High Contrast Imaging and Wavefront Control with a PIAA

Coronagraph: Laboratory System Validation

Olivier GuyonNational Astronomical Observatory of Japan, Subaru Telescope, Hilo, HI 96720

guyon@naoj.org

Eugene PluzhnikNASA Ames Research Center

Frantz Martinache, Julien Totems, Shinichiro TanakaNational Astronomical Observatory of Japan, Subaru Telescope, Hilo, HI 96720

Taro MatsuoNASA Jet Propulsion Laboratory, Pasadena, CA 91109

Celia BlainUniversity of Victoria, Victoria, BC, Canada V8W 3P6

Ruslan BelikovNASA Ames Research Center

ABSTRACT

In a Phase-Induced Amplitude Apodization (PIAA) coronagraph, the telescope pupil is geo-metrically remapped in a gaussian-like beam by a set of two highly aspheric optics (mirrors orlenses). Thanks to this largely lossless apodization, the PIAA coronagraph offers a nearly fullthroughput (94.3% for the configuration tested, excluding losses due to optics coatings), preservesthe angular resolution of the telescope, and can deliver high contrast very close to the opticalaxis. A prototype PIAA system including active wavefront control has been assembled in thelaboratory at Subaru Telescope. The system operates in air with monochromatic light. Coherentlight in this system has been suppressed to the dynamic speckle floor imposed by residual turbu-lence and vibrations, at 4.5e-8 contrast from 1.65 λ/D (inner working angle of the coronagraphconfiguration tested) to 4.4 λ/D (outer working angle). In addition, a static non-coherent back-ground, likely due to multiple reflections in the system, is observed at 1.6e-7 contrast. Pointingerrors are controlled at the 1e-3 λ/D level using a dedicated low order wavefront sensor. Theraw contrast achieved already exceeds requirements for a ground-based Extreme Adaptive Opticssystem aimed at direct detection of exoplanets. In a 4 hour long exposure, we have demonstratedcoherent averaging of speckles to the 3e-9 contrast level: this result is particularly encouragingfor ground based Extreme-AO systems seeking long term stability to recover planets much fainterthan the fast boiling speckle halo. This experiment validates technologies and algorithm whichwill be the core of the Subaru Coronagraphic Extreme-AO (SCExAO) system currently underassembly, and serves as a prototype for upcoming higher contrast experiments aimed at validatingPIAA technologies for direct imaging of Earth-like planets from space.

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

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1. Introduction

An imaging system aimed at dection or charac-terization (spectroscopy) of exoplanets must over-come the large contrast beween the planet and itsstar. This is particularly challenging for Earth-like planets, where the contrast is ≈ 10−10 in thevisible and the angular separation is 0.1′′for a sys-tem at 10pc. Many coronagraph concepts haverecently been proposed to overcome this challenge(see review by Guyon et al. (2006)). Among theapproaches suggested, Phase-Induced AmplitudeApodization (PIAA) coronagraphy is particularlyattractive. In a PIAA coronagraph, aspheric op-tics (mirrors or lenses) apodize the telescope beamwith no loss in throughput. A PIAA coronagraphcombines high throughput, small inner workingangle (2 λ/D at 10−10 contrast), low chromaticity(when mirrors are used), full 360 degree discoveryspace, and full 1λ/D angular resolution. This con-cept, orginally formulated by Guyon (2003), hassince been studied in depth in several subsequentpublications (Traub & Vanderbei 2003; Guyon etal. 2005; Vanderbei & Traub 2005; Galicher etal. 2005; Martinache et al. 2006; Vanderbei 2006;Pluzhnik et al. 2006; Guyon et al. 2006; Belikovet al. 2006; Guyon et al. 2009; Lozi et al. 2009),which the reader can refer to for detailed technicalinformation.

In the first laboratory demonstration of thePIAA concept (Galicher et al. 2005), lossless beamapodization was demonstrated, and the field aber-rations introduced by the PIAA optics were con-firmed experimentally. In this first prototype,the PIAA optics unfortunately lacked surface ac-curacy required for high contrast imaging, andsince this experiment did not include active wave-front control, the high contrast imaging potentialof the technique could not be demonstrated. Inthe present paper, we report on results obtainedwith a new PIAA prototype which includes wave-front control. Our prototype combines the mainelements/subsystems envisionned for a successfulPIAA imaging coronagraph instrument, with theexception of corrective optics required to removethe strong off-axis aberrations introduced by thePIAA optics. This last subsystem has been de-signed and built for another testbed, and its lab-oratory performance is reported in another paper(Lozi et al. 2009).

The overall system architecture adopted for ourtest is presented and justified in §2. The design ofthe main components of the coronagraphs (PIAAmirrors, masks) is also described in this section.Wavefront control and calibration are discussed in§3. Laboratory results are presented in §4.

2. Laboratory system architecture

2.1. Coronagraph architecture

The coronagraph architecture adopted is a hy-brid PIAA (Pluzhnik et al. 2006), where beamapodization is shared between the aspheric PIAAmirrors (described in §2.3) and a post-apodizer(described in §2.4). The PIAA mirrors performmost of the apodization, but leave a small amountof excess light at the edge of beam (left at 0.85% ofthe surface brightness at the center of the beam),which is then removed by the apodizer. Thanksto this hybrid approach, the PIAA mirrors aremore easily manufacturable (less aspheric) andthe apodizer tolerances are relaxed (the apodizeris not absorbing light in the bright parts of thebeam). The hybrid design also solves the prob-lem of propagation-induced chromaticity (Vander-bei 2006), which would otherwise limit contrast at≈ 10−7 in a non-hybrid system working in a 20%wide band. The cost in throughput and angularresolution due to the apodizer are small since theapodizer only removes light in the fainter edges ofthe remapped beam.

