Abstract

Single- and multiple-pulse detection statistics are presented for aperture-averaged direct detection optical receivers operating against partially developed speckle fields. A partially developed speckle field arises when the probability density function of the received intensity does not follow negative exponential statistics. The case of interest here is the target surface that exhibits diffuse as well as specular components in the scattered radiation. An approximate expression is derived for the integrated intensity at the aperture, which leads to single- and multiple-pulse discrete probability density functions for the case of a Poisson signal in Poisson noise with an additive coherent component. In the absence of noise, the single-pulse discrete density function is shown to reduce to a generalized negative binomial distribution. The radar concept of integration loss is discussed in the context of direct detection optical systems where it is shown that, given an appropriate set of system parameters, multiple-pulse processing can be more efficient than single-pulse processing over a finite range of the integration parameter n.

© 2000 Optical Society of America

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References

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  1. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1963).
    [CrossRef]
  2. D. G. Youmans, “Laser radar speckle and glint statistics: illustrations using AMOR data,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 43–57 (1990).
    [CrossRef]
  3. G. R. Osche, K. N. Seeber, Y. F. Lok, D. S. Young “Laser radar cross-section estimation from high-resolution image data,” Appl. Opt. 31, 2452–2460 (1992).
    [CrossRef] [PubMed]
  4. L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
    [CrossRef]
  5. S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), p. 98.
  6. R. Barakat, “First order probability densities of laser speckle patterns observed through finite sized scanning apertures,” Opt. Acta 20, 729–740 (1973).
    [CrossRef]
  7. J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
    [CrossRef]
  8. O. Kempthorne, L. Folks, Probability, Statistics, and Data Analysis (Iowa State U. Press, Ames, Iowa, 1975), pp. 171–173.
  9. J. Ohtsubo, T. Asakura, “Statistical properties of the sum of partially developed speckle patterns,” Opt. Lett. 1, 98–100 (1977).
    [CrossRef] [PubMed]
  10. V. S. Rano Gudimetla, “Two-point joint probability-density function for the sum of partially developed speckle patterns,” J. Opt. Soc. Am. 9, 1119–1123 (1992).
    [CrossRef]
  11. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Vol. 9, pp. 23 and 53.
  12. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of NBS Applied Mathematics Series (Dover, N.Y., 1964), p. 375, Eq. (9.6.10).
  13. See Ref. 12, p. 375, Eq. (9.6.7).
  14. See Ref. 8, pp. 89–93.
  15. J. V. DiFranco, W. L. Rubin, Radar Detection (Artech House, Norwood, Mass., 1980), p. 359.

1992 (2)

G. R. Osche, K. N. Seeber, Y. F. Lok, D. S. Young “Laser radar cross-section estimation from high-resolution image data,” Appl. Opt. 31, 2452–2460 (1992).
[CrossRef] [PubMed]

V. S. Rano Gudimetla, “Two-point joint probability-density function for the sum of partially developed speckle patterns,” J. Opt. Soc. Am. 9, 1119–1123 (1992).
[CrossRef]

1977 (1)

1973 (1)

R. Barakat, “First order probability densities of laser speckle patterns observed through finite sized scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

1972 (1)

1963 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1963).
[CrossRef]

1959 (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

Asakura, T.

Barakat, R.

R. Barakat, “First order probability densities of laser speckle patterns observed through finite sized scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

Dainty, J. C.

DiFranco, J. V.

J. V. DiFranco, W. L. Rubin, Radar Detection (Artech House, Norwood, Mass., 1980), p. 359.

Folks, L.

O. Kempthorne, L. Folks, Probability, Statistics, and Data Analysis (Iowa State U. Press, Ames, Iowa, 1975), pp. 171–173.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1963).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Vol. 9, pp. 23 and 53.

Kempthorne, O.

O. Kempthorne, L. Folks, Probability, Statistics, and Data Analysis (Iowa State U. Press, Ames, Iowa, 1975), pp. 171–173.

Lok, Y. F.

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

Ohtsubo, J.

Osche, G. R.

Rano Gudimetla, V. S.

