Comparison between a model-based and a conventional pyramid sensor reconstructor

Comparison between a model-based and a conventionalpyramid sensor reconstructor

Visa Korkiakoski,1,* Christophe Vérinaud,1 Miska Le Louarn,1 and Rodolphe Conan2

1European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching b. München, Germany2Adaptive Optics Laboratory, University of Victoria, Mechanical Engineering Lab Wing B133, P.O. Box 3055 STN CSC,

Victoria, British Columbia, Canada, V8W 3P6

*Corresponding author: [email protected]

Received 27 February 2007; revised 16 May 2007; accepted 13 June 2007;posted 18 June 2007 (Doc. ID 80486); published 16 August 2007

A model of a nonmodulated pyramid wavefront sensor (P-WFS) based on Fourier optics has beenpresented. Linearizations of the model represented as Jacobian matrices are used to improve the P-WFSphase estimates. It has been shown in simulations that a linear approximation of the P-WFS is sufficientin closed-loop adaptive optics. Also a method to compute model-based synthetic P-WFS command ma-trices is shown, and its performance is compared to the conventional calibration. It was observed that inpoor visibility the new calibration is better than the conventional. © 2007 Optical Society of America

OCIS codes: 010.1080, 070.2580.

1. Introduction

When celestial objects are imaged by ground-basedtelescopes, the atmospheric turbulence lowers thequality of the images. A widely used technique tocompensate these distortions in real time is adaptiveoptics (AO) [1].

Systems using AO need a method to relate thewavefront sensor (WFS) measurements to the corre-sponding phase shape. Currently, the most commonmethod assumes that a WFS responds linearly to thedeformable mirror (DM) and then creates an interac-tion matrix mapping the DM commands to the sensormeasurements.

The interaction matrix can be obtained theoreti-cally or more typically by measuring the responses ofknown commands. The wavefront reconstruction isthen essentially achieved by a single matrix vectormultiplication (MVM). The inverse of the interactionmatrix, hereafter called the command matrix, is mul-tiplied by the measurement signal represented in avector form.

This method works well with linear sensors such asShack–Hartmann or a pyramid WFS (P-WFS) with

modulation as described in [2,3]. However, it isknown that a nonmodulated P-WFS saturates whenthe measured phase distortions are large [4]. In poorvisibility, over 1 arcsec at 0.5 �m wavelength, thisoccurs even when the AO system is used in a closedloop.

In this paper we present an iterative reconstruc-tion method that can be used to evaluate the signif-icance of the nonlinearities in the nonmodulatedP-WFS. We also show how to compute a syntheticmodel-based interaction matrix for the P-WFS.

At first, in Section 2 we derive a model for anonmodulated P-WFS measurement signal usingFourier optics. We linearize the model for arbitraryphase position and represent the linearizations byJacobian matrices. Then we use these lineariza-tions iteratively—in the same manner as in New-ton’s method used to find zeros of a function—toobtain improved phase estimates for given mea-surements. In Subsection 2.B we describe the prop-erties of such Jacobian matrices, and in Subsection2.C we summarize how we use the Jacobian recon-struction (JR) method in an AO system.

In Section 3 we demonstrate the performance of theJR in complete end-to-end AO closed-loop simula-tions. We test how much the JR can decrease theresidual phase variance compared to the case where

0003-6935/07/246176-09$15.00/0© 2007 Optical Society of America

6176 APPLIED OPTICS � Vol. 46, No. 24 � 20 August 2007

the conventional MVM is used. In these simulationswe consider both the classical four-sided P-WFS anda two-sided P-WFS as proposed in [5].

2. Model-Based Reconstruction

The classical four-sided P-WFS is illustrated in Fig.1. The incoming wavefront is focused on a pyramidprism, and the prism divides the beam into four sub-beams. Finally an optical relay projects pupil imageson a detector.

The pyramid sensor measurement is formedfrom the continuous intensity patterns (I0,0�r�, I0,1�r�,I1,0�r�, I1,1�r�) at each detector plane quadrant as

�Sx�r�Sy�r��

I0,0�r� � I1,0�r� � I0,1�r� � I1,1�r�It

I0,0�r� � I0,1�r� � I1,0�r� � I1,1�r�It

�, (1)

where It is the average intensity over the whole de-tector plane and r � �x yT is the location in thedetector coordinate system. The explicit formula forthe signal has been derived in Appendix A and isshown in Eq. (A8).