A high contrast image is formed after theapodizer, where starlight is blocked by the focalplane mask. Since the upstream PIAA optics +apodizer have apodized the beam with little loss intelescope angular resolution, the focal plane maskis small, with a radius ranging from approximately1 λ/D on the sky for a 10−6 contrast goal to ap-proximately 2 λ/D on the sky for a 10−10 contrastgoal. The focal plane mask is also part of the loworder wavefront sensor (Guyon et al. 2009) brieflydescribed in §3.2 which uses starlight reflectedby the focal plane mask for accurate sensing ofpointing errors and defocus.

The optical layout of the laboratory experimentis shown in Figure 1. The light source is a sin-gle mode fiber fed by a HeNe laser (λ = 632.58nm), mounted on a x,y,z stage for control of theinput tip/tilt and focus. The PIAA system (mir-rors PIAA M1 and PIAA M2) creates a converg-

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Focal Plane Mask

l1

l2

l5

f = 200

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Light source

f = 100

Lyot Mask

Science Camera

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l4

l6

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Tip/Tilt mount

x,y,z stage

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x

z

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RemovablePinhole

Fig. 1.— Optical layout of the laboratory PIAA coronagraph system. The grey shaded area shows the rigidPIAA bench on which the two PIAA mirrors are mounted. The light source is at the upper corner of thefigure. The focal plane mask (near the bottom, center) separates light into the imaging channel and thecoronagraphic low order wavefront sensor (CLOWFS) channel.

ing apodized beam. PIAA M2 is chosen as thepupil plane for the system, and lens l1 creates asmall image of the pupil plane onto the apodizer.Lens l2 reimages the pupil plane on the deformablemirror, which is located in a weakly convergingbeam. Lenses l3 and l4 form a focal plane for thefocal plane mask. The reimaging lens l5 is usedto create a pupil plane and a focal plane. A Lyotmask is located in the pupil plane, but can beremotely moved out to allow the science camera(which is nominally in the focal plane) to moveforward and acquire a direct pupil plane image.The focal plane mask reflects some of the lightto the coronagraphic low order wavefront sensor(CLOWFS) camera. A detailed description of thisdevice is given in Guyon et al. (2009). This designwas numerically optimized to satisfy multiple con-straints: size of the pupil on the deformable mir-ror (driven by physical size of the DM), location ofcritical elements on the bench driven by mechan-ical constraints, size of the pupil plane of science

camera detector, plate scale in the focal plane im-age. The same optimization code was also used tocompute offsets in the position of several compo-nents during fine alignment of the system.

2.2. Deformable mirror location

The location of the deformable mirror in aPIAA system can have a large impact on the outerworking angle (OWA) of the system. The OWAis defined by the furthest (from the optical axis)speckles that the DM(s) can cancel in the sciencefocal plane. In non-PIAA coronagraph, the OWAis (N/2)× (λ/D), where N is the number of actu-ators along the diameter of the pupil. In a PIAAsystem, the relationship is more complicated, dueto the “phase slope amplification” factor βa > 1introduced at the center of the beam by the remap-ping (see Figure 2). While in a normal imagingsystem of focal length f , the image of a source atangular separation (on the sky) α from the opticalaxis will be at position x = fα from the focal plane

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center, the offset will be amplified to x = fαβa ina PIAA system. In the outer part of the beam,the phase slope has been multiplied by βr < 1 bythe remapping, explaining why in a off-axis PIAAimage, some of the light stays close to the opticalaxis even if the main lobe of the PSF is far fromthe center of the image. Typical values for thesequantities are βa = 3 and βr = 0.3.

These factors can quantitavely be used to es-timate the OWA for several PIAA+DM systemconfigurations (Figure 3):

• Configuration 1 (DM located after the PIAAoptics, no inverse PIAA). The DM can con-trol speckles up to r = f (N/2)λ/D from thecenter of the science image, correspondingto an OWA equal to α = (1/βa) × (N/2) ×(λ/D) on the sky. Beyond this separation,the DM does not have sufficient actuatordensity at the center of the pupil: if the ac-tuators were mapped to the entrance of thePIAA, they would be too large at the centerof the beam.

• Configuration 2 (DM located before thePIAA optics, no inverse PIAA). If the actua-tors were mapped to the output of the PIAA,we find that the actuator size at the edge ofthe apodized beam is divided by βr < 1 (theactuators are bigger), while they are consid-erably smaller in the center of the apodizedbeam. In the outer part of the beam, theDM sampling can only create a dark hole ofradius r = βr × (N/2)× (λ/D), correspond-ing to OWA = (βr/βa)× (N/2)× (λ/D) onthe sky.

In both configurations 1 and 2, the unaberratedfield of view is limited to r ≈ 5λ/D by the absenceof inverse PIAA optics. There is therefore littleadvantage to increasing the OWA much beyondthis radius. We note that the OWA and the un-aberrated field of view are equal for N ≈ 30 andN ≈ 100 in configurations 1 and 2 respectively.We now explore configurations including inversePIAA optics to provide a wide unaberrated field ofview. In these configurations, the field of view forhigh contrast observations is limited by the OWAof the wavefront control system.

• Configuration 3 (DM ahead of PIAA optics).In this configuration, the inverse PIAA can-

cels the effect of the forward PIAA, and theOWA is equal to what it would be in a non-remapped system: OWA = (N/2) × (λ/D).The only role of the remapping is to createan intermediate step where starlight is effi-ciently removed by the focal plane mask.

• Configuration 4 (DM after PIAA optics).The OWA can be found by remapping theDM geometry in the plane ahead of the for-ward PIAA, where its actuators will be mag-nified by βa at the center of the pupil. TheOWA, defined by the largest actuator in thisplane, is therefore OWA = (1/βa)×(N/2)×(λ/D).

