Abstract

Within the limitations of the generalized ray-matrix method, analytical expressions for the first-order intensity moments are obtained for arbitrary cylindrically symmetric ABCD optical systems, assuming beam illumination of reflective targets with arbitrary values of surface roughness whose heights are a Gaussian process. In contrast to previous work, the results presented here are valid for an arbitrary number of correlation areas of the target that contribute to the observed intensity. Also presented here are analytic closed-form results, as well as a highly accurate approximation based on elementary functions, for the heterodyne signal-to-noise ratio in situations where the scattered light is mixed with a strong local oscillator.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 63–75.
  2. E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
    [CrossRef]
  3. K. Creath, “Phase shifting speckle interferometry,” in InternationalConference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).
    [CrossRef]
  4. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 55–69.
  5. H. T. Yura, S. G. Hanson, L. Lading, “Laser-Doppler velocimetry: analytical solution to the optical systemincluding the effects of partial coherence of the target,” J. Opt. Soc. Am. A 12, 2040–2047 (1995).
    [CrossRef]
  6. For example, for a reflected power P it is straightforward to show that U0=2P/πrs2.
  7. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  8. A. E. Siegman, LASERS (University Science Books, Mill Valley, California, 1986), Chap. 20.
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 68.
  10. H. Fujii, T. Asakura, “A contrast variation of image speckle intensity under illuminationof partially coherent light,” Opt. Commun. 10, 32–38 (1974).
    [CrossRef]
  11. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signalwave front,” Proc. IEEE 57, 57–67 (1967).
    [CrossRef]
  12. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “The effects of target structure on the performance of laser time-of-flightvelocimeter systems,” Appl. Opt. 36, 518–533 (1997).
    [CrossRef] [PubMed]

1997 (1)

1995 (1)

1987 (1)

1977 (1)

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

1974 (1)

H. Fujii, T. Asakura, “A contrast variation of image speckle intensity under illuminationof partially coherent light,” Opt. Commun. 10, 32–38 (1974).
[CrossRef]

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signalwave front,” Proc. IEEE 57, 57–67 (1967).
[CrossRef]

Asakura, T.

H. Fujii, T. Asakura, “A contrast variation of image speckle intensity under illuminationof partially coherent light,” Opt. Commun. 10, 32–38 (1974).
[CrossRef]

Beckmann, P.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 55–69.

Creath, K.

K. Creath, “Phase shifting speckle interferometry,” in InternationalConference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).
[CrossRef]

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signalwave front,” Proc. IEEE 57, 57–67 (1967).
[CrossRef]

Fujii, H.

H. Fujii, T. Asakura, “A contrast variation of image speckle intensity under illuminationof partially coherent light,” Opt. Commun. 10, 32–38 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 68.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 63–75.

Hanson, S. G.

Imam, H.

Jakeman, E.

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

Lading, L.

Rose, B.

Siegman, A. E.

A. E. Siegman, LASERS (University Science Books, Mill Valley, California, 1986), Chap. 20.

Welford, W. T.

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

Yura, H. T.

Appl. Opt. (1)

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

Opt. Commun. (2)

H. Fujii, T. Asakura, “A contrast variation of image speckle intensity under illuminationof partially coherent light,” Opt. Commun. 10, 32–38 (1974).
[CrossRef]

E. Jakeman, W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72–79 (1977).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signalwave front,” Proc. IEEE 57, 57–67 (1967).
[CrossRef]

Other (6)

A. E. Siegman, LASERS (University Science Books, Mill Valley, California, 1986), Chap. 20.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 68.

K. Creath, “Phase shifting speckle interferometry,” in InternationalConference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).
[CrossRef]

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 55–69.

For example, for a reflected power P it is straightforward to show that U0=2P/πrs2.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 63–75.

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Figures (5)

Fig. 1
Fig. 1

(a) Variance of the real part (lower curve) σ r 2 and the imaginary part σ i 2 (upper curve) of the amplitude as a function of the standard deviation of phase σ ϕ for δ = 10 and n = 1 . (b) Same as (a) except that δ = 3 . (c) Same as (a) except that δ = 1 . (d) Same as (a) except that δ = 0 .