The derived model neglects all the interferenceeffects between the four subbeams. This is accept-able if the subbeams meet the detector far enoughfrom each other. Therefore, we will consider later on(also in simulations) that no interference effects arepresent.

The measurement signal in Eq. (1) is essentially asum of sinusoidal functions of the incoming phaseconvolved by 1�x and delta functions. Its direct in-version is very difficult. However, it is known that theP-WFS can be operated during good visibility (or us-ing long wavelengths) with a linear assumption �S�r�

is a linear function of ��r�]. If the simple linear ap-proximation is too inaccurate, additional methodshave to be considered for the inversion problem.

If the signal in Eq. (1) is smooth enough as a func-tion of ��r�, it can be linearized at all of its inputvalues. If each variable describing the measurementis a monotonous function of phase ��r�, an inverse ofS��r�� can be obtained using an iterative derivative-based algorithm as shown in Fig. 2 and described inthe following.

At first, the signal model is used to create a linearapproximation at � � 0, denoted as S�� J�0��.This approximation is used to find the first phaseestimate �0 given the known signal measurementS��r�. Then again the model is used to create a linearapproximation at �0, and the next phase estimate �1is obtained. The iteration can go on until the realphase �r is obtained at an arbitrary accuracy.

In reality, however, the P-WFS measurement sig-nal does not fulfill the monotonous requirementscompletely. The sinusoidal dependency of the signalwith respect to phase guarantees that if � � ��2, noinversion is possible. Nevertheless, as our simulationresults will show, at least some iterations can becarried out without problems, and an improved phaseestimate can be obtained.

A. Practical Implementation

To linearize the signal S�r� as a function of ��r�, weneed to know how a given point of the signal changesas a function of ��r�. For this purpose, we sample thephase with an N � N array of discrete points. N is atleast as big as the resolution of the WFS measure-ment grid.

As Fig. 3 illustrates, this approximation gives M� N2� phase values since the phase is zero outsidethe aperture. Also the signal S�r� needs to be dis-cretized.

The signal approximation can thus be representedas an M-dimensional vector with each element beinga function of the phase values over the grid

Fig. 1. Illustration of the nonmodulated pyramid sensor and itssignal composition. � is tangent of the pyramid divergence angle[8,9], f is the focal length, and f��2�� is half of the distancebetween the subbeam centers.

Fig. 2. Descriptive illustration of an iterative, derivative-based,numerical inversion of a monotonic nonlinear function.

20 August 2007 � Vol. 46, No. 24 � APPLIED OPTICS 6177

Sx�x, y� �Sx1��1, . . . , �M� · · · Sx

M��1, . . . , �M�,Sy�x, y� �Sy

1��1, . . . , �M� · · · SyM��1, . . . , �M�.

(2)

The elements of the signal vector can also be denotedas Sx

xi,yi�� meaning that the point �xi, yi� correspondsto the index n (see Fig. 3).

This enables making two M � M Jacobian matricesdescribing how the x and y signals change as a func-tion of the phase in the pupil area. The Jacobianmatrices can be computed elementwise given the co-ordinates �xi, yj� for the Jacobian row and �xj, yj� forthe Jacobian column,

Jn,m�� Jxn,m��

Jyn,m��

� ��Sx

xi,yi��1, . . . , �M��xj,yj

�Sxxi,yi��1, . . . , �M�

��xj,yj

�, (3)

where the row and column indexes n and m corre-spond to the coordinates as shown in Fig. 3. Theformulas for Eq. (3) have been derived in Appendix B.

B. Properties of the Jacobian

The Jacobian matrix in Eq. (3) has twice as manyrows as columns. The exact size depends on the gridpoints lying inside the aperture (at the borders anillumination threshold is used). Approximately thenumber of the Jacobian elements is

2��N2 �2��N

2 �2��18 �2N4, (4)

meaning that the computational requirements of theJacobian matrix increase proportionally to the fourthpower of the resolution N.