In order to optimize the use of a given num-ber of actuators, the DM(s) should therefore beplaced after the PIAA optics in a PIAA systemwithout inverse PIAA optics (configuration 1), orbefore the PIAA optics if inverse PIAA opticsare included (configuration 3). The configurationadopted in our laboratory demonstration is config-uration 1 (DM after PIAA optics, no inverse PIAAoptics).

2.3. Aspheric PIAA mirror design andfabrication

The geometric remapping is performed by twohighly aspheric mirrors, the PIAA mirrors. Therole of the first mirror is mostly to project on thesecond mirror the desired amplitude profile, whichis gaussian-like with a faint ”plateau” on the out-side of the beam (see Figure 4, left). This PIAAM1 mirror acts as a converging element in the cen-ter (to concentrate more light in the center of thebeam on PIAA M2) while light in the outside isdiluted in a wide area of the beam on PIAA M2.This behavior explains the peculiar aspheric sagshown in Figure 4 on the right. The PIAA M2mirror’s role is to re-collimate light to output abeam which is apodized but free of phase aberra-tions.

PIAA mirrors can be designed by solving a rel-atively simple differential equation when the in-put and output beams are collimated (the equa-tion is given in Guyon (2003), and also in Traub &Vanderbei (2003) in a different form) or when thesystem is on-axis. In our laboratory experiment,the PIAA mirrors are focusing elements and theaspheric remapping shapes are added to off-axis

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MASK

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Wavefront ControlOuter Working Angle (OWA)

ForwardPIAA

DM

InversePIAA

imageScience

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OWA = 1.0 x (N/2) x ( /D)λ

~ 5 /Dλ

~ 5 /Dλ

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Configuration 1

no field aberration

Configuration 3

no field aberration

Configuration 4

OWA = (1/3) x (N/2) x ( /D)λ

OWA = 0.1 x (N/2) x ( /D)λ

OWA = (1/3) x (N/2) x ( /D)λ

ForwardPIAA

DM

imageScience

imageScience

ForwardPIAA

DM

Fig. 3.— Four possible architectures for a PIAA coronagraph with wavefront control. For each configuration,the outer working angle of the wavefront control system and the field of view imposed by remapping aregiven. The PIAA slope amplification factor βa = 3 and slope reduction factor βr = 0.3 are considered here.Configurations shown in gray (configurations 2, 4 and 5) should be avoided (see text for details).

Beam diameter D

wavefront

slopeamplification

Post−PIAAwavefront

slopereduction

Pre−PIAA

Fig. 2.— Slope amplification and reduction fac-tors in a PIAA system. The remapping introducedby a PIAA system amplifies the wavefront slopeat the center of the apodized beam (where mostof the light is located) and reduces the wavefrontslope at the edges of the beam.

parabolas. In this configuration, the PIAA mirrorshapes cannot be derived from a simple differentialequation, and they were designed by an iterativealgorithm:

1. Initialization: The PIAA mirror shapes arecomputed by solving the differential equa-tion for an on-axis system.

2. A constant slope is added to each of thePIAA mirror. If there were no apodiza-tion, the mirrors obtained in step 1 wouldbe on-axis parabolas, and this slope wouldturn them into off-axis parabolas. In thePIAA system, however, adding this slopeonly leads to an approximation to the off-axis PIAA system

3. A 3 dimensional raytracing code is used tocompute the beam phase and amplitude onthe surface of a sphere centered on the out-

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Mirror 1 shape (deviation from off-axis parabola)Mirror 2 shape (deviation from off-axis parabola)

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radius (m)

sag

(m)

Beam radius (1 = edge of the beam = 0.0375m)

surf

ace

brig

htne

ss

Fig. 4.— Left: Beam apodization profile used for the design of the PIAA system. In this hybrid design, theapodization is not complete, and the beam surface brightness at the edge of the beam is left at 0.85% ofthe center surface brightness. This extra light will need to be removed by a conventional apodizer. Right:Apodization radial sag term for PIAA M1 and PIAA M2 in an on-axis configuration. The narrow region atthe edge of PIAA M1 has a strong localized curvature, and is the most challenging feature for manufacturingthe PIAA system optics.

put focus of the system immediately afterreflection on PIAA M2.

4. The difference between the measured anddesired beam amplitude on the sphere isused to update PIAA M1’s shape by lin-ear decomposition of this residual (usingpre-computed residuals obtained by addingZernike polynomials on PIAA M1’s shape).

5. The residual phase error measured on thesphere is compensated by changing PIAAM2’s shape.

6. Return to Step 3 with the new mirror shapes.

This algorithm converges because changing PIAAM2’s shape has little effect on the amplitude pro-file of the beam on the sphere, which is almostentirely a function of PIAA M1’s shape.

The PIAA shapes were computed for a 75mmbeam diameter at the PIAA mirrors, a 1.125m sep-aration from the center of PIAA M1 to the cen-ter of PIAA M2, and a 190mm offset between thePIAA M1 to PIAA M2 centerline and the inputand output of the PIAA system (see Figure 1).In the coordinate system shown in Figure 1, eachmirror shape can be written as:

z(x, y) = OAP (x, y) + sag(r) + ΣjiαiZ

ji (1)

• OAP (x, y) is the off-axis parabola whichwould be the PIAA mirror shapes if noapodization was performed. It is a 1133 mmfocal length OAP with a 190 mm off-axis dis-tance from the center of the optical element.This shape is identical for the two PIAA mir-rors although the orientation is different.

• sag(r) is the apodization radial sag on eachmirror. It is computed for an on-axis system.

• A corrective term is added to account for thefact that the system is off-axis (tends to 0 foran on-axis system), decomposed as a sumof Zernikes polynomials up to radial order7. For both PIAA mirrors, this correction is≈95 nm RMS (excluding tip-tilt).

