Abstract

A pulsed, dual CO2 laser lidar was used to measure return signal statistics as a function of the number of speckles integrated by the lidar receiver per laser pulse. A rotating target generated statistically independent speckle patterns on each laser pulse. Data were collected for a wide range of receiver aperture sizes. A statistical model is developed that predicts the probability density of the return lidar pulse energy, which includes speckle, depolarization by the target, and albedo sampling. The predictions of this model are compared with the measured probability density function of the return pulse energies. Very good agreement is found between the geometrically calculated number of integrated speckles and the number predicted by the model.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  35. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986), p. 668.
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  37. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  38. A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
    [CrossRef]
  39. R. Barakat, “First-order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 729–740 (1973).
    [CrossRef]

1992

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

1990

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 49, 339–342 (1990).
[CrossRef]

1989

1988

1987

1985

1984

1983

1982

1980

1976

1974

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optic 39, 258–267 (1974).

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

1973

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

1971

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

1965

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

Ahl, J. L.

Ariel, E. D.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Barakat, R.

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

Bogan, J. R.

Boreman, G. D.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 49, 339–342 (1990).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Brothers, A. M.

Busch, G. E.

G. E. Busch, “Speckle error in transmitter energy measurement,” , pp. 1–6, Feb.6, 1996 (Los Alamos National Laboratory, Los Alamos, N.M.).

Churnside, J. H.

Czuchlewski, S.

S. Czuchlewski, “Speckle effects with integrating spheres,” , pp. 1–6, Feb.20, 1996 (Los Alamos National Laboratory, Los Alamos, N.M.).

Dainty, J. C.

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 2–45.

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (IEEE, New York, 1987), p. 96.

Flamant, P. H.

Fox, J. A.

Fried, D. L.

Gautier, C. R.

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 Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, Ed. (Springer-Verlag, New York, 1984), Chap. 2.

Grant, W. B.

Hallerman, G. R.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Haner, D. A.

Holmes, J. F.

James, A. B.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 49, 339–342 (1990).
[CrossRef]

Kavaya, M. J.

Kerr, J. R.

Killinger, D. K.

Lee, M. H.

Lehman, F.

MacKerrow, E. P.

E. P. MacKerrow, M. J. Schmitt, D. C. Thompson, “The effect of speckle on lidar pulse-pair ratio statistics,” Appl. Opt. (to be published).

McIntyre, C. M.

McKechnie, T. S.

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optic 39, 258–267 (1974).

Measures, R. M.

R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

Menyuk, C. R.

Menyuk, N.

Menzies, R. T.

Meyer, S. L.

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975).

Milton, M. J. T.

Nemzek, R. J.

R. J. Nemzek, “Polarization of CO2 lidar hard-target returns,” Aug.10, 1995 (Los Alamos National Laboratory, Los Alamos, N.M.).

Oppenheim, U. P.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 3.

Payson, H. C.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Rao Gudimetla, V. S.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (IEEE, New York, 1987), p. 96.

Schmitt, M. J.

E. P. MacKerrow, M. J. Schmitt, D. C. Thompson, “The effect of speckle on lidar pulse-pair ratio statistics,” Appl. Opt. (to be published).

Scribot, A. A.

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

Shirley, L. G.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986), p. 668.

Sun, Y.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 49, 339–342 (1990).
[CrossRef]

Thompson, D. C.

E. P. MacKerrow, M. J. Schmitt, D. C. Thompson, “The effect of speckle on lidar pulse-pair ratio statistics,” Appl. Opt. (to be published).

Vivilecchia, J. R.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Warren, R. E.

Wiesemann, W.

Woods, P. T.

Appl. Opt.

W. B. Grant, “Effect of differential spectral reflectance on DIAL measurements using topographic targets,” Appl. Opt. 21, 2390–2394 (1982).
[CrossRef] [PubMed]

N. Menyuk, D. K. Killinger, C. R. Menyuk, “Limitations of signal averaging due to temporal correlation in laser remote-sensing measurements,” Appl. Opt. 21, 3377–3383 (1982).
[CrossRef] [PubMed]

M. J. Kavaya, R. T. Menzies, D. A. Haner, U. P. Oppenheim, P. H. Flamant, “Target reflectance measurements for calibration of lidar atmospheric backscatter data,” Appl. Opt. 22, 2619–2628 (1983).
[CrossRef] [PubMed]

N. Menyuk, D. K. Killinger, “Assessment of relative error sources in IR DIAL measurement accuracy,” Appl. Opt. 22, 2690–2698 (1982).
[CrossRef]

P. H. Flamant, R. T. Menzies, M. J. Kavaya, “Evidence for speckle effects on pulsed CO2 lidar signal returns from remote targets,” Appl. Opt. 23, 1412–1417 (1984).
[CrossRef]

