Abstract

The propagation of an optical beam through atmospheric turbulence produces wave-front aberrations that can reduce the power incident on an illuminated target or degrade the image of a distant target. The purpose of the work described here was to determine by computer simulation the statistical properties of the normalized on-axis intensity—defined as (D/r0)2 SR—as a function of D/r0 and the level of adaptive optics (AO) correction, where D is the telescope diameter, r0 is the Fried coherence diameter, and SR is the Strehl ratio. Plots were generated of (D/r0)2 〈SR〉 and σSR/SR, where 〈SR〉 and σSR are the mean and standard deviation, respectively, of the SR versus D/r0 for a wide range of both modal and zonal AO correction. The level of modal correction was characterized by the number of Zernike radial modes that were corrected. The amount of zonal AO correction was quantified by the number of actuators on the deformable mirror and the resolution of the Hartmann wave-front sensor. These results can be used to determine the optimum telescope diameter, in units of r0, as a function of the AO design. For the zonal AO model, we found that maximum on-axis intensity was achieved when the telescope diameter was sized so that the actuator spacing was equal to approximately 2r0. For modal correction, we found that the optimum value of D/r0 (maximum mean on-axis intensity) was equal to 1.79Nr+2.86, where Nr is the highest Zernike radial mode corrected.

© 2004 Optical Society of America

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References

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  1. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  2. J. Y. Wang, “Optical resolution through a turbulent medium with adaptive phase compensations,” J. Opt. Soc. Am. 67, 383–391 (1977).
    [CrossRef]
  3. J. B. Shellan, “An examination of the properties of partially corrected target illuminator beam following transmission through a turbulent atmosphere,” (the Optical Sciences Company, Anaheim, Calif., 2003).
  4. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  5. J. B. Shellan, “Turbulence phase screen generation using a technique based on FFT-KL-Legendre polynomial analysis,” (the Optical Sciences Company, Anaheim, Calif., 2000).
  6. G. M. Cochran, “Phase screen generation,” (the Optical Sciences Company, Anaheim, Calif., 1985).
  7. B. J. Herman, L. A. Stugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 183–192 (1990).
    [CrossRef]
  8. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  9. E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 435–478.
  11. M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 98–103.
  12. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  13. E. P. Wallner, “Optimal wavefront correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  14. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comp. Elec. Eng., 18, 451–466 (1992).
    [CrossRef]
  15. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).
  16. In our computations the mean phase gradient was computed over all SAs that were at least 50% illuminated. For cases where the SA was fully illuminated, this computation is equivalent to taking the difference between the mean phases along opposite SA edges (average G tilt). The matrix inversion needed for the least-squares reconstructor was implemented by Matlab’s pseudoinverse algorithm pinv.
  17. The computer runs described in this paper took approximately two full weeks to complete on a 1.2-GHz machine, so it was impractical to increase significantly the number of phase-screen realizations or the phase grid point resolution.

1992

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

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comp. Elec. Eng., 18, 451–466 (1992).
[CrossRef]

1990

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

1983

1977

1976

1966

Cochran, G. M.

G. M. Cochran, “Phase screen generation,” (the Optical Sciences Company, Anaheim, Calif., 1985).

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.

Gavel, D. T.

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[CrossRef]

Glindemann, A.

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, Statistical Optics (Wiley, New York, 1985), pp. 435–478.

Herman, B. J.

B. J. Herman, L. A. Stugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 183–192 (1990).
[CrossRef]

Johansson, E. M.

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[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]

Noll, R. J.

Roddier, N.

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

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comp. Elec. Eng., 18, 451–466 (1992).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 98–103.

Shellan, J. B.

J. B. Shellan, “Turbulence phase screen generation using a technique based on FFT-KL-Legendre polynomial analysis,” (the Optical Sciences Company, Anaheim, Calif., 2000).

J. B. Shellan, “An examination of the properties of partially corrected target illuminator beam following transmission through a turbulent atmosphere,” (the Optical Sciences Company, Anaheim, Calif., 2003).

Stugala, L. A.

B. J. Herman, L. A. Stugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 183–192 (1990).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

Wallner, E. P.

Wang, J. Y.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 98–103.

Comp. Elec. Eng.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comp. Elec. Eng., 18, 451–466 (1992).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

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

Waves Random Media

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

Other

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE2200, 372–383 (1994).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 435–478.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996), pp. 98–103.

J. B. Shellan, “Turbulence phase screen generation using a technique based on FFT-KL-Legendre polynomial analysis,” (the Optical Sciences Company, Anaheim, Calif., 2000).

