Wavefront sensor for millimeter/submillimeter-wave adaptive optics based on aperture-plane interferometry
Yoichi Tamura, Ryohei Kawabe, Yuhei Fukasaku, Kimihiro Kimura, Tetsutaro Ueda, Akio Taniguchi, Nozomi Okada, Hideo Ogawa, Ikumi Hashimoto, Tetsuhiro Minamidani, Noriyuki Kawaguchi, Nario Kuno, Yohei Togami, Masato Hagimoto, Satoya Nakano, Keiichi Matsuda, Sachiko Okumura, Tomoko Nakamura, Mikio Kurita, Tatsuya Takekoshi, Tai Oshima, Toshikazu Onishi, Kotaro Kohno
WWavefront sensor for millimeter/submillimeter-wave adaptiveoptics based on aperture-plane interferometry
Yoichi Tamura a , Ryohei Kawabe b , Yuhei Fukasaku c , Kimihiro Kimura a , Tetsutaro Ueda a , AkioTaniguchi a , Nozomi Okada d , Hideo Ogawa d , Ikumi Hashimoto e , Tetsuhiro Minamidani b,f ,Noriyuki Kawaguchi b , Nario Kuno c , Yohei Togami a , Masato Hagimoto a , Satoya Nakano a ,Keiichi Matsuda a , Sachiko Okumura g , Tomoko Nakamura g , Mikio Kurita h , TatsuyaTakekoshi j,i , Tai Oshima b , Toshikazu Onishi d , and Kotaro Kohno ia Nagoya University, Nagoya, 464-8602 Japan b National Astronomical Observatory of Japan, Mitaka, Tokyo, 181-8588 Japan c University of Tsukuba, Tsukuba, Ibaraki, 305-8573 Japan d Dept. Physical Science, Osaka Prefecture University, Sakai, Osaka, 599-8531 Japan e Dept. Aerospace Engineering, Osaka Prefecture University, Sakai, Osaka, 599-8531 Japan f Nobeyama Radio Observatory, Nagano, 384-1305 Japan g Japan Women’s University, Bunkyo, Tokyo, 112-8681 Japan h Kyoto University, Kyoto, 606-8501 Japan i Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo, 181-0015 Japan j Kitami Institute of Technology, Kitami, Hokkaido, 090-8507 Japan
ABSTRACT
We present a concept of a millimeter wavefront sensor that allows real-time sensing of the surface of a ground-based millimeter/submillimeter telescope. It is becoming important for ground-based millimeter/submillimeterastronomy to make telescopes larger with keeping their surface accurate. To establish ‘millimetric adaptive optics(MAO)’ that instantaneously corrects the wavefront degradation induced by deformation of telescope optics, ourwavefront sensor based on radio interferometry measures changes in excess path lengths from characteristicpositions on the primary mirror surface to the focal plane. This plays a fundamental role in planed 50-m classsubmillimeter telescopes such as LST and AtLAST.
Keywords:
Submillimeter, single-dish telescope, adaptive optics, aperture-plane interferometry
1. INTRODUCTION
The increase in size of telescopes, i.e., the acquisition of larger collecting area and spatial resolution, is the veryhistory of observational astronomy. For millimeter and submillimeter-wave telescopes (mm/submm telescopes,hereafter), increasing the antenna diameter with keeping the mirror surface accurate is important in developingnew astronomical fields. The advent of an aperture synthesis interferometer has resulted in significant improve-ment in collecting area and angular resolution. For example, the Atacama Large Millimeter/submillimeter Array(ALMA) realized the performance by keeping the size of the array element antennas moderate ( D = 7 and 12 m)and increasing stiffness of them to keep their surface down to 20 µ m r.m.s or even better.However, aperture synthesis interferometry is not a panacea in realizing a larger-scale mm/submm telescope.The focal plane instruments of radio interferometers are limited to coherent receivers (e.g., heterodyne receivers)that can detect the phase of celestial signals. On the other hand, large-format arrays of direct photon detectors(e.g., cameras and integrated superconducting spectrometers ), which have prospered in recent years, cannotbe accommodated as receivers for an aperture synthesis interferometer, because phase detection is not possible. Further author information: (Send correspondence to Y.T.)Y.T.: E-mail: [email protected], Telephone: +81 (0)52 789 2846 a r X i v : . [ a s t r o - ph . I M ] F e b hus, it becomes increasingly important to have a large-aperture single-dish mm/submm telescope (e.g., LargeSubmillimeter Telescope and Atacama Large Aperture Submillimeter Telescope ), especially in the field inwhich direct photon detector arrays play a crucial role.A limiting factor in size and operating frequency of mm/submm telescopes is the deterioration of opticalperformance due to changes in the environment surrounding the telescope, such as wind load and thermaldeformation in addition to gravitational deformation. ∗ This is unlike the visible/near-infrared (NIR) regime,where the major cause of wavefront degradation is due to temperature inhomogeneities of the troposphericatmosphere on the order of tens of cm. Mm/submm telescopes have been manufactured on the premise of large( >
