Chapter 4

Chapter 4, The Point Design

Section Design of a Narrow Field AO Coronagraph

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The collecting aperture and angular resolution potentially provided by GSMT will yield significant new opportunities for directly detecting extra-solar planets and studying circumstellar disks (see Science Case for a GSMT) if the blurring induced by atmospheric turbulence can be compensated by an adaptive optics (AO) system. The requirements for this system are at once highly conventional and entirely revolutionary. Narrow field compensation using a bright, on-axis natural guide star (NGS) has been achieved. However, achieving the science goals for high contrast imaging outlined in Chapter 2 will require Strehl ratios and an order of correction well beyond what has yet been demonstrated in currently operating AO systems.

This section summarizes our initial thoughts on the requirements, design, and technology issues for a Very High Order AO (VHOAO) coronagraph for GSMT. Section reviews the top- level science requirements, derives a preliminary set of the implied requirements for the AO system and its component parts, and outlines some of the interface requirements associated with implementing such a system as part of GSMT. Section describes the conceptual opto- mechanical design, which in comparison with the multi-conjugate adaptive optics (MCAO) design is relatively compact and almost straightforward. The same cannot be said for the AO components and control algorithms. Section summarizes some of the key issues for near-term work in these areas.

Assuming we can achieve the performance outlined in this section, we expect to be able to achieve contrast ratios of 107 and 108 at angular distances from a central source of 0.1" to 0.4" and 0.4" to 1.2" at wavelengths 2.2 microns and 10 microns respectively (see Section, High Dynamic Range Imaging). PERFORMANCE AND IMPLEMENTATION REQUIREMENTS Science Requirements Summary

Table 1 Draft science requirements for VHOAO point design.
Wavelength range, µm1.0-5.0
Field of view diameter, arcsec2
Mean on-axis Strehl
(with a bright on-axis guide star and
median seeing conditions)
0.9 in K (initially)
0.9 in J(potential upgrade)
PSF stability and predictabilityTBD (excellent)
Optical transmittance0.80
Emissivity< 20% (2.0-5.0 µm)
Atmospheric dispersion compensationTBD
Coronagraphic elementsOcculting disk
Lyot stop

Table 1 summarizes the top-level science requirements we have assumed to develop a conceptual design for an AO coronagraph. Faint companions can only be detected if nearly all of the light from the central star is blocked by an occulting disk, and this in turn requires that an AO system deliver extremely high Strehl ratios. The lower the Strehl ratio, the longer the integration time that will be needed to average out the time-varying speckles in the uncompensated halo of the point spread function (PSF). Additional study based on field tests will be necessary to map out the relationship between Strehl ratio, integration time, and PSF uniformity, and to revise current idealized models of AO performance.

This being said, it is clear that very high Strehl ratios are desirable for an AO coronagraph. We take a Strehl of 0.9 as a lower limit for acceptable performance; even at this Strehl, 10% of the starlight will reside in the seeing-limited halo, where any residual non-uniformities in the PSF can be confused with a planet. Once this Strehl ratio has been specified, the free parameters that determine the necessary order of correction for the AO system are the atmospheric seeing conditions and the wavelength of operation. We will assume average seeing at Cerro Pachón (r0 = 0.16 m in V-band), and we will investigate the AO requirements necessary to achieve a Strehl of 0.9 in J, H, or K band. The performance of any specific design will vary significantly with the seeing, but these sample cases give an indication of the type of systems that will be required.

The overall spectral band is restricted to the range of 1-5 µm. Very high Strehls at longer wavelengths can be achieved using the adaptive secondary alone (see Section 4.2), while the AO parameters needed to achieve high Strehl ratios at shorter wavelength appear too extreme to consider at this time. Other implementation issues, such as atmospheric dispersion compensation and the quality of the telescope optics, also become very difficult at shorter wavelengths.

The optical transmittance and emissivity values listed in Table 1 are for the AO system and coronagraph taken together, and allow for the additional surfaces necessary for atmospheric dispersion compensation, wavefront sensing (WFS), and an intermediate image plane. The requirements for atmospheric dispersion compensation are not determined at this time, but excellent compensation is feasible in this spectral passband (see the MCAO design described in Section 4.6.2). AO Requirements Flowdown AO System Parameters

Table 2 VHOAO system parameters vs. wavelength (Strehl = 0.9).
Wavelength, µm1.25 (J)1.65 (H)2.2 (K)5.0
sigmaOPD, nm64.685.2113.7258.3
Degrees of freedom, Nact65.6e333.7e316.9e32.36e3
Order of correction290220721472552

