Abstract

To achieve its full diffraction limit in the infrared, the primary mirror of the Keck telescope (now telescopes) must be properly phased: The steps or piston errors between the individual mirror segments must be reduced to less than 100 nm. We accomplish this with a wave optics variation of the Shack–Hartmann test, in which the signal is not the centroid but rather the degree of coherence of the individual subimages. Using filters with a variety of coherence lengths, we can capture segments with initial piston errors as large as ±30 μm and reduce these to 30 nm—a dynamic range of 3 orders of magnitude. Segment aberrations contribute substantially to the residual errors of ∼75 nm.

© 1998 Optical Society of America

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References

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  1. G. A. Chanan, J. E. Nelson, T. Mast, “Segment alignment for the Keck telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
    [CrossRef]
  2. G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
    [CrossRef]
  3. G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).
  4. 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, 82–93 (1994).
    [CrossRef]
  5. 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]
  6. G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.
  7. J Nelson, “Segment mechanical phasing,” Keck Observatory Tech. Note 389 (W. M. Keck Library, Kamuela Hawaii, July1993).
  8. J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Report 90 (W. M. Keck Library, Kamuela Hawaii, 1985), pp. 5-1 to 5-44.
  9. R. W. Cohen, T. S. Mast, J. E. Nelson, “Performance of the W. M. Keck telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 105–116 (1994).
    [CrossRef]
  10. G. Chanan, T. Mast, J. Nelson, “Keck telescope primary mirror segments: initial alignment and active control,” in Proceedings of the European Southern Observatory Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (Garching, Germany, 1988), pp. 421–428.
  11. G. A. Chanan, “Design of the Keck Observatory alignment camera,” in Precision Instrument Design, T. C. Bristow, A. E. Hatheway, eds., Proc. SPIE1036, 59–70 (1988).
    [CrossRef]
  12. G. A. Chanan, “Mask/pupil registration for the Keck telescope,” Keck Observatory Tech. Note 253 (W. M. Keck Library, Kamuela Hawaii, 1988).
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969), pp. 395–398.
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.
  15. T Mast, J Nelson, G Chanan, “Sampling segment phases: minimal spanning sets of zero-sum triplets,” Keck Observatory Tech. Note 208 (W. M. Keck Library, Kamuela Hawaii, 1986).
  16. See Ref. 14, pp. 502–515.
  17. T Mast, “Systematic errors in the alignment camera phase measurement,” Keck Observatory Tech. Note 299 (W. M. Keck Library, Kamuela Hawaii, 1990).
  18. See Ref. 14, pp. 52–64, 515–520.
  19. 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]

1966

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969), pp. 395–398.

Chanan, G

T Mast, J Nelson, G Chanan, “Sampling segment phases: minimal spanning sets of zero-sum triplets,” Keck Observatory Tech. Note 208 (W. M. Keck Library, Kamuela Hawaii, 1986).

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Chanan, G.

G. Chanan, T. Mast, J. Nelson, “Keck telescope primary mirror segments: initial alignment and active control,” in Proceedings of the European Southern Observatory Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (Garching, Germany, 1988), pp. 421–428.

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

Chanan, G. A.

G. A. Chanan, J. E. Nelson, T. Mast, “Segment alignment for the Keck telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[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]

G. A. Chanan, “Design of the Keck Observatory alignment camera,” in Precision Instrument Design, T. C. Bristow, A. E. Hatheway, eds., Proc. SPIE1036, 59–70 (1988).
[CrossRef]

G. A. Chanan, “Mask/pupil registration for the Keck telescope,” Keck Observatory Tech. Note 253 (W. M. Keck Library, Kamuela Hawaii, 1988).

Cohen, R. W.

R. W. Cohen, T. S. Mast, J. E. Nelson, “Performance of the W. M. Keck telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 105–116 (1994).
[CrossRef]

Dekens, F

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

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 Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Faber, S. M.

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Report 90 (W. M. Keck Library, Kamuela Hawaii, 1985), pp. 5-1 to 5-44.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.

