Abstract

The end-to-end performance achieved by an adaptive optical (AO) imaging system is determined by a combination of the residual time-varying phase distortions associated with atmospheric turbulence and the quasi-static unsensed and uncorrectable aberrations in the optical system itself. Although the effects of these two errors on the time-averaged Strehl ratio and the time-averaged optical transfer function (OTF) of the AO system are not formally separable, such an approximation is found to be accurate to within a few percent for a range of representative residual wave-front errors. In these calculations, we combined static optical system aberrations and time-varying residual phase distortion characteristics of a deformable mirror fitting error, wave-front sensor noise, and anisoplanatism. The static aberrations consist of focus errors of varying magnitudes as well as a combination of unsensed and uncorrectable mirror figure errors derived from modeling by the Gemini 8-Meter Telescopes Project. The overall Strehl ratios and OTF’s that are due to the combined effect of these error sources are well approximated as products of separate factors for the static and time-varying aberrations, as long as the overall Strehl ratio that is due to both errors is greater than approximately 0.1. For lower Strehl ratios, the products provide lower bounds on the actual values of the Strehl ratio and the OTF. The speckle transfer function is also well approximated by a product of two functions, but only where AO compensation is sufficiently good that speckle imaging techniques are usually not required.

© 1999 Optical Society of America

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References

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  4. M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. F. Rigaut, E. Gendron, “Dual adaptive optics, a solution to the tilt determination problem using laser guide star,” in Laser Guide Star Adaptive Optics Workshop, R. Fugate, ed. (Starfire Optical Range, Albuquerque, N.M., 1992), pp. 582–590.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. D. W. Tyler, B. L. Ellerbroek, “Sky coverage calculations for spectrometer slit power coupling with adaptive optics compensation,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 201–209 (1998).
    [CrossRef]
  17. G. A. Tyler, D. L. Fried, “Image position error associated with a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
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  18. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
    [CrossRef]

1998 (2)

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North Telescope,” Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

D. W. Tyler, B. L. Ellerbroek, “Spectrometer slit-power-coupling calculations for natural and laser guide-star adaptive optics,” Appl. Opt. 37, 4569–4576 (1998).
[CrossRef]

1997 (2)

1994 (1)

1993 (1)

T. Nakajima, C. A. Haniff, “Partial adaptive compensation and passive interferometry with large ground-based telescopes,” Astron. Soc. Pac. 105, 509–520 (1993).
[CrossRef]

1992 (1)

1982 (1)

1964 (1)

Born, M.

M. Born, E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics, 6 ed. (Pergamon, Sydney, Australia, 1989), pp. 459–490.

Delabre, B.

N. Hubin, B. Theodore, P. Petitjean, B. Delabre, “Adaptive optics system for the Very Large Telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 34–45 (1994).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North Telescope,” Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

D. W. Tyler, B. L. Ellerbroek, “Spectrometer slit-power-coupling calculations for natural and laser guide-star adaptive optics,” Appl. Opt. 37, 4569–4576 (1998).
[CrossRef]

F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of adaptive-optics technologies for large astronomical telescopes,” Appl. Opt. 36, 2856–2864 (1997).
[CrossRef] [PubMed]

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]

D. W. Tyler, B. L. Ellerbroek, “Sky coverage calculations for spectrometer slit power coupling with adaptive optics compensation,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 201–209 (1998).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Fried, D. L.

Gendron, E.

F. Rigaut, E. Gendron, “Dual adaptive optics, a solution to the tilt determination problem using laser guide star,” in Laser Guide Star Adaptive Optics Workshop, R. Fugate, ed. (Starfire Optical Range, Albuquerque, N.M., 1992), pp. 582–590.

Goodman, J. W.

J. W. Goodman, “Frequency analysis of optical imaging systems,” in Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 101–133.

Haniff, C. A.

T. Nakajima, C. A. Haniff, “Partial adaptive compensation and passive interferometry with large ground-based telescopes,” Astron. Soc. Pac. 105, 509–520 (1993).
[CrossRef]

Hubin, N.

N. Hubin, B. Theodore, P. Petitjean, B. Delabre, “Adaptive optics system for the Very Large Telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 34–45 (1994).
[CrossRef]

Hufnagel, R. E.

Mahajan, V. N.

V. N. Mahajan, “Random aberrations,” in Aberration Theory Made Simple, vol. TT6 of SPIE Tutorial Text Series (SPIE, Bellingham, Wash., 1991).

Maitre, H.

Matson, C. L.

Nakajima, T.

T. Nakajima, C. A. Haniff, “Partial adaptive compensation and passive interferometry with large ground-based telescopes,” Astron. Soc. Pac. 105, 509–520 (1993).
[CrossRef]

Northcott, M. J.

Pennington, T. L.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Petitjean, P.

N. Hubin, B. Theodore, P. Petitjean, B. Delabre, “Adaptive optics system for the Very Large Telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 34–45 (1994).
[CrossRef]

Rigaut, F.