Control bandwidth, Hz82.759.342.015.7
sigmanoise, nm32.342.656.8129.2
sigmaimplementation, nm32.342.656.8129.2

Table 2 is a top/level summary of the VHOAO system's wavefront error budget and the associated AO parameters, including allocations for: (1) fitting error (order of correction); (2) servo lag (control loop bandwidth); (3) WFS measurement noise; and (4) higher-order effects and implementation errors. Note that anisoplanatic wavefront errors have not been considered, because we are interested in the Strehl ratio for the bright on-axis star that must be occulted rather than the Strehl for the slightly off-axis companion. The total allowable root mean square (RMS) wavefront error for each case has been computed using the Marechal approximation, S = exp [-(2pisigmaOPD/lambda)2], for an overall Strehl S of 0.9. The overall value of sigmaOPD has then been divided (in an root sum squares (RSS) sense) into four equal allocations for the different error sources, with a corresponding Strehl allocation of 0.91/4 = 0.974. This exercise provides at least an initial estimate of the VHOAO system parameters, but more detailed trade studies must eventually be performed to optimally balance these allocations. We recommend that these studies be based upon a complete model of companion detection, not simply on maximizing the Strehl ratio.

As for the MCAO system described in Section 4.6.2, the required number of degrees of freedom Nact is based on the usual fitting error formula, sigmafit = lambda[0.28 (d/r0)5/3]>1/2/(2pi), where d = D/(4Nact/pi)1/2 is the actuator pitch for a square actuator geometry. The results for lambda = 5 µ indicate that a separate VHOAO system will not be required beyond this wavelength, because approximately this number of actuators will be provided by the adaptive secondary. The trend from lambda = 2.2 µm to lambda = 1.25 µm suggests the difficulty of implementing a GSMT VHOAO system for visible wavelengths, and we will restrict ourselves to the J, H, and K spectral bands for the remainder of this discussion.

The control loop bandwidth requirement is based on the formula sigmaservo = lambda(fg/f)5/6/(2pi), where fg is the Greenwood frequency (about 25 Hz in V-band for Cerro Pachón) and f is the -3dB closed- loop servo bandwidth. The results obtained range from 42 to 83 Hz and are not exceptional, except that they imply very high pixel read rates and signal processing requirements due to the order of the AO system, as described further in Section below.

The RMS OPD (optical path difference) allocated for WFS measurement noise effects ranges from 32 to 57 nm. The WFS signal levels necessary to achieve this level of accuracy will depend upon the details of the sensor design and the AO control loop parameters. Sample values for one particular design are given in Table 3 below.

Finally, the RMS OPD for implementation error sources is allocated the identical range from 32 to 57 nm. This value includes all of the implementation error sources listed in the lower section of the error budget presented previously for the MCAO system, with the sole exception of the term for laser guide star (LGS) focus tracking. We have not attempted to allocate values for the individual error sources at this time, because to do so meaningfully would require a much better understanding of the static and dynamic wavefront errors associated with the telescope. We also hypothesize that more detailed simulations may indicate looser tolerances for low spatial frequency errors such as tilt and focus, because intuitively these low-order aberrations will scatter less starlight outside of the occulting disk. Supervisory algorithms will be necessary to minimize calibration errors by tracking the long-term variations in a variety of gains and biases associated with the AO control loop. Some of these quantities include the WFS tilt measurement gain and DM (deformable mirror)-to-WFS misregistration. AO Component Parameters

Beyond the top-level AO system parameters summarized in Table 2, some of the significant first- order AO component parameters for VHOAO include (1) the control loop update rate, (2) the WFS detector array size and pixel read rate, (3) signal processing requirements for wavefront reconstruction, and (4) WFS signal level requirements for a 0.9 Strehl. These quantities are less fundamental than the order of the AO system and the closed-loop bandwidth, because their exact values will depend upon choices made in the design of the WFS and the AO control loop. Initial estimates based upon a somewhat conservative and conventional point design are still useful as an indication of the advances in AO component technologies that will be required for VHOAO on GSMT.

Table 3 summarizes a set of component parameters that have been computed for the case of a Shack-Hartmann WFS and a classical least squares wavefront reconstruction algorithm implemented using sparse matrix techniques. The following subsections describe the derivation of these values, together with an indication of how the values might vary for other design concepts.