The most challenging feature of the system is thesmall radius of curvature in the outer part of thebeam on PIAA M1. While the hybrid designadopted mitigates this problem, the radius of cur-vature still reaches a minimum of 155 mm near theedge of the mirror, at 36.1mm from the center ofthe mirror.

The PIAA mirrors were fabricated by AxsysImaging Technologies. The mirror substrates wereinitially diamond turned according to the 3-Dprescriptions described above, and then polished

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against computer generated holograms (CGHs).PIAA M1 and PIAA M2 were then assembled ona rigid aluminum bench, aligned and permanentlyfixed to the bench. The residual system wavefronterror was then reduced to 0.04 waves RMS by fig-uring PIAA M2. Two sets of PIAA mirrors (4mirrors total) were manufactured.

2.4. Post-apodizer and system throughput

In order to ease manufacturing, the PIAA op-tics were designed to perform most, but not all, ofthe beam apodization required for high contrastimaging. A more conventional apodizing schemeis therefore necessary to transform the beam pro-file at the output of the PIAA optics (solid curvein Figure 5) into the desired beam profile (dashedcurve in Figure 5).

The post apodizer was designed in transmis-sion, with a series of narrow opaque rings block-ing light. The position and width of the ringsis optimized to best approximate the ideal con-tinuous apodization profile shown in Figure 5 asthe “Apodizer transmission” curve. Several con-traints were imposed on the design to ensure man-ufacturability: no ring should be less than 0.8µm wide and the gap between consecutive opaquerings should be no less than 5 µm. The result-ing design is composed of 109 opaque rings for atotal apodizer diameter of 3.815 mm (defined bythe outer edge of the last opening between opaquerings). The apodizer was manufactured by lithog-raphy on a transmissive substrate.

The post-apodizer throughput over the 3.815mmdiameter is 96.9 %, but due to the narrow ringsin the apodizer, some of the light transmitted isdiffracted at large angles. The “effective” through-put of the apodizer is 94.3%, and would be equal tothe “raw” throughput if the apodizer were contin-uous instead of binary. The full system through-put can therefore reach 94.3% (excluding lossesdue to coating) provided that the telescope pupilsize on PIAA M1 is adjusted to the apodizer di-ameter. In practice, the telescope pupil shouldhowever be made slightly larger to allow for pupilcentration errors, and in very high contrast appli-cations (space coronagraphy), to mitigate possibleedge “ringing” effects due to Fresnel propagation.

The apodizer throughput was mesured by in-serting a pinhole in the PIAA output focus and

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DesignMeasured

squa

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oot o

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am s

urfa

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tnes

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Beam radius (1 = edge of nominal PIAA beam)

Fig. 6.— Laboratory measurement of the apodizerthroughput. The designed and measured radialthroughputs are compared.

moving the science camera in the pupil plane. The1 µm pinhole is used to de-apodize the beam atthe expense of a very low throughput. As shown inFigure 6, the measured apodizer profile agrees verywell with the designed apodization. The resid-ual difference between the two curves is due tothe finite size of the pinhole (the beam before theapodizer is slightly apodized, so the measured pro-file is slightly too bright in the center) and the fi-nite angular resolution of the pupil re-imaging (thesharp edge of the apodizer is blurred).

2.5. Focal plane and Lyot masks

opaque (transmission ~ 1e−5)

reflective

clear

Fig. 7.— Microscope image of the focal planemask

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Beam radius (1 = 37.5 mm on PIAA M2)

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face

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transparent

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ary

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dize

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Fig. 5.— Left: Apodizer design. Right: Microscope image of the outer part of the apodizer (the imagecovers approximately 1mm vertically). The outer edge of the apodizer (last transmissive ring) is at 1.9mmradius. The width of the individual opaque rings is 0.8µm.

The focal plane mask, shown in Figure 7, is usedin transmission, and serves three purposes:

• Blocks the bright central PSF core. The ra-dius of the non-transmissive central zone ofthe focal plane mask defines the inner work-ing angle of the coronagraph, which is 1.65λ/D in our experiment.

• Transmits the science field to the sciencecamera. The shape of the clear zone of thefocal plane mask is chosen to exclude re-gions of the focal plane where the wavefrontcontrol system cannot remove diffractedlight. Since our experiment uses a singledeformable mirror, diffracted light can onlybe controlled over half of the field of view.The clear opening in the mask is therefore“D”-shaped with an outer radius imposedby the DM actuator sampling. A slightlylarger rectangular zone could also have beenadopted, but would have imposed the rota-tion angle of the focal plane mask.

• Reflects some of the starlight to the low or-der wavefront sensor (LOWFS) camera. Areflective annulus, extending from 0.8 λ/D

to 1.65λ/D, is used for this purpose.

The “opaque” zones of the mask are not fullyopaque: their transmition and reflection are re-spectively ≈ 10−5 and ≈ 10%.

A Lyot mask is located in the pupil plane be-tween the focal plane mask and the science focalplane. This mask is designed to block all light out-side the geometrical pupil and transmit all lightwithin the pupil. It therefore has no effect onthe nominal system throughput, and its role is toensure that the scattered light reaching the focalplane camera does not contain light outside thepupil. Although correcting for such light is the-oretically possible if it is coherent, it requires anaccurate model of the coronagraph which can pre-dict how light outside the pupil is affected by DMactuator positions. The Lyot mask was made bydrilling a small hole in an aluminum plate, and itsdiameter is slightly smaller than the pupil size toaccount for alignment tolerances.

Both the focal and Lyot masks are on motorizedstages and can be removed from the beam.

