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The GMT active optics control strategies

R. Conana, A. Boucheza, F. Quiros–Pachecoa, B. McLeodb, and D. Ashbya

aGMTO, 465 N. Halstead Avenue, Pasadana, CAbHarvard Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA

ABSTRACT

The Giant Magellan Telescope (GMT) has a Gregorian 25.4-meter diameter primary mirror composed of seven8.4-meter diameter segments. The secondary mirror consists of seven 1.1-meter diameter segments. In the activeand adaptive operation modes of the GMT, around a dozen wavefront sensors are selectively used to monitor theoptical aberrations across the focal plane. A dedicated wavefront control system drives slow and fast correctionsat the M1 and M2 mirrors to deliver image quality optimized for the field of view of the scientific instrumentin use. This paper describes the control strategies for the active optics mode of the GMT. Different wavefrontestimation algorithm are compared and the performance of the GMT is evaluated using the Dynamic OpticalSimulation package.

1. INTRODUCTION

The Giant Magellan Telescope3 (GMT) is 25.4m diameter aplanatic gregorian telescope. It is doubly segmentedmeaning that both M1 and M2 are made of 7 circular segments. M1 segments are 8.4m in diameter and M2segments are geometrically matched to M1 segment with a diameter of 1.1m. The primary focus of the GMTis located below M2 and the gregorian focus is below the M1 center segment where a science instrument canbe inserted. A third flat mirror (M3) can be deployed to fold the gregorian focus toward a suite of scienceinstruments arranged on a rotating platform below M1, the gregorian instrument rotator (GIR). Below the GIRsits the acquisition, guiding and wavefront sensing system2 (AGWS). The AGWS is a set of 4 deployable probespicking up guide stars up to 10’ off–axis. The guide stars light is split between a high–order Shack–HartmannWFS, a segment image stacker, a phasing camera and an acquisition and guiding camera.

Each M1 segment has 6 hard–point actuators to adjust the segment orientation and 165 actuators to controlthe segment figure. M2 comes in two flavors: (i) 7 fast steering mirrors (FSM) with 200Hz tip–tilt correctionscapability and (ii) 7 adaptive secondary mirrors (ASM) with 672 actuators each with a 1kHz update rate. Thedifferent telescope operating modes fall into two main categories: natural seeing and adaptive optics modes. TheFSMs are used only with the natural seeing mode whereas the ASMs can be used for either mode.

In the following, we will be focusing mainly on the control of M1 and M2 segment in the natural seeing modewith the AGWS and the FSM. Section 2 discusses the observable modes of the system in the natural seeingmode. The system model is described in Section 3 and in Section 4, the performance of the AGWS system ispresented. Section 5 evaluates the effect of detector and atmospheric noise of the performance of the system.

2. OBSERVABLE MODES

Each M1/M2 segment pair is forming one image and a GMT image is obtained by stacking coherently the 7images of the 7 segment pairs. The optical aberrations in each image are corrected by adjusting the orientationof both M1 and M2 segments and by deforming M1 segments. The controlled modes are rigid body translations(Tx,Ty,Tz) and rotations (Rx,Ry,Rz) for both M1 and M2 segments and the bending modes of M1.

Table 1 shows the wavefront distortions associated with the 6 rigid body transformations. A rotation aroundthe z–axis (Rz) of the center segment of both M1 and M2 has no effect on the wavefront aberrations. These modescan then be safely ignored for the center segments. A translation along the z-axis (Tz) of the center segment

Further author information: (Send correspondence to R. Conan)R. Conan: E-mail: [email protected], Telephone: +1 626 204 0507

Adaptive Optics Systems V, edited by Enrico Marchetti, Laird M. Close, Jean-Pierre Véran, Proc. of SPIE Vol. 9909, 99091T© 2016 SPIE · CCC code: 0277-786X/16/$18 · doi: 10.1117/12.2231578

Proc. of SPIE Vol. 9909 99091T-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 02/01/2017 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx

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Table 1. Wavefront distortions associated to the 6 rigid body transformations of each segment for both M1 and M2.

of either M1 or M2 will introduce a piston and a plate scale (focus) error. However, either center segment cancompensate the error, so we will pin M1 center segment to the origin of the z–axis by not controlling Tz for thissegment. The total number of controlled rigid body transformations are 6× 12 for the outer segments, 4 for M1center segment and 5 for M2 center segment leading to a total of 81 rigid body modes.

