Fabrication and alignment issues for segmented mirror telescopes Feenix Y. Pan, James H. Burge, Rene Zehnder, and Yanqui Wang
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 OCIS codes: 220.4840, 120.3940.
1. Introduction
Giant primary mirrors are now being designed for ground- and space-based telescopes. As the mirrors get larger, telescope designers are driven to designs that create the effect of a single continuous mirror by using a mosaic of smaller elements. The twin 10-m Keck telescopes1 on Mauna Kea in Hawaii are the best known example of this technology, creating 10-m apertures with an array of 1.8-m mirror segments. The new designs for giant telescopes require higher performance and use more steeply curved optics than those in the past; as a result, they are more sensitive to small errors in the fabrication and alignment of the segments. In this paper we explore these fundamental sources of error and quantify their effects on system performance. Errors in the rigid body position of the mirror segments cause the images to degrade from the ideal diffraction-limited case. Operational changes in phasing or pointing cause degradation in the system performance. For monolithic mirrors, piston is never important and tilt is benign. The problem with segmented mirrors is that each segment may
F. Y. Pan, J. H. Burge 共
[email protected]兲, and R. Zehnder are with the Optical Science Center, University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721-0001. Y. Wang is with the Department of Information Engineering, ShiJiaZhuang University of Economics, ShiJiaZhuang, China. Received 24 July 2003; revised manuscript received 15 December 2003; accepted 15 December 2003. 0003-6935兾04兾132632-11$15.00兾0 © 2004 Optical Society of America 2632
APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
have a different phase and a different tilt. For ground-based telescopes, a real-time servo is required to maintain these errors in the presence of wind buffeting and gravitational loading as the telescope is slewed. The primary source of pointing and phasing error comes from limitations in this servo system.2 Segmented mirrors are also sensitive to other errors that are easily compensated for the case of monolithic optics. A radius of curvature 共ROC兲 variation among segments causes aberrations in the system. If the aspheric profiles of the optical surface for the individual segment are randomly misaligned, higherorder aberrations result. Both of these effects are easily accommodated for a monolithic mirror by means of moving the mirror. Because the segments are required to fit together with matched phase, it is not possible to accommodate these errors so easily. Other errors such as temperature gradient error3 and conic constant error are beyond the scope of this paper. The drive to make modern telescopes shorter 共as fast as f兾0.5兲 leads to an increase in the aspheric departure of the segments. This makes the tolerances on the optical testing much tighter.4 In other papers5,6 we present a testing technique in which we use computer-generated holograms to achieve alignment precision that yields 兾100 wave-front accuracy. In this paper we quantify how the system degrades when segments are presented with five possible errors that are caused by position errors in the segments. In Section 2 we define a merit function and present the formalism used to analyze the effect of random errors on telescope performance. In Sections 3 and 4
we give expressions and graphics showing system performance degradation for each type of error. Finally, in Tables 1 and 2 we summarize results from Sections 3 and 4, and all derivations are included in Appendix A. 2. Analysis of Segmented Optics
Ideal segments that are perfectly aligned will provide a primary mirror that gives the same quality as a monolithic mirror. Here we treat a random mismatch of five most significant degrees of freedom for the individual segments, dividing them into two categories: • Operational error—phasing and tilt. These errors typically create limitations in a real servo. • Fabrication and alignment errors—ROC and segment in-plane positioning errors of radial position and clocking. These errors come about from the limitation in the segment fabrication and alignment.
To investigate how these errors affect the telescope performance, we calculate the root-mean-square 共rms兲 wave-front error for the system with the errors. All calculations are performed for circular segments and then numerical factors are computed to give the results for hexagonal segments. This is valid for the case in which the segment size is much smaller than the size of the complete mirror. We assume uncorrelated effects and combine errors from different sources as root sum squares. The rms wave front is used to determine the Strehl ratio 共SR兲, defined as the peak intensity at the Gaussian image point in the presence of aberration, divided by the intensity obtained without aberration. This provides an excellent figure of merit for near-diffraction-limited systems. Extra edge effects associated with the segmented aperture are not accounted for in this calculation. Mathematically, SR can be calculated by7 SR ⫽
1 2
冏兰
2
0
兰
1
冏
2
exp关i2⌬W共, 兲兴dd ,
0
(1)
where ⌬W, in units of waves, is the wave-front aberration relative to the reference sphere for the diffraction focus. For small rms wave-front errors, where SR is greater than 0.1, Eq. 共1兲 can be approximated by approximation 共2兲, and we use this approximation to quantify how fabrication and alignment errors affect the SR of the system. SR ⬇ exp关⫺共2 w兲 2兴,
(2)
where sw is the rms wave-front error in units of waves. In this paper we give expressions for the rms wave-front error from each of the error sources, and they are calculated in the usual way: wf2 ⬅ ⌬W 2 ⫺ 共⌬W兲 2,
(3)
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.
where
⌬W 2 ⬅
兰兰
共⌬W兲 2dA
兰兰
共dA is integrated over the dA primary mirror兲,
冤 冥 兰兰 兰兰
(4)
2
共⌬W兲dA
⌬W ⬅ 2
共dA is integrated over the
dA
primary mirror兲.