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3. Wavefront Control

3.1. Initial Calibration Loop (without fo-cal plane mask)

Initial calibration is performed using conven-tional phase diversity with no focal plane mask:images are acquired with the science camera insix positions regularly spaced from the focal planeto the pupil plane. An iterative Gerchberg-Saxonalgorithm is used to reconstruct the pupil planecomplex amplitude. As shown in Figure 8 (topleft), the beam quality is intially quite poor, witha large amount of astigmatism. The correspond-ing focal plane image is shown in Figure 8, bottomleft.

Fig. 8.— Pupil phase (top) and focal plane image(bottom) when the deformable mirror is poweredoff (left) and set to its nominal position after cal-ibration (right). A malfunctionning actuator isvisible on the right side of the beam.

The phase diversity routine described above isreapeated N times (N ≈ 10), with a different setof DM voltages applied for each phase diversitymeasurement sequence. The N phase maps ob-tained and the N DM voltage maps used to obtainthem are then used to constrain a model of the DMresponse which consists of seven parameters:

• Geometrical correspondance between theDM and the pupil image: x and y shift,scale, and rotation

• Physical constants describing the DM be-havior: width of the actuator influence func-tion, DM dispacement for 100V applied andpower index α in the displacement to voltagerelationship (displacement ∝ V α).

The result of the DM calibration can then be usedto flatten the wavefront measured and produce asharp focal plane image (Figure 8, right).

This initial calibration is a necessary prelimi-nary step for the high contrast wavefront control,which needs (1) a knowledge of the starting point(typically less than 1 radian error on the wave-front) and (2) a good understanding of how DMcommands affect the pupil plane phase.

3.2. Low order wavefront errors

Low order wavefront errors are measured by re-flecting a portion of the bright starlight maskedby the coronagraph focal plane into a dedictedcamera. A detailed description of this low or-der wavefront sensor (LOWFS) can be found inGuyon et al. (2009). The LOWFS signal is usedto simultaneously drive the deformable tip-tilt andthe source position ahead of the PIAA optics. Akey feature of the LOWFS is the ability to sepa-rate pointing errors (pre-PIAA tip-tilt) from post-PIAA tip-tilt, which is essential to maintain highcontrast: even a small pre-PIAA tip-tilt createsdiffraction rings outside the IWA of the coron-agraph, and pre-PIAA tip-tilt errors cannot becompensated for by post-PIAA tip-tilt.

When the low-order loop is closed, the mea-sured residual pointing error is 10−3λ/D, and istherefore small enough to be negligible in the scat-tered light error budget shown in §4.3. A more de-tailed description of the design, calibration, con-trol algorithm and performance of the LOWFS inour experiment is given in Guyon et al. (2009).

3.3. High order wavefront control loop

Coherent scattered light in the “clear” openingof the focal plane mask is measured by phase diver-sity introduced on the DM. A series of focal planeimages, each acquired with a slightly different DM

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shape

I 0k

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Wavefront stabilityestimate

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upda

te D

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Compute DMδ ik

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i

increment k

........

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compute Xi2

update Mrest tominimize Xi2

of solution

Solve for complex amplitude & coherence

Apply DM shapes & acquire images

acquireimage

Fig. 9.— High order wavefront control loop, showing both the main loop and the system response matrixoptimization loop (light shaded area). The two dark shaded boxes indicate image acquisition, which in the“simulation” mode, can be replaced with a simulated image acquisition using a model of the experiment andDM response.

shape, is used to reconstruct the complex ampli-tude and coherence of the scattered light. Thehigh order wavefront control loop uses a linearizedrepresentation of the system in focal plane com-plex amplitude, as described in the electric fieldconjugation (EFC) approach proposed by Give’onet al. (2007). The wavefront control loop is shownin Figure 9, and is built around the EFC approach.

Prior to starting the loop, a model of the coro-nagraph is used to compute how each actuatormotion affects the complex amplitude in the focalplane. Since this relationship is linear for smalldisplacements, this model is stored as a complexamplitude system response matrix Mresp (shownon the left part of Figure 9) of size n by m, where nis the number of DM actuators (ignoring actuatorsoutside the pupil) and m is the number of pixelsin the high contrast region of the focal plane.

3.3.1. Loop initialization

For each iteration k of the loop, the first step inthe wavefront sensing process is to acquire an im-age Ik

0 with the DM shape set at the best knownposition for high contrast imaging (upper left cor-ner of Figure 9). This first image is then used tochoose the shapes to apply on the DM to optimizemeasurement accuracy and sensitivity.

3.3.2. Compute wavefront sensing DM shapes

For the control loop iteration k, we denoteδDMk

i (u, v), with i = 1...N , the N DM displace-ment (compared to the “reference” position of theDM), and P k

i (x, y) = MrespδDMki (u, v) the corre-

sponding complex amplitude “probes” which areadded to the focal plane by each of these DM of-fets. As a guideline, it is best to choose these DMoffsets so that the additional light (the complexamplitude focal plane “probes”) is approximatelyas bright as the light which needs to be measured.If the probes are too strong, the measurement is

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too sensitive to errors in the DM calibration; ifthey are too weak, the measurement is contam-inated by photon noise, readout noise and smallvariations in the incoherent scattered light. Fi-nally, randomly modulating the probes can miti-gate the effect of calibration errors.

• The first probe P k1

is chosen to satisfy, foreach pixel (x, y):

|P k1 (x, y)|2 = α0 + α1I

k0 (x, y) (2)

where Ik0 (x, y) is the image acquired with

DM shape DMk0. If α1 = 0, this constraint

will force the DM shape to add a uniform co-herent background of contrast α0 in the focalplane, while if α0 = 0 and α1 = 1 it will drivethe DM shape to add a speckle map withthe same intensity as in the Ik

0image. The

phase of this probe is not constrained, and ischosen by an iterative to best satisfy equa-tion 2 using Mresp, with δDMk

1as the free

parameter. Since there is ususally no exactsolution to this equation, the DM commandwhich best satifies this criteria are chosenwith a regularisation parameter to preventexcessive DM displacements.

