PYRAMID WAVEFRONT SENSOR PERFORMANCE WITH LASER GUIDE STARS

Florence, Italy. Adaptive May 2013 Optics for Extremely Large Telescopes III ISBN: 978-88-908876-0-4 DOI: 10.12839/AO4ELT3.13138 PYRAMID WAVEFRONT SENSOR PERFORMANCE WITH LASER GUIDE STARS Fernando Quirós-Pacheco 1, a, Enrico Pinna 1, Alfio Puglisi 1, Lorenzo Busoni 1, Guido Agapito 1, Sebastian Rabien 2, Simone Esposito 1 1 INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy. 2 Max Planck Institute für extraterrestrische Physik, Gießenbachstraβe, 85748 Garching, Germany. Abstract. All the ELT projects foresee and deeply rely on Laser Guide Star Adaptive Optics (LGS AO) systems. These systems, already successfully employed on the 8-10m class of telescopes, show several limitations in performance linked to the three-dimensional extension of the artificial guide star, which causes a reduction in wavefront sensor sensitivity and generates errors in the wavefront measurements. All of these effects will be accentuated passing from the 8-10m class of telescope to the ELTs. Currently, only the Shack-Hartmann (SH) wavefront sensor has been employed in the existing LGS AO systems, and all the LGS AO systems under design for the ELTs foresee the same kind of sensor. Recently, the use of the pyramid sensor in LGS AO systems has been proposed too, suggesting multiple advantages over the SH-based solution, but the investigations available in the literature are just preliminary. In this paper, we first review the advantages offered by the pyramid wavefront sensor coupled to an LGS, and then we present the results of numerical simulations aimed at quantifying the impact of a three-dimensionally extended source on this sensor. The work presented is a required step before laboratory and on-sky experimentation. If the experimental work confirms the expected results, the pyramid wavefront sensor will become a very attractive alternative to the SH one, simplifying significantly the design of LGS AO systems for the ELTs, and improving their performances and reliability. 1. Introduction Extremely Large Telescopes (ELTs) currently in the design phase foresee the use of Laser Guide Star Adaptive Optics (LGS AO) systems based on Shack-Hartmann Sensors (SHS) [1-3]. The problem of spot elongation, related to the three-dimensional extension of the artificial guide star, becomes critical at these aperture sizes [4]. Extended reference sources reduce wave-front sensing sensitivity and, as a consequence, demand more laser power to reach a given Signal-to-Noise Ratio. In order to evaluate the use of Pyramid Sensor (PS) in LGS AO systems for ELTs and its comparison with SHS, a thorough understanding of its behavior with extended sources is still required. Previous works addressing this topic [5-7] have already suggested that PS and SH have equivalent sensitivities when coupled with an LGS. In this work, following a numerical approach, we have characterized the a e-mail address: fquiros@arcetri.astro.it

PS sensitivity with 2D and 3D extended sources in a modal way, providing Noise Propagation Coefficients (NPCs) for another point of comparison. 2. Pyramid sensitivity with 2D extended reference sources The loss in PS sensitivity with a 2D extended source can be easily compared with the well-understood case of a modulated point source. This analysis will be presented in Sec. 2.1 based on simple geometrical considerations. Then, after a brief description in Sec. 2.2 of our implementation of extended objects simulation with the PS, we will present in Sec. 2.3 the NPCs as a function of the reference object size. We will set up our simulation conditions to follow up our previous work on this topic presented in the last AO4ELT conference [6]. In that occasion, we presented the results of laboratory closed-loop experiments with the FLAO system using illumination fibers of different core sizes to emulate extended reference sources. The loss in PS sensitivity as a function of the object size was evaluated in terms of closed-loop modal residuals (using a reconstruction matrix calibrated with an unresolved fiber to close the loop). It was not possible time-wise before shipping the FLAO system, to acquire interaction matrices with different fibers from which NPCs could be directly computed. We have used our numerical model of the FLAO system to do this, and to verify the results obtained in the lab. 2.1. Sensitivity analysis: comparison with tilt modulation Pyramid sensors designed to operate with point reference sources require a sort of modulation (i.e. a mechanism to distribute the incoming light over the four facets of the pyramid) in order to convey quantitative information on the wave-front derivatives. For instance, the PS of the LBT s FLAO system makes use of a Tip-Tilt (TT) mirror conjugated to the pupil plane to provide dynamic (circular) modulation [8]. For this kind of modulation, based on geometrical optics approximations, it can be shown that the pyramid signal on the y axis direction is [9]: S (1) sin (2) where the intensities are computed as and (see Figure 1), is the amount of transverse ray aberration, and R TT is the modulation radius. The PS sensitivity defined as the derivative of the signal to the aberration is given by [10]: 1. (3) The modulation radius (R TT ) determines the sensitivity and the dynamic range of the PS. In the case of extended objects, no TT modulation is in principle required since the image in the focal plane already spans a fraction of the four pyramid s facets. Following the same development outlined in [10], it can be shown that the PS signal obtained with a uniformly-illuminated top-hat extended object (like the illumination sources used in our previous lab experiments) of radius R EO is given by: S sin 1 (4) and the corresponding PS sensitivity is:

1. (5) The PS sensitivity in this case is a function of the extended object radius R EO. By equalizing Eqs. (3) and (5) evaluated at 0, we can obtain an equivalent TT modulation providing the same sensitivity (in the neighborhood of the origin) as the one obtained with the extended object, resulting in an equivalent TT modulation radius of:. (6) The plot on the right of Figure 1 compares the PS signals as a function of the aberration amplitude (Eqs. (2) and (4)) when the condition in Eq. (6) is satisfied. Note that the dynamic range with the extended object doubles the one obtained with the equivalent TT modulation. In conclusion, the PS coupled to a top-hat reference source of radius R EO features the same sensitivity as the one obtained with a circularly modulated point source with R TT =R EO /2, and a doubled dynamic range (i.e. ±2R TT ). Figure 1. (Left) Coordinates of the pyramid sensor showing the four quadrants (A, B, C, D) and the transverse ray aberration (y 0 ). (Right) PS signal (S y ) as a function of the aberration (y 0 ) for two kinds of reference sources: (black continuous line) extended object with uniform illumination and a unitary radius (R EO =1); (red dashed line) point source circularly modulated with a modulation radius equal to R TT = R EO /2. 2.2. Simulation of pyramid sensor with extended reference sources In order to validate the results of the sensitivity analysis presented above, and to investigate the PS sensitivity to measurement noise, we have implemented a simulation of the PS with extended objects. As already stated in [5], this kind of simulation is computationally intensive because it requires the discretization of the extended object intensity distribution in n point sources, computing for each of these a full optical propagation down to the sensor s CCD plane. The final CCD frame (containing the four pupil images characteristic of the pyramid sensor) is simply found by summing the n CCD images. In order to reduce the computing time, we have migrated the pyramid module of PASSATA (PyrAmid Simulator Software for Adaptive OpTics Arcetri) to a GPU-based cluster which allowed us not only to speed up the FFT computations required to simulate the PS [11], but also to parallelize ( 2) the n computations characteristic of an extended object. For this study we have arbitrarily chosen to distribute the n sampling points in concentric circles with a separation of ~2λ/D. 2.3. Noise propagation coefficients We have used the PS numerical model to investigate its sensitivity to measurement noise when using an extended reference source on an 8.22m (LBT) telescope. The NPCs characterize the sensitivity of

the noise in the reconstructed phase to the noise in the PS measurements. Following a modal approach to represent the reconstructed phase, we have simulated the calibration of an Interaction Matrix (IM) between 500 KL modes and a non-modulated PS with 30 30 sub-apertures, for three types of extended objects with uniform intensity distributions (e.g. top-hat illumination patterns) and an angular diameter of 0.4, 0.8, and 1.6 arc-seconds, respectively. The NPCs (p i ) were computed as: (7) where M denotes the modal IM. Figure 2 shows in continuous lines the NPCs associated with the three considered extended sources. As expected, the sensitivity to measurement noise increases with the size of the reference source. In order to validate our results, we have also computed a modal IM between KL modes and a modulated PS using an unresolved reference source. The applied TT modulation complies with the condition in Eq. (6), resulting in equivalent modulation radii of 5.3, 10.6, and 21.2 λ/d (where λ=750nm and D=8.22m in our simulations) for the corresponding 0.4, 0.8, and 1.6 extended objects. The NPCs for the three equivalent TT modulations are shown with dashed lines in Figure 2. For comparison, the plot also shows the NPCs of the ±3.0λ/D modulation typically used in the FLAO system [8]. Note that there is a good agreement between the corresponding NPCs, confirming the analysis presented in Sec. 2.1. Also, as already noted in [6], the good agreement is found on the modes having equivalent angular frequencies smaller than the size of the extended source. Figure 2. Noise Propagation Coefficients (NPCs) of the pyramid sensor. Continuous lines show the NPCs obtained with 2D extended sources with a top-hat illumination pattern, and a diameter of 0.4 (red), 0.8 (green), and 1.6 (blue) arc-seconds. No TT modulation was applied to the sensor. Dashed lines show the NPCs obtained with point sources when applying a TT modulation of ±3.0 (black), ±5.3 (green), ±10.6 (blue), and ±21.0 (red) λ/d. 3. Pyramid sensitivity with 3D extended reference sources We will address in this section the case of 3D reference sources. In particular, we will consider the case of a Sodium laser launched from the center of the telescope pupil. As it is well known, the laser creates a reference source with a finite extension along the optical (z-) axis due to the thickness of the Na layer. On a SH sensor, this extension when imaged from an off-axis sub-aperture produces a spot elongated in the radial direction. On a pyramid sensor, as discussed in [6], a z-elongated reference source reimaged atop of the pyramid introduces a continuous distribution of focus errors. As seen from an off-axis subaperture of the PS, the focus errors result in a continuous distribution of tilts oriented in the radial direction that increase in amplitude Θ with the separation from the pupil center as: Θ r, h r (8)