N. Menyuk, D. K. Killinger, C. R. Menyuk, “Error reduction in laser remote sensing: combined effects of cross correlation and signal averaging,” Appl. Opt. 24, 118–131 (1985).
[CrossRef] [PubMed]

R. E. Warren, “Effect of pulse-pair correlation on differential absorption lidar,” Appl. Opt. 24, 3472–3475 (1985).
[CrossRef] [PubMed]

W. Wiesemann, F. Lehman, “Reliability of airborne CO2 DIAL measurements: schemes for testing technical performance and reducing interference from differential reflectance,” Appl. Opt. 24, 3481–3486 (1985).
[CrossRef]

R. E. Warren, “Detection and discrimination using multiple-wavelength differential absorption lidar,” Appl. Opt. 24, 3541–3545 (1985).
[CrossRef] [PubMed]

M. J. T. Milton, P. T. Woods, “Pulse averaging methods for a laser remote monitoring system using atmospheric backscatter,” Appl. Opt. 26, 2598–2603 (1987).
[CrossRef] [PubMed]

J. A. Fox, C. R. Gautier, J. L. Ahl, “Practical consideration for the design of CO2 lidar systems,” Appl. Opt. 27, 847–855 (1988).
[CrossRef] [PubMed]

W. B. Grant, A. M. Brothers, J. R. Bogan, “Differential absorption lidar signal averaging,” Appl. Opt. 27, 1934–1938 (1988).
[CrossRef] [PubMed]

D. A. Haner, R. T. Menzies, “Reflectance characteristics of reference materials used in lidar hard target calibration,” Appl. Opt. 28, 857–864 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

Lincoln Lab. J.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, J. R. Vivilecchia, “Advanced techniques for target discrimination using laser speckle,” Lincoln Lab. J. 5, 367–440 (1992).

Opt. Acta

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[CrossRef]

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

Opt. Commun.

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

Opt. Eng.

G. D. Boreman, Y. Sun, A. B. James, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 49, 339–342 (1990).
[CrossRef]

Optic

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optic 39, 258–267 (1974).

Proc. IEEE

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

Other

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, Ed. (Springer-Verlag, New York, 1984), Chap. 2.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 3.

J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 14, pp. 2–45.

R. M. Measures, Laser Remote Sensing, Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

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

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (IEEE, New York, 1987), p. 96.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

R. J. Nemzek, “Polarization of CO2 lidar hard-target returns,” Aug.10, 1995 (Los Alamos National Laboratory, Los Alamos, N.M.).

G. E. Busch, “Speckle error in transmitter energy measurement,” , pp. 1–6, Feb.6, 1996 (Los Alamos National Laboratory, Los Alamos, N.M.).

S. Czuchlewski, “Speckle effects with integrating spheres,” , pp. 1–6, Feb.20, 1996 (Los Alamos National Laboratory, Los Alamos, N.M.).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986), p. 668.

E. P. MacKerrow, M. J. Schmitt, D. C. Thompson, “The effect of speckle on lidar pulse-pair ratio statistics,” Appl. Opt. (to be published).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975).

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Figures (11)

Fig. 1
Fig. 1

Schematic of the lidar system used for this experiment. BS, Beam splitter (R = 90%, T = 10%); W, wedge; OT, optical trigger; EM, energy monitor; P, periscope; BC, beam combiner (90% in main beam, 10% for diagnostics); PC, pyroelectric camera (for beam far-field profile and pointing); BE, beam expander (to 6-in. diameter); RO, relay optics; D, detector (LN2 cooled MCT, 1 mm × 1 mm square).

Fig. 2
Fig. 2

Rotating drum target used to generate independent speckle patterns. The cylindrical drum had a length of 2 m, a diameter of 2 m, and a angular rotation rate of 2 rpm. The drum was coated with aluminum roofing compound to approximate a Lambertian reflector.

Fig. 3
Fig. 3

Probability density function for measuring a single-wavelength speckled lidar return intensity of I, i.e., the Gamma distribution function, relation (17). Here M is approximately the number of speckle integrated by the receiver aperture, given by Eq. (C7). For M > 10 the Gamma distribution function approaches a Gaussian probability distribution function.

Fig. 4
Fig. 4

Measured full-aperture (therefore less sensitive to speckle) reflectance distribution of the rotating drum for both laser 0 (L0) and laser 1 (L1) at the 12C16O2 (10P16, 10.55 µm) and 12C16O2 (10P20, 10.59 µm) laser lines. Here IN is the ratio of the incoming lidar pulse energy to the outgoing pulse energy, and ρ is the mean normalized value of IN. Differences in the distributions for L0 and L1 are caused by slight differences in the spatial modes between the two lasers or differences in the beam pointing direction between the lasers.