• The second probe is chosen so that, at eachpoint (x, y) in the focal plane, its amplitudeis identical to the first probe, but its phaseis offset by π/2:

P k2 (x, y) ≈ i P k

1 (x, y) (3)

This π/2 phase offset maximizes the WFSsensitivity if the dominant sources of noisesare photon noise and readout noise (Guyonet al. 2005). We note that if all DM ac-tuators are functionning, there is a perfectsolution to this equation, which can be ob-tained by shifting each spatial frequency ofthe DMk

1map by π/2. Images acquired

with these first two probes, together withthe image Ik

0, would be sufficient to solve for

wavefront errors if light in the focal plane iffully coherent, but at least one more probe isneeded to unambiguously measure light co-herence, and more probes can also providethe redundancy required for implementationof the diagnostic tools described in §3.3.3and §3.3.4.

• Two additional probes can be chosen to beP k

3= −P k

1and P k

4= −P k

2, with exact

solutions δDMk3 = −δDMk

1 and δDMk4 =

−δDMk2

respectively.

• In our laboratory experiment, we chose toalso add 5 more probes with random uncor-related DM shapes of similar amplitude thanthe DM displacements obtained for probes 1to 4 above.

3.3.3. Solving for complex amplitude and coher-

ence

For each pixel (x, y) of the focal plane detec-tor, the complex amplitude A(x, y) of the coherentlight leak and the intensity I(x, y) of the incoher-ent light leak is estimated by solving the followingset of equations:

Iki (x, y) = |A(x, y) + P k

i (x, y)|2 + I(x, y) (4)

for i = 0...N , with P k0 (x, y) = 0 (image acquired

with nominal DM shape).

This set equation has three unknowns and istherefore overconstrained for N > 2. With N = 9adopted in our experiment, we can measure thefit quality by computing the fit χ2, which includeserrors due to:

• Photon and readout noise in the Iki (x, y)

measurements

• Variations in the light leaks during the mea-surement sequence. When solving for thisset of equation, we assume A(x, y) andI(x, y) are static, but if they vary, χ2 willincrease.

• Errors in Mresp, which lead to errors in theestimation of the values of P k

i (x, y). If Mresp

is wrong, then the DM command sent willnot produce the expected P k

i (x, y).

For convenience, we have scaled χ2 in corona-graphic contrast unit. This scaling is performedby measuring the uncorrelated “noise” that wouldneed to be added to A(x, y) between frames to re-produce the observed value of χ2. In this unit, theobserved χ2 is approximately 4.5 10−8 (see §4).

The first contribution (photon + readout noise)has been computed to be a small part of the χ2 ob-served. We observed that increasing α1 (see equa-tion 2) above 1 has little effect on the residual χ2,

11

also independantly suggesting that χ2 is not dueto detector readout noise.

3.3.4. System response matrix optimization

One key output of the wavefront control loopis the estimation of coherent light leaks (whichshould be used to compute the DM correctionto apply for the next iteration), incoherent lightleaks (which the DM can do nothing about) andthe measurement of the wavefront stability dur-ing the measurement sequence. All these quanti-ties depend upon a reliable estimation of Mresp.It is for example possible, if Mresp is wrong, toobtain a low value of the residual coherent lightand think the system has converged to a goodcontrast value, while in fact a significant amountof coherent light remains. This last issue is miti-gated, but not entirely addressed, by continuouslyvarying the probes (as this error is a function ofthe probes chosen, and will average to zero if theprobes are “randomly” chosen). An error in Mresp

would first appear as a large χ2 value for the so-lution of equations 4.

To address this, we have added a Mresp op-timization loop within our control loop. For eachiteration k, the derivative of χ2 with Mresp is com-puted (this is a total of 2 × n × m derivatives, asthe derivative is computed for the real and imag-inary parts of each element of the Mresp matrix).With careful regularization, Mresp is then slightlymodified in order to reduce the sum of the χ2 overall pixels of the image. This algorithm was firsttested on simulated data with an initial Mresp es-timate which was different from the actual Mresp

used in the simulation for computing the images.This test showed that Mresp did converge towardthe “true” Mresp, and that the χ2 value decreasesas a result. Convergence is very slow, due to thelarge number of coefficients in the Mresp matrix,requiring several hundred iterations before a sig-nificant improvement in χ2 is observed.

3.3.5. Correction applied to the DM

The linear electric field conjugation (EFC) al-gorithm (Give’on et al. 2007) is used to cancel co-herent scattered light. This algorithm uses the lin-earized coronagraph model which is also used forthe measurement step described above. The sys-tem response matrix is inverted to build a control

matrix which is multiplied to the coherent light es-timate to yield the DM shape offset to be applied.Regularization schemes proposed by Give’on et al.(2007) were used to improve the loop stability andconvergence speed.

4. Laboratory results

4.1. Alignment and Apodization measure-ment in the pupil plane

The pupil plane apodization map is measuredby placing the science camera in the pupil plane.In our experiment, the pupil is conjugated simul-taneously to the PIAA M2 mirror, the apodizer,and the DM. Alignment is necessary to ensure thatthese three planes are conjugated and that theirrelative scales are correct. The camera positionsfor which conjugation to these planes is achievedare measured and the corresponding pupil scalesare derived from the images. These six numbersare then fed to an optimization routine which com-putes the offsets to be applied to all movable op-tical elements after PIAA M2 to meet the conju-gation and scale requirements. A few iterations ofthis sequence were sufficient to converge.

