Imaging through turbulence with a quadrature …authors.library.caltech.edu/5589/1/KERao05.pdfImaging through turbulence with a quadrature-phase optical interferometer Brian Kern,

Imaging through turbulence with a quadrature-phaseoptical interferometer

Brian Kern, Paul E. Dimotakis, Chris Martin, Daniel B. Lang, and Rachel N. Thessin

We present an improved technique for imaging through turbulence at visible wavelengths using a rotationshearing pupil-plane interferometer, intended for astronomical and terrestrial imaging applications. Whileprevious astronomical rotation shearing interferometers have made only visibility modulus measurements,this interferometer makes four simultaneous measurements on each interferometric baseline, with phasedifferences of ��2 between each measurement, allowing complex visibility measurements (modulus andphase) across the entire input pupil in a single exposure. This technique offers excellent wavefront resolu-tion, allowing operation at visible wavelengths on large apertures, is potentially immune to amplitudefluctuations (scintillation), and may offer superior calibration capabilities to other imaging techniques. Theinterferometer has been tested in the laboratory under weakly aberrating conditions and at PalomarObservatory under ordinary astronomical observing conditions. This research is based partly on observa-tions obtained at the Hale Telescope. © 2005 Optical Society of America

OCIS codes: 120.3180, 010.1080, 350.1260.

1. Introduction

Propagation through turbulence degrades the angu-lar resolution that can be obtained by direct imag-ing.1 Under excellent astronomical conditions, theangular resolution at visible wavelengths is limitedto the neighborhood of 0.5 arc sec, far worse than acomparable diffraction limit, ��D � 0.010 arc sec (for� � 0.5 �m, D � 10 m). Several techniques have beendeveloped to obtain visible and near-IR imagesthrough turbulence, with resolution approaching thediffraction limit. These techniques include adaptiveoptics (AO), long-baseline interferometry, speckle im-aging, nonredundant masking, blind deconvolution,and pupil-plane shearing interferometry.

The past decade has seen extraordinary progress inthe field of visible and near-IR AO on large ground-based telescopes.2 AO systems obtain a referencewave-front phase map using a natural or artificialguide star, and apply phase corrections to the incom-ing light with deformable mirrors or other real-timephase modulators. Deformable mirrors have a finitenumber of actuators, N, which sets an upper limit to

the ratio of the telescope diameter, D, to the phasecoherence length,3 r0, of approximately D�r0 � N1�2.Deformable mirrors with hundreds to approximatelya thousand actuators enable AO systems to operateat good astronomical sites in near-IR wavelengthswith 10 m apertures,4 or visible wavelengths with3.6 m apertures.5

A handful of long-baseline interferometers, whichcombine light from separate apertures, are capable ofproducing images with resolution ��B, where B is thebaseline (separation) between apertures.6 Individualapertures must either have diameters comparable tor0, or else have AO systems operating on each apertureto produce flat wavefronts for interference. The abilityof an interferometer to form images of complicatedfields is determined by the number of baselines overwhich interference is measured, and whether complexvisibilities or only visibility moduli are measured.Long-baseline interferometers tend to be quite com-plex (and expensive), with the complexity increasingfor systems capable of producing high-quality images.Imaging long-baseline interferometers have only re-cently begun to explore angular resolutions that areunobservable using single (large) apertures. The sen-sitivity of such systems depends primarily on the in-dividual aperture sizes and the wavelength.

Alternative passive methods (methods with no ac-tively moving components) for obtaining diffraction-limited images from single apertures have beendeveloped over the past three decades. These meth-

The authors are with the California Institute of Technology,1200 East California Boulevard, Pasadena, California. B. Kern’se-mail address is [email protected].

Received 12 November 2004; revised manuscript received 21May 2005; accepted 24 May 2005.

0003-6935/05/347424-15$15.00/0© 2005 Optical Society of America

7424 APPLIED OPTICS � Vol. 44, No. 34 � 1 December 2005

ods include speckle imaging,7 nonredundant mask-ing,8 blind deconvolution,9 and pupil–plane shearinginterferometry.10–13 In their simplest forms, thesetechniques use short exposures to freeze turbulentaberrations, and obtain images by a variety of post-processing techniques. In speckle imaging, short-exposure focal-plane images retain some informationcontent (greatly attenuated) at high angular frequen-cies, which can be restored by calibrating the Fouriertransforms of the images using concurrent observa-tions of unresolved (pointlike) sources. In nonredun-dant masking, the input pupil is masked so that onlya small number of subapertures transmit light; theirinterference patterns are observed in the focal plane.Light interfering from each pair of subapertures canbe separately identified in Fourier transforms, fromwhich closure phase relationships can be determined.Closure phases are unaffected by turbulent aberra-tions, and allow images to be reconstructed, subject tosome uncertainties regarding symmetries anduniqueness. Blind deconvolution is similar to speckleimaging but requires essentially no calibration. It issubject to large systematic uncertainties and unique-ness issues.

The most efficient form of pupil-plane shearing in-terferometry, when attempting to reconstruct imagesusing a large number of baselines, is rotation shear-ing interferometry. A rotation shearing interferome-ter combines two identical copies of the input pupil,rotated with respect to one another, and records theresulting interferograms (in the pupil plane) on adetector. The rotation shear results in a large numberof independent baselines being observed simulta-neously. The first rotation shearing interferometer tohave been used for astronomy was designed10 andoperated14 in the early 1970’s. This visible-light in-terferometer obtained visibility amplitude measure-ments but no phase information. An interferometer ofthe same design, with the addition of a phase plate tocompensate for polarization-dependent phase shiftsand improve visibility modulus measurements, wasused to obtain high-resolution images of Betel-geuse.11,15 An infrared rotation shearing interferom-eter was constructed that used fringe scanning toattempt to measure the visibility phase from a timeseries of exposures,12 but the magnitude limit wasquite bright and the atmosphere changed on timescales faster than the fringe-scanning exposures. Anadditional pupil-plane interferometer produced pre-liminary fringes but does not appear to have beenpursued further.13 The design and construction ofother rotation shearing interferometers have beendiscussed in the literature, but astronomical obser-vations using these interferometers have not beenpublished. To date, the use of rotation shearing in-terferometers in astronomy has been largely limitedto observations of Betelgeuse.

Rotation shearing interferometry is also used in aFizeau arrangement, where the detector lies in thetelescope focal plane rather than the pupil plane. Thisarrangement may be used to provide nulling of on-axislight, allowing searches for exozodiacal disks and plan-

ets around nearby stars, for example.16 This inter-ferometer arrangement, while optically similar topupil-plane rotation shearing interferometry, operatesin a fundamentally different manner, and provides nocapability for improving image resolution beyond theseeing limit. Focal plane rotation shearing interferom-etry will not be discussed further in this paper.

We have developed a technique of rotation shear-ing interferometry that obtains phase informationusing an instantaneous phase-shifting interferome-ter optical arrangement. This arrangement measuresphase in four separate interferograms, ideally withinstrumental phase shifts of ��2 rad �90°� betweeneach (i.e., phase shifts of 0, ��2, �, and 3��2 rad),making this interferometer a quadrature-phase in-terferometer (QPI). This technique allows instanta-neous determination of the complex visibility (modulusand phase) using only a single exposure that, afterremoving turbulent phase aberrations, allows imagereconstruction at the diffraction limit of the system,subject to signal-to-noise limitations.

Because this is a pupil-plane interferometry tech-nique, the interferograms recorded on the detector aresuperposed, rotated images of the input pupil. Thenumber of pixels on the detector then determines thewave-front spatial resolution. We have developed ahigh-speed charge-coupled device (CCD) camera to op-erate with the QPI, which offers a relatively largenumber of pixels (1024 � 1024) and short read times�5–10 ms full-frame readout). The spatial resolution inthe input pupil is then approximately D�512, where Dis the input pupil diameter. Generally speaking, thisallows operation up to D�r0 � 100, i.e., finer than thescales at which typical AO systems can make phasecorrections (as limited by number of actuators).

