Abstract

Average characteristics of a partially coherent Bessel–Gauss optical beam are treated by using the paraxial approximation. The intensity distribution, as well as the coherence properties of the source, is assumed to vary in the azimuthal direction. The transport of the mean-squared width of the beam in the propagation direction is determined. The variation of the mean squared width of the beam in the azimuthal direction is investigated. The anisotropy of the intensity distribution can be balanced against the anisotropic coherence properties to obtain an optical beam whose mean-squared width in the far zone is a constant in the azimuthal direction.

© 1999 Optical Society of America

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References

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  1. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  2. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
    [CrossRef] [PubMed]
  3. N. A. Ansari, M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
    [CrossRef]
  4. M. S. Zubairy, J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
    [CrossRef] [PubMed]
  5. M. Zahid, M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
    [CrossRef]
  6. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  7. E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  8. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  9. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  10. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  12. M. Zahid, M. S. Zubairy, “Directionality of partially coherent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
    [CrossRef]
  13. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  14. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  15. Information on a manuscript by S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” is available from the author at 109 North Whitney Way, Madison, Wisconsin 53705-2718.
  16. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  17. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
    [CrossRef]
  18. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  19. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]

1999 (1)

1998 (1)

1997 (1)

1990 (1)

M. Zahid, M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[CrossRef]

1989 (1)

M. Zahid, M. S. Zubairy, “Directionality of partially coherent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[CrossRef]

1987 (3)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

M. S. Zubairy, J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

1986 (1)

N. A. Ansari, M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

1983 (1)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

1982 (2)

Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
[CrossRef] [PubMed]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1979 (1)

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1978 (2)

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

1977 (1)

Ansari, N. A.

N. A. Ansari, M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

Borghi, R.

Carter, W. H.

Cincotti, G.

Collett, E.

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Foley, J. T.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gori, F.

Guattari, G.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Li, Y.

Mandel, L.

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

McIver, J. K.

M. S. Zubairy, J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

Information on a manuscript by S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” is available from the author at 109 North Whitney Way, Madison, Wisconsin 53705-2718.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Vahimaa, P.

Wolf, E.

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
[CrossRef] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

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

Zahid, M.

M. Zahid, M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[CrossRef]

M. Zahid, M. S. Zubairy, “Directionality of partially coherent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[CrossRef]

Zubairy, M. S.

M. Zahid, M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[CrossRef]

M. Zahid, M. S. Zubairy, “Directionality of partially coherent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[CrossRef]

M. S. Zubairy, J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

N. A. Ansari, M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

Opt. Acta (1)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[CrossRef]

Opt. Commun. (7)

M. Zahid, M. S. Zubairy, “Directionality of partially coherent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[CrossRef]

N. A. Ansari, M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[CrossRef]

M. Zahid, M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[CrossRef]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. S. Zubairy, J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

Other (3)

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

Information on a manuscript by S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” is available from the author at 109 North Whitney Way, Madison, Wisconsin 53705-2718.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Normalized rms width of the m=1 mode as a function of the azimuthal angle φ0 for w0=1 mm,γ=1,σgx=0.50 mm, and σgy=0.47 mm: (a) z=0, (b) z=0.4b0, (c) z=0.8b0, (d) z=1.2b0.

Equations (119)

