Abstract

We show that the filtered white noise process applied to logarithmically sampled turbulence spectra executed in cylindrical spatial frequency coordinates produces accurate phase screens that are free of the shortcomings associated with uniform sampling schemes. Decoupled from the sampling requirements of the wave-optics computational mesh in which they are used, the screens have isotropic statistics, do not require low spatial frequency augmentation, have prescribed resolution with more optimum sampling for the simulation, and feature an economical method of achieving screen motion that minimizes memory requirements.

© 2013 Optical Society of America

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  1. A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  2. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  3. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010), pp. 149–184.
  4. D. Voelz, Computational Fourier Optics (SPIE, 2011), pp. 178–182.
  5. G. Cochran, “Phase screen generation,” (The Optical Sciences Company, 1985).
  6. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1989).
    [CrossRef]
  7. B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
    [CrossRef]
  8. J. Xiang, “Accurate compensation of the low-frequency components for FFT-based turbulent phase screens,” Opt. Express 20, 681–687 (2012).
    [CrossRef]
  9. R. J. Eckert, “Polar phase screens: a comparative analysis with other methods of random phase screen generation,” Proc. SPIE 6303, 630301 (2006).
    [CrossRef]
  10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), pp. 66–71.
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.
  12. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994), p. 164.
  13. D. L. Fried, “Optical resolution through randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  14. V. Strauss, “Functional conversion of signals in the study of relaxation phenomena,” Signal Process. 45, 293–312 (1995).
    [CrossRef]
  15. F. J. Beutler, “Error-free recovery of signals from irregularly spaced samples,” SIAM Rev. 8, 328–335 (1966).
    [CrossRef]

2012

2006

R. J. Eckert, “Polar phase screens: a comparative analysis with other methods of random phase screen generation,” Proc. SPIE 6303, 630301 (2006).
[CrossRef]

1995

V. Strauss, “Functional conversion of signals in the study of relaxation phenomena,” Signal Process. 45, 293–312 (1995).
[CrossRef]

1990

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

1989

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1989).
[CrossRef]

1983

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1976

A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1966

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), pp. 66–71.

Beutler, F. J.

F. J. Beutler, “Error-free recovery of signals from irregularly spaced samples,” SIAM Rev. 8, 328–335 (1966).
[CrossRef]

Cochran, G.

G. Cochran, “Phase screen generation,” (The Optical Sciences Company, 1985).

Eckert, R. J.

R. J. Eckert, “Polar phase screens: a comparative analysis with other methods of random phase screen generation,” Proc. SPIE 6303, 630301 (2006).
[CrossRef]

Feit, M. D.

A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.

Fleck, A. J.

A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fried, D. L.

Gavel, D. T.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1989).
[CrossRef]

Herman, B. J.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

Johansson, E. M.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1989).
[CrossRef]

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Morris, J. R.

A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), pp. 66–71.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994), p. 164.

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010), pp. 149–184.

Strauss, V.

V. Strauss, “Functional conversion of signals in the study of relaxation phenomena,” Signal Process. 45, 293–312 (1995).
[CrossRef]

Strugala, L. A.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.

Voelz, D.

D. Voelz, Computational Fourier Optics (SPIE, 2011), pp. 178–182.

Xiang, J.

Appl. Phys.

A. J. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Proc. IEEE

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Proc. SPIE

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1989).
[CrossRef]

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

R. J. Eckert, “Polar phase screens: a comparative analysis with other methods of random phase screen generation,” Proc. SPIE 6303, 630301 (2006).
[CrossRef]

SIAM Rev.

F. J. Beutler, “Error-free recovery of signals from irregularly spaced samples,” SIAM Rev. 8, 328–335 (1966).
[CrossRef]

Signal Process.

V. Strauss, “Functional conversion of signals in the study of relaxation phenomena,” Signal Process. 45, 293–312 (1995).
[CrossRef]

Other

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), pp. 66–71.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), p. 289.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994), p. 164.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010), pp. 149–184.