10 m) structures in order to ensure the large collecting area and high spatial resolution. For this reason,building a huge dome to cover it would require enormous costs. Therefore, it was an issue to construct an‘exposed’ antenna outside and ensure its mirror surface with an accuracy of ∼ µ m.With the existing technology, it is possible to measure the mirror surface shape in advance by the photogram-metry or radio holography methods, and to correct the mirror surface down to the level of ∼ µ m (r.m.s.) byadjusting the positions of the primary mirror panels. In recent years, the weight reduction and enhanced stiffnessof the primary mirror structure and the active surface control of the primary mirror panels by the motorizedactuators have been used to keep the primary mirror an ideal shape with respect to changes in telescope elevationangle. The problem is, however, the deformation of the primary mirror surface and the secondary mirror supportstructure due to wind load and thermal deformation. This is because the time-scale of the deformation is fairlyshort compared with a typical time-scale of astronomical observations. It is difficult to measure the mirrorsurface shape in real time during the observations. It is likely that most of the deformation across the primarymirror are dominated by lower-order deformation modes. Their typical spatial scale is found to be a fractionof the aperture diameter (1–10 m). The time-scale of the deformation is determined by the natural frequencyof the primary mirror structure to the wind load and is typically ∼ − –1 s (e.g., (cid:39) . ). In this case, the control technology of the adaptive optics does not become a majorobstacle. The spatial and time-scales of adaptive optics control, which is already established in NIR astronomy,are on the order of ∼
10 cm and ∼ how to measure the wavefront in real time .Wavefront sensing using a Shack-Hartmann sensor is the mainstream in the optical and NIR. A similar approach,however, is not applicable to MAO, since no large-format detector array that can be manufactured cheaply isavailable in the mm/submm. Instead, radio astronomy can exploit radio interferometry as a native wavefrontsensing technology, which measures the difference between the arrival times of the wavefronts coming throughtwo independent optical paths.In this paper, we present a concept of a mm wavefront sensor that allows real-time sensing of the primarymirror surface of a ground-based large-aperture mm/submm telescope.
2. APERTURE-PLANE INTERFEROMETRY FOR WAVEFRONT SENSING
As seen in the previous section, what needs to be measured first is the relative time variation of the deformation ofthe optical system, rather than wavefronts of celestial signals coming through the atmosphere. This is equivalentto the errors in excess path length (EPL) from the focus to arbitrary points on the primary surface. Therefore,the wavefront sensor employs multiple reference microwave/mm sources placed on the aperture of the telescopeas a phase standard. The goal of the surface accuracy we measure with a prototype sensor (see later sections) ∗ In the millimeter and submillimeter regime, the inhomogeneities in column density of water vapor in the tropospherealso cause wavefront degradation. However, the spatial scale at which the water vapor fluctuations dominate the wavefrontdegradation is typically much longer than the telescope’s aperture, for example, (cid:38) m at the ALMA site. For thisreason, in most cases, the Earth’s atmosphere is not a major source of wavefront degradation for single-dish mm/submmtelescopes. EFERENCECORRELATOR REFERENCECORRELATOR REFERENCE CORRELATOR (a) (b) (c)
Primary mirrorSecondary mirror
REFERENCECORRELATORSWITCH
Delay (d)
