Chapter 4

Chapter 4, The Point Design

Section 4.2: Requirements Flowdown and Error Budgets

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Now that we have set performance requirements on the telescope as a system, we will examine some of the implications of these requirements on subsystems of the telescope in the following areas:

  1. Image quality
  2. Throughput and emissivity
  3. Environmental issues
  4. Operability

4.2.1 IMAGE QUALITY

Since the beginning of the development of the 8-m-generation of telescopes, it has been clear that the delivered image quality of a telescope is no longer a consequence simply of precise fabrication and stiff structure. With the growth in telescope size, it has become impractical (and probably impossible) to deliver good performance based on precision and stiffness alone. With the growth in the technology of controls, it is no longer necessary. In examining the flowdown of science requirements, we will (1) identify points where precision is indispensable; (2) recognize limits to what can be done with classical techniques; and (3) identify the requirements on control technologies to make up the difference. In the end, we expect to extend this principle, already amply demonstrated by active optical systems on telescopes of the current generation, to systems of wider adaptive control bandwidths. The goal is to achieve diffraction-limited performance on a significantly larger aperture than telescopes of this generation.

4.2.1.1 Main Axes—Drives

The drives of the telescope optics about the azimuth and elevation axes provide the motions that enable tracking of the science target. To achieve images measured in milliarcsec, the rigid body motions of the optics about these axes must be accomplished to comparable precision.

The open-loop performance of the azimuth and elevation axis drives, which produce the motions of the telescope, is not the ultimate source of precision in the movement about those axes. Those drives will be used open-loop to point to the target with sufficient accuracy to be able to acquire a guide star and initiate the active/adaptive control loops. These control loops will then have the task of ensuring that the telescope is pointing at the science target. The open-loop pointing requirement from the science requirements flows down unmodified. The principal additional requirement on the drives coming from the imaging requirement is that they not produce disturbances outside of the control bandwidths of the systems designed to lock the axis of the telescope on the target. This then will translate into a drive capacity to correct for wind torques on the telescope, and into a drive vibration specification that will (1) limit the amplitude of the low frequency errors to the range allowed by the control loop, and (2) limit the upper frequency of the deflections of the mirror, either rigid-body or modal, driven by vibrations associated with the motions of the axes.

The final specifications will be developed in a trade between drive characteristics and control loop characteristics. Table 1 gives a hypothetical allocation that meets the requirements:

FrequencyDrive noise Amplitude
f > 1 Hz< 0.01 arcsec
0.1 < f < 1 Hz< 0.1 arcsec
f < 0.1 Hz< 1 arcsec
Table 1 Main axis drive requirements

4.2.1.2 Image Quality—Primary Mirror

The control of the optical figure of the primary to a precision of tens of nanometers is perhaps the most challenging of the requirements of the telescope. We begin by identifying the disturbances to the primary that threaten the image quality. The following table lists the disturbances and brackets the amplitude range and spatial and temporal scales of those influences. Although there is some overlap in the regimes in which these disturbances operate, they tend to segregate; the larger amplitude effects tend to occur more slowly and have large spatial scales. This is of great importance in designing the control system because it implies that the dynamic range and bandwidth of the controls can be limited. The following section will discuss the disturbances in more quantitative detail.

DisturbanceFrequency range Spatial scale Amplitude range Sensor/LookupCorrection
Surface figure errorsStatic1mm-1m< 100 nmHigh order calibration WFSFabrication, bending adjustments
Temperature(gradients in tel. structure)< 0.001 Hz1-30m10 µ - 10mmEdge, LO WFSCoarse Actuators
Temperature (gradients in segments)<0.01 Hz.01-1.5m<500 nmAO WFSAO DM's (partial correction)
Gravity (on tel. Structure)< 0.1 Hz5-30m10 µ - 10mmLookup, Edge, LO WFSCoarse Actuators
Gravity (on segments)< 0.1 Hz.01-1.5m<200 nmAO WFSAO DM's (partial correction)
Wind loading< 0.1 Hz> 5m< 20 µEdge, LO WFSFine actuators, main axis drives
Wind loading0.1 - 5 Hz5-1m5 µEdge, AO WFSFine actuators, tip-tilt mirror, Deformable secondary
Wind loading> 5 Hz~ 1m<100 nmAO WFSDeformable secondary,AO DMs
Turbulence0.1 - 5 Hz30-5m
Turbulence5 - 50 Hz5-0.2 m
Table 2 Primary mirror-layers of control

4.2.1.2.1 Dynamic Disturbances

The requirement on correction amplitude is set by the external disturbances. As discussed above, the largest of these are quasi-static and will be corrected by actuators acting at very low frequencies. The greater challenge is in the frequency range from about 0.5 Hz and above. Here the principal effects are wind-buffeting and atmospheric turbulence, estimated quantitatively below:

4.2.1.2.2 The Effects of Turbulence

The turbulent statistics of the atmosphere is well characterized, over scales of a few millimeters up to a few meters, by Kolmogorov theory. Median Kolmogorov turbulence at a good site is characterized by a Fried parameter, r0 of about 15 cm (at 550 nm). To be conservative, in our illustrations we will adopt a value of r0 of 12 cm. At such a level of turbulence, the total wavefront phase variance including all spatial frequencies is:

Equation 1

which evaluates to 8.86 microns RMS (root mean square) wavefront error computed over a 30-m aperture. Assuming Kolmogorov theory, it is possible to decompose the RMS wavefront error into a linear superposition of orthogonal functions, the Zernike polynomials, and to describe the statistics of the error in terms of the RMS coefficients of the polynomials in the superposition. The first few terms are tabulated below.

The highest Zernike amplitudes are found at the lowest spatial frequencies, and the amplitudes steadily decrease as the spatial frequency increases. This fact is behind the principle of most adaptive optics (AO) systems: attack the problem by eliminating the terms in order of importance, i.e., modally, in order of increasing spatial frequency.

