Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor

Calibrating the interaction matrix forthe LINC-NIRVANA high layer

wavefront sensor

Xianyu Zhang, 1,2,3,4,∗ Carmelo Arcidiacono,5,6 Albert R. Conrad,1

Thomas M. Herbst,1 Wolfgang Gaessler,1 Thomas Bertram,1 RobertoRagazzoni,7 Laura Schreiber,8 Emiliano Diolaiti,5 Martin Kuerster,1

Peter Bizenberger,1 Daniel Meschke,1 Hans-Walter Rix,1 ChanghuiRao, 2,3 Lars Mohr,1 Florian Briegel,1 Frank Kittmann,1 Juergen

Berwein,1 and Jan Trowitzsch1

1 Max Planck Institute for Astronomy, Koenigstuhl 17, D-69117 Heidelberg,Germany2The Laboratory on Adaptive Optics, Institute of Optics and Electronics, Chinese Academy of Sciences,

Chengdu 610209, China3The Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu 610209, China

4Graduate School of Chinese Academy of Sciences, Beijing 100039, China5INAF - Osservatorio Astronomico di Bologna,via Ranzani 1, I-40127 Bologna, Italy

6INAF - Arcetri Astrophysical Observatory, Largo Enrico Fermi 5, I-50125 Firenze, Italy7INAF - Astronomical Observatory of Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy

8Alma Mater Studiorum, Universita di Bologna, Astronomy Department, Via Ranzani 1, I-40127 Bologna, Italy∗[email protected]

Abstract: LINC-NIRVANA is a near-infrared Fizeau interferometricimager that will operate at the Large Binocular Telescope. In preparationfor the commissioning of this instrument, we conducted experimentsfor calibrating the high-layer wavefront sensor of the layer-orientedmulti-conjugate adaptive optics system. For calibrating the multi-pyramidwavefront sensor, four light sources were used to simulate guide stars. Usingthis setup, we developed the push-pull method for calibrating the interactionmatrix. The benefits of this method over the traditional push-only methodare quantified, and also the effects of varying the number of push-pullframes over which aberrations are averaged is reported. Finally, we discussa method for measuring mis-conjugation between the deformable mirrorand the wavefront sensor, and the proper positioning of the wavefront sensordetector with respect to the four pupil positions.

© 2012 Optical Society of America

OCIS codes: (010.1080) Active or adaptive optics; (010.7350) Wave-front sensing;(050.1960) Diffraction theory; (120.4640) Optical instruments; (350.1260) Astronomical op-tics; (280.7060) Turbulence; (120.0280) Remote sensing and sensors.

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1. Introduction

Recently, pyramid-based wavefront sensor adaptive optics [1] established a new level of perfor-mance, in terms of both wavefront reconstruction (77% Strehl ratio with 8.5 magnitude star inthe R band [2]) and reference star magnitude limit (MR = 16.5 [2]). The Large Binocular Tele-scope (LBT) Interferometric Camera and Near-InfraRed/Visible Adaptive iNterferometer forAstronomy (LINC-NIRVANA) [3] exploits the layer oriented multi-conjugate adaptive optics(LO-MCAO) approach [4–6] in its multiple field of view (MFoV) mode. The MCAO approachwas developed to achieve larger sky coverage with respect to competing solutions, such as thestar oriented approach for MCAO. These two techniques have been successfully proven on skywith the Multi-conjugate Adaptive Demonstrator at the VLT [7,8].

As seen in Fig. 1, LINC-NIRVANA implements the MFoV technique using two differentfields of view, one for the ground layer and the other for the high layer. For the ground layer, theadaptive secondary mirror [9], with 672 actuators, applies the correction sensed by a pyramidwavefront sensor (PWS) with up to 12 guide stars in a 2-6 arc-minute annular field of view(FoV). For the high layer (adjustable, but typically conjugated at 7.1km) [10], the PWS usesup to 8 guide stars in a 2 arc-minute FoV and applies correction via a Xinetics-349 actuatordeformable mirror mounted on the LINC-NIRVANA optical bench. Finally, the differentialpiston between the two LBT arms is compensated by a delay line which is realized by a pistonmirror mechanism. The piston mirror removes this optical path difference (OPD) based onfringe measurements taken from a reference star imaged on the infrared focal plane [11]. Onthis focal plane, Fizeau interferometry (imaging a FoV of ≈ 10 arcsecond×10 arcsecond with10 milliarcsecond resolution on a Hawaii-II infrared detector) is then realized.

