Abstract

Modelling phase fluctuations due to Kolmogorov turbulence is important in many areas of applied optics such as simulating adaptive optics configurations, prediction of the performance of laser designators and simulation of infrared (IR) scenes in the presence of atmospheric turbulence. The computational performance of algorithms implementing this model is an important issue because in many situations a large number of phase screens is required. For example, in IR scene simulation a different phase screen is required for each pixel in the scene, and in other situations there exists a need for many thousands of phase screens to be calculated to obtain a statistical average. Whilst there have been previous attempts to increase the computational speed of these algorithms, the computation time required for a large number of phase screens still remains an issue. In this paper, we apply linear and statistical properties to improve the performance of the previous best published algorithm by 60 times when implemented on a sequential processor in software. Because the new algorithm is now trivially parallelizable, a further 20 times speedup can easily be achieved through a parallel software or hardware implementation.

© 2007 Optical Society of America

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References

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  1. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  2. D. L. Fried, "Statistics of a Geometric Representation of Wavefront Distortion," J. Opt. Soc. Am. 55, 1427-1435 (1965).
    [CrossRef]
  3. C. M. Harding, R. A. Johnston, and R. G. Lane, "Fast Simulation of a Kolmogorov Phase Screen," Appl. Opt. 38, 2161-2170 (1999).
    [CrossRef]
  4. V. Sriram and D. Kearney, "High Speed High Fidelity Infrared Scene Simulation using Reconfigurable Computing," 2006 IEEE International Conference on Field Programmable Logic and Applications (FPL), Spain, (IEEE Press).
  5. E. P. Wallner, "Optimal wave-front correction using slope measurements," J. Opt. Soc. Am. 73, 1771-1776 (1983).
    [CrossRef]
  6. W. Rugh, "Linear Time-Invariant Systems and Convolution: An Interactive Lecture" http://www.jhu.edu/signals/lecture1/main.html#spot2>.
  7. E. Weisstein, "Normal Sum Distribution," From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/NormalSumDistribution.html>.
  8. V. Sriram and D. Kearney, "A high throughput area time efficient uniform random number generator based on the TT800 algorithm," in Proc of IEEE International Conference on Field Programmable Logic and Applications, Amsterdam, Netherlands, August 2007, (IEEE Press).

1999 (1)

1983 (1)

1965 (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (5)

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

V. Sriram and D. Kearney, "High Speed High Fidelity Infrared Scene Simulation using Reconfigurable Computing," 2006 IEEE International Conference on Field Programmable Logic and Applications (FPL), Spain, (IEEE Press).

W. Rugh, "Linear Time-Invariant Systems and Convolution: An Interactive Lecture" http://www.jhu.edu/signals/lecture1/main.html#spot2>.

E. Weisstein, "Normal Sum Distribution," From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/NormalSumDistribution.html>.

V. Sriram and D. Kearney, "A high throughput area time efficient uniform random number generator based on the TT800 algorithm," in Proc of IEEE International Conference on Field Programmable Logic and Applications, Amsterdam, Netherlands, August 2007, (IEEE Press).

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Figures (2)

Fig. 1.
Fig. 1.

Comparison of the structure function derived from the transformed algorithm as compared to the ideal function. The x axis is the distance between points in the phase screen non dimensionalized by the Fried parameter. (a) shows the ideal structure function (lower trace) as compared with the simulated structure function (upper trace). (b) shows the relative error between the ideal and simulated structure functions as a fraction of the ideal. Note that the errors reported in (b) are attributed to the original approximations made by Harding et al. and not to the transformations reported in this paper.

Fig. 2.
Fig. 2.

Number of standard deviations from the ideal structure function. (Lower trace) Harding’s original algorithm, (upper trace) transformed algorithm.

Equations (20)

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C S = U Λ U ,
Φ ̂ l = U x ˜ ,
E [ Φ ̂ l Φ ̂ l T ] = E [ U x ˜ x ˜ T U T ] = UE [ x ˜ x ˜ T ] U T = U Λ U = C S .
Φ h 1 = Φ h 0 I 4 x 4 + ε res 1 × x 1 ,
Φ h 2 = Φ h 1 I 4 x 4 rotated + ε res 2 × x 2 ,
I 4 x 4 rotated = [ 0 0 0 0.0017 0 0 0 0 0 0.0341 0 0.0341 0 0 0 0.0341 0 0.3198 0 0.0341 0 0.0017 0 0.3198 1 0.3198 0 0.0017 0 0.0341 0 0.3198 0 0.0341 0 0 0 0.0341 0 0.0341 0 0 0 0 0 0.0017 0 0 0 ] .
Φ h 2 = ( Φ h 0 I 4 x 4 + ε res 1 × x 1 ) I 4 x 4 rotated + ε res 2 × x 2
Φ h 2 = ( Φ h 0 I 4 x 4 ) I 4 x 4 rotated + ε res 1 ( x 1 I 4 x 4 rotated ) + ε res 2 × x 2
y = i = 1 n α i g i ,
μ y = i = 1 n α i μ g i
σ y 2 = i = 1 n α i 2 σ gi 2 .
x 1 4,4 α = x 1 1,1 α 1 + x 1 1,2 α 2 + x 1 1,3 α 3 + .. + x 1 7,7 α 49
Φ h 2 = ( Φ h 0 I 4 x 4 ) I 4 x 4 rotated + ε res 1 × x 3 + ε res 2 × x 2
Φ h 2 = ( Φ h 0 I 4 x 4 ) I 4 x 4 rotated + x .
D Φ ( r ) = 2 [ B Φ ( 0 ) B Φ ( r ) ] ,
B Φ ( r ) = ( Φ ( x ) Φ ( x r ) )
D Φ ideal ( r ) = 6.88 × ( r ) 5 3
r = r actual r 0 , [ 3 ]
σ ( r ) = { 1 ( N 1 ) i [ D ϕ ( i ) ( r ) 1 N D ϕ ( i ) ( r ) ] 2 } 1 2
Error ( standard deviation ) = D ϕ ideal ( r ) [ 1 N ] D ϕ i ( r ) σ ( r ) .

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