Table 3 VHOAO component parameters vs. wavelength, Strehl=0.9. Please see the text for a description of the underlying assumptions. PDE denotes photodetection events.
Wavelength, µm1.25 (J)1.65 (H)2.20 (K)
Control loop update rate, Hz250018001250
WFS array size, digitized pixels580241422942
WFS read rate, Mpixels/sec652

Signal processing requirements, Gflops

22.9 (TBR)7.48 (TBR)2.34 (TBR)
DM actuator dynamic range1.0-1.5 µm (TBR)
Guide star signal level, PDEs/m2/sec12.7e63.8e61.2e6
Guide star magnitude7.99.210.4

Control loop update rate

The closed-loop bandwidth of AO systems is limited by the signal processing latency in reading WFS measurements and computing DM actuator commands. For high-order AO systems that employ CCD arrays for wavefront sensing detectors, the latency associated with sequentially reading and digitizing each pixel of the array is typically slightly less than one WFS integration time. The additional latency in computing the DM actuator commands is small in comparison, because a conventional matrix multiply reconstruction algorithm may be implemented in pipeline mode with nearly all of the computations completed by the time the final WFS pixel is digitized. The total signal processing latency in the control loop is consequently somewhat less than two complete WFS integration times: the sum of the one integration time intrinsic for any sampled data control system, plus a little less than one integration time spent reading the WFS CCD array. The closed-loop bandwidth of the AO system is limited by this latency. For a fairly conservative filter design with 45 degrees of phase margin, for example, the ratio between the loop update rate and the -3dB closed-loop bandwidth is about 20:1.

We believe that this ratio may need to be further increased for VHOAO. As described in the section on signal processing requirements below, implementing the wavefront reconstruction algorithm for an VHOAO system as a conventional matrix multiply may be prohibitively expensive due to the number of degrees of freedom involved. More computationally efficient reconstruction methods are available, but all of the algorithms with which we are familiar require the full WFS measurement vector to be available before the majority of the computations can begin. The amount of latency in such a system is likely to be on the order of three WFS integration times: one for the sample-and-hold in any sampled data control system, approximately one for reading out the WFS CCD array, and approximately one for performing the wavefront reconstruction. The third contributor can only be reduced below one WFS integration time if the signal processor is idle for the remainder of each cycle, which suggests that more processing power has been purchased than is absolutely necessary.

With increased latency, the ratio between the control loop update rate and closed-loop bandwidth must also be increased to maintain the same level of loop stability. Three cycles of latency and a ratio of 30:1 correspond to about 42 degrees of phase margin, which is approximately the same as the 45 degrees achieved with two cycles of latency at a ratio of 20:1. We have consequently used a ratio of 30:1 to compute the control loop update rates listed in Table 3. This has obvious implications for the WFS CCD pixel read rate and wavefront reconstruction signal processing requirements as described further below.

WFS array size and pixel rate

Shack-Hartmann WFSs can be successfully implemented with a minimum of four pixels per subaperture, provided that the sensor is not required to work off null (i.e., the noncommon path wavefront errors in the WFS optical path are not too large) and that the individual Shack- Hartmann spots are approximately diffraction-limited (i.e., the subaperture size is not greater than about 2r0 at the wavefront sensing wavelength). Both of these conditions should apply for VHOAO on GSMT, so the WFS pixel values in Table 3 are simply twice the corresponding orders of correction appearing in Table 2. The WFS CCD array may contain additional undigitized guard pixels used to increase the separation between the Shack-Hartmann spots and to prevent optical cross talk. The required array sizes range from about 3002 to 6002. These values are significantly larger than the current maximum of 1282 for high-speed, low-noise CCD arrays, but they do not appear to be infeasible.

The WFS sensor pixel read rates are simply the product of these pixel values times the AO control loop update rate. These values are in the range from 85 to 652 megapixels per second. The current maximum read rate for low-noise digitization is about 0.5 to 1 megapixel per second per digitizer, so it is clear that each array must include a large number of output channels.

Other wavefront sensing concepts that may be considered include the shearing interferometer and the point diffraction, or Smartt, interferometer. Either of these approaches would require the same total number of pixel measurements per second as the Shack-Hartmann sensor, but would provide additional options for how these measurements could be obtained. These include (1) four smaller CCD arrays with the same total number of pixels, each of which is read at the AO control loop update rate, or (2) one such array read out at four times this rate.

Signal processing requirements

It appears to be very unlikely that wavefront reconstruction algorithms for VHOAO systems can be implemented as conventional matrix multiplies. The number of operations (adds and multiplies) needed to compute a single wavefront reconstruction using this approach is approximately equal to four times the square of the number of AO degrees of freedom. This corresponds to from 1.4 to 43 trillion operations per second for the VHOAO system parameters outlined in Table 2 and Table 3, values which are up to about 13 thousand times larger than the computational load for the Gemini South MCAO system.

