Abstract

Speckle-field long- and short-exposure spatial correlation characteristics for target-in-the-loop (TIL) laser beam propagation and scattering in atmospheric turbulence are analyzed through the use of two different approaches: the conventional Monte Carlo (MC) technique and the recently developed brightness function (BF) method. Both the MC and the BF methods are applied to analysis of speckle-field characteristics averaged over target surface roughness realizations under conditions of “frozen” turbulence. This corresponds to TIL applications where speckle-field fluctuations associated with target surface roughness realization updates occur within a time scale that can be significantly shorter than the characteristic atmospheric turbulence time. Computational efficiency and accuracy of both methods are compared on the basis of a known analytical solution for the long-exposure mutual correlation function. It is shown that in the TIL propagation scenarios considered the BF method provides improved accuracy and requires significantly less computational time than the conventional MC technique. For TIL geometry with a Gaussian outgoing beam and Lambertian target surface, both analytical and numerical estimations for the speckle-field long-exposure correlation length are obtained. Short-exposure speckle-field correlation characteristics corresponding to propagation in “frozen” turbulence are estimated using the BF method. It is shown that atmospheric turbulence–induced static refractive index inhomogeneities do not significantly affect the characteristic correlation length of the speckle field, whereas long-exposure spatial correlation characteristics are strongly dependent on turbulence strength.

© 2006 Optical Society of America

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  1. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  2. M. S. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, 1989).
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    [CrossRef]
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    [CrossRef]
  7. V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).
  8. O. Korotkova, L. C. Andrews, and R. L. Phillips, "Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source," in Proc. SPIE 4821, 98-109 (2002).
    [CrossRef]
  9. M. A. Vorontsov and G. W. Carhart, "Adaptive phase distortion correction in strong speckle-modulation conditions," Opt. Lett. 27, 2155-2157 (2002).
    [CrossRef]
  10. S. A. Kokorowski, M. E. Pedinoff, and J. E. Pearson, "Analytical, experimental, and computer-simulation results on the interactive effects of speckle with multidither adaptive optics systems," J. Opt. Soc. Am. 67, 333-345 (1977).
    [CrossRef]
  11. M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).
  12. M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
    [CrossRef]
  13. V. I. Polejaev and M. A. Vorontsov, "Adaptive active imaging system based on radiation focusing for extended targets," in Proc. SPIE 3126, 216-220 (1997).
    [CrossRef]
  14. There are several other effects that can contribute to return-wave coherence: molecular scattering, Doppler spectra broadening, dispersion, nonlinear effects, etc.
  15. N.Ageorges and C.Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, 2000).
  16. Numerical solution (integration) of the propagation equations is typically performed using the well-known split-operator method (fast Fourier transformation-based computations of wave propagation with turbulence representation as a set of phase screens).
  17. M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing," J. Opt. Soc. Am. A 22, 126-141 (2005).
    [CrossRef]
  18. F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).
  19. V. V. Tamoikin and A. A. Fraiman, "Statistical properties of field scattered by rough surface," Radiophys. Quantum Electron. 11, 56-74 (1966).
  20. V. V. Vorob'ev, "Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index," Radiophys. Quantum Electron. 13, 1053-1060 (1970).
  21. V. V. Kolosov and A. V. Kuzikovskii, "On phase compensation for refractive distortions of partially coherent beams," Sov. J. Quantum Electron. 8, 490-494 (1981).
  22. P. DuChateau and D. Zachmann, Applied Partial Differential Equations (Dover, 2002).
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    [PubMed]
  24. J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beam through the atmosphere," Appl. Phys. 11, 329-335 (1977).
  25. S. M. Flatte, G. Y. Wang, and J. Martin, "Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment," J. Opt. Soc. Am. A 10, 2363-2370 (1993).
    [CrossRef]
  26. S. M. Flatte, "Calculations of wave propagation through statistical random media, with and without a waveguide," Opt. Express 10, 777-804 (2002).
    [PubMed]
  27. A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers," in Turbulence, Classic Papers on Statistical Theory, S.K.Friedlander and L.Topper, eds. (Wiley-Interscience, 1961), pp. 151-155.
  28. D. L. Fried, "Statistics of a geometric representation of wavefront distortion," J. Opt. Soc. Am. 55, 1427-1435 (1965).
    [CrossRef]
  29. J. C. Ricklin, W. B. Miller, and L. C. Andrews, "Effective beam parameters and the turbulent beam waist for convergent Gaussian beams," Appl. Opt. 34, 7059-7065 (1995).
    [CrossRef] [PubMed]