Figure 1. A schematic diagram of aperture-plane interferometry. (a) A simple interferometer with a correlator whichmeasures a difference in arrival time between two signals coming from a common reference source. (b) The same as (a)but one of the optical paths coming through the free space as shown with the dashed line. (c) The same as (b) butthe path through the free space goes through the telescope optics. The lengths of the other path is adjusted by a delayline shown as a loop. (d) The same as (c) with a switch followed by an array of radiators which enables to measure thedistances from arbitrary positions to the focal point. is 40 µ m r.m.s. This is close to ≈ µ m r.m.s., which is routinely achieved by holographic measurements ofthe Nobeyama 45 m mm-wave telescope. This measurement accuracy is equivalent to the phase accuracy of 1 ◦ r.m.s. when operating the wavefront sensor at 20 GHz.Figure 1 shows the schematic diagram representing the principle of aperture-plane interferometry. A simpleinterferometer comprising a correlator (figure 1a, b) allows us to measure a difference in arrival time betweentwo signals generated by a common reference source. The interferometer in which one of the paths goes throughthe telescope optics (figure 1c) measures the EPL from a certain point on the primary surface to the focus ofthe telescope. This is similar to what is used for phase calibration (the p-cal method, REF) of phase-referencingreceivers used in VLBI Exploration of Radio Astronomy (VERA ). The method is a scalable technique and canbe time-multiplexed by an intervening switch which is followed by a series of radiators placed across the surfaceof the primary mirror (figure 1d).The required number of reference points (i.e., radiators) depends on to what order of deformation modesneeds to be characterized. As the wavefront sensor detects the relative deformation with respect to the idealsurface realized when the telescope structure is static, the surface needs to be adjusted by a conventional methodsuch as the radio holography in advance. The relative deformation of the structure induced by wind and thermalloads emerges mostly on large spatial scales (e.g., a fraction of an aperture size). Therefore, a couple of tensreferences across the aperture will be good enough.We opt broadband noise as the common reference source, rather than a continuous wave (CW) or narrowbandsignals. This is because in the case of CW or narrowband signals, the stray lights through multiple paths caninterfere with each other, resulting in considerable systematic errors in measurements of EPLs. On the otherhand, broadband noises are more robust against the multi-path interference, and thus they are often used fordelay calibration in aperture-synthesis interferometers such as ALMA. The EPL is imprinted as a phase slope ina cross power spectrum (CPS) of the broadband noise. Let C ( ν ; EPL) = A ( ν ) exp [ iφ ( ν ; EPL)] be a measuredCPS. Then, a calibrated CPS is obtained as C ( ν ; EPL) /C ( ν ; 0) = exp [ i { φ ( ν ; EPL) − φ ( ν ; 0) } ]. The phase ofa calibrated CPS, ∆ φ ( ν ; EPL) ≡ φ ( ν ; EPL) − φ ( ν ; 0), induced by a certain EPL is expressed as a function offrequency ν as ∆ φ ( ν ) = 2 πτ ν = 2 π EPL c ν, (1)where τ is the time delay induced by EPL. This means that the slope of the CPS phase immediately gives theEPL. 3 eference Noise Generator 16 −
24 GHz CorrelatorOptical ModulatorSM Fiber (70 m)MEMS Optical Switch 20 GHz Receiver at Focal PointOptical Demodulator)Optical Demodulator)Optical Demodulator)O/E (Optical Demodulator) OpticsPrimary MirrorSM Fibers (45 −
48 m) Anti Aliasing Filter (16 −
24 GHz)Power AmplifierFeedRECEIVER CABINTELESCOPESwitching Pattern Generator 1 PPS/10 MHz Ref.
Accelerometers (b)(a)
Receiver CabinO/E − Feed Optical Switch
Figure 2. (a) The system block diagram of the prototype wavefront sensor for millimetric adaptive optics. (b) Theconfiguration of the wavefront sensor system mounted on the Nobeyama 45 m telescope.
3. SYSTEM
Here we describe the system of the prototype wavefront sensor for demonstration. Although this prototype onlyhas several elements of radiators operating at 20 GHz, which is relatively low frequency and easy to handle, theaperture-plane interferometry is scalable up to tens to hundreds of elements at higher frequencies.