Alternative algorithms do exist that achieve dramatic reduction in computation requirements by exploiting the sparse structure of the DM-to-WFS influence matrix in a conventional NGS AO system. One possible method reduces the number of computations required for one wavefront reconstruction from 4 Nact2 to 12 Nact log (Nact). This corresponds to a range of 2.3 to 23 billion operations per second for the VHOAO systems considered here, equivalent to from 0.7 to 7 Gemini South reconstructors. This particular algorithm is not particularly parallelizable, however, and it would need to be implemented using a single CPU capable of this processing rate.

Developing new reconstruction approaches that are both computationally efficient and suitable for parallel implementation is now a very active area of research. An iterative algorithm developed at the University of Montana requires about 60 Nact log (Nact) operations per reconstruction, and is based upon sparse matrix multiply and Fourier transform operations that should be easily implemented in parallel. See Appendix 4.6.B for further details.

More work is needed on this and similar advanced algorithms, and also on the use of advanced signal processing hardware such as field programmable gate arrays (FPGAs).

DM actuator dynamic range

The requirement on DM actuator dynamic range is relatively modest, provided that the low frequency, high amplitude wavefront errors in the atmosphere and telescope have already been compensated by the adaptive secondary mirror. Assuming an adaptive M2 with at least 700 actuators, the residual aberrations to be corrected by the VHOAO DM will be similar to the tilt- removed wavefront errors on a 1-m aperture. This error is no more than about 0.3 µm RMS, even with 1 arcsecond seeing and an airmass of 2. An actuator dynamic range of about 1.0 to 1.5 µm should therefore be adequate.

Guide star signal level and magnitude

The VHOAO system will be used primarily to detect planets around nearby bright stars. It is still useful to determine the guide star signal levels corresponding to values of sigmanoise listed in Table 2, because this will indicate (1) if the equal error budget allocations in this table should be rebalanced, and (2) if the VHOAO system might be capable of other types of observations involving somewhat fainter guide stars. Table 3 summarizes the results of first-order calculations for the required guide star signal levels and magnitudes, which range from about magnitude 8 to 10 for the WFS and photometric parameters and assumptions outlined below. The nearest stars that provide the largest angular separation to a prospective planet will have magnitudes no dimmer than about 5 or 6, which suggests that the answer is "yes" to questions (1) and (2) above.

To compute the required WFS signal level, we have inverted the relationships

Equation Set 1

Here gs is the noise gain of the AO temporal filter, gr is the noise gain of the wavefront reconstruction algorithm, d is equal to the actuator pitch and subaperture width, and sigmatheta is the 1- axis, 1-sigma error in a single WFS subaperture tilt measurement. The formulas for gs and gr are approximations that are valid for a type I control loop, a least squares wavefront reconstruction algorithm, and either a Shack-Hartmann or shearing interferometer WFS. The quantity thetaB is the effective blur diameter of a Shack-Hartmann spot, SNR is the signal-to-noise ratio for a single WFS measurement, lambdas is the mean wavefront sensing wavelength, Npde is the number of photodetection events per WFS subaperture per measurement, and sigmae is the RMS detector read noise in units of electrons. The formula for thetaB is an approximation that is asymptotically correct for either very large or very small values of d and is modestly conservative in between. Sky background and dark current have been neglected in the formula for SNR, based upon the assumption that we are considering very bright guide stars and short integration times. The noise performance of the shearing interferometer WFS is roughly similar, assuming that the subaperture size and shear width have been optimized for the seeing conditions.

Solving for Npde in terms of the other variables yields the result

Equation Set 2

The WFS signal level L in units of photodetection events per second per square meter of collecting aperture is then given by

Equation Set 3

and the corresponding guide star magnitude M is

Equation Set 4

where tau is the end-to-end throughput of the telescope, AO system, and WFS detector, and L0 is zeropoint (the signal level for a zero magnitude guide star). Table 3 summarizes the results of these calculations for the parameters sigmae2 = 5, lambdas = 0.7 µm, tau = 0.4, and L0 = 4.4e10. All of these values must be iterated as the designs for the telescope and VHOAO system progress. Implementation and Interface Requirements Opto-mechanical Interfaces

The VHOAO system should be located on the optical axis of the telescope, immediately following the Cassegrain focus. The telescope pupil plane should not rotate with respect to the planes of the DM or the WFS. This will eliminate PSF variations with time due to the rotation of uncorrectable mirror figure errors, and also (perhaps more importantly) simplify the wavefront reconstruction algorithms for simultaneous control of the adaptive secondary mirror and the VHOAO DM. The simplest way to achieve this requirement is for the VHOAO system to rotate with the telescope in elevation. The science detector must then be derotated (or be preceded by a K mirror) to compensate for image rotation in the focal plane. Detector Studies