This paper describes the design, testing, and pre-liminary operation of the QPI. We present data fromboth laboratory and astronomical observations.

2. Design

A. Interferometric Imaging

The goal of any imaging technique, direct or interfero-metric, is to estimate the two-dimensional objectbrightness map, B��, ��, as a function of the angles �and �. The fundamental connection between inter-ferometry and imaging is the van Cittert–Zernike theorem,17,18 which states that the Fourier transform ofthe object brightness map, B��, ��, is the mutual co-herence function, ��u, v�,

��B��, �� u, v�. (1)

The mutual coherence function is also known as thecomplex visibility,

��u, v� � V�u, v� exp �i�u, v��, (2)

where V�u, v� � �0, 1� is the visibility modulus, and�u, v� is the visibility phase (both V and � are realvalued). Visible-light interferometers measure inten-sities modulated by the complex visibility,

1 December 2005 � Vol. 44, No. 34 � APPLIED OPTICS 7425

I�u, v��I0 � 1 � ��meas�u, v��,

�1 Vmeas�u, v� cos�meas�u, v��. (3)

The measured visibility modulus, Vmeas, and visibilityphase, meas, contain contributions from various in-strumental and aberrating conditions, as describedbelow. Forming an image with an interferometer canthen be described as obtaining �meas over a range of�u, v� points, estimating � from �meas, and obtaining Bthrough Eq. (1). In all cases considered here, it isassumed that the emission from the object is mutu-ally incoherent at the emission location, which is anecessary condition for use of the van Cittert–Zerniketheorem.

The coordinates u and v in the preceding equationsare angular frequencies, which arise in the Fouriertransform from angles � and � of the object bright-ness map B��, ��. Denoting spatial coordinates of theinput pupil by � and �, the interference of light frompoints ��A, �A� and ��B, �B� corresponds to an inter-ference baseline ��B �A, �B �A), and an angularfrequency coordinate

�u, v� � ��B �A��, ��B �A��, (4)

for quasi-monochromatic light at a wavelength �. Theintensity observed for the interference of the twopoints ��A, �A� and ��B, �B� from the input pupil isgiven by Eq. (3), using Eqs. (1), (2), and (4). A sche-matic representation of interference of two points inthe input pupil is shown in Fig. 1.

The object brightness map, B��, ��, is always realvalued, which makes the complex visibility, ��u, v�,Hermitian, so that

��u, v� � �*� u, v�,

V�u, v� � V� u, v�,

�u, v� � � u, v�. (5)

One consequence of this is that ��u, v� is fully definedif it is measured over only half of the (u, v) plane, e.g.,for u � 0.

B. Rotation Shearing Interferometry

Rotation shearing interferometers interfere light bymaking two copies of the input pupil, rotating the twocopies with respect to one another, and recombiningthem to form an interferogram. The QPI is a 180 degrotation shearing interferometer, meaning that thetwo copies of the input pupil are sent into two arms ofthe interferometer that rotate the input pupil copiesby 180 deg with respect to one another before beingrecombined. The interferometer can also introducean instrumental path-length difference (and there-fore an instrumental phase) between the opticalpaths of light through the two arms. A schematicrepresentation of this rotation shearing operation isshown in Fig. 2, and the mirror configuration used to

accomplish the rotation shear is shown in Fig. 3. TheQPI is arranged in a Mach–Zehnder geometry, sothat two full-pupil interferograms are output, as op-posed to a Michelson geometry, in which one inter-ferogram is reflected back toward the input pupil.Sample interferograms are shown in Fig. 4. The twointerferograms differ in the instrumental phase, asdescribed below.

By virtue of the rotation shear, each point in theinput pupil is combined with its diametrically op-posed point. Locations in the interferograms define athird coordinate system, designated by �x, y�, whichcorrespond to detector pixels and are distinct fromthe input pupil coordinates, ��, ��, and the angularfrequencies �u, v�. Because each interferogram isthe superposition of two copies of the input pupil, theinterferogram �x, y� coordinate systems have thesame physical dimensions as the input pupil ��, ��coordinate system, with two rotations applied. Thereare two �x, y� coordinate systems, one for each inter-ferogram, with origins at the center of each full-pupilinterferogram. A point �x, y� in an interferogramrecords the interference between points ��A, �A�� y, x� and ��B, �B� � � y, x�, corresponding to anangular frequency of

�u, v� � �2y��, 2x��. (6)

A comparable �x, y� point exists in each of the twofull-pupil interferograms, each corresponding to in-terference of the same two points in the input pupil,and therefore to the same �u, v� location.

C. Measured Visibility Phase Components

The measured visibility phase contains contributionsfrom the object under study, the instrument, and theturbulent aberrations, i.e.,

Fig. 1. Schematic description of interference between two pointsin the input pupil. Light from two subapertures of the input pupilis superposed (with the superposition shown as Q). The outputintensity is modulated by the complex visibility, evaluated at afrequency defined by the separation between the two subapertures.In the case of a rotation shearing interferometer, the subaperturesare defined by individual pixels on the detector, and all points inthe input pupil are simultaneously interfered pairwise to form atwo-dimensional interferogram.


meas � obj inst turb. (7)

Each of the terms in Eq. (7) can be expressed as afunction of either �x, y� interferogram coordinates or�u, v� angular frequencies.

Some conventions are useful when describing thecoordinate systems associated with QPI. Any quan-tity expressed in terms of interferogram coordinates,�x, y�, is due to the interference of light from twopoints in the input pupil. For example, the turbulentphase term, turb�x, y�, is the contribution of turbulentphase aberrations to the phase difference betweeninterfering points in the input pupil. This contrastswith the typical definition of phase aberrations,�turb��, ��, that are measured relative to a planarreference (such as the input pupil plane). While everyinterfering pair of ��, �� points in the input pupildefines a pair of �x, y� points in each interferogram,and the locations of ��, �� points and the resulting�x, y� interference are related by simple rotations, theindividual contributions of the two ��, �� points toany measured parameter cannot be disentangled bymeasurements in the interferogram. As such, theconvention adopted here is that any parameter ex-

pressed in �x, y� coordinates represents a product ofinterference, which is not uniquely defined in ��, ��coordinates. There is no conceptual difference be-tween expressions in �x, y� interferogram coordinatesand �u, v� angular frequencies (which are rotated andscaled by �), but, by convention, (u, v) coordinates areonly used after removal of instrumental terms, aswhen performing the Fourier transform to estimateB��, �� from ��u, v�.

The instrumental phase term in Eq. (7), inst, is deter-mined by the optical path-length differences betweenthe two arms of the interferometer and by the beam-splitter reflections and transmissions in the two

Fig. 2. Schematic rotation shearing geometry and coordinate sys-tems. Light is incident on the input pupil at the top of the figure.Arms A and B each receive a copy of the input pupil, and rotate thecopies 90 deg in opposite directions. When the two copies of theinput pupil are recombined (through a beam splitter) they form twointerferograms. Instrumental path-length differences, �L, shownhere as a constant with respect to (�, �), effectively retard oradvance the wavefronts in one arm. Vertical displacements in thefigure represent path-length differences. The two output interfero-grams differ by rad (180°) in phase, by virtue of conservation ofenergy, shown here as a ��2 path-length difference in Interfero-gram 2. While two detectors are shown in this figure, the QPI iscurrently configured so that the two interferograms land side byside on the same detector.

Fig. 3. Mirror arrangement giving rotation shear. Each arm of theinterferometer contains three mirrors, arranged to fold the lightpath by 90 deg, and rotate the field of view by 90 deg about thepropagation direction. This figure depicts a fan of light rays, initiallyoriented vertically as it exits the first beam splitter (propagating inthe z direction in the displayed coordinate frame), being folded topropagate in the �x direction and rotated to a horizontal orientationbefore entering the second beam splitter. The other arm of the in-terferometer performs the same rotation, in the opposite sense (�90deg instead of �90 deg). The angles of incidence of the chief ray oneach of the three mirrors are the same.

