Abstract

A previously presented method for modeling Kolmogorov phase fluctuations over a fin!ite aperture is both formalized and improved on. The method relies on forming an initial low-resolution Kolmogorov phase screen from direct factorization of a covariance. The resolution of the screen is then increased by a randomized interpolation to produce a Kolmogorov phase screen of the desired size. The computational requirement is asymptotically proportional to the number of points in the phase screen.

© 1999 Optical Society of America

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References

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  1. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  2. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  3. M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  4. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  5. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  6. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
    [CrossRef]
  7. E. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  8. H. Stark, J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).
  9. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  10. D. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
  11. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
    [CrossRef]

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)

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

1983 (1)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

1965 (1)

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Dainty, J. C.

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

Fried, D. L.

Glindemann, A.

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

Lane, R. G.

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

Luenberger, D.

D. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

McGlamery, B. L.

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

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Roddier, F.

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

Roddier, N.

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

Roggemann, M. C.

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

Stark, H.

H. Stark, J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).

Wallner, E.

Welsh, B.

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

Woods, J. W.

H. Stark, J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).

J. Opt. Soc. Am. (2)

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

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[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 (6)

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

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

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

H. Stark, J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).

D. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

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

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Figures (12)

Fig. 1
Fig. 1

Interpolation steps for generating a higher-resolution phase screen where the known initial points are marked by X’s. (a) First step of interpolation. The points determined at this stage are marked by circles. (b) Second step of interpolation. The points determined at this stage are marked by asterisks. (c) Third step of interpolation. The points determined are marked by pluses. In this case is reduced by a factor of 2, because the spacing is equivalently reduced.

Fig. 2
Fig. 2

Interpolation steps for generating a higher-resolution phase screen where the initial low-resolution points are marked by X’s and l is the size of the screen. (a) Circles and asterisks are the interpolated points. (b) The effect of removing the edge points for a 2 × 2 interpolator and rescaling the phase screen.

Fig. 3
Fig. 3

Illustration of the possible choices of 4 × 4 regions of low-resolution points, marked by X’s, capable of interpolating a high-resolution point (circle). The choice marked by the solid line is used except during interpolation near edges.

Fig. 4
Fig. 4

Residual variance on the estimated screen points for a 4 × 4 interpolator, = 1. The residual is zero at the initial screen points, since these are exact.

Fig. 5
Fig. 5

Interpolators change as their size increases; however, after 4 × 4 they are similar, and little is to be gained by increasing past this size. (a) 2 × 2 interpolator, (b) 4 × 4 interpolator, (c) 6 × 6 interpolator.

Fig. 6
Fig. 6

4 × 4 interpolators used for the (a) first and (b) second levels of interpolation. The structure of the filter depends on the spatial relationship between the known points and the point to be estimated as in Figs. 1(a) and 1(c).

Fig. 7
Fig. 7

Same initial phase screen as number of successive iterations is increased. (a) Initial 15 × 15 exact screen, (b) one iteration, (c) two iterations, (d) five iterations.

Fig. 8
Fig. 8

For each speckle a D/ r 0 of 10 was used, for a phase screen of (a) three iterations, (b) four iterations, (c) five iterations.

Fig. 9
Fig. 9

Phase structure functions for the phase screens generated (solid curves) are compared with the ideal phase structure function (dashed curves) as given by Eq. (5) (a) for the 2 × 2 interpolator and (b) for the 4 × 4 interpolator. In both cases edges are removed at each iteration.

Fig. 10
Fig. 10

Number of standard deviations from ideal structure functions for structure functions generated from an ensemble of 100 for (a) a 2 × 2 interpolator and (b) a 4 × 4 interpolator. Note that the structure functions for the 4 × 4 interpolator were generated from an initial 15 × 15 phase screen with six iterations, whereas for the 2 × 2 interpolator only four iterations were used. This is due to the greater edge effects in the 4 × 4 case.

Fig. 11
Fig. 11

Phase structure functions for the phase screens (solid curves) generated with different techniques for the edge points are compared with the ideal phase structure function (dashed curves) as given by Eq. (5). A 2 × 2 interpolator was used for these results. (a) Edge points are interpolated, (b) edge points are removed.

Fig. 12
Fig. 12

CPU time to produce phase screens as the number of samples generated increases.

Equations (29)

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Pk, l=-+-+ ψu, vexp-i2πuk+vldudv,
ψu, v=-+-+ Pk, lexpi2πuk+vldkdl.
Dpu1, v1, u2, v2=Dpu1-u2, v1-v2,
=ψu1, v1-ψu2, v22,
=6.88|u1-u2, v1-v2|r05/3,
WAu, v=cover the region of simulation0outside the region of simulation,
 WAu, vdudv=1.
ϕu, v=ψu, v- WAu, vψu, vdudv.
Cu1, v1, u2, v2=ϕu1, v1ϕu2, v2= ψu1, v1-ψu1, v1×WAu1, v1du1dv1 ψu2, v2-ψu2, v2WAu2, v2du2dv2= ψu1, v1-ψu1, v1ψu2, v2-ψu2, v2WAu1, v1×WAu2, v2du1du2dv1dv2.
Dpu1, v1, u2, v2=ψu1, v12-2ψu1, v1×ψu2, v2+ψu2, v22
Cu1, v1, u2, v2=-12 Dpu1, v1, u2, v2+12  Dpu1, v1, u2, v2×WAu1, v1du1dv1+12  Dpu1, v1, u2, v2×WAu2, v2du2dv2-12  Dpu1, v1, u2, v2WAu1, v1×WAu2, v2du1dv1du2dv2
ΦmN+n=RϕmΔu, nΔv,
CS=ΦΦT,
CS=UΛUT,
Λ=λ1000λ200λN2
Φ=Ux,
ΦΦT=UxxTUT=C.
ϕui, vi=1/4ϕu1, v1+ϕu2, v2+ϕu3, v3+ϕu4, v4+i.
ϕue, ve=1/2ϕu1, v1+ϕu2, v2+γe,
Φh=k=1M2 akUku, v,
Φh=UA,
Φl=ΘA,
Aˆ=ΘTΛ-1Θ-1ΘTΛ-1Φl,
Residual covariance=EΨ-ΨˆΨ-ΨˆT
=UΛUT-2UΛΘTΔTUT+UΔΘΔTUT,
I2×2=0.2500.250100.2500.25.
I4×4=-0.00170-0.03410-0.03410-0.00170000000-0.034100.319800.31980-0.03410001000-0.034100.319800.31980-0.03410000000-0.00170-0.03410-0.03410-0.0017,
σu, v=1N-1  Dpiu, v-1N Dpiu, v21/2.
Dpu, v-1N  Dpiu, vσu, v.

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