 Zernike mode FormMean square phase error [rad2]RMS OPD contribution Peak wave-front error RMS residual higher-order
1Tiprho sin (theta).449(D/r0)5/35.84 µ11.7 µ6.66 µ
2Tiltrho cos (theta).449(D/r0)5/35.8411.73.20
3Astigmatism-xrho2 sin (2theta).0232(D/r0)5/31.333.252.91
4Astigmatism-yrho2 cos (2theta).0232(D/r0)5/31.333.252.59
5Focus2rho2 -1.0232(D/r0)5/31.332.302.22
Table 3 The above table states the amplitude, in microns, of the lowest Zernike modes in worse-than-median seeing (r0=12 cm). After correcting only these five, the residual wavefront error has been reduced by a factor of 4 (from 8.86 to 2.22 microns). The contribution of these modes to the wavefront will evolve slowly, making it practical to consider a two-stage correction where a large part of the correction is done by a large stroke, but relatively low speed actuator. Specifically, the largest amplitude correction will be done by a tip-tilt mirror. The peak wavefront error states the maximum wavefront displacement corresponding to the mode that has to be corrected. From these data we can determine the throw requirements of the adaptive element. The residual shows the wavefront error that is left to be corrected in all higher-order terms, after the correction of that mode and all lower-order modes.

It should be pointed out that we have ignored the effects of finite outer scale. There is good evidence that the outer scale is of order 25 m.1 This will tend to reduce the low spatial frequency correlations in phase over large separation and reduce the tip-tilt amplitude in the turbulence- aberrated wavefront. At this outer scale, the amplitude of the wavefront disturbances is already attenuated by about 20% for a spatial period of 10 m.

Although (neglecting outer scale) the large primary diameter drives the RMS tilt amplitude to larger values as D1.67, the timescale on which the mean tilt of the wavefront occurs slows as D/v, where v is a characteristic velocity of the turbulence pattern (depending on the entire turbulence profile with height).

We proceed directly to an examination of the servo error on a tip-tilt compensating mirror to get an estimate of the requirements on that system. The RMS error residual tilt error, in radians, for a tip-tilt servo control system of control bandwidth fc is given by

Equation 2

where fT is the fundamental tilt tracking frequency given by

Equation 3

To take the Pachón turbulence profile as an example, this gives fT= 2.82 Hz. For a control bandwidth of 5 Hz, this results in a tilt error of 2.1 milliarcsec, which is about the level required at 1.6 microns.

4.2.1.2.3 The Effects of Wind

The deformations of the wavefront caused by wind loading on the primary or secondary require similar scrutiny. Our analysis of the interaction between wind and telescope structure is just beginning, but from our current design we can draw some conclusions that serve to set the scale of the problem.

The engineering design for the point design structure conducted by Simpson, Gumpertz, and Heger (SGH) included modal analysis and random response analysis. The structure in question did not detail the final mounting of the mirror segments to the structure, but the motions of the points of support for the mirrors represent the largest scale deflection modes that will affect the figure of the mirror. The Gemini wind loading study (see Section 5.5) offers a unique data set that allows us to see what real wind loadings are on a large telescope mirror. We are studying the extension of these results to larger apertures, and SGH has already applied dynamic loading to the finite element model of the point design structure based on the measured Gemini windloading. These preliminary results for the displacements of the support structure where the mirror is attached show a smooth power spectrum proportional to -2 on which is superimposed the resonant modes of the structure (see Figure 1).

Figure 1 Square root of the power spectrum of displacements of mirror support structure, high wind case (vent gates open).

The lowest resonance corresponds to a deformation in which the entire elevation structure nods back and forth about the elevation axis. The next resonance has an amplitude which is decreased by about a factor of 10. The RMS displacement amplitude of points on the mirror surface is about 18 microns worst case, in the high wind model computed by SGH. The wavefront error is a factor of two greater. We see that the motions are strongly dominated by displacements in the bandwidth below 3 Hz. In this, the high wind case, the wavefront errors caused by buffeting are larger than, but of the same order of magnitude as, the errors caused by turbulence. An inspection of the modes involved shows that the motion is overwhelmingly dominated by rigid body motions of the mirror, i.e., tip-tilt and piston. (Substantial - 40 to 100 micron - translations of the secondary structure accompany the motions of the primary, but that is another part of the story.) With the ventilation of the enclosure decreased, one can expect the corrections required to be dominated by turbulence.

A summary of the amplitudes of the turbulence- and wind-driven disturbances and further discussion of the implications of controlling them with the deformable secondary are presented in Table 6 and the discussion of the adaptive secondary.

4.2.1.2.4 The Control Hierarchy

The subject of controls is addressed in detail in Section 4.8. Here, in looking at the flowdown of the primary mirror performance requirements, we restrict ourselves to proposing a hierarchy of control systems and associated tolerances whose combination fulfills the performance requirements.

Passive support - In the absence of any active control, the primary will distort, owing to flexure of the primary support structure under gravity loading and thermal gradients in that structure. The point design exhibits a worst-case z-deflection of the surface of about 4 mm. This we take as a specification.

Active control (coarse) - The coarse control system, working with inputs from lookup tables alone and using only the coarse actuators, should correct the surface to an RMS surface error of less than 0.5 mm. (The segments mounted on the rafts are assumed not to contribute to this figure.)

Active control (fine) - The fine control system, using information from the edge sensors alone, will control the continuity of the surface at the joints between the segments on the rafts to an RMS surface error of 7 nm.

Active control (fine with WFS (wavefront sensor)) - The edge sensors are not sensitive to the lowest-order modes of the segmented array (e.g., focus, astigmatism). With a low-order natural guide star (NGS) sensor, integrating to filter out atmospheric errors, the fine control system will correct these lowest-order modes, leaving an RMS surface error of 50 nm.

4.2.1.2.5 Figure Error Sources and Residuals

The state of the primary figure is a product of fabrication errors at the shortest scales and multi- level active control at longer scales. The contributions of the primary mirror to the delivered RMS wavefront error documented in this subsection are those deformations beyond the spatial resolution of the aO and AO systems further down the control chain. The baseline spatial resolution of the multi-conjugate adaptive optics (MCAO) and direct Cassegrain AO systems is about 0.5 m (projected on the primary mirror), while corrections on somewhat finer scales will be achieved by the very-high-order AO system. Other residuals attributable to errors in these control systems (WFS measurement noise, servo lag, etc.) will be budgeted against those systems in later sections of this report.

Segment surface quality (fabrication) - The segments will be fabricated by methods that will leave their surfaces conforming to the corresponding part of the desired paraboloid with an RMS wavefront error of 40 nm; this error will be dominated by lower-order errors, specifically curvature and astigmatism relative to the design paraboloid. This estimate is based on the experience of the Keck telescope, extrapolated to a GSMT by Mast et al.2 This is somewhat better than the 60 nm allocated for the Gemini error budget, based on the earlier results with the polishing of the WIYN primary, but is consistent with the polished figures of the Gemini primary mirrors.