(5)
And for segmented optics,
兰
N
dA共⌬W兲 l ⫽
兺 i⫽1
兰
dz i 共⌬w i 兲 l,
for l ⫽ 1, 2, . . . , (6)
where dzi is the element integration area of the ith segment, ⌬wi is the wave-front aberration of the ith segment, and N is the total number of segments in the primary mirror. For each of the five errors, we show the effect graphically and give an analytic solution, validated with a Monte Carlo simulation. We obtained the closed-form expression for overall rms wave-front error by substituting Eqs. 共4兲–共6兲 into Eq. 共3兲. We performed the Monte Carlo analysis by analyzing the case in which each segment has a random 共Gaussian兲 variation. The effect of phasing, tilt, and ROC errors are relatively straightforward to model compared with the in-plane errors of segment rotation and translation errors. These two errors cause wave-front error because of the aspheric shape of the segments 共spherical segments look the same when they are moved this way兲. The segments are sensitive to radial and clocking motion, but to the first order they are not sensitive to the tangential motion 共see Fig. 1兲. To model the rotation and translation errors, we first find the aspherical departure Ai of the segments 1 May 2004 兾 Vol. 43, No. 13 兾 APPLIED OPTICS
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in the local segment coordinates 共r, 兲 and then use Eq. 共7兲 to calculate the induced wave-front error due to these displacements.
into Eq. 共7兲 to compute the effect of rotation and translation errors on the telescope performance.
冉
radial positioning error兲, ⌬w i ⫽ 2
冉 冊
(7)
A i 共, 兲 ⬇ ⫺
(8)
3
ka b ka b 3 2 共 cos 兲 3 共 cos 2兲 ⫺ 4R 2 R3
⫹ higher-order terms,
冉 冊
(11)
where Ai 共r, 兲 is the aspherical departure of the ith segment, 共r, 兲 is the local coordinate system for segments 共the segment surface is centered on the segment itself and the z axis is normal to the segment surface兲, ⌬bi is the random radial translation error of the ith segment, and ⌬i is the random rotation error of the ith segment. The aspheric departure for each segment has been well studied by Nelson et al.8 for arbitrary conic constant k. Ai 共, 兲 can be approximated: 2 2
冉 冊
A i ka 2b i2 2 ka 3b i 3 ⬇ 共 sin 2兲 ⫹ 共 sin 兲. 3 2R 2 R3
A i 共⌬兲 i 共wave-front error for clocking error兲,
冊
ka 2b i 2 ka 3 A i 共 3 cos 兲, ⬇ ⫺ 3 共 cos 2兲 ⫹ ⫺ b 2R 2 R3 (10)
A i ⌬w i ⫽ 2 共⌬b兲 i 共wave-front error for b
(9)
where 共, 兲 is the normalized local coordinate system 共 ⫽ r兾a兲, a is the segment size 共half-diameter兲, bi is the off-axis distance of the ith segment from the center of the parent mirror, R is the ROC of the parent primary mirror, and k is the conic constant. This approximation is justified when R, the ROC of the parent mirror, is much greater than a, the halfdiameter of the segment. The error introduced under this approximation is of the order of 共1兾8兲共a3兾R2兲. Furthermore, in a typical telescope configuration with segmented optics, the first term and its corresponding derivative dominate 关approximations 共9兲– 共11兲兴, so only the first astigmatic term is substituted
For subsequent analysis, we use only the first term 共astigmatism兲 and neglect the second 共coma兲. This is justified for the case in which the segment size is small compared with the size of the primary mirror. 共The error from this approximation goes as the ratio of segment size to the overall diameter.兲 For a mirror with a diameter of D that has M rings of segments and each segment has a half-diameter of a, D ⫽ 2共M ⫹ 1兲共2a兲, so the approximation error is of the order of D兾2a ⫽ 1兾关2共M ⫹ 1兲兴. 3. System Degradation Due to Operational Errors
System performance of a segmented mirror telescope critically depends on operational errors such as piston 共or phasing兲 and tilt 共or pointing兲. For example, a 兾40 piston error decreases the Keck Telescope modulation transfer function by 10% and the corresponding SR is 0.91.9,10 This sets tight tolerances on piston errors for segmented telescopes. Our analysis shows that controlling tilt is equally important. A.