In a general AO system, the wavefront sensor (WFS) measures the incoming distorted wave-front and a deformable mirror (DM) applies the opposite aberrations to flatten it. The interactionmatrix describes the control relationship between the DM and the WFS and hence is very im-portant. In order to derive the IM, the AO system needs to be calibrated [19], which meansdifferent shapes must be applied to the DM and then the corresponding pattern must be meas-ured on the WFS. Of all these many aspects of an AO system, in this paper we focus on thecalibration of the IM.

There are many factors which may add noise to the calibration, thus decreasing the qualityof the interaction matrix. These include, for example, the read out noise (RON) of the CCD, thestatic aberrations in the optical system, and the turbulence within the instrument. Other errorsources in the DM-WFS coupling relation include mis-conjugation between the DM and theWFS, and projection differences between the individual guide stars on the DM and WFS. Forthe multi-pyramid wavefront sensor, another error source is the lateral displacement betweenthe pyramid and the focal plane image of the corresponding reference star (i.e., decentering).If calibrate an adaptive secondary AO system on sky, turbulence also introduces noise. AsAO systems advance, calibration becomes more time consuming because of the large numberof actuators on the DM. This becomes severe when a natural guide star has to be used as areference. So fast calibration in noisy environments is important.

Section 2 of this paper describes the laboratory setup of the LINC-NIRVANA high layerwavefront sensor. In order to quantify the quality of the interaction matrix, we present the in-teraction matrix quality parameters in section 3. The interaction matrix acquisition is given insection 4. We discuss the laboratory results for the calibration in section 5, and the balancebetween sensitivity and linearity range of calibrating the pyramid wavefront sensor in section6. Finally, in section 7, we draw lessons for achieving a better interaction matrix during cali-


Fig. 1. Schematic of LINC-NIRVANA multiple field of view and layer oriented multi-conjugate adaptive optics system.

bration.

2. Laboratory setup

For understanding and testing the LINC-NIRVANA high layer wavefront sensor, we set up alaboratory experiment (see Fig. 2 and paper [14]), located at the MPIA in Heidelberg, Germany.Figure 3 shows a schematic of the optical path for the laboratory setup. For calibration, twocomponents are critical: a fiber plate to simulate the reference stars and science targets, and amulti-layer turbulence generator, called MAPS [15] (MAPS stands for Multiple AtmosphericPhase screens and Stars), with phase screens to simulate real seeing conditions.

The MAPS simulator also generates the F/15 focal plane of LBT and its collimator opticsproject each star footprint onto the DM, thus mimicking their partial overlap at the correspond-ing atmospheric altitude (see Fig. 4). In front of the focal plane inside the PWS, an optical relay(called the FP20 optics) generates the F/20 focal plane with a 2 arcminute field of view. On thebench is installed a Xinetics-349 actuator PZT deformable mirror (mechanical stroke is 5.9 μm,interactuator stroke is 2 μm, actuator coupling is 10%). Just before the F/20 focal plane, a large,flat, beam-splitter divides the light between the wavefront sensor and a patrol camera [16]. Therole of the patrol camera is to offer a quick view of the focal plane, in order to identify and tocalibrate the positions of the reference stars. The light of up to 8 stars is projected onto separatepyramids via individual optical relays, called star enlargers [17]. Each star enlarger multipliesthe focal ratio by a factor of 11.25, from F/20 to F/225. The purpose of this enlargement is topermit a subsequent shrinking, by the same factor, of the pupil image projection of each ref-erence star on the detector, an E2V CCD39 with 80 × 80 pixels. In fact, were it not for thefast re-imaging optics in front of the detector, the four pupil images generated by the pyramids


Fig. 2. The laboratory setup of LINC-NIRVANA high layer wavefront sensor [18].