Several coronagraphic applications would benefit from either a super-conducting tunnel junction or a transition edge-sensing detector. Such detectors can determine the spectral energy of a photon with a resolving power of a few. Such spectral sensitivity may enhance the ability to discern real objects from the residual scattered light of the occulted target. The relatively small field of view (FOV) produced by the coronagraph is an excellent match to the current state-of-the- art with these detectors, which have relatively small pixel formats of a few tens of pixels. Further exploration of a coronagraphic instrument should include a comparable exploration of these new types of detectors. Optical Design

The optical design should feature a small pupil diameter in the plane of the DM, because micro electrical mechanical systems (MEMS) technology will probably be used to implement the very large number of actuators required. A pupil diameter of 30 mm corresponds to an actuator pitch from 100 to 200 µm, which is in the range considered for many current MEMS concepts.

The optical design must be relayed. This provides an intermediate image plane for the occulting disk, and two pupil planes for the DM and the Lyot stop. The focal ratio at both the intermediate and final focal planes should be about f/39.5 to yield reasonable values for the diameter of the occulting disk and the size of the science detector pixels.

The entire optical design should be small enough to be placed in a dewar, in order to minimize thermal emission for observations at wavelengths between 2 and 5 µm.

Figure 1 30-m f/18.75 Cassegrain GSMT with narrow field VHOAO module. OPTO-MECHANICAL DESIGN CONCEPT Optical Design Approach

A relatively simple and compact optical system (especially in comparison with the MCAO design) suffices to implement the necessary AO and coronagraphic functions. The 30-m GSMT overshadows the VHOAO module, as shown in Figure 1. The field aberrations of the 30-m telescope are completely insignificant over the 2 arcsec FOV. Furthermore, although the local field angles are multiplied in the ratio 30 m/30 mm = 1000x defined by the pupil diameters, the optics within the VHOAO module typically operate at only 17 arcminutes off-axis. This is still small enough that comparatively simple transfer mirrors provide diffraction-limited images over flat, untilted, and undistorted focal planes. Finally, there are no complications associated with the use of LGSs at finite ranges.

The VHOAO science path is shown in Figure 2, beginning at the Cassegrain focus of the telescope. The module consists of two finite conjugate relays, each containing collimated space between the pairs of mirrors (T1, T2) and (T3, T4). The first collimated space is used for atmospheric dispersion correction and AO functions, while the second provides a location for the Lyot stop.

Figure 2 VHOAO optical module.

The first element in the space between T1 and T2 is the fast tip-tilt mirror (TTM), followed quickly with the MEMS DM. The DM is placed at the image of the telescope pupil or aperture stop, which coincides with the secondary mirror. The TTM is placed prior to the DM so that fast tip-tilt corrections do not degrade the registration between the DM and WFS.

Following the MEMS DM is a plane-parallel dichroic beamsplitter that transmits the IR science light and reflects the visible light out-of-plane (and upwards) to the WFS optical path, which will be described later. In order to transmit wavelengths to at least 5 microns, this beamsplitter will be made from calcium fluoride or other material selected for low absorbance at wavelengths longer than 2.5 microns.

The reflected beam need only be tipped enough to avoid mechanical interference with science optics. An angle of incidence of about 17 degrees appears more than adequate, which is sufficiently low to facilitate the fabrication of a highly efficient dichroic coating.

Next is the science atmospheric dispersion corrector (ADC), which presently consists of two zero- deviation doublets made from calcium fluoride and water-free fused silica. Non-exotic, non- hygroscopic materials suitable for a near-IR ADC are extremely limited. Primary considerations in material selection are the widely different wavelength dispersive properties that the two materials have, and the absence of significant absorption features. Single crystal calcium fluoride, whose cubic structure results in isotropic optical behavior, has negligible affinity for water when AR coated. CaF2 is the low dispersion "glass," and it is favorably matched with a water-free variety of fused silica, which has relatively high dispersion. Technology for coating both calcium fluoride and fused silica is highly developed, and the cost of the optical materials themselves is quite low. Their transparency in the visible facilitates fabrication and alignment by means of helium-neon lasers.

