Abstract

A theory is developed for analyzing the structure of a focused random field in systems of large Fresnel numbers. The analysis provides a general expression for the cross-spectral density of the focused field in terms of the cross-spectral density of the random field on a spherical reference surface in the region of the diffracting aperture. The results are illustrated by considering the structure of the focal region in a field produced by focusing a partially coherent wave whose correlation function is frequently encountered in practice.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. For a review of this subject see, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986).
  2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Part III.
  3. R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
    [Crossref]
  4. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [Crossref]
  5. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [Crossref]
  6. Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
    [Crossref]
  7. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
    [Crossref]
  8. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [Crossref]
  9. E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
    [Crossref] [PubMed]
  10. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [Crossref]
  11. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981), Eq. (25).
    [Crossref]
  12. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

1988 (2)

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[Crossref]

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

1987 (1)

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[Crossref]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

1982 (1)

1981 (2)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981), Eq. (25).
[Crossref]

1980 (1)

1972 (1)

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[Crossref]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Bodner, S. E.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[Crossref]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Part III.

Collett, E.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[Crossref]

Friberg, A. T.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[Crossref]

Goodman, J. W.

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

He, Q.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Lehmberg, R. H.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[Crossref]

Li, Y.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Marchand, E. W.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Schmitt, A. J.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[Crossref]

Stamnes, J. J.

For a review of this subject see, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986).

Turunen, J.

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[Crossref]

Wolf, E.

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Ann. Physik (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[Crossref]

J. Appl. Phys. (1)

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (3)

Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981), Eq. (25).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Other (3)

For a review of this subject see, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Part III.

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Illustration of notation.

Fig. 2
Fig. 2

Illustration of the correspondence between the coherence patch of the field on the reference sphere S in the aperture and the correlation solid angle of the field behind the aperture.

Fig. 3
Fig. 3

Partially coherent cylindrical wave converging toward a focal line OX, diffracted at a slit aperture. S is a portion of a cylindrical reference surface with axis along OX.

Fig. 4
Fig. 4

Normalized intensity at the geometrical focus as function of the scaled correlation angle γ of the focused partially coherent cylindrical wave, for several values of the variance σ2 of the phase. The angular semiaperture α was taken to be 30°. The normalization constant N=kfB/2π.

Fig. 5
Fig. 5

Normalized intensity distribution I(0, z)/I(0, 0) along the z axis. Dotted curves represent the coherent limit (σ2 = 0). The normalized intensity is symmetric with respect to the plane z = 0. The angular semiaperture α was taken to be 30°.

Fig. 6
Fig. 6

Normalized intensity distribution I(y, 0)/I(0, 0) in the geometrical focal plane z = 0. Dotted curves represent the coherent limit (σ2 = 0). The normalized intensity is symmetric with respect to the plane y = 0. The angular semiaperture α was taken to be 30°.

Fig. 7
Fig. 7

Intensity contours in the neighborhood of the geometrical focus for three values of the scaled correlation angle γ: (a) γ = 2, (b) γ = 1, (c) γ = 0.5. The intensity at the geometrical focus is normalized to unity. The intensity distribution is symmetric with respect to both the y = 0 plane and the z = 0 plane. The dashed lines indicate the boundary of the geometrical shadow. The angular semiaperture α was taken to be 30°, and the variance of the phase was taken to be σ2 = 3.

Equations (27)

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

US(-fs)=a(s)exp(-ikf)fwhens Ω,=0whensΩ.
fλ,Rλ,andR2/λf1,
U(r)=-ik2πΩa(s)exp(iks·r)dΩ.
U(r)=-ikf2πexp(ikf)ΩUS(-fs)exp(iks·r)dΩ.
W(r1, r2, ω)U*(r1, ω)U(r2, ω),
W(r1, r2)=(k2π)2ΩΩ A(s1, s2)×exp[ik(s2·r2-s1·r1)]dΩ1dΩ2.
A(s1, s2)a*(s1)a(s2)
A(s1, s2)=f2WS(-fs1,-fs2),
WS(-fs1,-fs2)US*(-fs1)US(-fs2)
W(r1, r2)=kf2π2ΩΩWS(-fs1,-fs2)×exp[ik(s2·r2-s1·r1)]dΩ1dΩ2.
WS(-fs1,-fs2)=WS*(-fs2,-fs1)=WS(-fs2,-fs1).
W(-r1,-r2)=W(r2, r1)=W*(r1, r2).
I(r)W(r, r),
I(-r)=I(r).
μ(r1, r2)W(r1, r2)I(r1)I(r2),
μ(-r1,-r2)=μ(r2, r1)=μ*(r1, r2).
W(ρ1, ρ2)=kf2π -ααWS(-fs1,-fs2)×exp[ik(s2·ρ2-s1·ρ1)]dθ1dθ2,
WS(-fs1,-fs2)=BU*(θ1)U(θ2).
U(θ)=exp[iϕ(θ)],
WS(-fs1,-fs2)=B exp{-σ2[1-g(θ2-θ1)]2},
σ2=[Δϕ(θ)]2,
g(θ2-θ1)=Δϕ(θ1)Δϕ(θ2)σ2.
g(θ)=exp(-θ2/2δ2).
WS(-fs1,-fs2)B exp-(θ2-θ1)22(δ/σ)2.
I(ρ)=kf2π -ααWS(-fs1,-fs2)×exp[ik(s2-s1)·ρ]dθ1dθ2.
γδ/α,
Icoh(0, 0)=(2α)2N.

Metrics