The top-level requirement on the system is to instantaneously measure the deviation from the ideal mirrorsurface with an accuracy of 40 µ m r.m.s. with a time-resolution of 100 ms, well below the natural frequencyof the primary mirror structure ( ≈ ∼
10 s) stability of EPL measurements, whereas long term stability needs to be taken into account in theactual implementation of the system. Table 1 summarizes the specifications of the prototype wavefront sensor.
Table 1. Specifications of the prototype wavefront sensor.
Item ValueFrequency (GHz) 16–24Number of elements 5
Figure 2 shows the schematic block diagram and configuration describing how the wavefront sensor works. Thesystem is threefold; (1) the transmitter subsystem, which generates and transfers a reference signal, and injectsit into the telescope optics from multiple positions across the primary surface; (2) the receiver subsystem, whichcollects the signal coming through the telescope optics and sends it to the correlator; and (3) the correlator sub-system, which differentiates the arrival time of the reference signals coming directly from the reference generatorand through the telescope optics.
The transmitter subsystem comprises a reference noise generator, optical transmitter and switch, and radiatorson the primary mirror.
Reference noise generator.
For the reference broadband noise, we use Johnson-Nyquist noise of a microwaveterminator amplified by a cascade of power amplifiers which is followed by a bandpass filter (17.3–23.6 GHz).The excess noise ratio (ENR) is ≈
70 dB. The reference noise is divided, one of which goes directly into thecorrelator while the other is sent to the telescope primary mirror through the optical fiber.4 eference Noise Generator 16 −
24 GHz CorrelatorOptical ModulatorSM Fiber (70 m)O/E (Optical Demodulator) Anti Aliasing Filter (17.3 − (a) Feed (Receiver)Free Space (2 m)Micrometer Stage
Micrometer StageFeed Horn (b)(c)
Low Noise Amplifier
Figure 3. (a) The setup of laboratory evaluation. (b) A feed horn mounted on a micrometer stage. (c) The OCTAD-Mcorrelator.
Optical fiber transmission and distribution.
The reference signal is transmitted on optical fiber, sincethe distances to the radiators are too long to maintain the phase stability and gain of the reference signal. Weuse a radio-frequency on fiber (RFoF) system (Optilab, RFLL-20-H), a pair of optical modulator (E/O) anddemodulator (O/E). We use a phase-stabilized single-mode fiber (Sumitomo Electric) with a linear expansioncoefficient of 4 . × − K − . The optical signal leaving the E/O is coupled with one of the five O/Es placedon the backup structure of the primary mirror by switching over the fibers with a MEMS-based optical switchplaced closed to the central hub of the primary mirror. Feed horn on the primary mirror.
The reference signal is demodulated with the O/E and is radiated byone of five identical feed horns placed on the surface of the primary mirror. We choose a linear-polarized feedhorn as a reference signal radiator as the operating frequency is broad (∆ f /f = 0 . ◦ , respectively. The feed horns are placed at a radius of 16 m withposition angles of − ◦ (top), 90 ◦ (left), 180 ◦ (bottom), and 270 ◦ (right), and at r = 5 m with a position angleof 0 ◦ (center; see also figure 2b). We use the H22 receiver, † a science-grade cryogenic coherent receiver operating at 20 GHz. The typical systemnoise temperature is T sys = 100 K. H22 detects two circular polarization, while we only use one of them forphase measurements. The receiver output is amplified and the bandwidth is constrained by the anti-aliasingfilter before the signal goes into the correlator. We develop a FX-type digital correlator, OCTAD-M (ELECS Inc.), which is based on the architecture initiallydeveloped for the FPGA-based fast Fourier transform (FFT) spectrometer OCTAD-S. OCTAD-M is equippedwith two 3-bit analog-to-digital converters with 16.384 GSa/s. We opt the third-order mode (16.384–24.576 GHz)of analog input signals, allowing us to directly sample the H22 receiver output with no down conversion. Twoinput signals are FFTed first and are multiplied to obtain the cross-power spectrum. The spectra are accumulatedfor 5 or 10 msec. OCTAD-M has a digital delay capability with up to 2 times sampling clock time, allowing todigitally insert an instrumental delay of 0–2 µ sec, corresponding to 0–599.585 m for free-space geometrical delay. To synchronize the optical switch and the correlator sampling, we use common 1 PPS and 10 MHz referencesignals fed by the observatory GPS server. System control and data acquisition are performed by a single Linuxserver. † EP L m ea s u r ed b y pha s e s l ope ( mm ) EPL induced by micrometer stage (mm)MeasuredPredicted -40-20 0 20 40 0 2 4 6 8 10 12 14 16 E rr o r i n EP L ( µ m ) EPL induced by micrometer stage (mm) C r o ss po w e r s pe c t r u m pha s e ( deg ) Frequency (GHz) 15 mm14 mm13 mm12 mm11 mm10 mm9 mm8 mm7 mm6 mm5 mm4 mm3 mm2 mm1 mm