Piston or Phasing Error
To achieve diffraction-limited performance, a segmented mirror telescope must be phased, that is, the piston errors between segments must be reduced to a small fraction of wavelength. This enables individual images from each segment to be superimposed coherently. A poorly phased telescope can achieve only the resolution limit set by the size of the segment, not the full parent aperture. Previous
Fig. 2. Graphical depiction of the effect of random piston error: 共a兲 phase map showing 0.15- rms piston errors, 共b兲 simulated interferogram for 0.05- rms piston errors, 共c兲 same as 共b兲 but for 0.15- rms piston error. 2634
APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
where ␣i is the piston error of the ith segment in units of micrometers. Furthermore, because we assumed that random piston error ␣ is a Gaussian random variable with zero mean and a standard deviation of p, Eq. 共13兲 is reduced to wf2 ⫽ 4 p2,
Fig. 3. SR as a function of rms piston error at the mirror surface. The thin curve is plotted with approximation 共15兲 and the dashed curve is from the Monte Carlo calculation. Both results agree with previous theoretical result established by Chanan and Troy9 关Eq. 共16兲兴.
studies9 –12 show that control of piston error is vital to good telescope performance. We quantify the effect of piston error as a wavefront error across the segment given by ⌬w i 共 x, y兲 ⫽ 2␣ I,
wf2 ⫽ 4
再
2
i
i
N
冋 册冎 兺 共␣ 兲 i
N
SR ⬇ exp关⫺共22 p兲 2兴.
(15)
The SR analysis on phasing is in excellent agreement with an analytical expression established by Chanan and Troy9 shown in Eq. 共16兲. As the number N of segments gets large, large N SR O ¡ exp关 ⫺ 共22 p兲 2兴, which is the same as approximation 共15兲. SR ⫽
1 ⫹ 共N ⫺ 1兲exp关2 2共2兲 2 p2兴 , N
,
(13)
(16)
where N is the total number of mirror segments and p is the rms piston error at the segment 共measured in units of waves兲. Figure 3 shows that the comparisons of the Monte Carlo calculation 关Eq. 共13兲兴, the closed form 关Eq. 共14兲兴, and the theoretical form by Chanan and Troy9 关Eq. 共16兲兴 are excellent. B.
Tilt Error
The individual segments must be matched in their tilt. We quantify the effect of tilt error as a wavefront error across the segment given by ⌬w i ⫽ 2关 i 共 x兾a兲 ⫹ ␥ i 共 y兾a兲兴,
2
i
⫺
where p is the standard deviation of the piston error at the mirror surface, measured in units of waves. From here, we obtain an expression for SR by substituting Eq. 共14兲 into approximation 共2兲:
(12)
where ⌬wi 共x, y兲 is the wave-front error 共in units of waves兲 of the ith segment in the local pupil coordinates 共x, y兲 and ai is the ith segment’s random piston error at the segment surface 共measured in waves兲. Piston error is illustrated below where ai’s are selected from a random Gaussian distribution with zero mean and a standard deviation of s. Figure 2 shows how random piston error affects the full aperture wave fronts and depicts the interferograms of the wave front corresponding to a rms piston error of 0.05- and 0.15-, respectively. By substituting the definition of phasing error expressed in Eq. 共12兲 into Eq. 共3兲, we obtain an analytical expression to numerically calculate the square of rms wave-front error wf:
兺 共␣ 兲
(14)
(17)
where ⌬wi 共x, y兲 is the wave-front error of the ith segment in local pupil coordinates 共x, y兲 and i , ␥i are
Fig. 4. Graphical depiction of the effect of random tilt error: 共a兲 phase map showing 0.15- rms tilt errors, 共b兲 simulated interferogram for 0.05- rms tilt errors, 共c兲 same as 共b兲 but for 0.15- rms tilt error. 1 May 2004 兾 Vol. 43, No. 13 兾 APPLIED OPTICS
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Gaussian distribution with a standard deviation of  ⫽ ␥ ⫽ . Because we assumed that the random tilt errors are two independent Gaussian random variables, each with zero mean and a standard deviation of t, Eq. 共18兲 is reduced to wf2 ⫽ 2 t2,
Fig. 5. SR as a function of rms tilt error at the mirror surface. The dotted curve is a plot of Eq. 共19兲, and the solid curve is a plot of the Monte Carlo calculation.
