Abstract

A simulation study is presented that evaluates the performance of Hartmann wave-front sensors with measurements obtained with the Fried geometry and the Hutchin geometry. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and the weaknesses of Hartmann sensors used in each of the two geometries in the two-dimensional space of the Fried parameter r 0 and the Rytov parameter. We found that the performance of Hartmann sensors degrades severely when the Rytov number exceeds 0.2 and the ratio l/ r 0 exceeds 1/4 (where l is the subaperture side length) because of the presence of branch points in the phase function and the effect of amplitude scintillation on the measurement values produced by the Hartmann sensor.

© 2002 Optical Society of America

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References

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  1. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
    [CrossRef]
  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–379 (1977).
    [CrossRef]
  3. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
    [CrossRef] [PubMed]
  4. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  5. D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
    [CrossRef]
  6. V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” National Science Foundation Technical Report TT-68-50464 (National Technical Information Service, Springfield, Va., 1968).
  7. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  8. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  9. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
    [CrossRef]
  10. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001); available (with associated software) from D. L. Fried or J. D. Barchers.
  11. D. L. Fried, “Adaptive optics wave-function/wave-front reconstruction: problems and solutions,” in Proceedings of OSA Annual Meeting (Optical Society of America, Washington, D.C., 2000), invited talk WR-1.
  12. M. C. Roggemann, A. C. Koivunen, “Wave-front sensing and deformable-mirror control in strong scintillation,” J. Opt. Soc. Am. A 17, 911–919 (2000).
    [CrossRef]
  13. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
    [CrossRef]
  14. D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
  15. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  16. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  17. We computed the formulation error in the Hutchin geometry by following the same development as previously published for the formulation error in the Fried geometry.5
  18. G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

2001

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001); available (with associated software) from D. L. Fried or J. D. Barchers.

2000

1998

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

1995

1994

1992

1977

1974

1965

Clifford, S. F.

Ellerbroek, B. L.

Fried, D. L.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001); available (with associated software) from D. L. Fried or J. D. Barchers.

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
[CrossRef]

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[CrossRef]

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

D. L. Fried, “Adaptive optics wave-function/wave-front reconstruction: problems and solutions,” in Proceedings of OSA Annual Meeting (Optical Society of America, Washington, D.C., 2000), invited talk WR-1.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

Hudgin, R. H.

Koivunen, A. C.

Lawrence, R. S.

Ochs, G. R.

Roggemann, M. C.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” National Science Foundation Technical Report TT-68-50464 (National Technical Information Service, Springfield, Va., 1968).

Tyler, G. A.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

Vaughn, J. L.

Welsh, B. M.

Appl. Opt.

Atmos. Oceanic Opt.

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001); available (with associated software) from D. L. Fried or J. D. Barchers.

Other

D. L. Fried, “Adaptive optics wave-function/wave-front reconstruction: problems and solutions,” in Proceedings of OSA Annual Meeting (Optical Society of America, Washington, D.C., 2000), invited talk WR-1.

We computed the formulation error in the Hutchin geometry by following the same development as previously published for the formulation error in the Fried geometry.5

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” National Science Foundation Technical Report TT-68-50464 (National Technical Information Service, Springfield, Va., 1968).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Measurements and points at which the field is to be reconstructed in the (a) Fried geometry and (b) Hutchin geometry. The arrows indicate the measurements produced by the subaperture in which it is located or centered. The squares define the subapertures. For the Hutchin geometry there are two sets of (partially) overlapping subapertures: one set for the x measurements and the other set for the y measurements.

Fig. 2
Fig. 2

Measured log-amplitude variance of the fields used in this paper shown as a function of the Rytov number. As expected, the agreement is excellent for small values of the Rytov number, a slight amplification is observed in the intermediate regime, and saturation occurs for large values of the Rytov number.

Fig. 3
Fig. 3

Estimation accuracy of the Hartmann sensor in the (a) Fried geometry and (b) Hutchin geometry by use of a least-squares (solid curve) or a complex exponential reconstructor (dashed curve) for l/ r 0 = 1/4 (circle), l/ r 0 = 1/2 (square), and l/ r 0 = 1 (left triangle). When l/ r 0 = 1/4, a significant improvement in estimation accuracy is noted by use of the complex exponential reconstructor when the Rytov number is large. However, when l/ r 0 = 1, there is actually a degradation in performance when the complex exponential reconstructor is applied.

Fig. 4
Fig. 4

Estimation accuracy of the Hartmann sensor in the Fried geometry in the absence of intensity fluctuations shown as a function of Rytov number for l/ r 0 = 1/4, 1/2, and 1 in (a), (b), and (c), respectively. Each line corresponds to the ability of either the least-squares (solid curve) or the noise weighted complex exponential reconstructor (dashed curve) to form an estimate of the least-squares (no symbol) or hidden (diamonds) phase at the prime coordinates. Also shown is the Strehl loss that is due to hidden phase S (dash-dot curve). In (a), when coherence length r 0 is much larger than a subaperture, it is clear that the CNW (defined in Subsection 2.B) successfully forms a good estimate of the hidden phase when the Rytov number is large (i.e., many branch points are present). The least-squares reconstructor ignores the hidden phase in the same conditions. However, as the coherence length shrinks relative to a subaperture [(b) and (c)], the nonlinearities in the CNW lead to significantly reduced performance. In fact, by ignoring the hidden phase, the least-squares reconstructor actually forms a better estimate of the hidden phase when l/ r 0 = 1 [(c)].

Fig. 5
Fig. 5

Strehl loss (or gain) that is due to scintillation shown as a function of the Rytov number for l/ r 0 = 1/4 (circle) and l/ r 0 = 1 (left triangle) for the Hartmann sensor in the Fried geometry by use of the (a) least-squares and (b) complex exponential reconstructors. Independent of the reconstructor, while estimation of the least-squares phase (solid curve) is degraded by scintillation, the estimation accuracy of the hidden phase (dashed curve) actually improves with scintillation.

Equations (10)

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

Ûr¯=ΩwfsrecUr¯.
SU1r¯, U2r¯ = kK U1r¯kU2*r¯k2kK U1r¯kU1* r¯kkK U2r¯kU2* r¯k.
sC=dr¯Wr¯Ir¯ϕr¯dr¯Wr¯Ir¯,
sG=dr¯Wr¯ϕr¯dr¯Wr¯.
Sϕ=GTG-1GTPVGϕ,
ϕ=PVargexpiϕexp-iϕ.
Sexpiϕr¯, expi argΩwfsrecexpiϕr¯,
Sexpiϕr¯, expi argΩwfsrecexpiϕr¯
Qscint,=Sexpiϕr¯, expi argΩwfsrecAr¯expiϕr¯Sexpiϕr¯, expi argΩwfsrecexpiϕr¯,
Qscint,=Sexpiϕr¯, expi argΩwfsrecAr¯expiϕr¯Sexpiϕr¯, expi argΩwfsrecexpiϕr¯.

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