Abstract

There is a great demand for new telescopes that use larger primary mirrors to collect more light. Because of the difficulty in the fabrication of mirrors larger than 8 m as a single piece, they must be made with numerous smaller segments. The segments must fit together to create the effect of a single mirror, which presents unique challenges for fabrication and testing that are absent for monolithic optics. This is especially true for the case of a highly aspheric mirror required to make a short two-mirror telescope. We develop the relationship between optical performance of the telescope and errors in the manufacture and operation of the individual segments.

© 2004 Optical Society of America

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References

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  1. J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985).
  2. M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
    [CrossRef]
  3. P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
    [CrossRef]
  4. J. Burge, “Efficient testing of off-axis aspheres with test plate and computer-generated holograms,” in Optical Manufacturing and Testing III, H. P. Stahl, ed., Proc. SPIE3782, 348–357 (1999).
    [CrossRef]
  5. F. Pan, J. Burge, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. I. Theory and system optimization,” Appl. Opt. (to be published).
  6. F. Pan, J. Burge, D. Anderson, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. II. Case study, error analysis, and experimental validation,” Appl. Opt. (to be published).
  7. R. Shannon, J. Wyant, eds., Applied Optics and Optical Engineering (Academic, New York, 1992).
  8. J. Nelson, G. Gabor, L. Hunt, J. Lubliner, T. Mast, “Stressed mirror polishing: fabrication of an off-axis section of a paraboloid,” Appl. Opt. 19, 2341–2352 (1980).
    [CrossRef] [PubMed]
  9. G. A. Chanan, M. Troy, “Strehl Ratio and modulation transfer function for segmented mirror telescope as functions of segmented phase error,” Appl. Opt. 38, 6642–6647 (1999).
    [CrossRef]
  10. G. Chanan, M. Troy, F. 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]
  11. G. Chanan, M. Troy, E. Sirko, “Phasing discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. 38, 704–713 (1999).
    [CrossRef]
  12. G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments for the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–202 (2000).
    [CrossRef]

1999 (2)

1998 (1)

1980 (1)

Adams, M. T.

P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
[CrossRef]

Anderson, D.

F. Pan, J. Burge, D. Anderson, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. II. Case study, error analysis, and experimental validation,” Appl. Opt. (to be published).

Burge, J.

F. Pan, J. Burge, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. I. Theory and system optimization,” Appl. Opt. (to be published).

J. Burge, “Efficient testing of off-axis aspheres with test plate and computer-generated holograms,” in Optical Manufacturing and Testing III, H. P. Stahl, ed., Proc. SPIE3782, 348–357 (1999).
[CrossRef]

F. Pan, J. Burge, D. Anderson, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. II. Case study, error analysis, and experimental validation,” Appl. Opt. (to be published).

Chanan, G.

G. Chanan, M. Troy, E. Sirko, “Phasing discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. 38, 704–713 (1999).
[CrossRef]

G. Chanan, M. Troy, F. 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]

M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
[CrossRef]

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments for the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–202 (2000).
[CrossRef]

Chanan, G. A.

Dekens, F.

Faber, S. M.

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985).

Fowler, J. R.

P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
[CrossRef]

Gabor, G.

Hunt, L.

Kirkman, D.

Leffert, E.

M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
[CrossRef]

Lubliner, J.

Mast, T.

Mast, T. S.

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985).

Michaels, S.

Nance, C. E.

P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
[CrossRef]

Nelson, J.

Nelson, J. E.

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985).

Ohara, C.

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments for the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–202 (2000).
[CrossRef]

Pan, F.

F. Pan, J. Burge, D. Anderson, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. II. Case study, error analysis, and experimental validation,” Appl. Opt. (to be published).

F. Pan, J. Burge, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. I. Theory and system optimization,” Appl. Opt. (to be published).

Sirko, E.

G. Chanan, M. Troy, E. Sirko, “Phasing discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. 38, 704–713 (1999).
[CrossRef]

M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
[CrossRef]

Troy, M.

G. Chanan, M. Troy, E. Sirko, “Phasing discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. 38, 704–713 (1999).
[CrossRef]

G. A. Chanan, M. Troy, “Strehl Ratio and modulation transfer function for segmented mirror telescope as functions of segmented phase error,” Appl. Opt. 38, 6642–6647 (1999).
[CrossRef]

G. Chanan, M. Troy, F. 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]

M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
[CrossRef]

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments for the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–202 (2000).
[CrossRef]

Worthington, P. T.

P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
[CrossRef]

Appl. Opt. (4)

Other (8)

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments for the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–202 (2000).
[CrossRef]

J. E. Nelson, T. S. Mast, S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985).

M. Troy, G. Chanan, E. Sirko, E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical/IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE3352, 307–317 (1998).
[CrossRef]

P. T. Worthington, J. R. Fowler, C. E. Nance, M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE4010, 267–278 (2000).
[CrossRef]

J. Burge, “Efficient testing of off-axis aspheres with test plate and computer-generated holograms,” in Optical Manufacturing and Testing III, H. P. Stahl, ed., Proc. SPIE3782, 348–357 (1999).
[CrossRef]

F. Pan, J. Burge, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. I. Theory and system optimization,” Appl. Opt. (to be published).