As seen in Eq. (B8), the Jacobian matrix can bedivided into two parts. The first part is due to themeasurement signal of the two-sided pyramid sensor�mx, my� and the latter comes from the cross terms�cx, cy�.

The main content of the Jacobian matrix comesfrom the terms mx and my. For instance, with a di-mension N � 40 giving a 2464 � 1232 Jacobian therelative energy difference between a complete Jaco-bian and a Jacobian without the cross terms,

elements

�Jfull�0� � Jm�0��2 ⁄ elements

�Jfull�0��2,

is 7.0%. In addition, if the cross terms were neglected,the resulting Jacobian would be sparse. For instance,a Jacobian of 2464 � 1232 would have 2.7% of theelements as nonzero. A system with N � 160 and aJacobian of 40,216 � 20,108 would have 0.68% of theelements as nonzero. It can also be noted that thecross terms contribute only to the Jacobian elementswhere the contribution of mx (or my) is zero.

Therefore, it is possible to consider the sparse Ja-cobian of the two-sided P-WFS (computed consider-ing only the terms mx and my) as an approximation ofthe complete four-sided P-WFS Jacobian. In our sim-ulations this approximation typically gave a 1–3 per-centage point lower Strehl ratio compared to thecases with the full Jacobian.

Yet another interesting question is whether theJacobian matrix can be inverted. As the matrix J��has more rows than columns, the linear equation

S � J��w (5)

is invertible given that J�� is well conditioned. Wecomputed 2464 � 1232-sized Jacobians for severaltypical residual phases in an AO loop. It was foundthat the condition numbers were always fewer than50. A theoretical study has not been carried out, butour simulations indicate that Eq. (5) is indeed solv-able with all Jacobians created in any meaningfulphase input �.

C. Application in the Adaptive Optics Loop

Figure 4 illustrates the AO system we used in oursimulations. The WFS measures the incoming wave-front �w using an intensity pattern at the detectorplane. Equation (1) shows how the measured inten-sity is interpreted as two two-dimensional signals Sx

and Sy.In a system having four Ns � Ns pixel CCD grids, Sx

and Sy are sampled by Ns � Ns matrices �Smx and Smy)where the elements outside aperture are zero. Thenonzero elements of Smx and Smy can be organized intoa measurement vector S. In the linear regime theWFS can be approximated as a linear mapping S� Bm�w. The phase reconstruction can be accom-

Fig. 3. Illustration of the phase discretization for the model. Anillumination threshold is required to choose which grid elementsbelong to the aperture and which are left outside.


plished using either the conventional MVM (apseudoinverse of Bm is used) or the JR.

To use the JR two parameters must be set—theJacobian resolution (N) and the number of Jacobianiteration steps �Nit�. The iteration steps are illus-trated in Fig. 2. If the signal is smooth, the moreJacobian iterations, the more accurate the recon-struction can be. The Jacobian resolution, on theother hand, defines the accuracy of the model—thereconstruction is not restricted to the resolution ofthe measurement grid.

If only one Jacobian iteration is used, the JR iscompletely a linear process. The first phase estimate(in a vector form with a resolution of N � N) can becomputed as

� � �J�0�†XmS, (6)

where �J�0�† is a pseudoinverse of the Jacobian at� � 0 and Xm is an interpolation matrix defining thebilinear interpolation relation Sb � XmS, where thevector Sb represents the measurement signal with aresolution of N � N instead of the original CCD pixelresolution.

The nth element of � represents the phase value atthe nth position on the pupil as illustrated in Fig. 3.This corresponds to the phase �ngn�r�, where gn�r� isone inside the nth grid element �r � r�xi, yi�) and zerootherwise. The actuator commands letting the DMshape to best fit in a least-squared sense to the phase�ngn�r�, are computed as

vn � Cm�1an�n, (7)

where matrix Cm describes the overlapping of theactuator influence functions fi�r� and vector an de-scribes the effect of the influence functions at the nthgrid element,

Cm�i, j� �� fi�r�fj�r�dr, (8)

an�i� �� gn�r�fi�r�dr. (9)

The sum of all commands vn �n � 1, . . . , M� is thebest DM fit for the JR phase estimate �. It is com-

puted as

c2 � Cm�1�an · · · aM�J�0�†XmS, (10)

where the matrix Cm�1�an · · · aM�J�0�†Xm is a syn-

thetic command matrix having the same dimensionsas the command matrix used in the conventionalMVM.

