Abstract

We address the problem of using adaptive optics to deliver power from an airborne laser platform to a ground target through atmospheric turbulence under conditions of strong scintillation and anisoplanatism. We explore three options for creating a beacon for use in adaptive optics beam control: scattering laser energy from the target, using a single uncompensated Rayleigh beacon, and using a series of compensated Rayleigh beacons. We demonstrate that using a series of compensated Rayleigh beacons distributed along the path provides the best beam compensation.

© 2008 Optical Society of America

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References

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  1. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  2. J. W. Hardy and R. W. Hudgin, “A compensation of wavefront sensing systems.” Proc. SPIE 141, 67-72 (1972)
  3. M. C. Roggemann and W. R. Reynolds, “A block matching algorithm for mitigating aliasing effects in undersampled image sequences,” Opt. Eng. 41, 359-369 (2002).
    [CrossRef]
  4. M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
    [CrossRef]
  5. J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).
  6. R. R. Beland, “Propagation through the atmospheric turbulence,” in IR/EO Handbook, F. G. Smith, ed. (SPIE Press, 1993), Vol. 2, pp. 157-232.
  7. R. R. Parenti and R. J. Sasiella, “Laser-guide-star system for astronomical applications,” J. Opt. Soc. Am. A 11, 288-309(1994).
    [CrossRef]
  8. M. Belen'kii and K. Hughes, “Beacon anisoplanatism,” Proc. SPIE 5087, 69-82 (2003).
    [CrossRef]
  9. M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  11. M. C. Roggemann and T. J. Schulz, “Algorithm to increase the largest aberration which can be reconstructed from Hartmann sensor measurements,” Appl. Opt. 37, 4321-4329 (1998).
    [CrossRef]
  12. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowski, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089-2101 (1995).
    [CrossRef] [PubMed]
  13. T. J. Brennan and P. H. Roberts, “AOTools: the adaptive optics toolbox (for use with Matlab),” AOTOOLS software package, http://cfao.ucolick.org/software/aotools.php
  14. G. Cochran, “Phase screen generation,” Tech. Rep. TR-663(Optical Sciences, 1985).
  15. 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]
  16. V. V. Dudorov, M. A. Vorontsov, and V. V. Kolosov, “Specklefield propagation in frozen turbulence: brightness function approach,” J. Opt. Soc. Am. A 23, 1924-1936 (2006).
    [CrossRef]
  17. P. Piatrou and M. C. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46, 6831-6842 (2007).
    [CrossRef] [PubMed]
  18. D. A. Long, Raman Spectroscopy (McGraw-Hill, 1977).
  19. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

2007 (1)

2006 (1)

2005 (2)

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[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]

2003 (1)

M. Belen'kii and K. Hughes, “Beacon anisoplanatism,” Proc. SPIE 5087, 69-82 (2003).
[CrossRef]

2002 (1)

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

2000 (2)

M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).

1998 (2)

1996 (1)

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

1995 (1)

1994 (1)

1993 (1)

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

1985 (1)

G. Cochran, “Phase screen generation,” Tech. Rep. TR-663(Optical Sciences, 1985).

1977 (1)

D. A. Long, Raman Spectroscopy (McGraw-Hill, 1977).

1972 (1)

J. W. Hardy and R. W. Hudgin, “A compensation of wavefront sensing systems.” Proc. SPIE 141, 67-72 (1972)

1968 (1)

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

Beland, R. R.

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

Belen'kii, M.

M. Belen'kii and K. Hughes, “Beacon anisoplanatism,” Proc. SPIE 5087, 69-82 (2003).
[CrossRef]

Brennan, T. J.

T. J. Brennan and P. H. Roberts, “AOTools: the adaptive optics toolbox (for use with Matlab),” AOTOOLS software package, http://cfao.ucolick.org/software/aotools.php

Cochran, G.

G. Cochran, “Phase screen generation,” Tech. Rep. TR-663(Optical Sciences, 1985).

Coles, W. A.

Dudorov, V. V.

Filice, J. P.

Frehlich, R. G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).

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

Hardy, J. W.

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

J. W. Hardy and R. W. Hudgin, “A compensation of wavefront sensing systems.” Proc. SPIE 141, 67-72 (1972)

Hudgin, R. W.

J. W. Hardy and R. W. Hudgin, “A compensation of wavefront sensing systems.” Proc. SPIE 141, 67-72 (1972)

Hughes, K.

M. Belen'kii and K. Hughes, “Beacon anisoplanatism,” Proc. SPIE 5087, 69-82 (2003).
[CrossRef]

Klein, T. L.

M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
[CrossRef]

Kolosov, V.

Kolosov, V. V.