F. Pan, J. Burge, D. Anderson, “Efficient testing of segmented aspherical mirrors by use of a reference plate and computer-generated holograms. II. Case study, error analysis, and experimental validation,” Appl. Opt. (to be published).

R. Shannon, J. Wyant, eds., Applied Optics and Optical Engineering (Academic, New York, 1992).

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Figures (11)

Fig. 1
Fig. 1

Three in-plane displacement errors: rotation, radial displacement, and tangential displacement. Because of symmetry, only the first two types introduce lower-order aberrations.

Fig. 2
Fig. 2

Graphical depiction of the effect of random piston error: (a) phase map showing 0.15-λ rms piston errors, (b) simulated interferogram for 0.05-λ rms piston errors, (c) same as (b) but for 0.15-λ rms piston error.

Fig. 3
Fig. 3

SR as a function of rms piston error at the mirror surface. The thin curve is plotted with approximation (15) and the dashed curve is from the Monte Carlo calculation. Both results agree with previous theoretical result established by Chanan and Troy9 [Eq. (16)].

Fig. 4
Fig. 4

Graphical depiction of the effect of random tilt error: (a) phase map showing 0.15-λ rms tilt errors, (b) simulated interferogram for 0.05-λ rms tilt errors, (c) same as (b) but for 0.15-λ rms tilt error.

Fig. 5
Fig. 5

SR as a function of rms tilt error at the mirror surface. The dotted curve is a plot of Eq. (19), and the solid curve is a plot of the Monte Carlo calculation.

Fig. 6
Fig. 6

Graphical depiction of the effect of random sag error: (a) phase map showing 0.15-λ rms sag errors, (b) simulated interferogram for 0.05-λ rms sag errors, (c) same as (b) but for 0.15-λ rms sag error.

Fig. 7
Fig. 7

SR as a function of rms sag error at the mirror surface. The thin curve is a plot of Eq. (23), and the heavy curve is a plot of the Monte Carlo calculation.

Fig. 8
Fig. 8

Graphical depiction of the effect of random translation error: (a) phase map showing 710-nm peak-to-valley wave-front error for a 1.5-mm rms translation error [parent mirror is a f/1.0, 10-m paraboloid consisting of 36 1.8-m (point-to-point) hexagonal segments], (b) simulated interferogram for a 0.27-mm translation, (c) same as (b) but for a 1.5-mm rms translation error.

Fig. 9
Fig. 9

SR as a function of rms translation error. The dotted curve is a plot of Eq. (26), and the solid curve is a plot of Monte Carlo calculation. (f/1, 30-m primary with 17 rings of 1-m-diameter segments.)

Fig. 10
Fig. 10

Graphical depiction of the effect of random rotation or clocking error: (a) phase map showing 166-nm peak-to-valley wave-front error for a 0.15-mrad rms translation error (parent mirror is a f/1.0, 10-m paraboloid consisting of 36 1.8-m hexagonal segments), (b) simulated interferogram for a 0.15-mrad rms rotational error, (c) same as (b) but for a 1.5-mrad rms rotation error.

Fig. 11
Fig. 11

SR as a function of rms translation error. The thin curve is a plot of Eq. (29) and the heavy curve is a plot of Monte Carlo calculation [multiple application of Eq. (28) with different sets of θ i ’s]. (f/1, 30-m primary, 1-m-diameter segments.)

Tables (2)

Tables Icon

Table 1 rms Wave Front as a Function of Alignment and Fabrication Errors

Tables Icon

Table 2 Segment Geometry Factors Dependent on Alignment and Fabrication Error Type

Equations (45)