where h 0 is the altitude of the Na layer to which the PS is focused, and h i is the distance from h 0 of the i th out-of-focus Na layer. This sort of radial tilt modulation produces a loss of PS sensitivity in the radial direction that increases with the separation from the pupil center; an effect that is equivalent to the one suffered by SH sensors. Aided by numerical simulations, we have verified this effect on the PS and quantified it for the case of a 39m telescope. After outlining our simulation implementation in Sec. 3.1, we will present the characterization of the radial sensitivity loss on the PS in Sec. 3.2. Finally, Noise Propagation Coefficients (NPCs) will be discussed in Sec. 3.3. 3.1. Simulation of pyramid sensor with laser guide stars Following the same simulation strategy described in Sec. 2.2, an LGS can be modelled as a 3D extended object for which the vertical Na distribution is sampled with a discrete number of layers. For each one of these layers, the resulting 2D intensity distribution is sampled with a discrete number of points, as before. The LGS intensity distribution that we will consider in our simulations consists in a 2D Gaussian extension with a FWHM of 0.8 arcsec (assumed to arise from the up-link propagation of the Na laser) and a vertical (Gaussian) intensity distribution centered on 90 km. Regarding the sampling, points sources in a layer were arbitrarily spaced at ~32λ/D (where λ=589nm and D=39m), whereas vertical layers were spaced at 75m. We have simulated a 39m telescope with 28% central obstruction, and a 78 78 PS providing a spatial sampling of 0.5m/sub-aperture. No TT modulation has been applied when using extended objects. The simulated Field-of-View (FoV) of the sensor (limited by the number of pixels sampling the telescope pupil) was 3.6 arcsec in diameter. As we will discuss below, a larger FoV is actually required to properly deal with the full Na layer thickness, and this will be implemented in our future simulations. Assuming that the PS has been focused to the mean height of the Na layer 90 km, the RMS value of the focus error coming from a layer at a distance h from h 0 is: a (9) where D denotes the telescope diameter. The vertical extension of the LGS is taken into account in our simulations by simply adding a focus term with RMS value defined in Eq. (9) to the phase in the pupil resulting from the propagation from each one of the point sources sampling the 3D reference source. No atmospheric turbulence in the downward propagation will be considered in the present work in order to solely evaluate the effects of the 3D extension on the PS sensitivity. The plot on the left of Figure 3 shows the focus error RMS value as a function of the layer separation ( h). Considering a typical Na layer thickness of ~10km 5km, the PS will have to deal with a continuous distribution of focus errors ranging from -38 to +30 µm RMS, which can be large enough to saturate the sensor. The intensity distributions in the focal and CCD planes resulting from different layers of the LGS vertical distribution are represented on the right of Figure 3. Note that the required FoV is driven by the focus error associated with the extremes of the Na layer vertical distribution. With our currently simulated FoV of 3.6 arcsec we were only able to properly take into account an extension of ~ 3km (Gaussian FWHM of 2.3km). The final CCD frame (from which PS slopes are computed) is simply found by summing the CCD frames resulting from the different layers (properly scaled by the corresponding flux).

