1. INTRODUCTION

All of the structure in the universe grew from small primordial perturbations, which can be observed as $\sim 100\mathrm{\mu K}$ temperature fluctuations in the 3 K cosmic microwave background (CMB) [1]. The CMB has $\sim 10\%$ linear polarization, with an E-mode (zero curl) pattern on the sky, generated by scattering of quadrupole temperature fluctuations when the CMB formed [2]. If the Big Bang started with inflation [3], gravitational waves should imprint a unique B-mode (zero divergence) polarization pattern in the CMB on degree angular scales [4]. There is no lower bound on the level of the B-mode signals, but many models of inflation predict signals at the 10 nK level. CMB observations have the potential to make a definitive test of inflation, through a detection of the primordial B-mode signal, and to measure many other effects that leave an imprint on the CMB, e.g., weakly interacting, light particles beyond the Standard Model, neutrino mass, the distribution of baryons in clusters of galaxies, the total mass distribution in the universe, the momentum field of large-scale structure, and the effect of feedback mechanisms on galaxy formation [5–7].

Sensitivity is the driving requirement for CMB experiments. Ground-based CMB detectors are background limited, so high sensitivity requires large numbers of detectors. Existing CMB telescopes have a few $\times 10\mathrm{k}$ detectors, but a few $\times 100\mathrm{k}$ will be needed for a measurement at the nK level in a few years of observing time. In this context, new telescope designs that can support large-format cameras are of considerable interest, but high raw sensitivity is useful only if systematic errors are also small, so new designs must include low pickup and small polarization errors. The angular resolution must be a few tens of arcminutes for measurements of primordial B-mode polarization [8], $1-2^{\prime }$ to resolve clusters of galaxies [9], and a few arcminutes to measure and remove the effects of gravitational lensing of the CMB [10], which converts E-mode polarization to B-mode, at the level of a few hundred nK on $\sim {10}^{\prime }$ scales [11]. The highest angular resolution requirement corresponds to a $\sim 5\mathrm{m}$ telescope at $\lambda =2\mathrm{mm}$.

This paper describes a three-mirror anastigmat (TMA) that combines wide field of view (FOV), with good control of systematic errors (low scattering, boresight rotation, and a full, comoving shield or baffle), in a 5 m telescope design. The combination of features is new, and it will allow CMB measurements over a wide range of angular scales with a single large-aperture telescope. For structural design purposes, the telescope is assumed to be located either at high altitude in the Atacama Desert, or at the South Pole; both sites offer excellent conditions for CMB observations.

The paper is divided into three parts: systematic errors and associated constraints on the general configuration of the telescope; optical design of the TMA; and mechanical design of a practical mirror support structure and mount. The body of the paper focuses on concepts and results; technical details and supporting calculations are in the appendices.

2. SYSTEMATIC ERRORS

A. Pickup

Pickup is a critical issue for CMB telescopes; it drives the design of shields, baffles, and mirrors, all of which impact the choice of telescope layout. Pickup is caused by telescope sidelobes, which are generated by diffraction, blockage, and scattering from optical surfaces. The ground, Sun, and Moon are the brightest sources for pickup. The Milky Way is also a concern for sensitive polarization measurements. All modern CMB telescopes are unblocked designs, to avoid the large diffraction sidelobes that come from placing a mirror in the telescope beam, but some scattering from optical components is inevitable because the camera must have a vacuum window and heat blocking filters, which typically scatter $\sim 1\%$ of the beam.

Large telescopes have segmented mirrors, which are a particularly serious source of pickup because the regular array of gaps between segments generates sharp grating sidelobes [12]. For $\sim 1\mathrm{m}$ segments, the gaps are typically a few millimeters (set by thermal expansion and manufacturing tolerances), so the fractional gap area is $<1\%$, but the sharp sidelobes can modulate bright features, e.g., the horizon, generating false signals that are much larger than the CMB polarization. A design with no segment gaps is highly desirable, but the size of a monolithic mirror is limited by thermal deformation and manufacturing errors. The thermal deformation is primarily overall expansion due to a change in soak temperature and cupping due to a temperature gradient through the mirror. Detailed calculations of the surface error contributions for a monolithic mirror are given in Appendix A. The thermal deformation problem can be at least partly addressed by refocusing based on temperature measurements. In this case, a monolithic, machined, aluminum mirror that is diffraction-limited at $\lambda =1\mathrm{mm}$ can be up to $\sim 5\mathrm{m}$ in diameter. A 5 m mirror is what is needed for cluster and lensing observations, so the TMA design in this paper has monolithic mirrors.

Even with monolithic mirrors, reflective shields or absorptive baffles are needed to control scattering from camera components. Shields direct scattered radiation to the cold sky, while absorbing baffles provide a termination that is hot but comoving with the beam. Reflective shields reduce the optical loading on the detectors, resulting in higher sensitivity. The entire surface of a reflective shield must be stable to avoid modulating residual pickup, but a perfectly absorbing baffle requires only stable edges; the baffle walls can move around as long as the scattered radiation is absorbed somewhere. The less stringent mechanical stability requirements and well-defined termination of scattered radiation are compelling reasons to use a baffle.

Current searches for weak, primordial, B-mode polarization use small $(\sim 0.5\mathrm{m})$ telescopes [13,14], primarily because it is easier to shield and baffle smaller optical systems. However, there is no fundamental reason why a large telescope with well-controlled pickup cannot make measurements on large angular scales; all CMB experiments with large telescopes push in this direction. The TMA design in this paper includes a support structure for a comoving shield or baffle around the entire telescope, with the goal of minimizing pickup to increase sensitivity on large angular scales.