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W0(ρ1, ϕ1; ρ2, ϕ2)=W0(x1, y1; x2, y2)=fr(ρ1)fa(ϕ1)fr(ρ2)fa(ϕ2)×g(x1, y1; x2, y2),
fr(ρ)=2w0[Im(γ)]1/2 Jm(βρ)expγ2-ρ2w02,
fa(ϕ)=(πm)-1/2 cos mϕ,
γ=β2w024,
m=2form=01form1.
x=ρ cos ϕ,y=ρ sin ϕ,
g(x1, y1; x2, y2)=exp-(x1-x2)2σgx2+(y1-y2)2σgy2,
0dρ ρ02πdϕ W0(ρ, ϕ; ρ, ϕ)=1,
02πdϕ fa2(ϕ)=1.
0dρ ρfr2(ρ)=4 exp(γ)w02Im(γ) 0dρ ρJm2(βρ)exp-2ρ2w02=1.
02πdϕ fa2(ϕ)(cos ϕ, sin ϕ)=(0, 0),
(u¯)0=002πdρdϕ ρuSz(ρ, ϕ; 0)=0foru=x, y.
(σu2)0=002πdρdϕ ρu2Sz(ρ, ϕ; 0)foru=x, y,
02πdϕ fa2(ϕ)(cos2 ϕ, sin2 ϕ)=(Em, Fm),
Em=3Fm=34form=1Fm=12form1.
(σx2)0Em=(σy2)0Fm=4 exp(γ)w02Im(γ) 0dρ ρρ2×Jm2(βρ)exp(-αρ2),
α=2w02.
0dρ ρρ2Jm2(βρ)exp(-αρ2)
=-ddα 0dρ ρJm2(βρ)exp(-αρ2)
=w048 exp(-γ)Im(γ)G-(γ, m),
G±(γ, m)=1+m±γ+γ Im+1(γ)Im(γ).
(σx2)0Em=(σy2)0Fm=w022 G-(γ, m).
(σx2)0=(σy2)0=w024 G-(γ, m)form13(σy2)0=3w028 G-(γ, m)form=1.
(σxy)0=002πdρdϕ ρxySz(ρ, ϕ; 0).
02πdϕ fa2(ϕ)cos ϕ sin ϕ=0,
(σxy)0=0.
(σφ2)0=002πdρdϕ ρ(x cos φ0+y sin φ0)2Sz(ρ, ϕ; 0)
=(σx2)0 cos2 φ0+(σxy)02 cos φ0 sin φ0+(σy2)0 sin2 φ0.
(σφ2)0=w024 G-(γ, m)form1w028 G-(γ, m)(3 cos2 ϕ+sin2 ϕ)form=1.
Sz(ρ, ϕ; z)=k24π2z2 002π002πdρ1dϕ1 ρ1 dρ2 dϕ2 ρ2×W0(ρ1, ϕ1; ρ2, ϕ2)×exp-ik2z (ρ12-ρ22)expikz ρ[ρ1 cos(ϕ-ϕ1)-ρ2 cos(ϕ-ϕ2)].
002πdρdϕ ρ×expikz ρ[ρ1 cos(ϕ-ϕ1)-ρ2 cos(ϕ-ϕ2)]
=4π2z2k2 δ(ρ1-ρ2)ρ2 δ(ϕ1-ϕ2),
002πdρdϕ ρSz(ρ, ϕ; z)
=002πdρ1dϕ1 ρ1W0(ρ1, ϕ1; ρ1, ϕ1)=1,
(x¯)z=(x¯)0+izk 002πdρdϕ×ρx1 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ.
W0(ρ1, ϕ1; ρ2, ϕ2)
=W0c(ρ1, ϕ1; ρ2, ϕ2)g(x1, y1; x2, y2),
x1 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ
=x1 W0c(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ.
x=(cos ϕ) ρ-(sin ϕ) 1ρ ϕ,
y=(sin ϕ) ρ+(cos ϕ) 1ρ ϕ.
x1 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ=ρ fr(ρ)fr(ρ)fa2(ϕ)cos ϕ-1ρ fr2(ρ)ϕ fa(ϕ)×fa(ϕ)sin ϕ.
02πdϕϕ fa(ϕ)fa(ϕ)sin ϕ
=-m2πm 02πdϕ sin 2mϕ sin ϕ=0,
(xˆ)z=0.
(y¯)z=(y¯)0+izk 002πdρdϕ×ρy1 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ=0.
(σx2)z=--dxdy x2Sz(x, y; z)=k24π2z2 ----dx1dy1dx2dy2×W0(x1, y1; x2, y2)exp-ik2z (x12-x22+y12-y22) z2k2 2x1x2×--dxdy expikz [x(x1-x2)-y(y1-y2)].