Fig. 4. Sample laboratory interferograms of a pinhole. Eachpoint in the interferogram shows the interference of two pointsin the input pupil (see Fig. 1). The pinhole has a diametersmaller than the diffraction limit of the interferometer, giving anear-uniform visibility modulus. Corresponding points in thetwo interferograms differ in interferometric phase by rad.The constant phase gradient (manifested as fringes) shows thatthe object is off axis. No turbulent aberrations are present.


arms. The optical path-length difference, �L�x, y�, isa function of position in the interferogram (althoughit is shown as a constant in Fig. 2). For a given �x, y�point, the effect of �L�x, y� is the same in both inter-ferograms.

In addition to �L, the instrumental phase containsa term representing the sequence of reflections andtransmissions as light travels through the beamsplitters, which is shown schematically as a ��2 path-length difference in Interferogram 2 in Fig. 2. Thisdifference in the instrumental phase provides theonly difference between the measured quantities inInterferograms 1 and 2. The instrumental phase dif-ference between corresponding locations in the twofull-pupil interferograms is exactly � rad �180°�. Thisphase difference can be interpreted as a consequenceof conservation of energy (assuming no losses in thebeam splitter) in that the intensity sum of the twofull-pupil interferograms must represent all of thelight entering the interferometer. At an �x, y� pointwhere one interferogram is bright, the same �x, y�point on the other interferogram must be dark. It ismisleading to represent this phase term as a ��2path-length difference in Fig. 2, as this phase shift isachromatic, unlike the instrumental path-length dif-ference �L shown in Fig. 2. The rad phase shift,which is a consequence of the conservation of energy,has no intrinsic relationship to the 180 deg rotationshear, which is a geometric design parameter. The rad phase shift is exact and would exist for any choiceof rotation shearing angle.

The instrumental phase terms, incorporating boththe path-length difference and the number of reflec-tions, are

inst,1(x, y) � 2��L(x, y)��,

inst,2(x, y) � 2��L(x, y)�� , (8)

where �L is the instrumental path-length difference,which is the same for both Interferograms 1 and 2,and the represents the phase shift due to an inequalnumber of reflections and transmissions.

D. Quadrature Phase Interferometry

The instrumental phase, inst, has no intrinsic sym-metry properties with respect to the �x, y� location.This differs from the object visibility phase and theturbulent phase (measured in the interferograms),

both of which are antisymmetric about the centers ofthe interferograms. The antisymmetry of obj andturb arises because they are phase differences ofpoints in the input pupil, which appear with an op-posite sign for diametrically opposed points in theinterferograms.

The simplest quadrature-phase arrangement usesan instrumental path-length difference, �L�x, y�,that is equal to ��8 everywhere. In this arrangement,for each point �x, y� and its diametric opposite,( x, y), we define

0(x, y) obj(x, y) turb(x, y) ��4. (9)

This implies,

meas,1(x, y) � obj(x, y) turb(x, y) ��4� 0(x, y),

meas,1( x, y) � obj(x, y) turb(x, y) ��4� �0(x, y) ��2�,

meas,2(x, y) � obj(x, y) turb(x, y) 5��4� 0(x, y) �,

meas,2( x, y) � obj(x, y) turb(x, y) 5��4� �0(x, y) 3��2�, (10)

where meas,1�x, y� is the measured phase at �x, y� inInterferogram 1, and meas,2�x, y� is the measuredphase at the same �x, y� point in Interferogram 2. Themeasured quantities in the interferograms are fluxes,which are given by Eq. (3) expressed in �x, y� coordi-nates,

I1(x, y) � I0(x, y)�1 Vmeas(x, y) cos�0(x, y)��,

I1( x, y) � I0(x, y)�1 Vmeas(x, y) sin�0(x, y)��,

I2(x, y) � I0(x, y)�1 Vmeas(x, y) cos�0(x, y)��,

I2� x, y� � I0�x, y��1 Vmeas�x, y� sin�0�x, y��,(11)

where I1 is the measured in Interferogram 1 and I2 inInterferogram 2. These measured quantities are usedto estimate I0, 0, and Vmeas, using

I0�x, y� � �I1�x, y� I2�x, y��2

� �I1� x, y� I2� x, y��2,

Vmeas(x, y) ��I1�x, y� I2�x, y��2 �I1� x, y� I2� x, y��2�1�2

2I0�x, y�,

obj(x, y) turb(x, y) � tan 1I1� x, y� I2� x, y�

I1�x, y� I2�x, y� ��4. (12)


By convention, Vmeas is nonnegative and the signs ofthe numerator and denominator of tan 0 unambig-uously determine 0 (modulo 2�). Equations (10)–(12) describe the case when �L�x, y� � ��8 for all�x, y�, which will not be generally true. In general, Eq.(12) takes on a more complicated form to account forthe actual values of �L�x, y�.

The quadrature-phase measurement is enabled bythe point-antisymmetry of obj and turb, and by thefact that the cosine function in Eq. (3), which deter-mines the observed intensity, is even. For a given�x, y� point, a quadrature-phase measurement uti-lizes the corresponding �x, y� points from both inter-ferograms, and the opposing � x, y� points fromboth interferograms. Because obj and turb are anti-symmetric, all of the available information is ob-tained by evaluating Vmeas and obj turb over justhalf of the �x, y� plane. It is then convenient to divideartificially each full-pupil interferogram into twohalves at the y � 0 line. In this sense, it can beconsidered that the interferometer produces four in-terferograms, which provide four measurements toestimate Vmeas and obj turb. A sample set of inter-ferograms is shown in Fig. 5, artificially manipulated(i.e., rotated by 180°) to align the � x, y� axes fory � 0 to the �x, y� axes for y � 0. In �u, v� coordinates,the visibility is estimated for all u � 0, which containsall of the available imaging information.

E. Optical Design

Because the interferograms and the image are re-lated by a Fourier transform, the image resolution isdetermined by the overall size of the interferograms(i.e., by the pupil diameter, D), and the image field ofview is determined by the interferogram resolution.Implicit in this statement is the fact that the 180 degrotation shear creates baselines whose lengths arerelated to interferogram location according to Eq. (6).The resulting image resolution is ��D.

For objects far off center in the field of view, thespacing of the resulting fringes becomes smallenough that the fringes are unresolved. To ensureNyquist sampling of the fringes, the field of viewmust be limited to ��D��Npix�2�, where Npix is thediameter of the interferogram measured in pixels (as-suming the pixel size limits the interferogram reso-lution). As described in Subsection 3.C, Npix istypically either 512 or 256, so the ratio of field of viewto image resolution is either 256 or 128. As an exam-ple, at 700 nm with a 5 m telescope, the diffractionlimit is 0.03 arc sec, and the field of view is typicallymasked off at 3 arc sec—smaller than the more strin-gent Nyquist limit of interferogram resolution. Thefield of view is defined by a circular field stop so thatlight from outside the usable field of view does notappear in the interferograms and add noise. The lim-itation on image resolution and field of view does notdepend on the turbulent conditions.

To facilitate object acquisition and tracking, abeam splitter is placed in the optical path before lightenters the interferometer, to allow a direct image tobe acquired simultaneously with the interferograms.This beam splitter reflects only 10% of the light toform the direct image. The detector (one of the CCDsdescribed below) is square, and the two interfero-grams (which are circular) land side by side on thedetector (see Fig. 4). This interferogram arrangementleaves much of the detector area unused, and so thedirect image is also fed onto the same detector. Thisensures that the direct image and the interferogramsare synchronous, and that they are observed throughidentical turbulent aberrations.