Fig. 3. The optical path of the laboratory setup of LINC-NIRVANA. The correspondingcomponents appear in Fig. 2.

would not fit the CCD39 geometry (see Fig. 5).The design of the high layer wavefront sensor (HWS) is optimized for the wavelength range

0.6−0.9μm. Other fundamental characteristics of the HWS appear in Table 1.The guide star distribution appears in Fig. 6. The Xinetics DM was conjugated to a 4 km at-

mospheric altitude. For these experiments, a detailed description of the laboratory setup appearsin a previous paper [18].

3. Quantifying the interaction matrix

Ideally, the interaction matrix is a perfect representation of the DM shape, as seen by the WFS.But in the real world, noise in the system disturbs the measurement. The interaction matrixrepeatability provides one quantitative measure of noise in the system. Another way to evaluate


Fig. 4. The optical co-added pupil image of guide stars on the wavefront sensor (CCD39).Here, four guide stars are used. For each guide star, the corresponding four quadrant pupilimages are marked by the same colored circle, while the big circle encompasses the meta-pupil, as seen in the left figure. For clarity, the right figure only shows the circles. The meta-pupil shown here appears smaller than that in Fig. 5, since the lab experiment has the WFSconjugated at 4 km and Fig. 5 shows the meta-pupil for 15 km conjugation. The quasi-collimator optical design [28] causes the meta-pupil size to be a function of conjugationaltitude. The final instrument configuration will place the high layer DM at 7.1 km, therebyproducing a larger metapupil than shown here and taking advantage of more of the CCDarea.

Fig. 5. The EEV39 detector format on the HWS. The circle shows a meta-pupil in thewavefront sensor. Each detector quadrant is filled by a meta-pupil (Fig. 4), leaving at leasta nominal 1 pixel thick margin around each. With this solution, the meta-pupil is mappedonto 38×38 CCD pixels, corresponding to 19×19 sub-apertures for binning factor 2 andapproximately 10×10 sub-apertures for binning factor 4.


Table 1. Basic requirements and constraints on the HWS design.

Item SpecificationInput focal plane F/20 telecentric, flatInput FoV 2 arcminuteWorking wavelength range 0.6−0.9μmMaximum number of reference stars 8Minimum distance between two reference stars ≤ 20 arcsecondConjugation range 4–15km (possible extension to 0 km)Star Enlarger FoV ≥ 1 arcsecond

Fig. 6. The location of the four guide stars in the fiber plate.

the quality of the interaction matrix is the sensitivity; that is, the ability of the PWS to sensedifferent modes. In order to quantify the repeatability, we define for a set of IM matrices (eachIM A of dimension I× J), the total variance value var:

var =I

∑i=1

J

∑j=1

(σ2i j) (1)

Here, I is the number of modes and J the number of slopes, twice the number of usefulsubapertures over the pupil. In this paper, we fix I to 100 (the first 100 Zernike modes [12],which we determined as sufficient for comparing the relative merits of different calibrationmethods) and J to 768 (this corresponds to 22 × 22 pixels in the circular pupil (see Fig. 4); thisgives 384 x-slopes and 384 y-slopes, or a total of 768 slopes). The parameter, σ , refers to thestandard deviation of each element in the interaction matrix A.

In order to have a measure of the sensitivity, we define the power p of the interaction matrixas:

p =I

∑i=1

J

∑j=1

(A2i j), (2)

where A, I, J have the same definitions as those given for Eq. (1).For comparison, a popular method for evaluating the condition number, as defined in Eq. (3)

below, is also introduced here. Smaller values indicate higher quality.

κ =|λmax||λmin| (3)


where λmax and λmin are, respectively, the maximum and minimum eigenvalues of the interac-tion matrix.

Since the power p is the sum of the squares of the elements, and the slopes in the IM can benegative, any additive noise will bias this metric. Therefore, for quantifying the quality of theinteraction matrix in the push pull experiment (section 5.1), we used only the var metric andnot the power. On the other hand, for the CCD positioning experiment described in section 5.2,the relatively large mis-positioning and mis-conjugation signal dominates this bias, and so weused the power p for this test.