(σx2)z=(σx2)0+izk 002πdρdϕ×ρxx1 W0(ρ1, ϕ1; ρ2, ϕ2)-x2 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ+z2k2 002πdρdϕ 
×ρ2x1x2 W0(ρ1, ϕ1; ρ2, ϕ2)ϕ1=ϕ2=ϕρ1=ρ2=ρ.
(σx2)z=(σx2)0+z2k2 002πdρdϕ×ρx [fr(ρ)fa(ϕ)]2+2σgx2.
x [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 exp-2ρ2w02 1πm×Jm-12(βρ)cos2(m-1)ϕ+Jm+12(βρ)cos2(m+1)ϕ+16ρ2β2w04 Jm2(βρ)cos2 mϕ cos2 ϕ-2Jm-1(βρ)Jm+1(βρ)cos(m-1)ϕ×cos(m+1)ϕ-2Jm-1(βρ)Jm(βρ) 4ρβw02×cos(m-1)ϕ cos mϕ cos ϕ+2Jm+1(βρ)Jm(βρ) 4ρβw02×cos(m+1)ϕ cos mϕ cos ϕ.
02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)I0(γ) β2w02 exp-2ρ2w022×J12(βρ)+4ρβw02 J0(βρ)J1(βρ)+4ρ2β2w04 J02(βρ).
02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 exp-2ρ2w02×Jm-12(βρ)+Jm+12(βρ)+8ρ2β2w04 Jm2(βρ)-4ρβw02 Jm-1(βρ)Jm(βρ)+4ρβw02 Jm(βρ)Jm+1(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 0dρ ρ exp-2ρ2w02×Jm-12(βρ)+Jm+12(βρ)+8ρ2β2w04 Jm2(βρ)-4ρβw02 Jm-1(βρ)Jm(βρ)+4ρβw02 Jm(βρ)Jm+1(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 12 0dρ ρ exp-2ρ2w02×Jm-12(βρ)+Jm+12(βρ)+1+4β2w022Jm2(βρ).
02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)I1(γ) β2w02 exp-2ρ2w02×2J02(βρ)+J22(βρ)+12ρ2β2w04 J12(βρ)-8ρβw02 J0(βρ)J1(βρ)+4ρβw02 J1(βρ)J2(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)I1(γ) β2w02 0dρ ρ exp-2ρ2w02×2J02(βρ)+J22(βρ)+12ρ2β2w04 J12(βρ)-8ρβw02 J0(βρ)J1(βρ)+4ρβw02 J1(βρ)J2(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)I1(γ) β2w02 34 0dρ ρ exp-2ρ2w02×J02(βρ)+J22(βρ)+1+4β2w022J12(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 Em0dρ ρ exp-2ρ2w02×Jm-12(βρ)+Jm+12(βρ)+1+4β2w022Jm2(βρ).
0dρ ρ02πdϕx [fr(ρ)fa(ϕ)]2=Em 2w02 G+(γ, m).
(σx2)z=Em w022 G-(γ, m)+z2k2 Em 2w02 G+(γ, m)+2σgx2.
(σy2)z=(σy2)0+z2k2 002πdρdϕ×ρy [fr(ρ)fa(ϕ)]2+2σgy2.
y [fr(ρ)fa(ϕ)]2=exp(γ)Im(γ) β2w02 exp-2ρ2w02 1πm×Jm-12(βρ)sin2(m-1)ϕ+Jm+12(βρ)×sin2(m+1)ϕ+16ρ2β2w04 Jm2(βρ)cos2 mϕ sin2 ϕ+2Jm-1(βρ)Jm+1(βρ)sin(m-1)ϕ sin(m+1)ϕ+2Jm-1(βρ)Jm(βρ) 4ρβw02 sin(m-1)ϕ cos mϕ×sin ϕ+2Jm+1(βρ)Jm(βρ) 4ρβw02×sin(m+1)ϕ cos mϕ sin ϕ.
02πdϕy [fr(ρ)fa(ϕ)]2=exp(γ)I1(γ) β2w02 exp-2ρ2w02J22(βρ)+4ρ2β2w04 J12(βρ)+4ρβw02 J1(βρ)J2(βρ).