the x-tilt and y-tilt errors at the segment surface for the ith segment in units of waves across the segment’s half-diameter. Tilt error is illustrated below where i’s and ␥i’s are selected from two independent random Gaussian distributions with zero mean and the same standard deviation s ⫽ s␥ ⫽ s. Here we assume that the x-tilt and y-tilt errors are uncorrelated to simplify the calculations. Figure 4 shows how random tilt errors affect the full aperture wave fronts for a rms tilt error of x ⫽ y ⫽ 0.15 and depicts the interferograms of the wave front corresponding to x ⫽ y ⫽ 0.05 and 0.15 , respectively. By substituting 共see Appendix A兲 the definition of tilt error expressed in Eq. 共17兲 into Eq. 共3兲, we derive an analytical expression to numerically calculate the square of the rms wave-front error wf: wf2 ⫽
兺 i
共 i 兲 2 共␥ i 兲 2 ⫹ , N N
(18)
where ␥i is the x-tilt error of the ith segment drawn from a zero-mean Gaussian distribution with a standard deviation of ␥ and i is the y-tilt error of the ith segment drawn from another independent zero-mean
(19)
where t is the standard deviation of the tilt error at the mirror surface, measured in units of waves per radius. Figure 5 shows the excellent agreement between the Monte Carlo calculation and the closed form. 4. Fabrication and Alignment Errors
In addition to operational errors such as phasing and tilt errors, limitations in the segment fabrication and alignment also degrade the system performance because segments must fit together with matched phase. A.
Relative Radius of Curvature Mismatch
ROC must also be matched in all the segments. The ROC error varies as the square of the radial position in the segment. We quantify the effect of this error as a wave-front error across the segment given by ⌬w i ⫽ 2S i 共r兾a兲 2,
(20)
where ⌬wi 共r, 兲 is the wave-front error of the ith segment in local pupil coordinates 共r, 兲 and Si is the sag error for the ith segment. For segmented optics, the ROC error relates to mismatching of the relative ROC ⌬Ri through Eq. 共21兲: Si ⫽ ⫺
a i2 ⌬R i , 2 R2
(21)
where ⌬Ri is the error in ROC for the ith segment. The effect of the ROC error is illustrated below, where the magnitude of the error in each segment ⌬Si is generated through a Gaussian random variable with zero mean and a standard deviation of . Fig-
Fig. 6. Graphical depiction of the effect of random sag error: 共a兲 phase map showing 0.15- rms sag errors, 共b兲 simulated interferogram for 0.05- rms sag errors, 共c兲 same as 共b兲 but for 0.15- rms sag error. 2636
APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
B.
Translation or Off-Axis Distance Error for Segments
When segments are misaligned in the radial direction, lower-order aberrations are present. We quantify the effect of this radial translation error as a wave-front error across the segment given by approximation 共24兲, where we took only the first term of approximation 共10兲 as justified above.
Fig. 7. SR as a function of rms sag error at the mirror surface. The thin curve is a plot of Eq. 共23兲, and the heavy curve is a plot of the Monte Carlo calculation.
ure 6 shows how a random rms sag error of 0.15 affects the full-aperture wave fronts, and it depicts interferograms of the wave front corresponding to rms sag errors of ⫽ 0.05 and 0.15 , respectively. Note that when we express relative ROC mismatch in terms of sag difference, the results are independent of segment size and ROC of the primary mirror. For a 30-m, f兾1 primary mirror with a segment diameter of 1 m, to match the relative ROC to better than 1 part in 105 requires that the relative sag difference be less than 35 nm. Following similar steps, the rms wave-front and SR calculations resulted in the following formulas: 4 wf2 ⫽ 3 wf2 ⫽
冋 册 兺 共S 兲
冉
⌬w i ⬇ ⫺
冉 冊冋
2
2
N
,
4 2 s 共closed form兲, 3
(22)
(23)
where s is the standard deviation of the sag error at the mirror surface measured in units of waves. Figure 7 verifies that the closed form 关Eq. 共23兲兴 produces the same result as the Monte Carlo calculation 关multiple application of Eq. 共22兲兴.
(24)
where ⌬wi 共r, 兲 is the wave-front error of the ith segment when radial translation error is present and ⌬bi is the magnitude of the radial translation error for the ith segment. Radial translation error is illustrated below where ⌬bi is drawn from a Gaussian distribution with a standard deviation of . Simulation of the fullaperture phase map is shown in Fig. 8 with a rms translation error of 1.5 mm for a f兾1.0, 10-m primary with a segment diameter of 1.8 m. It also depicts interferograms corresponding to a rms translation error of 0.15 and 1.5 mm, respectively. Much like the results for the rotation case, the results depend on the size of the segment. The rms wave-front calculations are detailed in Appendix A, and we simply report the results here.
i
i
冊
ka 2b i 2 共 cos 2兴⌬b i , 2 R3
4 ka 2 wf2 ⫽ 6 2 R3 wf
2
冋
N
兺 共⌬b 兲 共b 兲 2
i
i
i⫽1
N
3 k 2M 2共M ⫹ 1兲 2 ⫽ 4 N
册冉 冊
2
册
,
(25)
6
a ⌬b2, R
(26)
where M is the maximum number of rings in the primary and ⌬b is the rms translation error. Figure 9 verifies that the closed form 关Eq. 共26兲兴 produces the same result as the Monte Carlo calculation.