Segment thermal gradients - The largest effect of an axial temperature gradient in a segment produces a change in the curvature of the segment. The wavefront error associated with this is, to first order given by:

sigmawvf=CTE·r2·DeltaT .

For a typical CTE of 20 x 10-9 C-1 and r = 0.65 m, and DeltaT = 10 C·m-1, the raw error is 85 nm. However, the effect will be corrected to considerable extent (not yet modeled) by the adaptive mirrors, so we allocate 64 nm.

Segment support - The maximum wavefront error allocated to gravitational distortions of the segment against the finite number of support points is 40 nm.

Segment warping - Static, adjustable warping harnesses on each segment can reduce residual focus, astigmatism, and coma on each segment. We allocate 40 nm for the wavefront residual after adjustment.

Vibration - Wind and irregularities on the axis drive surfaces can generate high frequency vibrations of the structure that will couple to the mirror segments. We allocate 25 nm to this source, based on some estimates of structure stiffness and estimates by CELT.

Error source RMS Wavefront Error
Edge discontinuities

49 nm

Polishing residuals

40

Segment thermal gradients

64

Segment support print-through

40

Vibration

25

Segment warping residual

40

  
Total (RSS)

109

Table 4 Primary Surface Errors (uncorrectable with DM actuators of ~0.5m spacing)

Edge discontinuities - These can arise in a number of ways. This category involves very high spatial frequency contributions to the wavefront error and errors of the segment control itself, so by definition these effects are impossible for the active and adaptive systems to correct. They can arise because of:

  • Errors in edge control systems - vibration, sensor noise, etc., 43 nm allocation (from CELT analysis)
  • Figure errors - changes in radius of curvature or conic, segment to segment, 10 nm allocation
  • Segment positioning errors - 20 nm allocation

Total allocation: 49 nm

Finally, the budget for the uncorrectable figure errors of the primary is summarized in Table 4.

4.2.1.3 Image Quality-Prime Focus

The f/1 beam from the primary passes to either:
  • A prime focus corrector that produces a final f/1 beam at its focal plane
  • A deformable secondary mirror that produces an f/18.75 beam for use at Cassegrain and Nasmyth foci

These two cases are discussed in turn in this section.

The prime focus provides a convenient scale (6.875 arcsec/mm) for a relatively wide field of 20 arcmin, corresponding to 175 mm in diameter. This focus is designed for optical, seeing-limited observations. In the point design concept, the primary mirror figure will not be fully corrected without reference to a wavefront sensor. At prime focus, the concept is for the WFS to use multiple NGSs to derive information about wavefront errors resulting from the combined effects of the ground layer of the atmosphere and low-order deformations of the primary. This mode is referred to as "enhanced seeing" because it effectively improves r0 by eliminating the contribution of an atmospheric layer. See Appendix 4.6.A for a full discussion.

Figure 2  Prime focus corrector for  spectrograph fiber feed (see Section 4.7.1, MOMFOS

The correction to achieve enhanced seeing is accomplished by means of an adaptive mirror built into a multi-element catadioptric prime focus corrector (see Figure 2). When complete, it consists of:

  • 1 fixed mirror
  • 1 deformable mirror
  • 4 lenses
  • 2 ADC prisms
  • 2 mirror surfaces
  • 12 air-glass surfaces

The design image quality of the prime focus enables the fiber feed of the spectrograph to capture the light from seeing-limited images with good efficiency. The RMS geometrical blur degrades from 0.22 to 0.59 arcsec from the center to the edge of the 20 arcmin field. For enhanced-seeing performance of 0.5 arcsec obtained under median atmospheric conditions, over 90% of the energy is captured even at the field edge in the 0.72 arcsec field stop at the entrance to each fiber.

In this mode of use, the requirement on telescope and corrector performance is to preserve the design value of 90% throughput to the fibers of seeing-limited images. The enemies of performance come in two types: attenuation by absorption, and throughput losses due to image degradation at the entrance to the fiber. We require the telescope to degrade the images delivered to the entrance field stop by no more than 10%.

Contributions to image D80 D80 [microns] D80[arcsec]
Corrector design D80600.41
Primary high-frequency surface errors200.14
Corrector flexure, alignment errors- 
Corrector high-frequency surface errors200.14
Table 5 Prime focus image quality allocations

The assumption of Table 5 is that the adaptive corrector will compensate for the low-order errors associated with alignment and flexure errors, as well as the lower-order figure errors in the primary and corrector. The value of 0.14 arcsec is allocated to both mirrors to meet the requirement on degradation. This is comparable to the value used by Gemini and corresponds to models of residuals with RMS surface amplitudes of 21 nm. The air glass surfaces are near focal planes and have very low leverage on image quality. A quick check shows that surfaces with surface irregularity of lambda/4 PP have a completely negligible effect on the images.

4.2.1.4 Image Quality-Secondary Mirror

The secondary mirror contributes to the imaging error budget at all Cassegrain and Nasmyth foci through the quality of its surface finish and other high-spatial frequency residuals arising from stresses in its support and actuation. As an adaptive element, it is capable of correcting its own aberrations (as well as those of the primary and atmosphere), but only up to certain limits in spatial and temporal frequency. As in the case of the primary, we will focus here on the aberrations that will fall outside of the spatial passband of either the secondary itself or other adaptive elements to correct them, because we can immediately label these errors as uncorrectable. Residuals left uncorrected due to other limitations of the aO and AO control systems will be budgeted to the control systems, not to the primary or secondary themselves. However, because the secondary is used both with and without further correction, the size of the allocations for the secondary may vary accordingly.

4.2.1.4.1 Adaptive Secondary-Requirements and Error Budget

The secondary is active/adaptive. The number of actuators is chosen to be Nact = 2400, although this number requires further study. This value is chosen so as not to stress current technology too severely; although the mirror is larger, the density of actuators on the secondary will be smaller than on current secondaries under test at the University of Arizona for the MMT (Multiple Mirror Telescope), and those under development for the LBT (Large Binocular Telescope). At the same time, we set the number of actuators to as high a value as necessary to give robust performance in the application where the secondary is the only adaptive element. (See Example - Direct Cassegrain AO in this section, and Section 4.7.3 on instruments based on this mode.)

