Abstract

A new method, to our knowledge, allowing one to simulate correlated random processes is suggested. Structure (or correlation) functions of the processes under simulation are assumed to be given. The method is based on the generation of random wave vectors that allows one to simulate processes for a wide class of structure functions. The validity of the method proposed is illustrated by simulations of the turbulence-induced log-amplitude and phase distortions.

© 1997 Optical Society of America

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References

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  1. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing Pacific Grove, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
  2. N. Roddier, “Atmospheric wavefront simulation and Zernike polynomials,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckenridge, ed., Proc. SPIE, 1237, 668–679 (1990).
  3. N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  4. N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
    [CrossRef]
  5. M. C. Roggermann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
    [CrossRef]
  6. D. Kouznetsov, R. Ortega-Martinez, “Simulation of random field with given structure function,” Rev. Mex. Fis. 41, 563–571 (1995).
  7. R. C. Cannon, “Optimal bases for wave-front simulation and reconstruction on annular apertures,” J. Opt. Soc. Am. A 13, 862–867 (1995).
    [CrossRef]
  8. H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. 35, 1561–1565 (1995).
    [CrossRef]
  9. I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1, p. 363.
  10. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.
  11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  12. R. E. Hufnagel, “Variation of atmospheric turbulence,“ in Digest of OSA Topical Meeting Optical Propagation through Turbulence, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.
  13. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  14. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.
  15. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 1, p. 446.
  16. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.
  17. R. G. Paxman, B. J. Thelen, J. H. Seldin, “Phase diversity correction of turbulence-induced space blur,” Opt. Lett. 19, 1231–1233 (1994).
    [CrossRef] [PubMed]

1995 (4)

1994 (2)

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

R. G. Paxman, B. J. Thelen, J. H. Seldin, “Phase diversity correction of turbulence-induced space blur,” Opt. Lett. 19, 1231–1233 (1994).
[CrossRef] [PubMed]

1990 (1)

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

1966 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 1, p. 446.

Cannon, R. C.

Charnotskii, M. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.

Fried, D. L.

Gel’fand, I. M.

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1, p. 363.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.

Hufnagel, R. E.

R. E. Hufnagel, “Variation of atmospheric turbulence,“ in Digest of OSA Topical Meeting Optical Propagation through Turbulence, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

Iye, M.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

Jakobsson, H.

Kouznetsov, D.

D. Kouznetsov, R. Ortega-Martinez, “Simulation of random field with given structure function,” Rev. Mex. Fis. 41, 563–571 (1995).

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 1, p. 446.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing Pacific Grove, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).

Montera, D.

Ortega-Martinez, R.

D. Kouznetsov, R. Ortega-Martinez, “Simulation of random field with given structure function,” Rev. Mex. Fis. 41, 563–571 (1995).

Paxman, R. G.

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 1, p. 446.

Rhoadarmer, T. A.

Roddier, N.

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

N. Roddier, “Atmospheric wavefront simulation and Zernike polynomials,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckenridge, ed., Proc. SPIE, 1237, 668–679 (1990).

Roggermann, M. C.

Seldin, J. H.

Shilov, G. E.

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1, p. 363.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

Takato, N.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

Tatarskii, V. I.

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

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.

Thelen, B. J.

Welsh, B. M.

Yamaguchi, I.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

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

Opt. Lett. (1)

Pub. Astron. Soc. Pac. (1)

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Pub. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

Rev. Mex. Fis. (1)

D. Kouznetsov, R. Ortega-Martinez, “Simulation of random field with given structure function,” Rev. Mex. Fis. 41, 563–571 (1995).

Other (9)

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 360.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 1, p. 446.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation in theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1993), Vol. 32, pp. 203–266.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing Pacific Grove, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).

N. Roddier, “Atmospheric wavefront simulation and Zernike polynomials,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckenridge, ed., Proc. SPIE, 1237, 668–679 (1990).

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1, p. 363.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

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

R. E. Hufnagel, “Variation of atmospheric turbulence,“ in Digest of OSA Topical Meeting Optical Propagation through Turbulence, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

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Figures (3)

Fig. 1
Fig. 1

Theoretical (solid curve) and simulated (dashed curve) structure functions of the phase.