F. Imaging Extended Sources

Interferometric imaging is performed in a fundamen-tally different manner from direct (focal-plane) imag-ing. The visibility modulus, which is directly mea-sured interferometrically, can be interpreted as thedeviation of the imaged scene from that of a pointsource. A point source gives a visibility modulus ofunity everywhere, while a “flat field” (a uniformlyilluminated field of view) gives a delta-function visi-bility modulus (unity at zero baseline, zero at allother baselines). Because the quadrature-phase in-terferometer is a homodyne interferometer operatingat visible electromagnetic frequencies, Poisson noisefrom all light in the field of view is present in themeasurement of the visibility modulus. This meansthat as the number of point sources in the field of viewincreases, the visibility modulus decreases and themeasurement noise increases.

The signal to noise ratio for extended sources cal-culated from measurements of the real part of thecomplex visibility was analyzed by Ribak et al.19 Thatanalysis included consideration of aliasing problemsdue to the incomplete visibility information, in thatthe resulting image formed from only the real part ofthe visibility is forced to be symmetric. The QPI mea-sures the full complex visibility, and so is not subjectto this same difficulty. Apart from that difference, thesignal-to-noise analysis for the QPI follows that of

Fig. 5. Sample laboratory interferograms of a pinhole, dividedinto four interferograms. The upper panels are identical to theupper halves of the interferograms in Fig. 4, while the bottompanels have been rotated 180 deg to emphasize the correlationbetween diametrically opposed points (x, y) in the upper panels and(�x, �y) in the lower panels. Four points that compose aquadrature-phase measurement are identified by crosses. The in-terferometric phases of the corresponding points in the left andright panels differ by rad, while those in the upper and lowerpanels differ by �2 rad.


Ribak et al. In practice, while extended images lead tolower signal-to-noise ratios, a QPI laboratory demon-stration yielded high-quality images of a complicated,extended source (see Subsection 4.A).

G. CCD Cameras

This interferometer must capture short-exposure im-ages ��1 s� to freeze the turbulent variations. In low-light conditions, detector performance is determinedby the read noise and the pixel readout rate. Thesetwo performance criteria are generally at odds withone another, requiring a compromise to be struck fora given observation. Two CCD cameras have beenused with the QPI: a low-speed, relatively low-noisecamera and a high-speed, relatively high-noisecamera.

The high-speed camera was custom designed withmultiple output amplifiers to minimize the per-amplifier bandwidth and reduce the read noise asmuch as possible. It is a front-illuminated design,divided into 32 segments, each with its own outputamplifier, as shown in Fig. 6. This enables framerates up to 1000 frames per second, giving the high-speed CCD the name KFS–CCD, short for kiloframeper second CCD.

Each of the 32 segments consists of 512 64 opti-cally active pixels, with each pixel measuring12 �m � 12 �m, with a full well of 100 000 e . Eachsegment contains an on-chip, three-stage, source-follower amplifier capable of supporting 40 Mpixel�s(1000 frames per second) data rates. The maximumpower dissipated by the 32 on-chip amplifiers is10 W, which is typically cooled by LN2 to operate at�0 °C.

The read noise of the KFS–CCD depends on thereadout rate, temperature, and pixel binning. Theshortest read time at which the performance has beenoptimized is 5 ms �200 fps�, which gives 26 e rmsnoise unbinned and 19 e rms when binned 2 � 2. At10 ms read times �100 fps�, the noise performance is20 e unbinned, 15.5 e binned 2 � 2. These numbersare higher by 2–4 e when uncooled.

The low-noise CCD is an off-the-shelf SITeSI502A back-illuminated, AR-coated 512 � 512pixel CCD. The read noise is 5 e rms when cooled(nearly independent of readout rate), with a readtime of 1.5 s, which will be reduced to 0.5 s with thepurchase of a new analog-to-digital converter (ADC)board. While the low-noise CCD readout is 2 ordersof magnitude slower than the KFS–CCD and thenumber of pixels smaller by a factor of 4, its quan-tum efficiency is higher by a factor of 2 and readnoise lower by a factor of 3, making its per-framesensitivity significantly better. The KFS–CCD,however, is the preferred camera in cases where theaddition of separate frames is efficient and thenumber of frames the KFS–CCD can acquire is suf-ficiently larger than the number the low-noise cam-era can acquire.

3. Effects of Turbulence

A. Phase and Amplitude Variations

Index-of-refraction inhomogeneities due to turbu-lence add phase variations to the incident wave front.As the aberrated wave front propagates, the phasevariations give rise to amplitude variations (scintil-lation). The turbulent phase variations are measuredalong with the object visibility phase as described byEq. (12), while the amplitude fluctuations are mea-sured as variations in I0�u, v�. As can be seen inEq. (12), the measured visibility modulus and phaseare not affected by amplitude fluctuations, to firstorder.

B. Visibility Modulus

In the absence of noise, the measured visibility mod-ulus is the product of several factors,

Vmeas � VbandVtempVspatVampVobj, (13)

where Vband is the visibility loss due to finite band-width (coherence), Vtemp is temporal smearing (expo-sure time), Vspat is spatial resolution (pixellation),Vamp is the second-order visibility loss due to ampli-tude fluctuations (scintillation), and Vobj is the objectvisibility modulus. All of the terms in Eq. (13) arepotentially functions of �u, v�, and all are in the range[0, 1], which means the cumulative effect is always toreduce the measured visibility modulus (althoughmeasurement noise will bias Vmeas to higher values).

Equation (13) establishes several order-of-magnitudeobserving parameters. Vband limits �� 2��rms,

Fig. 6. Architecture of the KFS–CCD. The device is divided into32 equivalent segments, each with a high-bandwidth floating dif-fusion amplifier.


where �� is the spectral bandwidth and rms is therms phase difference between points being interfered.In practice, for a typical D�r0 � 50 in Kolmogorovturbulence, this limits the fractional bandwidth to�1�10 at the longest baselines. Vtemp limits exposuretime to �x�U, where �x is the pixel spacing (or theeffective spatial resolution) and U is the bulk windspeed. Vspat will be discussed in Section 5. Vamp is asecond-order effect due to the fact that when scintil-lation is present, the wavefront amplitudes of a pairof points in the input pupil being interfered are notequal, while the visibility modulus calculation as-sumes they are equal. In conditions of strong scintil-lation, this limitation can be eliminated by separatelyrecording pupil images (without interference), whichcan be done using a small fraction of the incidentlight, as the Fresnel zone sizes are likely to be large,allowing significant binning of the pupil image.

Under typical astronomical seeing conditions, theamplitude variations should be small, even at wave-lengths as short as 400 nm. However, in horizontalpath propagation, or if a high-accuracy determinationof V�u, v� is required, the QPI can record pupil imagesusing a parallel imaging channel (fed by a 90�10beam splitter in front of the interferometer) that isnormally configured as a direct-imaging channel fortarget acquisition and guiding. This mode of opera-tion, recording noninterfered pupil images, allows di-rect calibration of Vamp in each exposure.

C. Visibility Phase

To form an accurate image from the measured data,the object visibility phase, obj, must be distinguishedfrom the turbulent-phase, turb. This problem is sim-ilar to those encountered in speckle imaging, which isnormally solved using a Knox–Thompson algorithm20

or a phase-gradient algorithm.21 We use a phase-gradient technique, which makes use of the correla-tion of turbulent-phase variations across shortdistances.

Turbulent-phase fluctuations are traditionally ex-pressed as deviations from a planar wave front, mea-sured in the input pupil. Phase fluctuations in theinput pupil are expressed as �turb��, ��, which deter-mine turb�x, y� when the corresponding ��, �� pointsare interfered. A Kolmogorov spectrum of phase vari-ations yields a structure function of phase fluctua-tions at the input pupil,3

D�� turb��, �� turb�� , � ��2�

� 6.88��r0�5�3 rad2, (14)

where D�� is the mean square of the phase differ-ence between two points in the input pupil separatedby a distance � � ��2 ��2�1�2, averaged over all��, �� and all ��, ��. The turbulent phase measuredfor a given �x, y� point, turb�x, y�, is the differencebetween the input phase fluctuations at the twopoints in the input pupil being interfered, so that

turb2�x, y�� D��x2 y2�1�2�, (15)

using Eqs. (4) and (14).The phase sum estimated by Eq. (12) is only known

modulo 2�, making it a “circular” quantity. The sam-ple mean of a circular quantity whose variance islarger than approximately 1 rad2 has extremely poorstatistics, making it impractical to simply take themean of obj turb to eliminate turb (which has amean value of zero). Instead, we form gradients bytaking the difference in the phase in neighboring�x, y� locations, so that ��turb�2� is small and can beaveraged out (here, �. . .� denotes a temporal averageover successive exposures).

