Abstract

We derive the Strehl ratio for a segmented mirror telescope as a function of the rms segment phase error and the observing wavelength, with and without the effects of the atmosphere. A simple analytical expression is given for the atmosphere-free case. Although our specific results are in the context of the Keck telescope, they are presented in a way that should be readily adaptable to other segmented geometries. We also derive the corresponding modulation transfer functions. These results are useful in determining how accurately a segmented mirror telescope needs to be phased for a variety of observing applications.

© 1999 Optical Society of America

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References

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  1. G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
    [CrossRef]
  2. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1384 (1966).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969).
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).
  6. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  7. P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
    [CrossRef]
  8. J. E. Nelson, P. R. Gillingham, “Overview of the performance of the W. M. Keck Observatory,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 92–93 (1994).
  9. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, New York, 1980).

1998 (1)

1976 (1)

1966 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969).

Chanan, G.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Chanan, G. A.

G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

Dekens, F. G.

DiVittorio, M.

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

Djorgovski, G.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Fried, D. L.

Gillingham, P. R.

J. E. Nelson, P. R. Gillingham, “Overview of the performance of the W. M. Keck Observatory,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 92–93 (1994).

Gleckler, A.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, New York, 1980).

Kirkman, D.

Kulkarni, S.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Mast, T.

G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Mast, T. S.

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

Max, C.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Michaels, S.

Nelson, J.

G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck Telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Nelson, J. E.

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

J. E. Nelson, P. R. Gillingham, “Overview of the performance of the W. M. Keck Observatory,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 92–93 (1994).

Noll, R. J.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, New York, 1980).

Troy, M.

Wizinowich, P.

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

Wizinowich, P. L.

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969).

G. Chanan, G. Djorgovski, A. Gleckler, S. Kulkarni, T. Mast, C. Max, J. Nelson, P. Wizinowich, “Adaptive Optics for Keck Observatory,” (W. M. Keck Observatory, Kamuela, Hawaii, Jan.1996).

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

P. L. Wizinowich, T. S. Mast, J. E. Nelson, M. DiVittorio, G. A. Chanan, “Optical quality of the W. M. Keck Telescope,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 94–104 (1994).
[CrossRef]

J. E. Nelson, P. R. Gillingham, “Overview of the performance of the W. M. Keck Observatory,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 92–93 (1994).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, New York, 1980).

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Figures (7)

Fig. 1
Fig. 1

MTF for the Keck telescope at a wavelength of 1 µm for various piston errors. Except for the piston errors, the telescope optics are assumed to be perfect and the atmosphere is ignored. Curves are labeled with the rms piston error p rms in micrometers, which is related to the rms phase error by σrms = 2kp rms. The horizontal axis is the spatial frequency in cycles of modulation per arcsecond in the image plane. The x axis is parallel to (any) segment edge; the y axis is perpendicular.

Fig. 2
Fig. 2

Same as Fig. 1, but at a wavelength of 2 µm.

Fig. 3
Fig. 3

Same as Fig. 1, but at a wavelength of 5 µm.

Fig. 4
Fig. 4

Same as Fig. 1, but at a wavelength of 10 µm.

Fig. 5
Fig. 5

Keck Strehl ratio as a function of rms piston error p rms for various wavelengths. Except for the piston errors, the telescope optics are assumed to be perfect. The nominal Keck theoretical atmosphere [r 0(0.5 μ) = 20 cm] has been included in the long-exposure limit. Therefore this figure corresponds to normal observing.

Fig. 6
Fig. 6

Same as Fig. 5, but the atmosphere has been included in the short-exposure limit. Therefore this figure corresponds to observing with fast guiding. The solid curves are exact; the dashed curves utilize the Marechal approximation. The Marechal approximations for λ = 2 µm and λ = 1 µm, which lie far below the corresponding exact curves, have been suppressed for clarity.

Fig. 7
Fig. 7

Same as Fig. 5, but the atmosphere has been neglected entirely. Therefore this figure corresponds approximately to speckle or adaptive optics observing (or exactly to space-based observing). As in Fig. 6, the solid curves are exact and the dashed curves utilize the Marechal approximation.

Equations (29)

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fρ=eiϕinside the aperture0outside the aperture,
fˆw= eikρ·wfρρdρdθ,
Iw=|fˆw|2,
 Iwwdwdψ=λ2A.
τα=1λ2A  e2iπα·wIwwdwdψ,
Iw=λ2A  e-2iπα·wτααdαdϕ.
S=λ2A  τααdαdϕ.
τlα=exp-3.44λαr05/3
τsα=exp-3.44λαr05/31-1.0433λαD1/3,
τcα=2πarccosu-u1-u21/2u10u>1,
Sl=1-1.0324Dr05/3,
Ss=1-0.1346Dr05/3.
fρ=i fiρ,
fiρ=hρ-ρieiδi,
δi=2kpi.
fˆw=hˆwi eikρi·w+δi.
x=32 ks, y=32 ls,
Iw=|hˆw|2n+2 i>jcoskρi-ρj·w×cosδi-δj,
Iw=|hˆw|2n+2e-σ2i>jcoskρi-ρj·w,
S=1+e-σ2n-1n.
limσ Iw=n|hˆw|2,
limσ0 Iw=n2|hˆw|21-σ21-1n.
σn=σn-1n1/2,
hˆ0w=4 032 sdy 0s-33 ydx coskuxcoskvy,
hˆ0u, v=332 s2.
hˆ0u, v=2skvsin3kvs2+433k2v21-cos3kvs2.
hˆ0u, v=43k2u2coskus2-coskus.
hˆ0u, v=3skusinkus+3k2u21-coskus.
hˆ0u, v=23k2ucos3kvs2+kus2-coskusu-3v+cos3kvs2-kus2-coskusu+3v.

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