Abstract

Average characteristics of partially coherent electromagnetic beams are treated with the paraxial approximation. Azimuthally or radially polarized, azimuthally symmetric beams and linearly polarized dipolar beams are used as examples. The change in the mean squared width of the beam from its value at the location of the beam waist is found to be proportional to the square of the distance in the propagation direction. The proportionality constant is obtained in terms of the cross-spectral density as well as its spatial spectrum. The use of the cross-spectral density has advantages over the use of its spatial spectrum.

© 2000 Optical Society of America

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References

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  2. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  3. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  4. M. Zahid, M. S. Zubairy, “Directionality of partially co-herent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
    [CrossRef]
  5. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  6. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  7. F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  8. 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]
  9. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  10. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [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. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of a general partially coherent beam propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  13. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [CrossRef]
  14. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  15. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  16. 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]
  17. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]

1999 (2)

1997 (1)

1991 (3)

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

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

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

1989 (1)

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

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]

1983 (2)

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

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

1982 (1)

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 (3)

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]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[CrossRef]

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

Borghi, R.

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]

Gori, F.

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]

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

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

Guattari, G.

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

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).

Martinez-Herrero, R.

Mejias, P. M.

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]

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.

Serna, J.

Seshadri, S. R.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

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

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]

Sona, A.

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[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]

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]

Vahimaa, P.

Wolf, E.

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

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[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]

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

Zahid, M.

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

Zubairy, M. S.

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

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

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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 co-herent Bessel–Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[CrossRef]

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

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[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]

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

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

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

Opt. Lett. (2)

Other (2)

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

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 (2)

Fig. 1
Fig. 1

Normalized rms width of the symmetric beam as a function of the azimuthal angle φ for w0=1 mm, σgx=0.5 mm, and σgy=2 mm: (a) z=0, (b) z=0.6b, (c) z=1.2b.

Fig. 2
Fig. 2

Normalized rms width of the dipolar beam as a function of the azimuthal angle φ for w0=1 mm, σgx=2 mm, and σgy=1/2 mm: (a) z=0, (b) z=0.6b, (c) z=1.2b.

Equations (95)