This approach is satisfactory for transmission to 2.5 microns wavelength. For beam diameters of about 30 mm, wedge angles of about 6 degrees suffice for observations in J, H, and K to a 45- degree zenith angle, provided that the ADC is adjusted separately for each band. IR absorption bands occur in CaF2 in the vicinity of 2.5 microns, making it undesirable for this range of wavelengths. Based upon the atmospheric dispersion model included in the ZEMAX optical design program, atmospheric dispersion is so small beyond about 2.5 microns that the ADCs may be removed from the collimated beam, with no appreciable loss of performance.

Placing the WFS pickoff beamsplitter after the DM allows us to do the wavefront sensing in closed loop. Placing the science ADC after the beamsplitter keeps this element away from the visible light sent to the WFS, which will contain its own independent ADC.

The f/18.75 Cassegrain focus is relayed by T1 and T2 at about f/39 to the occulting plane focus, which is then relayed again to a final f/40 detector focus. Lying between the two mirrors forming this second relay is the Lyot stop, which is placed at a second pupil plane optically conjugate to the MEMS DM.

The mirrors forming each relay are "toroidal," and have been chosen over off-axis paraboloids due to their ease of fabrication. Nuances in minimizing distortion and reducing aberrations lead to the slightly non-integer f-number multipliers, an issue that could be remedied in a second design iteration if it became a concern. The large intermediate f-number was chosen solely to provide a more manageable occulting aperture, which may be only several Airy disc diameters in size. The final f-number was selected based on detector element size, and could be redesigned for almost any f-number desired.

As will be seen, the toroidal mirrors provide diffraction-limited performance over the small VHOAO FOV at both the occulting focus and the final detector focus. The mapping of the stop and the DM/MEMS to the Lyot stop and WFS pupil planes is also predictable and well behaved. Science Path Performance Summary

Image Quality: The image quality of the telescope is illustrated in Figure 3, which is a spot diagram at the Cassegrain focus. The Cassegrain focus is relayed to the occulting focus, whose optical quality is shown in Figure 4. This image is then relayed to the final detector plane, as shown in Figure 5. Curiously, the spot diagrams at the final detector focus are clearly superior to those at the occulting plane. This is no mistake! We find that the second mirror relay has largely corrected the field aberrations of the first mirror relay. In fact, if the relays operated at unit magnification and were separated such as to provide telecentric input and output, the field aberrations would be almost perfectly corrected (except for field curvature, which is inherently determined by all mirrors being concave). The present design combines one relay operating at 2X magnification, the other at 1X. Neither is spaced for telecentricity, and still the correction is remarkable.

Figure 3 Geometrical spot diagrams at Cassegrain focus. The circles represent Airy disc diameter
at 2.2 microns wavelength Figure 4 Geometrical spot diagrams at the occulting focus plane. The circles represent Airy disc
diameters at 2.2 microns wavelength.

Figure 5 Geometrical spot diagrams at the final detector focal plane. The circles represent the
Airy disc diameter at 2.2 microns wavelength.

Additionally, the object to image mapping is almost perfectly linear, a condition seldom found in an unobstructed reflecting optical system. The maximum distortion is no more than 0.037%. Although the large f-numbers make the issue somewhat moot, the image planes are perpendicular to the "axis."

The RMS wavefront error over the FOV (sampled in just one meridian; but it is, as may be surmised from the spot diagrams, similar in any meridian) is shown in Figure 6. The Strehl ratio for the paper design is about 98% in J-band at the edge of the field.

Figure 6 RMS wavefront error at the final (detector) focal plane, in a representative azimuth.

Atmospheric Dispersion Compensation: By the Lagrange optical invariant, the angular dispersion required at the ADC is 1000x that occurring at the input to the 30-m telescope. The ZEMAX code includes an "atmospheric dispersion" surface that we have tested and found reliable. Setting this to a 45-degree zenith angle, with the altitude and other properties of a site like Cerro Pachón (which enables us to reuse work we performed earlier for the Gemini South AO system), we find that zero deviation prism pairs made from calcium fluoride and fused silica have acceptable angles for the J-band and longer, as shown in a preliminary design presented in Figure 7. Figure 8 illustrates the worst-case performance of the ADC in J-band. Note that the spots for 1.15 microns are hidden under those for 1.35 microns, with the separation between 1.25 and 1.15 microns representing the secondary (uncorrectable) dispersion.

Figure 7 Science path ADCs set for J-band at a 45-degree zenith angle.

Figure 8 Geometric spot diagrams for science ADCs set for J-band at a 45-deg zenith angle. The
circles indicate diameter of the Airy disc at 1.35 microns wavelength.