Defining the x gradient as �x�x, y� � ��x �x, y� �x, y��x, the variance of the mean gra-dient is approximately

��xturb�x, y��2� � 2D��x��x2, (16)

with an equivalent relationship for ��yturb�x, y��2�.Keeping �x2��xturb�2� below 1 rad2 sets a limit on the�x, y� resolution,

�x � r0�5. (17)

Using Palomar as an example, if r0 � 10 cm, themaximum interferogram pixel spacing is 2 cm, re-quiring 250 pixels to span the 5 m diam mirror. Thisallows both interferograms to be recorded, side byside, in a 512 � 256 pixel region, which in turn allowsthe KFS–CCD (with 1024 � 1024 pixels) to bin pixels2 � 2 to reduce the total read noise and shortenthe read time, while retaining sufficient spatialresolution.

As long as the condition on Eq. (17) is satisfied, theturbulent phase terms can be eliminated by averag-ing over a number of individual exposures, withoutfirst unwrapping the phase. This leaves a map of�xobj and �yobj, assuming that the object visibilityphase does not change with time. The visibilityphase, obj, can then be recovered from its gradients,e.g., by constructing �2obj and solving Poisson’sequation. The phase gradient technique can recover aphase map over contiguous regions of the �x, y� plane,but cannot connect the disjoint regions. Depending onthe size of the gaps between contiguous regions, itmay be possible to relax the constraint set in Eq. (17)to connect disjoint regions of the �x, y� plane. Thepractical limits of Eq. (17) have not been explored.

D. Self-calibration

Optimal interferometer alignment, giving �L�x, y�� 8 as described in Eqs. (10)–(12), is not alwayspossible. The surfaces of the mirrors that make upthe interferometer, for instance, are not perfectly flat,so that the interferometer may be properly aligned atsome �x, y� locations but not at others. In addition,path-length differences can only equal ��8 for onevalue of �, so if the interferometer is used at multiple


wavelengths without realignment, it will only beproperly aligned for one of those wavelengths.

If the instrumental phase between the two halvesof each interferogram does not sum to ��2, i.e.,�L�x, y� �L� x, y� � ��4, the equations used toestimate Vmeas and obj turb incorporate the instru-mental phase offset, off �x, y� � �2��L�x, y� �L� x, y��, defined for y � 0. For a given level ofnoise, the estimation of V and obj turb are mostaccurate when off is near ��2, and least accuratewhen off is near 0 or �. Separately from the deter-mination of off, deviations of �L�x, y� from ��8 (fory � 0) act as an additive term to the estimate ofobj turb using Eq. (12). It is useful to separateconceptually the determination of instrumentalphases into the determination of inst,1�x, y� for y� 0, and the determination of off �x, y�.

Both of these terms can be determined using aninternal calibration lamp, that collimates light froma pinhole and sends it through the instrument. Al-ternately, if off �x, y� is known a priori, external cali-bration using observations of a point source (orany object with known structure) can determineinst,1�x, y�.

In the presence of turbulent-phase fluctuationswith amplitudes much larger than 2�, off �x, y� canbe determined, up to a sign, from any data set.This self-calibration makes use of the identity1��2�� 0

2� cos � cos�� d� � cos��2 to say that

This equation, while seemingly complicated, involvesonly the measured intensities of sets of correspondingpoints in the four virtual interferograms, and can beobtained from observations of any object. Moreover,this calibration is independently determined for ev-ery �x, y� point (for y � 0).

Using this self-calibration technique, the quadra-ture-phase measurements can always be estimatedusing instrumental phase terms determined from theobservational data itself. In other words, no separatecalibration data need to be taken to accurately deter-mine off because it can be determined from any dataduring analysis, provided that turb has sufficientlylarge variance and that obj does not change on sim-ilar timescales.

4. Performance

The interferometer has been tested in the laboratoryunder controlled conditions, and at Palomar Obser-vatory under ordinary nighttime observing condi-tions. In the laboratory tests, a complicated object

was imaged under weakly aberrating conditions�D�r0 � few�. In the astronomical tests, simple objectswere imaged under moderately aberrating conditions�D�r0 � 40�. Observations of Vega (an unresolvedpoint source) and of Capella (a binary star) provideduseful data during the astronomical tests.

A. Laboratory Test

In the laboratory tests, a short-arc mercury lampilluminates the test object, the light from which iscollimated, passed through a pupil stop, and fed intothe interferometer, as shown in Fig. 7. A turbulentHe–air jet is then passed through the collimatedbeam, introducing turbulent phase and amplitudevariations. A chopper wheel limits exposure times to200 �s, short enough to freeze the turbulent aberra-tions in each exposure. An optical filter limits thespectral bandpass to a single Hg spectral line at577 nm. Direct images and interferograms are re-corded simultaneously to compare the imaging per-formance of the interferometer to that of directimaging.

Without the turbulent He–air jet, the imaging per-formance of the direct and interferometric images iscomparable, as seen in the top panels of Fig. 8. Thepixel scale in the direct image does not Nyquist-sample the diffraction limit, resulting in some loss ofdirect image resolution. When the turbulent He–airjet is introduced, the interferometric image quality is

only slightly degraded, while the aberrations dramat-ically blur the direct images.

In the configuration used, the diameter, D, of thepupil stop where the He-air jet was Introduced is8 mm. This small D yields D�r0 on the order of 5–10,which does not constitute as demanding an aberrat-ing environment as the interferometer is designed for�D�r0 � 50–100�. Rather than using the full phasegradient technique described in Subsection 3.C, theinterferometric phase correction employed to correctthese low-order aberrations is to subtract the spa-tially averaged phase gradient from the measuredphases. This is equivalent to performing a tip-tiltcorrection.

The quality of the laboratory interferometric im-ages, given that only a first-order correction tech-nique was employed, demonstrates an intrinsicstrength of the rotation shearing interferometer. The180 deg rotation shear ensures that point-symmetricaberrations are eliminated from the measurements.This prevents defocus, spherical aberration, astigma-

cos off �x, y� � 4��I1�x, y� I2�x, y��I1� x, y� I2� x, y��

�I1�x, y� I2�x, y��I1� x, y� I2� x, y��I1�x, y� I2�x, y��2

�I1�x, y� I2�x, y��2��I1� x, y� I2� x, y��2

�I1� x, y� I2� x, y��2� . (18)


tism, etc. from affecting the interferometric images.In an environment where D�r0 is large, this insensi-tivity has relatively little effect, but when D�r0 issmall, this point-symmetric insensitivity, combinedwith a tip-tilt correction, allows good imaging perfor-mance. The aberrated image quality can be furtherimproved over that in Fig. 8 by making full use of thephase gradient technique described in Subsection 3.Crather than only the first-order correction shown.

The complexity of the reconstructed image showsthat while the signal-to-noise ratio of an extendedscene is reduced because of the low visibility modulusand large amount of Poisson noise (see Subsection2.F), complicated objects can still be imaged with thistechnique. The equivalent number of diffraction-limited point sources that combine to make the imageshown in Fig. 8 is approximately 1000. This experi-ment is likely to be made more difficult when appliedto real imaging environments of interest, where D�r0may be quite large, but the signal-to-noise reductiondue to the extended nature of the source does notpreclude the use of this technique on such objects.

