Abstract

Closed-form integral expressions are developed for the mean and variance of power and energy received from a diffusely reflective object upon illumination by laser radiation with partial temporal coherence. Expressions are presented in dimensionless form and analytic approximations to the integrals are given for signal variations at a receiver caused by fully developed laser speckle. Results are presented in terms of three parameters: the mutual Fresnel number of the receiver and object, the number of longitudinal modes of the illuminating source, and the dimensionless mode spacing of the illuminating source. The calculations assume high light levels and free-space geometry.

© 2003 Optical Society of America

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  1. J. D. Rigden, E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).
  2. B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
    [CrossRef]
  3. R. V. Langmuir, “Scattering of laser light,” Appl. Phys. Lett. 2, 29–30 (1963).
    [CrossRef]
  4. L. Allen, D. G. C. Jones, “An analysis of the granularity of scattered optical maser light,” Phys. Lett. 7, 321–323 (1963).
    [CrossRef]
  5. G. Gould, S. F. Jacobs, J. T. La Tourrette, M. Newstein, P. Rabinowitz, “Coherent detection of light scattered from a diffusely reflecting surface,” Appl. Opt. 3, 648–649 (1964).
    [CrossRef]
  6. J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  7. M. von Laue, “Die Beugunserscheinungen an vielen unregelmaßig verteilten Teilchen,” Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften (Berlin) 44, 1144–1163 (1914).
  8. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).
  9. G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
    [CrossRef]
  10. G. Parry, “Some effects of temporal coherence on first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
    [CrossRef]
  11. H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
    [CrossRef]
  12. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [CrossRef]
  13. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.
  14. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 1.
  15. T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 4.
  16. David L. Fried, “Statistics of the laser radar cross section of a randomly rough target,” J. Opt. Soc. Am. 66, 1150–1160 (1976).
    [CrossRef]
  17. J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
    [CrossRef]
  18. H. T. Yura, B. Rose, S. G. Hanson, “Dynamic laser speckle in complex ABCD systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
    [CrossRef]
  19. Peter A. Bakut, Valery I. Mandrosov, “Properties of the Fourier-telescopic images of remote rough objects,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 49–57 (1999).
    [CrossRef]
  20. A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
    [CrossRef]
  21. O. Svelto, Principles of Lasers (Plenum, New York, 1982), Chap. 4.
  22. See, for example, J. W. Goodman, Statistical Optics (Wiley, New York, 1985) p. 207, Eq. 5.5-21.

1998 (1)

1994 (1)

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

1976 (1)

1975 (2)

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

1974 (2)

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[CrossRef]

G. Parry, “Some effects of temporal coherence on first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

1965 (1)

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

1964 (1)

1963 (3)

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[CrossRef]

R. V. Langmuir, “Scattering of laser light,” Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

L. Allen, D. G. C. Jones, “An analysis of the granularity of scattered optical maser light,” Phys. Lett. 7, 321–323 (1963).
[CrossRef]

1962 (1)

J. D. Rigden, E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

1958 (1)

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

1914 (1)

M. von Laue, “Die Beugunserscheinungen an vielen unregelmaßig verteilten Teilchen,” Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften (Berlin) 44, 1144–1163 (1914).

Allen, L.

L. Allen, D. G. C. Jones, “An analysis of the granularity of scattered optical maser light,” Phys. Lett. 7, 321–323 (1963).
[CrossRef]

Bakut, Peter A.

Peter A. Bakut, Valery I. Mandrosov, “Properties of the Fourier-telescopic images of remote rough objects,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 49–57 (1999).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

Fried, David L.

Genack, A. Z.

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 1.

See, for example, J. W. Goodman, Statistical Optics (Wiley, New York, 1985) p. 207, Eq. 5.5-21.

Gordon, E. I.

J. D. Rigden, E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Gould, G.

Hanson, S. G.

Jacobs, S. F.

Jones, D. G. C.

L. Allen, D. G. C. Jones, “An analysis of the granularity of scattered optical maser light,” Phys. Lett. 7, 321–323 (1963).
[CrossRef]

La Tourrette, J. T.

Langmuir, R. V.

R. V. Langmuir, “Scattering of laser light,” Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

Li, J. H.

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

Mandrosov, Valery I.

Peter A. Bakut, Valery I. Mandrosov, “Properties of the Fourier-telescopic images of remote rough objects,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 49–57 (1999).
[CrossRef]

McKechnie, T. S.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 4.

Newstein, M.

Oliver, B. M.

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[CrossRef]

Parry, G.

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[CrossRef]

G. Parry, “Some effects of temporal coherence on first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.

Pedersen, H. M.

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Rabinowitz, P.