0dρ ρ02πdϕy [fr(ρ)fa(ϕ)]2
=exp(γ)I1(γ) β2w02 0dρ ρ exp-2ρ2w02×J22(βρ)+4ρ2β2w04 J12(βρ)+4ρβw02 J1(βρ)J2(βρ).
0dρ ρ02πdϕy [fr(ρ)fa(ϕ)]2
=exp(γ)I1(γ) β2w02 14 0dρ ρ exp-2ρ2w02×J02(βρ)+J22(βρ)+1+4β2w022J12(βρ).
0dρ ρ02πdϕy [fr(ρ)fa(ϕ)]2
=exp(γ)Im(γ) β2w02 Fm0dρ ρ exp-2ρ2w02×Jm-12(βρ)+Jm+12(βρ)+1+4β2w022Jm2(βρ).
0dρ ρ02πdϕy [fr(ρ)fa(ϕ)]2=Fm 2w02 G+(γ, m).
(σy2)z=Fm w022 G-(γ, m)+z2k2 Fm 2w02 G+(γ, m)+2σgy2.
(σxy)z=002πdρdϕ ρxySz(ρ, ϕ; z)=(σxy)0+z2k2 002πdρdϕ ρx [fr(ρ)fa(ϕ)]×y [fr(ρ)fa(ϕ)].
x [fr(ρ)fa(ϕ)]y [fr(ρ)fa(ϕ)]
=-exp(γ)Im(γ) β2w02 1πm exp-2ρ2w02×12 Jm-12(βρ)sin 2(m-1)ϕ-12 Jm+12(βρ)sin 2(m+1)ϕ-8ρ2β2w04 Jm2(βρ)cos2 mϕ sin 2ϕ+Jm-1(βρ)Jm+1(βρ)sin 2ϕ-Jm-1(βρ)Jm(βρ) 4ρβw02 cos mϕ sin(m-2)ϕ-Jm+1(βρ)Jm(βρ) 4ρβw02 cos mϕ sin(m+2)ϕ.
02πdϕx [fr(ρ)fa(ϕ)]y [fr(ρ)fa(ϕ)]=0.
(σxy)z=0.
(σφ2)z=(σx2)z cos2 φ0+(σxy)z2 cos φ0 sin φ0+(σy2)z sin2 φ0.
(σφ2)z=w022 G-(γ, m)(Em cos2 φ0+Fm sin2 φ0)+z2k2 2w02 G+(γ, m)(Em cos2 φ0+Fm sin2 φ0)+2 cos2 φ0σgx2+2 sin2 φ0σgy2.
(σx2)z+(σy2)z
=w022 G-(γ, m)+z2k2 2w02 G+(γ, m)+2σgx2+2σgy2
σ02=(σx2)0+(σy2)0=w022 G-(γ, m).
σ2=sp2+sq2=12π2 1w02 G+(γ, m)+1σgx2+1σgy2.
ρ2=σ021+zd2 4π2d2σ2k2σ02,
M2=2πdσkσ0.
d=kσ02=12 kw02G-(γ, m),
M2=2πσ0σ.
(σx2)z+(σy2)z=ρ2=2(σφ2)z.
bγ,m=12 kw02G-(γ, m)G+(γ, m)1/2.
b0=12 kw02
(σφ2)z=w022 G-(γ, m)(Em cos2 φ0+Fm sin2 φ0)×1+z2(b0/Mp2)2,
Mp2=[G-(γ, m)(Em cos2 φ0+Fm sin2 φ0)]-1/2×G+(γ, m)(Em cos2 φ0+Fm sin2 φ0)+w02 cos2 φ0σgx2+w02 sin2 φ0σgy21/2.
Mf2=G+(γ, m)G-(γ, m)1/2,
MBS2=[G-(γ, m)G+(γ, m)]1/2,
12w02 G+(γ, 1)=-1σgx2+1σgy2.
0dt t exp(-p2t2)Jm(at)Jm(bt)
=12p2 exp-a2+b24p2Imab2p2,
Im(z)=Im+1(z)+mz Im(z),
Jm(z)=12 [Jm-1(z)-Jm+1(z)],
mz Jm(z)=12 [Jm-1(z)+Jm+1(z)],
J-m(z)=(-1)mJm(z).
Q1=1β2w02 0dρ ρ exp-2ρ2w02βρJm-1(βρ)Jm(βρ).
Jm(z)=Jm-1(z)-mz Jm(z),
Jm-1(z)=-Jm(z)+m-1z Jm-1(z),
Q1=0dρ ρ exp-2ρ2w0214 Jm-12(βρ)-14 Jm2(βρ).
Q2=1β2w02 0dρ ρ exp-2ρ2w02βρJm(βρ)Jm+1(βρ),
Q2=0dρ ρ exp-2ρ2w0214 Jm2(βρ)-14 Jm+12(βρ).
Q3=1β2w04 0dρ ρ exp-2ρ2w02ρ2Jm2(βρ).
Q3=0dρ ρ exp-2ρ2w0212β2w02 Jm2(βρ)+116 Jm-12(βρ)+116 Jm+12(βρ)-18 Jm2(βρ).
Im-1(z)=Im+1(z)+2mz Im(z).

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