B. Polarization

Measuring CMB B-mode polarization at the nK level is challenging. Current CMB polarization experiments separate two orthogonal linear polarizations, measure the total power in each, and then take the difference. The key issue in these experiments is instrumental polarization, which converts bright CMB temperature fluctuations into polarization fluctuations [15–17]. A gain difference between the two beams in a polarimeter converts temperature on some angular scale to polarization on the same angular scale. Higher-order errors, e.g., pointing and ellipticity differences, convert temperature gradients on the angular scale of the telescope beam to polarization [18]. The power in CMB temperature fluctuations falls rapidly with decreasing angular scale below $\sim {10}^{\prime }$, so a telescope with a small beam sees less power on the angular scale of the beam to convert to polarization. Smaller temperature to polarization conversion is an important advantage of using a small beam for measurements of polarization on large angular scales. In such a measurement, it is the polarized response of the large synthesized beam that matters, but the response of the small telescope beam sets a lower bound. CMB temperature fluctuations are a few tens of μK on degree scales, so a polarization measurement at the nK level requires differential gain errors $\lesssim {10}^{-5}$. On arcminute scales, CMB temperature gradients are of order 1 μK, so a nK measurement on degree scales, using a $\sim 1^{\prime }$ beam, requires higher-order differential errors $\lesssim {10}^{-3}$. Cross-polarization also causes polarization errors, by moving power between polarizations, but is less important because CMB polarization fluctuations are weak, just a few μK. Cross-polarization $\lesssim {10}^{-3}$ is good enough for polarization measurements at the nK level and is easily achieved.

Many polarization errors can be at least partly corrected, in which case what matters is the residual polarized response. There are three general techniques for dealing with polarization errors. (1) If the telescope beams can be measured accurately, and the beams are stable, polarization errors can be calculated and subtracted from the CMB maps. (2) Polarization errors can be removed by filtering the maps using templates that capture the expected effects of beam errors [19,20]. Filtering must be applied carefully because some beam errors, e.g., differential ellipticity, behave like polarization on the sky, in which case filtering removes both the error and some real CMB signal. Filtering works well for differential gain, beam size, and pointing errors [21,22]. (3) Measurements at different boresight angles can measure some polarization errors, e.g., a 90° boresight rotation moves the CMB signal from one detector in a polarimeter to the other, so combining 0° and 90° boresight measurements cancels the stable component of differential gain errors. A similar cancellation occurs for any beam difference that changes sign with 180° rotation, e.g., a pointing offset between the two beams in a polarimeter. Boresight rotation does not cancel polarization errors that change sign with 90° rotation, e.g., differential ellipticity, and it does not cancel errors that change when the telescope rotates, e.g., changes in differential beam shape due to gravitational deflection of the mirror support structure or baffle. The only experiments that have produced interesting limits on primordial B-mode polarization have used boresight rotation, so it is included in the TMA design in this paper. Boresight rotation does not have to be continuous or fast.

3. OPTICAL DESIGN

A. Why a TMA?

In most existing CMB experiments, the telescope FOV has been completely filled with detectors. Future experiments will need far more detectors, which could be on multiple copies of existing telescopes or on a new telescope with larger FOV. Several experiments have pursued the two-mirror, cross-Dragone telescope [23–28], which has a few times larger FOV than a Cassegrain or Gregory design (e.g., a 5 m, $f/3$ cross-Dragone telescope can achieve $\sim 5/7/9°$ FOV at $\lambda =1/2/3\mathrm{mm}$, cf. $\sim 3/4/5°$ for a Cassegrain or Gregorian telescope). A TMA offers an even larger FOV because it has enough surfaces to correct all the major Seidel aberrations [29,30]. TMA designs are widely used at short wavelengths (e.g., the Large Synoptic Survey Telescope [31], the European Extremely Large Telescope [32], and the James Webb Space Telescope [33] are all of this type), but the TMA has received little attention for longer-wavelength experiments, primarily because three mirrors are needed and the off-axis form has an awkward mechanical configuration.

Off-axis forms of the TMA have an aperture offset, to move the secondary out of the incoming beam, and a field offset, to move the image away from the parent optical axis. The excellent image quality of the TMA allows large aperture and field offsets, which are needed for a large, unblocked FOV. Several design variations are possible: the Wetherell–Womble form [34] is an off-axis modified Paul–Baker design, with no intermediate foci; other forms have a focus between the primary and secondary, or between the secondary and tertiary [29]. An intermediate focus generally results in less space for the image, because more clearance between the incoming beam and the secondary pushes the image closer to the parent optical axis, so the TMA design in this paper is based on the Wetherell–Womble form, where the image shift is in the opposite direction. All the off-axis TMA designs can be improved by tilting and decentering the secondary and tertiary [35].

To illustrate what is possible with a TMA, Fig. 1 shows a 5 m Wetherell–Womble design with $32\times 12°$ FOV at $\lambda =3\mathrm{mm}$. The design is optimized for maximum FOV, with no limit on the sizes of the secondary and tertiary mirrors. Both the secondary and tertiary are tilted and decentered, and have a 4th-order aspheric correction; the field is convex to improve the image telecentricity, in the sense that the principal ray for each field angle is roughly normal to the image surface. Design details are given in Appendix B. The throughput is $13359/17245/17685{\mathrm{cm}}^2\mathrm{sr}$ at $\lambda =1/2/3\mathrm{mm}$, so the telescope can accommodate up to 1700 k/549 k/250 k $F\lambda$ diameter detector pixels, where $F$ is the image focal ratio, and the number of pixels is calculated as $(\text{field area})/{(\text{telescope full beamwidth at}\text{half-maximum})}^2$ to account for some lost space between pixels. The design in Fig. 1 has an enormous tertiary, but ray bundles for different field angles are well separated on the mirror, so there are clearly more compact designs with smaller FOV.