The first 36 M1 bending modes are depicted in Fig. 1. The bending modes correspond to the piston andtip–tilt free eigen modes of an outer segment. Using a finite element model of a M1 segment, the deformationof the segment is computed when one actuator is poked while the stress on the segment is minimized with theother actuators. The singular value decomposition applied to the set of segment deformations (one per actuator)gives the segment eigen modes. Piston, tip and tilt are projected out of these eigen modes which are then re–normalized to give the final eigen modes of an M1 segment. Zernike polynomials5 and disk harmonics4,8 circularmodes have been compared to M1 bending modes. Fig. 2 shows the fitting error for both modal basis. As theaberrations of most optical systems are low order, only the first 45 bending modes will be controlled. Fig. 2shows that Zernike polynomials provides the best fit of the first 50 bending modes, consequently we will use theZernike basis in our models.

Table 2 summarizes the system observables in terms of wavefront aberrations and the corresponding M1/M2segment modes that contributes the most to the observables. It is obvious that with a single wavefront sensor(WFS), one cannot control all M1/M2 segment modes and minimize the wavefront error at the same time. In

Figure 1. M1 bending modes.



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Bending ModesZernike fit errorDisk harmonics fit error

0 20 40 60 80 100

Bending modes120 140 160

Figure 2. M1 bending modes fitting error with Zernike polynomials and disk harmonics as a function of mode order.

Observables M1/M2 segments modetip and tilt Tx, Ty, Rx, Ry

piston, focus Tz, Zernike modeastigmatism Rz, Zernike modehigher orders Zernike modes

Table 2. GMT observable modes.

Section 4, we will show that a subset of M1/M2 segment modes is enough to minimize the wavefront error.

3. SYSTEM MODEL

The GMT and AGWS system is modeled with the Cuda–Engined Optics (CEO) simulator.6 CEO computes theoptical paths of rays through the telescope that bounces off M1 and M2 segments up to the GMT exit pupil.In the exit pupil, the optical path differences of the rays are converted into the phase of the wavefront. Then aFourier propagation is performed to either the focal or pupil plane of a sensor. CEO implements the 6 rigid bodymotion transformations of each segment of M1 and M2 and uses Zernike polynomials to shape the segments. Thecoordinate systems in which the transformation are defined are shown in Fig. 3. GMT’s Shack–Hartmann andpyramid wavefront sensors, segment piston sensor and edge-sensors are all modeled in CEO. CEO can import aZemax optical prescription and convert it into its own optical system description. CEO can also be integratedwith Simulink using custom Simulink block and the Zeromq message passing library. CEO has two applicationprogram interfaces (API): a low level API written in C++/CUDA and a high level (API) for Python writtenwith Cython. Fig. 4 describes the CEO design with the relationships between all CEO C++ structures andPython classes.

In the following, the AGWS is modeled with up to 3 probes, each pointing towards a guide star 8’ off–axis;the guide stars are evenly arranged on a ring. Each Shack–Hartman WFS is a 30 × 30 lenslet array measuringthe geometric centroid i.e. the lenslet–average of the finite difference of the phase of the wavefront.

4. CONTROL

In the natural seeing mode of the GMT, the AGWS drives M1 and M2 segments such that the segment positionand figure errors do not degrade the point–spread–function (PSF) of the telescope. The PSF degradation ismeasured with respect to the PSF with ideal segments. The comparison between PSFs is given in terms ofthe size of the 80% ensquared energy patch (EE80). Fig. 5 compares the telescope wavefront error (WFE) foran on–axis guide star and the 8’ off–axis guide star. In both cases, on and off axis, the segments are at their



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Figure 3. M1 and M1 rigid body transformations coordinate systems.

nominal location and without any aberrations. The reference slopes of the WFS are set to cancel the telescopefield aberration. The diffraction limited PSF is also shown in Fig. 5. The EE80 for the diffraction limited PSFis 28mas.

