ is the wavelength of the laser light.

Tables (2)

Tables Icon

Table 1 Real and Imaginary Parts of ξ2k,lη2m,n (and ξ2k,lη2m,n*, If Necessary) up to Fourth Order

Tables Icon

Table 2 Moment Combinations Undergoing a Rotation of the Form of Eqs. (32) up to Fourth Ordera

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

Wf(x, u; y, v)=--f(x+12x, y+12y)×f*(x-12x, y-12y)×exp[-j2π(ux+vy)]dxdy.
f(x+12x, y+12y)f*(x-12x, y-12y).
Af(x, u; y, v)=--f(x+12x, y+12y)×f*(x-12x, y-12y)×exp[-j2π(ux+vy)]dxdy.
Af(x, u; y, v)=----Wf(x, u; y, v)×exp[-j2π(ux-ux+vy-vy)]dxdudydv.
E=----Wf(x, u; y, v)dxdudydv=Af(0, 0; 0, 0)=--|f(x, y)|2dxdy.
μpqrsE=----Wf(x, u; y, v)×xpuqyrvsdxdudydv
=(-1)p+r(j2π)p+q+r+s p+q+r+sAf(x, u; y, v)xqupysvrx=u=y=v=0=1(j2π)q+s×--xpyrq+sξqηsf(x+12ξ, y+12η)×f*(x-12ξ, y-12η)|ξ=η=0 dxdy=1(j4π)q+s×--xpyrx1-x2qy1-y2s×f(x1, y1)f*(x2, y2)|x1=x2=x,y1=y2=y dxdy,
μp0r0E=----Wf(x, u; y, v)xpyrdxdudydv=--xpyr|f(x, y)|2dxdy,
Cf(x, u; y, v)=----Φ(xo, uo; yo, vo)×Wf(x-xo, u-uo; y-yo, v-vo)dxoduodyodvo,
C¯f(x, u; y, v)=Φ¯(x, u; y, v)Af(x, u; y, v),
exp[jπ(xu+yv)],η(12x, 12y)η*(-12x,-12y),
Aγ(-x,-u;-y,-v),
γpqrs=----Cf(x, u; y, v)xpuqyrvsdxdudydv=(-1)p+r(j2π)p+q+r+s p+q+r+sC¯f(x, u; y, v)xqupysvrx=u=y=v=0=(-1)p+r(j2π)p+q+r+s p+q+r+sΦ¯(x, u; y, v)Af(x, u; y, v)xqupysvrx=u=y=v=0=(-1)p+r(j2π)p+q+r+s k=0pl=0qm=0rn=0rpkqlrmsn×k+l+m+nΦ¯(x, u; y, v)xlukynvm p+q+r+s-k-l-m-nAf(x, u; y, v)xq-lup-kys-nvr-mx=u=y=v=0=E k=0pl=0qm=0rn=0spkqlrmsn (-1)k+m(j2π)k+l+m+n k+l+m+nΦ¯(x, u; y, v)xlukynvmx=u=y=v=0μp-k,q-l,r-m,s-n.
2k+2mSx(xu)Sy(yv)xkukymvmx=u=y=v=0=2kSx(xu)xkukx=u=02mSy(yv)ymvmy=v=0=k!m!Sx(k)(0)Sy(m)(0),
γpqrs=Ek=0min(p,q)m=0min(r,s)pkqkrmsm×(2π)-(2k+2m)k!m!Sx(k)(0)×Sy(m)(0)μp-k,q-k,r-m,s-m.
Fαβ(u, v)=--Kα(x, u)Kβ(y, v)f(x, y)dxdy,
Kα(x, u)=expj 12αj sin α expjπ (x2+u2)cos α-2uxsin α.
xoyouovo=AxxAxyBxuBxvAyxAyyByuByvCuxCuyDuuDuvCvxCvyDvuDvvxiyiuivi=cos α0sin α00cos β0sin β-sin α0cos α00-sin β0cos βxiyiuivi,