The computational requirements of the JR dependon Nit. When Nit � 1, the synthetic command matrixcan be computed offline as shown before. This makesthe JR computationally equivalent to the MVM. How-ever, if Nit � 1, the amount of computation increasesdramatically—at each iteration one must recomputethe Jacobian, compute a new measurement estimateand solve the linear equation Sb � J��i�1��i �

�i�1� � Sbi�1. This means a 50–1000 times slower

reconstruction time depending on the used solver al-gorithms.

3. Simulations

A. Scope

We made a set of end-to-end AO closed-loop simula-tions using the parameters shown in Table 1. Wesimulated an 8 m telescope with turbulence havingthe Fried parameter r0 � 8 cm at a wavelength of0.5 �m. The simulation of the P-WFS was imple-mented by a straightforward application of Eq. (A8).Thus, the simulated system was a perfect realizationof the used model.

The simulations were based on parallel softwaredescribed in more detail in [6]. The software imple-ments Monte Carlo simulations using a layer modelof the turbulent atmosphere and geometrical wave-front propagation. Each layer is a random realizationof the Von Karman power spectrum. The softwaremodels no DM dynamics nor phase evolution duringthe WFS integration. It can simulate only a mono-chromatic case.

We made most of our simulations using only asingle turbulence realization, as our goal is only to

Table 1. Simulation Parameters

Telescope diameter 8 m�imaging 1.6 �m�WFS 0.5, 0.7, 1.65 �mNoise NoMeasurement resolution (Ns) 40Frame rate 1 kHzInfluence functions Linear spline, no cross couplingServo lag 2 framesController Simple integratorNumber of layers 2Wind speeds 6, 15 m�sOuter scale L0 26 mr0 at 0.5 �m 0.08 mSimulated frames 336Simulation resolution 320 � 320

Fig. 4. Illustration of the simulated system.


demonstrate the feasibility of JR. Comparisons be-tween different models and reconstruction methodsare done by using identical phase screens. However,as described later, in selected cases we also made astatistical analysis with five different turbulence re-alizations to show the error range of our results.

In all of our simulations the Strehl ratio reached asteady state after �100 simulated frames. The addi-tional frames were simulated to ensure that no slowlydeviating instabilities were present.

We optimized the MVM reconstruction (in terms ofthe best long exposure Strehl ratio) by choosing theoptimal number of controlled modes. The system had1347 degrees of freedom, and we controlled 998 mir-ror modes [Karhunen–Loeve (K–L) approximations].This way it was not necessary to truncate any singu-lar values when inverting the modal interaction ma-trix. The modal interaction matrix was made bymeasuring the mirror modes at the linear regime andeach mode was normalized to have the same maxi-mum value. We also tested the MVM by controllingmore modes (up to 1346) and truncating the optimalnumber of singular values when inverting the inter-action matrix. Also the zonal interaction matrix wastested. These tests, however, did not improve theperformance. As a controller, we used a simple inte-grator with a constant loop gain individually opti-mized for each simulation parameter set.

The P-WFS simulation results are compared withresults obtained using a linear sensor. The linearWFS is simulated by replacing cos ��r� by 1 andsin ��r� by ��r� in the signal shown in Eq. (A8). Thislinear WFS is then controlled with the same MVMreconstruction as the original P-WFS. This enablesobtaining a clear measure of how much the nonlin-earities decrease the P-WFS performance.

B. Results

At first, we tested the JR performance in a closed-loop simulation with a static phase distortion (nowind) using a sensing wavelength of 0.5 �m. Thisway we could see how much of the P-WFS nonlinear-ity can ideally be compensated. It was noted that theStrehl ratio saturated with a linear sensor at 0.860and with the conventional MVM at 0.845 (two-sidedP-WFS) or 0.825 (four-sided P-WFS). The JR withNit � 1 gave equally good results as the MVM, andthe help of the additional Jacobian iterations wasnegligible. This loss of 0.015–0.035 in Strehl ratiomay be explained by a higher sensitivity to aliasingfor the P-WFS in the nonlinear regime. We also notedthat one Jacobian iteration was enough in all ourfurther simulations. Therefore, we will consider onlyJR with Nit � 1.

