Abstract

Monte Carlo (MC) simulation of phase front perturbations by atmospheric turbulence finds numerous applications for design and modeling of the adaptive optics systems, laser beam propagation simulations, and evaluating the performance of the various optical systems operating in the open air environment. Accurate generation of two-dimensional random fields of turbulent phase is complicated by the enormous diversity of scales that can reach five orders of magnitude in each coordinate. In addition there is a need for generation of the long “ribbons” of turbulent phase that are used to represent the time evolution of the wave front. This makes it unfeasible to use the standard discrete Fourier transform-based technique as a basis for the MC simulation algorithm. We propose a new model for turbulent phase: the sparse spectrum (SS) random field. The principal assumption of the SS model is that each realization of the random field has a discrete random spectral support. Statistics of the random amplitudes and wave vectors of the SS model are arranged to provide the required spectral and correlation properties of the random field. The SS-based MC model offers substantial reduction of computer costs for simulation of the wide-band random fields and processes, and is capable of generating long aperiodic phase “ribbons.” We report the results of model trials that determine the number of sparse components, and the range of wavenumbers that is necessary to accurately reproduce the random field with a power-law spectrum.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. T. A. ten Brummelaar, “Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike,” Opt. Commun. 132, 329–342 (1996).
    [CrossRef]
  28. R. J. Mathar, “Karhunen–Loève basis of Kolmogorov phase screens covering a rectangular stripe,” Waves Random Complex Media 20, 23–35 (2010).
    [CrossRef]
  29. C. Schwartz, G. Baum, and E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  30. G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
    [CrossRef]
  31. D. G. Perez, L. Zunino, and M. Garavaglia, “Modeling turbulent wave-front phase as a fractional Brownian motion: A new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
    [CrossRef]
  32. D. G. Perez and L. Zunino, “Generalized wave front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
    [CrossRef]
  33. A. Beghi, A. Cenedese, and A. Masiero, “Multiscale stochastic approach for phase screens synthesis,” Appl. Opt. 50, 4124–4133 (2011).
    [CrossRef]
  34. M. Charnotskii, “Sparse spectrum model of the sea surface,” in Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, H. R. Riggs, ed. (ASME, 2011), p. 49958.
  35. M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” in Proceedings of the 22 International Offshore and Polar Engineering Conference.J. S. Chung, ed. (ISOPE, 2012), pp. 655–659.
  36. M. I. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29, 711–721 (2012).
    [CrossRef]
  37. M. Charnotskii and G. Baker, “Long and short-term scintillations of focused beams and point spread functions in the turbulent atmosphere,” Proc. SPIE 8517, 85170L (2012).
    [CrossRef]

2012

2011

2010

2009

2008

D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A 25, 463–468 (2008).
[CrossRef]

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens.” J. Opt. Soc. Am. A 25, 515–525 (2008).
[CrossRef]

D. G. Perez and L. Zunino, “Generalized wave front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
[CrossRef]

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

V. Sriram and D. Kearney, “Multiple parallel FPGA implementations of a Kolmogorov phase screen generator,” J. Real-Time Image Proc. 3, 195–200 (2008).
[CrossRef]

2007

2006

2004

1999

1998

1997

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).

G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
[CrossRef]

1996

T. A. ten Brummelaar, “Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike,” Opt. Commun. 132, 329–342 (1996).
[CrossRef]

H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. 35, 1561–1565 (1996).
[CrossRef]

1995

1994

C. Schwartz, G. Baum, and E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
[CrossRef]

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

1993

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

1992

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

1990

1986

J. Wallace and F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” Proc. SPIE 642, 261–268 (1986).
[CrossRef]

1976

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

Aitken, G. J. M.

G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
[CrossRef]

Assemat, F.

Baker, G.

M. Charnotskii and G. Baker, “Long and short-term scintillations of focused beams and point spread functions in the turbulent atmosphere,” Proc. SPIE 8517, 85170L (2012).
[CrossRef]

Barakat, R.

Baum, G.

Beghi, A.

Beletic, J. W.

Cain, S.

Carbillet, M.

Cenedese, A.

Charnotskii, M.

M. Charnotskii and G. Baker, “Long and short-term scintillations of focused beams and point spread functions in the turbulent atmosphere,” Proc. SPIE 8517, 85170L (2012).
[CrossRef]

M. Charnotskii, “Sparse spectrum model of the sea surface,” in Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, H. R. Riggs, ed. (ASME, 2011), p. 49958.