Figure 3. (Left) Focus error RMS value vs. layer separation ( h) given by Eq. (9) for D=39m. (Right) Intensity distributions in the PS focal and CCD planes resulting from different layers of the LGS vertical distribution (no atmospheric turbulence introduced). The images are not scaled to the flux associated with each layer. The focal plane image at h=0km is shown in a log stretch to put in evidence the point sources sampling the 2D Gaussian extension. 3.2. Effect of 3D elongation: radial sensitivity loss As a first verification, let us simulate the PS response to a simple tilt when using 3D extended objects. Of course, no LGS would be used to sense tip/tilt; this is simply a tutorial case to characterize the radial sensitivity loss experienced by the PS. According to Eq. (8), the radial modulation implies that all subapertures in a column (same x-coordinate) or row (same y-coordinate) will feel the same modulation component in the x- or the y-axis, respectively. The left of Figure 4 shows the simulated PS response to 0.1 arcsec x- and y-tilts when using an LGS composed of a 2D Gaussian (0.8 FHWM), and a Gaussian vertical Na distribution with a FWHM of 2.3km ( h max = ±3.0km). As expected from our heuristic analysis, the sensitivity on the x- and y- slopes decreases gradually for subapertures away from the y- and x- axis, respectively. The y-tilt signal profile is shown with a red line on the right of Figure 4. Note the linear reduction in sensitivity for subapertures away from the central column (x=0). Figure 4. (Left) Pyramid signals [Sx, Sy] produced by x- and y-tilts of 0.1 arcsec and a 3D extended object (2D Gaussian of 0.8 FHWM, z-elongation of h max =±3km). (Right) y-signal profile (input WF: 0.1 of y-tilt) produced by four different extended reference sources: (black line) 2D Gaussian spot with 0.8arcsec FWHM; (blue line) Refocused LGS: 2D Gaussian spot as above, and a z-elongation of h max =±0.5km; (green line) 2D Gaussian spot as above, h max =±1.5km; (red line) 2D Gaussian spot as above, h max =±3.0km. All vertical Na profiles are Gaussian.

This plot also shows the signal profiles obtained when reducing the vertical extension of the LGS to h max = ±1.5km (green line), and ±0.5km (blue line). The latter case, corresponding to a refocused LGS [4], is almost identical to the constant signal profile obtained with a 2D extended object (black line). In other words, the sensitivity becomes dominated by the 2D extension. 3.3. Noise propagation coefficients We have used our simulation tool to compute the NPCs of the PS, following the same approach outlined in Sec. 2.3. We have simulated the calibration of an IM between a subset of 4000 KL modes (chosen to sample the range of radial orders) and our PS with 78 78 sub-apertures. Figure 5 shows the NPCs obtained when using different reference sources: a modulated diffraction-limited source, a 2D extended object, a refocused LGS ( h max = ±0.5km), and an elongated LGS with h max = ±3.0km. As expected, noise propagation increases with the effective size of the reference source, being the lower order modes the most affected. A further analysis of the gain in sensitivity attained by the use of dynamical refocusing on the LGS is presented in [4]. Figure 5. Noise propagation coefficients for KL modes measured with a PS. Four different reference sources considered: (white crosses) Diffraction-limited source (with a tilt modulation of ±5λ/D); (red rhombs) 2D Gaussian spot with 0.8arcsec FWHM; (purple squares) Refocused LGS: 2D Gaussian spot as above, and a z- elongation of h=±0.5km; (green triangles) 2D Gaussian spot as above, h=±3.0km. No tilt modulation has been applied when using extended objects. 4. Conclusions and future work The use extended reference sources with the pyramid sensor decreases its sensitivity. In this work we evaluated the impact of source extension in the focal plane (2D) and along the optical axis. The loss of sensitivity due to a 2D source is quantitatively comparable to the one obtained with dynamical tilt modulation, being spatially uniform on the SAs but differing in saturation and response to high spatial order aberrations. On the other hand, a z-elongated reference source reimaged atop of the pyramid introduces defocus errors which, in turn, cause a sensitivity loss that increases linearly with the radial distance from the pupil center, similarly to what experienced by a SHS because of the spot elongation.

As part of our future work on this topic, we will perform a quantitative comparison of sensitivity with the SHS under the same LGS conditions. In particular, we will study the effects of vertical Na distribution variability (e.g. multiple peaks profile) on both sensors. Finally, we will evaluate the practical implementation advantages offered by a pupil plane WFS such as PS when coupled to an LGS. 5. References 1. N. Hubin, This conference, (2013) 2. B. Ellerbroek, This conference, (2013) 3. A. Bouchez, This conference, (2013) 4. S. Rabien, F. Quirós-Pacheco, E. Pinna, L. Busoni, S. Esposito, This conference, (2013) 5. B. Le Roux, Proc. SPIE, 7736, 773657, (2010) 6. E. Pinna, A. T. Puglisi, J. Argomedo, et al., 2 nd AO4ELT Conference, (2011) 7. E. Gendron, D. Gratadour, Proc. SPIE, 8447, 844713, (2012) 8. S. Esposito, A. Riccardi, F. Quirós-Pacheco, et al., Applied Optics, 49, 31, pp. G174, (2010) 9. A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, P. Stefanini, Proc. SPIE, 3353, pp. 941, (1998) 10. R. Ragazzoni, E. Dioaliti, E. Vernet, Optics Communications, 208, pp. 51, (2002) 11. M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, L. Fini, MNRAS, 356, pp. 1263, (2005)