 

Fig. 1. (top) Layout and (bottom) FOV for a wide-field TMA with a 5 m primary. Marginal rays in the layout are at field angles (yellow) $x=+16°$, (red) $y=-6°$ and (green) $y=+6°$; the orange surface is the image. The solid black/blue/red curves in the bottom plot are 80% Strehl ratio contours at $\lambda =1/2/3\mathrm{mm}$, ignoring vignetting. The dashed, $32\times 12°$ ellipse is the unvignetted FOV set by the size of the tertiary.

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B. TMA with 5 m Mirrors

Figure 2 shows a 5 m TMA in which all the mirrors are small enough to be monolithic. In this design, the $\sim 5\mathrm{m}$ tertiary limits the FOV to $12\times 8°$ at $\lambda =3\mathrm{mm}$. The secondary and tertiary are tilted and decentered, all the mirrors have a 4th-order aspheric correction, and the image has been made flat and telecentric (within $\sim 4°$) to simplify coupling to arrays of detectors with directional antennas, e.g., feedhorns. Design details for the telescope are given in Appendix B. The throughput is $3330/4268/4421{\mathrm{cm}}^2\mathrm{sr}$ at $\lambda =1/2/3\mathrm{mm}$, so the design can accommodate up to 424 k/136 k/63 k $F\lambda$ pixels. For comparison, the South Pole Telescope has a throughput of $434{\mathrm{cm}}^2\mathrm{sr}$ for $\lambda \ge 1\mathrm{mm}$ [36].

 

Fig. 2. (top) Layout, (center) FOV, and (bottom) polarization errors for a TMA with 5 m primary and tertiary mirrors. Marginal rays in the layout are at field angles (red) $y=-4°$ and (green) $y=+4°$. The solid black/blue/red curves in the center plot are 80% Strehl ratio contours at $\lambda =1/2/3\mathrm{mm}$, ignoring vignetting. The dashed, $12\times 8°$ ellipse is the unvignetted FOV set by the size of the tertiary. Numbers in the bottom plot are cross-polarization (in dB, negative numbers) and instrumental polarization (in %) for uniform aperture illumination.

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To reduce pickup and noise, the detectors must see only signals from the telescope entrance pupil. Any spillover must be absorbed at a cold stop, so some reimaging optics are needed to generate a pupil inside the camera. Reimaging is generally done with cold lenses, making for a compact camera, but the telescope image in Fig. 2 is $3.6\times 2.7\mathrm{m}$, which is much larger than practical lenses. In this case, the field must be broken up into smaller pieces, each with its own reimaging optics [37–40]. The design in Fig. 2 has some distortion in the focal plane, resulting in an exit pupil with $\sim 10\%$ ellipticity that will require elliptical stops in the camera.

C. Alignment Tolerances

In the design of Fig. 2, decentering the secondary/tertiary by 5/25 mm degrades the Strehl ratio at the edge of the field by 5%–10%, so the mirror alignment should be maintained at the level of a few millimeters. This drives the choice of material in the telescope structure. At the South Pole, the annual soak temperature variation is $\sim 60\mathrm{K}\mathrm{p}\text{-}\mathrm{p}$; at lower latitudes the variation is smaller. A thermal deformation $<1\mathrm{mm}$ due to a 30 K change in the temperature of a 10 m structure requires a coefficient of thermal expansion $\alpha <3\mathrm{ppm}/\mathrm{K}$, which is a factor of a few better than steel, so the telescope structure should be made of carbon fiber reinforced polymer (CFRP). Inexpensive CFRP, with modest thermal performance, would be sufficient [41]. Despace and overall expansion of the mirrors are less important because they can be compensated by moving the camera to refocus. For a 5%–10% degradation in Strehl ratio at the edge of the field, after refocusing, the design can tolerate a 20/100 mm despace of the secondary/tertiary, or a 1% change in the curvature of all the mirrors, with the vertices fixed.

D. Polarization Errors

Figure 2 shows the results of polarization ray tracing, in which the effects of reflection loss are captured by coating the mirrors with a thin layer of complex refractive index, $n$. The loss at each surface is assumed to be 0.25%, based on measurements of aluminum reflector panels for the Atacama Large Millimeter Array [42]. The power reflectivity of a surface at normal incidence is $\mathcal{R}={(n-1)}^2/{(n+1)}^2$ [43], and for metals at millimeter wavelengths, $\mathrm{Im}(n)=-\mathrm{Re}(n)$ [44], so 0.25% loss corresponds to $n=800-i800$.

Cross-polarization, which is determined by the geometry of the surfaces, is $\le -38\mathrm{dB}$, and instrumental polarization, which is set by both the geometry and conductivity of the surfaces, is $\le 0.3\%$. The uncorrected cross-polarization due to the telescope is good enough for B-mode measurements, but the overall cross-polarization will probably be dominated by the detectors. The small millimeter-wavelength horns used to couple CMB detectors to telescopes typically achieve $-30\mathrm{dB}$ cross-polarization [45,46]; planar antennas are usually worse [47,48]. The uncorrected instrumental polarization due to the telescope represents all the differential beam effects combined, and the total is already close to what is needed for high-order differential beam errors after correction. The differential gain error is too large by a factor 100, but filtering or boresight rotation can easily correct differential gain errors at this level.

