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LINC-NIRVANA is a near-infrared Fizeau interferometric imager that will operate at the Large Binocular Telescope. In preparation for the commissioning of this instrument, we conducted experiments for calibrating the high-layer wavefront sensor of the layer-oriented multi-conjugate adaptive optics system. For calibrating the multi-pyramid wavefront sensor, four light sources were used to simulate guide stars. Using this setup, we developed the push-pull method for calibrating the interaction matrix. The benefits of this method over the traditional push-only method are quantified, and also the effects of varying the number of push-pull frames over which aberrations are averaged is reported. Finally, we discuss a method for measuring mis-conjugation between the deformable mirror and the wavefront sensor, and the proper positioning of the wavefront sensor detector with respect to the four pupil positions.
© 2012 Optical Society of America
1. Introduction
Recently, pyramid-based wavefront sensor adaptive optics [1] established a new level of performance, in terms of both wavefront reconstruction (77% Strehl ratio with 8.5 magnitude star in the R band [2]) and reference star magnitude limit (MR = 16.5 [2]). The Large Binocular Telescope (LBT) Interferometric Camera and Near-InfraRed/Visible Adaptive iNterferometer for Astronomy (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 approach was developed to achieve larger sky coverage with respect to competing solutions, such as the star oriented approach for MCAO. These two techniques have been successfully proven on sky with the Multi-conjugate Adaptive Demonstrator at the VLT [7, 8].
As seen in Fig. 1, LINC-NIRVANA implements the MFoV technique using two different fields of view, one for the ground layer and the other for the high layer. For the ground layer, the adaptive secondary mirror [9], with 672 actuators, applies the correction sensed by a pyramid wavefront 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 uses up to 8 guide stars in a 2 arc-minute FoV and applies correction via a Xinetics-349 actuator deformable mirror mounted on the LINC-NIRVANA optical bench. Finally, the differential piston between the two LBT arms is compensated by a delay line which is realized by a piston mirror mechanism. The piston mirror removes this optical path difference (OPD) based on fringe measurements taken from a reference star imaged on the infrared focal plane [11]. On this focal plane, Fizeau interferometry (imaging a FoV of ≈ 10 arcsecond×10 arcsecond with 10 milliarcsecond resolution on a Hawaii-II infrared detector) is then realized.
Fig. 1 Schematic of LINC-NIRVANA multiple field of view and layer oriented multi-conjugate adaptive optics system.
In a general AO system, the wavefront sensor (WFS) measures the incoming distorted wavefront and a deformable mirror (DM) applies the opposite aberrations to flatten it. The interaction matrix describes the control relationship between the DM and the WFS and hence is very important. In order to derive the IM, the AO system needs to be calibrated [19], which means different shapes must be applied to the DM and then the corresponding pattern must be measured on the WFS. Of all these many aspects of an AO system, in this paper we focus on the calibration of the IM.
There are many factors which may add noise to the calibration, thus decreasing the quality of the interaction matrix. These include, for example, the read out noise (RON) of the CCD, the static aberrations in the optical system, and the turbulence within the instrument. Other error sources in the DM-WFS coupling relation include mis-conjugation between the DM and the WFS, and projection differences between the individual guide stars on the DM and WFS. For the multi-pyramid wavefront sensor, another error source is the lateral displacement between the 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. As AO systems advance, calibration becomes more time consuming because of the large number of actuators on the DM. This becomes severe when a natural guide star has to be used as a reference. So fast calibration in noisy environments is important.
Section 2 of this paper describes the laboratory setup of the LINC-NIRVANA high layer wavefront sensor. In order to quantify the quality of the interaction matrix, we present the interaction matrix quality parameters in section 3. The interaction matrix acquisition is given in section 4. We discuss the laboratory results for the calibration in section 5, and the balance between sensitivity and linearity range of calibrating the pyramid wavefront sensor in section 6. Finally, in section 7, we draw lessons for achieving a better interaction matrix during calibration.
2. Laboratory setup
For understanding and testing the LINC-NIRVANA high layer wavefront sensor, we set up a laboratory 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, two components are critical: a fiber plate to simulate the reference stars and science targets, and a multi-layer turbulence generator, called MAPS [15] (MAPS stands for Multiple Atmospheric Phase screens and Stars), with phase screens to simulate real seeing conditions.
Fig. 3 The optical path of the laboratory setup of LINC-NIRVANA. The corresponding components appear in Fig. 2.