Prisms that correct the I-band at a 45-degree zenith angle would have large angles of about 17 degrees, but the present design will allow work in the I-band to a Zenith angle of about 15 degrees. This pair of doublet prisms also serves for the H and K bands when counter-rotated by a smaller angle, with similar high quality. Because atmospheric dispersion decreases with increasing wavelength, we find that no ADC is required for wavelengths longer than the K-band, and the ADCs are removed for that situation.

Used in collimated space, prismatic ADCs introduce no image aberrations, and only slightly disturb the quality and location of the system exit pupil. Doublets are less perfect than symmetrical triplet prisms is this regard, but the latter are more expensive and may absorb more light. Before a design is considered final, the tradeoff between perfect image quality and a shifted exit pupil should be studied and a decision made as to whether to compromise one for the other. Airspaced prisms must be used in either case, due to the difficulties of using optical cement at cryogenic temperatures.

Science path throughput and emissivity: Table 4 presents our current calculations for science path throughput and emissivity. There are seven reflections, one beamsplitter transmission, and two ADC doublets with a total of eight air-glass interfaces. The effects of the occulting disk and Lyot stop are not included.

Table 4 Throughput and emissivity estimates for the GSMT VHOAO science path.
Wavelength, µm1.001.652.202.20 (emissivity)
Per reflection0.9790.9860.9870.013
Per air-glass interface0.9910.9870.9890.011
Overall for 7 reflections0.8620.9060.9120.091
Overall without ADC0.8020.8700.8810.113
Overall for 8 ADC air-glass interfaces0.9300.9010.9150.088
Overall with ADC0.7460.7840.8060.201 NGS WFS Path Design and Performance

A beamsplitter takes light between 500 and 900 nm from the first toroidal collimating mirror, and reflects it at an angle of about 34 degrees relative to the input science path. Both in-plane or out- of-plane folds are under consideration at this time. The in-plane baseline is illustrated in Figure 9 and described below.

Figure 9 Layout of the NGS WFS optics, folded into a single plane for clarity.

The ADCs for the NGS WFS are made from ordinary optical glass, using the "buried surface" between a crown and a flint glass that have the same nominal refractive index but different dispersive powers. An angle of about 9.6 degrees permits compensation to a zenith angle of 45 degrees. The thickness of the prisms is unimportant. The preliminary design is shown in Figure 10.

Figure 10 NGS WFS ADCs at a 45-degree zenith angle. Figure 11 Wavefront at NGS exit pupil, for an on-axis source at 600nm wavelength.

Figure 11 illustrates the wavefront from an on-axis star. The peak-to-valley wavefront error for this preliminary design is about 300 nm. Figure 12 illustrates the performance of the NGS WFS ADC over a spectral band from 500 to 700 nm at a zenith angle of 45 degrees. Note that only the on- axis case need be considered, and that dispersion compensation is really only required to the diffraction limit of a subaperture. In fact, the level of performance achieved is nearly diffraction- limited for the full GSMT aperture.

Figure 12 NGS WFS spot diagrams at focus prior to collimator lens.

The use of toroidal mirrors seems to have created a ZEMAX analysis conflict in trying to map the DM onto the final exit pupil plane, so we are presently unable to evaluate the pupil mapping between the DM and the WFS. The excellence of the mapping between the aperture stop (secondary mirror) and the science path exit pupil suggests that similar results will be obtained between DM and the WFS, except that the mapping will be distorted by the cosine of the tip on the DM. This is correctible with a cylindrical afocal singlet lens placed after the collimator lens and the exit pupil, as was done for the Gemini South MCAO WFS design. Lower quality pupil mapping is permitted as long as it is stable, and it has been suggested that the elaborate measures taken for the Gemini South design are not required for the 30-m GSMT.

Finally, the NGS WFS path contains six reflections, two ADCs, and a doublet refractive collimating lens. This is very similar in complexity and element count to the science path optical system. The transmittance estimates presented in Table 1 for the science path are representative of what we expect here.

In Figure 13, we provide a three-dimensional layout for the AO-fed coronagraph outlined above.