4. Interaction matrix acquisition

For the basis, we selected Zernike modes [12]. The DM influence function is used to calculatethe corresponding voltages to command the DM to each mode. To avoid the overshoot [2], theshape sent to the DM is always kept below 2% (RMS) of the mechanical actuator stroke. Thepyramid wavefront sensor [1] is used in our setup and the slopes are calculated from the fourquadrants (I1(x,y),I2(x,y),I3(x,y),I4(x,y), x and y describe the corresponding slope position) asshown in Eqs. 4 and 5 for the x- and y-direction, respectively.

Sx(x,y) =I1(x,y)+ I3(x,y)− [I2(x,y)+ I4(x,y)]I1(x,y)+ I2(x,y)+ I3(x,y)+ I4(x,y)

(4)

Sy(x,y) =I1(x,y)+ I2(x,y)− [I3(x,y)+ I4(x,y)]I1(x,y)+ I2(x,y)+ I3(x,y)+ I4(x,y)

(5)

For the push-only method, after applying the Zernike basis shape on the DM, the correspond-ing slopes are calculated from the WFS measurements. Finally the IM is built by concatenatingthe measured slopes on the WFS as the rows of the matrix. For details of the push-pull method,see section 5.1.1.

5. Calibration results

5.1. Noise effects when calibrating the pyramid wavefront sensor

Although the Shack-Hartmann WFS and pyramid WFS have many different characteristics,they are both sensitive to the first derivative (slope) of the wavefront. The calibration errorcontribution stems from the following sources:

• the difference between the applied and the real shape of the DM, it includes: DM relatedbias, the non-linearities of DM, etc.;

• noise introduced by the WFS (e.g. the read out noise, saturation issues);

• photon noise;

• vibration introduced by the optical bench;

• atmospheric turbulence;

• static aberrations in the optical system.

Each component of the noise has a different spectrum and different amplitude, and affects theIM in a different way. Generally speaking, we may divide the error sources into two families:those that are additive and those that are multiplicative. The former adds noise signals on topof the signature of the mode to be measured and the latter affects the sensitivity. In the case of


the PWS, the two families are inter-related, since the sensitivity is directly related to the size ofthe spot on the tip of the pyramid, and additive aberrations increase that size.

In section 5.1.1 we discuss three of the six error sources listed above: turbulence, vibration,and static aberrations. Photon noise and DM stability are addressed in section 5.1.2. The seconderror source, noise introduced by the WFS, is negligible compared to the other sources and isnot discussed in this paper.

5.1.1. Dealing with turbulence, vibration, and static aberrations

In order to minimize the noise contributions from turbulence, vibration and static aberrations,we use the push-pull method [19]. With this method, for each mode, we apply two oppositeshapes to the DM. The two shapes should be applied to the DM rapidly enough to freeze theturbulence and vibration. Each shape corresponds to the same optical mode with the sameamplitude, but with opposite sign. We then get the two corresponding sets of slope vectorsfrom the WFS. Half the difference between the two vectors gives us the mean value of theslope signals for the given mode and thereby cancels out those aberrations which are due tosystematic measurement error in the slope vectors. This result is then multiplied by a constantwhich reflects the scale factor between counts sensed on the WFS and the voltages applied tothe DM. That result is then used to fill in the corresponding row in the interaction matrix.

This push-pull method, used by the first light AO team at LBT [2], is effective in cancelingout the static aberrations superimposed on the mode signature applied. The measured slopevectors should, in this way, be free of static aberrations, but residuals arise from differences inthe actual opposite-amplitude shape achieved on the DM, turbulence evolution, and also fromnon-linearity effects. Those non-linearity effects may be large, if static aberrations and turbu-lence are large. Static aberration can be minimized by flattening the wavefront using the DM.We close the loop without turbulence and after a few iterations of the loop, the DM convergesto the shape that delivers the flattest wavefront. On top of this “flat,” we then apply the modesand measure the corresponding signal.

As a demonstration, we measured interaction matrices by the push-pull and push-only meth-ods. Figure 7 shows the ratio between the push-only and push-pull methods in condition numberand var (see Eq. (1)). As seen in Fig. 7, the condition number of the interaction matrix is ap-preciably decreased by the push-pull method in all the measurements. The var value was alsoappreciably decreased. In short, the push-pull method can improve the quality of the interactionmatrix dramatically.