Fig. 8. Graphical depiction of the effect of random translation error: 共a兲 phase map showing 710-nm peak-to-valley wave-front error for a 1.5-mm rms translation error 关parent mirror is a f兾1.0, 10-m paraboloid consisting of 36 1.8-m 共point-to-point兲 hexagonal segments兴, 共b兲 simulated interferogram for a 0.27-mm translation, 共c兲 same as 共b兲 but for a 1.5-mm rms translation error. 1 May 2004 兾 Vol. 43, No. 13 兾 APPLIED OPTICS
2637
as a wave-front error across the segment given by approximation 共27兲 where we took only the first term of approximation 共11兲, as justified above.
⌬w i ⬇
Fig. 9. SR as a function of rms translation error. The dotted curve is a plot of Eq. 共26兲, and the solid curve is a plot of Monte Carlo calculation. 共 f兾1, 30-m primary with 17 rings of 1-mdiameter segments.兲
C.
Rotation or Orientation Error for Asphere
When segments are misaligned rotationally, we adopt an approach similar to that discussed in Subsection 4.B. Here we quantify the effect of this error
冉 冊
ka 2b i2 2 共 sin 2兲⌬ i , R3
(27)
where ⌬wi 共r, 兲 is the wave-front error of the ith segment when radial translation error is present and ⌬i is the magnitude of the rotational error for the ith segment. Rotation error is illustrated below where ⌬i’s are drawn from a Gaussian distribution with zero mean and a standard deviation of . Simulation of the full-aperture phase map is shown in Fig. 10 with a rms rotation error of 0.15 mrad for the case of a f兾1.0, 10-m primary with 1.8-m segments 共diameters兲. It also depicts interferograms corresponding to rms rotation errors of ⫽ 0.15 and 1.5 mrad, respectively.
Fig. 10. Graphical depiction of the effect of random rotation or clocking error: 共a兲 phase map showing 166-nm peak-to-valley wave-front error for a 0.15-mrad rms translation error 共parent mirror is a f兾1.0, 10-m paraboloid consisting of 36 1.8-m hexagonal segments兲, 共b兲 simulated interferogram for a 0.15-mrad rms rotational error, 共c兲 same as 共b兲 but for a 1.5-mrad rms rotation error.
Table 1. rms Wave Front as a Function of Alignment and Fabrication Errors
Error
Description
Wave-Front Errora
Perturbation p: rms piston variation for segment position tilt: rms variation for segment tilt s: rms sag variation between the segments
wf2 ⫽ 4p2
Piston
Cophasing the segments
Tilt
Copointing of the segments
ROC
Matching the curvature
Translation
Segment radial shift
⌬b: rms variation for segment position
wf2 ⫽
Clocking
Segment rotation about the local center
⌬: rms segment rotation 共in radians兲
wf2 ⫽ 9
wf2 ⫽ ⫻ 2tilt2 wf2 ⫽ ⫻ 4兾3s2
冋 册冉 冊 冋 冉 冊冉 冊 兺 册 3 4
k 2M 2共M ⫹ 1兲 2 N ⌬2 N
a R
a R
6
⌬b2
M
6
共ka兲 2
J 5 ⌬2
j⫽1
R, primary mirror ROC; a, segment size 共half-diameter兲; k, conic constant; N, total number of segments; M, total number of rings.
a
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APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
Table 2. Segment Geometry Factors Dependent on Alignment and Fabrication Error Type
Circular Segments
Hexagonal Segments 共obtained numerically兲
Tilt ROC Translation Clocking
1.0 1.0 1.0 1.0
0.83 0.70 0.70 0.70
Appendix A: rms Wave-Front Calculation
Using Eq. 共6兲 in Eq. 共3兲, and because all segments are identical in size, we have
冢 冣冢 冣 冤 兰兰 冥 兰兰 冢 兰兰 冣 兰兰 兰兰 兺 兰兰
dSW 2
⌬W 2 ⫺ 共⌬W兲 2 ⫽
⫺
dS
2
wf
2
4 ⫽ 6
冉 冊冋 2
ka 4R 3
N
兺 共⌬ 兲 共b 兲 2
i
i⫽1
冋 冉 冊冉 冊
wf2 ⫽ 9
⌬2 N
i
N 6
a 共ka兲 2 R
M
兺 j⫽1
4
册
,
册
J 5 ⌬2,
⫽
dz i 共⌬w i 兲 2
i⫽1
N
兺
dz i
i⫽1
(28)
2
N
(29)
⫺
where ⌬ is the rms rotation error in radians. Figure 11 verifies that the closed form 关Eq. 共29兲兴 produces the same result as the Monte Carlo calculation. 5. Conclusion
Segmenting primary mirrors into a mosaic of smaller pieces enables ever-larger telescopes to be constructed. This technology brings unique challenges to the fabrication and testing of the segments because it requires all segments to fit together to form a continuous optical surface. In this paper we quantify five types of alignment and fabrication errors, which are benign to monolithic optics and degrade the overall telescope performance for segmented optics. Results obtained for the case of piston, tilt, and ROC are independent of segment size and the primary mirror configurations. For the case of translation and rotation misalignments, results are in a simple form that can be used to assist the determination of the fundamental telescope design parameter such as the size of the segment.