Fig. 2
Fig. 2

Theoretical (solid curve) and simulated (dashed curve) log-amplitude structure functions.

Fig. 3
Fig. 3

Theoretical (solid curve) and simulated (dashed curve) log-amplitude–phase structure functions.

Equations (29)

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fx=n=1M Fpncospn·x+ϕn,gx=n=1M Gpncospn·x+ϕn+ψn,
Bfρ=Bfeρ=k,l=1MFpkFpl×cospk·x1+ϕkcospl·x2+ϕl,Bgρ=Bgeρ=k,l=1MGpkGpl×cospk·x1+ϕk+ψkcospl·x2+ϕl+ψl,Bfgρ=Bfgeρ=k,l=1MFpkGpl×cospk·x1+ϕkcospl·x2+ϕl+ψl,
Bfρ=12n=1MF2pncospn·ρ+12n=1MF2pncospn·x1+x2+2ϕn,Bgρ=12n=1MG2pncospn·ρ+12n=1MG2pncospn·x1+x2+2ϕn+2ψn,Bfgρ=12n=1MFpnGpncospn·ρcosψn+12n=1MG2pncospn·x1+x2+2ϕn+ψn.
Bfρ=12n=1MF2pncospn·ρ,Bgρ=12n=1MG2pncospn·ρ,Bfgρ=12n=1MFpnGpncospn·ρcosψn.
Hκ=12π  d2ρ hρexpiκ·ρ.
Wfκ=πn=1MF2pnδpn-κ+δpn+κ,Wgκ=πn=1MG2pnδpn-κ+δpn+κ,Wfgκ=πn=1MFpnGpnδpn-κ+δpn+κcosψn,
d2ρ expiκ·ρ=4π2δκ.
ξu= duξuϒu,
Wfκ=πn=1M  d2PnΩpnF2pnδpn-κ+δpn+κ,Wgκ=πn=1M  d2PnΩpnG2pnδpn-κ+δpn+κ,Wfgκ=πn=1M  d2Pn  dψnηψn,pnΩpncosψn×FpnGpnδpn-κ+δpn+κ,
Wfκ=2πMΩκF2κ,Wgκ=2πMΩκG2κ,Wfgκ=2πMΩκFκGκ  dψηψ, κcosψ,
Fκ=Wfκ2πMΩκ,Gκ=Wgκ2πMΩκ,  dψηψ, κcosψ=WfgκWfκWgκ.
ηψ, κ=12πακexp-ψ24ακ,ακ=logWfκWgκWfgκ.
μκ= μκ Ωκd2κ=2π K1K2dκ κ μκ Ωκ.
K1K2dκ κ μκΩκ=logK1/K0logK2/K0dy μK0 expy×ΩK0 expyK0 expy2.
Ωκ=12πκ2 logK2/K1.
Bsρ=2π2k2×0.033 0dzCn2z0dp p-8/3J0pρ×1+coszkp2,
Bχρ=2π2k2×0.033 0dzCn2z 0dp p-8/3J0pρ×1-coszkp2,
BχSρ=2π2k2×0.033 0dzCn2z0dp p-8/3×J0pρsinzk p2,
Dρ=2B0-Bρ.
Cn2z=C0r0-5/3k-2zz010 exp-zz1+exp-zz2,
WSqz2k=B0q-11/3μz1z01010!1+cos11 arctanμq2μ2q4+111/2+1+cosarctanq2q4+11/2,
Wχqz2k=B0q-11/3μz1z01010!1-cos11 arctanμq2μ2q4+111/2+1-cosarctanq2q4+11/2,
WχSqz2k=B0q-11/3μz1z01010!sin11 arctanμq2μ2q4+111/2+sinarctanq2q4+11/2,
Iρ, z=0dp p J0pρfp,z.
J0pρ=12π02πdφ exp-ip·ρ,
Iρ, z=12π  d2p fp, zexp-ip·ρ.
Ĩκ, z=12π d2p Iρ, zexpiκ·ρ=12π2  d2p fp, z  d2ρ expiρ·κ-p.
Ĩκ, z=fκ, z.
0dx xα-1 exp-pxsinbxcosbx=Γαb2+p2α/2sinccosc,c=α arctanb/p,

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