Long, D. A.

D. A. Long, Raman Spectroscopy (McGraw-Hill, 1977).

Parenti, R. R.

Piatrou, P.

Reynolds, W. R.

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

Roberts, P. H.

T. J. Brennan and P. H. Roberts, “AOTools: the adaptive optics toolbox (for use with Matlab),” AOTOOLS software package, http://cfao.ucolick.org/software/aotools.php

Roggemann, M. C.

P. Piatrou and M. C. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46, 6831-6842 (2007).
[CrossRef] [PubMed]

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[CrossRef]

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

M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
[CrossRef]

M. C. Roggemann and T. J. Schulz, “Algorithm to increase the largest aberration which can be reconstructed from Hartmann sensor measurements,” Appl. Opt. 37, 4321-4329 (1998).
[CrossRef]

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

Sasiella, R. J.

Schulz, T. J.

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[CrossRef]

M. C. Roggemann and T. J. Schulz, “Algorithm to increase the largest aberration which can be reconstructed from Hartmann sensor measurements,” Appl. Opt. 37, 4321-4329 (1998).
[CrossRef]

Sergeyev, A. V.

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[CrossRef]

Soehnel, G.

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[CrossRef]

Vorontsov, M. A.

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
[CrossRef]

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

Yadlowski, M.

Appl. Opt. (3)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

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

Proc. SPIE (4)

M. C. Roggemann, B. M. Welsh, and T. L. Klein, “Algorithm to reduce anisoplanatism effects on infrared images,” Proc. SPIE 4125, 140-149 (2000.
[CrossRef]

M. Belen'kii and K. Hughes, “Beacon anisoplanatism,” Proc. SPIE 5087, 69-82 (2003).
[CrossRef]

M. C. Roggemann, T. J. Schulz, A. V. Sergeyev, and G. Soehnel, “Beacon creation and characterization for beam control in strong turbulence,” Proc. SPIE 5895, 589506 (2005).
[CrossRef]

J. W. Hardy and R. W. Hudgin, “A compensation of wavefront sensing systems.” Proc. SPIE 141, 67-72 (1972)

Other (8)

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

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

T. J. Brennan and P. H. Roberts, “AOTools: the adaptive optics toolbox (for use with Matlab),” AOTOOLS software package, http://cfao.ucolick.org/software/aotools.php

G. Cochran, “Phase screen generation,” Tech. Rep. TR-663(Optical Sciences, 1985).

D. A. Long, Raman Spectroscopy (McGraw-Hill, 1977).

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

J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).

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

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Figures (14)

Fig. 1
Fig. 1

Novel bootstrap beacon generation technique.

Fig. 2
Fig. 2

Structure constant C n 2 as a function of altitude.

Fig. 3
Fig. 3

(a) Fried parameter r 0 and (b) isoplanatic angle θ 0 as a function of laser platform altitude.

Fig. 4
Fig. 4

Isoplanatic patch radius ρ I and mean square instantaneous spot radius in the target plane ρ s 2 as a function of laser platform altitude.

Fig. 5
Fig. 5

Rytov variance σ χ 2 as a function of laser platform altitude.

Fig. 6
Fig. 6

Isoplanatic patch radius ρ I in the target plane for different altitudes of the laser platform as a function of the slant range.

Fig. 7
Fig. 7

Fried geometry WFS/DM mutual layout. Crosses in the vertices of the square grid represent DM actuator locations, and crosses in the centers represent WFS subapertures locations.

Fig. 8
Fig. 8

(a) Uncompensated beacon created at the target plane and (b) beacon created using the target plane compensation technique. Laser platform altitude was 300 m .

Fig. 9
Fig. 9

Beacon formation process along the optical path using the bootstrap compensation technique under strong turbulence strength conditions. Laser platform altitude was 300 m . (a)–(e) Five beacons created at 2.75, 3.25, 3.75, 4.25, and 5.0 km , respectively, from the aperture. Circles drawn on the top represent isoplanatic patch radii for the current location of the beacon. f shows the final beacon generated at the target plane.

Fig. 10
Fig. 10

Number of PDEs per WFS subaperture per laser pulse for different altitudes of the laser platform as a function of the laser pulse energy. Altitude of the laser platform is 300 m .

Fig. 11
Fig. 11

Number of PDEs per WFS subaperture per laser pulse for different altitudes of the laser platform as a function of the laser pulse energy. Altitude of the laser platform is 1500 m .

Fig. 12
Fig. 12

Number of PDEs per WFS subaperture per laser pulse for different altitudes of the laser platform as a function of the laser pulse energy. Altitude of the laser platform is 500 m .