System requirement (± 40 μ m) Mean: 4.8 μ mStd. dev.: 17.4 μ m (a) (b) (c) Figure 4. (a) The cross power spectrum (CPS) phases measured for different excess path lengths (EPLs) induced bymechanical deviation of the micrometer stage z . (b) The EPL measured by the phase slope as a function of z . (c) Theresidual, EPL − z , as a function of z . The error bar on each plot accounts for the 1 σ error in linear regression of theCPS phase. The average of the residuals is 4.8 µ m with a standard deviation of 17.4 µ m. The grey region represents thesystem requirement ( ± µ m). To help the wavefront sensing experiment, we use ancillary devices placed on and around the Nobeyama 45 mtelescope. We have six piezoelectric accelerometers attached to the backup structure of the primary mirror, which are placed close to the edge of the primary mirror with position angles of 0 ◦ (top), 45 ◦ , 90 ◦ (left), 180 ◦ (bottom), 270 ◦ (right), and close to the central hub (see figure 2b). We also use a weather monitor placed at thetop of a 50-m tall meteorological tower, which is located 75 m north of the telescope. ‡
4. DEMONSTRATION4.1 Laboratory evaluation of linearity and accuracy in excess path length measurements
In advance to the on-site demonstration, we evaluate the linearity, accuracy, and stability of the system withrespect to mechanical change in EPL. Although neither the actual telescope optics nor the H22 receiver isavailable in the lab, we use almost the same configuration of the other subsystems which are used for the on-sitemeasurements. The configuration of the lab experiment is shown in figure 3. Here we use a feed horn followedby a power amplifier and a bandpass filter instead of the H22 receiver. We placed the two feed horns of theradiator and receiver ≈ µ m.We measure the CPS phase at z = 0, 1, 2, ..., 15 mm, where z is the EPL induced by the mechanical deviationof the micrometer stage. The CPS is calibrated for the complex bandpass (i.e., amplitude and phase) by dividingthe CPS by that of z = 0. Figure 5a shows the calibrated CPS phases obtained at z = 1, 2, 3, ..., 15 mm.The CPS is running-averaged with a rectangle window of 300 MHz. We see a linear slope for each CPS phasealthough a small fluctuation remains around the best-fitting line. Since the statistical error of the CPS phase isalmost negligible, the fluctuation could be dominated by systematic errors such as standing waves, multi-pathinterference, or phase drift of the system. As shown in figure 5b, the EPL derived from equation 2 is linear withrespect to the mechanical deviation z . Figure 5c shows the residual of the measured EPL subtracted by z as afunction of z . The average of the residuals is 4.8 µ m with a standard deviation of 17.4 µ m, which is well belowthe system requirement of < µ m, confirming the system linearity and accuracy. ‡ V w i nd ( m / s ) Time (s)-50 0 50 A cc . ( mm / s ) topcenter-20-10 0 10 20 C PS pha s e a t G H z ( deg ) Phase acc.Phase V w i nd ( m / s ) Time (s)-50 0 50 A cc . ( mm / s ) topcenter-20-10 0 10 20 C PS pha s e a t G H z ( deg ) Phase acc.Phase (a) (b)
Figure 5. (a) The cross power spectrum (CPS) phase obtained under a very windy (5–10 m s − ) condition and (b) undera moderately windy ( ∼ − ) condition. The time origins are 3:21:00 on 2020 November 22 UTC and 2:09:00 on 2020November 23 UTC, respectively. (Top) The CPS phase measured at the ‘top’ position of the 45 m primary mirror (bluecurve). As the CPS is measured at 20 GHz ( λ ≈
15 mm), a phase shift of 10 deg approximately corresponds to a surfacedeviation of 400 µ m. The grey curve represents its second-order differential in arbitrary units. (Center) The accelerationmeasured with two accelerometers attached to the top edge (blue) and the center (purple) of the 45 m primary mirror.(Bottom) The wind speed measured with a weather monitor placed atop the 50-m tall tower located 75 m north of the45 m telescope. We carried out a commissioning campaign of the MAO wavefront sensor with the Nobeyama 45 m telescopein 2020 November. The goals are to confirm that the wavefront sensor actually works as designed and todemonstrate that aperture plane interferometry allows us to measure an EPL in real time. In this campaign,only 2 radiators were installed at the ‘top’ and ‘center’ positions.The top pannels of figures 5a and 5b show the temporal changes in CPS phase at 20 GHz, which were takenunder very windy (5–10 m s − ) and moderate ( ∼ − ) conditions, respectively. A large phase drift andsmall ∼ ∼ ∼ .
