Abstract

A comprehensive study of the interactive effects of multidither adaptive optics (COAT) systems with the spurious signals induced by speckle modulations is presented. An analysis based on a statistical model of a COAT system is developed in order to predict convergence levels in the presence of such modulations. A computer simulation study of these effects is also presented. Good agreement is found between the data gathered in both of these studies. In order to further corroborate these results, experimental data is then presented. These data were gathered using a laboratory model of a COAT system interacting with a real target that produced speckle modulations.

© 1977 Optical Society of America

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References

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  1. James E. Pearson, S. A. Kokorowski, and M. E. Pedinoff, “Effects of speckle in adaptive optical systems,” J. Opt. Soc. Am. 66, 1261 (1976).
    [CrossRef]
  2. The acronym “COAT” stands for coherent optical adaptive techniques and is applied generally to self-adaptive phased arrays deriving active phase correction information from target return radiation.
  3. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, “Coherent Optical Adaptive Techniques,” Appl. Opt. 13, 291 (1974).
    [CrossRef] [PubMed]
  4. J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, “Coherent Optical Adaptive Techniques: Design and Performance of an 18-Element, Visible, Multidither COAT System,” Appl. Opt. 15, 611 (1976).
    [CrossRef] [PubMed]
  5. James E. Pearson, “Atmospheric Turbulence Compensation Using Coherent Optical Adaptive Techniques,” Appl. Opt. 15, 622 (1976).
    [CrossRef] [PubMed]
  6. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145 (1976).
    [CrossRef]
  7. This computation as described is very cumbersome, involving the solution of transcendental equations. Rather than calculating α from Eq. (28), what one has to do is compute and plot 〈ID2〉 and 〈IM2〉 using Eqs. (20) and (21). Then for a given value of 〈ID2〉 and 〈IM2〉, α can be read from the plot.
  8. J. C. Erdmann and R. I. Gellert, “Speckle field of curved, rotating surfaces of Gaussian roughness illuminated by a laser spot,” J. Opt. Soc. Am. 66, 1194 (1976).
    [CrossRef]
  9. Convergence time is defined as the time required for the intensity on a target glint to increase from 10% to 90% of maximum.

1976 (5)

1974 (1)

Bridges, W. B.

Brown, W. P.

Brunner, P. T.

Erdmann, J. C.

Gellert, R. I.

Goodman, J. W.

Hansen, S.

Kokorowski, S. A.

Lazzara, S. P.

Nussmeier, T. A.

O’Meara, T. R.

Pearson, J. E.

Pearson, James E.

Pedinoff, M. E.

Sanguinet, J. A.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

Other (3)

Convergence time is defined as the time required for the intensity on a target glint to increase from 10% to 90% of maximum.

The acronym “COAT” stands for coherent optical adaptive techniques and is applied generally to self-adaptive phased arrays deriving active phase correction information from target return radiation.

This computation as described is very cumbersome, involving the solution of transcendental equations. Rather than calculating α from Eq. (28), what one has to do is compute and plot 〈ID2〉 and 〈IM2〉 using Eqs. (20) and (21). Then for a given value of 〈ID2〉 and 〈IM2〉, α can be read from the plot.

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Figures (20)

FIG. 1
FIG. 1

Simple statement of the speckle problem. A, COAT laser array; B, target; C, COAT servo; D, photomultiplier receiver.

FIG. 2
FIG. 2

Phaser diagram of a partially converged, N-element COAT array.

FIG. 3
FIG. 3

Statistical expectation value of the convergence level vs α for an 18-channel multidither COAT system (dither amplitude = 20°), calculated from Eq. (11).

FIG. 4
FIG. 4

Statistical expectation value of the rms dither amplitude vs α for an 18-channel multidither COAT system (dither amplitude = 20°).

FIG. 5
FIG. 5

Statistical expectation value of the rms dither amplitude vs convergence level for an 18-channel multidither COAT system (dither amplitude = 20°).

FIG. 6
FIG. 6

Power spectrum of an arbitrary modulation function.

FIG. 7
FIG. 7

Analytically predicted convergence level for an 18-channel COAT array (dither amplitude = 20°).

FIG. 8
FIG. 8

Typical time histories from computer simulations. Signal-to-noise ratio = 1010. (a) COAT performance with no speckle, Cs = 0.0. (b) COAT performance with speckle, Cs = 0.127. (c) Speckle modulation function used to produce (b).