Fig. 5
Fig. 5

Time series examples of the return signals from the rotating drum target with (a) a large-aperture receiver and (b) a small-aperture receiver. Speckle noise on the small-aperture signal dwarfs the reflectivity variations of the target, which are seen as periodic fluctuations in (a). The normalized standard deviation (normalized to the mean) values of these data are indicated by σ.

Fig. 6
Fig. 6

Example of the probability density function of the reflected intensity from the target. The solid curve is the prediction of Eq. (11). (a) Width of the output laser distribution is 5% (dotted curve), typical of our laser. For this case the probability density function of the reflected intensity is smeared out significantly by the reflectance probability density function. (b) The same information is shown for a narrower laser output distribution (e.g., 0.5% standard deviation). Here the probability density function of the reflected intensity is determined mainly by the target reflectance probability density function. The histogram is the measured reflectance distribution of the target for the full receiver aperture (integration of many correlation cells).

Fig. 7
Fig. 7

Measured probability density function from the rotating target with the smallest receiver apertures used in the experiment. Also shown is the best fit (solid curve), in a least-squares sense, of the probability density function to the measured data. The best-fit values of the number of speckles integrated, M, are also given along with the aperture diameter and the measured signal-to-noise ratio, S/N. In most cases the best-fit curve is hidden by the data.

Fig. 8
Fig. 8

The same as Fig. 7 except for larger receiver apertures. As the aperture size, and therefore M, increases, the distribution approaches a Gaussian.

Fig. 9
Fig. 9

Fitted values of M versus the ratio of the aperture area to the estimated correlation area (Sc ≈ 70 mm2, which we estimated by fitting Eq. (C13) to the measured values of M versus Sm/Sc). Also shown are the plots of M predicted by Eq. (C13) for a Gaussian target illumination with a circular receiver aperture and a Gaussian target illumination with square receiver aperture. SM is the area of the receiver aperture, and Sc is the average area of a speckle.

Fig. 10
Fig. 10

S/N ratio for single-wavelength speckle patterns versus M (obtained from numerical curve fits of Eq. (28) to measured probability density, shown in Figs. 7 and 8. A least-squares fit of Eq. (20) was made to our predicted values of M. The best fit returned a value of ℘ ≈ 0.70 for the degree of polarization as compared with our estimate of ℘ ≈ 0.70 from previous depolarization measurements.

Fig. 11
Fig. 11

Spatial profile of the laser beam from a 1-cm-diameter pole located at the target location. Modulations in the profile were caused by diffraction of the beam after clipping in the beam expander. The solid line is a fit of a Gaussian TEM00 mode to the beam profile. The spot size obtained from fitting a Gaussian TEM00 mode to the measured beam profile was w(115 m) = 6.4 cm. Using this spot size in Eq. (A12) gives a correlation area of Sc = 115 mm2. The best fit of Eq. (C13) to the data shown in Fig. 9 was found for a correlation area of Sc = 70 mm2.

Equations (78)

