Abstract

We propose a technique for the accurate modeling and simulation of scintillation patterns that are due to Kolmogorov statistics without assuming periodic boundary conditions. We show how the more physically justifiable assumption of smoothness results in a propagation kernel of finite extent. This allows the phase screen dimensions for an accurate simulation to be determined, and truncation can then be used to eliminate the unwanted spectral leakage and diffraction effects usually inherent in the use of finite apertures. A detailed outline of the proposed technique and comparison of simulations with analytic results are presented.

© 2000 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics (Elsevier, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  2. V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
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  4. J. M. Martin, S. M. Flatte, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
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  5. S. M. Flatte, G.-Y. Wang, 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).
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  8. M. J. Adcock, N. Jones, “Atmospheric propagation simulation,” http://op.ph.ic.ac.uk/users/miles/see/see.html .
  9. J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
    [CrossRef]
  10. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  11. A. Glindemann, R. G. Lane, J. C. Dainty, “Simulation of time evolving speckle using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
    [CrossRef]
  12. J. Vernin, F. Roddier, “Experimental determination of two dimensional spatiotemporal power spectra of stellar light scintillation—evidence for a multilayer structure of the air turbulence in the upper troposphere,” J. Opt. Soc. Am. 63, 270–273 (1973).
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  13. C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
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1999 (2)

J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
[CrossRef]

C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
[CrossRef]

1998 (1)

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

1995 (1)

1994 (2)

1993 (2)

1992 (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1990 (1)

1988 (1)

1973 (2)

Adcock, M. J.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

Baum, G.

Belmonte, A.

J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Coles, W. A.

Comeron, A.

J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
[CrossRef]

Dainty, J. C.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

A. Glindemann, R. G. Lane, J. C. Dainty, “Simulation of time evolving speckle using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Davis, C. A.

Filice, J. P.

Flatte, S. M.

Frelich, R. G.

Glindemann, A.

A. Glindemann, R. G. Lane, J. C. Dainty, “Simulation of time evolving speckle using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Harding, C. M.

Horowtiz, P.

Johnston, R. A.

Kluckers, V. A.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

Lane, R. G.

C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
[CrossRef]

A. Glindemann, R. G. Lane, J. C. Dainty, “Simulation of time evolving speckle using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Martin, J.

Martin, J. M.

Munro, I.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

Nicholls, T. W.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

Ribak, E. N.

Roddier, F.

Rubio, J. A.

J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
[CrossRef]

Schwartz, C.

Vdovin, G.

G. Vdovin, LightPipes: Beam Propagation Toolbox, Version 1.1 (Electronic Instrumentation Laboratory, Delft University of Technology, 1993–1996).

Vernin, J.

Walters, D. L.

Wang, G.-Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Wooder, N. J.

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

Yadlowsky, M.

Appl. Opt. (4)

Astron. Astrophys. Suppl. (1)

V. A. Kluckers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, J. C. Dainty, “Profiling of atmospheric turbulence strength and velocity using a generalized scidar technique,” Astron. Astrophys. Suppl. 130, 141–155 (1998).
[CrossRef]

J. Mod. Opt. (1)

A. Glindemann, R. G. Lane, J. C. Dainty, “Simulation of time evolving speckle using Kolmogorov statistics,” J. Mod. Opt. 40, 2381–2388 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

J. A. Rubio, A. Belmonte, A. Comeron, “Numerical simulation of long-path spherical wave propagation in three-dimensional random media,” Opt. Eng. 38, 1462–1469 (1999).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (5)

M. J. Adcock, N. Jones, “Atmospheric propagation simulation,” http://op.ph.ic.ac.uk/users/miles/see/see.html .

G. Vdovin, LightPipes: Beam Propagation Toolbox, Version 1.1 (Electronic Instrumentation Laboratory, Delft University of Technology, 1993–1996).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics (Elsevier, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Periodic assumption. Note that the propagated wave front at Q depends equally on the initial wave front at points P1 and P2. (b) Use of an extended initial wave front showing the extra screen dimensions required for an accurate scintillation simulation. A phase screen of dimensions (D + 2d) m × (D + 2d) m is required for an accurate scintillation representation of Dm × D m at the telescope aperture, after propagation over the height z.

Fig. 2
Fig. 2

Comparison of real parts of (a) original and (b) smoothed Fresnel kernels.

