Abstract

We apply a target-in-the-loop strategy to the case of adaptive optics beam control in the presence of strong atmospheric turbulence for air-to-ground directed energy laser applications. Using numerical simulations we show that in the absence of a cooperative beacon to probe the atmosphere it is possible to extract information suitable for effective beam control from images of the speckled and strongly turbulence degraded intensity distribution of the laser energy at the target. We use a closed-loop, single-deformable-mirror adaptive optics system driven by a target-in-the-loop stochastic parallel gradient descent optimization algorithm minimizing a mean-radius performance metric defined on the image of the laser beam intensity distribution formed at the receiver. We show that a relatively low order 25-channel zonal adaptive optical beam control system controlled in this way is capable of achieving a high degree of turbulence compensation with respect to energy concentration if the tilt can be corrected separately.

© 2007 Optical Society of America

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References

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2007 (1)

2006 (3)

2005 (2)

2002 (2)

M. A. Vorontsov, "Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion," J. Opt. Soc. Am. A 19, 356-368 (2002).
[CrossRef]

M. C. Roggemann and W. R. Reynolds, "Block matching algorithm for mitigating aliasing effects in undersampled image sequences," Opt. Eng. 41, 359-369 (2002).
[CrossRef]

2001 (1)

2000 (1)

1998 (3)

1997 (1)

1996 (2)

1995 (1)

1994 (1)

1992 (1)

J. C. Spall, "Multivariate stochastic optimization using a simultaneous perturbation gradient approximation," IEEE Trans. Autom. Control 37, 332-341 (1992).
[CrossRef]

1990 (1)

1982 (1)

1978 (1)

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651-697 (1978).
[CrossRef]

1974 (1)

Appl. Opt. (5)

IEEE Trans. Autom. Control (1)

J. C. Spall, "Multivariate stochastic optimization using a simultaneous perturbation gradient approximation," IEEE Trans. Autom. Control 37, 332-341 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field," J. Opt. Soc. Am. A 13, 1456-1466 (1996).
[CrossRef]

V. P. Sivokon and M. A. Vorontsov, "High-resolution adaptive phase distortion suppression based solely on intensity information," J. Opt. Soc. Am. A 15, 234-247 (1998).
[CrossRef]

M. A. Vorontsov and V. P. Sivokon, "Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction," J. Opt. Soc. Am. A 15, 2745-2758 (1998).
[CrossRef]

M. A. Vorontsov, "Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion," J. Opt. Soc. Am. A 19, 356-368 (2002).
[CrossRef]

M. A. Vorontsov and 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, G. W. Carhart, M. Cohen, and G. Cauwenberghs, "Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration," J. Opt. Soc. Am. A 17, 1440-1453 (2000).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field," J. Opt. Soc. Am. A 13, 1456-1466 (1996).
[CrossRef]

J. M. Martin and S. M. Flatte, "Simulation of point-source scintillation through three-dimensional random media," J. Opt. Soc. Am. A 7, 838-847 (1990).
[CrossRef]

V. V. Dudorov, M. A. Vorontsov, and V. V. Kolosov, "Speckle-field propagation in 'frozen' turbulence: brightness function approach," J. Opt. Soc. Am. A 23, 1924-1936 (2006).
[CrossRef]

R. R. Parenti and R. J. Sasiela, "Laser-guide-star systems for astronomical applications," J. Opt. Soc. Am. A 11, 288-309 (1994).
[CrossRef]

Opt. Eng. (1)

M. C. Roggemann and W. R. Reynolds, "Block matching algorithm for mitigating aliasing effects in undersampled image sequences," Opt. Eng. 41, 359-369 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. IEEE (1)

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651-697 (1978).
[CrossRef]

Other (9)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

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

R. R. Beland, "Propagation through atmospheric optical turbulence," in IR/EO Handbook, F. G. Smith, ed. (SPIE, 1993), Vol. 2, pp. 157-232.

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

M. Le Louarn and M. Tallon, "3D mapping of turbulence: theory," in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, 1066-1073 (2000).

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

L. Hasdorff, Gradient Optimization and Nonlinear Control (Wiley, 1976).

T. J. Brennan and P. H. Roberts, AOTOOLs: the adaptive optics toolbox. (Optical Sciences Company, 1999).

G. Cauwenberghs, "A fast stochastic error-descent algorithm for supervised learning and optimization," in Advances in Neural Information Processing Systems (Morgan Kaufman, 1993), Vol. 5, pp. 244-251.

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Figures (8)

Fig. 1
Fig. 1

Optical scheme of a target-in-the-loop laser AO system.

Fig. 2
Fig. 2

(Color online) Air-to-ground laser beam propagation through 5 km Hufnagel–Valley turbulence: (a) long exposure, ζ = 45 ° ; (b) short exposure, ζ = 45 ° ; (c) long exposure, near horizontal path; and (d) short exposure, near horizontal path. For reference, the diffraction-limited intensity distribution is also shown.

Fig. 3
Fig. 3

(Color online) Short exposure averaged intensity distributions on the target obtained with ideal phase conjugation for (a) ζ = 45 ° and (b) near-horizontal propagation paths. For reference, the diffraction-limited intensity distribution is also shown.

Fig. 4
Fig. 4

(Color online) Actuator geometries for (a) 109-, (b) 25-, (c) 16-, and (d) 9-actuator DMs filling a 50   cm system aperture.