The interaction matrices for all the controlled mode of M1 and M2 (Sec. 2) have been computed. Fig. 6shows the eigen values corresponding to different calibration matrices. In the top–left are the eigen values of thecalibration matrix of Rx, Ry and Rz for all the segments (Rz has been removed for the center segment of M1and M2). The 14 first eigen values correspond to tip and tilt aberrations, followed by 6 weaker eigen values thatare associated to astigmatism, the other modes are degenerated ones. The top–right graph is a plot of the eigenvalues of the concatenated matrix of the former calibration matrix with the calibration matrix of Tx, Ty and Tzfor all the segments. One can recognize the 14 tip–tilt modes as a combination of Rx, Ry, Tx and Ty, followedby the 7 focus modes associated to the Tz motions and the 6 Rz astigmatisms. The bottom graph correspondsto the eigen values of the calibration matrix resulting of the concatenation of the former calibration matrix withthe calibration matrix of the Zernike modes. The shape of each M1 segment is controlled with 42 Zernike modesfrom focus (mode #4) to mode #45 leading to a total of 294 modes for M1. Out of the 42 Zernike modes persegment, focus and x–axis astigmatism are combined with similar modes from the rigid body motions, leadingto 40 ”controllable modes” per segment. Combined with the 27 unique rigid body modes, we obtain 307 uniqueeigen modes for the M1/M2 system as seen in Fig. 6. The command matrix of the system is derived from thetruncated singular value decomposition of the former calibration matrix. All the modes with eigen values smallerthan the eigen value #307 are discarded.

The system is initially perturbed by applying random errors to all the degrees of freedom. The random errorsare drawn from a normal distribution. The standard deviations of the errors are summarized in Table 3. They

Tx,y[µm] Tz[µm] Rx,y[arcsec] Rz[arcsec] Zernike[µm]M1 75 160 (0 on segment #7) 0.38 40 (radial order)−1

M2 75 170 3.00 330 N.A.Table 3. Standard deviations of M1 and M2 initial perturbations.

correspond to the required accuracy of the telescope metrology system. The first correction is applied with thesegment image stacker by canceling segment tip and tip with M2 Rx and Ry motions. Then the loop is closedbetween all the segment actuators and the SH–WFS until steady state is reached.



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Figure 4. CEO C++ structures (ellipses) and Python classes (rounded boxes) connections. Solid lines links C++ structurewhereas dash lines connects Python classes.



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Figure 5. Telescope diffraction limited PSF (left), on–axis telescope WFE (middle) and 8’ off-axis telescope WFE (right).WFE units are nanometers.

First, a single, 8’ off–axis, guide star is used to control the orientations and aberrations of M1 and M2segments. A steady state PSF, WFE and segment piston removed WFE is shown in Fig. 10 (i). Segment pistonsare the dominant sources of wavefront error because the WFS is not sensitive to piston. The on–axis EE80 forthat particular example is 103mas. The experiment was repeated 100 times with refreshed random draws eachtime. The histogram of the EE80 of the 100 PSFs is plotted in Fig. 7. The histogram is fitted with a normaldistribution with a 121ms mean and 31mas standard deviation. Fig. 10 (ii) shows a steady state PSF and WFEfor an on–axis guide star. The dominant error is still the piston error, but compared to the off–axis guide starcase, this is now the sole error. The mean EE80 across the 100 steady state PSF is 30mas with a 1mas standarddeviation. Fig. 10 (iii) shows a steady state PSF and WFE for 3 guide stars on a ring of 8 arcminute radius.The mean EE80 across the 100 steady state PSF is 30mas.