xouoyovo=AxxBxuAxyBxvCuxDuuCuyDuvAyxByuAyyByvCvxDvuCvyDvvxiuiyivi=cos αsin α0-sin αcos α0000cos βsin β00-sin βcos βxiuiyivi.
WFαβ(x, u; y, v)=Wf(x cos α-u sin α,x sin α+u cos α;y cos β-v sin β,y sin β+v cos β),
AFαβ(x, u; y, v)=Af(x cos α-u sin α,x sin α+u cos α;y cos β-v sin β,y sin β+v cos β).
|Fαβ(x, y)|2=--WFαβ(x, u; y, v)dudv,
|Fαβ(x, y)|2=--AFαβ(0, u; 0, v)×exp[j2π(ux+vy)]dudv.
μpqrs(α, β)E=----WFαβ(x, u; y, v)xpuqyrvsdxdudydv=----Wf(x cos α-u sin α,x sin α+u cos α;y cos β-v sin β,y sin β+v cos β)xpuqyrvsdxdudydv=----Wf(x, u; y, v)(x cos α+u sin α)p×(-x sin α+u cos α)q(y cos β+v sin β)r×(-y sin β+v cos β)sdxdudydv,
μpqrs(α, β)=k=0pl=0qm=0rn=0spkqlrmsn×(-1)l+nμp-k+l,q-l+k,r-m+n,s-n+m×(cos α)p-k+q-l(sin α)k+l(cos β)r-m+s-n(sin β)m+n,
μp0r0(α, β)=k=0pm=0rpkrmμp-k,k,r-m,m×cosp-kα sink α cosr-m β sinm β.
E=----WFαβ(x, u; y, v)dxdudydv=----Wf(x, u; y, v)dxdudydv.
μpq00(α, β)=μpq00(α, 0),
μ00rs(α, β)=μ00rs(0, β).
μp0r0(α, β)E=--xpyr|Fαβ(x, y)|2dxdy;
μpqrs(α, β)=(-1)p+rμqpsr(α+12π, β+12π),
ξ=x+ju,
η=y+jv,
ξ(α)=exp(-jα)ξ,
η(β)=exp(-jβ)η,
ξ2k,l(α)=[ξ(α)]k+l[ξ*(α)]k=|ξ(α)|2k[ξ(α)]l,
η2m,n(β)=[η(β)]m+n[η*(β)]n=|η(β)|2m[η(α)]n,
ξ2k,l(α)η2m,n(β)=exp[-j(lα+nβ)]ξ2k,lη2m,n,
ξ2k,l(α)η2m,n*(β)=exp[-j(lα-nβ)]ξ2k,lη2m,n*,
R{ξ2k,l(α)η2m,n(β)}I{ξ2k,l(α)η2m,n(β)}=R(lα+nβ)R{ξ2k,lη2m,n}I{ξ2k,lη2m,n},
R{ξ2k,l(α)η2m,n*(β)}I{ξ2k,l(α)η2m,n*(β)}=R(lα-nβ)R{ξ2k,lη2m,n*}I{ξ2k,lη2m,n*},
R(α)=cos αsin α-sin αcos α,
----WFαβ(x, u; y, v)ξ2k,lη2m,ndxdudydv=----Wf(x, u; y, v)×ξ2k,l(α)η2m,n(β)dxdudydv=exp[-j(lα+nβ)]----Wf(x, u; y, v)×ξ2k,lη2m,ndxdudydv
----WFαβ(x, u; y, v)ξ2k,lη2m,n*dxdudydv=exp[-j(lα-nβ)]----Wf(x, u; y, v)×ξ2k,lη2m,n*dxdudydv,
μ0040(α, β)-6μ0022(α, β)+μ0004(α, β)4μ0031(α, β)-4μ0013(α, β)=R(4β)μ0040-6μ0022+μ00044μ0031-4μ0013.
μ1000(α, β)μ0100(α, β)=R(α)μ1000μ0100,
μ0010(α, β)μ0001(α, β)=R(β)μ0010μ0001,
μ1000(α, β)=μ1000 cos α+μ0100 sin α,
μ0010(α, β)=μ0010 cos β+μ0001 sin β,
μ10002+μ01002,
μ00102+μ00012,
μ2000(α, β)+μ0200(α, β)=μ2000+μ0200,
μ2000(α, β)-μ0200(α, β)2μ1100(α, β)=R(2α)μ2000-μ02002μ1100,