Roggemann, M. C.

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Rouan, D.

Stanley, N. R.

Theodore, B.

N. Hubin, B. Theodore, P. Petitjean, B. Delabre, “Adaptive optics system for the Very Large Telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 34–45 (1994).
[CrossRef]

Tyler, D. W.

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North Telescope,” Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

D. W. Tyler, B. L. Ellerbroek, “Spectrometer slit-power-coupling calculations for natural and laser guide-star adaptive optics,” Appl. Opt. 37, 4569–4576 (1998).
[CrossRef]

D. W. Tyler, B. L. Ellerbroek, “Sky coverage calculations for spectrometer slit power coupling with adaptive optics compensation,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 201–209 (1998).
[CrossRef]

Tyler, G. A.

Veran, J. P.

Welsh, B. M.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics, 6 ed. (Pergamon, Sydney, Australia, 1989), pp. 459–490.

Appl. Opt. (2)

Astron. Soc. Pac. (2)

B. L. Ellerbroek, D. W. Tyler, “Adaptive optics sky coverage calculations for the Gemini-North Telescope,” Astron. Soc. Pac. 110, 165–185 (1998).
[CrossRef]

T. Nakajima, C. A. Haniff, “Partial adaptive compensation and passive interferometry with large ground-based telescopes,” Astron. Soc. Pac. 105, 509–520 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Other (9)

L. M. Stepp, ed., Advanced Technology Optical Telescopes V, SPIE Proc.2199, (1994).

M. A. Ealey, F. Merkle, eds., Adaptive Optics in Astronomy, SPIE Proc.2201, (1994).

J. W. Goodman, “Frequency analysis of optical imaging systems,” in Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 101–133.

D. W. Tyler, B. L. Ellerbroek, “Sky coverage calculations for spectrometer slit power coupling with adaptive optics compensation,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 201–209 (1998).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

F. Rigaut, E. Gendron, “Dual adaptive optics, a solution to the tilt determination problem using laser guide star,” in Laser Guide Star Adaptive Optics Workshop, R. Fugate, ed. (Starfire Optical Range, Albuquerque, N.M., 1992), pp. 582–590.

N. Hubin, B. Theodore, P. Petitjean, B. Delabre, “Adaptive optics system for the Very Large Telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 34–45 (1994).
[CrossRef]

M. Born, E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics, 6 ed. (Pergamon, Sydney, Australia, 1989), pp. 459–490.

V. N. Mahajan, “Random aberrations,” in Aberration Theory Made Simple, vol. TT6 of SPIE Tutorial Text Series (SPIE, Bellingham, Wash., 1991).

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Figures (13)

Fig. 1
Fig. 1

Representative unsensed and uncorrectable mirror figure errors for the Gemini-North Telescope. This phase error profile yields a Strehl ratio of approximately 0.66 at a wavelength of 1.65 µm, which corresponds roughly to the Gemini specification for mirror figure errors and misalignments. The profile is a sum of contributions that are due to primary and secondary mirror polishing and mounting errors, together with noncommon path aberrations between the WFS and the science instrument. The mirror figure errors have been studied in considerable detail by Gemini, but the noncommon path errors used here are merely typical values selected to match the overall Strehl ratio given above. A more benign profile corresponding to an overall Strehl ratio of approximately 0.82 was also considered.

Fig. 2
Fig. 2

Strehl ratio calculations for representative AO system parameters and wave-front errors. These curves plot the exact Strehl ratio and its multiplicative approximation computed for the AO system parameters and static and varying wave-front errors summarized for cases 1 through 3 of Table 1. (a) Results for wave-front errors dominated by DM fitting error with d/ r 0= 1 (top curve), 2, and 4 (bottom curve). (b) The case of WFS noise with σθ/(λ/d) = 0 (top curve), 0.1, 0.2, and 0.4 (bottom curve). (c) Anisoplanatic wave-front errors with θ/θ0 = 0 (top curve), 0.5, 1, and 2 (bottom curve). In (a), the results for the multiplicative approximation overlay the exact Strehl ratios to within the resolution of the plot. See the footnote to Table 1 for the definitions of these variables.

Fig. 3
Fig. 3

OTF reduction that is due to a static focus aberration of 0.7 rad rms.

Fig. 4
Fig. 4

OTF calculations for representative AO system parameters and wave-front errors. These OTF’s correspond to the AO system parameters and turbulence-induced wave-front errors described in the caption to Fig. 2, combined with a static focus error with a magnitude of 0.7 rad rms.

Fig. 5
Fig. 5

STF calculations for wave-front fitting error with d/ r 0 = 1, 2, and 4, and the AO system parameters listed for case 1 of Table 1. Exact STF’s and multiplicative approximations are shown for a static focus error of (a) 0.2 rad rms, (b) 0.7 rad rms, and (c) 1 rad rms.