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, 82–93 (1994).
[CrossRef]

Gleckler, A

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Kulkarni, S

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Leffert, E

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

Mast, T

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

T Mast, J Nelson, G Chanan, “Sampling segment phases: minimal spanning sets of zero-sum triplets,” Keck Observatory Tech. Note 208 (W. M. Keck Library, Kamuela Hawaii, 1986).

T Mast, “Systematic errors in the alignment camera phase measurement,” Keck Observatory Tech. Note 299 (W. M. Keck Library, Kamuela Hawaii, 1990).

Mast, T.

G. Chanan, T. Mast, J. Nelson, “Keck telescope primary mirror segments: initial alignment and active control,” in Proceedings of the European Southern Observatory Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (Garching, Germany, 1988), pp. 421–428.

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

G. A. Chanan, J. E. Nelson, T. Mast, “Segment alignment for the Keck telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[CrossRef]

Mast, T. S.

R. W. Cohen, T. S. Mast, J. E. Nelson, “Performance of the W. M. Keck telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 105–116 (1994).
[CrossRef]

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Report 90 (W. M. Keck Library, Kamuela Hawaii, 1985), pp. 5-1 to 5-44.

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 Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Nelson, J

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

J Nelson, “Segment mechanical phasing,” Keck Observatory Tech. Note 389 (W. M. Keck Library, Kamuela Hawaii, July1993).

T Mast, J Nelson, G Chanan, “Sampling segment phases: minimal spanning sets of zero-sum triplets,” Keck Observatory Tech. Note 208 (W. M. Keck Library, Kamuela Hawaii, 1986).

Nelson, J.

G. Chanan, T. Mast, J. Nelson, “Keck telescope primary mirror segments: initial alignment and active control,” in Proceedings of the European Southern Observatory Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (Garching, Germany, 1988), pp. 421–428.

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

Nelson, J. E.

G. A. Chanan, J. E. Nelson, T. Mast, “Segment alignment for the Keck telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[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, 82–93 (1994).
[CrossRef]

R. W. Cohen, T. S. Mast, J. E. Nelson, “Performance of the W. M. Keck telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 105–116 (1994).
[CrossRef]

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Report 90 (W. M. Keck Library, Kamuela Hawaii, 1985), pp. 5-1 to 5-44.

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]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.

Schaefer, B.

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

Sirko, E

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.

Troy, M

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.

Wizinowich, P

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

Wizinowich, P.

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

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), pp. 395–398.

J. Opt. Soc. Am.

Other

G Chanan, G Djorgovski, A Gleckler, S Kulkarni, T Mast, C Max, J Nelson, P Wizinowicheds., “Adaptive optics for Keck Observatory,” Keck Observatory Report 208 (W. M. Keck Library, Kamuela Hawaii, 1996), pp. 3-26 to 3-36.

J Nelson, “Segment mechanical phasing,” Keck Observatory Tech. Note 389 (W. M. Keck Library, Kamuela Hawaii, July1993).

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Report 90 (W. M. Keck Library, Kamuela Hawaii, 1985), pp. 5-1 to 5-44.

R. W. Cohen, T. S. Mast, J. E. Nelson, “Performance of the W. M. Keck telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 105–116 (1994).
[CrossRef]

G. Chanan, T. Mast, J. Nelson, “Keck telescope primary mirror segments: initial alignment and active control,” in Proceedings of the European Southern Observatory Conference on Very Large Telescopes and their Instrumentation, M.-H. Ulrich, ed. (Garching, Germany, 1988), pp. 421–428.

G. A. Chanan, “Design of the Keck Observatory alignment camera,” in Precision Instrument Design, T. C. Bristow, A. E. Hatheway, eds., Proc. SPIE1036, 59–70 (1988).
[CrossRef]

G. A. Chanan, “Mask/pupil registration for the Keck telescope,” Keck Observatory Tech. Note 253 (W. M. Keck Library, Kamuela Hawaii, 1988).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1969), pp. 395–398.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, New York, 1989), pp. 484–487.