2005 (1)

2002 (3)

1997 (1)

V. I. Polejaev and M. A. Vorontsov, "Adaptive active imaging system based on radiation focusing for extended targets," in Proc. SPIE 3126, 216-220 (1997).
[CrossRef]

1995 (1)

1993 (1)

1984 (1)

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

1981 (1)

V. V. Kolosov and A. V. Kuzikovskii, "On phase compensation for refractive distortions of partially coherent beams," Sov. J. Quantum Electron. 8, 490-494 (1981).

1980 (1)

1978 (1)

1977 (2)

1976 (1)

1970 (1)

V. V. Vorob'ev, "Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index," Radiophys. Quantum Electron. 13, 1053-1060 (1970).

1966 (1)

V. V. Tamoikin and A. A. Fraiman, "Statistical properties of field scattered by rough surface," Radiophys. Quantum Electron. 11, 56-74 (1966).

1965 (1)

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source," in Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

J. C. Ricklin, W. B. Miller, and L. C. Andrews, "Effective beam parameters and the turbulent beam waist for convergent Gaussian beams," Appl. Opt. 34, 7059-7065 (1995).
[CrossRef] [PubMed]

Banakh, V. A.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).

Bass, F. G.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

Carhart, G. W.

DuChateau, P.

P. DuChateau and D. Zachmann, Applied Partial Differential Equations (Dover, 2002).

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beam through the atmosphere," Appl. Phys. 11, 329-335 (1977).

Flatte, S. M.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beam through the atmosphere," Appl. Phys. 11, 329-335 (1977).

Fossey, M. E.

Fraiman, A. A.

V. V. Tamoikin and A. A. Fraiman, "Statistical properties of field scattered by rough surface," Radiophys. Quantum Electron. 11, 56-74 (1966).

Fried, D. L.

Fuks, I. M.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

Holmes, J. F.

Karnaukhov, V. N.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Kerr, J. R.

Kokorowski, S. A.

Kolmogorov, A. N.

A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers," in Turbulence, Classic Papers on Statistical Theory, S.K.Friedlander and L.Topper, eds. (Wiley-Interscience, 1961), pp. 151-155.

Kolosov, V. V.

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing," J. Opt. Soc. Am. A 22, 126-141 (2005).
[CrossRef]

V. V. Kolosov and A. V. Kuzikovskii, "On phase compensation for refractive distortions of partially coherent beams," Sov. J. Quantum Electron. 8, 490-494 (1981).

Korotkova, O.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source," in Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

Kravtsov, Yu. A.

M. S. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, 1989).

Kuzikovskii, A. V.

V. V. Kolosov and A. V. Kuzikovskii, "On phase compensation for refractive distortions of partially coherent beams," Sov. J. Quantum Electron. 8, 490-494 (1981).

Kuz'minskii, A. L.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Lee, M. H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martin, J.

Miller, W. B.

Mironov, V. L.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beam through the atmosphere," Appl. Phys. 11, 329-335 (1977).

Pearson, J. E.

Pedinoff, M. E.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source," in Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

Pincus, P. A.

Polejaev, V. I.

V. I. Polejaev and M. A. Vorontsov, "Adaptive active imaging system based on radiation focusing for extended targets," in Proc. SPIE 3126, 216-220 (1997).
[CrossRef]

Ricklin, J. C.