Fig. 13
Fig. 13

Standard deviation of shot noise for a Rayleigh beacon generated at a distance of 4500 m from the transmitter as a function of the laser pulse energy. Solid curve denotes the case of the laser platform located at the altitude of 300 m . Solid-dotted curve denotes the case of the laser platform located at an altitude of 1500 m . Dashed curve denotes the case of the laser platform located at an altitude of 500 m .

Fig. 14
Fig. 14

Geometry of Rayleigh beacon.

Tables (2)

Tables Icon

Table 1 Strehl Ratios for Various Beam Compensation Scenarios as a Function of the Laser Platform Altitude a

Tables Icon

Table 2 Optimal Receiver Gate Range as a Function of the Beacon Position Along the Optical Path

Equations (34)

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r 0 = [ 0.423 sec ( ϕ ) k 2 0 Z d z C n 2 ( z ) ( z / Z ) 5 / 3 ] 3 / 5 ,
θ 0 = [ 2.914 k 2 [ sec ( ϕ ) ] 8 / 3 0 Z d z C n 2 ( z ) z 5 / 3 ] 3 / 5 ,
σ χ 2 = 0.563 k 7 / 6 [ sec ( ϕ ) ] 11 / 6 0 Z d z C n 2 ( z ) z 5 / 6 ( z / Z ) 5 / 6 ,
C n 2 ( z ) = 5.94 × 10 53 ( υ / 27 ) 2 z 10 e z / 1000 + 2.7 × 10 16 e z / 1500 + A e z / 100
ρ s 2 = 4 L 2 ( k D ) 2 + 4 L 2 ( k ρ 0 ) 2 [ 1 0.62 ( ρ 0 D ) 1 / 3 ] 6 / 5 ,
ρ I = L θ 0 2 .
U t ( x , z n ) = T ( x , z n ) U i ( x , z n ) ,
U ( x , z n + 1 ) = F 1 { F [ T ( x , z n ) U ( x , z n ) ] H ( f , z n + 1 z n ) } , x = ( x , y ) ,
H ( f , z n + 1 z n ) = exp { j 2 π ( z n + 1 z n ) λ ( 1 λ 2 | f | 2 ) 1 / 2 }
T l ( x ) = exp ( j 2 π λ | x | 2 2 f )
Δ x i 2 λ f 3 D ,
Δ x min [ l 1 turb 3 , ... , l P turb 3 , 2 λ f 1 3 D 1 , ... , 2 λ f K 3 D K ] ,
N 3 λ ( Δ x ) 2 max i ( z i + 1 z i ) ,
τ s τ r τ a ,
T wfs = 1 d 2 i = 1 S rect ( x x i d ) exp { j k 2 f l | x x i | 2 } ,
s x = ( I 1 + I 2 I 3 I 4 ) ( I 1 + I 2 + I 3 + I 4 ) , s y = ( I 2 + I 3 I 1 I 4 ) ( I 1 + I 2 + I 3 + I 4 ) ,
ϕ ^ dm ( x ) = u ^ T r i ( x ) ,
r i ( x ) = tri [ x x i d ] tri [ y y i d ] ,
tri ( x ) = { 1 | x | if | x | 1 0 otherwise
u ^ = arg arg u s G u 2 ,
T i ( x ) = T wfs ( x ) r i ( x ) , i = 1 , ... , M
σ ( θ ) = σ 0 1 + cos 2 ( θ ) 2 ,
σ 0 = 4 π 2 ( n 0 1 ) 2 N 0 2 λ 4 ,
β ( h ) = β 0 ρ ( h ) ρ 0 ,
β 0 = 8 π 3 N 0 σ 0 .
η ( z , ψ ) = exp [ β 0 ρ 0 0 z d h ρ ( h ) ] .
P ( z , t ) = P 0 ( t z / c ) η ( z , ψ ) ,
P r ( t ) = 0 d z P 0 ( t 2 z / c ) η 2 ( z , ψ ) β r ( z , ψ ) ,
β r ( z , ψ ) = σ 0 N 0 A r z 2 ρ ( z cos ψ ) ρ 0 ,
E = 2 Z / c Δ z / 2 2 Z / c + Δ z / 2 d t P r ( t ) = 0 d z 2 Z / c Δ z / 2 2 Z / c + Δ z / 2 d t P 0 ( t 2 z / c ) η 2 ( z , ψ ) β r ( z , ψ )
Δ z = 2 Δ α z 2 D p [ D p 2 ( z Δ α ) 2 ] ,
Δ α = 2.44 λ r 0 .
Δ z 4.88 λ z 2 D p r 0 .
σ n = 0.74 π η ( K ¯ w ) 1 / 2 d ,

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