4. MECHANICAL DESIGN

A. Requirements

All ground-based CMB telescopes scan rapidly, so that celestial signals appear in the detector timestreams at frequencies above most of the noise due to atmospheric brightness fluctuations, and above the detector $1/f$ noise. The usual approach is to scan fast enough to freeze the motion of the atmosphere; scanning faster does not help because it just moves the atmospheric brightness fluctuations to higher timestream frequencies. Most of the water vapor that causes atmospheric brightness fluctuations is in the first $\sim 2\mathrm{km}$ of the atmosphere and is moving along at a typical wind speed of $\sim 10{\mathrm{ms}}^{-1}$, so CMB telescopes scan at $\sim 1°{\mathrm{s}}^{-1}$ [49,50]. Scanning has two important consequences for the mechanical design of the telescope. (1) The scan axis (or axes) must have bearings that are suitable for continuous back and forth motion. (2) The time to turn around and settle at the end of a scan must be a small fraction of the scan time. For scans of tens of degrees at $1°{\mathrm{s}}^{-1}$, the total turnaround time must be of an order a second, requiring a telescope structure with a natural frequency of at least a few hertz.

Observations of bright sources are needed to measure focus, mirror alignment, surface errors, and beam shapes. Planets are typically used to measure focus, beam shapes, and low-order surface and alignment errors [51–53], but measurements of higher-order surface errors require a much brighter source, e.g., an oscillator on a tower [54]. A similar setup is needed for measurements of far sidelobes [21]. To see an artificial source, the telescope must be able to point near the horizon.

B. Mirror Support

Figure 3 shows a mirror support structure for the optical layout of Fig. 2. Each monolithic mirror is mounted on a spaceframe cone with a post running from the apex to the center of the base. The post is supported at each end, but there are no other connections to the cone or mirror, so any deformation of the telescope structure causes rigid-body motion of the cone but no deformation of the mirror. The cone and mirror are both made of aluminum to minimize differential thermal expansion.

 

Fig. 3. (top) Mirror support structure and (bottom) baffle frame for the design of Fig. 2. Mirror rims and the camera are shown in magenta (primary at bottom left and camera at top right), aluminum spaceframe cones that support the mirrors are red, CFRP spaceframe struts are black, and the steel cone that supports the boresight rotation bearing is blue. Thicker lines indicate struts with larger cross-section. Dotted black lines in the bottom plot represent the beam envelope.

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All the cones are connected with a CFRP spaceframe, which maintains the alignment over a wide temperature range (see Section 3.C). The CFRP spaceframe also connects to a stiff, steel cone behind the boresight bearing. The struts that form the steel cone are sized to put the center of gravity of the mirror support structure at the tip of the cone, making this a natural support point. Spaceframes are used throughout, because they are efficient, and they allow a 3D structure with predictable performance to be built using struts that have well-controlled properties in only the axial direction. The latter makes for an easier design with anisotropic materials like CFRP.

C. Mount

The TMA layout is awkward for a fork-style elevation over azimuth mount because it results in a large structure that wraps around the mirrors; an unconventional approach is needed to keep the overall size and mass of the telescope reasonable. Figure 4 shows a design with a hexapod [55,56] for the elevation axis and slewing ring bearings for azimuth and boresight rotation. The hexapod provides a stiff connection between the slewing rings and allows the telescope to kneel forward at lower elevation, so there is enough clearance for full rotation in azimuth and boresight over the full elevation range. Since the hexapod is used only for elevation motion, at least some of the joints can be simpler and stiffer than in a conventional hexapod. The hexapod configuration is particularly appropriate for observations from the South Pole, where scanning is generally in azimuth, with small steps in elevation at the ends of azimuth scans and occasional steps in boresight rotation.

 

Fig. 4. Mount with a hexapod elevation drive and slewing rings for azimuth and boresight rotation, pointing at (top) zenith and (bottom) horizon. The boresight slewing ring is at the base of the upper steel cone, shown in blue, and the azimuth slewing ring is at the base of the steel tower, also shown in blue. The hexapod ball screws are orange, and the green structure is the counterweighted lever on which the mirror support structure floats. Access to the camera is in the bottom configuration, but with 180° boresight rotation to put the camera close to the ground.

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Nonrepeatable position errors due to backlash and play in the hexapod joints can be a serious problem [57], so the design in Fig. 4 includes a scheme that allows high preload in the joints. A counterweighted lever carries the weight of the mirror support structure, without constraining its position, much like the flotation systems used in optical telescope mirrors [58]. The connection between the lever and the mirror support structure is a hydraulic cylinder, supplied at the appropriate pressure to support the weight of the structure. The cylinder stroke takes up any misalignment to ensure that the hexapod is not over constrained. The hexapod sets the position of the mirror support structure, but the only forces on the hexapod are wind and seismic loads, so the joints can have high preload. The lever in Fig. 4 also maintains the overall center of gravity of the telescope above the center of the azimuth bearing, so the mount is balanced in all three axes.

At the South Pole, winter temperatures can drop to $-80°\mathrm{C}$, which is too cold for most bearings and drive components, so the mechanisms in Fig. 4 are kept warm by enclosing them in a flexible, insulated sock that extends from the boresight bearing down to the azimuth bearing; the mirrors and their associated support structures are all at ambient temperature. Drive motors and amplifiers are mounted below the boresight bearing to simplify cable management. The elevation cable wrap is just a loop; the azimuth and boresight cable wraps have slip rings for power, rotary joints for compressed helium for the camera, and small cable chains for network connections. The sock provides covered, warm access to all the mechanisms from below, through the azimuth bearing and hexapod. An arrangement of this type is essential for operation at the South Pole, but enclosed access to the mechanisms is useful for any telescope in a harsh environment.