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W0(x1, y1; x2, y2)
=4π(x1x2+y1y2)w04exp-x12+y12w02×exp-x22+y22w02g(x1, y1; x2, y2),
g(x1, y1; x2, y2)=exp-(x1-x2)2σgx2+(y1-y2)2σgy2,
W0(x1, y1; x2, y2)
=2πw021-2x12w021-2x22w02+4x1y1x2y2w04×exp-x12+y12w02×exp-x22+y22w02g(x1, y1; x2, y2),
--d xd yW0(x, y; x, y)=1,
(u¯)0=--dxdyuW0(x, y; x, y)=0
foru=x, y.
(σu2)0=--dxdyu2W0(x, y; x, y)foru=x, y,
(σx2)0=(σy2)0=w022,
(σx2)0=58w02,(σy2)0=38w02.
(σxy)0=--dxdyxyW0(x, y; x, y).
(σxy)0=0
(σφ2)0=--dxdy(x cos φ+y sin φ)2W0(x, y; x, y)
=(σx2)0 cos2 φ+(σxy)02 cos φ sin φ+(σy2)0 sin2 φ.
(σφ2)0=w022,
(σφ2)0=w028(5 cos2 φ+3 sin2 φ).
Wz(x3, y3; x4, y4)
=----dx1dy1dx2dy2W0(x1, y1; x2, y2)×G*(x3-x1, y3-y1; z)G(x4-x2, y4-y2; z),
G(x, y; z)=-ik2πzexpik2z(x2+y2).
Sz(x, y; z)=k24π2z2----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z[x12-x22+y12-y22-2x(x1-x2)-2y(y1-y2)].
--dxdySz(x, y; z)
=----dx1dy1dx2dy2W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×δ(x1-x2)δ(y1-y2).
--dxdySz(x, y; z)
=--dx1dy1W0(x1, y1; x1, y1)=1.
(x¯)z=--dxdyxSz(x, y; z)=k24π2z2----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×--dxdy-izkx1expikxz(x1-x2)×expikyz(y1-y2)=-izk----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×x1δ(x1-x2)δ(y1-y2).
(x¯)z=(x¯)0+izk--dx1dy1×x1×W0(x1, y1; x2, y1)x2=x1.
(x¯)z=0.
(y¯)z=--dxdy ySz(x, y; z)=k24π2z2----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×--dxdyexpikxz(x1-x2)×-izky1expikyz(y1-y2)=0.
(σx2)z=--dxdy x2Sz(x, y; z)=k24π2z2----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×--dxdyz2k22x1x2expikxz(x1-x2)×expikyz(y1-y2)=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×exp-ik2z(x12-x22+y12-y22)×z2k22x1x2δ(x1-x2)δ(y1-y2).
(σx2)z=z2k2--dy1dy2×exp-ik2z(y12-y22)δ(y1-y2)×--dx1dx2δ(x1-x2)×x1exp-ik2zx12 x2expik2zx22×W0(x1, y1; x2, y2).
(σx2)z=--dx1dy1x12W0(x1, y1; x1, y1)+izk--dx1dy1x1×x1W0(x1, y1; x2, y1)x2=x1-x2W0(x1, y1; x2, y1)x2=x1+z2k2--dx1dy1×2x1x2W0(x1, y1; x2, y1)x2=x1.
W0(x1, y1; x2, y2)
=W0c(x1, y1; x2, y2)g(x1, y1; x2, y2),
2u1u2W0(x1, y1; x2, y2)x1=x2=xy1=y2=y=2u1u2W0c(x1, y1; x2, y2)x1=x2=xy1=y2=y+2σgu2W0c(x, y; x, y)foru=x, y,
2x1y2W0(x1, y1; x2, y2)x1=x2=xy1=y2=y=2x1y2W0c(x1, y1; x2, y2)x1=x2=xy1=y2=y.
(σx2)z=(σx2)0+z2k2--dxdy×2x1x2W0c(x1, y; x2, y)x1=x2=x+2σgx2.
(σy2)z=--dxdyy2Sz(x, y; z)
=(σy2)0+z2k2--dxdy×2y1y2W0c(x, y1; x, y2)y1=y2=y+2σgy2,
(σxy)z=--dxdyxySz(x, y; z)
=(σxy)0+z2k2--dxdy×2x1y2W0c(x1, y1; x2, y2)x1=x2=xy1=y2=y.
(σφ2)z=(σx2)z cos2 φ+(σxy)z2 cos φ sin φ+(σy2)z sin2 φ.
(σu2)z=w022+z2k22w02+2σgu2foru=x, y,
