High Sensitivity Wavefront Sensing with a non-linear Curvature

Wavefront Sensor

Olivier GuyonSubaru Telescope, 650 N. A’ohoku Place, Hilo, 96720 HI, USA

Steward Observatory, University of Arizona, 933 N Cherry Ave., Tucson, 85721 AZ, USA

guyon@naoj.org

ABSTRACT

A new wavefront sensing approach, derived from the successful curvature wavefront sensingconcept but using a non-linear phase retrieval wavefront reconstruction scheme, is described.The non-linear curvature wavefront sensor (nlCWFS) approaches the theoretical sensitivity limitimposed by fundamental physics by taking full advantage of wavefront spatial coherence in thepupil plane. Interference speckles formed by natural starlight encode wavefront aberrations withthe sensitivity set by the telescope’s diffraction limit λ/D rather than the seeing limit of moreconventional linear WFSs. Closed-loop adaptive optics simulations show that with a nlCWFS,a 100 nm RMS wavefront error can be reached on a 8-m telescope on a mV = 13 naturalguide star. The nlCWFS technique is best suited for high precision adaptive optics on brightnatural guide stars. It is therefore an attractive technique to consider for direct imaging ofexoplanets and disks around nearby stars, where achieved performance is set by wavefront controlaccuracy, and exquisite control of low order aberrations is essential for high contrast coronagraphicimaging. Performance gains derived from simulations are shown, and approaches for high speedreconstruction algorithms are briefly discussed.

Subject headings: instrumentation: adaptive optics — techniques: high angular resolution

1. Introduction

Adaptive Optics (AO) systems are now an es-sential part of ground-based telescopes, and rou-tinely deliver diffraction limited images at near-IR and mid-IR wavelengths on 8 to 10 m tele-scopes. For several key science applications, therequired wavefront quality is however still well be-yond what current systems deliver. For example,obtaining high quality diffraction-limited imagesat visible wavelengths requires residual wavefronterrors to be below 100 nm RMS. Direct imagingof exoplanets or circumstellar debris disks with ahigh contrast coronagraphic camera is even moredemanding, and requires exquisite control of loworder aberrations and mid-spatial frequencies.

In the last decade, several high performancecoronagraphs concepts have been developed in thelaboratory, and full AO+coronagraph laboratory

systems have demonstrated that both deformablemirror and coronagraph technologies are now suf-ficiently mature to allow direct imaging of plan-ets down to Earth mass provided that wavefrontaberrations are small: for example, the vacuumHigh Contrast Imaging Imaging Testbed (HCIT)a NASA JPL has achieved better than 10−9 rawPSF contrast at 4 λ/D separation (Trauger &Traub 2007). This high contrast value is orders ofmagnitude better than, at a few λ/D, the ≈ 10−3

best raw PSF contrast currently achieved in thenear-IR on 8 m to 10 m telescopes and the ex-pected ≈ 10−4 raw PSF contrast for Extreme-AO systems currently under assembly such asSPHERE (Beuzit et al. 2008) on the Very LargeTelescope (VLT) and Gemini Planet Imager (Mac-intosh et al. 2008). The large gap between labo-ratory demonstrations and on-sky performance isentirely due to wavefront control accuracy. A key

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difference between laboratory systems and on-skyconditions is the timescale of wavefront variations,which requires fast wavefront sampling and there-fore fundamentally limits the number of photonsavailable for sensing on ground-based systems.

The wavefront information which can be ex-tracted from a given number of photons is knownfrom first principles, and accurately defines the”ideal” performance which can be reached byAO systems on ground-based telescopes (Guyon2005). Direct comparison between this ideal limitand the sensitivity offered by current wavefrontsensing schemes reveals a huge performance gap.Commonly used wavefront sensors such as Shack-Hartmann and Curvature are very robust and flex-ible, but are poorly suited to high quality wave-front measurement: for low-order aberrations on8 to 10 m telescopes, they require ≈ 100 to 1000times more photons than more optimized wave-front sensing techniques. While they offer nearlyideal sensitivity at the spatial frequency definedby the subaperture spacing, they suffer from poorsensitivity at low spatial frequencies, which aremost critical for direct imaging of exoplanets anddisks.

This paper describes a high sensitivity wave-front sensing approach which is optically derivedfrom the curvature wavefront sensing technique,and which shares the same non-linear reconstruc-tion algorithm as used for phase diversity. Inphase diversity, images near the focal plane areused for wavefront reconstruction with a non-linear algorithm, while curvature wavefront sens-ing uses images near the pupil plane and a linearwavefront reconstruction algorithm. Phase diver-sity is theoretically more sensitive than linear cur-vature wavefront sensing (Fienup et al. 1998), but,for ground-based adaptive optics applications, suf-fers in practice from smearing of focal plane speck-les due to chromatic effects and time averaging ofthe fast-changing PSF. I propose in this paper asolution to improve curvature WFS sensitivity byusing non-linear signals in the near-pupil images,and offer a solution to optically mitigate the chro-matic smearing of the speckles which would other-wise reduce sensitivity. Unlike phase diversity, theproposed scheme is robust against time averagingof wavefront aberrations, which gradually bringsthe proposed technique closer to a less sensitivebut stable linear curvature wavefront sensor.

The non-linear curvature wavefront sensor (nl-CWFS) principle is described in §2. The wavefrontreconstruction algorithm for the nlCWFS is de-scribed in §3. The WFS performance is evaluatedthrough numerical simulations in §4 and discussedin §5.

2. nlCWFS Principle

Figure 1 shows the proposed optical imple-mentation for the non-linear Curvature WFS (nl-CWFS). In this section, the main differences froma conventional curvature WFS are explained:

• The nlCWFS achieves its sensitivity by us-ing non-linear signal in images that are ac-quired in planes which are considerably fur-ther away from the pupil, where non-lineareffects are strong (§2.1)

• While curvature systems use a variable fo-cal length vibrating membrane, here, beamsplitters and lenses are used to produce therequired conjugation planes on the detec-tor(s) array (§2.2)

• Chromatic lenses are used in the sensor’s fo-cal planes for compensation of chromatic ef-fects ( §2.3 )

• Four conjugation planes are created insteadof two in conventional curvature (§2.4)

2.1. Non-linear signal and large defocus

distances

The performance of conventional CurvatureWFS (cCWFS) is strongly constrained by the re-quirement that wavefront sensor signals must bea linear function of the input wavefront aberra-tions (Guyon et al. 2008). This linear constraintkeeps the defocus distance much smaller than itshould be for optimal conversion of phase aberra-tions into intensity fluctuations. The dual strokelinear scheme proposed in Guyon et al. (2008)only slightly mitigates the negative effects of theseconstraints, but linearity is still enforced.