The deformable secondary will be used in three distinct modes:

  • (Very) High-Order AO-This is an extrapolation of classical AO. A single NGS is used. Very high correction is the goal (over a very small field). In this case, the secondary will be the first level of correction in a two level system, with another deformable mirror providing higher-order correction.
  • Direct Cassegrain AO-Here the secondary is the only adaptive element. This mode will be well-adapted to thermal IR spectroscopy because, with no additional mirror, background will be minimized, and excellent Strehl ratios can be achieved at intermediate order of correction at these wavelengths.
  • MCAO-The secondary will be the first stage in a multi-DM (deformable mirror) system. The secondary will provide relatively low-order, high-stroke correction for low-altitude turbulence. Higher-order correction will be provided by the three DMs conjugated to different heights in the atmosphere.

Although these modes are very different, the role of the secondary is similar in each, and a single set of requirements can stand for all three modes. The secondary has to satisfy two very different requirements:

  • Excellent surface finish (low polishing residuals)
  • High correction amplitude and rate for low spatial frequency corrections

4.2.1.4.2 Turbulence, Wind, and Dynamic Range Requirements

The secondary is capable of rigid-body tip-tilt correction, and the face sheet can deform to compensate for higher-order wavefront distortions. The throw of the tip-tilt correction must be able to handle the combined tip-tilt errors of the Kolmogorov turbulence and the pointing error caused by wind-buffeting. The throw of the facesheet deformations must be sufficient to handle the combined amplitude of the higher-order Kolmogorov turbulence and the residual distortions of the primary (from wind buffeting, etc.) We use the results from Table 3 for the turbulence contributions (RMS microns). We have derived the results for wind-buffeting by computing the Zernike decomposition of the structural modes analyzed by SGH. Using their figures on the excitation of these modes by windloading, we are able to translate the structural-modal excitations into optical, Zernike-mode excitations. A full description of this analysis is found in Section 4.8. From the Zernike analysis, we can separately compute tip-tilt and higher-order corrections.

The RMS wavefront disturbance contributions from the two sources are summarized in Table 6.

 Tip-tiltHigher order
Turbulence5.8 µ3.2 µ
Wind15.0 µ3.2 µ
RSS Total16.1 µ4.5 µ
Table 6 Summary: turbulence and wind buffeting amplitudes: [RMS wavefront errors]

To handle outliers, we set the requirement at about five times the RSS (root sum squares) total of the two contributions, and divide the requirement by two because we are using a mirror to correct. Thus the throw for the tip-tilt correction corresponds to about 40 microns at the edge of the 2 m secondary, or about 8 arcsec secondary tilt amplitude. For the higher-order terms, we require a throw of about 11 microns peak. The latter number is a reasonable fraction of the current spacing between the deformable secondary face sheet and the reference surface (~ 50 microns) in the deformable secondary being developed by the University of Arizona. Furthermore, the deflection is mostly low-order, so the stiffness of the sheet should not be a problem.

It is worth pointing out some other differences between the effects of the two disturbance sources. The temporal frequency of the tip-tilt from wind-buffeting will be primarily that of the lowest modes of the telescope structure, near 2 Hz. The turbulence-induced tip-tilt will be of lower frequency, in addition to having lower amplitude. Thus in budgeting for the residuals from these sources, the turbulence contribution will be negligible compared to that of wind-buffeting in normal conditions.

Conversely, the amplitude of the contributions of wind-buffeting appear to be dropping steeply with spatial frequency. Almost all of the wind-driven perturbations are tip-tilt and astigmatism. Thus the AO residuals in spatial frequency (fitting errors) in the case of wind-buffeting are expected to be negligible compared to those of turbulence, though we have allocated a small amount for segment edge discontinuities because these will be inevitably uncorrectable.

4.2.1.4.3 The Control Hierarchy

Like the primary, the figure of the secondary can be thought of as the product of several layers of definition and control:

Passive - This is the figure of thin mirror under the effect of gravity and with all position actuators off. The secondary will not be used in this mode except at startup. But startup is also important, and in particular, as the actuators are powered up, the stresses on the mirror must be safe, and the final state of the mirror must be within the capture range of the wavefront sensors so that active/adaptive correction can begin.

Active (local DC control) - The actuators are enabled and receive signals generated by look-up tables or some other source. In principle, the performance is comparable in this mode to a classical solid secondary. In this mode, the errors are high spatial frequency polishing errors, and errors arising from imperfect calibration, hysteresis, etc. of the local control.

Active (WFS control) - In this mode, signals from a WFS command the actuator to correct for wavefront errors that may be arising in the secondary itself, but that more normally arise from the atmosphere or from the distortions of the primary figure. Here, the wavefront errors allotted to the primary are reduced to the high frequency polishing errors.

Adaptive (secondary alone) - This is the way the secondary is used in the Direct Cassegrain AO observing mode. In this mode, the residual wavefront error is the combination of high spatial frequency polishing errors, residuals after applying commanded deformations (i.e., non-ideal response of the secondary), and errors due to limited bandwidth in response to high temporal frequency components of the atmospheric wavefront. This mode is analyzed in some detail in the control section in this chapter.

Adaptive (secondary and DM) - This combination is seen in both MCAO and Very-High-Order AO. The wavefront allocations to the secondary are identical in both cases; just the high frequency polishing errors: 40 nm. All other errors should be correctable, in principle, by the DM.

4.2.1.4.4 Figure Error Sources and Residuals

As in the case of the primary, the secondary (and subsequent DMs) can be assumed to correct for low spatial frequency aberrations, provided that they are within the dynamic range of those elements. This means that the image quality allocation we will assign to the secondary is that of high spatial frequency polishing errors. Again, the failures of the control system to completely correct for low spatial frequency errors arising in the secondary (due to WFS noise, servo lag, etc.) will be assigned to the control systems, and budgeted against them in the following sections.

The allocation for polishing errors will be set to 40 nm wavefront error RMS, the goal set for the fabrication of the MMT adaptive secondary.3 In addition, based somewhat on the experience with the MMT prototype, we allocate an additional 20 nm RMS for wavefront errors arising because of print through of the actuators onto the facesheet.

4.2.1.4.5 Summary of Requirements on Secondary

We summarize the active control and figure quality requirements for the secondary in Table 7:

CorrectionFrequencyAmplitude
Tip-tilt (entire secondary structure)0-3 Hz8.0 arcsec (peak)
Higher order corrections0-30 Hz11 microns (peak)
High Frequency Surface errors0< 40 nm RMS
Table 7 Adaptive secondary requirements

4.2.1.5 Alignment Effects

Alignment should not seriously diminish the design image quality of each focus. In the case of those instruments operating at the diffraction-limited foci, alignment should not increase the FWHM (full width half-maximum ) of the diffraction-limited core by more than 5%. In the case of the instruments, the flowdown of this requirement on the instrument will depend on the design.