In the fine alignment step, the apodizer aloneis moved. The pupil image is compared to a sim-ulated pupil image where the relative scale andlateral offset between the PIAA apodization andthe apodizer transmission map are free parame-ters. The values of these three parameters whichgive the smallest residual difference is then used toguide fine alignment of the apodizer. Fine tuningof the scale between the apodizer and the PIAAapodization is possible because the beam at theapodizer is non collimated: apodizer motion alongthe optical axis changes this scale. As shown inFigure 10, the pupil apodization profile measuredafter alignment is in good agreement with the the-oretical profile.

4.2. Imaging with a non-corrected PIAAsystem

Figure 11 shows the system on-axis PSF in“imaging” mode (no coronagraph focal planemask). The on-axis PSF is similar to an Airyfunction without the Airy rings beyond 1.22 λ/D.

While the on-axis image is sharp and exibitshigh contrast, our laboratory system did not in-

12

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Measureddesign

Bea

m s

urfa

ce b

righ

tnes

s

Beam radius (1 = edge of the beam)

Fig. 10.— Measured apodization profile, com-pared the to the beam profile the experiment wasdesigned to deliver. A 2-D image of the beam isalso shown.

clude the ”inverse” PIAA optics necessary to cor-rect for the strong off-axis aberrations introducedby the ”forward” PIAA optics. These inverse op-tics do not need to be of coronagraphic quality,and can be a small set of lenses. A laboratorydemonstration of wide field correction with inversePIAA optics is described in a separate paper (Loziet al. 2009).

Without PIAA optics, the image of an off-axissource rapidly changes shape as the source movesaway from the optical axis. As shown in Figure12, measured off-axis PSFs are in good agreementwith simulations using a remapping of the beamphase (Guyon 2003). With such a strong fieldaberration, measuring the focal plane plate scale ischallenging and its value is a function of the met-ric used. In this paper, we choose to adopt thenon coronagraphic PSF photocenter to measureplate scale. In Figure 11, the 4 λ/D ring thereforeshows where the photocenter of the PSF would beif the source was 4 λ/D from the optical axis ofthe entrance telescope. The bright PSF core atthis separation would be slightly outside the ring,but the fainter assymetric diffraction arcs of theoff-axis PSF would be inside the ring. The samephotocenter metric is used to measure the angularsizes on the focal plane mask given in §2.5 .

2 /D

λ

λ

4 /D

Fig. 11.— On-axis PSF without focal plane mask(log scale indicated at the bottom of the figure).The vertical “bleeding” feature extending down-ward of the PSF core is a camera artefact, andis removed when the focal plane mask is used. Afaint ghost due to the entrance window of the cam-era is visible just beyond 2 λ/D.

4.3. Coronagraphic performance: mea-surements

Coronagraphic performance is measured withthe focal plane and Lyot masks in the beam andboth the LOWFS loop and high order wavefrontcontrol loop closed. Figure 13 (upper left) showsa raw image from the science camera. Most of thelight reaching the camera is due to partial trans-mittance (at the 10−5 level) of the core of the focalplane mask, which produces a central peak in theimage. The clear 1.65λ/D to 4.4λ/D opening inthe focal plane mask is visible below this centralpeak.

Residual light is decomposed in two compo-nents by the wavefront sensing sytem, using theapproach described in §3.3:

• An incoherent component composed of lightwhich does not interfere with light extracteddirectly from the central PSF core. Thiscomponent appears to be mostly stable in

13

Raw ImageAverage Contrast = 2.27e−7

Incoherent portionAverage Contrast = 1.63e−7

Coherent portionAverage Contrast = 4.48e−8

Complex Amplitude averageof coherent portion over 1300loop iterationsAverage Contrast = 3.5e−9

0 1e−7 2e−7 3e−7 4e−7

(Contrast scale x10 in image)

Fig. 13.— Top left: Raw coronagraphic image. A decomposition of the scattered light into a coherentcomponent (top right) and incoherent component (bottom left) shows that the raw contrast is dominatedby incoherent light. The coherent “bias” over a long period of time is shown in the bottom right.

14

Simulation Laboratory Experiment

Fig. 12.— Off-axis PSF without focal plane mask,for a 1.7 λ/D off-axis distance. A cross indicatesthe optical axis in the simulated image (left). Theimage obtained in the laboratory (right) is in verygood agreement with the simulation. Both imagesare shown to the same brightness scale (bottom).

structure and varies from iteration to itera-tion between 1.510−7 and 210−7 in contrast.

• A coherent component which is used to com-pute the DM shape for the next iteration.This component is at approximately 510−8

contrast is well decorrelated on timescalesabove the response time of the wavefrontcontrol loop. The estimation of this com-ponent varies greatly from iteration to itera-tion, and is sometimes below 110−8 contrast.

We also measured the χ2 of the solution, and foundit to correspond to a change in coherent light at the4.510−8 contrast level during the measurement se-quence, which is 10s long. The Mresp optimizationloop described in §3.3.4 gave no noticable improve-ment in χ2, suggesting that the inital calibrationled to a good estimate of Mresp and that the ob-served χ2 is indeed dominated by fluctuations incoherent light at the 4.5 10−8 contrast level.

Visual inspection of the coherent and incoher-ent portions of the light strongly suggests that thisdecomposition was successful:

• The images obtained show very little highspatial frequency noise. Although the re-construction is performed independantly foreach pixel, both the coherent and incoher-ent components show speckles and featurescovering several pixels in these oversampledimages.