Fig. 2
Fig. 2

(a) Variance of the real part σ r 2 (lower curve) and the imaginary part σ i 2 (upper curve) of the amplitude as a function of the standard deviation of phase σ ϕ for δ = 10 and n = 0.1 . (b) Same as (a) except that δ = 3 . (c) Same as (a) except that δ = 1 . (d) Same as (a) except that δ = 0 .

Fig. 3
Fig. 3

Comparison of the approximate (lower curve) and the exact (upper curve) expressions for I rel for n = 0.1 and various values of δ.

Fig. 4
Fig. 4

Comparison of the approximate (lower curve) and the exact (upper curve) expressions for I rel for n = 1 and various values of δ.

Fig. 5
Fig. 5

Comparison of the approximate (lower curve) and the exact (upper curve) expressions for I rel for n = 10 and various values of δ.

Equations (33)

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B ( r ) exp { i [ ϕ ( r 1 ) - ϕ ( r 2 ) ] } = exp { - σ ϕ 2 [ 1 - ρ h ( r ) ] } ,
σ ϕ 2 = [ k ( 1 + cos   β ) ] 2 σ h 2 ,
ρ h ( r ) = exp ( - r 2 / r h 2 ) ,
U 0 ( r ) = U 0   exp ( - r 2 / r s 2 ) exp [ i ϕ ( r ) ] ,
U ( 0 ) = input plane U 0 ( r ) G ( r ,   0 ) d r ,
G ( r ,   0 ) = - ik 2 π B   exp - ik 2 B   Ar 2 ,
U ( 0 ) = U d u ,
u = - ik 2 π BU d   U 0 ( r ) exp - r 2 1 w e 2 + ik 2 z e d r ,
w e = - 2 / k   Im ( A / B ) ,
z e = 1 / Re ( A / B ) ,
u r = exp ( - σ ϕ 2 / 2 ) ,
u i = 0 .
| u | 2 = 1 F 1 1 ,   1 + 1 + δ 2 2 n ;   - σ ϕ 2 ,
u 2 = 1 + i δ 2 n   Γ 1 + i δ 2 n ,   0 ,   σ ϕ 2 ( σ ϕ 2 ) ( 1 + i δ ) / 2 n   exp ( - σ ϕ 2 ) ,
δ = a 1 + b 2 , a = kw e 2 2 z e , b = w e r s ,
n = ( r w / r h ) 2 ,
1 r w 2 = 1 r s 2 + 1 w e 2 .
u r 2 = 1 2 ( | u | 2 + Re u 2 ) ,
u i 2 = 1 2 ( | u | 2 - Re u 2 ) ,
u r   u i = 1 2   Im u 2 .
δ = a 1 + b 2 1 .
SNR = SNR 0 I rel ,
I rel = | u | 2 ,
B ( r ) [ 1 - exp ( - σ ϕ 2 ) ] exp - c ( σ ϕ ) r 2 r h 2 + exp ( - σ ϕ 2 ) ,
c ( σ ϕ ) = σ ϕ 2 1 - exp ( - σ ϕ 2 ) .
I rel 1 - exp ( - σ ϕ 2 ) 1 + 2 nc ( σ ϕ ) 1 + δ 2 + exp ( - σ ϕ 2 ) ,
U d = - ikU 0 2 π B   exp   - r 2 r w 2   ( 1 + i δ ) d r = - ikU 0 2 π B   π r w 2 1 + i δ ,
Re [ U ( 0 ) ] Im [ U ( 0 ) ] = Re ( U d ) exp ( - σ ϕ 2 / 2 ) Im ( U d ) exp ( - σ ϕ 2 / 2 ) .
| U ( 0 ) | 2 = | U d | 2 | u | 2 ,
[ Re   U ( 0 ) ] 2 [ Im   U ( 0 ) ] 2 = 1 2 [ | U ( 0 ) | 2 ± Re U 2 ( 0 ) ] ,
Re   U ( 0 ) Im   U ( 0 ) = 1 2   Im U 2 ( 0 ) ,
Re U 2 ( 0 ) Im U 2 ( 0 )
= Re ( U d 2 ) [ u r 2 - u i 2 ] - 2   Im ( U d 2 ) u r u i 2   Re ( U d 2 ) u r u i + Im ( U d 2 ) [ u r 2 - u i 2 ] .

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