Abstract

We study screening of optical singularities in random optical fields with two widely different length scales. We call the speckle patterns generated by such fields speckled speckle, because the major speckle spots in the pattern are themselves highly speckled. We study combinations of fields whose components exhibit short- and long-range correlations and find unusual forms of screening.

© 2008 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. London, Ser. A 336, 165-190 (1974). For additional sources see online citation databases for the numerous papers that reference this work.
    [CrossRef]
  2. J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).
  3. J. W. Goodman, Speckle Phenomena In Optics (Roberts, 2007).
  4. B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R.Balian, M.Kleman, and J.-P.Poirier, eds. (North-Holland, 1981), pp. 814-857.
  5. F. Liu and G. F. Mazenko, “Defect-defect correlation in the dynamics of first-order phase transitions,” Phys. Rev. B 46, 5963-5971 (1992).
    [CrossRef]
  6. B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
    [CrossRef]
  7. I. Freund and M. Wilkinson, “Critical-point screening in random wave fields,” J. Opt. Soc. Am. A 15, 2892-2902 (1998).
    [CrossRef]
  8. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
    [CrossRef]
  9. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 3048 (2000) (erratum).
  10. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
    [CrossRef]
  11. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  12. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
    [CrossRef]
  13. G. Foltin, “Signed zeros of Gaussian vector fields--density, correlation functions, and curvature,” J. Phys. A 36, 1729-1742 (2003).
    [CrossRef]
  14. M. R. Dennis, “Correlations and screening of topological charges in Gaussian random fields,” J. Phys. A 36, 6611-6628 (2003).
    [CrossRef]
  15. M. Wilkinson, “Screening of charged singularities of random fields,” J. Phys. A 37, 6763-6771 (2004).
    [CrossRef]
  16. G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
    [CrossRef]
  17. B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).
  18. I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
    [CrossRef]
  19. R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
    [CrossRef] [PubMed]
  20. D. A. Kessler and I. Freund, “Short- and long-range screening of optical phase singularities and C points,” Opt. Commun. 281, 4194-4204 (2008).
    [CrossRef]
  21. I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Commun. 33, 479-481 (2008).
  22. I. Freund and D. A. Kessler, “Singularities in speckled speckle: statistics,” Opt. Commun. (to be published). doi:10.1016/j.optcom.2008.09.029
  23. J. W. Goodman, Statistical Optics (Wiley, 1985).
  24. M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27-37 (1978).
    [CrossRef]

2008 (3)

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

D. A. Kessler and I. Freund, “Short- and long-range screening of optical phase singularities and C points,” Opt. Commun. 281, 4194-4204 (2008).
[CrossRef]

I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Commun. 33, 479-481 (2008).

2007 (1)

I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
[CrossRef]

2006 (1)

B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).

2004 (2)

M. Wilkinson, “Screening of charged singularities of random fields,” J. Phys. A 37, 6763-6771 (2004).
[CrossRef]

G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
[CrossRef]

2003 (2)

G. Foltin, “Signed zeros of Gaussian vector fields--density, correlation functions, and curvature,” J. Phys. A 36, 1729-1742 (2003).
[CrossRef]

M. R. Dennis, “Correlations and screening of topological charges in Gaussian random fields,” J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

2002 (2)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
[CrossRef]

2001 (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

2000 (2)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 3048 (2000) (erratum).

1998 (1)

1996 (1)

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
[CrossRef]

1992 (1)

F. Liu and G. F. Mazenko, “Defect-defect correlation in the dynamics of first-order phase transitions,” Phys. Rev. B 46, 5963-5971 (1992).
[CrossRef]

1978 (1)

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27-37 (1978).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. London, Ser. A 336, 165-190 (1974). For additional sources see online citation databases for the numerous papers that reference this work.
[CrossRef]

Anache, D.

B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).

Berry, M.

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27-37 (1978).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 3048 (2000) (erratum).

J. F. Nye and M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. London, Ser. A 336, 165-190 (1974). For additional sources see online citation databases for the numerous papers that reference this work.
[CrossRef]

Bodenschatz, E.

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
[CrossRef]

Dennis, M. R.

M. R. Dennis, “Correlations and screening of topological charges in Gaussian random fields,” J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 3048 (2000) (erratum).

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

Egorov, R. I.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
[CrossRef]

Foltin, G.