5.1.2. Dealing with photon noise and the DM stability

A simple way to increase the signal to noise ratio of the slope measurement is to provide morelight. Once we reach the illumination limit fixed by saturation of the CCD39 (as seen in Fig.4, the meta-pupil is not equally illuminated, and some pixels can be saturated), we can onlysum more frames to increase the signal to noise ratio and hence decrease the effects of photonnoise. We attempt this by averaging the frames taken by the WFS for each DM shape (mode).Note, however, that the repeatability of the DM is another source of noise for the interactionmatrix. To assess the value of this noise reduction strategy, we repeated the push-pull sequence1, 2, 4 and 8 times, while each time averaging the slope vectors obtained for each push-pullcouple. In the end, we obtain four interaction matrices corresponding to the average of the 1,2, 4 and 8 push-pull sequences, respectively.

Of course, to disentangle the possible effect of the DM, the slope vectors need to be meas-ured in the lowest possible noise conditions, with special attention given to reducing photonnoise. For this reason, each set of push-pull sequences was taken with 1, 2, 4 and 8 frames


Fig. 7. Improvement in var and condition number with the push-pull method. Each trialwas initiated with a different starting wavefront condition. For each initial condition, theinteraction matrix was calibrated 8 times: 4 with push-pull and 4 with push-only. We thencomputed the var and condition number for these two sets of 4 interaction matrices (thecondition number was computed from the average of the 4). The ratios for both are plottedhere. For example, for trial number 2, the condition number of the average of the interac-tion matrices calibrated with push-pull is approximately 8 times smaller than the matricescalibrated with push-only, for the same initial conditions. And for this case, the var is ap-proximately 50 times smaller.

Fig. 8. The var of the interaction matrix changes with different number of frames per push-pull and total push-pulls taken during calibration.


Table 2. Same data as shown in Fig. 8.

push-pull repeats 1 2 4 81 frame 204.6 107.5 57.9 32.52 frames 108.7 57.0 34.0 18.94 frames 59.3 35.1 21.4 14.18 frames 40.1 26.8 16.9 14.1

Fig. 9. The var decreases with increasing number of images. (The number of images equalsthe number of push-pulls times the number of frames per push-pull.) The data is the sameas shown in Fig. 8.

for each push and pull of the mode commanded to the DM. Finally, to test the repeatability ofthe measurements, we repeated the full set of 4× 4 interaction matrices 4 times. We show theresulting var values in Fig. 8 and Table 2.

Figure 9 plots the data from Fig. 8, using for the abscissa the total actual number of framestaken for each mode (the number of frames averaged times the number of push-pulls). Figure9 shows that the var of the interaction matrix decreases with the total number of images takenwith the WFS. Thus, we have demonstrated that averaging frames improves the quality of theinteraction matrix, as expected, by increasing the signal and thereby minimizing the effects ofphoton noise and DM stability. In particular, we found the number of frames needed in oursetup to minimize these effects.

Figure 8 and Table 2 show that the quality of the interaction matrix increases with the numberof push-pulls. They also show the quality of the interaction matrix increasing with the numberof frames per push-pull. Figure 8 and Table 2 also demonstrate that averaging the frames foreach mode addresses the noise introduced by the WFS and/or DM. But what is most interestingis the effect we see when we disentangle DM noise from the other sources of error. This can beseen by considering the results in Fig. 8 and Table 2, and fixing to 8 the total number of framestaken (for example, 8 frames taken on each side of one push-pull, or 2 frames taken on eachside of 4 push-pulls, etc). This is the diagonal on the 3D plot in Fig. 8. In this case, we see thatthe var increases in the direction of the number of frames per push-pull taken from the WFSand decreases in the direction of the number of push-pulls realized. In other words, the leftmostbar in Fig. 8 is higher than the rightmost (by about 23%). Figure 8 and Table 2 also demonstratethat the DM errors can be averaged out by a number of different DM realizations [19].