A far more serious limitation is that linear re-construction fully ignores the non-linear couplingbetween high and low order aberration in the de-focused pupil images. This is best visualized bydirect comparison of linear vs. non-linear mea-surement of tip-tilt from defocused pupil images.

2

+dz1

−dz1

+dz2

−dz2

pupil plane position (with defocus)

Pupil plane(without defocus)

chromatic defocus lens

BS

BS BS

light fromtelescope

45deg Mirrorssend beams out of the plane of

this figure

Beam splitting assembly produces 4 identical copies ofbeam from the telescope

outp

ut o

f bea

m s

plitt

ing

asse

mbl

y

(without defocus)Pupil planePupil plane

L1+

L1−

L2+

L2−

Detectorplane

Fig. 1.— Conceptual layout of a non-linear Curvature wavefront sensor. See text for details.

In cCWFS, tip-tilt is measured as an overalldisplacement of the pupil in the defocused im-ages: only the edges of the defocused pupil im-ages are used to extract signal. The displace-ment is proportional to the defocus distance dz.If the wavefront is otherwise perfectly flat, tip-tilt is theoretically best sensed by an extremelylarge (infinite) defocusing distance (van Dam &Lane 2002b), where the defocused pupil image isin fact close to a focal plane image, and the tip-tiltmeasurement sensitivity is equivalent to what isachieved when measuring tip-tilt from a properlysampled λ/D-wide focal plane diffraction-limitedimage. This is hardly realistic, as higher orderwavefront aberrations also need to be sensed, andthe defocus distance must therefore be kept small,and not all photons should be allocated to tip-tilt sensing. Even if the defocus distance werevery large, wavefront aberrations would precludediffraction-limited imaging at the wavefront sens-ing wavelength. The defocused pupil plane imagesare therefore convolved by the λ/r0-wide atmo-spheric seeing kernel (were r0 is the Fried param-eter quantifying the strength of the atmosphericturbulence): unless the AO correction is quitegood, the tip-tilt sensitivity is at best equivalent tomeasuring the centroid of a λ/r0-wide spot. It is infact not as good as that, as the defocus distance isfinite, and tip-tilt is therefore only measured fromdetectors at the edge of the pupil.

If non-linear effects are to be taken into accountfor tip-tilt measurement, the situation is radicallydifferent. First, larger defocus distances are al-lowed, which would improve the tip-tilt measure-ment sensitivity even if it were derived with the

simple “pupil photocenter” diagnostic (which islinear). The real benefit however comes from thefact that the defocused pupil images are “speck-led” due to edge (pupil edge, central obstruction,spiders) diffraction effects and high order aberra-tions. These highly contrasted speckles, visiblein figure 2, are λ/D wide (which means they arephysically smaller closer in to the pupil plane, andinfinitely small - invisible - at the pupil plane).Provided that the defocused pupil image is ac-quired with proper spatial sampling, a tip-tilt cantherefore be measured as the displacement of a“cloud” of λ/D-wide speckles. High order aberra-tions must be known to some degree to be able torecover this highly sensitive tip-tilt measurement.Similarly, low order aberrations need to be knownto measure high order aberrations from the defo-cused pupil images.

2.2. Beam splitters

In conventional curvature systems, the requireddefocus is achieved sequentially with a resonantvibrating membrane (VM) in the system’s focalplane. Because the high performance of the pro-posed wavefront sensing technique relies on re-solving speckles in the defocused pupil images,care must be given to avoid smearing them, whichwould happen with a sine-wave motion of themembrane. If a VM were used, its motion shouldtherefore be “squared”, and the short transitionperiod during which the defocus altitude is rapidlychanging should not be used toward wavefrontsensing (this time could conveniently be availablefor detector readout).

A better solution is to simultaneous image the

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with

cor

rect

ion

(0.4

to 0

.8 m

icro

n)

poly

chro

mat

ic(0

.4 to

0.8

mic

ron)

0 0.5 1 1.5 2

−3500 km −300 km +3500 km+300 km

8 m diameter pupil

mon

ochr

omat

ic

371 nm RMS(0.6

mic

ron)

poly

chro

mat

ic

Fig. 2.— Simulated frames obtained by the nlCWFS in monochromatic light (top), in polychromatic lightwith no chromatic compensation (center) and in polychromatic light with chromatic compensation (bottom).The wavefront error across the 8m diameter pupil used for this simulation is shown on the right.

four defocused pupil images on a single detectorarray (or on four smaller detector arrays) with asmall number of beam splitters and mirrors. Thedesired beam size on the WFS detector array ison the order of 1mm (100 pixels accross the pupilfor 10 µm pixels). In this scaled down copy ofthe telescope beam, a 1000 km vertical propaga-tion distance is scaled down to approximately 1cm. In the non-linear curvature wavefront sens-ing scheme described in this paper, there is alsono need to adapt the defocus altitudes as a func-tion of atmoshperic conditions, so such a “static”implementation appears suitable.

2.3. Chromatic compensation with the fo-

cal plane lenses

The primary purpose of lenses L1+, L1−, L2+and L2− in the WFS’s focal planes is to achievethe desired altitude conjugation on the detectorplanes: if the lenses are removed, four identicalpupil plane images would be acquired. In the nl-CWF, these lenses are designed to be chromaticso that, in each of the 4 channels, the conjugation

altitude is inversely proportional to wavelength.Figure 2 illustrates this chromatic compensation.Without compensation, speckles in the polychro-matic images are slightly blurred. This is moreapparent in with larger propagation distances, forwhich the finest-sized speckles disappear. Withthe chromatic compensation, polychromatic defo-cus pupil images are very close to monochromaticimages, and the blurring of the smallest specklesis very minimal. For very large propagation dis-tances, defocused pupil images are close to a focalplane image, and the proposed chromatic compen-sation is equivalent compensating for the λ scalingof the position of PSF diffraction features in thefocal plane.