G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
[CrossRef]

G. Foltin, “Signed zeros of Gaussian vector fields--density, correlation functions, and curvature,” J. Phys. A 36, 1729-1742 (2003).
[CrossRef]

Freund, I.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

D. A. Kessler and I. Freund, “Short- and long-range screening of optical phase singularities and C points,” Opt. Commun. 281, 4194-4204 (2008).
[CrossRef]

I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Commun. 33, 479-481 (2008).

I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
[CrossRef]

I. Freund and M. Wilkinson, “Critical-point screening in random wave fields,” J. Opt. Soc. Am. A 15, 2892-2902 (1998).
[CrossRef]

I. Freund and D. A. Kessler, “Singularities in speckled speckle: statistics,” Opt. Commun. (to be published). doi:10.1016/j.optcom.2008.09.029

Ghysels, A.

B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).

Gnutzmann, S.

G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena In Optics (Roberts, 2007).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Halperin, B. I.

B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R.Balian, M.Kleman, and J.-P.Poirier, eds. (North-Holland, 1981), pp. 814-857.

Kessler, D. A.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Commun. 33, 479-481 (2008).

D. A. Kessler and I. Freund, “Short- and long-range screening of optical phase singularities and C points,” Opt. Commun. 281, 4194-4204 (2008).
[CrossRef]

I. Freund and D. A. Kessler, “Singularities in speckled speckle: statistics,” Opt. Commun. (to be published). doi:10.1016/j.optcom.2008.09.029

Liu, F.

F. Liu and G. F. Mazenko, “Defect-defect correlation in the dynamics of first-order phase transitions,” Phys. Rev. B 46, 5963-5971 (1992).
[CrossRef]

Mazenko, G. F.

F. Liu and G. F. Mazenko, “Defect-defect correlation in the dynamics of first-order phase transitions,” Phys. Rev. B 46, 5963-5971 (1992).
[CrossRef]

Mokhun, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. London, Ser. A 336, 165-190 (1974). For additional sources see online citation databases for the numerous papers that reference this work.
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

Roberts, B. W.

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
[CrossRef]

Sethna, J. P.

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
[CrossRef]

Smilansky, U.

G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
[CrossRef]

Soskin, M. S.

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
[CrossRef]

van Tiggelen, B. A.

B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).

Wilkinson, M.

M. Wilkinson, “Screening of charged singularities of random fields,” J. Phys. A 37, 6763-6771 (2004).
[CrossRef]

I. Freund and M. Wilkinson, “Critical-point screening in random wave fields,” J. Opt. Soc. Am. A 15, 2892-2902 (1998).
[CrossRef]

Electron. Lett. (1)

B. A. van Tiggelen, D. Anache, and A. Ghysels, “Role of mean free path in spatial phase correlation and nodal screening,” Electron. Lett. 74, 999-1005 (2006).

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

J. Phys. A (4)

G. Foltin, “Signed zeros of Gaussian vector fields--density, correlation functions, and curvature,” J. Phys. A 36, 1729-1742 (2003).
[CrossRef]

M. R. Dennis, “Correlations and screening of topological charges in Gaussian random fields,” J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

M. Wilkinson, “Screening of charged singularities of random fields,” J. Phys. A 37, 6763-6771 (2004).
[CrossRef]

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27-37 (1978).
[CrossRef]

J. Phys. A: Math. Gen. (1)

G. Foltin, S. Gnutzmann, and U. Smilansky, “The morphology of nodal lines--random waves versus percolation,” J. Phys. A: Math. Gen. 37, 11363-11372 (2004).
[CrossRef]

Opt. Commun. (4)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial fields,” Opt. Commun. 208, 223-253 (2002).
[CrossRef]

D. A. Kessler and I. Freund, “Short- and long-range screening of optical phase singularities and C points,” Opt. Commun. 281, 4194-4204 (2008).
[CrossRef]

I. Freund and D. A. Kessler, “Singularities in speckled speckle,” Opt. Commun. 33, 479-481 (2008).

Opt. Lett. (1)

I. Freund, R. I. Egorov, and M. S. Soskin, “Umbilic point screening in random optical fields,” Opt. Lett. 22, 2182-2184 (2007).
[CrossRef]

Phys. Rev. B (1)

F. Liu and G. F. Mazenko, “Defect-defect correlation in the dynamics of first-order phase transitions,” Phys. Rev. B 46, 5963-5971 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

R. I. Egorov, M. S. Soskin, D. A. Kessler, and I. Freund, “Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields,” Phys. Rev. Lett. 100, 103901 (2008).
[CrossRef] [PubMed]

Physica D (1)

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, “A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations,” Physica D 99, 252-268 (1996).
[CrossRef]

Proc. R. Soc. London, Ser. A (4)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. London, Ser. A 336, 165-190 (1974). For additional sources see online citation databases for the numerous papers that reference this work.
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. London, Ser. A 456, 3048 (2000) (erratum).