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Rytov, M. S.

M. S. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, 1989).

Shmalhauzen, V. I.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Tamoikin, V. V.

V. V. Tamoikin and A. A. Fraiman, "Statistical properties of field scattered by rough surface," Radiophys. Quantum Electron. 11, 56-74 (1966).

Tatarskii, V. I.

M. S. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, 1989).

Vorob'ev, V. V.

V. V. Vorob'ev, "Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index," Radiophys. Quantum Electron. 13, 1053-1060 (1970).

Vorontsov, M. A.

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing," J. Opt. Soc. Am. A 22, 126-141 (2005).
[CrossRef]

M. A. Vorontsov and G. W. Carhart, "Adaptive phase distortion correction in strong speckle-modulation conditions," Opt. Lett. 27, 2155-2157 (2002).
[CrossRef]

V. I. Polejaev and M. A. Vorontsov, "Adaptive active imaging system based on radiation focusing for extended targets," in Proc. SPIE 3126, 216-220 (1997).
[CrossRef]

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Wang, G. Y.

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Zachmann, D.

P. DuChateau and D. Zachmann, Applied Partial Differential Equations (Dover, 2002).

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beam through the atmosphere," Appl. Phys. 11, 329-335 (1977).

J. Opt. Soc. Am. (5)

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

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source," in Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

V. I. Polejaev and M. A. Vorontsov, "Adaptive active imaging system based on radiation focusing for extended targets," in Proc. SPIE 3126, 216-220 (1997).
[CrossRef]

Radiophys. Quantum Electron. (2)

V. V. Tamoikin and A. A. Fraiman, "Statistical properties of field scattered by rough surface," Radiophys. Quantum Electron. 11, 56-74 (1966).

V. V. Vorob'ev, "Narrowing of light beam in nonlinear medium with random inhomogeneities of the refraction index," Radiophys. Quantum Electron. 13, 1053-1060 (1970).

Sov. J. Quantum Electron. (2)

V. V. Kolosov and A. V. Kuzikovskii, "On phase compensation for refractive distortions of partially coherent beams," Sov. J. Quantum Electron. 8, 490-494 (1981).

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle-effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Other (12)

P. DuChateau and D. Zachmann, Applied Partial Differential Equations (Dover, 2002).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers," in Turbulence, Classic Papers on Statistical Theory, S.K.Friedlander and L.Topper, eds. (Wiley-Interscience, 1961), pp. 151-155.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, 1987).

There are several other effects that can contribute to return-wave coherence: molecular scattering, Doppler spectra broadening, dispersion, nonlinear effects, etc.

N.Ageorges and C.Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, 2000).

Numerical solution (integration) of the propagation equations is typically performed using the well-known split-operator method (fast Fourier transformation-based computations of wave propagation with turbulence representation as a set of phase screens).

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

M. S. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, 1989).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

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Figures (11)

Fig. 1
Fig. 1

Target-in-the-loop wave propagation configurations. Transmitted (outgoing) wave with complex amplitude A propagates in an optically inhomogeneous medium along the optical axis (z direction) toward a target and, after scattering off the target surface at the plane z = L , the return wave ψ propagates back to the receiver. The propagation directions of the outgoing A and return ψ waves are shown by dashed lines with arrows.

Fig. 2
Fig. 2

Schematic representation of TIL propagation for a layered phase-distorting medium model (three layers are shown). Large arrows are associated with TIL wave propagation as described by Eqs. (1, 2) in the MC technique. Small (dashed) arrows are associated with brightness function trajectories (trajectories L and L 1 ) as described by ray equations (16). Optical wave propagation through a phase screen results in a change in the outgoing wave’s wavefront phase. In the case of the brightness method the phase-distorting layer changes the brightness function trajectory angle by the factor Δ θ j , j = 1 , 2 , 3 .