2.4. Number of defocus planes

In Guyon et al. (2008), the dual stroke sensingscheme was justified by the need to mitigate someof the constraints brought by the requirement ofpreserving a linear relationship between WFS sig-nals (curvature) and input wavefront phase. Witha non-linear wavefront reconstruction scheme, it

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would therefore seem that the need for dual strokeis removed.

With the non-linear reconstruction algorithmand simulation parameters presented in this pa-per, the AO loop could not be closed using twosymmetric detector planes. No attempt has beenmade to explore other configurations (for exam-ple, 2 non symmetric planes, or 3 planes) or sta-bilize the loop by modifying or constraining thereconstruction algorithm, so this does not excludethe possibility that nlCWFS may work with fewerthan 4 planes.

While the reasons for the loop instability in asymmetric 2 planes nlCWFS have not been stud-ied, it may be due to a fundamental property ofdiffractive propagation. The nlCWFS achieves itssensitivity by acquiring images at large propaga-tion distances z from the pupil plane. In such animage, there is a spatial frequency f =

√

2/(zλ)for which z is the Talbot propagation distance atwhich the intensity modulation vanishes. It maytherefore be important to measure the pupil in-tensity at two different values of z to ensure thatevery spatial frequency within the control range ofthe AO system produces an intensity signal in theWFS.

2.5. Detector sampling

The effect of detector sampling for nlCWFS hasnot been numerically quantified in this work, butphysical considerations suggest it is driven by tworequirements:

1. The non-linear dual stroke curvature achievesits high sensitivity for low order aberrationsby “tracking” λ/d wide speckles in the defo-cused pupil images. The sampling thereforeneeds to be sufficiently fine to image thesespeckles with 2 pixels per λ/D for maxi-mum sensitivity. The linear number of pix-els across the pupil should therefore be atleast N = 2D2/(λz) where z is the defocusdistance and D is the telescope diameter.With D = 8m, λ = 0.8µm and z = 2000km,we find that the pupil should be 80 pixelsacross. Coarser sampling can be achieved bypushing the defocus altitudes higher, withlittle or no loss in performance since the re-construction algorithm can accurately takeinto account non-linear effects.

2. As the defocus distance is increased, Fresneldiffraction “blurrs” the pupil image, whichbecomes larger than the geometric pupil.This effect is visible in fig. 2. This is es-pecially important for the highest spatialfrequencies which the WFS must measure,since high spatial frequencies will diffractat larger distances away from the geomet-ric pupil. The “blurred” pupil size becomesDblurred = D + 2zα where α is the greaterof two values: α1, the angular “control ra-dius” of the AO system at the sensing wave-length, or α2, half the angular size of thePSF at the wavefront sensing wavelength. Inclosed loop, α2 is usually much smaller thanthe natural seeing, and α2 is usually smallerthan α1. With D = 8m, z = 2000km,λ = 0.8µm, and α = 0.2′′= 10 λ/d (ap-proximately 300 modes sensed), the detectorneeds to extend almost 2 m in every direc-tion around the edge of the 8 m geometricpupil size.

When accounting for both effects, the linear num-ber of pixels across the detector should be:

N =2D(D + 2zα)

λz. (1)

This equation clearly shows that smaller values ofz require more pixels. To minimize the numberof pixels read, it is best to chose z equal to orgreater than z0 = D/(2α), which is the propaga-tion distance for which the “blurred” pupil size istwice the geometric pupil size. For D = 8m andα = 0.2′′, z0 = 4000km. For z = z0, N = N0 =4Dα/λ, and for z >> z0, N = N∞ = N0/2.

With z >> z0, the total number of pix-els required per conjugation plane is N2

∞π/4 =

πD2α2/λ2, which is equal to the number of λ/D-wide speckles within the control radius α of theAO system. Since each of these focal plane speckescorresponds to two degrees of freedom (amplitudeand phase), a pair of defocused pupil images con-tains as many pixels as there are modes sensed bythe nlCWFS. With two such pairs of defocusedimages read, and if z > z0, the nlCWFS thereforerequires at least 2 pixels to be read per WFS mode.Compared to a Shack-Hartmann WFS with 2x2pixels cells or a pyramid WFS designed to sensethe same total number of modes, the non-lineardual stroke curvature WFS is therefore expected

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to require as many or slightly more pixels to beread.

Equation 1 has not been verified by numericalsimulations, and the non-linear iterative nature ofthe reconstruction may produce a more compli-cated picture than the one presented in this sec-tion. Both the required detector sampling and therequired number of conjugation planes are impor-tant drivers for system cost and complexity, andwill need to be explored in the future by compar-ing closed loop numerical simulations for severalcombinations of defocus distances, pixel scale foreach plane, and detector size for each plane. Re-construction algorithms should also be chosen tomitigate coarse detector sampling or limited num-ber of conjugation planes if possible.

3. Wavefront reconstruction algorithm

In this section, an iterative algorithm whichsuccessfully reconstructs the wavefront from thefour defocused pupil images is identified. Whilemore efficient algorithms may exist for this prob-lem, this algorithm is then used in the rest of thispaper to estimate the nlCWFS performance.

3.1. Phase retrieval

The wavefront may be reconstructed from sev-eral defocused pupil images through a phase di-versity algorithm (Gonsalves & Chidlaw 1979). AGershberg-Saxon (Gerchberg & Saxton 1972) al-gorithm is chosen in this work for simplicity andflexibility, but many other options and variationson this algorithm are available (Fienup 1982), asmuch work has been devoted to phase retrievalalgorithms in the last few decades. Similar phaseretrieval algorithms have been successfully used tomeasure optical aberrations in the Hubble SpaceTelescope (Fienup et al. 1993; Roddier & Roddier1993; Krist & Burrows 1995). The algorithm usedfor this work is illustrated in figure 3.

Using this algorithm, nearly optimal closedloop performance has been obtained in simulationswith as few as 10 GS loops by (1) fast forwardingthe convergence thanks to the fact that in con-secutive GS iterations, the solution moves in thesame direction, and (2) running a few lower res-olution GS iterations on binned data to rapidlyapproach the solution prior the full size slowerGS iterations. Essentially all of the computation

time is taken by the full-size FFTs in the propa-gations, while the phase unwrapping described inthe next section is very quick, as it is both lin-ear and non-iterative (see §3.2). Execution speedis a steep function of sampling of the defocusedpupil images. Eight complex 64x64 pixels 2DFFTs are required per GS iteration (assuming a45 pixel diameter pupil). On a single modernCPU, each FFT takes 0.088 ms (number adoptedfrom http://www.fftw.org/speed/ for a 3 GHz In-tel Core Duo CPU), corresponding to 7ms totalFFT time. On a single CPU, the iterative nl-CWFS discussed in this section can therefore runat ≈100 Hz with current hardware.