D. Baffle

The design in Fig. 3 includes a baffle that wraps around the mirrors and incoming beam. On the tertiary side of the telescope, the baffle is attached to the truss that supports the secondary and camera; the primary side has a dedicated spaceframe just for the baffle. Ideally, the baffle would be supported on a completely independent frame, attached only to the boresight bearing cone, so wind forces on the baffle do not couple into the mirror support structure; however, an independent frame is impractical in Fig. 3 because of the density of the structure below the primary and tertiary mirrors, so the baffle is connected directly to the stiff, CFRP struts that support the mirrors. The bottom of the baffle could be completely closed or covered with overlapping panels to allow snow to fall through.

Wind-induced motion of the baffle relative to the telescope beam modulates the spillover, resulting in additional noise in the detector timestreams. A ground-based, millimeter-wavelength bolometer typically has a noise equivalent temperature of $\sim 500{\mathrm{\mu Ks}}^{1/2}$ [59], and the time scale for motion of a $\sim 10\mathrm{m}$ baffle in a typical $10{\mathrm{ms}}^{-1}$ wind is $\sim 1\mathrm{s}$, so changes in spillover due to wind forces should be limited to $\sim 100\mathrm{\mu K}$. If the baffle intercepts the beam at $-20\mathrm{dB}$, 100 μK change in spillover corresponds to $\sim 1\mathrm{mm}$ displacement at the baffle rim. Baffle motion also causes a small change in beam shape. A 1 mm shift in a 5 m aperture could give a beam error at the $\sim {10}^{-4}$ level, but this is already small compared to the differential beam error required for B-mode polarization measurements (see Section 2.B).

The baffle frame must provide stiff edges, so that spillover on the baffle is stable, but as long as the baffle is a good absorber, the walls can be flexible, e.g., absorber glued to thin, foam-core panels with CFRP facesheets, or absorber stitched to architectural fabric. Some flexing of the baffle walls in the wind will discourage buildup of snow, a major concern for a large baffle.

E. Performance

Finite element model details for the design in Figs. 3 and 4 are given in Appendix C. The mirror support structure in Fig. 3 has a total mass of $38t$, which includes $5t$ for the mirrors and $3t$ for the camera. With the base of the boresight bearing cone fixed, the natural frequency is 9.8 Hz. The complete telescope in Fig. 4 has a total mass of $148t$, including the counterweight and lever, and the natural frequency is 6.6/5.6 Hz at zenith/horizon, so the design is capable of the fast turnarounds typical in CMB scanning observations (see Section 4.A). In a $10{\mathrm{ms}}^{-1}$ steady wind, the rim of the baffle around the incoming beam deflects 1.2 mm, so the stability of the baffle is consistent with the noise and beam error estimates in Section 4.D. The camera and secondary deflect 0.26 mm, giving a pointing error of $5^{\prime \prime }$, but the steady component of the pointing can be measured using sources in the maps, leaving an unsteady residual of just a few arcseconds that is negligible for B-mode polarization measurements [15]. Accurate measurements of the telescope beam shape may require smaller pointing errors, but such measurements could be restricted to when the wind speed is low. Differential deflection of the mirror support structure is $\sim 1\mathrm{mm}$ due to gravity and $\sim 2\mathrm{mm}$ due to a 30 K soak temperature change; both are within the alignment tolerance for a high Strehl ratio over the full FOV (see Section 3.C).

F. Survival

Survival during earthquakes and high winds is a significant concern for a telescope in the Atacama Desert. The telescope must not collapse under the combined loading due to gravity, soak temperature changes, which could be 30 K, and either seismic acceleration, which could be 5 g in the telescope structure, or wind, which could be $70{\mathrm{ms}}^{-1}$ but with 70% of the air density at sea level. For the combination with seismic loading, the peak stress in the structure in Fig. 4 is 144 MPa, located in the hexapod arms; with wind loading, the peak stress is 156 MPa, located in the baffle frame around the incoming beam. In both cases, the peak stress is well below the yield strength of the structure.

Survival at the South Pole is generally less of a concern because there are no earthquakes and the wind speed is much lower. The worst-case load combination is gravity, $35{\mathrm{ms}}^{-1}$ wind, and 60 K soak temperature change, giving a peak stress of only 55 MPa, located in the baffle frame.

5. CONCLUSION

We have developed a concept for a 5 m diameter, off-axis TMA for future CMB observations. The design has high throughput (424 k/136 k/63k $F\lambda$ pixels at $\lambda =1/2/3\mathrm{mm}$) and incorporates proven techniques from small-aperture CMB experiments to control systematic errors (monolithic mirrors, so there is no scattering from panel gaps, boresight rotation for measuring polarization errors, and a comoving shield or baffle around the entire telescope to reduce pickup). The telescope mirrors are supported by a CFRP spaceframe, which sits on a compact mount with a hexapod elevation axis and slewing rings for azimuth and boresight rotation. The structure is stiff ($\sim 7\mathrm{Hz}$ natural frequency and $\sim 1\mathrm{mm}$ deformation due to wind and gravity) and fairly light (tipping mass $\sim 40t$ and total mass $\sim 150t$). The design is interesting because the combination of high throughput, small systematic errors, and a large aperture diameter will allow sensitive measurements over a wide range of angular scales.

APPENDIX A: MONOLITHIC MIRRORS

The size of a monolithic mirror is limited by image quality degradation due to surface profile errors. This appendix gives estimates of the various surface error contributions for an aluminum, monolithic mirror, machined from a single billet, that is likely to be the least expensive option.