5. CONCLUSIONS
We present a concept of a millimeter wavefront sensor which allows real-time sensing of the primary mirror surfaceof a future 50-m class mm/submm telescope, such as LST and AtLAST. To establish millimetric adaptive optics(MAO) that instantaneously corrects the wavefront degradation induced by deformation of telescope optics dueto wind and thermal loads, our wavefront sensor employs aperture-plane interferometry to measure real-timechanges in EPLs from characteristic points on the primary surface to the focal plain. The proposed wavefrontsensor operates at 16–24 GHz, which is cost-effective and is accurate enough to measure the EPL down to thelevel of 40 µ m r.m.s. as demonstrated in the laboratory evaluation. Although the verification is in progress, wehave demonstrated that a 2-element pilot wavefront sensor with the Nobeyama 45 m telescope worked properlyand detected the EPL change induced by wind load on the telescope structure.7 PPENDIX A. SENSITIVITY
In general, the variance of the cross power spectrum is given by σ = ( S + N ) · ( S + N ) + S S ν ∆ t , where S i and N i ( i = 1 ,
2) are the correlated and non-correlated components of the cross power spectrum of twoincident signals, respectively. Therefore, the S/N of the cross power spectrum isSNR = √ S S √ ν ∆ t (cid:112) ( S + N ) · ( S + N ) + S S . Here, the correlated and non-correlated components input directly to the correlator from the reference signalsource (reference continuum source) are S , N , and components input to the correlator via the transmissionsystem and optical system are S , N , then S (cid:29) N , so the S/N per spectral channel of the cross power spectrumcan be approximated asSNR ≈ √ ν ∆ t (cid:112) N /S + 2 ∼ (cid:112) ν ∆ t ( S /N ) ≈ . (cid:18) ∆ ν (cid:19) . (cid:18) ∆ t
10 ms (cid:19) . (cid:18) N /S (cid:19) − . , where N /S ∼ O (10 ) (cid:29) n s spectral channels is scaled with √ n s for S/N per channel, thenSNR tot ≈ (cid:18) ∆ ν (cid:19) . (cid:18) ∆ t
10 ms (cid:19) . (cid:18) N /S (cid:19) − . . Assuming that thermal noise dominates N and the cross power spectrum has isotropic dispersion on thecomplex plane, the uncertainty of the phase θ is σ θ ∼ / SNR tot (radian). Therefore, σ θ ≈ .
078 (deg) × (cid:18) ∆ ν (cid:19) − . (cid:18) ∆ t
10 ms (cid:19) − . (cid:18) N /S (cid:19) . If the value is fiducial, sufficient phase measurement accuracy can be achieved. In other words, it is necessaryto obtain a correlator input signal that satisfies N /S < ACKNOWLEDGMENTS
We acknowledge K. Handa, C. Miyazawa, T. Kanzawa, T. Wada, and M. Saito for their support. This work wassupported by JSPS KAKENHI (Grant No. 17H06206) and NAOJ Research Coordination Committee, NINS.
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