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

S/N=NpulseNspec,
Irt=IotRet.
Ret=RtgttRsys.
ρtRetRet=RtgttRtgtt.
PIrIr=IoRe<IrpIo, RedIodRe
=IoRe<IrpIoIopReRedIodRe,
pReRe=1Repρρ,
R¯eReIrtIot.
PIrIr=01/R¯edρpρρ0Iret/R¯eρdIopIoIo.
pIrIr=ddIrPIrIr
=1R¯e01/R¯e1ρdρpρρpIoIrR¯eρ.
Irx=I¯r21+,  Iry=I¯r21-,
J=uxux*uyux*uxuy*uyuy*,
PJP=λ100λ2.
λ1-λ2λ1+λ2.
pIsIs, Ir=1Irexp-IsIr,
pIsIs1ΓMMIrMIsM-1 exp-MIsIr  if Ir>0.0  otherwise
PIs|IrxIs|Irx=MxIrxMxIsMx-1ΓMxexp-MxIsIrx,  PIs|IryIs|Iry=MyIryMyIsMy-1ΓMyexp-MyIsIry,
S/Nrms=21+2Mx+1-2My-1/2.
S/Nrms=2M1+1/2
S/Nrms2M,
pIsIrIs, Ir=pIrIrpIs|IrIr|Ir.
pIsiIsi=-+dIirpIsiIriIsi,Iri,=-+dIirpIriIripIsi|IriIsi|Iri.
pIsxIsx=MxIsxMx-1ΓMx-+dIrxpIrxIrx×1Irxexp-MxIsxIrx,  pIsyIsy=MyIsyMy-1ΓMy-+dIrypIryIry×1Iryexp-MyIsyIry,
Is=Isx+Isy.
pIsIs=pIsxIsx*pIsyIsy=-+dIsxpIsxIsxpIsyIs-Isx.
Id=Is-Ib.
pIdId=-+dIpIsIpIbId+I.
χ2=j=1KpmodelIdataj-pdataIdatajσNj2.
RIx1, y1; x2, y2=Ix1, y1I*x2, y2,
JAx1, y1; x2, y2=Ax1, y1A*x2, y2.
Ax, y=1λzexp-iπλzx2+y2×-dξdηαξ, ηexp×-iπλzξ2+η2expi2πλzxξ+yη,
μAx1, y1; x2, y2JAx1, y1; x2, y2JAx1, y1; x1, y1JAx2, y2; x2, y21/2.
Jαξ1, η1, ξ2, η2Pξ1, η1P*ξ2, η2δξ1-ξ2, η1-η2,
μAΔx, Δy=-Pξ, η2dξdη-1-Pξ, η2×expi2πξΔx+ηΔyλzdξdη,
RIΔx, Δy=I21+μAΔx, Δy2.
Sc=-μAΔx, Δy2dΔxdΔy.
Pξ, η2=IOT exp-2ξ2+η2wT2,
μAΔx, Δy=exp-π2wT22λ2z2Δx2+Δy2,
RIΔx, Δy=I21+μAΔx, Δy2,  =I21+exp-π2wT2λ2z2Δx2+Δy2.
Sc=πρc2=λ2z2/πwT2,
ρc=λz/πwT=λ/πθdiv.
wz=w01+z/zR21/2,
zRπw02/λ,
ρc=w01+zR/z2-1/2
μAΔx, Δy=λzπaΔx2+Δx21/2×J12πaλzΔx2+Δx21/2,
Sc=2λ2z2πa20dyJ1y2y=λ2z2πa2.
rc=λz/πaλ/πθdiv.
μAΔx, Δy=sincπΔxλzWxsincπΔyλzWy,
Sc=-Wx/2+Wx/2dx sinc2πΔxλzWx×-Wy/2+Wy/2dy sinc2πΔyλzWy,
Sc=λ2z2/π2WxWy.
Ax, y=1Sn=0bnϕnx, y,
I0=1SAx, yA*x, ydxdy.
Sdx2dy2JAx1, y1; x2, y2  ϕnx2, y2=λnϕnx1, y1,
I0=1S2n=0bn2.
Mxiν=-+pxexp+iνxdx,
px=12π-+Mxiνexp-iνxdν.
MI0iν=n=011-iλn,
pI0I0=n=0m=0mnexp-I0/λnλn-λm.
JAΔx, Δy=κITwT2πλ2z2exp-k2wT2Δx2+Δy28z2.
Δx=x1-x2=ρ1 cos θ1-ρ2 cos θ2,  Δy=y1-y2=ρ1 sin θ1-ρ2 sin θ2,
JAρ1, ρ2, θ1, θ2JOA exp-kOAρ12+ρ22-2ρ1ρ2 cosθ1-θ2.
πa2JOA02πdθ20aρ2dρ2 3xp2kOAρ1ρ2 cosθ1-θ2ϕnρ2, θ2=λnρ1, θ1ϕnρ1, θ1,
Imeas=1SM-+dxdyΥx, yIsx, y,
SM=-+dxdyΥx, y
Imeas=1SM-+dxdyΥx, yIs=Is,
Imeas2=1SM2-+-+dx1dy1dx2dy2Υx1, y1×Υx2, y2Isx1, y1Isx2, y2=1SM2-+RSΔx, ΔyRIsΔx, ΔydΔxdΔy,
σ2=Imeas2-Imeas2,
S/NImeasσ=1SM2-+RSΔx, ΔyμAΔx, Δy2dΔxdΔy-1/2,
S/N=M,  M1SM2-+RSΔx, ΔyμAΔx, Δy2dΔxdΔy-1.
RSΔx, Δy=Lx-ΔxLy-Δy0for ΔxLx and ΔyLyotherwise.
RSΔx, Δy=½DA2cos-1Δr-Δr1-Δr20for Δr1otherwise,
Δr1DAΔx2+Δy2.
Mrectrect=1LxLy2-Lx+LxdΔx sincΔxLcxLx-Δx-Ly+LydΔy sincΔyLcyLy-Δy-1,
Lcxλz/πwx,  Lcyλz/πwy,
McircGauss=π1601dyy cos-1y-y21-y2exp×-4SMScy2-1,
Sc=πρC2=λ2z2/πwT2
MrectGaus=ScSMerfπSMSc-ScM1-exp-MSc-2,

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