1. Gravitational Deformation

The rms surface error for a plate of area $A$ supported on $N$ points is [60]

(A1)$${\delta }_g=\gamma \frac{Q}{D}{\left(\frac{A}{N}\right)}^2,$$

where $\gamma \approx 2\times {10}^{-3}$ is the support efficiency, $Q$ is the pressure on the plate, and $D$ is the flexural rigidity. For a lightweighted plate, the pressure is

(A2)$$Q=\rho \eta gt,$$

where $\rho$ is the density of the plate material, $\eta$ is the lightweighting factor, $g$ is the acceleration due to gravity, and $t$ is the thickness of the plate. The flexural rigidity is

(A3)$$D\approx \frac{E{t}^3\eta /3}{12(1-{\nu }^2)},$$

where $E$ and $\nu$ are Young’s modulus and Poisson’s ratio for the plate material, and the effective modulus for the lightweighted plate is taken to be $E\eta /3$. Combining Eqs. (A1)–(A3) gives

(A4)$${\delta }_g\approx \frac{9{\pi }^2}{4}\frac{\gamma g\rho (1-{\nu }^2)}{E}{\left(\frac{d^2}{Nt}\right)}^2,$$

where $d={(4A/\pi )}^{1/2}$ is the plate diameter. Small, aluminum mirrors generally have $t=d/10$ and $N=3$, in which case ${\delta }_g\approx 4.3\mathrm{\mu m}\times {[d/(5\mathrm{m})]}^2$, assuming typical values of $\nu$, $\rho$, and $E$ for aluminum. The gravitational deformation is small, so the size of an aluminum mirror for millimeter wavelengths will be limited by thermal deformation or manufacturing errors.

2. Thermal Deformation

A temperature gradient $\mathrm{\Delta }{T}_z$ through the mirror causes the mirror to cup with radius of curvature

(A5)$$R=\frac{t}{\alpha \mathrm{\Delta }{T}_z},$$

where $\alpha$ is the coefficient of thermal expansion of the mirror material. The height of the cupped surface at radius $r$ is

(A6)$$z=\frac{r^2}{2R},$$

and the rms surface error is

(A7)$${\delta }_{\mathrm{\Delta }{T}_z}=\frac{1}{16\sqrt{3}}\frac{d^2}{t}\alpha \mathrm{\Delta }{T}_z.$$

An overall expansion of the mirror due to a change in soak temperature $\mathrm{\Delta }T$ changes the radius of curvature from $R$ to $R+\mathrm{\Delta }R$, where

(A8)$$\mathrm{\Delta }R=R\alpha \mathrm{\Delta }T.$$

The change in height of the surface is

(A9)$$z=\frac{r^2}{2R}\frac{\mathrm{\Delta }R}{R},$$

and the rms surface error is

(A10)$${\delta }_{\mathrm{\Delta }T}=\frac{1}{16\sqrt{3}}\frac{d^2}{R}\alpha \mathrm{\Delta }T.$$

A radial temperature gradient $\mathrm{\Delta }{T}_r$ causes the thickness of the mirror to vary with radius. For a linear radial gradient, the change in height of the surface is

(A11)$$z=\frac{2r}{d}\alpha \mathrm{\Delta }{T}_{r}t,$$

and the rms surface error is

(A12)$${\delta }_{\mathrm{\Delta }{T}_r}=\frac{1}{3\sqrt{2}}\alpha \mathrm{\Delta }{T}_{r}t.$$

The surface error due to a linear temperature gradient across the mirror will be similar.

3. Manufacturing Errors

Manufacturing errors are caused by deformation of the mirror and tool during machining, and since many deformations scale as $d^2$, it is likely that manufacturing errors also scale in this way, probably with some minimum error that depends on the details of the machining setup. Machined, aluminum mirrors up to $\sim 1\mathrm{m}$ in diameter have been made with $\sim 3\mathrm{\mu m}\mathrm{rms}$ surface error (e.g., panels for the Atacama Large Millimeter Array and the South Pole Telescope), and a precision milling machine with control based on laser tracker measurements should achieve 25 μm rms accuracy on a 5 m part [61], so the rms surface error due to manufacturing errors is modeled as

(A13)$${\delta }_{\mathrm{fab}}^2={[25\mathrm{\mu m}\times {\left(\frac{d}{5\mathrm{m}}\right)}^2]}^2+{(3\mathrm{\mu m})}^2.$$

4. Mirror Size

A telescope with rms surface error $\delta$ delivers a Strehl ratio [62], as follows:

(A14)$$S=\exp [-{\left(2\pi \frac{2\delta }{\lambda }\right)}^2]\mathrm{for}\delta \ll \lambda .$$

In the usual definition of the diffraction limit, $S>0.8$ and $\delta <\lambda /27$, so a telescope operating at $\lambda =1/2/3\mathrm{mm}$ must have $<38/75/113\mathrm{\mu m}\mathrm{rms}$ surface error. Figure 5 shows the various surface error contributions for a monolithic, aluminum mirror with a 30 K soak temperature change, which is half the p–p annual temperature variation at the South Pole, and 1 K temperature gradients, which are typical for a radio telescope mirror at night [63]. Overall thermal expansion and thermal cupping are the largest errors, but both are just a change in curvature, so they can be at least partly corrected by refocusing the telescope.

 

Fig. 5. Surface error due to gravitational deformation [green, Eq. (A4)], manufacturing errors [magenta, Eq. (A13)], thermal cupping [solid black, Eq. (A7) with $\mathrm{\Delta }{T}_z=1\mathrm{K}$], overall thermal expansion [red, Eq. (A10) with $\mathrm{\Delta }T=30\mathrm{K}$ and $R=5/10/15\mathrm{m}$ dotted/solid/dashed], and radial temperature gradient [blue, Eq. (A12) with $\mathrm{\Delta }{T}_r=1\mathrm{K}$] for a machined, aluminum mirror with $t=d/10$. The horizontal, black, dashed line is the surface error for 80% Strehl ratio at $\lambda =1\mathrm{mm}$ [Eq. (A14)].