Actually, given a limited time (e.g., in order to freeze quasi-static effects like vibration) to


Fig. 10. The var decreases with the number of push-pulls ( For each push-pull shape, onlyone frame was taken from the WFS).

take IM measurements, as seen in Fig. 8 and Table 2, it is better to take one frame for each push-pull. So for the LINC-NIRVANA high layer wavefront sensor, given the limited time availablefor calibration, we should take one frame from the WFS for each push and pull of the modeand increase the number of push-pull sequences. To this end, we repeated our test using thislatter scenario: we took IM measurements for different push-pull sequences (1, 2, 4, 8, 16, 32and 64 average push-pulls) taking only one frame for each push or pull. Again, we performedthe measurement four times to get var values. We plot the results in Fig. 10, which shows thatby taking less then 32 push-pull frames, we can achieve a stable interaction matrix (which alsomatches the results in Fig. 9).

5.2. Calibrating layer oriented multi conjugated AO system

5.2.1. Conjugation between the WFS and the DM

The mis-conjugation between the WFS and DM altitude will smear out the DM shape on theWFS. In the Layer-Oriented sensor, the conjugation to an atmospheric layer is realized byadjusting the focus until the star footprints’ have the correct overlap geometry. Actually, thiscondition is realized also on the DM: the reference footprints overlap according to that geome-try.

Another error in mis-conjugation comes about when signals from stars are not properly su-perimposed onto the corresponding footprint on the DM. From the DM calibration point ofview, this means that the signals introduced by pushing an actuator on the DM and measuringit in direction of different guide stars, will exactly overlap on the WFS only if the proper con-jugation between the WFS and the DM has been realized. On the other hand, if the conjugationis not correct, the same signal is seen as a blurred measurement due to the imperfect superposi-tion. This situation corresponds to a decrease of the slope modulus and therefore to a decreasein the power merit figure.

For the HWS, the CCD is mounted on three linear stages which allow movement alongthree orthogonal axes. Moving along the optical axis, the CCD can be conjugated to differentatmospheric altitudes. Scanning along a few hundreds of microns range corresponds to a fewkilometers in the atmosphere. By measuring an interaction matrix at each linear stage position,we obtain a direct measure of the overlap of the pupil footprints on the CCD. The result of onesuch scan is shown in Fig. 11. Clearly, the power factor p (see Eq. (2)) of the interaction matrixdecreases with the mis-conjugation between the DM and WFS.


Fig. 11. The power, p, of the interaction matrix as a function of mis-conjugation betweenthe DM and the WFS. The mis-conjugation unit used here is the corresponding km distancein the atmosphere once the instrument will be installed on the LBT telescope. The zeroposition refers to the starting point of the calibration. The curve is a simple Gaussian fitand the dotted line indicates the maximum value.

Fig. 12. The power of the interaction matrix with CCD stage x direction. The optimal Xposition is 0.03 pixel. The zero position refers to the starting point of the calibration. Thecurve is a simple Gaussian fit and the dotted line indicates the maximum value.

5.2.2. Mapping the CCD pixels

The slope computation is very sensitive to proper definition of the meta-pupil positions on theCCD, which is often called WFS-DM misregistration [25]. Before calibration, the best meta-pupil positions should be found. Since the geometry of the four meta-pupils (their centers anddiameters) is fixed, the position on the CCD can be optimized in order to center the meta-pupilto the equipped accuracy. Using a method similar to that which we used to establish the properconjugation altitude, we moved the CCD linear stages in the x and y directions, measuring thepower changes of the IM at each step of the scan along the two axes. The plots in Figs. 12 and13 show that the sensor is sensitive to a shift corresponding to ≈ 1/10 of pixel.


Fig. 13. The power of the interaction matrix with CCD stage y direction. The optimal Yposition is 0.29 pixel. The zero position refers to the starting point of the calibration. Thecurve is a simple Gaussian fit and the dotted line indicates the maximum value.