Next we consider the real closed-loop simulationswith the temporal evolution of the turbulence byshifting the phase screens based on the wind speedvalues. In Fig. 5 we demonstrate the effect of loopgain when using the P-WFS. The long exposureStrehl ratio is plotted for the conventional MVM andJR with both two- and four-sided P-WFS. These re-

sults are averaged from five different turbulence re-alizations.

It can be noted that the two-sided P-WFS performsbetter that the four-sided classical P-WFS [the opti-mal Strehl ratios are 0.73 and 0.53 (MVM) or 0.81and 0.73 (JR)]. This result is also confirmed by [5] andsuggests that the cross terms of Eq. (B1) significantlycontribute to the nonlinear behavior of the four-sidedpyramid sensor.

Concerning the dependency on loop gain, we needto draw attention to the fact that the optimum Strehlratios are obtained at gains higher than 1, even forthe JR method, which should in principle better de-scribe the measurement accuracy. Indeed, the opti-mum gains of the JR (1.8 and 2.6) are highercompared with the optimum gains of the MVM (1.0and 1.65). However, the optimal Strehl ratios of JRare always 0.05–0.15 better than that of the MVMreconstruction.

This peculiar behavior suggests that a normaliza-tion issue subsists in the pyramid model. Among pos-sible reasons, the fraction of light diffracted by thevery sharp prism edges out of the four geometricalpupils may not be described with sufficient accuracyby the model and could explain this normalizationissue. This definitely needs a deeper analysis.

It was noted that the higher the Jacobian resolu-tion, the better the performance. A resolution of N� 80 gave approximately 0.015 worse Strehl ratios(compared to N � 160) and N � 40 was too low tomake the AO loop stable.

Next we simulated both two- and four-sided P-WFSfor three different sensing wavelengths (0.5, 0.7, and1.65 �m). The optimum Strehl ratios of those simu-lations are shown in Table 2.

It can be seen that when using a two-sided P-WFSwith any reconstruction method, the optimal Strehlratio of 0.85 can almost be obtained at high wave-lengths. However, at �WFS � 0.5 �m the Strehl ratio isalready 0.12 lower (with MVM reconstruction) and

Fig. 5. Long exposure Strehl as a function of loop gain. Solidcurves represent two-sided P-WFS, dotted curves the four-sidedP-WFS. Dot markers show the conventional MVM, curves withoutmarkers the JR. The error bars are computed using the standarderror from five sets of simulations. The simulations are made with�WFS � 0.5 �m, N � 160, Nit � 1.


0.04 lower (with JR). The four-sided P-WFS giveseven worse results. The Strehl ratio at �WFS � 0.5�m is with MVM 0.32 lower and with JR 0.12 lowerthan the optimum.

Thus, the JR is indeed most useful at the nonlinearregime of the P-WFS. At �WFS � 1.65 �m the JR gives0.007 worse Strehl ratio compared with the MVMreconstruction. This can be explained by the approx-imate nature of the JR. Correctly calibrated MVM isby definition the optimal way of reconstruction, if thesensor is fully linear. JR, on the other hand, uses anapproximated model to compute the actuator com-mands, and slight inaccuracies are inevitable.

In high contrast imaging it is also important tooptimize the power spectral density (PSD) being de-fined as PSD�f� � � ��r�� 2�, where ��r�� is theFourier transform of the residual phase and �·� de-notes temporal averaging. The radially averagedPSDs of the two-sided P-WFS simulation using theoptimal loop gains at �WFS � 0.5 �m are shown inFig. 6.

It can be seen that the JR mainly helps to re-duce the residual energy at low spatial frequencies�1.0 m�1�, which is particularly important for thedetection of exoplanets at small angular separationsfrom their host star [7]. The reduction at the lowestspatial frequencies is equal to one third of the casewhere MVM reconstruction was applied. However,with larger frequencies JR gives practically the sameresults as the MVM.