M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” in Proceedings of the 22 International Offshore and Polar Engineering Conference.J. S. Chung, ed. (ISOPE, 2012), pp. 655–659.

Charnotskii, M. I.

Clark, T.

Dainty, J. C.

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

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

Formwalt, B.

Fried, D. L.

Garavaglia, M.

Gavel, D. T.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

Gebhardt, F. G.

J. Wallace and F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” Proc. SPIE 642, 261–268 (1986).
[CrossRef]

Gendron, E.

Glindemann, A.

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

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

Harding, C. M.

Jakobsson, H.

Johansson, E. M.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

Johnston, R. A.

Kearney, D.

V. Sriram and D. Kearney, “Multiple parallel FPGA implementations of a Kolmogorov phase screen generator,” J. Real-Time Image Proc. 3, 195–200 (2008).
[CrossRef]

V. Sriram and D. Kearney, “An ultra fast Kolmogorov phase screen generator suitable for parallel implementation,” Opt. Express 15, 13709–13714 (2007).
[CrossRef]

Lane, R. G.

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

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

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

Li, S. S.

Li, X. Y.

Masiero, A.

Mathar, R. J.

R. J. Mathar, “Karhunen–Loève basis of Kolmogorov phase screens covering a rectangular stripe,” Waves Random Complex Media 20, 23–35 (2010).
[CrossRef]

McGaughey, D. R.

G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
[CrossRef]

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

Montera, D.

Paramonov, P. V.

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Perez, D. G.

Phillips, R.

Rhoadamer, T. A.

Ribak, E. N.

Riccardi, A.

Roddier, N.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rossille, D.

G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
[CrossRef]

Schwartz, C.

Sedmak, G.

Sriram, V.

V. Sriram and D. Kearney, “Multiple parallel FPGA implementations of a Kolmogorov phase screen generator,” J. Real-Time Image Proc. 3, 195–200 (2008).
[CrossRef]

V. Sriram and D. Kearney, “An ultra fast Kolmogorov phase screen generator suitable for parallel implementation,” Opt. Express 15, 13709–13714 (2007).
[CrossRef]

Tallon, M.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

ten Brummelaar, T. A.

T. A. ten Brummelaar, “Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike,” Opt. Commun. 132, 329–342 (1996).
[CrossRef]

Thiebaut, E.

Valley, M. T.

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Vorontsov, A.

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Vorontsov, M. A.

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Wallace, J.

J. Wallace and F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” Proc. SPIE 642, 261–268 (1986).
[CrossRef]

Welsh, B. M.

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).

M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadamer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
[CrossRef]

Welsh, G.

Wilson, R. W.

Wu, H. L.

Xiang, J.

Yan, H. X.

Zunino, L.

Appl. Opt.

J. Mod. Opt.

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

J. Opt. Soc. Am. A

J. Real-Time Image Proc.

V. Sriram and D. Kearney, “Multiple parallel FPGA implementations of a Kolmogorov phase screen generator,” J. Real-Time Image Proc. 3, 195–200 (2008).
[CrossRef]

Opt. Commun.

T. A. ten Brummelaar, “Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike,” Opt. Commun. 132, 329–342 (1996).
[CrossRef]

Opt. Eng.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

M. Charnotskii and G. Baker, “Long and short-term scintillations of focused beams and point spread functions in the turbulent atmosphere,” Proc. SPIE 8517, 85170L (2012).
[CrossRef]

G. J. M. Aitken, D. Rossille, and D. R. McGaughey, “Filtered fractional Brownian motion as a model for atmospherically induced wavefront distortions,” Proc. SPIE 3125, 310–317 (1997).
[CrossRef]

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
[CrossRef]

J. Wallace and F. G. Gebhardt, “New method for numerical simulation of atmospheric turbulence,” Proc. SPIE 642, 261–268 (1986).
[CrossRef]

Waves Random Complex Media

M. A. Vorontsov, P. V. Paramonov, M. T. Valley, and A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

R. J. Mathar, “Karhunen–Loève basis of Kolmogorov phase screens covering a rectangular stripe,” Waves Random Complex Media 20, 23–35 (2010).
[CrossRef]

Waves Random Media

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

Other

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

M. Charnotskii, “Sparse spectrum model of the sea surface,” in Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, H. R. Riggs, ed. (ASME, 2011), p. 49958.