The MAPS simulator also generates the F/15 focal plane of LBT and its collimator optics project each star footprint onto the DM, thus mimicking their partial overlap at the corresponding 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 the bench 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]. The role of the patrol camera is to offer a quick view of the focal plane, in order to identify and to calibrate the positions of the reference stars. The light of up to 8 stars is projected onto separate pyramids via individual optical relays, called star enlargers [17]. Each star enlarger multiplies the focal ratio by a factor of 11.25, from F/20 to F/225. The purpose of this enlargement is to permit a subsequent shrinking, by the same factor, of the pupil image projection of each reference star on the detector, an E2V CCD39 with 80 × 80 pixels. In fact, were it not for the fast re-imaging optics in front of the detector, the four pupil images generated by the pyramids would not fit the CCD39 geometry (see Fig. 5).
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 pupil images 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 WFS conjugated 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 conjugation altitude. The final instrument configuration will place the high layer DM at 7.1 km, thereby producing a larger metapupil than shown here and taking advantage of more of the CCD area.
Fig. 5 The EEV39 detector format on the HWS. The circle shows a meta-pupil in the wavefront sensor. Each detector quadrant is filled by a meta-pupil (Fig. 4), leaving at least a nominal 1 pixel thick margin around each. With this solution, the meta-pupil is mapped onto 38 × 38 CCD pixels, corresponding to 19 × 19 sub-apertures for binning factor 2 and approximately 10 × 10 sub-apertures for binning factor 4.
The design of the high layer wavefront sensor (HWS) is optimized for the wavelength range 0.6 – 0.9 μm. Other fundamental characteristics of the HWS appear in Table 1.
Table 1. Basic requirements and constraints on the HWS design.
The guide star distribution appears in Fig. 6. The Xinetics DM was conjugated to a 4 km atmospheric altitude. For these experiments, a detailed description of the laboratory setup appears in 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 matrix repeatability provides one quantitative measure of noise in the system. Another way to evaluate the quality of the interaction matrix is the sensitivity; that is, the ability of the PWS to sense different modes. In order to quantify the repeatability, we define for a set of IM matrices (each IM A of dimension I × J), the total variance value var:
Here, I is the number of modes and J the number of slopes, twice the number of useful subapertures 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 calibration methods) and J to 768 (this corresponds to 22 × 22 pixels in the circular pupil (see Fig. 4); this gives 384 x-slopes and 384 y-slopes, or a total of 768 slopes). The parameter, σ, refers to the standard 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 matrix as:
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.
where λmax and λmin are, respectively, the maximum and minimum eigenvalues of the interaction matrix.Since the power p is the sum of the squares of the elements, and the slopes in the IM can be negative, any additive noise will bias this metric. Therefore, for quantifying the quality of the interaction matrix in the push pull experiment (section 5.1), we used only the var metric and not 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 we used 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 calculate the corresponding voltages to command the DM to each mode. To avoid the overshoot [2], the shape sent to the DM is always kept below 2% (RMS) of the mechanical actuator stroke. The pyramid wavefront sensor [1] is used in our setup and the slopes are calculated from the four quadrants (I1(x, y),I2(x, y),I3(x, y),I4(x, y), x and y describe the corresponding slope position) as shown in Eqs. 4 and 5 for the x- and y-direction, respectively.
For the push-only method, after applying the Zernike basis shape on the DM, the corresponding slopes are calculated from the WFS measurements. Finally the IM is built by concatenating the 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 error contribution stems from the following sources:
- the difference between the applied and the real shape of the DM, it includes: DM related bias, 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 the IM 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 top of 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 of the 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 second error source, noise introduced by the WFS, is negligible compared to the other sources and is not 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 opposite shapes to the DM. The two shapes should be applied to the DM rapidly enough to freeze the turbulence and vibration. Each shape corresponds to the same optical mode with the same amplitude, but with opposite sign. We then get the two corresponding sets of slope vectors from the WFS. Half the difference between the two vectors gives us the mean value of the slope signals for the given mode and thereby cancels out those aberrations which are due to systematic measurement error in the slope vectors. This result is then multiplied by a constant which reflects the scale factor between counts sensed on the WFS and the voltages applied to the 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 canceling out the static aberrations superimposed on the mode signature applied. The measured slope vectors should, in this way, be free of static aberrations, but residuals arise from differences in the actual opposite-amplitude shape achieved on the DM, turbulence evolution, and also from non-linearity effects. Those non-linearity effects may be large, if static aberrations and turbulence 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 converges to the shape that delivers the flattest wavefront. On top of this “flat,” we then apply the modes and measure the corresponding signal.