Fig. 5
Fig. 5

(Color online) Spot-on-target SPGD adaptation result for 109-actuator DM and 500 two-sided SPGD iterations. Examples of (a) initial and (b) final intensity distributions on the target for a single atmospheric turbulence realization. (c) Final target intensity cross sections through the spot's center of gravity versus uncompensated and diffraction-limited ones short exposure averaged over 100 atmospheric turbulence realizations. (d) Examples of the SPGD optimization convergence histories for J = 0.02 and different values of descent parameter γ 0 .

Fig. 6
Fig. 6

(Color online) Exemplary spot-on-target SPGD adaptation results after 100 double iterations for 9-, 16-, and 25-actuator DMs and a single atmospheric turbulence realization. (a) Final target intensity cross sections through the spot's center of gravity versus uncompensated and diffraction-limited ones. (b) Typical SPGD optimization convergence histories for γ 0 = 100 and J = 0.02 .

Fig. 7
Fig. 7

Exemplary spot-on-receiver SPGD adaptation result after 100 two-sided SPGD iterations for 25-actuator DM and a single atmospheric turbulence realization. γ 0 = 60 , J = 0.03 , SR achieved is 0.5. (a), (b) initial and (c), (d) final intensity distributions on the target and on the receiver, respectively.

Fig. 8
Fig. 8

(Color online) Spot-on-receiver SPGD adaptation result after 100 two-sided iterations for 25-actuator DM, γ 0 = 60 , and J = 0.03 . (a) Target intensity cross section short exposure averaged over 100 atmospheric turbulence realizations versus the short exposure averaged uncompensated and diffraction-limited ones. (b) A typical SPGD optimization convergence history.

Equations (38)

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

u n + 1 = A [ u n , δ J ( δ u ) ] ,
Φ n ( κ ) = 0.033 C n 2 ( z ) | κ | 11 / 3 ,
C n i 2 = sec ζ z i z i 1 z i 1 z i C n 2 ( z ) d z ,
C n 2 ( z ) = 5.94 × 10 53 ( 21 27 ) 2 z 10 exp ( z 10 3 ) + 2.7 × 10 16 exp ( z 1.5 × 10 3 ) + 1.7 × 10 14 exp ( z 10 2 ) ,
r 0 = [ 0.423 k 2 sec ζ 0 L C n 2 ( z ) ( z / L ) 5 / 3 d z ] 3 / 5 ,
f G = [ 0.102 k 2 sec ζ 0 L C n 2 ( z ) v 5 / 3 ( z ) ( z / L ) 5 / 3 d z ] 3 / 5 ,
σ χ 2 = 0.56 k 7 / 6 ( sec ζ ) 11 / 6 0 L C n 2 ( z ) z 5 / 6 ( z / L ) 5 / 6 d z ,
θ 0 = [ 2.914 k 2 ( sec ζ ) 8 / 3 0 L C n 2 ( z ) z 5 / 3 d z ] 3 / 5 ,
U a ( x , y ) = U t * ( x , y , z = L ) ,
U a ϕ ( x , y ) = W ( x , y ) U * ( x , y , z = L ) | U ( x , y , z = L ) | ,
W ( x , y ) = { 1 ( x , y ) aperture 0 otherwise ,
U i + 1 ( x ) = 1 { [ t i ( x ) U i ( x ) ] H i ( f ) } , x = ( x , y ) ,
t i ( x ) = { t l ( x ) P i ( x ) i = 1 P i ( x ) i > 1 ,
t l ( x ) = exp ( j 2 π λ | x | 2 2 f l ) ,
H i ( f , z i + 1 z i ) = exp { j 2 π ( z i + 1 z i ) sec ζ λ ( 1 λ 2 | f | 2 ) 1 / 2 } ,
I t ( x ) = | U t ( x ) | 2 ,
Δ x i l turb i / 3 .
Δ x i 2 λ f l 3 D ,
Δ x i min ( l i turb 3 , 2 λ f l 3 D ) ,
Δ x = min i Δ x i .
N 3 λ ( Δ x ) 2 max i ( z i + 1 z i ) .
U t ( x ) = U t ( x ) T ( x ) ,
T ( x ) = γ 0 exp ( i k ξ ( x ) + i k S ) ,
τ s τ r < τ a .
P DM ( x ) = u T r ( x ) ,
r m ( x ) = B ( x x m ) = tri ( x x m d ) tri ( y y m d ) ,
tri ( x ) = { 1 | x | | x | 1 0 | x | > 1 ,
      t r ( x ) = exp { j P DM ( x ) + j 2 π λ 1 2 ( 1 f p + 1 f r ) | x | 2 } .
u t + 1 = u t γ t J u ,
J / u σ 2 δ J δ u .
u t + 1 = u t γ t σ 2 δ J t δ u t ,
J / u ( δ J + δ J ) δ u .
SR = I ( 0 ) / I d l ( 0 ) ,
EE = I ( x ) d 2 x ,
I S m n = | m + n I ( x ) m x n y | 2 d 2 x , ( m + n ) = 0 , 1 , .
MR = D | x x ¯ | I ( x ) d 2 x D I ( x ) d 2 x ,     x ¯ = D x I ( x ) d 2 x D I ( x ) d 2 x ,
γ t = γ 0 J t + C ,
γ t = γ 0 J t J t J ,

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