dS
N
The rms wave-front calculations are detailed in Tables 1 and 2 and we simply report the results here. 2
兰兰 兰兰
2
dSW
兺
dz i ⌬w i
i⫽1
,
N
兺
(A1)
dz i
i⫽1
where W is the total wave front of the aspherical departure of the parent mirror, wi is the wave-front aberration in each segment, and dzi is the elemental area of the segment. Here we distinguish the two cases of 共1兲 spherical segment used to approximate the hexagonal ones and 共2兲 actual hexagonal segments. The following notations are used for derivations: 共r, 兲 is the local coordinates of the segments, 共, 兲 is the normalized local coordinates of the segments, N is the total number of segments, a is the segment half-diameter 共for hexagonal segments, the diameter equals the point-to-point dimension of the hexagon兲, segc ⫽ cir at a means that the segment is a circle with half-diameter a, segh ⫽ hex at a means that the segment is a hexagon with half-diameter a, and seg0 means that the segment size is normalized 共halfdiameter of 1兲. 1. Piston Case ⌬w i ⫽ 2␣ i ,
(A2)
where ␣i is the piston error of the ith segment. N
兺 i⫽1
Fig. 11. SR as a function of rms translation error. The thin curve is a plot of Eq. 共29兲 and the heavy curve is a plot of Monte Carlo calculation 关multiple application of Eq. 共28兲 with different sets of i’s兴. 共 f兾1, 30-m primary, 1-m-diameter segments.兲
⌬W 2 ⫽
兰兰 seg N
兺 i⫽1
dz i 共⌬w i 兲 2
兰兰
dz i
seg
1 May 2004 兾 Vol. 43, No. 13 兾 APPLIED OPTICS
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N
22
兺 共␣ 兲
2
i
i⫽1
⫽
dz i
冉 冊 N
兺␥
4
seg
兰兰
N
兰兰
⫹
dz i
i
i
i⫽1
兰兰
dz共sin cos 兲 2
seg0
兰兰
N
seg
N
兺 共␣ 兲
2
i
⫽4
i⫽1
冉 冊 N
N
共⌬W兲 2 ⫽ 2 2
兺␣
兺
4
⫹
2
i
i⫽1
N
⫽ 0 because random variable ⫽
␣ has a zero mean, i.e.,
冉兺 冊 N
␣ i 兾N
冉
冋 册
 i2
i⫽1
2
兺␥
N
⫽ 4 ␣ . 2
(A3) wf ⫽ 2
These results of piston are the same for the circular and hexagonal cases.
⫹
i⫽1
再
N
冊
xp yp ⌬w i ⫽ 2 ␥ i , ⫹ i a a
兺 i⫽1
⌬W 2 ⫽
兰兰 冉 seg
兰兰 兰兰
N
冉 冊 N
4
⫽
兺 i⫽1
N
␥ i2
冊
N
冊
dz
seg0
冉
d cos ⫽
N
N
兺
兺
␥ i2 ⫹
i⫽1
N
i⫽1
N
(A4)
再
再
2 i
冊再
d sin ⫽ 0.
0
1 seg ⫽ cir at a , 0.83 seg ⫽ hex at a (A5)
1 seg ⫽ cir at a 0.83 seg ⫽ hex at a
1 seg ⫽ cir at a . 0.83 seg ⫽ hex at a
⌬w i ⫽ 2S i
(A6)
冉冊
2
兰兰 冋 冉 冊 册 兰兰 兺 兰兰 冉 冊 兰兰
N
rp dz i 2S i a
兺 i⫽1
⌬W 2 ⫽
seg
N
dz
seg
dz共cos2 兲 2
APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
dz 4
N
S i2
⫽4 dz
2
rp , a
(A7)
where Si is the sag error of the ith segment drawn from a zero-mean Gaussian random variable with a standard deviation of .
dz
兰兰
兰
2
3. Radius of Curvature Case
seg0
2640
兰
2
0
seg
seg0
兰兰
1 seg ⫽ cir at a 0.83 seg ⫽ hex at a 共0.83 is
⫽ 2 2
where ␥i is the x-tilt error of the ith segment drawn from a zero-mean Gaussian distribution with a standard deviation of ␥, and i is the y-tilt error of the ith segment drawn from another independent zero mean Gaussian distribution with a standard deviation of  ⫽ ␥ ⫽ . xp yp dz i 2␥ i ⫹ 2 i a a
2 i
i⫽1
wf2 ⫽ 共 ␥2 ⫹ 2兲
2. Tilt Case
N
seg0
兺
2 i
共⌬W兲 2 ⫽ 0 because
i⫽1
冉
dz共sin2 兲 2
numerically integrated兲
i
wf2 ⫽ 2 2
兰兰
N
N
兺 共␣ 兲
seg0
N
N
⫻
i⫽1
⫽ 0.