• The ghost to the lower left of the opticalaxis was properly analysed by our algorithm.This ghost, also visible in figure 11, is dueto internal reflection in the window of theCCD camera. Because the two surfaces ofthe window are not perfectly parallel, thisghost is “fringed”. Although the fringes arecoherent, they are too thighly spaced for thewavefront control system to remove: theyare the only known feature in the image thatshould be seen in the coherent image, shouldnot be seen in the incoherent image, andcannot be removed by the wavefront controlsystem. As shown in figure 13, our coher-ent/incoherent analysis correctly identifiesthese fringe as coherent, and only a minutefraction of the fringes is present in the in-coherent portion of the image (likely due tosmall motion of the fringes during the mea-surement sequence).

4.4. Analysis

As summarized in table 1, together, these mea-surements show that:

• During the measurement sequence, coherentlight is varying by ≈ 4.510−8 in contrast dueto turbulence or vibrations in the system, asshown by the χ2 analysis.

• The raw contrast is dominated by a verystable incoherent component which is mostlikely a ghost. Given the high number ofair-glass surfaces (twenty), including somewhich are not anti-reflection coated, this re-sult is not too surprising. The variations ob-served in the estimate of the incoherent lightare due to the coherent light variation dur-ing the measurement, which affects the inco-herent estimate. The data obtained is com-patible with a fully static incoherent back-ground, as would be expected from a ghost.

• The coherent light leak estimate is ≈ 510−8,which is at the level expected from the4.510−8 variations shown by the χ2 analysis.The large variation, from iteration to itera-tion, observed in the coherent light residualis due to the turbulence/vibrations in thesystem. We note that the “lucky” iterations

15

Table 1

Contrast budget (Averaged over 1.6 to 4.4 λ/D)

Term Origin Value Calibration

Incoherent light, static optical ghost 1.63 10−7 Incoherent portion of WFS dataCoherent, variable in t < 10s turbulence, vibrations 4 10−8 residual from WFS reconstructionCoherent residual (in 10s) 5 10−8 (typical) Coherent portion of WFS data

7 10−9 (best)Coherent, static (in 4hr) uncorrectable coherent light < 3 10−9 time-averaged coherent light

where the coherent light is estimated be-low 10−8 are artefacts of the time averagingduring the measurement period: even dur-ing these “lucky” periods the coherent lightdid vary by ≈ 4.5 10−8.

• The wavefront control loop successfully re-moves static coherent speckles. Over a 4hrperiod of time, we have measured the staticcoherent speckles to be at the 310−9 contrastlevel. Except for a known ghost on the cam-era window, we could not detect any residualbias in the residual coherent light.

5. Conclusion

The results obtained in this experiment are es-pecially encouraging for ground-based coronagra-phy. The 2 10−7 raw contrast we have achievedalready exceeds by two orders of magnitudes theraw contrast that can be hoped for in even antheoretically ideal Extreme-AO system (Guyonet al. 2005). We note that with a more care-ful optical design and anti-reflection coated op-tics, our experiment could probably have reached5 10−8 raw contrast. More importantly, we havedemonstrated that with the coronagraph + wave-front control architecture adopted in our exper-iment, static speckles can be pushed down verylow (3 10−9) in long exposures. Our system suc-cessfully removed long term correlations in the co-herent speckles, and their averaged level in longexposure was reduced with a 1/

√T law. The com-

bination of a high performance PIAA coronagraphand a focal-plane based wavefront control there-fore appears extremely attractive for ground-basedExtreme-AO. In that regard, our experiment hasbeen a successful validation of the key technolo-gies and control algorithms of the Subaru Corona-

graphic Extreme-AO (SCExAO) system currentlyin assembly. The major differences between theSCExAO PIAA coronagraph and our laboratoryprototype are (1) the need to design and oper-ate a PIAA coronagraph on a centrally obscuredpupil with thick spider vanes and (2) the need forcorrective optics to recover a wide field of view.These two requirements have been validated ina separate laboratory experiment using the finalSCExAO coronagraph optics (Lozi et al. 2009).

Our experiment was limited at the 2 10−7 con-trast by an optical ghost and at the 510−8 contrastby turbulence or vibrations. The PIAA corona-graph could therefore not be tested to the con-trast level required for direct imaging of Earth-like planets from space (approximately 10−10), al-though several key concepts were demonstrated,including simultaneous operation of a low-orderwavefront sensor using starlight in the PSF coreand high-order wavefront sensor using scatteredlight in the science focal plane. New calibrationschemes which will be very useful for high contrastcoronagraphy were also developped and validated,such as the system response matrix optimizationloop, which can slowly run in the “background” tofine-tune the system.

PIAA coronagraph technologies for high con-trast space applications are now being activelydevelopped and tested at NASA Ames ResearchCenter and NASA Jet Propulsion Laboratory. Anew set of PIAA mirrors was recently manufac-tured to higher surface accuracy than the oneswe used, and is being integrated within the HighContrast Imaging Testbed (HCIT) vacuum cham-ber at NASA Jet Propulsion Laboratory. We notethat the HCIT chamber has already demonstratedstability to the 1e-10 contrast with a Lyot-typecoronagraph (Trauger & Traub 2007). The exper-

16

iment described in this paper served as a precur-sor to this new step, which is aimed at reachinghigher contrast (minimum goal of 1e-9) in broad-band light using a two-DM wavefront correction.In parallel to this effort, a highly flexible high sta-bility air testbed at NASA Ames Research Centeris coming online to explore technology and archi-tecture trades for PIAA systems.

This research was conducted with funding fromNASA JPL and the National Astronomical Ob-servatory of Japan. Technical input and advicefrom the members of NASA Jet Propulsion Lab-oratory’s High Contrast Imaging Testbed (HCIT)team, NASA Ames Research Center’s coronaraphteam, and Princeton University’s coronagraphteam have been of considerable help to conductthis work, both for design/simulations and labo-ratory implementation. In addition to providinglaboratory space and infrastructure, Subaru Tele-scope made this research possible through majorcontributions from its technical staff (electronics,hardware, software).

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