Fig. 3
Fig. 3

Numerical simulation results obtained using the MC technique for TIL propagation in a vacuum (upper row) and in a frozen layered phase-distorting medium with D r 0 = 8 (bottom row). Gray-scale images correspond to (a), (e) target and (b), (f) receiver plane intensity distributions, (c), (g) speckle-average intensity ψ ( r , z = 0 , t ) 2 s , and (d), (h) the instantaneous MCF modulus Γ ψ ( ρ , R = 0 , z = 0 , t ) . The picture size corresponds to 4.0 a 0 for (a) and (e), and 8.0 a 0 otherwise.

Fig. 4
Fig. 4

Computer simulation of (a)–(c) speckle-field instantaneous brightness function B ( θ , R = 0 , 0 ) and (d)–(f) the MCF Γ ψ ( ρ , R = 0 , 0 ) for propagation of a Gaussian beam of size b s in a layered phase-distorting medium after scattering off the Lambertian surface (one-way propagation from target to receiver). Functions B ( θ x , θ y ) = B ( θ , R = 0 , 0 ) and Γ ψ ( ρ x , ρ y ) = Γ ψ ( ρ , R = 0 , 0 ) are calculated using the brightness function method (a), (d) for an optically homogeneous medium; (b), (e) for one; and (c), (f) for a different set of phase screens. The propagation distance is L = 10 k b s 2 and a sp r 0 = 2.8 , where a sp = 2 L ( k b s ) is the characteristic correlation distance for a speckle field originated from a Gaussian beam scattered off a Lambertian surface in a vacuum. The gray-scale image size corresponds to the angular range 14.0 ( b s L ) , in (a)–(c) and the coordinate range 130.0 b s in (d)–(f).

Fig. 5
Fig. 5

TIL propagation of the outgoing Gaussian beam with scattering off a Lambertian surface. The images in the columns are calculated using the brightness function method for propagation in a homogeneous medium (left column), and for propagation in a layered phase-distorting medium (central and right columns) with D r 0 = 2.8 . Images in rows correspond to target-plane intensity distribution (upper row), instantaneous brightness function B ( θ , R = 0 , 0 , t ) (central row), and the MCF modulus Γ ψ ( ρ , R = 0 , 0 , t ) (lower row). The propagation conditions and image scales (in the central and right columns) are the same as in Fig. 4. Image (a) corresponds to the diffraction-limited beam of size b s = L ( k a 0 ) = 0.1 a 0 . The image scales in (a)–(c) are identical.

Fig. 6
Fig. 6

Long-exposure normalized MCF modulus γ ψ LE for one-way propagation computed using MC (solid curves) and brightness function BF (dotted curve) methods. The dashed curve corresponds to the analytical solution (21). Atmospheric averaging is performed using a set of instantaneous MCFs corresponding to M at different distorting medium realizations. For the MC technique, M updates of surface roughness and phase-distorting medium realizations are performed simultaneously. The transversal coordinate ρ is normalized by the characteristic speckle size a sp = 2 L ( k b s ) in a vacuum. Calculations are performed on a 256 × 256 numerical grid for a sp r 0 = 0.7 ( r 0 is the Fried parameter) and propagation distance L = 2.5 k b s 2 ( b s is the radius of the Gaussian beam scattered off the Lambertian surface).

Fig. 7
Fig. 7

Numerical simulation errors ϵ MC ( ρ , M = 400 ) and ϵ BF ( ρ , M at = 100 ) [see Eqs. (22)] for computation of the normalized long-exposure MCF modulus γ ψ LE ( ρ ) using the brightness function (solid curves) and MC (dashed curves) methods as functions of the parameter a sp r 0 characterizing the atmospheric (Kolmogorov) turbulence strength for two characteristic points of ρ 1 [ ϵ MC ( 1 ) and ϵ BF ( 1 ) ] and ρ 2 [ ϵ MC ( 2 ) and ϵ BF ( 2 ) ] of function γ ψ LE ( ρ ) . The propagation distances are (a) L = 0.4 k b s 2 and (b) L = 2.5 k b s 2 .