The alignment effects between the primary and secondary are of low-order and can be corrected to a high degree by different means of compensation: decenter by tilt of the secondary; and residual astigmatism by deformation of the adaptive secondary. In the cases of some of the implementations, further low-order compensation is available at DMs in the AO systems. The compensation on-axis will be virtually perfect. We allocate a small fraction of the raw misalignment error-based on estimates of secondary motion with wind-buffeting-to residual field dependent error in the wavefront. Allocation: 30 nm

4.2.1.6 Magnification (Focal Plane Scale)

The total magnification of the telescope at the focal plane is a parameter that must be controlled at a high level not only to permit astrometric observations, but also to assure good image quality at the edge of the field. Changes in the magnification will elongate images differentially from the point at which guiding is taking place. For images of .01 arcsec FWMH at the edge of a 2 arcmin field, limiting elongation to no more than 10% requires that the scale stay fixed to within 1 part in 60,000. Changes in any combinations of the radii of curvature and spacing of primary and secondary can change the scale. Suppose, for example, that the primary curvature changes and the secondary spacing compensates to refocus (while keeping the secondary curvature constant). To keep the images at the edge of the field from elongating by more than 10% in this example, the primary radius of curvature must not change by more than .55 mm (i.e., to 1 part in 105 during the exposure).

This level of control of the primary is unlikely to be possible directly. Multiple wavefront sensors in the focal plane will be required to detect dilation of the plate scale at this small but significant level.

4.2.1.7 Control System

The control system is discussed in Section 4.8. Here we restrict ourselves to identifying the chief error terms that the control system should minimize. The estimates and allocations are stated in the top-down budget.

The telescope is reconfigured to work in each of the operating modes. The numerical goals are different in each, but the nature of the goal and its implications for performance will be discussed in general here.

A control system can be thought of as containing three parts: sensing, amplifying, and correcting. Ideally, the system: (1) senses the current state of the telescope, detecting errors in its performance; (2) amplifies the detected signals, filtering it and otherwise transforming it to compute corrections that it is capable of making with the array of actuators at its disposal; and (3) commands the actuators to make the corrections.

At each stage, errors arise which limit the degree to which the correction can be achieved, such as: (1) sensor noise, gain errors, delay, blindness (in-sensitivity to certain errors); (2) processing delays and transformation errors; and (3) inadequacies of the actuators to fully correct noise, nonlinearities (including hysteresis), and fitting errors.

At this stage, it is convenient to divide the problem among the different observing modes, due to the very different levels of control required:

  • Prime focus (enhanced seeing and low-order primary figure correction)
  • Cassegrain (AO modes)

4.2.1.7.1 Prime Focus

There are two levels of sensors and two levels of actuators involved in the control of image quality at prime focus:

Sensors:
  • Edge sensors-measures edge-to-edge discontinuities at segment gaps
  • WFS-measures low-order deformations of the overall wavefront; senses low-order deformations of the primary that are not well detected by the edge sensors
Actuators:
  • Segment support actuators (may be two-stage but ultimately achieve ~20 nm resolution)
  • Deformable corrector mirror-(order 1000-2000, wavefront correction amplitude ~ 20 microns)
Contributions to wavefront error (in approximate order of decreasing importance):
  • Residual atmospheric seeing - This will be the dominant error source. In the enhanced seeing compensation scheme proposed (see Appendix 4.6.A), multiple NGSs are used to detect low-order deformations of the primary and ground-layer atmospheric contributions to the wavefront. The contributions of high altitude turbulence will be averaged over. Unfortunately, the typical ground layer is actually several kilometers thick, with substantial turbulence throughout. The contribution of this thick layer cannot be fully corrected with a deformable element conjugate to the ground only. The uncorrected ground layer turbulence, together with the high-altitude turbulence, leave a significant residual seeing which is the dominant contribution to image quality in this mode. In conditions of median seeing, it is anticipated that, neglecting any other errors, these residual contributions will produce image FHWM of about 0.5 arcsec in median seeing in the V-band.
  • Wavefront sensor noise - This requires full modeling to estimate properly. The WFS relies on NGSs. Because the corrections required are of relatively low order and low frequency (5 Hz update rate), it is expected that stars of adequate brightness can be found.
  • Servo lag - This is related to the wavefront noise specification and will be part of what has to be modeled with real stars. Computation is not the main issue; the servo lag will be set by the integration time on the guide star.
  • Edge sensor drift and noise - This aspect has been extensively studied by the CELT group and appears to be well understood. Sensor noise will contribute no more than 30 nm to the wavefront.
  • Edge actuator noise - 10 nm (from CELT).

4.2.1.7.2 Cassegrain and Nasmyth Foci

In the present study, all Cassegrain and Nasmyth modes are AO-corrected and are currently aimed at wavelengths longward of about 1.0 micron. This is not a requirement, but rather a reflection that this is where the priority is (given the prime focus capability in the visible).

The control system is again multi-level. There is some variation in the configuration, depending on the modes used. Here we list the common elements of the system and then, mode-by-mode, we enumerate the differences.

Control elements common to all modes
Sensors
Primary segment edge sensors
WFS (visible light) There will be multiple sensors-peripheral sensors of low bandwidth, plus higher-order AO WFS. (These may be different physical sensors with different properties in each mode.)
Actuators
Primary segment support actuators
Secondary tip-tilt and focus
Deformable secondary
Mode-specific control elements
Direct Cassegrain AO - Mid-IR spectroscopy
No additional DM or tip-tilt mirror; only deformable secondary
NGS WFS
Narrow field high-order AO
Tip-tilt mirror
Very-high-order DM (MEMS) (16,000-65,000 elements)
NGS WFS
MCAO-Imaging and Spectroscopy
Tip-tilt mirror
Multiple DMs conjugated to different atmospheric heights
WFS for multiple LGS and multiple NGS

4.2.1.7.3 Adaptive Optics-Contributions to Wavefront Error

As an example of the construction of an AO error budget, we will consider here the case of the Direct Cassegrain AO mode. In the present study, this mode is aimed at high Strehl for either high dynamic-range imaging (coronagraphy) or high spectral resolution IR spectroscopy. This is in most respects the simplest AO mode to analyze, and serves as an introduction to the subject.

The implementation of the other two AO modes, MCAO and near-IR high-order AO, are described in Sections 4.6.2 and 4.7.6.2. The flowdown from science requirements to error budget allocations is described there. Many of the concepts discussed there are introduced here.