Other approaches to this non-linear problem ex-ist, but have not been tested in this work. vanDam & Lane (2002c) have suggested a geometry-based algorithm to derive wavefront aberrationsfrom defocused image. The reconstruction canalso be linearized in a ≈ 1rad domain around anychosen wavefront, and it should therefore be pos-sible to effectively combine a coarse non-linear al-gorithm with a fine resolution linear algorithm.

3.2. Phase unwrapping and filtering

The output of the GS iterative algorithm isa phase map of the pupil, which then needs tobe unwrapped and filtered before commands aresent to the DM. Unwrapping may not be neces-sary if the closed loop performance is such thatthe peak-to-valley phase is less than 2π. Figure3 shows how unwrapping and filtering (in this ex-ample, projection on a set of Zernike polynomial)can be done together in one step. The phase slopeis insensitive to phase wrapping effects, as it ishere computed as difference between phase of ad-jacent pixels, and any value above π or below −πis brought back in the −π to π interval by addingor subtraction 2π. For each of the two axis (xand y) the resulting phase slope is decomposedas a linear sum of precomputed phase slopes (oneper Zernike polynomials). The coefficients of thisdecomposition are the Zernike coefficients of thewavefront, which can therefore simultaneously beunwrapped and filtered by being reconstructed asa sum of Zernike polynomials. If the output ofthe GS algorithm is sufficiently good, the differ-ence between the wrapped pupil phase and thefiltered/unwrapped pupil phase (this difference issmall enough to be free of phase wrapping effects)

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directions

Zernike decompositionof pupil phase

Zernike Polynomialsphase slopes in X and Y(pre−computed)

Impose Intensity from+dz1 WFS image

Impose Intensity from−dz1 WFS image

Impose Intensity from+dz2 WFS image

Impose Intensity from−dz2 WFS image

Reconstructed Pupil phase Reconstructed pupil OPD

Residual OPD (93 nm RMS)+dz1

WFSoptics

Propagation to −dz2

Propagation to +dz2

Propagation to −dz1

500 ph / frame, 2000 ph total

Pupil OPD (457 nm RMS)

WFS images @ 0.65 micron

Propagation to +dz1

Propagation to pupil planePhase Unwrapping & filtering

−dz1

−dz2+dz2

Impose pupilshape

Gerchberg−Saxton phase retrieval loop

Compute phaseslopes in X and Y

Fig. 3.— Proposed wavefront reconstruction algorithm for non-linear curvature wavefront sensing.

may be added to the final unwrapped phase mapto get back high spatial frequencies removed by thefiltering step. If however the WFS frames are verynoisy, some filtering may also be included withinthe GS loops to force the GS algorithm to producea smooth phase map, which will mitigate problemsassociated with phase unwrapping errors on noisywavefront data. The final step of the wavefrontestimation is to appropriately weigh the output ofour wavefront reconstruction algorithm with priorknowledge of the wavefront and knowledge of thestatistical properties of the wavefront (Gendron &Lena 1994; Irwan & Lane 1998) and its tempo-ral evolution. This optimization problem has notbeen explored in this paper, and is not specific tothe nlCWFS wavefront sensor (although the exact“tuning” of this optimization is wavefront sensor-dependent).

4. Performance: Numerical Simulations

In this section, numerical simulations are per-formed to verify and evaluate the nlCWFS’s per-formance in a closed loop adaptive optics system.In §4.1, reconstruction of a single wavefront istested using the algorithm shown in Fig. 3. Closed

loop simulations of nlCWFS and SH based AO sys-tems are presented in §4.2 to illustrate the sensitiv-ity gain offered by nlCWFS. Chromaticity effectsare investigated in §4.3 with and without the op-tical chromaticity compensation proposed in thiswork. In §4.4, the flux limit at which nlCWFSstarts to loose the ability to see individual speck-les is probed through numerical simulations. Fi-nally, in §4.5, closed loop simulations are shownfor sparse apertures.

4.1. Reconstruction of a single aberrated

wavefront

Figure 4 illustrates the performance of non-linear CWFS. For this simple simulation, the ini-tial wavefront aberrations were 609nm RMS acrossthe pupil. This level of aberration is, for a 8mtelescope, intermediate between what is expectedto be encountered in open loop (typically 2000nmRMS) and in closed-loop (around 100nm RMS fora bright guide star). The number of photons avail-able for wavefront sensing was chosen to be 2e4,which corresponds to approximately 0.5 ms inte-gration time on a mV = 10 source in a 0.5µm widebandwidth and a 20% system efficiency (prod-

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(b)

(d)

(e)

(f) (h)

(g)(a)

(c)

Fig. 4.— Wavefront reconstruction using the algorithm shown in fig. 3. Four noisy defocused pupil images(images (a), (b), (c) and (d)) are acquired to transform the pupil phase aberrations (e) into intensity signals.The input pupil phase is 609 nm RMS, yielding the PSF (f) before correction. After correction, the residualpupil phase aberration (g) is 34.4 nm RMS, allowing high Strehl ratio imaging (h). All images in this figurewere obtained at 0.65 µm. The total number of photons available for wavefront sensing in 2e4.

uct of optics throughput and detector quantumefficiency). This example is therefore very rep-resentative of situations commonly encounteredin ground-based Adaptive Optics when observingmoderately bright targets.

Results shown in figure 4 demonstrate the nl-CWFS’s good sensitivity: with only 2e4 photons(photon noise is very pronounced in the WFSframes), the wavefront is reconstructed to a 34nmRMS accuracy. The PSF obtained is thereforediffraction limited in the visible. In the next partof this paper, simulations are performed in condi-tions closer to a real AO system: in §4.3 the effectof chromaticity is evaluated, and in §4.2 and §4.4detailed closed-loop simulations are used to verifyperformance.