V. S. Rano Gudimetla, “Two-point joint probability-density function for the sum of partially developed speckle patterns,” J. Opt. Soc. Am. 9, 1119–1123 (1992).
[CrossRef]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), p. 98.

Rubin, W. L.

J. V. DiFranco, W. L. Rubin, Radar Detection (Artech House, Norwood, Mass., 1980), p. 359.

Seeber, K. N.

Youmans, D. G.

D. G. Youmans, “Laser radar speckle and glint statistics: illustrations using AMOR data,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 43–57 (1990).
[CrossRef]

Young, D. S.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
[CrossRef]

V. S. Rano Gudimetla, “Two-point joint probability-density function for the sum of partially developed speckle patterns,” J. Opt. Soc. Am. 9, 1119–1123 (1992).
[CrossRef]

Opt. Acta (1)

R. Barakat, “First order probability densities of laser speckle patterns observed through finite sized scanning apertures,” Opt. Acta 20, 729–740 (1973).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1963).
[CrossRef]

Proc. Phys. Soc. London (1)

L. Mandel, “Fluctuations of photon beams: the distribution of photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

Other (8)

S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), p. 98.

D. G. Youmans, “Laser radar speckle and glint statistics: illustrations using AMOR data,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 43–57 (1990).
[CrossRef]

O. Kempthorne, L. Folks, Probability, Statistics, and Data Analysis (Iowa State U. Press, Ames, Iowa, 1975), pp. 171–173.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Vol. 9, pp. 23 and 53.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of NBS Applied Mathematics Series (Dover, N.Y., 1964), p. 375, Eq. (9.6.10).

See Ref. 12, p. 375, Eq. (9.6.7).

See Ref. 8, pp. 89–93.

J. V. DiFranco, W. L. Rubin, Radar Detection (Artech House, Norwood, Mass., 1980), p. 359.

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Figures (8)

Fig. 1
Fig. 1

Probability density functions for the integrated intensity of a partially developed speckle field for the case of the mean intensity of the diffuse component 〈I s 〉 = 1, the intensity of the coherent component I c = 0.01, and several values of speckle count M.

Fig. 2
Fig. 2

Probability density functions for the integrated intensity of a partially developed speckle field for the case of the mean intensity of the diffuse component 〈I S 〉 = 1, the intensity of the coherent component I c = 1, and several values of speckle count M.

Fig. 3
Fig. 3

Probability density functions for the integrated intensity of a partially developed speckle field for the case of the mean intensity of the diffuse component 〈I s 〉 = 0.01, the intensity of the coherent component I c = 1, and several values of speckle count M.

Fig. 4
Fig. 4

Detection probability versus total signal consisting of diffuse and specular components for the case of κ = 0.1, where κ = N c / s represents the ratio of specular-to-diffuse photoelectron counts, n is the mean noise photoelectron count, and M is the number of speckles intercepted by the aperture.

Fig. 5
Fig. 5

Detection probability versus total signal consisting of diffuse and specular components for the case of κ = 5, where κ = N c / s represents the ratio of specular-to-diffuse photoelectron counts, n is the mean noise photoelectron count, and M is the number of speckles intercepted by the aperture.

Fig. 6
Fig. 6

Integrated detection probability versus signal photoelectron count for a fully developed speckle field (κ = 0), a speckle count of M = 2, a noise count of n = 1, and a false-alarm probability of P FA = 10-6.

Fig. 7
Fig. 7

Integrated detection probability versus total signal photoelectron count for a partially developed speckle field of various κ values, a speckle count of M = 2, a noise count of n = 1, and a false-alarm probability of PFA = 10-6.

Fig. 8
Fig. 8

Integration loss (decibels) versus integrated samples n for n = 1, κ = 0, P FA = 10-6, and several values of speckle count M and detection probability P D .