Fig. 8
Fig. 8

Characteristic long-exposure spatial correlation length a sp LE for a speckle field propagating in an inhomogeneous medium as described by the Kolmogorov turbulence model. The speckle field originates from scattering off a Lambertian surface: (a) a target-plane Gaussian beam with intensity distribution of width b s (single-pass propagation) and (b) an outgoing Gaussian beam of width a 0 (double-pass propagation). The correlation length a sp LE is normalized by a sp = 2 L ( k b s ) in (a) and the transmitter aperture diameter D = 2 a 0 in (b). Solid curves correspond to a sp LE computed using Eq. (21) [in (a)] and Eq. (26) [in (b)]. The parameter α = b s b s dif in (b) characterizes beam size on the target for propagation in a vacuum. The diamonds correspond to numerical simulation using the brightness function method for α = 2 . The dashed curves are calculated using approximations (28) in (a) and (27) in (b) for α = 1 .

Fig. 9
Fig. 9

Speckle-field correlation characteristics obtained for outgoing Gaussian beam TIL propagation over the distance L = 10 k b s 2 with scattering off a Lambertian target surface using the brightness function method: (a)–(c) instantaneous normalized speckle-field MCF modulus γ ψ ( ρ , R = 0 , z = 0 , t ) for different phase-distorting medium realizations. (d) The speckle-field MCF modulus γ ψ ( ρ , R = 0 , z = 0 ) corresponds to propagation in a vacuum. The long- and short-exposure normalized speckle-field MCF modulus γ ψ LE ( ρ , R = 0 , z = 0 ) and γ ψ SE ( ρ , R = 0 , z = 0 ) in (e) and (f) are obtained by averaging 100 instantaneous functions γ ψ ( ρ , R = 0 , z = 0 , t ) computed for different refractive index fluctuation realizations. The length of the horizontal bars in (d)–(f) correspond to the correlation distances a sp (speckle size in vacuum), a sp LE , and a sp SE . Refractive index inhomogeneities correspond to the Kolmogorov turbulence model with a sp r 0 = 8 .

Fig. 10
Fig. 10

Comparison of the short- and long-exposure correlation coefficients γ ψ SE ( ρ ) (solid curves) and γ ψ LE ( ρ ) (dashed curves) for different turbulence strengths as defined by the ratio a sp r 0 (one-way propagation from the target to receiver plane). Ratio a sp r 0 = 0 corresponds to propagation in a vacuum. The characteristic short- and long-exposure correlation distances a sp SE and a sp LE are defined as the e 1 falloff in the functions γ ψ SE ( ρ ) and γ ψ LE ( ρ ) . Calculations were performed for a Gaussian beam scattered off a Lambertian surface using the brightness function method (solid curves) and Eq. (21) (dashed curves). The beam width b s and propagation distance L = 10 k b s 2 are fixed.

Fig. 11
Fig. 11

Short-exposure correlation length a sp SE versus the target-plane beam radius b s for (a) propagation of a speckle field originated from a Gaussian beam scattered off a Lambertian surface (one-way propagation) and (b) TIL propagation of a Gaussian beam in atmospheric turbulence conditions described by the ratio D r 0 . The propagation distance is L = 0.1 k a 0 2 . The end points of the curve corresponding to D r 0 = 5.6 in (b) are marked by symbols A ( F = L ) and B ( F = 100 L ) .