(σxy)z=0,
(σφ2)z=w022+z2k22w02+2 cos2 φσgx2+2 sin2 φσgy2.
(σx2)z=58w02+z2k2921w02+2σgx2,
(σy2)z=38w02+z2k2321w02+2σgy2,
(σxy)z=0,
(σφ2)z=w028(5 cos2 φ+3 sin2 φ)+z2k2×321w02(3 cos2 φ+sin2 φ)+2cos2 φσgx2+sin2 φσgy2.
W0(x1, y1; x2, y2)
=----dp1dq1dp2dq2W˜0(p1, q1; p2, q2)×exp[2πi(p1x1+q1y1+p2x2+q2y2)],
W˜0(p1, q1; p2, q2)
=----dx1dy1dx2dy2W0(x1, y1; x2, y2)×exp[-2πi(p1x1+q1y1+p2x2+q2y2)].
Sz(x, y; 0)=W0(x, y; x, y)=----dp1dq1dp2dq2×W˜0(p1, q1; p2, q2)×exp{2πi[x(p1+p2)+y(q1+q2)].
--dxdy Sz(x, y; 0)=1=----dp1dq1dp2dq2W˜0(p1, q1; p2, q2)×δ(p1+p2)δ(q1+q2)=--dpdq W˜0(p, q;-p,-q).
p=k2πsin θ cos φ,q=k2πsin θ sin φ,
(p¯)=--dpdqpW˜0(p, q;-p,-q)
=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)--dpdq-12πix1×exp{-2πi[p(x1-x2)+q(y1-y2)]}
=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×-12πix1δ(x1-x2)δ(y1-y2).
(p¯)=12πi--dx1dy1x1W0(x1, y1; x2, y1)x2=x1.
(p¯)=0.
(q¯)=--dpdqqW˜0(p, q;-p,-q)=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×--dpdq-12πiy1×exp{-2πi[p(x1-x2)+q(y1-y2)]}=0.
(sp2)=--dpdqp2W˜0(p, q;-p,-q)=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×--dpdq14π22x1x2×exp{-2πi[p(x1-x2)+q(y1-y2)]}.
(sp2)=14π2--dxdy×2x1x2W0(x1, y1; x2, y2)x1=x2=xy1=y2=y.
(sq2)=--dpdqq2W˜0(p, q;-p,-q)
=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×--dpdq14π22y1y2×exp{-2πi[p(x1-x2)+q(y1-y2)]}=14π2--dxdy×2y1y2W0(x1, y1; x2, y2)x1=x2=xy1=y2=y,
(spq)=--dpdqpqW˜0(p, q;-p,-q)
=----dx1dy1dx2dy2×W0(x1, y1; x2, y2)×--dpdq14π22x1y2×exp{-2πi[p(x1-x2)+q(y1-y2)]}
=14π2--dxdy×2x1y2W0(x1, y1; x2, y2)x1=x2=xy1=y2=y.
W˜z(p1, q1; p2, q2)
=----dx3dy3dx4dy4×W0(x3, y3; x4, y4)×----dx1dy1dx2dy2×G*(x1-x3, y1-y3; z)×G(x2-x4, y2-y4; z)×exp[-2πi(p1x1+q1y1+p2x2+q2y2)]
=W˜0(p1, q1; p2, q2)×expi2π2zk(p12-p22+q12-q22).
W˜z(p, q;-p,-q)=W˜0(p, q;-p,-q).
(σx2)z=(σx2)0+4π2z2k2(sp2),
(σy2)z=(σy2)0+4π2z2k2(sq2),
(σxy)z=(σxy)0+4π2z2k2(spq).
W˜0(p, q;-p,-q)
=2π3DxDy1Dx2σgx21+2σgx2p2Dx2w02+1Dy2σgy21+2σgy2q2Dy2w02exp-2p2Dx2-2q2Dy2,
Dv2=1π21w02+2σgv2forv=x, y.
(sp2)=12π21w02+1σgx2,
(sq2)=12π21w02+1σgy2,
(spq)=0.
W˜0(p, q;-p,-q)
=2π5Dx3Dy3Dx2σgx4+12p2Dx4σgx2w02+4p4Dx6w04+1Dy2σgx2σgy2×1+2σgx2p2Dx2w02+2σgy2q2Dy2w02+4σgx2σgy2p2q2Dx2Dy2w04×exp-2p2Dx2-2q2Dy2.
(sp2)=14π2921w02+2σgx2,
(sq2)=14π2321w02+2σgy2,
(spq)=0.
(σx2)z=(σy2)z=(σφ2)z=w022+2z2k2w02.
(σx2)z=(σy2)z=(σφ2)z=w0221+z2b2.
(σx2)z=58w0x2+9z22k2w0x2,
(σy2)z=38w0y2+3z22k2w0y2,
Av[(σφ2)z]=wφ22+3z2k2wφ2.
(σu2)z=w0221+z2(b/Mu2)2foru=x, y,
Av[(σφ2)z]=w0221+z2(b/Mφ2)2,
Mx2=1.667,My2=0.750,Mφ2=1.225.

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