Abstract

The characteristics of highly directional electromagnetic beams generated by planar sources of different states of spatial coherence are investigated by using variations of Gaussian Schell-model beams. Azimuthally symmetric beams with the electric vector polarized in the azimuthal direction and dipolar beams with linearly polarized electric fields are treated. The effect on the shape of the beam of decreasing the coherence length of the source distribution is discussed.

© 1999 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), pp. 263–287.
  2. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  3. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  4. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  5. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [CrossRef]
  6. W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
    [CrossRef]
  7. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  8. 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]
  9. E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  10. M. W. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
    [CrossRef]

1998

1993

1983

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

1982

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

1979

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

1978

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]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
[CrossRef]

Bertolotti, M.

Carter, W. H.

Collett, E.

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (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]

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.

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.

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

Kowarz, M. W.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

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]

Seshadri, S. R.

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]

Wolf, E.

M. W. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[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]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

Zubairy, M. S.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Commun.

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]

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]

Opt. Lett.

Other

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

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Figures (4)

Fig. 1
Fig. 1

Transverse distributions of the normalized Poynting vector for the azimuthally symmetric beam as functions of the coherence length σg of the source. The physical parameters are P=1, w0=1 mm, and z=0.2b, where b=4.965 m is the Rayleigh distance. (a) σg=, (b) σg=0.5 mm, (c) σg=0.3 mm. (d) The distributions of the normalized Poynting vector in the plane y=0 for the same three coherence lengths: (a) σg=, (b) σg=0.5 mm, (c) σg=0.3 mm.

Fig. 2
Fig. 2

Transverse distributions of the normalized Poynting vector for the dipolar beam as functions of the coherence length σg of the source. The physical parameters are the same as in Fig. 1. (a) σg=, (b) σg=1 mm (c) σg=0.5 mm, (d) σg=0.3 mm. (e) The distributions of the normalized Poynting vector in the plane y=0 for the same four coherence lengths: (a) σg=, (b) σg=1 mm, (c) σg=0.5 mm, (d) σg=0.3 mm.

Fig. 3
Fig. 3

Normalized rms beam radius br/w0 in the radial direction and normalized rms beam widths bx/w0 and by/w0 in the x and the y directions, respectively, as functions of w0/σg for z=0.2b. (a) br/w0, (b) bx/w0, (c) by/w0.

Fig. 4
Fig. 4

Ratio by/bx of the normalized beam width in the y direction to that in the x direction as a function of w0/σg for z=0.2b.

Equations (50)

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E˜(r, t)=Re[E˜(r)exp(-iωt)],
H˜(r, t)=Re[H˜(r)exp(-iωt)],
Ht(r)=zˆ×Et(r)=-[tF0(r)]exp(ikz),
2x2+2y2+2ikzF0(x, y, z)=0.
F0(x, y, z)=--dx1dy1F0(x1, y1; z1=0)×G(x-x1, y-y1; z),
G[x, y, z]=-ik2πzexpik2z(x2+y2).
tF0(x, y, z)=--dx1dy1[t1F0(x1, y1; z1=0)]×G(x-x1, y-y1; z),
ti=xˆxi+yˆyi,i=1, 2.
S(r, t)=c2Re E(r)×H*(r),
Sz(r, t)=c2(tF0*)·(tF0)
Sz(r)=----dx1dy1dx2dy2W(x1, y1; x2, y2)×G*(x-x1, y-y1; z)G(x-x2, y-y2; z),
W(x1, y1; x2, y2)=(c/2)t1F01*(x1, y1; z1=0)·t2F02(x2, y2; z2=0).
F0(x, y; z=0)=N exp[-(x2+y2)/w02],
F0(x, y, z)=N(1+iz˜)exp-(x2+y2)w02(1+iz˜),
z˜=z/b,b=12kw02,
Hx=-Ey=N(1+iz˜)22xw02exp-(x2+y2)w02(1+iz˜)exp(ikz),
Hy=Ex=N(1+iz˜)22yw02exp-(x2+y2)w02(1+iz˜)exp(ikz).
Sz(r)=2cN2(x2+y2)wfc4(z)exp-2(x2+y2)wfc2(z),
wfc2(z)=w02+4z2k21w02
Pfc=--Sz(r)dxdy=c2πN2.
W(x1, y1; x2, y2)=2cN2(x1x2+y1y2)w04exp-x12+y12w02×exp-x22+y22w02g(x1, y1; x2, y2),
g(x1, y1; x2, y2)=exp-1σg2[(x1-x2)2+(y1-y2)2],
Sz(r)=2cN21w4(z)1σg24z2k2+(x2+y2)wfc2(z)w2(z)×exp-2(x2+y2)w2(z),
w2(z)=w02+4z2k21w02+2σg2.
Ppc=--Sz(r)dxdy,
Ppc=c2πN2,
(br)2=--dxdy(x2+y2)Sz(r)--dxdySz(r).
br=w02+4z2k21w02+1σg21/2.
F0(x, y; z=0)=Nx exp[-(x2+y2)/w02].
F0(x, y, z)=N(1+iz˜)2x exp-(x2+y2)w02(1+iz˜).
Hx=-Ey=-N(1+iz˜)21-2x2w02(1+iz˜)×exp-(x2+y2)w02(1+iz˜)exp(ikz),
Hy=Ex=N(1+iz˜)22xyw02(1+iz˜)exp-(x2+y2)w02(1+iz˜)exp(ikz).
Sz(r)=c2N2w04wfc4(z)exp-2(x2+y2)wfc2(z)×1-4x2wfc2(z)+4(x2+y2)x2w02wfc2(z).
Pfc=--Sz(r)dxdy=c4πN2w02.
W(x1, y1; x2, y2)=c2N21-2x12w021-2x22w02+4x1y1x2 y2w04exp-x12+y12w02×exp-x22+y22w02g(x1, y1; x2, y2),
Sz(r)=c2N2w02w2(z)exp-2(x2+y2)w2(z)×[Q0+Q1x2+Q2y2+Q3(x2+y2)x2],
Q0=w02w2(z)1+64z4k4σg4w02w2(z),
Q1=4w2(z)-112z4k4σg4w4(z)+4z2k2w2(z)1w02+112σg2-1,
Q2=8z2k2σg2wfc2(z)w6(z),
Q3=4wfc4(z)w8(z).
Ppc=--Sz(r)dxdy.
Ppc=c4πN2w02Q0+(Q1+Q2)w2(z)4+Q3w4(z)4.
Ppc=c4πN2w02,
br2=12w2(z)Q0+12(Q1+Q2)w2(z)+34Q3w4(z).
(bv)2=--dxdy(v2)Sz(r)--dxdySz(r),v=x, y.
bx2=14w2(z)Q0+34Q1+14Q2w2(z)+98Q3w4(z),
by2=14w2(z)Q0+14Q1+34Q2w2(z)+38Q3w4(z).
br2=w02+z2k26w02+4σg2,
bx2=58w02+z2k292w02+2σg2,
by2=38w02+z2k232w02+2σg2.

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