4. Conclusions

It has been shown that the performance of an AOsystem using a nonmodulated P-WFS with a conven-tionally calibrated MVM gets worse in high turbu-lence. The results can be improved by using adiffraction model with a synthetic Jacobian-based re-construction matrix as described here.

We demonstrated the JR in difficult conditions�r0 � 0.08 m and �WFS � 0.5 �m) where the optimalStrehl ratio given by a linear sensor was approxi-mately 0.85. The Strehl ratio of a classical four-sidedP-WFS can be improved from 0.53 up to 0.73 and theratio of a two-sided P-WFS from 0.73 up to 0.81.

It can be concluded that the two-sided P-WFS ismore linear compared to the four-sided P-WFS. Thismeans that the cross terms not present in the two-sided P-WFS contribute most of the classical P-WFSnonlinearity properties.

In addition, it was seen that most of the JR Strehlratio improvement is achieved by only one Jacobianiteration, the increase of the additional iterations in-crease is negligible. This implies that the P-WFS in-deed saturates rather quickly and the nonlinearitiescannot be efficiently compensated in the conditionsconfronted in a typical AO system.

However, we demonstrated that the linear approx-imation of the JR gives better results than the con-ventionally calibrated MVM. This can be explainedby problems in the conventional calibration. For in-stance, the relative modal gains might become non-optimally assigned when the calibration is done atthe linear regime, but the command matrix is appliedfor high phase distortions.

The simulations are done in the monochromaticcase. Introducing multichromaticity needs significantdevelopment of our code and is for the moment post-poned until a later date. The effect of multichroma-ticity is at the first order a difference in the Strehlratio dependency on the loop gain.

In a real system, if the wavefront-sensing spectralbandwidth is significantly increased, especially to-ward shorter wavelengths, the pyramid model in thestrong perturbation case used to compute the Jaco-bians will need to be adapted to multichromaticity byaveraging Jacobians at different wavelengths cover-ing the spectral domain.

The study presented in this paper is a first attemptto deal with the nonlinearity of the P-WFS. In a forth-coming paper, we will present how to optimize thisnew method for real systems with the usual con-straints of noise, instrumental errors such as mis-alignments, and computing power limitations ingreater detail.

Appendix A: Pyramid Signal

The complex amplitudes at the detector plane (asillustrated in Fig. 1) can be written as

�p�r�� r�� 1�Tpyr�f��, (A1)

where ��r�� P�r��exp�i��r�� is the complex ampli-tude in the pupil plane, [P�r�� is the aperture func-

Table 2. Optimal Strehl Ratios

�WFS

(�m)

Two-Sided P-WFS Four-Sided P-WFS

1.65 0.7 0.5 1.65 0.7 0.5

Linear sensor 0.849 0.849 0.849 0.852 0.852 0.852MVM 0.849 0.837 0.729 0.848 0.804 0.530JRa 0.842 0.837 0.807 0.840 0.809 0.728

aN � 160, Nit � 1.

Fig. 6. Radially averaged power spectral densities of the residualphases in two-sided P-WFS simulations with optimal loop gains at�WFS � 0.5 �m. JR was used with N � 160 and Nit � 1.


tion], and Tpyr�f�� is the pyramid transmittancefunction given as

Tpyr�f�� n�0

1

m�0

1

Tn,m�f� � n�0

1

m�0

1

H��1�nf�x, ��1�mf�y��exp��i��1�nf�x � c� ��1�mf�y � c�,

where H�x, y� is the two-dimensional Heaviside func-tion �H�x, y� � 1, when x and y � 0 and zero other-wise], � is the tangent of the pyramid divergenceangle (as defined in [8,9]), and c is half of the distancebetween two opposite pyramid corners. Thus, thecomplex amplitudes can be written as

�p�r�� n�0

1

m�0

1

�n,m�r��, (A2)

where

�n,m�r�� x� � ��1�n

2�, y� � ��1�m

2�� 1�H�f��1�nx�, ��1�my��, (A3)

where the Fourier transform of the Heaviside func-tion is

��1�H�f��1�nx, ��1�my� �14 ��x, y� �

��1�n�m

4�2 p.v.1xy

�i

4��1�n�p.v.1x ��y��

� ��1�m��x�p.v.1y��,

(A4)

with p.v. denoting the principal value. The intensityat the detector plane is

Ip�r�� r��*�r��

� n�0

1

m�0

1

�n,m�r��n,m* �r��

� 2 n�0

1

m�0

1

n��0n��n

1

m��0m��m

1

Re��n,m�r��n�,m�* �r��.