冉 冊 N
,
dz
i⫽1
seg0
N
dz
seg0
2 2
冉 冊 再 N
兺S
4 ⫽ 3
2
i
i⫽1
N
⫻
共⌬W兲 2 ⫽
冤
兰兰 冉 冊
N
2
兺
dz i S i
i⫽1
rp a
seg
兰兰
N
dz i
seg
冥
dz i 4 cos2共2兲
seg0
兰兰
N
1 seg ⫽ cir at a ⫻ 0.70 seg ⫽ hex at a 共0.70 is numerically calculated兲
兰兰
冉 冊冋
dz
seg0
,
2
2
4 ka ⫽ 6 2 R3
2
⫻
再
N
兺 共⌬b 兲 共b 兲 2
i
2
i
i⫽1
N
册
1 seg ⫽ cir at a 0.70 seg ⫽ hex at a 共0.70 is numerically calculated兲
⫽ 0 because random variable
共⌬W兲 ⫽ 2
S has a zero mean, i.e.,
冉兺 冊
冤
N
2
兺 i⫽1
兰兰
dz i 共⌬b i 兲共c˙ 22i兲共 cos 2兲 2
Na 2
冥
2
⫽ 0. (A11)
N
S i 兾N ⫽ 0.
i⫽1
冉 冊再
To further simplify Eq. 共B11兲, we note that, for the circular segment case,
N
wf
2
4 ⫽ 3
wf2 ⫽
兺S
2
i
i⫽1
N
1 seg ⫽ cir at a , 0.70 seg ⫽ hex at a (A8)
再
4 2 1 seg ⫽ cir at a . S 0.70 seg ⫽ hex at a 3
4 ka wf2 ⫽ 6 2 R3 ⫻
(A9)
再
冋 册 冉 冊 c 22i共b i , 兲 ⌬b i
2 cos 2
⫹ n 2 b 22
kb i a , 2 R3
⌬W 2 ⫽
dz i 共⌬b i c˙ 22i 2 cos 2兲 2
seg0
N
冉 冊 冋兺
ka 2 ⫽4 2 R3
2
兰兰
兺 共⌬b 兲
2
i
i⫽1
n1
兺 共⌬b 兲
seg0
共⌬b i 兲 2共b i 兲 2
2
⫹...
i⫽1
n2
⫽
兺
2
n last ring
冉 冊冉 冊
4 ka 2 6 2 R3
2
冉 冊冉 冊兺 冉 冋 册 冉冊 再
4 ka 2 ⫽ 6 2 R3
2
冥
⌬b2 共n 1 b 12 ⫹ n 2 b 22 ⫹ . . . N
⫹ n last ringb last ring2兲
wf2 ⫽
共⌬b i 兲 2
i⫽1
⌬b2 N
M
共6j兲
j⫽1
冑3a 2
2j
冊
2
3 k 2M 2共M ⫹ 1兲 2 4 N 6
dz
N
i⫽1
册
n1
1 ⫻ n 1 b 12 N
⫹ n last ringb last ring
(A10)
where ⌬bi is the translation error 共mm兲 of the ith segment drawn from a zero-mean Gaussian random variable with a standard deviation of ⌬b.
兰兰
N
nlast ring
⫽ 2⌬b i 共c˙ 22i兲 2 cos 2 where c˙ 22i ⫽
i⫽1
i⫽1
i
2
兺
2
i
n2
kb i a ⫽ 2⌬b i 2 cos 2 2 R3
4
2
i
冉 冊冉 冊 冤 2
2
N
N
兺 共⌬b 兲 共b 兲
1 seg ⫽ cir at a 0.70 seg ⫽ hex at a
4 ka 2 ⫽ 6 2 R3
4. Translation Error
⌬w i ⬇ 2⌬b i
冉 冊冋 2
2
⫻
册
a 1 seg ⫽ cir at a ⌬b2 , 0.70 seg ⫽ hex at a R
where M is the number of rings in the primary. 1 May 2004 兾 Vol. 43, No. 13 兾 APPLIED OPTICS
2641
5. Rotation Error ⌬w i ⬇ 2⌬ i
冋
共c 22i 2 cos 2兲
册 2
kb i a , 4R 3
⫽ 2⌬ i 共c 22i兲2 2 cos 2, where c 22i ⫽
(A12)
⫻
where ⌬i is the rotation error 共rad兲 of the ith segment drawn from a zero-mean Gaussian random variable with a standard deviation of ⌬. All others are in meters.