Rigden, J. D.

J. D. Rigden, E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Rose, B.

Schawlow, A. L.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, New York, 1982), Chap. 4.

Townes, C. H.

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

von Laue, M.

M. von Laue, “Die Beugunserscheinungen an vielen unregelmaßig verteilten Teilchen,” Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften (Berlin) 44, 1144–1163 (1914).

Yura, H. T.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. V. Langmuir, “Scattering of laser light,” Appl. Phys. Lett. 2, 29–30 (1963).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (3)

G. Parry, “Some effects of temporal coherence on first order statistics of speckle,” Opt. Acta 21, 763–772 (1974).
[CrossRef]

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Opt. Commun. (1)

G. Parry, “Some effects of surface roughness on the appearance of speckle in polychromatic light,” Opt. Commun. 12, 75–78 (1974).
[CrossRef]

Phys. Lett. (1)

L. Allen, D. G. C. Jones, “An analysis of the granularity of scattered optical maser light,” Phys. Lett. 7, 321–323 (1963).
[CrossRef]

Phys. Rev. (1)

A. L. Schawlow, C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Phys. Rev. E (1)

J. H. Li, A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[CrossRef]

Proc. IEEE (2)

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE 51, 220–221 (1963).
[CrossRef]

J. W. Goodman, “Some effects of target induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Proc. IRE (1)

J. D. Rigden, E. I. Gordon, “The granularity of scattered optical maser light,” Proc. IRE 50, 2367–2368 (1962).

Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften (Berlin) (1)

M. von Laue, “Die Beugunserscheinungen an vielen unregelmaßig verteilten Teilchen,” Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften (Berlin) 44, 1144–1163 (1914).

Other (7)

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

Peter A. Bakut, Valery I. Mandrosov, “Properties of the Fourier-telescopic images of remote rough objects,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 49–57 (1999).
[CrossRef]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 1.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 4.

O. Svelto, Principles of Lasers (Plenum, New York, 1982), Chap. 4.

See, for example, J. W. Goodman, Statistical Optics (Wiley, New York, 1985) p. 207, Eq. 5.5-21.

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Figures (6)

Fig. 1
Fig. 1

Physical picture of the effect of partial coherence on signal variance. Partially coherent laser light of mean wavelength λ is reflected from an object O to a receiver R of width DR at a range z from the object. The resolvable cell C on the object is of diameter λz/DR. The length scale over which coherence decays is lc=c/2ΔνB and the length scale over which coherence recurs is lr=c/2Δνm, where c is the speed of light, ΔνB is the laser bandwidth in Hz, and Δνm is the mode spacing in Hz.

Fig. 2
Fig. 2

Normalized standard deviation of received energy into a square receiver versus mutual Fresnel number γ for different values of normalized laser coherence length Nβ for N=10 longitudinal laser modes: circles, Nβ=12.6; x’s, Nβ=50; pluses, Nβ=100; squares, Nβ=200. Curves without symbols are the corresponding curves using the analytic formula, Eq. (18).

Fig. 3
Fig. 3

Normalized standard deviation of received energy versus normalized laser bandwidth μy=Nβ/γy for different values of mutual Fresnel number γ in a rectangular receiver with long axis parallel to axis of nonzero depth slope on object and for 10 longitudinal laser modes: circles, γ(γx, γy)=(0.5, 10); x’s, γ=(1, 20); pluses, γ=(1.5, 30); squares, γ=(2, 40).

Fig. 4
Fig. 4

Normalized standard deviation of received energy versus normalized laser bandwidth μy for different values of mutual Fresnel number γ in a rectangular receiver and for 50 longitudinal laser modes: circles, γ(γx, γy)=(0.5, 10); x’s, γ=(1, 20); pluses, γ=(1.5, 30); squares, γ=(2, 40).

Fig. 5
Fig. 5

Normalized standard deviation of received energy versus normalized laser bandwidth μx=Nβ/γx for different values of mutual Fresnel number γ in a rectangular receiver rotated 90° viewing the original object and for 10 longitudinal laser modes: circles, γ(γx, γy)=(10, 0.5); x’s, γ=(20, 1); pluses, γ=(30, 1.5); squares, γ=(40, 2).

Fig. 6
Fig. 6

Normalized standard deviation of received energy versus normalized laser bandwidth μx=Nβ/γx for different values of mutual Fresnel number γ in a rectangular receiver rotated 90° viewing the original object and for 50 longitudinal laser modes: circles, γ(γx, γy)=(10, 0.5); x’s, γ=(20, 1); pluses, γ=(30, 1.5); squares, γ=(40, 2).