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Overall thermal expansion is relatively easy to correct, because $d{\delta }_{\mathrm{\Delta }T}/d\mathrm{\Delta }T$ is small. A correction based on measurements of ambient temperature should reduce the surface error by an order of magnitude, likely limited by uniformity of the expansion coefficient of the aluminum. Thermal cupping is more difficult to correct because $d{\delta }_{\mathrm{\Delta }{T}_z}/d\mathrm{\Delta }{T}_z$ is much larger and the effect is driven by thermal gradients, not by a simple change in average temperature. In this case, a factor 2 improvement based on temperature measurements seems more realistic. With these corrections, a 5(4) m mirror has $\sim 32(22)\mathrm{\mu m}\mathrm{rms}$ surface error, so a telescope with two 5 m mirrors and one 4 m mirror (as in Fig. 2) should achieve $S=0.67/0.91/0.96$ at $\lambda =1/2/3\mathrm{mm}$. Reducing the manufacturing error by a factor 2 would give $S=0.81/0.95/0.98$.

APPENDIX B: TMA DESIGN

This appendix gives design details for the telescopes in Section 3. The starting point is an on-axis, classical design, following the approach and notation of Korsch [29], leading to an off-axis form that is optimized using ray tracing. Design details in Appendices B.1. and B.2. are from Korsh; Appendix B.3. is new.

1. Layout

A TMA is fully defined by seven parameters: the object distance, which is usually infinite for a telescope; the mirror separations, $d_1$ and $d_2$ (where $d_i$ is the distance from mirror $i$ to mirror $i+1$); the paraxial ray height ratios, ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ (where ${\mathrm{\Omega }}_i$ is the ray height at mirror $i+1$ divided by the ray height at mirror $i$); the system focal length, $f^{\prime }$; and the entrance pupil to primary separation, $t_1$. The ray height ratios are a rough proxy for the relative sizes of the mirrors; ${\mathrm{\Omega }}_i<0$ indicates a focus between mirrors $i$ and $i+1$. If the entrance pupil is located at the primary (e.g., to minimize the size of the primary), $t_1=0$.

The layouts in Figs. 1 and 2 were generated using the following parameters: inverse distance from a mirror to its object,

(B1)$$v_1=0\text{for an object at}\infty ,v_i=v_{i-1}^{\prime }/{\mathrm{\Omega }}_{i-1}\mathrm{for}i=\{2,3\},$$

inverse distance from a mirror to its image,

(B2)$$\begin{gathered} v_i^{\prime }=(1-{\mathrm{\Omega }}_i)/d_i\mathrm{for}i=\{1,2\}, \\ {v}_3^{\prime }=-1/({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{f}^{\prime })f^{\prime }\mathrm{is}-\mathrm{ve}, \end{gathered}$$

vertex curvature ($+\mathrm{ve}$ is concave for $i=\{1,3\}$, convex for $i=2$),

(B3)$$c_i=\frac{1}{2}(v_i+v_i^{\prime }),$$

magnification,

(B4)$$m_i=-v_i/v_i^{\prime },$$

eccentricity,

(B5)$${\epsilon }_i=\frac{1+m_i}{1-m_i},$$

distance from a mirror to its exit pupil,

(B6)$$t_i^{\prime }=-t_i/(1-2c_i{t}_i),$$

and distance from a mirror to its entrance pupil,

(B7)$$t_i=t_{i-1}^{\prime }-d_{i-1}\mathrm{for}i=\{2,3\}.$$

2. Aberrations

The conic constants of the mirrors are given by

(B8)$${\delta }_i=\mathrm{\Delta }{\delta }_i-{\left(\frac{1+m_i}{1-m_i}\right)}^2,$$

and for a design with no spherical aberration, coma, or astigmatism,

(B9)$$\begin{gathered} \mathrm{\Delta }{\delta }_3=\frac{2{\mathrm{\Omega }}_1(v_1+v_3^{\prime })-v_1^2{d}_1-{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{(v_3^{\prime })}^2({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{d}_1-2d_2)}{4{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2^2{d}_2({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{d}_1-d_2)c_3^3}, \\ \mathrm{\Delta }{\delta }_2=\frac{v_1^2-{\mathrm{\Omega }}_1^2{\mathrm{\Omega }}_2^2{(v_3^{\prime })}^2+4{\mathrm{\Omega }}_1^2{\mathrm{\Omega }}_2^3({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{d}_1-d_2)c_3^3\mathrm{\Delta }{\delta }_3}{4{\mathrm{\Omega }}_1^3{d}_1{c}_2^3}, \\ \mathrm{\Delta }{\delta }_1={\mathrm{\Omega }}_1^4(c_2^3\mathrm{\Delta }{\delta }_2-{\mathrm{\Omega }}_2^4{c}_3^3\mathrm{\Delta }{\delta }_3)/c_1^3. \end{gathered}$$

For an anastigmat, the image curvature is just the Petzval curvature,

(B10)$$c_P=2(c_1-c_2+c_3),$$

so a flat image requires $c_P=0$, in which case at least one mirror must be convex; there are no solutions with two intermediate foci and a flat image. Combining Eqs. (B10), (B3), (B1), and (B2), and $c_P=0$ gives