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

Other (5)

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

J. W. Goodman, Speckle Phenomena In Optics (Roberts, 2007).

B. I. Halperin, “Statistical mechanics of topological defects,” in Physics of Defects, R.Balian, M.Kleman, and J.-P.Poirier, eds. (North-Holland, 1981), pp. 814-857.

I. Freund and D. A. Kessler, “Singularities in speckled speckle: statistics,” Opt. Commun. (to be published). doi:10.1016/j.optcom.2008.09.029

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (6)

Fig. 1
Fig. 1

Vortex structures. Positive (negative) vortices are shown by white (black) solid circles. (a) Speckled speckle. A random diffuser is illuminated by two concentric disks of light, a and b. The diameter of disk b is ten times the diameter of a; the total optical power in a, however, is ten times that in b. Major (minor) speckle spots in the speckled speckle field are due primarily to beam a ( b ) . Vortices of the combined beam cluster in the dark regions between a field speckle spots because they require perfect destructive interference between the strong a and weak b fields. In normal speckle produced by a single disk, vortices tend to be uniformly distributed with only minor clustering. (b) Normal-speckle phase map produced by a random diffuser illuminated with a single ring; the vortices tend to form a lattice with a square unit cell.

Fig. 2
Fig. 2

Charge variance Q 2 for normal speckle obtained from numerical integration of Eq. (1). (a) Short-range screening. (b) Long-range screening. The dependence on the parameter p in Eqs. (5, 6, 7, 8) is here scaled out by plotting Q 2 ( p R ) vs. p R . As can be seen, for p R > 1 , the results in (a) quickly asymptote to the theoretical values in Eqs. (10, 11, 12), whereas the result in (b) asymptotes to the theoretical form (thin straight line) in Eq. (4). (c) Q 2 for small R. In all four cases (G, Gauss; D, Disk; S, Sinc; and R, Ring), the curves approach the R 0 limit given in Eq. (13).

Fig. 3
Fig. 3

Q 2 R vs. R for composite sources in which both the a and b beams have short-range correlations. The thick curves are the exact result in Eq. (1). The solid lines labeled B are the theory in Eqs. (20, 21, 22, 23); the dashed lines labeled A are the theory in Eqs. (24, 25, 26, 27), whereas the solid lines labeled C are the short-range screening result, Eq. (3). In all cases a = 1 , b = 100 . (a) GG, beams a and b are Gaussians. K = 0.01 . (b) SG, beam a is a Sinc, beam b is a Gaussian. K = 0.01 . (c) DG, beam a is a Disk, beam b is a Gaussian. K = 0.04 . In all three examples displayed here the b field was chosen to be a Gaussian in order to emphasize that the agreement with the theory represented by line B does not depend on the nature of the a beam. Other short-range choices for the b beam, Disk or Sinc, show equally good agreement.

Fig. 4
Fig. 4

Q 2 R vs. R for composite sources in which the a beam has long-range and the b beam short-range correlations. The thick curves are the exact result in Eq. (1). The solid lines labeled B show the short-range screening theory in Eqs. (20, 21, 22); the dashed lines labeled A represent the long-range screening result in Eq. (4). (a) RG, beam a is a Ring, b a Gaussian. (b) RD, beam a is a Ring, beam b a Disk. Beam parameters in both (a) and (b) are a = 1 , b = 100 , K = 0.01 .

Fig. 5
Fig. 5

Q 2 R vs. R for composite sources in which beam a is a Gaussian with short-range correlations and beam b a Ring with long-range correlations. Beam parameters are a = 1 , b = 100 . The thick curves are the exact result in Eq. (1). The solid lines labeled B are the long-range screening theory in Eqs. (29). (a) K = 0.015 . (b) K = 0.2 .

Fig. 6
Fig. 6

Q 2 R vs. R for a composite source in which both beam a and beam b are Rings. Beam parameters are a = 1 , b = 100 , K = 0.015 . The thick curve is the exact result in Eq. (1). The solid line labeled B is the long-range screening theory in Eqs. (29).