G. M. Cochran, “Phase screen generation,” (the Optical Sciences Company, Anaheim, Calif., 1985).

B. J. Herman, L. A. Stugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 183–192 (1990).
[CrossRef]

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

In our computations the mean phase gradient was computed over all SAs that were at least 50% illuminated. For cases where the SA was fully illuminated, this computation is equivalent to taking the difference between the mean phases along opposite SA edges (average G tilt). The matrix inversion needed for the least-squares reconstructor was implemented by Matlab’s pseudoinverse algorithm pinv.

The computer runs described in this paper took approximately two full weeks to complete on a 1.2-GHz machine, so it was impractical to increase significantly the number of phase-screen realizations or the phase grid point resolution.

J. B. Shellan, “An examination of the properties of partially corrected target illuminator beam following transmission through a turbulent atmosphere,” (the Optical Sciences Company, Anaheim, Calif., 2003).

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Figures (10)

Fig. 1
Fig. 1

Mean normalized on-axis intensity In=(D/r0)2 〈SR〉 is plotted as a function of D/r0 and the highest Zernike radial degree that has been corrected, Nr. Results are shown for Nr=020 (bottom to top).

Fig. 2
Fig. 2

Expanded view of Fig. 1 for Nr=120. It can be used to determine the optimum value of a primary mirror diameter in units of r0 for a specified level of AO modal correction. The result for Nr=1 (tilt correction only) is well known [(D/r0)opt 4.27].

Fig. 3
Fig. 3

Peak value of In, Inpeak, for each curve in Figs. 1 and 2 is plotted as a function of Nr. The plot shows, for example, that for Nr=20 (231 Zernike modes corrected) the mean on-axis intensity is approximately 450 times larger than that for an uncorrected beam.

Fig. 4
Fig. 4

The value of D/r0 corresponding to Inpeak in Fig. 2 is plotted as a function of Nr. The relationship is nearly linear and is well represented by (D/r0)opt1.79Nr+2.86.

Fig. 5
Fig. 5

σSR/SR, as estimated by using 1000 independent SR realizations, is plotted as a function of D/r0 for Nr=020 (top to bottom).

Fig. 6
Fig. 6

Similar to Fig. 1, but for the case of zonal AO correction. The mean normalized on-axis intensity, In=(D/r0)2SR, is plotted as a function of D/r0 and the number of actuators spanning the pupil diameter na. The number of subapertures in the Hartmann WFS spanning the pupil diameter is equal to na-1.

Fig. 7
Fig. 7

Expanded view of Fig. 6; it can be used to estimate the optimum value of a primary mirror diameter in units of r0 for a specified WFS–DM configuration.

Fig. 8
Fig. 8

Inpeak plotted as a function of na. The plot shows, for example, that if there are 32 actuators spanning the pupil diameter and the optimum primary mirror diameter is used, then the on-axis intensity will be ≈1000 times greater than that for an uncorrected beam.

Fig. 9
Fig. 9

The value of D/r0 corresponding to Inpeak in Fig. 7 is plotted as a function of na. The relationship is nearly linear and is well represented by (D/r0)opt2.08na-2.88.

Fig. 10
Fig. 10

σSR/SR as estimated by using 1000 independent realizations of the SR, plotted as a function of D/r0 and na. Results are shown for 2, 4, 6, 8, 12, 16, 20, 25, and 32 (top to bottom) actuators spanning the pupil diameter. When evaluated at (D/r0)opt, σSR/SR10% for na12.

Equations (17)

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

In=(D/r0)2SR.
RealImagϕs,t=RealImagm=1Nyn=1NxΦm,n1/2[qm,n(r)+iqm,n(i)]×exp[i2π(m-1)(s-1)/Ny]×exp[i2π(n-1)(t-1)/Nx].
qm,n(r)=qm,n(i)=0,
qm,n(r)qm,n(r)=qm,n(i)qm,n(i)=δm,mδn,n,
qm,n(r)qm,n(i)=0,
δm,m=1ifm=m0otherwise.
ϕs1,t1ϕs2,t2=12m=1Nyn=1NxΦm,n(exp{i2π[(m-1)×(s2-s1)/Ny+(n-1)×(t2-t1)/Nx]}+exp{-i2π[(m-1)(s2-s1)/Ny+(n-1)(t2-t1)/Nx]}).
Φ(κ)d2κ=0.02288r0-5/3d2κκ11/3,
r0Friedcoherencediameter.
Φm,n=0.02288ML(Lx/r0)5/3[ML2κm2+κn2]-11/6,
MLLx/Ly=Nx/Ny,
κmm-1form=1to 12Ny-Ny-1+mform=12Ny+1toNy,
κnn-1forn=1to 12Nx-Nx-1+nforn=12Nx+1toNx.
Φ0,0=0.
SR=1Nn=1Nexp(iϕn)2.
(D/r0)opt1.79Nr+2.86.
(D/r0)opt2.08na-2.88.

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