Figure 13 A three-dimensional model depicting the optical bench layout of the AO-fed coronagraph
described in this section. Key components are labeled. SUBJECTS FOR FUTURE WORK System Optimization and Trade Studies

More work is clearly necessary to balance the preliminary AO error budgets presented above and to obtain satisfactory levels of performance in the most cost-effective fashion. The principal error sources and effects in the overall system should be treated in an integrated fashion to correctly assess how performance is influenced by the various AO design parameters. Some of these effects and errors include: (1) the atmospheric turbulence profile, (2) static and dynamic wavefront errors introduced by the telescope, (3) the detailed characteristics of the wavefront sensing, (4) estimating and correcting elements of VHOAO, (5) the effect of the occulting disk and Lyot stop upon polychromatic image formation, (6) science detector characteristics, and (7) the image postprocessing algorithms that are utilized. Different classes of wavefront errors (for example, tip-tilt, segment phasing errors, DM waffle error, and residual high-spatial frequency turbulence-induced errors) may have different relative effects upon speckle characteristics in the PSF halo than they do upon the Strehl ratio. The overall system error budget and the top-level specifications for VHOAO should take this into account. Wavefront Sensing Concepts

Additional wavefront sensing designs (the shearing interferometer and the SMARTT (or point diffraction) interferometer, for example) should be considered as alternatives to the standard Shack-Hartman WFS. (It seems doubtful that a curvature sensor would be suitable for VHOAO.) Analysis and simulation results to date indicate that all three sensing approaches yield approximately the same fitting error coefficient for the case of good seeing conditions with little or no scintillation, so the order of the VHOAO system and the limiting guide star magnitude will probably not be dramatically influenced by the choice of WFS. The more significant differences would relate to the hardware requirements for the WFS CCD detector arrays and the wavefront reconstructor. Either the shearing or Smartt interferometer would reduce the linear dimension required for the WFS CCD by a factor of two or more. The Smartt interferometer is a phase measurement device that could conceivably eliminate the requirement for wavefront reconstruction entirely.

The shearing interferometer is by far the more mature of these two alternate approaches, and does not require any special development effort supported by GSMT. Success with the Smartt interferometer (for AO applications) has been much more limited, with limitations identified in the areas of initially closing the loop and fringe contrast. The approach may still be worth revisiting, because the potential for simplifying the signal processing requirements is so great. Real-Time Control Electronics and Algorithms

As described previously, work is already in progress to develop new wavefront reconstruction algorithms that are dramatically more efficient than the conventional matrix multiply methods in use today. This work should now begin investigating how to implement these techniques in a real- time control loop. The issues to be considered include (1) minimizing latency, (2) algorithm performance and computation requirements in a closed-loop system, (3) simultaneous control of the VHOAO DM and an adaptive secondary mirror, and (4) the choice of the signal processing hardware (FPGAs, for example). Work should begin relatively early to confirm that feasible solutions exist and to estimate the ROM cost for their development. MEMS Mirrors

The ongoing work on MEMS mirrors should be tracked and supported by GSMT in part. The essential performance requirements are (1) the total number of degrees of freedom, (2) excellent mirror figure quality and fill factor, and (3) sufficient stiffness to enable control bandwidths in the range from 40 to 80 Hz. Cryogenic operation is also very desirable. Hexagonal actuator geometries may be desirable, because they may reduce or eliminate the so-called "waffle mode" error and the associated "satellite images" experienced with the standard square geometry. Large dynamic range is not a requirement, assuming that an adaptive secondary mirror is available to compensate for the large amplitude, low spatial frequency wavefront errors. Calibration Techniques

The wavefront error budgets for VHOAO are so stringent that every effort must be made to adaptively track and compensate potential sources of calibration error. These include (1) variations in WFS tilt measurement gain due to changes in seeing, (2) drift in the sky background level and guide star signal level, and (3) drift in DM-to-WFS registration due to flexure and temperature, if these errors cannot be passively controlled. Existing algorithms for these purposes should be reviewed, and their estimated performance factored into the VHOAO error budgets. AO System Field Tests

At least two measurement programs are needed to anchor the theoretical models for AO system performance presently being used to develop the specifications for VHOAO on GSMT. In general, the performance of most real-world AO systems in use today, particularly very high-order systems, is still at least modestly below the predicted values. Field tests on existing AO systems should be performed with a goal of demonstrating that Strehl ratios of about 0.9 can indeed be achieved when they are predicted by theory, and that long-term variations in performance are properly correlated with the changes in seeing. Secondly, measurements are also needed to characterize the rate at which residual PSF speckles average out with increasing integration time, both with and without AO.

Existing predictions are based upon theoretical models for the long-range correlations in atmospheric turbulence that are poorly grounded in experiments and cannot be considered very precise.

Note that neither category of measurement requires a particularly high-order or large-aperture AO system to get started. It may be possible to begin studying the temporal behavior of PSF speckles without AO at all, which should place an upper bound upon the time required to achieve a particular level of PSF uniformity. Near-term measurements of either sort would be especially useful to help support the initial definition of the requirements for a GSMT VHOAO system.

March 2002