Fig. 6
Fig. 6

STF calculations for anisoplanatic wave-front errors with θ/θ0 = 0.5, 1, and 2, and the AO system parameters listed for case 2 of Table 1.

Fig. 7
Fig. 7

STF’s for diffraction-limited imaging, residual atmospheric wave-front error only, and static focus wave-front error only cases. The atmospheric wave-front error is due to anisoplanatism with (a) θ/θ0 = 1 and (b) θ/θ0 = 2. AO system parameters are as listed for case 3 of Table 1. The static error is due to a 0.7-rad rms focus error. Exact and approximate STF’s are also shown for the combination of the two effects.

Fig. 8
Fig. 8

OTF calculations for Gemini-North AO system parameters and wave-front errors. These OTF’s correspond to the Gemini AO system parameters and turbulence-induced phase errors summarized in the final column of Table 1, combined with typical and best-case unsensed and uncorrectable optical system aberrations. The wavelength is 1.65 µm. The OTF’s are well approximated by the product of terms computed separately for the two error sources.

Fig. 9
Fig. 9

STF’s for diffraction-limited imaging, residual atmospheric wave-front error only, and static telescope wave-front error only cases. The atmospheric wave-front error is due to nominal residual phase at 1.2 µm for the Gemini AO system parameters. The static error is due to the typical case Gemini optical aberrations. Exact and approximate STF’s are also shown for the combination of the two effects. Because the Gemini aberrations are not symmetric, all plots are radial averages of two-dimensional distributions.

Fig. 10
Fig. 10

Impact of uncorrectable telescope aberrations on sky coverage in terms of Strehl ratio for Gemini-North at an observing wavelength of 1.65 µm. Here sky coverage is defined as the fraction of the sky close enough to a sufficiently bright guide star for the AO system to deliver the indicated Strehl ratio. The uncorrectable telescope aberrations used for these calculations are as illustrated in Fig. 1, and the AO system and observing parameters are listed in column 4 of Table 2. Information on the guide star density model and tip and tilt sensor radiometry parameters can be found in a previous paper.5

Fig. 11
Fig. 11

Impact of uncorrectable telescope aberrations on slit power coupling sky coverage for Gemini-North. This figure is similar to Fig. 10, except that AO system performance is described in terms of the fraction of energy from a point source that is transmitted through a 0.1-arc sec spectrometer slit.

Fig. 12
Fig. 12

Effect of approximations on the estimated Strehl ratio sky coverage for Gemini-North. This figure illustrates how sky coverage predictions for Gemini-North are effected if the end-to-end performance of the AO system is estimated using approximations (2.8) or approximations (2.10). The observing, AO system, and guide star parameters used for these calculations are the same as for Fig. 9, along with the larger of the two values for uncorrectable telescope aberrations.

Fig. 13
Fig. 13

Effect of approximations on estimated slit power coupling sky coverage for Gemini-North. This figure is similar to Fig. 12, except that the AO system performance is described in terms of the fraction of energy from a point source that is transmitted through a 0.1-arc sec spectrometer slit. The results labeled multiplicative Strehl approximation were obtained by multiplying the slit power coupling without telescope aberrations by the Strehl ratio that were due to these aberrations alone.

Tables (2)

Tables Icon

Table 1 Parameter Values for OTF and Strehl Ratio Calculationsa

Tables Icon

Table 2 Strehl Ratio Results for Gemini-North Telescope and AO System Parametersa

Equations (14)

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OTFκ= dr UrU*r-λκ dr |Ur|2,
Ur=Prexpiϕvr,
Pr=P0rexpiϕsr,
OTFκ= dr P0rP0r-λκexpiϕsr-ϕsr-λκexpiϕvr-ϕvr-λκ drP02r
expiϕvr-ϕvr-λκ=exp-12 ϕvr-ϕvr-λκ2=exp-12 Dr, r-λκ,
OTFκ= dr P0rP0r-λκexpiϕsr-ϕsr-λκexp-12 Dr, r-λκ dr P02r.
S= dκ OTFκ.
OTFκϕs,D  OTFκ0,DOTFκϕs,0OTFκ0,0
S1-σ2,
Sϕs,D  1 - σv2 + σf2 exp-σv2 + σf2 S0,DSϕs,0,
S= dκ OTFκ=exp-σv2 dr P0rP0r-λκexpiϕsr-ϕsr-λκ dr P02r=S0,DSϕs,0,
STFκ=|OTFκ|2= dr  dr P0rP0r-λκ×P0rP0r-λκ×exp-iϕsr-ϕsr-λκ-ϕsr+ϕsr-λκ×exp-iϕvr-ϕvr-λκ-ϕvr+ϕvr-λκ.
STFκϕs,D  STFκ0,DSTFκϕs,0STFκ0,0.
PSFθ, R=PSFho * PSFttθ, R.

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