D. Voelz, Computational Fourier Optics (SPIE, 2011), pp. 178–182.

G. Cochran, “Phase screen generation,” (The Optical Sciences Company, 1985).

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Figures (7)

Fig. 1.
Fig. 1.

Cylindrical frequency mesh with logarithmic sampling. The sample pattern is obtained from a range factor, f, of 3.3×103 and 20% resolution, which results in 42 radial samples and eight azimuthal samples for a total of 336 samples.

Fig. 2.
Fig. 2.

Structure function test for ro=0.1m and Lo=10m. The theoretical answer is shown as a solid line, and simulated screens are shown as points. These screens were generated for 20% resolution and with 400 samples of spatial frequency. The vertical bars are the standard deviations.

Fig. 3.
Fig. 3.

Structure function tests for outer scales of 0.25 to 300 m for 20% resolution and ro of 0.1 m. The dashed line is the theoretical structure function for corresponding outer scales.

Fig. 4.
Fig. 4.

Wave-optics results for the case of an uncompensated beam with logarithmically sampled turbulence screens compared to theory. The open circles are the simulation results as average Strehl obtained with 80 realizations.

Fig. 5.
Fig. 5.

Wave-optics results for a tilt-compensated beam with logarithmically sampled turbulence screens compared to theory. The open circles are the simulation results as average Strehl obtained with 80 realizations.

Fig. 6.
Fig. 6.

Turbulence screen phase cache array includes the wave-optics computational mesh shown as dark lines. The cross-hatched elements are to accommodate beam motion through the air by allowing interpolation of phase from the phase screen grid onto the computational mesh.

Fig. 7.
Fig. 7.

Example computational grid path evolution in the course of an extended wave-optics simulation. The upper image represents a screen surrounding the entire problem space, and the lower image represents only the screen space required.

Equations (59)