4.2. Closed-loop simulation: performance

and direct comparison with SHWFS

Figure 5 compares closed loop AO performanceobtained with SHWFS and nlCWFS for a totalflux of 3x106 ph/s on a 8-m telescope (magnitude13 with 20% efficiency and a 0.5 µm wide band-

pass). Parameters adopted for this simulation aregiven in Table 1. Both the WFS and imagingwavelength are 0.85 µm. The WFS detector arrayis considered perfect (no readout noise, no darkcurrent). For the SH systems, each subaperturePSF is imaged on 64x64 pixels, and the DM isconsidered ideal, correcting all spatial frequenciesmeasured. Four SHWFS configurations were sim-ulated with D/d varying from 60 to 9 (D = tele-scope diameter, d = subaperture size). For thenlCWFS simulation, the detector acquires simul-taneously the four defocused pupil images with 90pixels across the pupil in each image. The DM isassumed to perfectly correct all modes up to 16cycles per aperture (CPA), and provides no cor-rection for higher spatial frequencies.

The simulations code was written in C lan-guage, and used the fftw library (Frigo & Johnson2005) for fast Fourier transforms. In each case, AOloop frequency was optimized to minimize resid-ual wavefront aberration (RMS phase error overthe pupil): this optimization was done by run-ning the loop at different frequencies in 20Hz in-

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Table 1

Default parameters used for simulations

nlCWFS SHWFS

Telescope diameter 8mSeeing 0.6′′FWHM at 0.5 µmWind speed 10 m.s−1

WFS wavelength 0.85 µmWFS number of subapertures not applicable from 9 to 60 subapertures across pupilWFS defocus distances -3500 km, -2500 km, 2500 km, 3500 km not applicableWFS detector sampling 90 pixels across pupil 64x64 pix. per subaper. (pix. size ≪ λ/d)WFS detector readout noise 0AO loop gain 1.0AO loop modal filtering none (gain = 1 for all modes within control range)Spatial frequency control range up to 16 cycles per aperture up to CPA = D/3dEdge subapertures not applicable active if > 70 % illuminated

crements and selecting the optimal solution. Fig-ure 5 illustrates the effect of noise propagation ina SHWFS: with a large number of subapertures(D/d = 60), sensing accuracy for low order aber-rations is poor, and the overall wavefront RMSerror is large even though a high number of modesis corrected. With poor wavefront sensing sensi-tivity, the optimal loop frequency is low (140 Hzfor D/d=60) to balance time lag and photon noise.With fewer larger subapertures (D/d = 36), cor-rection is better for low spatial frequencies, andresidual wavefront error drops even though highspatial frequencies are not corrected, as can beseen from the light just outside the darker squareindicating the control region of the AO system.Further reducing the number of subaperture how-ever fails to significantly improve correction of lowspatial frequencies as the SHWFS sensitivity isnow dominated by seeing size (the subaperturesare larger than r0). For a given guide star bright-ness, there is an optimal number (higher numberfor brighter stars) of subapertures in a SHWFS forwhich sensitivity loss on low order aberrations isbalanced against number of modes measured.

Thanks the nlCWFS’s high sensitivity acrossthe full range of spatial frequencies measured, thePSF halo is much fainter than could be achievedwith any of the SHWFS configurations tested.This improved sensitivity also allows the nlCWFS-based AO system to run faster (260 Hz). Increas-ing the number of wavefront modes measured doesnot lower performance for low order aberrations ina nlCWFS.

4.3. Chromaticity

As described in §2.3, the proposed design nl-CWFS uses chromatic lenses to improve the wave-front sensor’s ability to work in broadband light.This scheme does compensate for chromaticity in-duced by diffraction propagation, but will blur thedefocused pupil images in the geometrical opticsregime where aberrations are large. For example,a large amount of tip/tilt α will shift defocusedpupil images by αdz independently of wavelength.The chromatic lenses proposed in §2.3 will blurthe images since dz is now a function of λ. It istherefore important to verify that the use of chro-matic lenses does not prevent the loop from ini-tially closing when aberrations are large, and thatthe correction remains stable in time.

Simulation results shown in Figure 6 confirmthat the chromatic lenses scheme allows the nl-CWFS to both close the loop and maintain agood correction in broadband light with little lossof performance compared to the monochromaticcase. Simulations shown in Figure 6 use the sameparameters as used in the nlCWFS result shownin Figure 5, except for dz1, which was changedfrom 2500 km to 300 km. Reducing dz1 does helpwith loop stability in polychromatic light, at verylittle cost in performance. Without the chromaticlenses, the loop would not have been stable in a20% wide band (Figure 6, bottom).

These results are very encouraging, as they wereobtained by using a wavefront reconstruction al-gorithm which assumes the nlCWFS is working

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nlC, limit = 16 CPA

LOOP OFF

SH, D/d = 18

SH, D/d = 60 SH, D/d = 36

SH, D/d = 9

1537 nm RMS

195 nm RMS

227 nm RMS

315 nm RMS

183 nm RMS

101 nm RMS

Loop frequency = 160 HzLoop frequency = 140 Hz

Loop frequency = 180 Hz Loop frequency = 180 Hz Loop frequency = 260 Hz

Fig. 5.— Simulated 10 second exposures closed loop PSFs delivered by AO systems with SHWFS andnlCWFS.

in monochromatic light. It therefore appears thatpolychromatic operation of a nlCWFS requires noadditional computing power.

Broadband use of the nlCWFS also requires anatmospheric dispersion compensator (ADC) aheadof the nlCWFS. For nlCWFS to operate at full ef-ficiency, chromaticity introduced by atmosphericdispersion should be accurately compensated tokeep the lateral PSF shift below the diffractionlimit of the telescope across the wavefront sens-ing spectral coverage: even a small angular extentof the source used for wavefront sensing is highlydetrimental.

4.4. Coherence loss and optimal Wave-

front Sensing wavelength

Closed loop simulations using the nlCWFSshow a rapid drop in performance at about m ≈15(Figure 8). This limit is due to a loss of coher-ence in the pupil plane: the number of photonsbecomes too low to track the diffraction-limited

speckles which are essential to the nlCWFSs sen-sitivity. Beyond this limit, the nlCWFS operatesin a similar regime as conventional linear WFSs,and delivers partially corrected PSFs (Figure 7).