B. Astronomical Observations

The quadrature-phase interferometer operated at thecoudé focus of the Palomar 5 m telescope in July 2002under �1 arc sec R-band seeing conditions. Tele-scope flexure was greater than anticipated, and thelarge motions of the input pupil complicated the in-terferometer alignment. Usable data were obtainedon Vega, a bright unresolved, pointlike source, andCapella, a narrow-separation binary star system.These observations were made at a center wave-length of 700 nm with a 40 nm bandpass (this frac-tional bandwidth of 0.06 is conservative relative tothe order-of-magnitude bandwidth limit of �1�10 as

described in Subsection 3.B), in 12.5 ms exposuresfollowed by 12.5 ms readouts.

In the process of realigning the interferometer, therotation shear deviated from 180 deg. As such, aquadrature phase relationship did not exist in thedata taken on Capella. Without a quadrature phaserelationship, visibility modulus measurements weremade but no visibility phase measurements could bemade, and so true images could not be formed ofCapella. This misalignment was corrected for the ob-servations of Vega.

At the time of the observations, the KFS–CCD dataacquisition system filled internal memory buffers,then stopped acquisition and transferred the storeddata to disk. The buffers were capable of storing 400exposures, which at a rate of 40 Hz �12.5 ms expo-sures and 12.5 ms read times) gave 10 s contiguousdatasets (at a 50% duty cycle). In the time since theseobservations were made, this limitation has beenlifted.

1. Vega ObservationsA simple analysis has been performed on a single 10 sdata set of Vega interferograms. A sample interfero-gram is shown in Fig. 9. Phase gradients are aver-aged as described in Subsection 3.C and obj isestimated from the mean gradients. The complex ob-

Fig. 7. Optical layout for the laboratory test. A Hg lamp illumi-nates a mask, and the light is collimated and stopped down. Tur-bulent aberrations are introduced into the collimated light by aHe–air jet. A beam splitter breaks off 10% of the light as a directimage, shown here shaded as the DIRECT component, while therest of the light passes through the interferometer, shown here asthe PUPIL component. The CCD records the direct image and thetwo interferograms simultaneously (near the CCD, the direct im-age and one interferogram overlap as drawn, but are separatedvertically). The mirrors in the interferometer, shown in Fig. 3, arenot drawn here.

Fig. 8. Direct and interferometric laboratory images. The leftpanels are direct images, and the right panels are interferometricimages. Corresponding direct and interferometric images are ac-quired simultaneously through identical conditions. The top pan-els include no turbulent aberrations, and the bottom panels areimaged through a turbulent He–air jet that introduces phase andamplitude aberrations. Each of the bottom panels is an average ofthe same ten exposures. The interferometric image quality is onlyslightly affected by the aberrations, while the aberrated directimages are blurred beyond recognition. The diffraction limit of thesystem is approximately half the width of the line segments mak-ing up the letters.


ject visibility is then Fourier transformed to give animage. This raw image is shown in the middle panelof Fig. 10. This image has a full width at half maxi-mum (FWHM) of 0.5 arc sec, smaller than the directimage FWHM of 1.0 arc sec (seeing limited), but farlarger than the diffraction-limited resolution of0.029 arc sec. The instrumental phase offset, off, wascalculated from the data, ensuring proper quad-rature-phase relationships, but no estimate wasmade of inst,1 to remove instrumental phase termsfrom the quadrature measurements.

Using the same 10 s dataset, a simple estimate canbe made of the image quality due to phase noisealone, by subtracting the estimated object phase ofthe second 5 s of the data set from the object phase ofthe first 5 s, removing any instrumental phase con-tributions. The image formed from this difference isshown in the right-hand panel of Fig. 10, which dis-plays a FWHM of 0.2 arc sec. This is analogous tousing 5 s of data as an external calibration, whileobserving a point source, to correct the instrumentalphase terms in another 5 s data set. This is not a trueimage of Vega, of course, because the object phaseshould be constant over the entire 10 s, and there-fore this represents the image of an object withobj�u, v� � 0. To the extent that Vega is unresolved,this is a valid exercise.

An analysis of the temporal autocorrelation of thephase gradients shows long-lived correlations in turb.The temporal autocorrelation shows a roughly expo-nential decay with time lag, showing a characteristic

�1�e� time scale of 0.65 s. This characteristic timescale, when compared to a characteristic lengthscale of r0 � 14 cm, implies a characteristic speed of0.2 m/s. This low speed is indicative of dome seeing orother turbulent motions not related to the bulk windvelocity outside of the telescope dome (likely to belarger by an order of magnitude).

The variance of the mean phase gradient dependson the number of phase-gradient samples and thecorrelation between those samples. With long-time-scale correlations between phase-gradient samples,the effective number of independent samples is quitesmall, and the resulting mean phase gradient retainsa large variance, as seen by the FWHM of the recon-structed images. As an order of magnitude approxi-mation, 10 s of phase gradient data with a 0.65 scorrelation time scale yields �15 independent sam-ples, reducing the rms phase noise by a factor of �4,comparable to the decrease in FWHM from 1.0 to0.2 arc sec.

The interpretation of these results is that theturbulent-phase aberrations present were dominatedby dome-level atmospheric structures (i.e., slow-moving structures), which necessitates a longerobserving time span to separate obj from turb. Ac-cordingly, longer individual exposure times couldhave been used, increasing the sensitivity with littleloss of visibility.

2. Capella ObservationsCapella is a binary system, with a 100-day orbitalperiod and semimajor axis 0.05 arc sec. This separa-tion is unresolved under ordinary 1 arc sec seeingconditions, but well beyond the 0.03 arc sec diffrac-tion limit of the 5 m telescope at � � 700 nm.

As previously mentioned, we did not obtain quad-rature-phase measurements of Capella. The visibilitymodulus is estimated from 10 s of data. The visibilitymodulus of a binary star system is sinusoidal, and themeasured visibility modulus is fit to a sinusoid todetermine the vector separation of the binary pair.Because only the visibility modulus was used, theorientation is uncertain by 180 deg. The measuredseparation is shown in Fig. 11, overlaid on the orbit asmeasured by long-baseline interferometry.22 The dif-ference between the measured separation and the

Fig. 9. Sample differenced interferogram of Vega. This figure isthe difference between Interferogram 1 and 2 for a single expo-sure, removing scintillation effects. The visibility modulus isessentially uniform across the exposure. The input pupil was notcorrectly aligned with the interferometer center of rotation, lead-ing to an oversized central obscuration and noncircular outerboundary.

Fig. 10. Direct, raw interferometric, and differenced interfero-metric images of Vega. The left panel is the average of 10 s (400frames) of direct images of Vega. The middle panel is a raw inter-ferometric image, from the same 10 s of data as the left panel,formed without estimating the instrumental phase. The rightpanel is formed by calibrating one half of the 10 s dataset using theother half. The FWHM of these images is 1.0, 0.5, and 0.2 arc sec.


expected separation at the time of the observationwas 0.0012 arc sec (excluding the possibility that themeasurements disagree by 180 deg). This disagree-ment is considerably smaller than the semimajor axisof the binary orbit, 0.05 arc sec, and than the diffrac-tion limit of the telescope, 0.03 arc sec.

C. Expected Sensitivities

A simple estimate of the sensitivity of QPI can be madeby requiring an intensity signal-to-noise ratio of unity ineach pixel in each frame. For Palomar, assuming D� 5 m, � � 700 nm, �� 40 nm (this bandwidth ismore conservative than the estimate in Subsection3.B), r0 � 15 cm, and v � 10 m�s, the interferogrampixel scale could be �x � 3 cm, with an exposure timetexp � 15 ms. Using the KFS–CCD (the high-speedcamera), with a detection efficiency of 15% and a CCDread noise of 15.5 e �pix, the point-source sensitivitylimit is R � 5.5. At Keck with D � 10 m and r0� 30 cm, the sensitivity limit is R � 7.8. For terres-trial (ground-to-ground) imaging with D � 0.4 m,r0 � 8 mm, and v � 10 m�s, the point-sourcesensitivity limit is �104 phot cm 2 s 1 Å 1, which iswell below the flux of reflected sunlight ��106 photcm 2 s 1 Å 1 from a source the size of the telescopediffraction limit), but above the flux of scattered sun-light ��102 phot cm 2 s 1 Å 1 from a source the sizeof the telescope diffraction limit).