FIG. 9
FIG. 9

Convergence level vs Cs theory and computer simulation data for an 18-channel COAT array (20° dither amp.).

FIG. 10
FIG. 10

Summary of computer simulation results with realistic speckle modulations from a spherical target rotating at 0.4 rad/s at a range of 2 km from receiver. (a) Fourier spectrum of speckle modulations. (b) Time history of speckle modulations. (c) Time history of COAT performance.

FIG. 11
FIG. 11

Summary of computer simulation results with realistic speckle modulations from a spherical target rotating at 2.0 rad/s at a range of 2 km from receiver. (a) Fourier spectrum of speckle modulations. (b) Time history of speckle modulations. (c) Time history of COAT performance.

FIG. 12
FIG. 12

Summary of computer simulation results with realistic speckle modulations from a spherical target rotating at 10.0 rad/s at a range of 2 km from receiver. (a) Fourier spectrum of speckle modulations. (b) Time history of speckle modulations. (c) Time history of COAT performance.

FIG. 13
FIG. 13

Target speckle measurement apparatus. A, mirror; B, truncating iris; C, spatial filter and recollimating telescope; D, helium-neon laser 0.6328 μm; E, beam reduction optics; F, synchronous 100 Hz chopper; G, mirrors; H, 0.15-mm-diam iris; I, rotatable target; J, lock-in detector; K, strip chart recorder; L, monitor scope; M, spectrum analyzer.

FIG. 14
FIG. 14

Speckle patterns generated by rotation of a metalized sphere. (a) 0.15 mm receiver diameter. (b) 0.05 mm receiver diameter.

FIG. 15
FIG. 15

Speckle frequency spectra for stationary and rotating metalized sphere. (a) 0 rps; (b) 0.67 rps. Circles indicate digital FFT data.

FIG. 16
FIG. 16

Schematic of experimental apparatus used to study speckle effects on a multidither COAT system. Laser: 0.488 μm argon. Phaser matrix: optical elements where dither and correction phase changes are impressed on beam. L1, L2: recollimating telescope; BS: Beam splitter. D: pinhole detector. PMT: photomultiplier tube.

FIG. 17
FIG. 17

Experimental convergence data. Target: metal sphere with rotation axis perpendicular to transmitted beam; λ = 0.488 μm; upper pictures are outputs of spectrum analyzer looking at COAT receiver signal. The lower pictures show the peak target irradiance seen by pinhole detector as servo loop is closed. (a) Target rotation rate, ΩT = 0.0. Average power after convergence, P = 1.0. (b) ΩT = π rad/s, P = 0.70. (c) ΩT = 2π rad/s, P = 0.31.

FIG. 18
FIG. 18

Effects of orientation and loop gain in speckle measurements. Target: metal sphere; λ = 0.488 μm; rotation rate, ΩT = 2π rad/s; the upper pictures are outputs of spectrum analyzer looking at COAT receiver signal. Lower pictures show peak target irradiance seen by pinhole detector as servo loop is closed. (a) Rotation axis parallel to incident beam (θ = 0°); static optimization; average power after convergence, P = 0.92. (b) Rotation axis perpendicular to incident beam (θ = 90°); static optimization; P = 0.31. (c) Rotation axis perpendicular to incident beam (θ = 90°); dynamic optimization; P = 0.93.

FIG. 19
FIG. 19

Summary of experimental convergence level data. (a) Rotation axis parallel to incident beam (θ = 0°); static and dynamic optimization are identical. (b) Rotation axis oriented at 45° to incident beam (θ = 45°); ● ~ static optimization; ■ ~ dynamic optimization; (c) Rotation axis perpendicular to incident beam (θ = 90°); ● ~ Static optimization; ■ ~ dynamic optimization.

FIG. 20
FIG. 20

Summary of experimental convergence time data. (a) Rotation axis parallel to incident beam (θ = 0°); static and dynamic optimization are identical. (b) Rotational axis oriented at 45° to incident beam (θ = 45°); ● ~ static optimization; ■ ~ dynamic optimization; (c) Rotation axis perpendicular to incident beam (θ = 90°); ● ~ static optimization; ■ ~ dynamic optimization.

Tables (1)

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TABLE I Comparison of analytical speckle model with computer simulations for realistic speckle modulations.