As a demonstration, we measured interaction matrices by the push-pull and push-only methods. Figure 7 shows the ratio between the push-only and push-pull methods in condition number and var (see Eq. (1)). As seen in Fig. 7, the condition number of the interaction matrix is appreciably decreased by the push-pull method in all the measurements. The var value was also appreciably decreased. In short, the push-pull method can improve the quality of the interaction matrix dramatically.
Fig. 7 Improvement in var and condition number with the push-pull method. Each trial was initiated with a different starting wavefront condition. For each initial condition, the interaction matrix was calibrated 8 times: 4 with push-pull and 4 with push-only. We then computed the var and condition number for these two sets of 4 interaction matrices (the condition number was computed from the average of the 4). The ratios for both are plotted here. For example, for trial number 2, the condition number of the average of the interaction matrices calibrated with push-pull is approximately 8 times smaller than the matrices calibrated with push-only, for the same initial conditions. And for this case, the var is approximately 50 times smaller.
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 more light. 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 only sum more frames to increase the signal to noise ratio and hence decrease the effects of photon noise. 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 interaction matrix. To assess the value of this noise reduction strategy, we repeated the push-pull sequence 1, 2, 4 and 8 times, while each time averaging the slope vectors obtained for each push-pull couple. 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 measured in the lowest possible noise conditions, with special attention given to reducing photon noise. For this reason, each set of push-pull sequences was taken with 1, 2, 4 and 8 frames for each push and pull of the mode commanded to the DM. Finally, to test the repeatability of the measurements, we repeated the full set of 4 × 4 interaction matrices 4 times. We show the resulting var values in Fig. 8 and Table 2.
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.
Figure 9 plots the data from Fig. 8, using for the abscissa the total actual number of frames taken for each mode (the number of frames averaged times the number of push-pulls). Figure 9 shows that the var of the interaction matrix decreases with the total number of images taken with the WFS. Thus, we have demonstrated that averaging frames improves the quality of the interaction matrix, as expected, by increasing the signal and thereby minimizing the effects of photon noise and DM stability. In particular, we found the number of frames needed in our setup to minimize these effects.
Fig. 9 The var decreases with increasing number of images. (The number of images equals the number of push-pulls times the number of frames per push-pull.) The data is the same as shown in Fig. 8.
Figure 8 and Table 2 show that the quality of the interaction matrix increases with the number of push-pulls. They also show the quality of the interaction matrix increasing with the number of frames per push-pull. Figure 8 and Table 2 also demonstrate that averaging the frames for each mode addresses the noise introduced by the WFS and/or DM. But what is most interesting is the effect we see when we disentangle DM noise from the other sources of error. This can be seen by considering the results in Fig. 8 and Table 2, and fixing to 8 the total number of frames taken (for example, 8 frames taken on each side of one push-pull, or 2 frames taken on each side of 4 push-pulls, etc). This is the diagonal on the 3D plot in Fig. 8. In this case, we see that the var increases in the direction of the number of frames per push-pull taken from the WFS and decreases in the direction of the number of push-pulls realized. In other words, the leftmost bar in Fig. 8 is higher than the rightmost (by about 23%). Figure 8 and Table 2 also demonstrate that 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 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 available for calibration, we should take one frame from the WFS for each push and pull of the mode and increase the number of push-pull sequences. To this end, we repeated our test using this latter scenario: we took IM measurements for different push-pull sequences (1, 2, 4, 8, 16, 32 and 64 average push-pulls) taking only one frame for each push or pull. Again, we performed the measurement four times to get var values. We plot the results in Fig. 10, which shows that by taking less then 32 push-pull frames, we can achieve a stable interaction matrix (which also matches the results in Fig. 9).
Fig. 10 The var decreases with the number of push-pulls (For each push-pull shape, only one frame was taken from the WFS).
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 the WFS. In the Layer-Oriented sensor, the conjugation to an atmospheric layer is realized by adjusting the focus until the star footprints’ have the correct overlap geometry. Actually, this condition is realized also on the DM: the reference footprints overlap according to that geometry.
Another error in mis-conjugation comes about when signals from stars are not properly superimposed onto the corresponding footprint on the DM. From the DM calibration point of view, this means that the signals introduced by pushing an actuator on the DM and measuring it in direction of different guide stars, will exactly overlap on the WFS only if the proper conjugation between the WFS and the DM has been realized. On the other hand, if the conjugation is not correct, the same signal is seen as a blurred measurement due to the imperfect superposition. This situation corresponds to a decrease of the slope modulus and therefore to a decrease in the power merit figure.