T Mast, J Nelson, G Chanan, “Sampling segment phases: minimal spanning sets of zero-sum triplets,” Keck Observatory Tech. Note 208 (W. M. Keck Library, Kamuela Hawaii, 1986).

See Ref. 14, pp. 502–515.

T Mast, “Systematic errors in the alignment camera phase measurement,” Keck Observatory Tech. Note 299 (W. M. Keck Library, Kamuela Hawaii, 1990).

See Ref. 14, pp. 52–64, 515–520.

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]

G. A. Chanan, J. E. Nelson, T. Mast, “Segment alignment for the Keck telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[CrossRef]

G. Chanan, J. Nelson, T. Mast, P. Wizinowich, B. Schaefer, “The W. M. Keck telescope phasing camera system,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 1139–1150 (1994).
[CrossRef]

G Chanan, M Troy, F Dekens, E Sirko, E Leffert, “Residual misalignments of the Keck telescope primary mirror segments,” submitted to SPIE Symposium on Astronomical Telescopes and Instrumentation (1998).

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, 82–93 (1994).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Theoretical point response functions for the Keck telescope, assuming an atmospheric coherence diameter of r 0 (0.5 μm) = 20 cm for a variety of wavelengths and piston errors of the primary mirror segments. The profiles are averaged over an ensemble of primary mirror configurations that are consistent with the given rms piston error. All plots are to the same scale. FWHM of the images is 0.5 arc sec at λ = 0.5 μm. Note the devastating effects of phase errors at the longer wavelengths.

Fig. 2
Fig. 2

Geometry of the primary mirror of the Keck telescope showing the 78 circular subapertures that sample the intersegment edges in the phasing procedure. Each segment is 0.9 m on a side. The subapertures are 12 cm in diameter. The 35 peripheral subapertures are used for pupil registration. The six subapertures on the innermost edges (not shown) are obscured by the tertiary mirror and secondary baffles and are not used in the phasing procedure.

Fig. 3
Fig. 3

Theoretical diffraction patterns (monochromatic light) for a split circular subaperture of radius a with a physical step δ between the two halves given by = 0, π/11, 2π/11, … , 10π/11. The boxes are 15/ka rad on a side.

Fig. 4
Fig. 4

Theoretical diffraction pattern for a split circular subaperture of radius a with a step between the two halves in the incoherent limit: δ ≫ λ2/2Δλ. As in Fig. 3, the box is 15/ka rad on a side.

Fig. 5
Fig. 5

Typical broadband sequence of CCD subimages. The step size is 6 μm and the coherence length is 40 μm. The edge is in phase between the fourth and fifth subimages (see top panel of Fig. 7). Boxes are 5 arc sec on a side.

Fig. 6
Fig. 6

Theoretical curve of coherence parameter versus edge step height for a filter with a central wavelength of 891 nm, Δλ = 10 nm, and a coherence length of 40 μm. The solid curve is the best-fit Gaussian approximation.

Fig. 7
Fig. 7

Typical curves of coherence parameter versus edge position for a filter with a coherence length of 40 μm (Δλ = 10 nm) (top panel) and for a filter with a coherence length of 1.2 μm (Δλ = 200 nm) (bottom panel). The top panel corresponds to the particular sequence of subimages shown in Fig. 5.

Fig. 8
Fig. 8

Uncorrected ring mode in the primary mirror. The ring mode—normally corrected to a high degree in software—is a systematic effect caused by dispersion in the phasing camera optics. The edge height measurements are indicated by line segments drawn perpendicular to the intersegment edges. The length of the line segment is proportional to the edge height; the line is drawn on the lower mirror segment. The ring-mode component here corresponds to a rms edge height of 273 nm. Note that the outer mirror segments are systematically low with respect to the inner segments.

Fig. 9
Fig. 9

Surface of the Keck 2 primary mirror before and after phasing. These data represent the first time the procedure was run on this telescope.