Equations (30)

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S K I T ( no speckle ) ,
S = K M s ( t ) I T ( t ) ( with speckle ) .
M s ( t ) 1 ( 2 π ) 1 / 2 - d ω a ( ω ) e - i ω t .
I T I M + I D + { negligible terms } I M + I D ,
M s M M + M ac .
S = K [ M M I M + M M I D + M ac I M + M ac I D + ( higher - order terms ) ] .
E T ( t ) = n = 1 N E 0 e i Γ n ( t ) ,
I T = E T 2 = 1 N 2 n = 1 N m = 1 N exp i [ Γ n ( t ) - Γ m ( t ) ] .
I T = 1 N + 1 N 2 n = 1 n m N m = 1 N { cos ( β n - β m ) [ J 0 2 ( ψ ) + ] + sin ( β n - β m ) [ - 2 J 1 ( ψ ) J 0 ( ψ ) ( e i ω n t - e - i ω n t 2 i ) + 2 J 1 ( ψ ) J 0 ( ψ ) ( e i ω m t - e - i ω m t 2 i ) + ] } .
I M 1 N + J 0 2 ( ψ ) N 2 n = 1 n m N m = 1 N cos ( β n - β m )
I D - 4 J 0 ( ψ ) J 1 ( ψ ) N 2 n = 1 n m N m = 1 N sin ( β n - β m ) sin ω n t .
I T = I M + I D + higher - order products of Bessel functions ,
P ( Δ β ) = 1 α - Δ β α 2 .
I M = 1 N + J 0 2 ( ψ ) N 2 n = 1 n m N m = 1 N cos ( β n - β m ) .
cos ( β n - β m ) = - P ( Δ β ) cos ( Δ β ) d ( Δ β ) = 2 α 2 ( 1 - cos α ) .
I M = 1 N + J 0 2 ( ψ ) ( 1 - 1 N ) 2 α 2 ( 1 - cos α ) .
I D 2 = 1 2 ( 4 J 0 ( ψ ) J 1 ( ψ ) N 2 ) 2 × n = 1 N ( m = 1 m n N sin ( β n - β m ) ) 2 .
b n m = 1 n m N sin ( β n - β m ) .
b n 2 = d β 1 d β 2 d β N P ( β 1 ) P ( β 2 ) P ( β N ) × b n 2 ( β 1 , β 2 , , β N ) .
b n 2 = 1 2 ( N 2 - 3 N + 2 ) sin 2 ( α / 2 ) ( α / 2 ) 2 ( 1 - sin α α ) + 1 2 ( N - 1 ) ( 1 - sin 2 α α 2 ) .
I D 2 = 4 [ J 0 ( ψ ) J 1 ( ψ ) ] 2 × [ ( 1 N - 3 N 2 + 2 N 3 ) sin 2 ( α / 2 ) ( α / 2 ) 2 ( 1 - sin α α ) + ( 1 N 2 - 1 N 3 ) ( 1 - sin 2 α α 2 ) ] .
I M 2 = 1 N 2 + 2 J 0 2 ( ψ ) ( 1 N - 1 N 2 ) sin 2 ( α / 2 ) ( α / 2 ) 2 + J 0 4 ( ψ ) [ ( 1 - 6 N + 11 N 2 - 6 N 3 sin 4 ( α / 2 ) ( α / 2 ) 4 ) + 2 ( 1 N - 3 N 2 + 2 N 3 ) sin 2 ( α / 2 ) ( α / 2 ) 2 ( 1 + sin α α ) + ( 1 N 2 - 1 N 3 ) ( 1 + sin 2 α α 2 ) ] .
ρ = P D / P S ,
S D K M M I D .
P D = 1 2 T T T S D 2 d t = K 2 2 T - T T I D 2 M M 2 d t = K 2 I D 2 ,
C s { ± ω i ω i - Δ ω ω i + Δ ω a ( ω ) a * ( ω ) 2 T d ω } 1 / 2 .
P S = 1 2 T - T T S S 2 d t = K 2 I M 2 C s 2 ,
P T S = lim T 1 2 T - a ( ω ) a * ( ω ) d ω ,
I D 2 = C s 2 ρ I M 2 .
M s ( t ) 1 + k = 1 N j = 1 M a j k sin ( ω j k t + ϕ j k ) ,