PYRAMID WAVEFRONT SENSOR PERFORMANCE …Adaptive Optics for Extremely Large Telescopes III PYRAMID WAVEFRONT SENSOR PERFORMANCE WITH LASER GUIDE STARS Fernando Quirós-Pacheco 1, …ao4elt3.arcetri.astro.it/proceedings/fulltext_13138.pdf ·

Adaptive Optics for Extremely Large Telescopes III

PYRAMID WAVEFRONT SENSOR PERFORMANCE WITH

LASER GUIDE STARS

Fernando Quirós-Pacheco1, a

, Enrico Pinna1, Alfio Puglisi

1, Lorenzo Busoni

1, Guido Agapito

1,

Sebastian Rabien2, Simone Esposito1

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: [email protected]

Third AO4ELT Conference - Adaptive Optics for Extremely Large TelescopesFlorence, Italy. May 2013ISBN: 978-88-908876-0-4DOI: 10.12839/AO4ELT3.13138


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 RTT 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 (RTT) 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 REO is given by:

S� � � sin��

��()� �

� �

��()� "1 $ � ��

�()�%

�&' (4)

and the corresponding PS sensitivity is:

Third AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes


� !��

*��+, "1 $ � ��

�+,�%

�&'. (5)

The PS sensitivity in this case is a function of the extended object radius REO. 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:

.// � 0+, . (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 REO features the same sensitivity

as the one obtained with a circularly modulated point source with RTT =REO /2, and a doubled dynamic

range (i.e. ±2RTT).

Figure 1. (Left) Coordinates of the pyramid sensor showing the four quadrants (A, B, C, D) and the transverse ray

aberration (y0). (Right) PS signal (Sy) as a function of the aberration (y0) for two kinds of reference sources:

(black continuous line) extended object with uniform illumination and a unitary radius (REO =1); (red dashed line)

point source circularly modulated with a modulation radius equal to RTT = REO/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 (pi) were computed as:

12 � 3454622��

(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 sub-

aperture of the PS, the focus errors result in a continuous distribution of tilts oriented in the radial

direction that increase in amplitude Θ8 with the separation from the pupil center 9:; as:

Θ89r, ∆h8; � ∆@A@�' r (8)



where h0 is the altitude of the Na layer to which the PS is focused, and ∆hi is the distance from h0 of the

ith 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 B� � 90km . 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 9B� � 90km;, the RMS

value of the focus error coming from a layer at a distance ∆h from h0 is:

aGH � ∆@3√J6K�L��∆LM �'

(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 9∆BOPQ � R5km;, 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 ∆BOPQ~R 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 (∆hmax = ±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 ∆hmax=±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 ∆hmax=±0.5km; (green line) 2D

Gaussian spot as above, ∆hmax=±1.5km; (red line) 2D Gaussian spot as above, ∆hmax=±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

∆hmax = ±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 (∆hmax = ±0.5km), and an elongated LGS with ∆hmax = ±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., 2nd 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)


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PYRAMID WAVEFRONT SENSOR PERFORMANCE …Adaptive Optics for Extremely Large Telescopes III PYRAMID WAVEFRONT SENSOR PERFORMANCE WITH LASER GUIDE STARS Fernando Quirós-Pacheco 1, …ao4elt3.arcetri.astro.it/proceedings/fulltext_13138.pdf ·