(A5)

When assuming that the four waves are far from eachother, the interferences between them can be ne-glected. Then intensity becomes

Ip�r�� n�0

1

m�0

1

In,m�r��, (A6)

where

In,m�r�� n,m�r��n,m* �r�� (A7)

are the intensities in each quadrant. The pyramidsensor measurement is obtained by dividing the de-tector plane into four sections and combining the in-tensities as shown in Eq. (1). When substituting Eqs.(A3), (A4), and (A7) into Eq. (1), the pyramid mea-surement signal can be written as

ItSx�x, y� � �1��R��x, y��I��x, y� � �p.v.

1x ��y��

� I��x, y��R��x, y� � �p.v.1x ��y��

�1

�3��R��x, y� � p.v.1xy�

� �I��x, y� � ��x�p.v.1y��

� �I��x, y� � p.v.1xy�

��R��x, y� � ��x�p.v.1y��,

ItSy�x, y� � �1��R��x, y��I��x, y� � ��x�p.v.

1y��

� I��x, y��R��x, y� � ��x�p.v.1y��

�1

�3��R��x, y� � p.v.1xy�

� �I��x, y� � �p.v.1x ��y��

� �I��x, y� � p.v.1xy�

� �R��x, y� � �p.v.1x ��y��, (A8)

where R��x, y� � P�x, y�cos ��x, y�, I��x, y� � P�x, y�sin ��x, y�. P�x, y� is the field amplitude and ��x, y� isthe field phase value. The amplitude P�x, y� is as-sumed to be a constant P inside of the pupil and zerooutside.

Appendix B: Linearization

Next we derived the formulas for Eq. (3). The dis-cretized phase approximation vector ��1, . . . , �M isabbreviated in this appendix as �. At first, it can benoted that the measurement of the four-sided pyra-mid sensor is a sum of the two-sided pyramid sensormeasurement �mx, my� and cross terms �cx, cy�. Thus,the signal vector can be written as

Sxxi,yi�� mx

xi,yi�� cxxi,yi��,

Syxi,yi�� my

xi,yi�� cyxi,yi��, (B1)


where

mxxi,yi��

1��P�xi, yi�cos �xi,yi

Csx�xi, yi�

� P�xi, yi�sin �xi,yiCcx�xi, yi��,

myxi,yi��

1��P�xi, yi�cos �xi,yi

Csy�xi, yi�

� P�xi, yi�sin �xi,yiCcx�xi, yi��,

cxxi,yi��

1

�3�Ccxy�xi, yj�Csy�xi, yj�

� Csxy�xi, yj�Ccy�xi, yj��,

cyxi,yi��

1

�3�Ccxy�xi, yj�Csx�xi, yj�

� Csxy�xi, yj�Ccx�xi, yj��, (B2)

having the convolution matrices defined as

Csx�xi, yi� ��x�,y�

P�x�, y��sin �x�,y�

1xi � x�

� ��yi � y��dx�dy�,

Ccx�xi, yi� ��x�,y�

P�x�, y��cos �x�,y�

1xi � x�

� ��yi � y��dx�dy�,

Csy�xi, yi� ��x�,y�


1yi � y�

� ��xi � x��dx�dy�,

Ccy�xi, yi� ��x�,y�


1yi � y�

� ��xi � x��dx�dy�,

Csxy�xi, yj� ��x�,y�


�1

�xi � x��yi � y��dx�dy�,

Ccxy�xi, yj� ��x�,y�


�1

�xi � x��yi � y��dx�dy�. (B3)

The terms mx, my, cx, and cy can be derived in respectto the phase values at each grid element. The Jaco-bian matrices of mx and my become relatively sparse,

where the distance matrices Dmx and Dmy depend onlyon the horizontal and vertical distances of the gridelements. They are computed as