The results obtained in figure 8 also show thatfor faint guide stars (approximately 14 < mV < 17in the figure), it is preferable to choose a longerwavefront sensing wavelength, thanks to the re-covery of coherence in the pupil plane. For brightstars, a shorter sensing wavelength is better, sincethe same OPD will correspond to a larger phaseshift, which can be detected more easily. Figure8 does not show this last point because the RMSresidual wavefront error for bright stars is domi-nated by the number of modes corrected in this”low-order” AO system.

4.5. Sparse apertures, spiders and low or-

der aberrations

For both existing and future telescopes, thepupil is usually not a single continuous disk as

10

1e-07

1e-06

10000 100000 1e+06 1e+07

WFS @ 1 micronWFS @ 0.85 micron

WFS @ 0.7 micronLoop OFF

SR(1.6) = 54 %

SR(1.6) = 25 % SR(2.2) = 48 %

SR(2.2) = 72 %

m =

17

m =

16

m =

15

SR(2.2) = 27 %

m =

18

RM

S r

esid

ual w

avef

ront

err

or (

m)

Guide star brightness (ph/s)

Fig. 8.— Simulated performance of a low order nlCWFS-based system (first 200 Zernike modes corrected)as a function of sensing wavelength (0.7, 0.85 and 1.0 µm) and guide star brightness. The stellar magnitudesgiven in this figure assume a 20% efficiency in a 0.5µm wide band. Each point in this figure is a 1-secondaverage of the RMS residual wavefront error. See text for details.

assumed so far in this work. Narrow spider vaneshave a very small effect on most wavefront sensingschemes, but thicker spider vanes can be a moreserious problem, especially if a Shack-Hartmannsensor is used with subaperture size comparableto or smaller than the spider thickness. This sit-uation will likely occur on the next generation oflarge telescopes. This problem is even more se-rious for a large telescope composed a few largesegments, with large gaps between the segments,of which the Giant Magellan Telescope is the per-fect example. Incorporating statistical knowledgeof the phase error can help, but will not workfor large gaps or vibration-induced piston betweensegments. A separate scheme to measure differen-tial piston between segments is often required tosolve this problem.

To illustrate this problem, I now consider a tele-scope with 20cm thick spider vanes, and a Shack-Hartmann wavefront sensor. To improve the AOsystem performance for bright stars, a large num-ber of small subapertures is preferred. With 1msubapertures, the spiders would not be a seriousproblem, but with 20cm subapertures, the sub-apertures along the spiders are either fully darkor partially illuminated. No subaperture sees lightsimultaneously from both sides of the spider andthe WFS is therefore blind to an OPD between thetwo sides of the spider. Even if the subaperturesare twice the spider thickness, the amount of lightwhich is mixed between the two sides of the gap isrelatively small, and OPD measurement betweenthe two sides will be noisy. The ability to mea-sure wavefront across a gap is a function of how

11

Fig. 9.— Wavefront reconstruction with dual stroke non-linear CWFS on a sparse pupil. The pupil amplitude(a) and phase (b) yield the four defocused pupil images shown on the right. The recovered pupil phase (c)and the residual phase error (d) demonstrate that dual stroke non-linear CWFS can simultaneously measureOPD within and across segments. This polychromatic simulation was performed with 2e8 photons in adλ/λ = 0.5 wide band centered at 0.65 µm.

much light is mixed within the WFS between thetwo sides of the gap. Shack-Hartmann sensors,especially with small subapertures, are thereforepoorly suited for this task.

As illustrated in figure 9, with a nlCWFS theimages acquired are sufficiently far from the pupilplane to introduce significant mixing betweenpupil “segments”, therefore enabling high sensi-tivity measurement of OPD between segments. Afundamental problem however persists if the gapsare wide: no monochromatic wavefront sensingscheme can detect OPDs which are multiples ofλ. The solution to this problem (which has notyet been explored) is to intelligently use chro-matic effects: a trial and error iterative algorithmusing the focal plane science image (which, pre-sumably is at a different wavelength) may be usedto “lock” the inter-gap OPD within λ - once thisis achieved, the closed loop nlCWFS-based AOsystem will accurately maintain the correct inter-gap OPD. Alternatively, the broadband defocusedpupil images acquired within the WFS should besufficient to solve for this problem: the “inter-ference” fringes between the segments are highlycontrasted if the inter-gap OPD is close to zero,but much less contrasted if it is close to λ. Us-

ing this as a diagnostic could reveal the absolutevalue of the intergap OPD, not its sign: some it-erative trial-and-error algorithm seems necessaryto initially lock the AO control loop in the correctintergap OPD.

The fundamental gain (over a Shack-Hartmannsensor) offered by non-linear CWFS for sensinginter-gap OPD measurement is merely a demon-stration of its higher sensitivity for low-order aber-rations sensing. An inter-gap OPD is mostly alow-order aberration. In both cases (inter-gapOPD and low order aberration), the key to highsensitivity lies in being able to mix light from rel-atively distant parts of the pupil.

5. Discussion

5.1. Sensitivity comparison with Other

Adaptive Optics Wavefront Sensing

Techniques

Figure 10 shows for different wavefront sens-ing schemes how strong an optical signal is pro-duced when small changes are introduced in thewavefront of a 8 m diameter telescope. The WFSis operating at 0.85 µm, and all phases are mea-sured at this wavelength. The ”strength” of the

12

1

10

100

1000

1 10

(1) Shack-Hartmann, D/d =16(2) Shack-Hartmann, D/d = 32

(3) Pyramid (linear)(4) Curvature (linear)

(5) Dual stroke Curvature (linear)(6) Pyramid, non-linear signal

(7) Curvature, non-linear signal

1

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(1) Shack-Hartmann, D/d =16(2) Shack-Hartmann, D/d = 32

(3) Pyramid (linear)(4) Curvature (linear)

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(7) Curvature, non-linear signal

1

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(1) Shack-Hartmann, D/d =16(2) Shack-Hartmann, D/d = 32

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(7) Curvature, non-linear signal

1

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(3) Pyramid (linear)(4) Curvature (linear)

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(7) Curvature, non-linear signal

12 rad RMS

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Cycles per aperture (CPA) Cycles per aperture (CPA)

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)(7)

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(1)

(5)

(4)

(2)

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(1)

4 rad RMS

0 rad RMS

20 rad RMS

Fig. 10.— Optical sensitivity of several wavefront sensor options. The number of photon (y axis, relative)required to achieve a fixed wavefront sensing SNR is shown as a function of spatial frequency (x-axis).Sensitivities are shown for wavefront quality ranging from perfectly flat (top left) to 20 rad RMS (bottomright).