The single off–axis guide star case has more aberrations because the initial perturbations of the segments

(a) (b)

(c) (d)Figure 6. Calibration matrices for a single star on–axis (a, b and c) and for 3 guide stars at 8’ (d). The red dot indicatesthe first eigen mode to be truncated.



0.016

0.014

0.012

0.010TCV

j 0.008g°- 0.006

0.004

0.002

Figure 7. EE80 histogram for 1 guide star off–axis at 8 arc–minute.

induces additional field aberrations. The segments are driven to cancel any aberrations on the off–axis SH–WFS.Non–common path aberration between the on–axis and off–axis optical paths are then transfer from off–axis toon–axis, hence further degrading the on–axis PSF.

The command matrices are derived from the truncated singular value decomposition of the interaction matrixshown in Fig. 6(c) for a single guide star and Fig. 6(d) for 3 guide stars. The truncation occurs at mode #307for a single guide star and adding more eigen modes renders the system unstable. The residual WFE in the fieldis given in Fig. 8. This is the WFE after removing segment piston and telescope field aberrations. For 3 guidestars and with the same truncation than for 1 guide star, the performance on–axis is slightly degraded comparedto 1 guide star whereas the performance off–axis is similar. However with 3 guide stars, more eigen modes canbe included into the command matrix, up to mode #342, and the WFE is minimized across the whole field.

Fig. 9 shows the coefficients associated to the propagation of the noise into the system for an on–axis guidestar (left graph) and for 3 guide stars at 8’ (right graph). The coefficients q are given by

q = diag(MMT

), (1)

where M is the command matrix of the system:

M = V (S∗)−1UT . (2)

The matrices U , S and V are derived from the singular value decomposition of the calibration matrix D, e.g.

D = USV T . (3)

S∗ is the truncated version of S given by

(S∗)−1∣∣ii

= S−1∣∣ii∀i < N, (4)

= 0, otherwise. (5)

The first 80 coefficients corresponds to the rigid body motions, the others are related to the Zernike modesof M1 segments. As expected,7 the Zernike coefficients decrease with the Zernike orders. For the rigid bodymotions, the largest coefficients correspond to Rx and Ry motions following by Tx and Ty motions of M1 and M2segments. Depending on the value of N , the system transitions from an optimal and stable state to an unstablestate when one of the Rx or Ry mode is lifted up to the level of the coefficients associated to a command matrixthat is not truncated.



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No truncation

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Figure 8. WFE versus field for 1 guide star on-axis and 3 guide stars at 8 arc–minute.

5. NOISE PROPAGATION

5.1 Detector noise

Photon noise and readout noise were added to the centroids of the SH–WFSs. The variance of the centroidscorresponding to photon noise and read–out noise are respectively given by1

σ2cx,y

=

(λ

r0

)21

8 log(2)Nph, and (6)

σ2cx,y

=

(pσronNph

)2 N4px

12. (7)

λ = 640nm is the wavelength, r0 = 15cm is the Fried parameter, p = 0.38” is the detector pixel scale, σron = 0.5e−

is the detector read–out noise, Npx = 8 is the number of pixel per lenslet and Nph is the number of photon perlenslet per frame given by:

Nph = Qd2T (10.87× 109)10−0.4MR , (8)

where Q = 0.25 is the system optical throughput, d = 87cm is the lenslet size projected om M1, T is the exposuretime and MR is the guide star magnitude. The constant 10.87 × 109 is the number of photon in m−2.s−1 atmagnitude 0.

Fig. 11 (blue curve) plots the EE80 as a function of guide star magnitudes for the case of 3 guide starsat 8’ (Fig. 10 (iv)). For each magnitude, the experiment was repeated 100 times. The closed–loop integrator

Figure 9. Noise propagation coefficients for 1 guide star on–axis (left) and for 3 guide stars at 8’.



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PSF WFE Segment Piston Free

(i) 1 GS at 8’,〈EE80〉 = 121mas

(ii) 1 GS on–axis,〈EE80〉 = 30mas

(iii) 3 GSs at 8’,〈EE80〉 = 30mas

Figure 10. Steady state PSF (left) and WFE (middle) and segment piston free WFE (right) in microns (see text fordetails).