6. Sensitivity and linearity range of the pyramid wavefront sensor

Theory [1,24], numerical simulations [21], and experiments [22] all converge on demonstratingthat introducing mechanical modulation will increase the PWS’s linear range. But the benefitsof modulation come at the cost of increased opto-mechanical complexity and decreased sensi-tivity (as defined in Eq. (2)), due to the lower energy concentration that comes with modulation.In our case, the pyramids are not modulated, and the variation in the sensitivity comes from theeffective wavefront aberration that the reference light experiences. The power figure, p, definedin Eq. (2) above, is strictly related to this behavior. What we expect is an increase in the sensi-tivity with the fine centering of the pyramids with respect to the reference positions. In fact, amis-positioning of a pyramid on the focal plane introduces a fixed tip-tilt signal (for the multi-ple pyramids above the ground layer case, it is more complicated.), which erodes the linearityrange of the sensor and decreases the sensitivity.

For the LINC-NIRVANA adaptive optics system, the pyramid wavefront sensor is neithermodulated nor vibrated. Therefore, under typical calibration conditions, characterized by verysmall aberrations in the optical path (typically smaller than 100nm to give an example), thePWS is very sensitive. However at the same time, it only has a limited linearity range.

We have demonstrated that the push-pull technique actually improves the calibration, in-creasing both the sensitivity and the signal to noise ratio. In order to take full advantage ofthis technique, the pyramids should operate near the middle of their linearity range, to be asclose as possible to zero static aberrations [26]. The trick is to not use diffraction limited refer-ence sources; the core of the fiber corresponds to 0.1 arcsecond. In this way, the linear range isproportionally enlarged.

In other words, a way to change the sensitivity (or linearity range) is to use a reference fiberof different core size. Once the core size becomes larger than λ/D (D is the diameter of thetelescope), the sensitivity decreases [23]. An extended object presents itself as a collection ofpoint sources, simultaneously imaged by the PWS. This is similar to what happens when a flatmirror modulates the pyramid by rotating the focal image of the reference around the vertexduring a single CCD integration. It also corresponds to the real world, in which the stellarimages are not diffraction limited. In the AO loop, the sensitivity of the pyramid wavefrontsensor can be adjusted by altering modal sensitivity compensation coefficients [27].


7. AO performance

In this paper we have measured the quality of our interaction matrix calibration using statisticalmeasures only. Although we have started an analysis to determine what performance improve-ment might be achieved, this effort is in progress. Our preliminary analysis suggests that acarefully calibrated interaction matrix (using the push-pull method described here) could im-prove closed loop AO performance (in terms of Strehl ratio) by as much as 20%. This work isongoing and will appear in a future publication together with measured performance.

8. Conclusions

We have presented the first quantitative analysis of AO calibration. Based on our tests withthe LINC-NIRVANA high layer wavefront sensor, we quantified the benefits of the push-pullmethod over the traditional push-only method. We then discussed the inverse relationship be-tween sensitivity and the linear range of the pyramid wavefront sensor. Finally, we presentedour methods for conjugating the DM to the WFS and for mapping the wavefront sensor pixelsto the meta-pupil. The results are as follows:

1. The push-pull method is superior to the traditional push-only method, as shown in Fig.7. It is also a useful method for on-sky calibration.

2. More push-pulls are better than fewer, as shown in Figs. 8, 9, 10 and Table 2.

3. If you have limited opportunity for total frames (e.g., in order to freeze quasi-static effectslike vibration), then it is better to do more push-pulls and less frames per push-pull, asshown in Figs. 8 and Table 2.

4. In theory, the balance between the sensitivity and linearity range in pyramid wavefrontsensors can be achieved in the laboratory by selecting a suitable fiber size.

5. In a LO-MCAO system, good conjugation between the DM and WFS CCD, and also ac-curate CCD pixel mapping, can be achieved by adjusting the CCD position to maximizethe power of the interaction matrix, as shown in Figs. 11, 12 and 13.

Acknowledgments

Xianyu Zhang wishes to thank all the members of the LINC-NIRVANA team, as well as D.Peter for helpful discussions on the AO loop. Guido Agapito and Fernando Quiros-Pachecoprovided helpful discussion on forming the injection matrix. Xianyu Zhang has been supportedby a PhD scholarship provided by the Max-Planck Society and the Chinese Academy of Sci-ences.


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Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor

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