Dmx�i, j� � �x��r�xj,yj�

P�x�, y��1

xi � x�dx�,

Dmy�i, j� � �y��r�xj,yj�

P�x�, y��1

yi � y�dy�, (B5)

where r�xj, yj� is the region representing the grid el-ement �xj, yj�. Assuming that all the grid elements�xj, yj� lay completely inside the pupil (having a con-stant width d), the distance matrices become

�mxxi,yi��

��xj,yj

��

1��P cos �xi,yi

cos �xj,yjDmx�i, j� � P sin �xi,yi

sin �xj,yjDmx�i, j��, if xi � xj, yi � yj,

�1��P sin �xi,yi

Csx�xi, yi� � P cos �xi,yiCcx�xi, yi��, if xi � xj, yi � yj,

0, otherwise

�myxi,yi��

��xj,yj

��

1��P cos �xi,yi

cos �xj,yjDmy�i, j� � P sin �xi,yi

sin �xj,yjDmy�i, j��, if yi � yj, xi � xj,

�1��P sin �xi,yi

Csy�xi, yi� � P cos �xi,yiCcy�xi, yi��, if yi � yj, xi � xj,

0, otherwise, (B4)


Dmx�i, j� �1d��xi � xj � d � ��log xi � xj � d � � � �xi � xj � ��log xi � xj � � � ��xi � xj � �� log xi � xj � � � �xi � xj � d � �� log xi � xj � d � � �, (B6)

where approaches zero and xi, xj are the middlecoordinates of the grid elements �xi, yi� and �xj, yj�.

In the same way the derivatives of the cross termscan be computed:

Thus, the Jacobian is then formed by substitutingEqs. (B1), (B4), (B5)–(B7) into Eq. (3):

Jxn,m��

�mxxi,yi��

��xj,yj

��cx

xi,yi��xj,yj

,

Jyn,m��

�myxi,yi��

��xj,yj

��cy

xi,yi��xj,yj

. (B8)

References1. F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press,

1999).2. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillat-

ing prism,” J. Mod. Opt. 43, 289–293 (1996).3. C. Vérinaud, “On the nature of the measurements provided by a

pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).4. O. Wulff and D. Looze, “Nonlinear control for pyramid sensors

in adaptive optics,” Proc. SPIE 6272, 62721S (2006).5. D. W. Phillion and K. Baker, “Two-sided pyramid wavefront

sensor in the direct phase mode,” Proc. SPIE 6272, 627228(2006).

6. M. Le Louarn, C. Verinaud, V. Korkiakoski, and E. Fedrigo,

“Parallel simulation tools for AO on ELTs,” Proc. SPIE 5490,705–712 (2004).

7. C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet,“Adaptive optics for high-contrast imaging: pyramid sensor ver-sus spatially filtered Shack–Hartmann sensor,” Mon. Not. R.Astron. Soc. 357, L26–L30 (2005).

8. C. Arcidiacono, “Beam divergence and vertex angle measurementsfor refractive pyramids,” Opt. Commun. 252, 239–246 (2005).

9. A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefa-nini, “Laboratory characterization of a Foucault-like wavefrontsensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).

�cxxi,yi��

��xj,yj

��1

�3�Csxy�xi, yi�sin �xj,yjDmy�i, j� � Ccxy�xj, yj�cos �xj,yj

Dmy�i, j��, if xi � xj,

1

�3��cos �xj,yjDmx�i, j�Dmy�i, j�Ccy�xi, yi� � sin �xj,yj

Dmx�i, j�Dmy�i, j�Csy�xi, yi��, if xi � xj,

�cyxi,yi��

��xj,yj

��1

�3�Csxy�xi, yi�sin �xj,yjDmx�i, j� � Ccxy�xj, yj�cos �xj,yj

Dmx�i, j��, if yi � yj,

1

�3��cos �xj,yjDmx�i, j�Dmy�i, j�Ccx�xi, yi� � sin �xj,yj

Dmx�i, j�Dmy�i, j�Csx�xi, yi��, if yi � yj. (B7)


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