Fig. 3
Fig. 3

Simulation parameters as a function of smoothing and propagation distance. (a) Extra distance required versus smoothness for z = 10 km and D/ r 0 = 1. (b) Phase screen distortion measured as mean-square error (mse) between the original and the smoothed screens versus smoothness for z = 10 km and D/ r 0 = 1. (c) Envelope of the modified kernel as a function of the propagation distance (solid curve, 10 km; dashed curve, 7.5 km; dotted–dashed curve, 5 km; dotted curve, 2.5 km) for D/ r 0 = 1 and σ = 0.005 m.

Fig. 4
Fig. 4

Illustration of edge effects for an aperture of size 1 m × 1 m, propagation distance of 3 km, r 0 of 10 cm, σ = 0.002 m, and visible light. (a) 1 m × 1 m Scintillation pattern resulting from the convolution of a 1 m × 1 m nonperiodic phase screen with the unmodified Fresnel kernel. Edge effects are apparent only close to the edges in the image; however, they extend across the entire result. (b) Use of the smoothed Fresnel kernel and a starting phase screen of size of 1.8438 m × 1.8438 m is necessary to produce an undistorted 1 m × 1 m scintillation pattern. Although the introduced smoothing does not eliminate the edge effects, it does ensure that they do not extend into the valid result region. (c) Valid 1 m × 1 m portion of the scintillation from the 1.8438 m × 1.8438 m starting screen, which is clearly free from diffraction effects.

Fig. 5
Fig. 5

Real part of the regularized kernel in (a) the time domain, ĥ(x, y), and (b) the frequency domain, Ĥ(u, v).

Fig. 6
Fig. 6

Scintillation patterns for σ = 0.003 m and D/ r 0 = 4 phase screens propagated over (a) 2.5, (b) 5, (c) 7.5, and (d) 10 km.

Fig. 7
Fig. 7

Comparison of theoretical (solid) and simulated intensity variance results for propagation over 3 km, D = 1 m, r 0 = 25 cm. (a) σ = 0.002 m (dotted), (b) zoomed version for σ = 0.002 m (dotted), σ = 0.005 m (dotted–dashed), and σ = 0.008 m (dashed).

Fig. 8
Fig. 8

Comparison of theoretical (solid) and simulated covariance curves for D = 1 m and r 0 = 25 cm. (a) Propagation distances of 2.5, 5, 7.5, 10 km, σ = 0.005 m (dotted, 2.5 km, lowest; 10 km, highest). (b) σ values of 0.002 m (dotted), 0.005 m (dotted–dashed), 0.008 m (dashed) for a propagation distance of 2.5 km.

Fig. 9
Fig. 9

Comparison of theoretical zero inner scale (solid), theoretical nonzero inner scale (dotted), and simulated covariance curves (dashed) for D = 1 m, r 0 = 25 cm, σ = 0.008 and a propagation distance of 10 km. Note that the nonzero inner-scale covariance is a smoothed version of the ideal result obtained by convolution of the ideal result with the autocorrelation of the corresponding Gaussian smoothing term.

Equations (26)

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ψx, y, z-=ψx, y, z+expiϕx, y, z,
hx, y=expikziλzexpik2zx2+y2.
Upx, y=-- Uoξ, ηhx-ξ, y-ηdξdη,
Hu, v=expikzexp-i z2ku2+v2
 gzexpikfzdz,
sx, y=12πσ2exp-x2+y22σ2,
Upx, y=Uox, ysx, yhx, y.
hˆx, y=hx, ysx, y=K expikzexp-x2+y22σw2expik2zx2+y2β,
K=-iz+kσ2λz2+k2σ4,σw2=z2+k2σ4k2σ2,β=1+k2σ4z2
1-12πσw2-W/2W/2-W/2W/2exp-x2+y22σw2<X,
Hˆu, v=expikzexp-u2+v22σs2exp-iz2ku2+v2,
Δxλzβ/W.
σI2=I2-I2I2,
σI2=19.12λ-7/60 h5/6CN2hdh.
Wf=0.039k2f-11/30 CN2hsinπλhf2dh.
ψx, y, z-=ψx, y, z+expiϕx, y, z.
Ψu, v, z-=ψx, y, z-,
ψx, y, z-=-+-+ ψx, y, z-expi2πux+vydxdy.
Ψu, v, 0=Ψu, v, z-Hu, v.
ψx, y, 0=-1Ψu, v, 0.
ϕx-ϕx+ΔxΔxπ,
DϕΔx=Eϕx-ϕx+Δx2
=6.88Δx/r05/3,
Δx=r0/3,
rf=λz.
Δxλz/2.

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