μ1010(α, β)-μ0101(α, β)μ1001(α, β)+μ0110(α, β)=R(α+β)μ1010-μ0101μ1001+μ0110,
μ1010(α, β)+μ0101(α, β)-μ1001(α, β)+μ0110(α, β)=R(α-β)μ1010+μ0101-μ1001+μ0110,
μ0020(α, β)+μ0002(α, β)=μ0020+μ0002,
μ0020(α, β)-μ0002(α, β)2μ0011(α, β)=R(2β)μ0020-μ00022μ0011,
μ2000(α, β)=μ2000 cos2 α+2μ1100 cos α sin α+μ0200 sin2 α,
μ1010(α, β)=μ1010 cos α cos β+μ1001 cos α sin β+μ0110 sin α cos β+μ0101 sin α sin β,
μ0020(α, β)=μ0020 cos2 β+2μ0011 cos β sin β+μ0002 sin2 β.
μ2000+μ0200,
(μ2000-μ0200)2+4μ11002,
(μ1010-μ0101)2+(μ1001+μ0110)2,
(μ1010+μ0101)2+(-μ1001+μ0110)2,
μ0020+μ0002,
(μ0020-μ0002)2+4μ00112.
--c2x2+u2c2mWf(x, u)dxdum!--Wf(x, u)dxdu,
μ3000(ga, β)=μ3000 cos3 α+3μ2100 cos2 α sin α+3μ1200 cos α sin2 α+μ0300 sin3 α,
μ2010(α, β)=μ2010 cos2 α cos β+μ2001 cos2 α sin β+2μ1110 cos α sin α cos β+2μ1101 cos α sin α sin β+μ0210 sin2 α cos β+μ0201 sin2 α sin β,
μ1020(α, β)=μ1020 cos α cos2 β+2μ1011 cos α cos β sin β+μ1002 cos α sin2 β+μ0120 sin α cos2 β+2μ0111 sin α cos β sin β+μ0102 sin α sin2 β,
μ0030(α, β)=μ0030 cos3 β+3μ0021 cos2 β sin β+3μ0012 cos β sin2 β+μ0003 sin3 β.
μ4000(α, β)=μ4000 cos4 α+4μ3100 cos3 α sin α+6μ2200 cos2 α sin2 α+4μ1300 cos α sin3 α+μ0400 sin4 α,
μ3010(α, β)=μ3010 cos3 α cos β+μ3001 cos3 α sin β+3μ2110 cos2 α sin α cos β+3μ2101 cos2 α sin α sin β+3μ1210 cos α sin2 α cos β+3μ1201 cos α sin2 α sin β+μ0310 sin3 α cos β+μ0301 sin3 α sin β,
μ2020(α, β)=μ2020 cos2 α cos2 β+2μ2011 cos2 α cos β sin β+μ2002 cos2 α sin2 β+2μ1120 cos α sin α cos2 β+4μ1111 cos α sin α cos β sin β+2μ1102 cos α sin α sin2 β+μ0220 sin2 α cos2 β+2μ0211 sin2 α cosβ sin β+μ0202 sin2 α sin2 β,
μ1030(α, β)=μ1030 cos α cos3 β+3μ1021 cos α cos2 β sin β+3μ1012 cos α cos β sin2 β+μ1003 cos α sin3 β+μ0130 sin α cos3 β+3μ0121 sin α cos2 β sin β+3μ0112 sin α cos β sin2 β+μ0103 sin α sin3 β,
μ0040(α, β)=μ0040 cos4 β+4μ0031 cos3 β sin β+6μ0022 cos2 β sin2 β+4μ0013 cos β sin3 β+μ0004 sin4 β.
μ4000+2μ2200+μ0400,
μ0040+2μ0022+μ0004,
μ2020+μ2002+μ0220+μ0202,

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