兰兰
N
兺
42
i⫽1
⌬W 2 ⫽
冉 冊冋
2
4 ⫽ 6
共⌬W兲 2 ⫽ 0. 2
4 wf2 ⫽ 6 ⫻
2
2
再
⫻
dz i 共⌬ i c 22i 2 cos 2兲 2
ka 4R 3
Na 2 N
兺 共⌬ 兲 共b 兲 2
i
4
i
i⫽1
N
册
冉 冊冋
再
2
ka 4R 3
N
兺 共⌬ 兲 共b 兲 2
i
4
i
i⫽1
N
册
1 seg ⫽ cir at a 0.70 seg ⫽ hex at a 共0.70 is numerically averaged兲 .
(A13)
To further simplify Eq. 共B13兲, we note that
2
4 wf2 ⫽ 6
再
⫽
冉 冊冋 2
ka 4R 3
2
N
兺 共⌬ 兲 共b 兲 2
i
i
i⫽1
N
1 seg ⫽ cir at a 0.70 seg ⫽ hex at a
4
册
冉 冊冉 冊
4 2 ka 2 6 4R 3
冤
⫻ n1 b1
2
1 N
n1
兺
n2
共⌬ i 兲 2
4 i⫽1
n1
兺 共⌬ 兲
4 i⫽1
n2
nlast ring
⫹ . . . ⫹ n last ringb last ring ⫽
冉 冊冉 冊
4 2 ka 2 6 4R 3
2
2
i
⫹ n2 b2
4
兺
共⌬ i 兲 2
i⫽1
n last ring
冥
⌬2 共n 1 b 14 ⫹ n 2 b 24 ⫹ . . . N
⫹ n last ringb last ring4兲
2642
2
⌬2 N
6
a 共ka兲 2 R
M
共6j兲
j⫽1
冑3a 2
冊
4
2j ,
M
J 5 ⌬2
j⫽1
1 seg ⫽ cir at a , 0.70 seg ⫽ hex at a
where ⌬ is the standard deviation Gaussian random variable ⌬ 共zero mean兲. We thank E. Rudkevich, S. Errico, and D. Anderson for helpful editorial comments. Special thanks go to ZhenShan Yan for meticulously verifying the calculations. This research has been partially funded by NASA contract NGT5-50419, National Optical Astronomical Observatory contract C10360A. References
1 seg ⫽ cir at a , 0.70 seg ⫽ hex at a
2
9 N
wf2 ⫽
2
冉 冊冉 冊兺 冉 冋冉 冊冉 冊 兺 册 再
4 2 ka 2 6 4R 3
⫽
APPLIED OPTICS 兾 Vol. 43, No. 13 兾 1 May 2004
1. J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 共W. M. Keck Library, Kamuela, Hawaii, 1985兲. 2. M. Troy, G. Chanan, E. Sirko, and E. Leffert, “Residual misalignments of the Keck Telescope primary mirror segments: classification of modes and implications for adaptive optics,” in Advanced Technology Optical兾IR Telescopes VI, L. M. Stepp, ed., Proc. SPIE 3352, 307–317 共1998兲. 3. P. T. Worthington, J. R. Fowler, C. E. Nance, and M. T. Adams, “Thermal conditioning of the Hobby-Eberly Telescope,” in Observatory Operations to Optimize Scientific Return II, P. J. Quinn, ed., Proc. SPIE 4010, 267–278 共2000兲. 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. SPIE 3782, 348 –357 共1999兲. 5. F. Pan and 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. 共to be published兲. 6. F. Pan, J. Burge, and 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. 共to be published兲. 7. R. Shannon and J. Wyant, eds., Applied Optics and Optical Engineering 共Academic, New York, 1992兲. 8. J. Nelson, G. Gabor, L. Hunt, J. Lubliner, and T. Mast, “Stressed mirror polishing: fabrication of an off-axis section of a paraboloid,” Appl. Opt. 19, 2341–2352 共1980兲. 9. G. A. Chanan and M. Troy, “Strehl Ratio and modulation transfer function for segmented mirror telescope as functions of segmented phase error,” Appl. Opt. 38, 6642– 6647 共1999兲. 10. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140 –155 共1998兲. 11. G. Chanan, M. Troy, and E. Sirko, “Phasing discontinuity sensing: a method for phasing segmented mirrors in the infrared,” Appl. Opt. 38, 704 –713 共1999兲. 12. G. Chanan, M. Troy, and 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. SPIE 4003, 188 – 202 共2000兲.