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SR=1π202π01expi2πΔWρ, θρdρdθ2,
SRexp-2πσw2,
σwf2ΔW2¯-ΔW¯2,
ΔW2¯ ΔW2dA dAdA is integrated over the primary mirror,
ΔW¯2 ΔWdA dA2dA is integrated over the primary mirror.
 dAΔWl=i=1N  dziΔwil, for l=1, 2,,
Δwi=2 AibΔbi wave-front error for radial positioning error,
Δwi=2 AiθΔθi wave-front error for clocking error,
Aiρ, θ-ka2b24R3ρ2 cos 2θ-ka3b2R3ρ3 cos θ+higher-order terms,
Aib-ka2bi2R3ρ2 cos 2θ+-ka32R3ρ3 cos θ,
Aiθka2bi22R3ρ2 sin 2θ+ka3bi2R3ρ3 sin θ.
Δwix, y=2αI,
σwf2=4iαi2N-iαiN2,
σwf2=4σp2,
SRexp-2π2σp2.
SRlarge Nexp-2π2σp2,
SR=1+N-1exp222π2σp2N,
Δwi=2βix/a+γiy/a,
σwf2=iβi2N+γi2N,
σwf2=2σt2,
Δwi=2Sir/a2,
Si=-ai22R2 ΔRi,
σwf2=43iSi2N,
σwf2=43 σs2closed form,
Δwi-ka2bi2R3(ρ2 cos 2θ]Δbi,
σwf2=46ka22R32i=1NΔbi2bi2N,
σwf2=34k2M2M+12NaR6σΔb2,
Δwika2bi2R3ρ2 sin 2θΔθi,
σwf2=426ka24R32i=1NΔθi2bi4N,
σwf2=9σΔθ2NaR6ka2j=1M J5σΔθ2,
ΔW2¯-ΔW¯2= dSW2 dS- dSW dS2 =i=1N  dziΔwi2i=1N  dzi-i=1N  dziΔwii=1N  dzi2,
Δwi=2αi,
ΔW2¯=i=1NsegdziΔwi2i=1Nsegdzi=22i=1Nαi2segdziN segdzi=4 i=1Nαi2N,ΔW¯2=22i=1N αiN2 =0 because random variable α has a zero mean, i.e.,i=1N αi/N=0.σwf2=22i=1Nαi2N =4σα2.
Δwi=2γixpa+βiypa,
ΔW2¯=i=1Nsegdzi2γixpa+2βiypa2N segdz =4 i=1N γi2Nseg0dzcos2 θρ2seg0dz+4 i=1N γiβiNseg0dzsin θ cos θρ2seg0dz +4 i=1N βi2Nseg0dzsin2 θρ2seg0dz=i=1N γi2N+i=1N βi2N ×1seg=cir at a0.83seg=hex at a 0.83 is numerically integratedΔW¯2=0 because02πd θ cos θ=02πdθ sin θ=0.σwf2=i=1N γi2N+i=1N βi2N1seg=cir at a0.83seg=hex at a,
σwf2=σγ2+σβ21seg=cir at a0.83seg=hex at a=2σ21seg=cir at a0.83seg=hex at a.
Δwi=2Sirpa2,
ΔW2¯=i=1Nsegdzi2Sirpa22N segdz=4i=1N Si2Nseg0dzρ4seg0dz=43i=1N Si2N×1seg=cir at a0.70seg=hex at a 0.70 is numerically calculated,ΔW¯2=2 i=1NsegdziSirpaN segdzi2 =0 because random variable S has a zero mean, i.e.,i=1N Si/N=0.σwf2=43i=1N Si2N1seg=cir at a0.70seg=hex at a,
σwf2=43 σS21seg=cir at a0.70seg=hex at a.
Δwi2Δbic22ibi, θΔbiρ2 cos 2θ=2Δbikbia22R3ρ2 cos 2θ=2Δbiċ22iρ2 cos 2θ where ċ22i=kbia22R3,
ΔW2¯=4 i=1Nseg0dziΔbiċ22iρ2 cos 2θ2N seg0dz=4ka22R32i=1NΔbi2bi2×seg0dziρ4 cos22θN seg0dz=46ka22R32i=1NΔbi2bi2N×1seg=cir at a0.70seg=hex at a 0.70 is numerically calculatedΔW¯2=2 i=1N  dziΔbiċ22iρ2 cos 2θNπa22=0.
σwf2=46ka22R32i=1NΔbi2bi2N×1seg=cir at a0.70seg=hex at a=46ka22R321N×n1b12i=1n1Δbi2n1+n2b22i=1n2Δbi2n2++nlast ringblast ring2i=1nlast ringΔbi2nlast ring=46ka22R32σΔb2Nn1b12+n2b22++nlast ringblast ring2=46ka22R32σΔb2Nj=1M6j3a2 2j2σwf2=34k2M2M+12N×aR6σΔb21seg=cir at a0.70seg=hex at a,
Δwi2Δθic22iρ2 cos 2θθ =2Δθic22i2ρ2 cos 2θ, where c22i=kbi2a24R3,
ΔW2¯=42i=1N  dziΔθic22iρ2 cos 2θ2Nπa2=426ka24R32i=1NΔθi2bi4N×1seg=cir at a0.70seg=hex at a,ΔW¯2=0.σwf2=426ka24R32i=1NΔθi2bi4N×1seg=cir at a0.70seg=hex at a 0.70 is numerically averaged.
σwf2=426ka24R32i=1NΔθi2bi4N1seg=cir at a0.70seg=hex at a=426ka24R321N×n1b14i=1n1Δθi2n1+n2b24i=1n2Δθi2n2++nlast ringblast ring4i=1nlast ringΔθi2nlast ring=426ka24R32σΔθ2Nn1b14+n2b24++nlast ringblast ring4=426ka24R32σΔθ2Nj=1M6j3a2 2j4,σwf2=9NaR6ka2j=1M J5σΔθ2×1seg=cir at a0.70seg=hex at a,

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