(B11)$$v_3^{\prime }=\frac{{(1-{\mathrm{\Omega }}_1)}^2}{{\mathrm{\Omega }}_1{d}_1}-\frac{{(1-{\mathrm{\Omega }}_2)}^2}{{\mathrm{\Omega }}_2{d}_2},$$

and combining Eqs. (B11) and (B2) gives

(B12)$$d_2=\frac{{(1-{\mathrm{\Omega }}_2)}^2{\mathrm{\Omega }}_1{d}_1}{{(1-{\mathrm{\Omega }}_1)}^2{\mathrm{\Omega }}_2+d_1/f^{\prime }}.$$

3. Telecentricity

Millimeter-wavelength cameras generally use directional antennas to couple signals into the detectors, and efficient coupling requires a telecentric image, which can be achieved by making the tertiary exit pupil distance equal to $1/c_P$. In the case of a flat field, $c_P=0$, so $t_3^{\prime }=\infty$ and Eq. (B6) gives

(B13)$$t_3=\frac{1}{2c_3}.$$

If the entrance pupil is at the primary, $t_1=0$, and Eqs. (B7) and (B6) give

(B14)$$t_3=\frac{d_1}{1+2c_2{d}_1}-d_2.$$

Combining Eqs. (B13), (B14), (B3), (B1), and (B2) gives

(B15)$$\begin{gathered} 0=d_2^2{(1-{\mathrm{\Omega }}_1)}^2-d_2{d}_1{\mathrm{\Omega }}_1\{2[{\mathrm{\Omega }}_2(1-{\mathrm{\Omega }}_1)-1]+{\mathrm{\Omega }}_1^2{\mathrm{\Omega }}_2\} \\ +d_1^2{\mathrm{\Omega }}_1^2{(1-{\mathrm{\Omega }}_2)}^2. \end{gathered}$$

Equations (B12) and (B15) must be solved simultaneously for an image that is both flat and telecentric. Figure 6 shows an example solution.

 

Fig. 6. $d_2$ versus ${\mathrm{\Omega }}_2$ for a flat image [solid, Eq. (B12)] and exit pupil at infinity [dashed, Eq. (B15)] for a TMA with $d_1=6.5\mathrm{m}$, ${\mathrm{\Omega }}_1=0.33$, and $f^{\prime }=-15\mathrm{m}$ (starting point for the design in Fig. 2).

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For an off-axis telescope, the situation is more complicated. Placing the exit pupil at infinity ensures that the principal rays for different fields are parallel, but they are not necessarily normal to the image surface. The latter must be achieved by adjusting the field and aperture offsets.

4. Design Procedure

The designs in Section 3 were constructed using the following procedure: (1) place the entrance pupil at the primary ($t_1=0$) to minimize the diameter of the primary; (2) choose the focal length to give a reasonable compromise between image size and speed of the camera reimaging optics; (3) choose $d_1$ to set the overall length of the telescope; (4) choose ${\mathrm{\Omega }}_1$ to set the size of the secondary; (5) choose $c_P=0$ for a flat image, or make the image slightly curved to improve telecentricity for large FOV; (6) calculate $d_2$ and ${\mathrm{\Omega }}_2$ to make the tertiary exit pupil distance equal to $1/c_P$ [Eqs. (B12) and (B15) for the case of a flat image with the tertiary exit pupil at infinity]; (7) calculate the mirror curvatures [Eq. (B3)], conic constants [Eqs. (B8) and (B9)], and the tertiary to image distance [Eq. (B2)]; (8) choose the aperture and field offsets to maximize the unblocked FOV; (9) optimize the layout using ray tracing, allowing changes in mirror separation and curvature, decenter and tilt of the secondary, tertiary, and image, and aspheric terms in all the mirror surfaces. The addition of tilt and decenter results in a large, complicated design space that makes optimization challenging. For the designs in Section 3, the optimization merit function included wavefront error, angle of incidence at the image, and focal length. A laborious manual search was needed to minimize the size of the tertiary and maintain reasonable clearance between the beam and the secondary. Essentially all of the layout parameters were adjusted during optimization.

Tables 1 and 2 give the optical prescriptions for the designs in Section 3. The table entries and sign conventions follow ZEMAX [64]. Decenter and then tilt are applied immediately before a surface, and removed immediately after as −tilt then −decenter, before applying the distance to the next surface. The profile for the $i$th surface is

(B16)$$z_i(r)=\frac{c_i{r}^2}{1+{[1-(1+{\delta }_i)c_i^2{r}^2]}^{1/2}}+A_i{r}^4.$$

Values of $A_i$ in the tables are for $z$ and $r$ in millimeters; the change in profile due to the aspheric term is $\sim 1\mathrm{mm}$. Each table also shows the on-axis starting point for the design.

Table 1. Optical Prescription for the Design in Fig. 1^a

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Table 2. Optical Prescription for the Design in Fig. 2^a

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APPENDIX C: FINITE ELEMENT MODEL

The layout in Fig. 4 is a wireframe model of the telescope generated in MATLAB [65]. The wireframe model drives an ANSYS [66] finite element model that contains struts, point masses, universal joints, loads, and constraints. Model details are given in Table 3, materials properties are in Table 4, and Table 5 shows the mass distribution. The design is roughly optimized in the sense that the cross-section of the struts has been adjusted to give a reasonable compromise between natural frequency of the structure, total mass, deflections due to gravity and wind loads, and stress under survival conditions. CFRP properties in Table 4 represent realistic values for complete struts with metal end fittings and CFRP tubes made of commercial-grade, high-modulus fiber.

Table 3. Finite Element Model Details

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Table 4. Materials Properties

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Table 5. Telescope Mass

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Funding

University of Chicago (UChicago).

Acknowledgment

The author thanks Tom Crawford, Mark Devlin, Frederick Takayuki Matsuda, Jeff McMahon, Michael Niemack, and three reviewers for useful comments.

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