M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” in Proceedings of the 22 International Offshore and Polar Engineering Conference.J. S. Chung, ed. (ISOPE, 2012), pp. 655–659.

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Figures (12)

Fig. 1.
Fig. 1.

SS model structure functions for the various exponents α of the power-law [Eq. (1)]. Also shown are the best-fit power-law approximations.

Fig. 2.
Fig. 2.

Ratios of the SS model structure functions to the target structure functions for the various exponents α of the power-law [Eq. (1)] obtained from the SS MCmodel.

Fig. 3.
Fig. 3.

Dependence of the structure functions on the largest scale included in the SS model LMAX.

Fig. 4.
Fig. 4.

Dependence of the structure functions on the smallest scale included in the SS model LMIN.

Fig. 5.
Fig. 5.

Dependence of the SS model on the target structure functions ratio for 1 mm–1000 m range of scales, and various numbers of the components N.

Fig. 6.
Fig. 6.

Same as Fig. 5, but for the 1 mm–3000 m range of scales.

Fig. 7.
Fig. 7.

SS structure function for a 1 m × 5 m rectangle area.

Fig. 8.
Fig. 8.

Sample of the phase surface generated by the model over a 1 m × 5 m rectangle area.

Fig. 9.
Fig. 9.

Samples of the phase values along the y direction referenced at the phase at y=1m at the different positions along the “ribbon.”

Fig. 10.
Fig. 10.

Structure functions calculated based on the 3000 samples at different positions along the “ribbon.”

Fig. 11.
Fig. 11.

Strehl numbers calculated according to Eq. (26) using the SS phase with various numbers of components. Also shown are the results of numerical integration of Eq. (27) and asymptote Eq. (28).

Fig. 12.
Fig. 12.

Strehl number SI for SS model and numerical integrations.

Equations (28)

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

D(r⃗)=[S(R⃗+r⃗)S(R⃗)]2=(rrC)α,1<α<2
D(r⃗)=2d2KΦS(K⃗)[1cos(K⃗·r⃗)],
S(x,y)=n,m=anmexp[ik0(nx+my)],
anm=0,anmanm=snmδ(n+n)δ(m+m).
D(x,y)=2n,m=snm[1cos(k0nx+k0my)].
ΦS(Kx,Ky)=n,m=snmδ(Kxk0n)δ(Kyk0m),
snm=ΦS(k0n,k0m)k02.
S(xj,yl)=n,m=N/2N/21anmexp[i2πN(nj+ml)],
Y⃗=A¯¯X⃗+B¯¯N⃗,
A¯¯=Y⃗X⃗TX⃗X⃗T1,B¯¯B¯¯T=Y⃗Y⃗TA¯¯X⃗Y⃗T.
S(r⃗)=Ren=1Nanexp(iK⃗n·r⃗)
an=0,anam=0,anam*=snδmn.
P{K⃗nK⃗+dK⃗}=pn(K⃗)dK⃗.
D(r⃗)=[S(R⃗+r⃗)S(R⃗)]2{an,K⃗n}=n=1Nsn[1cos(K⃗n·r⃗)]{K⃗n}=d2Kn=1Nsnpn(K⃗)[1cos(K⃗·r⃗)].
n=1Nsnpn(K⃗)=2Φ(K⃗).
Φ(K⃗)=B(α)K2α,
B(α)=α2α2Γ(1+α2)πΓ(1α2)rCα.
pn(K⃗)d2K=pn(K,φ)kdkdφ=pn(K)dkdφ2π.
n=1Nsnpn(K)=4πB(α)K1α.
n=1Nsnpn(χ)=4πB(α)eαχ.
snpn(χ)=4πB(α)eαχ,χn1χχn,n=1,2,,N.
eαχ1α(χnχn1)(eαχn1eαχn),χn1χχn.
sn=4πB(α)1α(eαχneαχn+1),pn(χ)=1χn+1χn.
χn=χMIN+nN(χMAXχMIN),n=0,1,2,,N.
u(rIM)=Cd2rA(r)exp[ikr·rIM+iS(r)],
St#=1Σ2d2Rd2ρA(R⃗+12ρ⃗)A(R⃗12ρ⃗)exp[iS(R⃗+12ρ⃗)iS(R⃗12ρ⃗)],
St#=1Σ2d2ρ[d2RA(R⃗+12ρ⃗)A(R⃗12ρ⃗)]exp[12D(ρ⃗)].
St#Γ(1+2α)22+2α[D(2a)]2α.

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