For the HWS, the CCD is mounted on three linear stages which allow movement along three orthogonal axes. Moving along the optical axis, the CCD can be conjugated to different atmospheric altitudes. Scanning along a few hundreds of microns range corresponds to a few kilometers 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 one such scan is shown in Fig. 11. Clearly, the power factor p (see Eq. (2)) of the interaction matrix decreases with the mis-conjugation between the DM and WFS.
Fig. 11 The power, p, of the interaction matrix as a function of mis-conjugation between the DM and the WFS. The mis-conjugation unit used here is the corresponding km distance in the atmosphere once the instrument will be installed on the LBT telescope. The zero position refers to the starting point of the calibration. The curve 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 the CCD, 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 and diameters) is fixed, the position on the CCD can be optimized in order to center the meta-pupil to the equipped accuracy. Using a method similar to that which we used to establish the proper conjugation altitude, we moved the CCD linear stages in the x and y directions, measuring the power changes of the IM at each step of the scan along the two axes. The plots in Figs. 12 and 13 show that the sensor is sensitive to a shift corresponding to ≈ 1/10 of pixel.
Fig. 12 The power of the interaction matrix with CCD stage x direction. The optimal X position is 0.03 pixel. The zero position refers to the starting point of the calibration. The curve is a simple Gaussian fit and the dotted line indicates the maximum value.
Fig. 13 The power of the interaction matrix with CCD stage y direction. The optimal Y position is 0.29 pixel. The zero position refers to the starting point of the calibration. The curve 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 demonstrating that introducing mechanical modulation will increase the PWS’s linear range. But the benefits of modulation come at the cost of increased opto-mechanical complexity and decreased sensitivity (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 the effective wavefront aberration that the reference light experiences. The power figure, p, defined in Eq. (2) above, is strictly related to this behavior. What we expect is an increase in the sensitivity with the fine centering of the pyramids with respect to the reference positions. In fact, a mis-positioning of a pyramid on the focal plane introduces a fixed tip-tilt signal (for the multiple pyramids above the ground layer case, it is more complicated.), which erodes the linearity range of the sensor and decreases the sensitivity.
For the LINC-NIRVANA adaptive optics system, the pyramid wavefront sensor is neither modulated nor vibrated. Therefore, under typical calibration conditions, characterized by very small aberrations in the optical path (typically smaller than 100nm to give an example), the PWS 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, increasing both the sensitivity and the signal to noise ratio. In order to take full advantage of this technique, the pyramids should operate near the middle of their linearity range, to be as close as possible to zero static aberrations [26]. The trick is to not use diffraction limited reference sources; the core of the fiber corresponds to 0.1 arcsecond. In this way, the linear range is proportionally enlarged.
In other words, a way to change the sensitivity (or linearity range) is to use a reference fiber of different core size. Once the core size becomes larger than λ/D (D is the diameter of the telescope), the sensitivity decreases [23]. An extended object presents itself as a collection of point sources, simultaneously imaged by the PWS. This is similar to what happens when a flat mirror modulates the pyramid by rotating the focal image of the reference around the vertex during a single CCD integration. It also corresponds to the real world, in which the stellar images are not diffraction limited. In the AO loop, the sensitivity of the pyramid wavefront sensor 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 statistical measures only. Although we have started an analysis to determine what performance improvement might be achieved, this effort is in progress. Our preliminary analysis suggests that a carefully calibrated interaction matrix (using the push-pull method described here) could improve closed loop AO performance (in terms of Strehl ratio) by as much as 20%. This work is ongoing 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 with the LINC-NIRVANA high layer wavefront sensor, we quantified the benefits of the push-pull method over the traditional push-only method. We then discussed the inverse relationship between sensitivity and the linear range of the pyramid wavefront sensor. Finally, we presented our methods for conjugating the DM to the WFS and for mapping the wavefront sensor pixels to the meta-pupil. The results are as follows:
- 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.
- In theory, the balance between the sensitivity and linearity range in pyramid wavefront sensors can be achieved in the laboratory by selecting a suitable fiber size.
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-Pacheco provided helpful discussion on forming the injection matrix. Xianyu Zhang has been supported by a PhD scholarship provided by the Max-Planck Society and the Chinese Academy of Sciences.
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