Equations (46)

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Q 2 = 1 2 π 0 2 R 4 R 2 r 2 ( W ( r ) ) 2 1 W 2 ( r ) d r ,
η = W ( 0 ) 2 π .
Q 2 R π 0 ( W ( r ) ) 2 1 W 2 ( r ) d r .
Q 2 p R π 2 [ K + ln ( ρ R ) ] ,
K = π D + γ + 5 ln 2 3 2.81182 ,
D = 0 d x J 0 2 ( x ) J 1 2 ( x ) 1 J 0 2 ( x ) 0.563047 ,
S ( G ) ( u ) = I ( G ) exp ( [ u ( 2 p ) ] 2 ) ,
P ( G ) = 4 π p 2 I ( G ) ,
W ( G ) ( r ) = exp ( p 2 r 2 ) .
S ( D ) ( u ) = I ( D ) ϴ ( u p ) ,
P ( D ) = π p 2 I ( D ) ,
W ( D ) ( r ) = 2 J 1 ( p r ) ( p r ) ,
S ( S ) ( u ) = I ( S ) 1 ( u p ) 2 ϴ ( u p ) ,
P ( S ) = 2 π p 2 I ( S ) ,
W ( S ) ( r ) = I ( S ) sinc ( p r ) ,
S ( R ) ( u ) = I ( R ) ε δ ( u p ) ,
P ( R ) = 2 π p ε I ( R ) ,
W ( R ) ( r ) = J 0 ( p r ) ,
s ( u ) = ϴ ( u p ε 2 ) ϴ ( u p + ε 2 ) ,
lim ε 0 [ s ( u ) ε ] = δ ( u p ) .
Q 2 Gauss 1 4 2 π ζ ( 3 2 ) p R = 0.521093 p R ,
Q 2 Disk 0.227210 p R ,
Q 2 Sinc 0.305898 p R .
Q 2 R 0 η π R 2 = N ,
S ( T a T b ) ( u ) = S a ( T ) ( u ) + S b ( T ) ( u ) ,
W ( T a T b ) ( r ) = W a ( T ) ( r ) + K ( T a T b ) W b ( T ) ( r ) 1 + K ( T a T b ) ,
K ( T a T b ) = P b ( T ) P a ( T ) .
η ( T a T b ) = 1 2 π W a ( T ) ( 0 ) + K ( T a T b ) W b ( T ) ( 0 ) 1 + K ( T a T b ) .
S a ( G ) ( u ) = I a ( G ) exp ( [ u ( 2 a ) ] 2 ) ,
S b ( S ) ( u ) = I b ( S ) 1 ( u b ) 2 Θ ( u b ) ,
W ( GS ) ( r ) = exp ( a r ) + K ( GS ) sinc ( b r ) 1 + K ( GS ) ,
K ( GS ) = [ 2 b 2 I b ( S ) ] [ a 2 I a ( G ) ] ,
η ( GS ) = 2 a 2 + K ( GS ) b 2 3 1 + K ( GS ) ,
η η b K 2 π W b ( T ) ( 0 ) .
Q 2 b R 2 π K ( T a T b ) 0 [ d W b ( T ) ( r ) d r ] 2 1 W b ( T ) ( r ) d r .
Q 2 b ( G ) K ( T a G ) b [ ζ ( 3 2 ) 1 ] 2 π R , = 0.454843 K ( T a G ) b R .
Q 2 b ( D ) 0.187153 K ( T a D ) b R ,
Q 2 b ( S ) 0.238531 K ( T a S ) b R .
Q 2 Q 2 a + Q 2 b ;
Q 2 a ( G ) = Q 2 Gauss [ Eq. ( 10 ) ] ,
Q 2 a ( D ) = Q 2 Disk [ Eq. ( 11 ) ] ,
Q 2 a ( S ) = Q 2 Sinc [ Eq. ( 12 ) ] ,
Q 2 b K b 2 4 π 0 2 R 4 R 2 r 2 J 1 2 ( b r ) 1 J 0 ( b r ) d r .
Q 2 b K b R 2 π 2 [ F + ln ( b R ) ] ,
F = π I + γ + 5 ln 2 3 4.84258 ,
I = 0 J 0 ( x ) J 1 2 ( x ) 1 J 0 ( x ) d x 1.20946 .

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