Using the unity signal-to-noise ratio rule of thumb,near the faint end of the sensitivity limits the low-

noise CCD provides superior performance to that ofthe KFS–CCD, in the sense that individual exposuresretain adequate signal-to-noise ratios. With a detec-tion efficiency of 30% and read noise of 5 e �pix, thePalomar sensitivity limit is R � 7.8 (under the sameconditions described previously), and the Keck sensi-tivity limit is R � 10. For terrestrial imaging, thesensitivity limit is �103 phot cm 2 s 1 Å 1, whichwould be comparable to the limit for scattered sun-light off an extended scene, whose illuminated areacovers �10 diffraction-limited resolution elements (amuch simpler scene than used in the demonstrationin Fig. 8, which imaged �1000 diffraction-limitedresolution elements).

The high measured visibility modulus of the Vegaobservations implies that the exposure time, resolu-tion, and bandwidth can be increased beyond theirrule-of-thumb limits to increase sensitivity at the ex-pense of some additional calibration requirements.

The sensitivities quoted above are highly simpli-fied. The ultimate sensitivity of an observation de-pends on the ability to reduce the variance of themean phase gradient (and mean visibility modulus),which may depend on the atmospheric conditions, asstructures causing phase aberrations may persistover many exposures. Assuming that phase errorsdominate image resolution, image resolution in-creases with observing time. For objects brighterthan the limits above, the KFS–CCD is often pre-ferred because it allows many more exposures in agiven period of time, thereby reducing phase errors.

5. Discussion

Depending on the imaging application, the primarystrengths of quadrature-phase interferometry maybe its high wavefront resolution, passive operation,insensitivity to amplitude fluctuations, operation atshort wavelengths, or ability to perform internal cal-ibration. The interferometric �u, v� coverage obtainedby this technique is nearly complete, allowing forreconstruction of highly detailed, near diffraction-limited images.

The imaging performance of quadrature-phase in-terferometry can be compared to AO and speckle in-terferometry. Compared to AO, QPI is simple,inexpensive, and capable of working at visible wave-lengths (with D�r0 � 50), but is less sensitive andgenerally observes a narrower field of view. Assum-ing a similar wavelength of operation, AO is moresensitive because the AO wavefront sensors are notsubject to the same resolution criterion as QPI forphase-gradient recovery [Eq. (17)]. By using fewerpixels to record the same wavefront area, they incurless read noise per measurement. Once the AO con-trol loop is closed, the corrected image is sent to ascience camera that can take long exposures whereread noise is not a limiting factor. When imaging inthe presence of strong scintillation, however, QPIshould be insensitive to the wavefront amplitude fluc-tuations, while AO would suffer some image degra-

Fig. 11. Binary separation of Capella. The diamonds mark theseparation measured by the QPI, with two diamonds shown be-cause the separation is uncertain by 180 deg. The crosses mark theorigin and the expected binary separation at the time of the ob-servation, with the dotted curve tracing out an entire orbit. Theseparation is measured by fitting the measured visibility modulusfrom 10 s of data to a sinusoid, appropriate to a binary pair. Thedisagreement between the measured and expected binary separa-tions is 0.0012 arc sec.


dation. This potential advantage of QPI has not yetbeen verified.

Unlike AO, QPI does not make direct measure-ments of the turbulent-phase fluctuations at the in-put pupil, �turb��, ��. The turbulent-phase terms thatthe QPI does measure, turb�x, y�, are differences ofphase fluctuations at two points in the input pupil,�turb��, ��. It is not possible with the arrangementdescribed here to uniquely determine �turb��, �� usingQPI. While this precludes direct wavefront measure-ments, the QPI does directly measure the turbulent-phase structure function, D�, so while QPI candirectly determine the statistical properties of theturbulent-phase fluctuations, it cannot determineany individual realizations of ��, ��. The measure-ment of D� using QPI does not require any assump-tions about Kolmogorov turbulence or homogeneityor isotropy, only of translational invariance in the��, �� plane. It is interesting to note that rotationshearing interferometry is inherently unaffected bypoint-symmetric aberrations, such as spherical aber-ration, defocus, astigmatism, etc.

Compared to speckle interferometry, QPI has com-parable sensitivity and simpler calibration require-ments, but is more expensive and optomechanicallymore complex. The practical sensitivity of QPI islikely to be lower than that of speckle, because theQPI optical design involves more reflections, result-ing in a lower instrumental throughput. The calibra-tion requirements of speckle interferometry arehighly dependent on the turbulent conditions, as theyessentially involve measuring the modulation trans-fer function (MTF) degradation due to the turbulentaberrations, and statistically correcting the mea-sured MTF to its unaberrated level. The MTF ofpupil-plane interferometry, on the other hand, is es-sentially unity for all frequencies up to thediffraction-limited frequency �D��.23 QPI suffers aloss of visibility due to finite exposure times, wave-front resolution, and spectral bandwidth, as de-scribed in Eq. (13), but as seen from the observationsof Vega, the measured visibility modulus is still nearunity, requiring little calibration. The QPI instru-mental phase calibration can be done internally andthough the self-calibration of imaging data, althoughthe observations of Vega did not demonstrate thisability.

Previous rotation shearing interferometers wereinefficient primarily in three ways. First, the onlyway to measure visibility phase instantaneously wasto add a large phase gradient (wavefront tilt) to theinterferograms and measure deviations from the gra-dient, which limits the spectral bandwidth and gen-erates systematic uncertainties. If an interferometerdoes not measure phase instantaneously, the mea-surement is confused by phase changes between ex-posures.12 Second, all previous rotation shearinginterferometers used in astronomical observationshave used only half of the incident light, reflectingone interferogram back toward the telescope. As athird, minor point, some rotation shearing inter-ferometers used internal reflections in roof prisms to

generate rotation shear, which introduced a polariza-tion mismatch between the two arms, reducing themeasured visibility modulus unless a corrector platewas introduced in the system. The QPI avoids theselimitations; the first two problems are avoided bydesign, while the third (polarization mismatch givinglow-visibility moduli) is bounded by the raw mea-sured visibility modulus in interferograms such asthose in Fig. 4, which have V � 0.97 in unpolarizedlight.

Alternate modes of operation are possible withQPI. As demonstrated with observations of Capella,QPI can be operated in a modulus-only mode. Whilethe Capella observations did not make use of thequadrature-phase relationships of QPI, a plannedmodulus-only measurement would make quadrature-phase measurements of modulus and phase, butwould relax the phase-gradient resolution limit in Eq.(17) to �x � r0, which relaxes the magnitude limitslisted above by approximately 3.5 magnitudes, sothat the low-noise CCD could observe objects down toapproximately R � 11.3 at Palomar and R � 13.5 atKeck. No attempt would be made to separate obj fromturb, but the visibility modulus would be accuratelymeasured at all baselines in each exposure. Theseexpected sensitivities can be compared to the demon-strated speckle interferometry magnitude limit of V� 12.5 for binary star measurements at the U.S.Naval Observatory 0.66 m refractor.24 While it mayseem inappropriate to compare a sensitivity limit ona 0.66 m telescope to expected sensivities on 5 and10 m telescopes, the expected sensivities have no de-pendence on D, only on r0. Use of large telescopes,such as those at Palomar and Keck, greatly improvesthe diffraction limit for a given �, but does not changethe expected sensitivity limit for a given r0.