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ϕ(x,y)=h(κ⃗r)F12(κr)exp(iκ⃗·r⃗)dκ⃗r,
ϕ(x,y)=κxκyh(κx,κy)F12(κx,κy)exp[i(xκx+yκy)]ΔκxΔκy.
Δκx=Δκy=κmaxN,
κi=iΔκ,
κj=jΔκ.
h(κx,κy)=gr(κx,κy)+igi(κx,κy)A,
A=ΔκxΔκy,
N=LΔx;Δκ=2πL;κmax=2πΔx.
Lx=Lxo+vxτsim,
κx=κcosφ,κy=κsinφ,κ=[κx2+κy2]12,
dκxdκy=κdφdκ.
ϕ(x,y)=002πh(κ,φ)F12(κ)exp[iκ(xcosφ+ysinφ)]κdφdκ.
ϕ(x,y)=0F12(κ)0πh(κ,φ)exp[iκ(xcosφ+ysinφ)]κdφdκ+0F12(κ)π2πh(κ,φ)exp[iκ(xcosφ+ysinφ)]κdφdκ.
h(κ,φ)=h*(κ,φ+π)
ϕ(x,y)=0F12(κ)0π{h(κ,φ)exp[iκ(xcosφ+ysinφ)]+h*(κ,φ)exp[iκ(xcosφ+ysinφ)]}κdφdκ.
h(κ,φ)=ξ(κ,φ)exp[iη(κ,φ)],
ϕ(x,y)=0F12(κ)0πξ(κ,φ){exp[iη(κ,φ)]exp[iκ(xcosφ+ysinφ)]+exp[iη(κ,φ)]exp[iκ(xcosφ+ysinφ)]}κdφdκ,
ϕ(x,y)=20F12(κ)0πξ(κ,φ)cos[η(κ,φ)+κ(xcosφ+ysinφ)]κdφdκ.
ϕ(x,y)=20F12(κ){0π2ξ(κ,φ)cos[η(κ,φ)+κ(xcosφ+ysinφ)]dφ+0π2ξ(κ,φ+π2)cos[η(κ,φ+π2)+κ(xcos(φ+π2)+ysin(φ+π2))]dφ}κdκ,
ϕ(x,y)=20F12(κ)0π2{ξ(κ,φ)cos[η(κ,φ)+κ(xcosφ+ysinφ)]+ξ(κ,φ+π2)cos[η(κ,φ+π2)κ(xsinφycosφ)]}dφκdκ.
κmin<2πLo;κmax>2πlo.
f=κmaxκmin,
κs=κminfsNκ,
Δκs=κsκs1,
κ¯s=κs+κs12.
Δκs=κminfsNκ(1f1Nκ)
κ¯s=κmin2fsNκ(1+f1Nκ).
δ=Δκsκ¯s=2[f1Nκ1f1Nκ+1].
κ¯Δφ=Δκ.
Δφ=Δκκ¯=δ.
Nφ=π2Δφ=π2δ,
Nφ=π4[f1Nκ+1f1Nκ1].
Nκ=lnfln((2+δ)(2δ)).
δ=π2Nφ,
f=[4Nφ+π4Nφπ]Nκ.
ϕ(x,y)=2i=1Nκj=1NφF12(κ¯i){ξ(κ¯i,φj)cos[η(κ¯i,φj)+κ¯i(xcosφj+ysinφj)]+ξ(κ¯i,φj+π2)cos[η(κ¯i,φj+π2)κ¯i(xsinφjycosφj)]}κ¯iΔφΔκ,
φj=(j12)Δφ,
F(κi)=2π(0.033)k2Λ(κi)z=az=bCn2(z)dz,
Λ(κ¯i)=exp((κ¯iκl)2)[κo2+κ¯i2]116
ro=0.185[λ2z=az=bCn2(z)dz]35
F(κi)=0.49ro53Λ(κ¯i).
ϕ(x,y)=1.4ro56i=1Nκj=1NφΛ12(κ¯i){ζ(κ¯i,φj)cos[η(κ¯i,φj)+κ¯i(xcosφj+ysinφj)]+ζ(κ¯i,φj+π2)cos[η(κ¯i,φj+π2)κ¯i(xsinφjycosφj)]}κ¯iΔφΔκi.
ξ=[gr2+gi2]12A
η=tan1(gigr).
gr=[2lnβ1]12cos(2πβ2),
gi=[2lnβ1]12sin(2πβ2),
ξ=[2lnβ1]12A=ζA,
η=2πβ2.
ϕ(x,y)=1.4ro56i=1Nκj=1NφΓ(κ¯i){ζ(κ¯i,φj)cos[η(κ¯i,φj)+κ¯i(xcosφj+ysinφj)]+ζ(κ¯i,φj+π2)cos[η(κ¯i,φj+π2)κ¯i(xsinφjycosφj)]},
Γ(κ¯i)=[Λ(κ¯i)κ¯iΔκiΔφ]12.
ζ(κ¯i,φj)ζ1(i,j);ζ(κ¯i,φj+π2)ζ2(i,j);
η(κ¯i,φj)η1(i,j);η(κ¯i,φj+π2)η2(i,j),
ϕ(x,y)=1.4ro56i=1Nκj=1NφΓi(κ¯i){ζ1(i,j)cos[η1(i,j)+κ¯i(xcosφj+ysinφj)]+ζ2(i,j)cos[η2(i,j)κ¯i(xsinφjycosφj)]}.
D(ρ)=6.16ro53[35(Lo2π)53(ρLo4π)56Γ(116)K56(2πρLo)],
ro_scrn=nscrn35ro,
Strehl=245πn=0n=4(1)nn!Γ(32+5n6)Γ(n+65)Γ(3+5n6)Γ(n+115)[3.44(Dro)53]n,
Strehl=(roD)2245π3.4495n=0n=8(1)nn!(n+12)Γ(5n6+95)Γ(n+32)3.446n5(roD)3+2n.
Strehl=16π01x[cos1xx(1x2)12]exp[3.44(Drox)53](1x13)dx.
κ¯max2=κmin4f[1+f1Nκ]

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