Fig. 10
Fig. 10

Residual edge heights (after phasing) for the 78 intersegment edges for a series of 35 phasing-30 runs on Keck 2. Error bars represent the rms variation from run to run (not the error on the mean). The fact that the variations are relatively small shows that random errors in tip–tilt alignment are not the major contributor to the residuals, which are believed to arise from aberrations in the segments themselves.

Tables (3)

Tables Icon

Table 1 Broadband Phasing Parameters

Tables Icon

Table 2 Comparison of Theoretical and Actual Broadband Phasing Modes

Tables Icon

Table 3 Broadband Phasing Performance

Equations (33)

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

f ρ ;   k δ = exp ik δ exp - ik δ 0 η 0 ;   ρ a η < 0 ;   ρ a ρ > a ,
f ˆ ω ;   k δ = 1 π a 2 0 π 0 a exp ik δ exp ik ρ · ω ρ d ρ d θ + 1 π a 2 - π 0 0 a exp - ik δ exp ik ρ · ω ρ d ρ d θ ,
f ˆ ω ;   k δ = 2 π a 2 0 π 0 a cos k δ + k ρ · ω ρ d ρ d θ .
I ω ;   k δ = f ˆ 2 ω ;   k δ .
f ˆ ω ;   0 = 2 π a 2 0 π 0 a cos k ρ · ω ρ d ρ d θ .
f ˆ ω ;   0 = 2 a 2 0 a   J 0 k ρ ω ρ d ρ ,
I ω ;   0 = 2 J 1 ka ω ka ω 2 .
I tot = 0 2 π 0   I ω ω d ω d ψ = 4 π k 2 a 2 = λ 2 π a 2 .
f ˆ ω ;   π 2 = - 2 π a 2 0 π 0 a sin k ρ · ω ρ d ρ d θ ,
= 2 π 0 π u   cos   u - sin   u u 2 d θ ,
I ω ;   k δ = cos   k δ   f ˆ ω ;   0 + sin   k δ   f ˆ ω ;   π 2 2 .
y = - 16 3 π δ a .
g k = 1 2 π σ k 2 exp - k - k 0 2 2 σ k 2 .
σ k = π 8   ln 2 1 l = 1.334 l .
I ω ;   k δ = α 1 f ˆ 2 ω ;   0 + α 2 f ˆ ω ;   0 f ˆ ω ;   π 2 + α 3 f ˆ 2 ω ;   π 2 ,
α 1 = 1 2 1 + exp - 2 σ k 2 δ 2 cos   2 k 0 δ , α 2 = exp - 2 σ k 2 δ 2 sin   2 k 0 δ , α 3 = 1 2 1 - exp - 2 σ k 2 δ 2 cos   2 k 0 δ .
I ω ;   = 1 2 I ω ;   0 + I ω ;   π 2 ,
r = Σ i x i - x ¯ y i - y ¯ Σ i x i - x ¯ 2 1 / 2 Σ i y i - y ¯ 2 1 / 2 .
p i j - p i j = δ j ;   j = 1 , ,   78 ,
i   p i = 0 ;   i = 1 , ,   36 .
Δ δ j = δ j - p i j * - p i j * ,
ϕ = 2   δ λ cycles .
Δ ϕ = - 2 δ λ Δ λ λ ,
x = f n - 1 θ ,
Δ x = f θ   d n d λ   Δ λ .
β = θ b   d n d λ Δ λ λ .
Δ ϕ = κ β ,
δ = - 1 2   κ b θ λ   d n d λ = - 211 κ   nm ,
χ 2 δ = i frac 2 δ λ i - ϕ i + m i 2 σ i 2 .
λ 12 = λ 1 λ 2 2 | λ 1 - λ 2 | .
δ = n 1 + ϕ 1 λ 1 / 2 ,
δ = n 2 + ϕ 2 λ 2 / 2 ,
δ = λ 12 ϕ 2 - ϕ 1 + n 2 - n 1

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