Equations (22)

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P(t)=|ER(x, t)|2dx2,
ER(x, t)=EI(x1, t)R(x1)×exp(iko|x-x1|2/2Z)dx12/(21/2λZ)+c.c.
EI(x1, t)=j EIjexp(i{ωj[t-2z(x1)/c]})/21/2+c.c.,
P=dx12 dx22j,kEIjEIk*×exp{2i[ωjz(x1)/c-ωkz(x2)/c]}×exp[i(ωj-ωk)t]R(x1)R*(x2)×dx2exp{iko/2Z[2x(x1-x2)+|x1|2-|x2|2]}/(λZ)2.
dx2exp{iko/2Z[2x (x1-x2)+|x1|2-|x2|2]}
=DRxDRysinc[koDRx(x1-x2)/2Z]×sinc[koDRy(y1-y2)/2Z]×exp[iko/2Z(|x1|2-|x2|2)].
R(x1)R*(x2)=|R(x1)|2δ(x1-x2)λ2/π,
P=dx12j,kEIjEIk*exp[2i(ωj-ωk)z(x1)/c]×exp[i(ωj-ωk)t]|R(x1)|2DRxDRy/(πZ2).
P2=dx12dx22dx32dx42j1,j2,j3,j4EIj1EIj2*EIj3EIj4*×exp{2i[ωj1z(x1)/c-ωj2z(x2)/c]}×exp{2i[ωj3z(x3)/c-ωj4z(x4)/c]}×exp{i[(ωj1-ωj2)+(ωj3-ωj4)]t}×R(x1)R*(x2)R(x3)R*(x4)×(DRxDRy)2sinc2[koDR (x1-x2)/2Z]×sinc2[koDR (x3-x4)/2Z]/(λZ)4,
P2=P2+dx12dx22j1,j2,j3,j4EIj1EIj2*×EIj3EIj4*×|R(x1)|2|R(x2)|2exp{2i[(ωj1-ωj4)×z(x1)/c-(ωj2-ωj3)z(x2)/c]}×exp{i[(ωj1-ωj2)+(ωj3-ωj4)]t}(DRxDRy)2×|sinc2[koDR (x1-x2)/2Z]|2/(π2Z4).
W=dx12j,kEIjEIk*exp[2i(ωj-ωk)z(x1)/c]×sinc[(ωj-ωk)T/2]|R(x1)|2TDRxDRy/Z2/π.
W2=W2+dx12dx22j1,j2,j3,j4EIj1EIj2*EIj3EIj4*×|R(x1)|2|R(x2)|2exp{2i[(ωj1-ωj4)z(x1)/c-(ωj2-ωj3)z(x2)/c]}×sinc[(ωj1-ωj2)T/2]sinc[(ωj3-ωj4)T/2]×(DRxDRy)2|sinc2[koDR (x1-x2)/2Z]|2/(π2Z4).
ΔW2(W2-W2)/W2=Cdx12dx22j1, j2, j3, j4EIj1EIj2*EIj3EIj4*|R(x1)|2×|R(x2)|2exp{i[(j1-j4)βζ(x1)-(j2-j3)βζ(x2)]}×sinc[(j1-j2)π]sinc[(j3-j4)π]×|sinc2[γ (x1-x2)/L]|2,
ΔW2=Cdx12dx22j1,j3|EIj1|2|EIj3|2|R(x1)|2×|R(x2)|2exp{i(j1-j3)β[ζ(x1)-ζ(x2)]}×|sinc2[γ (x1-x2)/L]|2.
C=dx12j|EIj|2|R(x1)|2-2.
ΔW2=Cdx12dx22|R(x1)|2|R(x2)|2×|sinc2[γ (x1-x2)/L]|2×j1|EIj1|2exp{ij1β[ζ(x1)-ζ(x2)]}
×j2|EIj2|2exp{ij2β[ζ(x1)-ζ(x2)]}*,
ΔW2=C|EIm|4dx12dx22|R(x1)|2×|R(x2)|2|sinc2[γ (x1-x2)/L]|2×|sin{Nβ[ζ(x1)-ζ(x2)]/2}/N sin{β[ζ(x1)-ζ(x2)]/2}|2,
ΔW2=C|EIm|4|R|22dx12dx22×|sinc2[γ (x1-x2)/L]|2×|sin[Nβa (x1-x2)/2L]/ N sin[βa(x1-x2)/2L]|2,
ΔW2{1/[1+(γx/π)(1+(2/π)N×tan-1{[βz(x)/x/γx]1.8})]}×{1/[1+(γy/π)(1+(2/π)×N tan-1{[βz(x)/y/γy]1.8})]}.
ΔW2/W21/[(1+γx/π)(1+γy/π)(1+N)].

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