Figure 11. EE80 versus R magnitude for 3 guide stars at 8’.

was using a gain of 0.5 and 20 iterations. The on–axis PSF was integrated after the 10th iteration. The EE80was computed from the integrated PSF. In Fig. 11, the dots and the error bars correspond to the mean and totwo standard deviations, respectively, of the 100 EE80 sample for each magnitude. The thick solid black linecorresponds to the EE80 requirement imposed by the image quality requirements for the natural seeing operationmode of the GMT. The red dash line indicates the lower limit of the EEE80 for 3 GSs and without noise.



0.016

0.014

0.012

0.010

it 0.006

0.004

0.002

0.000-300 -200 -100 0 100

centroids [mas]200 300

Figure 12. Atmosphere centroid error distributions for 3 WFS exposure time: 1s, 5s and 10s.

5.2 Atmosphere

The distribution of WFS centroid errors due to the atmospheric turbulence is plotted in Fig. 12 for differentdetector exposure time. The centroid distribution is build–up by modeling the propagation of the light throughatmosphere phase screens and forming an image of a square subaperture of size d = 87cm. The images aresampled at 5ms and integrated for T=1s, 5s or 10s. The centroids of the images are accumulated over a timeperiod of 5mn. The standard deviation of the centroid errors are 83mas, 38mas and 28mas for T=1s, 5s and 10s,respectively.

The variance of the atmosphere centroid errors were added in quadrature to the variance of the SH-WFSsphoton and read–out noise errors. The resulting EE80 for exposure times of 5s and 10s has been added toFig. 11. The EE80 increases when the exposure time decreases because of both the larger atmosphere error andthe reduced number of photons.

6. CONCLUSION

We have reviewed the control of the AGWS system of the GMT in the natural seeing mode of operation of thetelescope. We have shown how the number of observables is limited and how the singular value decompositionof the observable allows to select a subset of controllable modes.

Using these modes, 3 guides stars are required to reduce the EE80 to a level compatible with AGWS per-formance requirements. Three guide stars are also necessary to maintain the same level of performance acrossthe whole field. The propagation of atmosphere and detector noise in the system have been evaluated. Theperformance requirements are met with WFS exposure time down to 5s for guide star magnitude up to 20 in Rband.

Further work will include the confirmation of these results using a full end–to–end model with a fast M2tip–tilt loop and with additional error sources like wind buffeting jitters.

REFERENCES

[1] J.M. Beckers, P. Lena, O. Lai, P.Y. Madec, G. Rousset, M. Sechaud, M.J. Northcott, F. Roddier, J.L. Beuzit,F. Rigaut, and D.G. Sandler. Adaptive Optics in Astronomy. Cambridge University Press, 1999.

[2] B. McLeod et al. The GMT active optics system: design and simulated end-to-end performance. Proceedingsof SPIE, 9906, 2016.

[3] P. McCarthy et al. Scientific Promise and Status of the Giant Magellan Telescope Project. In AdaptiveOptics for Extremely Large Telescopes III, May 2013.



[4] Norman Mark Milton and Michael Lloyd-Hart. Disk harmonic functions for adaptive optics simulations. InAdaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photon-ics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, page AWA3. Optical Society of America,2005.

[5] R.J. Noll. Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am., 66(3):207–211, March 1976.

[6] F. Quiros-Pacheco R. Conan, A. Bouchez and B. McLeod. The GMT Dynamic Optical Simulation. InAdaptive Optics for Extremely Large Telescopes, 2015.

[7] F. Rigaut and E. Gendron. Laser guide star in adaptive optics : The tilt determination problem. Astron.Astrophys., 261:677–684, 1992.

[8] Steven C. Verrall and Ramakrishna Kakarala. Disk-harmonic coefficients for invariant pattern recognition.J. Opt. Soc. Am. A, 15(2):389–401, Feb 1998.



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