Equations (32)

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2 i k A ( r , z , t ) z = 2 A ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) A ( r , z , t ) ,
2 i k ψ ( r , z , t ) z = 2 ψ ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) ψ ( r , z , t ) ,
A ( r , z = 0 , t ) = A 0 ( r ) exp [ i u ( r ) ] ,
A ( r , z = 0 , t ) = exp [ r 2 ( 2 a 0 2 ) i k r 2 ( 2 F ) ] ,
ψ ( r , z = L , t ) = T ( r , t ) A ( r , z = L , t ) ,
Γ ψ ( ρ , R , z , t ) = ψ ( R + ρ 2 , z , t ) ψ * ( R ρ 2 , z , t ) s .
Γ ψ LE ( ρ , R , z ) Γ ψ ( ρ , R , z , t ) at = ψ ( R + ρ 2 , z , t ) ψ * ( R ρ 2 , z , t ) s , at .
γ ψ LE ( ρ , R , z ) Γ ψ LE ( ρ , R , z ) Γ ψ LE ( 0 , R , z ) ,
γ ψ ( ρ , R , z = 0 , t ) Γ ψ ( ρ , R , z = 0 , t ) Γ ψ ( 0 , R , z = 0 , t ) .
γ ψ SE ( ρ , R , z = 0 ) γ ψ ( ρ , R , z = 0 , t ) at = Γ ψ ( ρ , R , 0 , t ) Γ ψ ( 0 , R , 0 , t ) at .
γ ψ SE ( ρ = a sp SE ) = e 1 .
B ( θ , R , z , t ) = 1 ( 2 π ) 2 Γ ψ ( ρ , R , z , t ) exp ( i k θ ρ ) d 2 ρ ,
n 1 ( R + ρ 2 , z , t ) n 1 ( R ρ 2 , z , t ) ρ R n 1 ( R , z , t )
B ( θ , R , z , t ) z + θ R B ( θ , R , z , t ) + R n 1 ( R , z , t ) θ B ( θ , R , z , t ) = 0 .
B ( θ , R , z = L , t ) = c V ( R ) I ( R , z = L , t ) ,
d R ( z , t ) d z = θ ( z , t ) , d θ ( z , t ) d z = R n 1 ( R , z , t ) .
Γ ψ ( ρ , R , z = 0 , t ) = k 2 B ( θ , R , z = 0 , t ) exp ( i k θ ρ ) d 2 θ .
R ( z ) = R j + θ j z and θ ( z ) = θ j = θ j 1 + Δ θ j ( R j , t ) = θ j 1 + R φ j ( R j , t ) k ,
I ( r , z = L , t ) = I 0 exp ( r 2 b s 2 ) ,
γ ψ LE ( ρ ) = exp { ( k b s ρ ) 2 ( 2 L ) 2 0.55 C n 2 k 2 L ρ 5 3 } .
γ ψ LE ( ρ ) = exp { ( ρ a sp ) 2 1.3 ( ρ r 0 ) 5 3 } .
ϵ MC ( ρ , M ) = [ γ MC LE ( ρ , M ) γ ψ LE ( ρ ) ] γ ψ LE ( ρ ) ,
ϵ BF ( ρ , M at ) = [ γ BF LE ( ρ , M at ) γ ψ LE ( ρ ) ] γ ψ LE ( ρ ) .
b s LE b s at = a 0 [ p F 2 + ( 1 + 5.52 a 0 2 r 0 2 ) ( b s dif a 0 ) 2 ] 1 2 ,
p F 2 = [ b s 2 ( b s dif ) 2 ] a 0 2 .
b s LE = b s dif [ α 2 ( F ) + 5.52 a 0 2 r 0 2 ] 1 2 ,
γ ψ LE ( ρ ) = exp [ α 2 ( F ) ( ρ 2 a 0 ) 2 ] exp [ 1.38 ( ρ r 0 ) 2 ] exp [ 1.3 ( ρ r 0 ) 5 3 ] .
( 1 a sp LE ) 2 α 2 ( F ) D 2 + 2.68 r 0 2
or ( 1 a sp LE ) 2 1 a sp 2 + 2.68 r 0 2 .
( 1 a sp LE ) 2 1 a sp 2 + 1.3 r 0 2 .
b s 2 = P 1 ( r r c ) 2 I ( r , z = L ) d 2 r ,
r c = P 1 r I ( r , z = L ) d 2 r ,

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