13

1e-07

1e-06

0 0.1 0.2 0.3 0.4 0.5

Monochromatic20% band

40% band

105.9 nm 107.6 nm

time (s)

wav

efro

nt r

esid

ual (

nm R

MS

)

No chromatic correction

20% band

monochromatic

monochromatic

Fig. 6.— Top: Closed loop 1 second expo-sure PSFs at 0.85 µm with a nlCWFS work-ing in monochromatic and broadband (dλ/λ =40%) light. In both simulations, the total num-ber of photons in the nlCWFS is 3e6 ph/s. Bot-tom: Without chromatic correction within the nl-CWFS (replacing the chromatic lenses by achro-matic lenses in Figure 1), the loop would be un-stable with dλ/λ=0.2.

signal is measured as signal to noise (SNR) fora fixed number of available photons, or, equiva-lently, how many photons are required to reacha fixed SNR. Although only photon noise is con-sidered, the highest sensitivity WFSs will also bemore robust against detector readout noise. TheWFS schemes compared in Figure 10 are:

• Two SHWFSs with different number of sub-apertures (D/d = 16 and D/d = 32), bothusing linear wavefront reconstruction.

• A curvature WFS with a dz = 1000km prop-agation distance on either side of the pupilplane and a linear wavefront reconstruction

• A linear dual-stroke curvature WFS (Guyonet al. 2008) with dz1 = 800km and dz2 =2000km

1"

LOOP OFF m~19.3

m~16.8m~18.03.3e4 ph/s 1e5 ph/s

1e4 ph/s

Fig. 7.— Simulated long exposure 1.6 µm PSFsobtained with a non-linear Dual stroke Curvature-based AO system. The sensing wavelength is 0.85µm for this simulation.

• A linear pyramid wavefront sensor where thepyramid position is fixed (Fixed PyramidWFS, or FPYRWFS).

• A FPYRWFS where non-linear signal istaken into account for wavefront reconstruc-tion sensitivity.

• A curvature WFS where non-linear signal istaken into account, with two defocus dis-tances dz1 = 2500km and dz2 = 3500km(4 pupil images total).

The y-axis shows how many photons are needed toachieve a fixed wavefront sensing signal-to-noiseratio (SNR) compared to the optimal sensitiv-ity wavefront sensor as defined in Guyon (2005).For example, the top left panel (0 rad RMS case)shows that at 1 cycle per aperture (CPA), both theSHWFS with D/d = 32 and the curvature WFSrequire 200 times more photons than an idealWFS to produce an intensity signal with the sameSNR. This flux factor is given as a function of spa-tial frequency (x axis), measured here in CPA. Fig-ure 10 was generated by (1) Computing the inten-sity response in the WFS, (2) Computing how thisintensity response changes when a small aberra-tion is added to the wavefront, and (3) Comparing

14

this change, pixel by pixel, to the photon noise andcombining the resulting SNRs into a single over-all SNR. This computation was done not only fora flat incoming wavefront, but also for aberratedwavefronts, and therefore quantifies optical sensi-tivity beyond the small perturbation linear regimeconsidered in Guyon (2005). A distinction is alsomade between linear schemes (SH, conventionalcurvature, conventional pyramid), where the in-tensity signals from a series of uncorrelated wave-fronts were added, and non-linear schemes, wherea single aberrated wavefront is used. This compu-tation step is necessary to average out non-linearsignals in linear WFSs: for example, a SHWFScannot use the shape and speckles of individualspots to unambiguously measure high order aber-rations within each subaperture; and the centroidsmeasurement error is driven by the overall sizeof the spots instead of the size of speckles withinthese spots

Figure 10 shows why a Curvature WFS in whichnon-linear signal is used appears to be the mostpromising approach. Other existing WFS designscannot as easily be modified to offer similar bene-fits:

• The SHWFS suffers from poor sensitivity atlow spatial frequencies. This poor sensitiv-ity is fundamentally due to the splitting ofthe pupil into subapertures which preventsinterferences between distant points in thepupil.

• The linear fixed pyramid WFS is very sen-sitive at all spatial frequencies in the smallaberration regime, but shows poor sensitiv-ity at low spatial frequencies when the wave-front quality is less than ideal. This orig-inates from both the linear constraint andan optical range limitation due to the factthat the inner parts of the PSF can spenda significant fraction of the time completelyin a single quadrant of the pyramid - in thisconfiguration, the WFS does not produce astrong optical signal for low order aberra-tions. If the non-linear part of the signalis used, sensitivity is improved, but is still,at low spatial frequencies, about an order ofmagnitude poorer than it would be for a flatwavefront.

• The linear curvature WFS lacks sensitivityat low spatial frequencies. The dual-strokescheme offers a modest (factor 2) sensitivityimprovement at low spatial frequencies whenwavefront errors are moderate. As shown inGuyon et al. (2008), in the defocused pupilimages, mixing of light between distant partsof the pupil occurs, but it requires large de-focus distances to be easily seen and it ishighly non-linear. If this non-linear signal istaken into account to derive the WFS sensi-tivity, the dual-stroke CWFS is highly sensi-tive at all spatial frequencies when the wave-front is highly distorted. Curiously, simula-tions show that sensitivity is poorer whenthe wavefront quality is good: in this case,the speckles which carry the non-linear in-formation in the defocused pupil images areweak or absent. In this regime, a staticknown aberration should be added in theWFS to maintain full sensitivity. A 2 λ RMSstatic aberration with a Kolmogorov powerspectral density has been included in Figure10 in the 0 rad and 4 rad cases.

This analysis therefore shows that the non-linear curvature wavefront sensor (nlCWFS) is,among all the approaches listed, the only one tooffers near-ideal optical sensitivity at all spatialfrequencies and all levels of wavefront aberrationswithin the 0 to 20 rad RMS range considered. Thesimulations shown in this section only consideredthe effect of a single wavefront mode at a time, andthe results obtained do not automatically meanthat the nlCWFS (or non-linear pyramid) signalscan unambiguously be used to accurately measurethe wavefront. For example, different wavefrontaberrations could produce identical nlCWFS sig-nals. The existence of a wavefront reconstructionalgorithm which can recover a complex wavefrontmap was shown in §3.