Equations (50)

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pk=0 pk|WpWdW,
pk|W=N¯n+bWkk!exp-N¯n+bW,
pW=aMWM-1 exp-aWΓM,W<0.   =0,W0,
pI=1Isexp-IIs.
ps+nk=MM+N¯sM exp-N¯nM-1!×j=0kk+M-j-1!j!k-j!N¯njN¯sM+N¯sk-j.
ps+nk=pMqkΓk+MΓMΓk+1×exp-N¯n1F1α, β; N¯nq,
1F1α, β; z=n=0kαnβnznn!,
pI=1Isexp-I+IcIsI02IIcIs,I0,  =0,I<0,
I=Is+Ic,
varI=IsIs+2Ic,
px=12exp-x2-λI02λx,
Giυ=exp-λ1-1-2iυ-11-2iυ.
Gmiυ=j=0λmj exp-λmj!1-2iυ-1+j,
Giυ=m=1M Gmiυ=j=0Mλmj exp-Mλm1-2iυM+j.
px=j=0MλmjxM+j-1 exp-x/2+Mλm2M+jj!ΓM+j.
IM-1y=y2M-1j=0y2/4jj!ΓM+j,
px=xMλmM-1/2exp-x/2+Mλm2M+1/2×IM-12Mxλm.
pI=MIsIIcM-1/2 exp-MI+IcIsIM-12MIIcIs.
I=Is+Ic,
varI=IsMIs+2Ic.
pW=aWWcM-1/2×exp-aW+WcIM-12aWcW,
limz0IM-1z/zM-12M-1M-1!-1,
ps+nk=a exp-N¯n+aWck!0N¯n+bWk×WWcM-1/2 exp-a+bW×IM-12aWWcdW.
ps+nk=a 0N¯n+bWkk!exp-N¯n+bW×j=0aWcjMM+jbWN¯sM+j-1×exp-aW+Wcj!ΓM+jdW.
ps+nk=exp-N¯n+aWck!j=0aWcjaM+jj!ΓM+j×0N¯n+bWk exp-a+bWWM+j-1dW.
N¯n+bWk=i=0kk!i!k-i! N¯nibWk-i
0 xs-1 exp-txdx=s-1!ts,
ps+nk=exp-N¯n+aWcj=0aWcjaM+jj!ΓM+ja+bM+j×i=0kM+k+j-i-1!i!k-i!N¯niN¯sM+N¯sk-i
ps+nk=pMqk exp-N¯n+pNc/q×j=0p2NcqjΓM+k+jΓM+jΓk+1×1F1α, βj; N¯nq,
limNc0j0 Ncj00=1,
psk=pMqk exp-pNcqj=0p2NcqjΓM+k+jΓM+jΓk+1=pMqkΓM+kΓMΓk+1exp-pNcq×2F1δ, ε; μ; p2Ncq,
2F1δ, ε; μ; z=n=0δnεn/μnzn/n!.
ps+nk=pMqk exp-N¯n+κM×j=0κpMjΓM+k+jΓM+jΓk+1×1F1α, βj; N¯nq.
PD=1-k=0Nt-1 ps+nk,
PFA1-k=0Nt-1N¯nkk!exp-N¯n,
Ps+nk=N¯n+N¯s+Nckk!exp-N¯n+N¯s+Nc.
Gmiυ=k=0 ps+nkexpiυk,
Giυ=pM1-q expiυ-M exp-N¯n1-expiυ,
Gniυ=i=1n Giiυ=p1-q expiυ- exp-N¯n1-expiυ,
ps+nnk=pqkΓk+ΓΓk+1exp-N¯n1F1α, β; N¯nq,
PD=1-k=0Nt-1 ps+nnk,
PFA1-k=0Nt-1N¯nkk!exp-N¯n.
Giυ=pM1-q expiυ-M exp-N¯n1-expiυ×exp-pNcq1-p1-q expiυ-1.
Giυ=p1-q expiυ- exp-N¯n1-expiυ×exp-pNcq1-p1-q expiυ-1,
ps+nnk=pqk exp-N¯n+pNc/q×j=0p2NcqjΓ+k+jΓ+jΓk+1×1F1α, βj; N¯nq,
ps+nnk=pqk exp-N¯n+κ×j=0κpjΓ+k+jΓ+jΓk+1×1F1α, βj; N¯nq.
ps+nk=N¯n+N¯s+Nckk!exp-N¯n+N¯s+Nc,
L=10 logn SNRnSNRc,
L=10 lognĒnpĒsp,
L=10 lognN¯npN¯sp,

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