A potential upgrade to QPI would allow differentialphase measurements as a new mode of operation.Differential phase measurements are enabled byadding spectral dispersion and reducing the inter-ferogram spatial resolution. The simplest way toachieve this is to add a dispersive grating and a len-slet array. The lenslets reduce the interferogram spa-tial resolution, and the grating spectrally dispersesthe light from each lenslet as it lands on the detector.Quadrature-phase relationships are independentlymeasured for each wavelength, and phase differencesbetween wavelengths can be accurately measured.When observing objects smaller than the diffractionlimit, the differential phase measurements deter-mine the relative displacements of the photocenter atdifferent wavelengths. This could be used to measurephysical scales in broad-line regions of active galacticnuclei or orbits of extrasolar planets with strong ab-sorption lines.

6. Conclusions

The fundamental operation of a quadrature-phaseinterferometer has been demonstrated, to varying de-grees, in laboratory and astronomical tests. Thecapabilities of this technique under the viewing con-ditions of interest, observing complicated objects un-


der astronomical viewing conditions or objects understrongly aberrating conditions, have not yet beendemonstrated.

This research was funded in part by the Air ForceOffice of Scientific Research (AFOSR) Grants No.F49620-94-1-0283, F49620-00-1-0036, F49620-03-1-0102, F49620-98-1-0052, and DoD-DURIP�AFOSRGrant No. F49620-95-1-0199. This material is basedon work supported by the National Science Founda-tion under Grant No. AST9618880. This work wasperformed in part under contract with the Jet Pro-pulsion Laboratory (JPL) funded by NASA throughthe Michelson Fellowship Program. JPL is managedfor NASA by the California Institute of Technology.Based partly on observations obtained at the HaleTelescope, Palomar Observatory, as part of a collab-orative agreement between the California Institute ofTechnology, its divisions Caltech Optical Observato-ries, and the Jet Propulsion Laboratory (operated forNASA), and Cornell University. Bryan Jacoby,Stephen Kaye, and Ryan McLean were invaluable infielding the QPI.

Appendix A: QPI Optical Implementation

The optical design described here for creatingquadrature-phase interferograms is not unique. Theoriginal implementation of the QPI had a more com-plicated optical arrangement.

In the original implementation of the QPI, one mir-ror in one arm of the interferometer was split into twohalves. The division between mirror halves was po-sitioned so that the split between the mirrors ranthrough the center of the interferograms. The instru-mental phase terms were independently adjustablefor each of the two halves. In this configuration, thedivision of the two full-pupil interferograms into fourinterferograms, shown graphically in Fig. 5, wasstraightforward, as the interferograms were alreadysplit in half.

The advantage of the split arrangement is that itallows any instrumental path-length difference �L,to be used on each half of the interferograms. Thesimplest choice for �L�x, y� is 0 on one half, i.e., fory � 0, and ��4 on the other (for y � 0), enablingquadrature-phase measurements. An additional ad-vantage of having independent alignment of the twohalves is that the adjustment resolution of each mir-ror half determines the adjustment resolution ofinst�x, y� inst� x, y�, while in the current config-uration, with just one set of adjustments, the resolu-tion of inst�x, y� inst� x, y� is twice as coarse.

The primary disadvantage of the split arrange-ment is that the split between the mirror halves re-sults in some loss of interferogram coverage, at thelowest angular frequencies. As a result, the recon-structed images are high-pass filtered, and have adifferent appearance from the direct images. A com-parison of the imaging performance of the split ar-rangement and the current arrangement is shown inFig. 12.

We determined that the improved resolution in in-

strumental phase adjustments from using a split mir-ror did not justify its increased complexity and theloss of (u, v) coverage. However, further improve-ments in the optical design are certainly possible.

References1. F. Roddier, “The effects of atmospheric turbulence in optical

astronomy,” in Progress in Optics, Vol. XIX, E. Wolf, ed.(North-Holland, 1981), pp. 281–376.

2. F. Roddier, Adaptive Optics in Astronomy (Cambridge Univer-sity Press, 1999).

3. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).

4. P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase,J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez,and J. Larkin, “First light adaptive optics images from theKeck II telescope: a new era of high angular resolution imag-ery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).

5. L. C. Roberts and C. R. Neyman, “Characterization of theAEOS adaptive optics system,” Pub. Astron. Soc. Pacific 114,1260–1266 (2002).

6. J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F.Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Night-ingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill,P. J. Warner, D. M. A. Wilson, and R. W. Wilson, “The firstimages from an optical aperture synthesis array: Mapping ofCapella with COAST at two epochs,” Astron. Astrophys. 306,L13–L16 (1996).

7. A. Labeyrie, “Attainment of diffraction limited resolution inlarge telescopes by Fourier analyzing speckle patterns in starimages,” Astron. Astrophys. 6, 85–87 (1970).

8. C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, andJ. E. Baldwin, “The first images from optical aperture synthe-sis,” Nature 328, 694–696 (1987).

9. G. R. Ayers and J. C. Dainty, “Iterative blind deconvolutionmethod and its applications,” Opt. Lett. 13, 547–549 (1988).

10. J. B. Breckinridge, “Coherence interferometer and astronom-ical applications,” Appl. Opt. 11, 2996–2998 (1972).

11. C. Roddier and F. Roddier, “High angular resolution observa-tions of Alpha Orionis with a rotation shearing interferome-ter,” Astrophys. J. 270, L23–L26 (1983).

12. J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, and A.Zadrozny, “Pupil plane interferometry in the near infrared I.Methodology of observation and first results,” Astron. Astro-phys. 255, 462–476 (1992).

Fig. 12. Comparison of imaging performance under two QPI con-figurations. The left panel is the image produced by a split mirrorQPI configuration, and the right panel is the image produced bythe current QPI configuration. The split mirror configuration re-sults in a loss of low-frequency information, producing a high-passfiltered image.


13. C. M. de Vos, J. D. Bregman, and U. J. Schwarz, “Pupil planeinterferometry: Some conclusions from SCASIS,” in Very HighAngular Resolution Imaging, J. G. Robertson and W. J. Tango,eds. (Kluwer, 1994), pp. 419–420.

14. J. B. Breckinridge, “Two-dimensional white light coherenceinterferometer,” Appl. Opt. 13, 2760–2762 (1974).

15. F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, andC. Aime. “New observations of Alpha Orionis with a rotationshearing interferometer,” Astrophys. J. 305, L77–L80(1986).

16. K. Wallace, G. Hardy, and E. Serabyn. “Deep and stable in-terferometric nulling of broadband light with implications forobserving planets around nearby stars,” Nature 406, 700–702(2000).

17. M. Born and E. Wolf, Principles of Optics (Cambridge Univer-sity Press, 1999).

18. A. R. Thompson, Interferometry and Synthesis in RadioAstronomy (Wiley, 1986).

19. E. Ribak, C. Roddier, F. Roddier, and J. B. Breckinridge,

“Signal to noise limitations in white light holography,” Appl.Opt. 27, 1183–1186 (1988).

20. K. T. Knox and B. J. Thompson, “Recovery of images fromatmospherically degraded short-exposure photographs,” As-trophys. J. 193, L45–L48 (1974).

21. N. Ageorges, J.-L. Monin, L. Desbat, and E. Tessier, “Phaseand image reconstruction from interferometric imaging: Inte-gration of the phasors,” Astron. Astrophys., Suppl. Ser. 112,163–171 (1995).

22. C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F.Buscher, D. Mozurkewich, N. M. Elias, and R. E. Wilson, “Veryhigh precision orbit of Capella by long baseline interferome-try,” Astron. J. 107, 1859–1867 (1994).

23. F. Roddier, “Interferometric imaging in optical astronomy.”Phy. Rep. 170, 99–166 (1988).

24. B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty,G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, andT. E. Corbin, “Speckle interferometery at the US Naval Ob-servatory VI,” Astron. J. 120, 1120–1132 (2000).


Documents

Imaging through turbulence with a quadrature …authors.library.caltech.edu/5589/1/KERao05.pdfImaging through turbulence with a quadrature-phase optical interferometer Brian Kern,