5.2. Comparison with phase diversity

In Phase Diversity (PD), images near the fo-cal plane are acquired to measure wavefront aber-rations. Similarly to nlCWFS, a non-linear re-construction algorithm is used to estimate pupilphase from several images acquired in differentplanes. The position of these planes is usuallygiven in pupil defocus phase for PD while it is

15

given in propagation distance for nlCWFS. Physi-cally, an image of the pupil plane acquired after afree space propagation is equivalent to a defocusedfocal plane image. The two quantities are relatedby the following equation:

dφ =πd2

4λl(2)

where dφ is the pupil defocus term in radian peak-to-valley, d is the telescope diameter and l is thepropagation distance.

In conventional linear curvature wavefront sens-ing, the propagation distance l is about 1000 kmor less, equivalent to at least 10 waves (peak tovalley) of defocus for PD. PD operates at smallerdefocus values, typically 1 to 5 waves, and it hasbeen shown that operating PD with larger defocusvalues reduces its sensitivity (Fienup et al. 1998).

In the nlCWFS example adopted for Figure 10,l is 3500 km, yielding dφ = 2.3 waves (peak tovalley). nlCWFS therefore acquires images whichare physically very similar to PD images, anderases much of the fundamental difference previ-ously identified between PD and curvature wave-front sensing (Fienup et al. 1998). The resultspresented in this paper, and especially Figure 10,are consistent with the sensitivity comparison be-tween PD and curvature previoulsy published inFienup et al. (1998). One key finding of theirstudy is that curvature WFS’s sensitivity is fun-damentally limited by the large defocus value (=small propagation distance in the pupil plane) atwhich it operates (as shown in Figure 5 in Fienupet al. (1998)). This limitation is in fact due tothe linearity constraint in conventional curvature,but is lifted in the nlCWFS scheme which allowsnon-linear signals to be used at large propagationdistances. Figure 5 of Fienup et al. (1998) showsthat optimal sensitivity is reached when the defo-cus is below ≈ 5 waves for a wavefront aberrationincluding the 52 lowest order Zernike polynomials.With higher order aberrations included and a wellsampled detector, the same figure would show thatthe defocus can be increased further (the propaga-tion distance can be reduced) because the acquiredimages would still contain the λ/D-wide specklesnecessary for high sensitivity wavefront sensing.

5.3. nlCWFS performance for Extreme-

AO on bright targets

5.3.1. nlCWFS linearity in small aberration

regime

When wavefront variations are small (less than1 rad), the defocused pupil images acquired bythe nlCWFS become less contrasted and they aredominated by diffraction ringing from the pupil’sedges: a known static aberration should then beadded to the beam in the WFS path to maintainfull sensitivity (see §5.1). If the wavefront aberra-tions around this static wavefront offset are small(

1e-06

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Airy1 kHz WFS sampling2 kHz WFS sampling4 kHz WFS sampling

no readout speed limit

D/d = 50Shack−Hartmann WFS

non linear Curvarure WFS

Fig. 11.— Achievable PSF contrast C2 (y-axis) as a function of angular separation (x-axis) for an AOsystem measuring wavefront aberrations at λ = 0.7µm with 7.8 107 ph.s−1 and acquiring scientific imagesat λi = 1.6µm on a 8-m diameter telescope. The PSF surface brightness is shown for several WFS framereadout frequencies for both a Shack-Hartmann WFS (left) and a non linear Curvature WFS (right).

Figure 11 shows, for a photon rate of 7.8 107

ph.s−1 in the WFS (mV ≈ 7 star observed ina 0.1 µm wide band with no loss in the opticsand an ideal detector), how the PSF halo surfacebrightness C2 varies with both spatial frequencyand WFS sampling time. With a SHWFS withD/d = 50, PSF halo surface brightness is domi-nated by photon noise in the WFS for samplingrates beyond 1 kHz (Figure 11): increasing theWFS sampling rate beyond 1 kHz offers no bene-fit. The nlCWFS’s high sensitivity allows perfor-mance (lower PSF halo surface brightness) withhigher WFS sampling rates. The performance dif-ference is especially large at small angular separa-tion, where a full order of magnitude can be gainedif the WFS can be sufficiently fast.

6. Conclusions

Simulations and analysis presented in this pa-per show that the non linear Curvature WFS tech-nique is potentially able to work close to the fun-damental theoretical sensitivity limit imposed byphoton noise even if the PSF is not diffraction lim-ited at the WFS wavelength. It is also a highlyflexible WFS choice, as it can efficiently sensewavefront on sparse pupils, and performs an ex-plicit wavefront reconstruction, therefore allow-ing open-loop operation and compensation of non-common path errors.

In the preliminary study presented in this pa-per, encouraging results were obtained on interme-diate brightness guide stars with closed loop simu-lations using a relatively simple control algorithm.Further performance gains are to be expected ifboth modal control (optimal weighting of wave-front modes according to their measurement SNRand variance in the atmospheric turbulence) andpredictive control are also included. A key chal-lenge to fully take advantage of this technique isto achieve sufficiently high AO loop frequency, andalgorithms with faster convergence than the itera-tive loop used in this paper should be developed.

The nlCWFS appears especially well suited forExtreme-AO systems aimed at delivering nearlyflat wavefronts to a high contrast imaging cam-era (coronagraph) for direct imaging of exoplanetsand disks. In these systems a nlCWFS can de-liver for low order aberrations on a 8m telescopethe same wavefront measurement accuracy as aSHWFS with ≈ 1000 times fewer photons. ThenlCWFS can take full advantage of the telescope’sdiffraction limit for wavefront sensing, with a sen-sitivity scaling as D4 (combination of a D2 gaindue smaller diffraction limit and a D2 gain due tothe telescope’s collecting power) while more con-ventional WFSs scale only as D2 (telescope’s col-lecting power). This difference is especially largefor the next generation of Extremely Large Tele-scopes (ELTs), where the use of high sensitivity

17

WFS techniques such as the nlCWFS may allowdirect imaging of exoplanets in reflected light.

The author is thankful to Roger Angel, PhilHinz, Michael Hart, Markus Kasper, Marcos vanDam and Christophe Verinaud for valuable com-ments on this manuscript.

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