A full system description for the mid-IR mode is not discussed elsewhere in this document, in part because it is very simple. A full discussion of the Infrared Echelle Spectrographs designed to work in this mode are discussed in Section 4.7.3.

4.2.1.7.4 Example: Direct Cassegrain AO

Beyond about 2.0 microns, the requirements on the AO system are relaxed enormously by the increase of r0 with wavelength. For median seeing, r0 is 15 cm at 0.55 microns, but 80 cm at 2 microns. Because r0 is proportional to the spacing, referenced to the primary, of the actuators required for a given level of phase correction, this immediately implies a decrease by a factor of 28 = (80/15)2 in the order of correction required.

ParameterCommentValue
DAperture diameter30 m
r0Fried parameter12 cm @ 550 nm
NDOFNumber of DM actuators2400
fsSample rate500 Hz
MMultiplying factor15
fgGreenwood frequency25 Hz @ 550 nm
theta0Isoplanatic angle2.75 arcsec @ 550 nm
MgsMagnitude at 550nm12
fBandwidth of WFS300 nm
lambdawfsWavelength of WFS600 nm
sigmarRead noise of WFS5 electrons/read
NbBackground of WFS1 electron/second
Table 8 Atmospheric and instrumental parameters used in the computations of the Direct Cassegrain AO system described here.

The thermal IR is thus a niche for very good AO correction with a single deformable element with an intermediate number of actuators, such as the deformable secondary. The operating configuration consists of the primary and secondary feeding the focal plane. Near the focal plane is a dichroic beamsplitter that sends the optical light to a WFS from a NGS. The order of correction provided by the secondary is a critical parameter whose final value will require detailed trade studies among technological feasibility, cost, and desired performance. Let us adopt 2400 actuators for this evaluation. (This is well below the spatial density-at the secondary-of actuators in the deformable secondary under development for the MMT by the University of Arizona /Arcetri collaboration, and thus is likely to be achievable on the timescale of a GSMT project.)

The Pachón seeing profile is characterized by a median value of r0 = 15 cm at 550 nm. For purposes of illustration, we take a somewhat more conservative value, r0 = 12 cm, and evaluate the contributions at the zenith. Other parameters used in the example are given in Table 8.

The formulas given below are for the phase error in radians. Recall that at high Strehl, the Strehl ratio is given by: S=exp(-sigma2) , where sigma is the phase error. However, expressed as optical path difference (OPD) in nanometers, the errors are achromatic. Thus, to make the results more general, we express them as an OPD and convert to phase when we know the wavelength and want to compute the Strehl.

The contributions to the AO control error budget are as follows, in approximate order of decreasing importance (Ellerbroek, private communication, Keck (1996)):4

  • Fitting error - This reflects the degree to which in an otherwise perfect system, the deformable element is unable to deform in such a way as to correct the incident wavefront, due to limitations in the number of degrees of freedom. The residual, expressed in terms of variance of the error in phase evaluated over the pupil, is:
    Equation 4
    where cf is the fitting error coefficient, dependent on actuator and WFS geometry and on deformable mirror influence function. 0.28 is a typical value for a continuous face- sheet DM with a square grid of actuators. The actuator spacing, referenced to the primary, is d. Of course, d is related to the total number of actuators or degrees of freedom by
    Equation 5

    The resulting RMS error, converted to a displacement-independent of wavelength, is 180 nm.

  • Servo lag - This describes the error in the application of the wavefront correction due to the finite bandwidth and time delay of the control system. The formula for the error is:
    Equation 6

    where tau is the time delay in the correction 1/2pif and is the bandwidth of the control system. The bandwidth and sampling frequency are related by f=fs / M where M is a multiplier of typical value 15. The timescale for variations in the atmospheric turbulence is expressed by the Greenwood frequency fg, which is given by an integral over height of the turbulence amplitude C2n(h) and velocity v2(h) profiles. The dependence on zenith distance is included via thetaz:

    Equation 7

    The Greenwood frequency has been evaluated for the Pachón seeing profile and is typically 25 Hz. With this result and a 500 Hz sampling rate producing a servo bandwidth of about 33 Hz, the servo lag error (expressed as a wavefront displacement) is 72.2 nm.

  • Anisoplanatism - This is not a wavefront error in the direction of the guide star, but rather a measure of the degradation of the wavefront as one considers objects an angular distance theta away from the guide star. As in the case of the Greenwood frequency, this error depends not on the level of turbulence alone but on its distribution in altitude. It can also be written in terms of an integral over the turbulence profile, here weighted by height:
    Equation 8 where theta is the angle from the guide star, and theta0 is the isoplanatic angle which can be computed from the turbulence profile as follows:
    Equation 9 .
    Note that if the turbulence were located at the telescope pupil, the isoplanatic angle would be infinite. Physically this is because the wavefronts of starlight from all directions would suffer the same phase distortion. Thus the wavefront correction for one field angle would work for all field angles. For a single layer of turbulence, the isoplanatic angle is inversely proportional to the height. It is also proportional to r0.

    At the short wavelength end of the thermal IR (2.2 microns), the isoplanatic angle-under conditions of median seeing with the Pachón profile-is 16.3 arcsec at the zenith (cf. 2.75 arcsec at 0.55 microns).

    For a field angle theta=1 arcsec, the OPD is 37.7 nm (independent of wavelength).

  • Natural vs. laser guide stars (LGSs) - In the present discussion, the observing mode will use NGSs. In the context of anisoplanatism, it is interesting to draw attention to two related LGS contributions to the error budget that using NGSs avoids. First, "focal anisoplanatism," which arises because the artificial guide star is not at an infinite distance and therefore does not sample the atmosphere exactly the same way the natural star does. This is more descriptively dubbed the "cone effect." A second effect is finite-source- size anisoplanatism, which introduces errors in the wavefront because the cones of light contributing to the wavefront measurement arrive from slightly different angles.

    The geometry factors are much more complex for focal than normal anisoplanatism, and are expressed as a series of integrals over the turbulence profile. The reader is referred to Tyler for a complete discussion.5 The RMS phase error for focal anisoplanatism can be written

    Equation 10 where d0 is given by: d0=2.49theta0Hsec(thetaz),
    approximately valid when the guide star is at a height H much greater than that of the turbulence. We note that this is problem that increases rapidly with aperture. The equivalent RMS wavefront displacement with a single sodium beacon would be 544 nm, overwhelming the other sources of error. This is another reason for the increasing importance of MCAO for large telescopes: MCAO reduces the field restriction of classical AO and permits use of artificial guide stars as well, using multiple beacons to more fully sample the turbulence volume.

    The effect of finite source size can be expressed as
    Equation 11

    where thetags is the angular diameter of the laser beacon.

    This is a small effect that does not increase with aperture. The effective wavefront error for this case is 19 nm.

  • Wavefront sensor noise - Returning to error sources that will affect us regardless of the nature of the guide star, we consider the effect of noise in the wavefront measurement process. The RMS phase noise arising in the WFS can be written
    Equation 12

    where ds is the width of a sub-aperture, lambda and lambdas are the science and WFS wavelengths respectively, and SNR is the signal-to-noise ratio of a single WFS measurement, given by the usual

    Equation 13

    where Ns and Nb are the number of signal and background photoevents respectively, and sigmar is the read noise per pixel. The WFS is in a servo loop, so the phase noise is modified by the reconstructor noise gain and the servo loop gains:

    GR=0.2 + 0.08·ln(NDOF)

    is the reconstructor gain, where NDOF is the number of degrees of freedom of the deformable mirror and

    Equation 14

    is the servo loop gain for a closed loop servo bandwidth of f, and fS is the WFS sample rate. Thus the final effect of the WFS noise on the RMS phase error is

    Equation 15

    For a guide star of V-magnitude 12, a visible light WFS of typical noise characteristics, 500 Hz sampling, and a 33 Hz servo bandwidth, the raw phase noise is equivalent to a wavefront error of 98 nm. Thus, the final wavefront error including the gain factors is 41 nm.

  • Telescope and implementation errors - These include sizeable errors from the telescope and a number of additional smaller implementation effects listed below.

    • Telescope - 135 nm. These are the same errors discussed in the context of the primary's and secondary's roles in image quality.
      • High spatial frequency polishing errors in the primary and secondary
      • Inter-segment discontinuities produced by limitations in the control of those discontinuities, e.g., sensor error and noise, finite gain, high frequency vibrations of the segments
      • Segment figure mismatches at the edges, clocking of segments
      • Print through of the segment support hardware (gravity dependent)
      • Alignment error
      • Self-induced seeing (thermal effects around mirror and telescope)
      • Acquisition mirror residuals

    • Instrument Limitations - 30 nm
      • Noncommon path errors in the AO system
      • Calibration errors
      • Uncorrected residuals in instrument optics (pre-slit foreoptics residual)

    • Additional AO system implementation errors - 113 nm
      • Wind shake residuals
      • Uncorrected internal optics
      • Calibration errors

The total allocation for the telescope and implementation errors is 179 nm RMS.

All error terms discussed above are summarized in the error budget (EB3) (see Image Quality Error Budgets in this section). The net wavefront error, combining terms in the RSS sense, is 279 nm RMS. We have expressed the errors as wavefront deformations or OPD errors, and as such they are independent of wavelength. The Strehl ratio, however, depends on phase error as given in the case of high Strehl, by the approximate formula:

Equation 16

We summarize the performance of the system under consideration as a function of wavelength in Table 9.

Lambda [microns] Strehl
1.2 0.14
1.6 0.33
2.2 0.56
5.0 0.89
10.0 0.97
20.0 0.99
Table 9 Strehl vs lambda

As the wavelength increases, we get performance that ranges from modest to excellent. By the time the wavelength is 5 microns, we achieve a delivered Strehl of 0.89, even with the limited correction provided by the deformable secondary alone. At 2.2 microns, the delivered Strehl is still 0.56, comfortably above the value of 0.3 assumed to derive the science case.

The error budgets for image quality are described under 4.2.5 in this chapter. The error budget tables themselves are appended to the end of the section.

4.2.2 THROUGHPUT AND EMISSIVITY

The requirements of throughput and emissivity have been discussed in Chapter 3. The basic throughput requirements at the Cassegrain focus require no further discussion. The prime focus instrument position requires additional optics of some complexity, so it is worth reviewing our performance goals for them.

4.2.2.1 Prime Focus

For the mirror surfaces, we assumed a protected silver coating that will have a reflectivity at 600 nm of 0.97. The anti-reflection coated air-glass surfaces will each have a transmission of 0.99. We make preliminary allocations of the throughput (at 600 nm) in Table 10:

ContributionThroughput to fibers
Transparency: 
Primary reflectance0.97
Corrector baffle0.9
Corrector element transmissions0.83
Imaging: 
Primary high-frequency surface errors0.95
Design imaging losses0.9
Corrector flexure, alignment errors-
Corrector high-frequency surface errors0.97
Final throughput 0.58
Table 10 Prime Focus Throughput.

The prime focus is used only for nonthermal wavelengths, so emissivity is not an issue.

4.2.2.1.1 Throughput and Emissivity in the Near-IR - MCAO

For the MCAO system, we have followed the requirements developed by Gemini, as presented in Table 11.

 Wavelength [microns]
Throughput/Emissivity1.01.652.22.2 (emiss.)
Per reflection0.9790.9860.9870.013
Per air-to-glass interface0.9910.9870.9890.011
Beamsplitter0.9300.9600.9650.022
Overall for 9 reflections0.8260.8810.8890.117
Overall (no ADC)0.7680.8450.8580.139
For 4 ADC air-glass interfaces0.9650.9490.9570.044
Overall with ADC0.7420.8020.8210.183
Table 11 MCAO throughput and emissivity

With aggressive use of AR coatings, good performance can be had despite the large number of warm optics.

4.2.3 ENVIRONMENTAL ISSUES

Here is a partial list of the site-related characteristics that have to be evaluated (Section 5.2 discusses the site testing program in greater detail):

  • Cloud cover
  • Turbulence, wind profile (seeing)
  • Surface wind (wind-buffeting)
  • Precipitable water vapor (PWV) (thermal background and transparency)
  • Latitude
  • Light pollution (present and projected)
  • Seismic effects
  • Accessibility

It is worth mentioning here that the prioritization of the site issues is not complete, specifically because there are tensions that will only be resolved when the prioritization among science goals is complete. Specifically, the need for low precipitable water vapor (PWV) is a driver for a high altitude site, whereas the desire to minimize wind-buffeting will be a driver for a lower altitude site.

A similar conflict exists in the specifications of the dome. Is wind-flushing (good) more important than wind-buffeting (bad)? It is clear that the use of adjustable wind-gates will be required to balance these criteria empirically.

Windloading is a key factor in the design of the GSMT. Section 5.5 of this book is devoted entirely to our progress in understanding this issue.

4.2.4 OPERABILITY

The ability to operate the GSMT productively depends on many factors. Prioritizing them will be driven by a model for the use of the telescope. In this section, we restrict ourselves to identifying a few aspects of the design to which we give high priority.

Flexibility - The ability to support diverse observing modes is one of the hallmarks of the point design.

Reconfigurability - The ability of the telescope to be transformed into another data-taking mode without rebuilding. Specifically, GSMT will provide for the simultaneous co-location of instruments at various foci. In the point design, we have included provision for instruments at:

  • Prime focus
  • Co-moving Cassegrain focus
  • Fixed Cassegrain focus (room for more than one interchangeable instrument)
  • Nasmyth focus
  • MCAO focus (at Nasmyth)

Maintainability - Because of the complexity of the GSMT, maintainability will be one of the key design criteria throughout the design. Most of the complexity of the system will reside in the interconnected multiplicity of active modular units (segments, actuators, sensors, and local controllers). Manual troubleshooting and repair of a system of 600+ segments each with a substantial number of internal active elements is simply not practical. It is also not practical to think about how to accomplish this after the system is designed. However, because of the modularity of these units, it is relatively easy to provide spares, and the extra cost of designing maintainability will be amortized over a large number of virtually identical units. The other key elements are the elimination of single point failure modes and the requirement that the system monitor and diagnose itself. We think this leads to a redundant networked architecture wherein the failure of either a module (sensor, actuator) or a signal pathway (1) does not cause the system to crash, and (2) permits identification of the failure which has occurred.

Single component failures will, if the system is properly designed, produce a small degradation in performance only. That degradation will likely be invisible to the operator, so it is essential that the detection and reporting of these failures be automatic.

Finally, the now familiar concept of multiple "hot" instruments will be a part of the GSMT philosophy, which increases the robustness against failure of the system overall.

As discussed in the Science Requirements, the contribution of operations to lifetime costs is enormous and should play an important role in guiding design decisions.

4.2.5 IMAGE QUALITY ERROR BUDGETS

Summary top-down error budgets follow for the residual errors presented by the primary and secondary mirrors, and then for the four operating modes:

  • EB1 - Primary and secondary mirror residual wavefront error budget
  • Image quality error budget for:
    • EB2 - Prime focus AO
    • EB3 - Direct Cassegrain AO
    • EB4 - Narrow field high-order AO
    • EB5 - MCAO

4.2.5.1 Explanatory Notes

The supporting material for the different parts of the error budget is located in different sections of this GSMT "book." In addition, we draw attention here to various differences in approach between the different error budgets.

EB1 - To simplify the later budgets, we present a separate error budget for the contributions to the image quality by the primary and secondary mirrors. These contributions will be very similar in the error budgets EB2, EB3, and EB5. The supporting material for this section is found in this chapter under 4.2.1.2.5. In this same budget, we also present adjusted estimates for the mirrors to use with EB4, where higher-order corrections are available.

EB2 - Although the prime focus AO mode uses adaptive techniques, the image quality is dominated by the atmosphere and by the optics of the wide field prime focus corrector. The image quality data are presented in terms of arcsec measures of the 50% energy diameter. Recall that the issue for this mode is efficient coupling of the light into fibers, rather than point spread functions (PSFs) on a scientific focal plane.

EB3 - The Direct Cassegrain AO uses the error budget EB1 for the primary and secondary. The other contributions are discussed in considerable detail in 4.2.1.7.4.

EB4 - The narrow field high-order AO mode looks very similar to that of EB3, except that a deformable element of extremely high-order is placed near the end of the optical train. This adaptive element performs correction, using a NGS, in the classical way over an extremely narrow field. The form of the error budget is therefore very similar to that of EB3. The supporting material for this section is in Section 4.7.6.2, where a number of hypothetical configurations are discussed.

For the purpose of presenting an error budget, we have chosen the most conservative of the configurations discussed in Section 4.7.6.2. We work from the column corresponding to a wavelength of 2.2 microns in Table 2 in Section 4.7.6.2. The deformable mirror is projected to be a MEMS (Micro Electro Mechanical System) mirror with a square array of 147 x 147 mirror segments. We adopt also adopt 42 Hz for the control bandwidth and a guide star magnitude of 10.4 (see Table 3).

With the circular pupil of the GMST mapped onto it, the MEMS mirror will provide 16900 degrees of freedom. This corresponds to 20 cm spacing of the "actuators" on the primary mirror and permits correcting the primary and secondary mirror errors to a much greater degree than, for example, the Direct Cassegrain AO mode in which the actuator spacing is 54 cm. With over seven times as many actuators available per segment with the MEMS DM, we project a factor of three reduction in the RMS error for the larger spatial scale aberrations, and a factor of two for the smaller scale ones. These numbers are presented in the second column of EB1 and are used for the telescope aberrations in EB4. It should be pointed out that this adjustment cannot be made with authority until the spatial spectrum of the aberrations is characterized in more detail. For the purposes of this presentation they should be regarded as plausible allotments. Similar adjustments are made for other terms in the error budget. We have also dropped an allocation for tip-tilt error because with a bright guide star and a small tip-tilt mirror, the tip-tilt error can be made negligible.

Note that we do not achieve the full 90% Strehl that is set as a goal in Section 4.7.6.2, even after some aggressive improvement in the implementation errors. At 2.2 microns and the given error allocations, the achieved Strehl is 83%.

EB5 - The MCAO mode is discussed in detail in Section 4.6.2 and the data in the error budget are extracted from that section.

REFERENCES

  1. Martin, F., Tokovinin, A., Ziad, R., Conan, R., Borgnino, J., Avila, R., Agabi, A., Sarazin, M., "First statistical data on wavefront outer scale at La Silla observatory from the GSM instrument", A&A, 336, L49-L52 (1998).

  2. Mast, T.S., Nelson, J.E., Sommargren, G., "Primary Mirror Segment Fabrication for CELT", Proceedings of the SPIE, 4003, 2000a.

  3. Martin, H.M., Burge, J.H., Vecchio, C., Dettman, L.R., Miller, S.M., Smith, B., Wildi, F., "Optical Fabrication of the MMT adaptive secondary mirror", SPIE Proc. 4007, 502 (2000).

  4. Keck Observatory Report #208, "Adaptive Optics for Keck Observatory", revised January, 1996.

  5. Tyler, G, "Rapid evaluation of d0: the effective diameter of a laser guide star adaptive optics system", JOSA A11, 325-338 (1994).


October 2002