monolithic achromatic nulling interference coronagraph: design and performance

Monolithic achromatic nulling interferencecoronagraph: design and performance

Brian Hicks,1,* Timothy Cook,1 Benjamin Lane,2 and Supriya Chakrabarti1

1Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston,Massachusetts 02215, USA

2Draper Laboratory, 555 Technology Square, Cambridge, Massachusetts 02139, USA

*Corresponding author: [email protected]

Received 27 March 2009; revised 16 June 2009; accepted ;posted 4 August 2009 (Doc. ID 109333); published 2 September 2009

We present the design of the monolithic achromatic nulling interference coronagraph (MANIC), a nullinginterferometer consisting of optically contacted prisms and a symmetric beam splitter. The optic is de-signed to enable the direct detection of nearby Jupiter-like exoplanets, and may be extended to enableEarth-like system detection. The monolithic nature of the optic improves on the current state-of-the-artin nulling interferometers by providing built-in alignment and stability, as well as a reduction in size andmass. These qualities make the MANIC extremely robust and simple to integrate, and an excellentcandidate for space-based applications. © 2009 Optical Society of America

OCIS codes: 120.6085, 120.3180, 260.3160, 030.1640, 080.2208.

1. Introduction

While a few low-contrast direct exoplanet detectionsat large separation angles have recently been re-ported, notably those of β Pictoris b [1], Fomalhautb [2], and HR 8799 b, c, and d [3], considerable tech-nological advances must be made to access themultitude of higher-contrast systems at smaller se-paration angles (see Fig. 1). A multitude of stellarcoronagraph designs have been proposed that arethought to be capable of delivering the >107 com-bined Airy fall-off and coronagraphic contrast reduc-tion at small separation angles that is required toenable direct imaging and spectroscopic study of so-lar systemlike extrasolar planets. The principle ofoperation and performance trade-offs of some ofthese designs are reviewed and compared in [4]. Afamily of coronagraph designs that has shown pro-mise is that of the interferometric coronagraphs,which are essentially nulling interferometers, or“nullers” that rely on coherent combination of dis-crete, on-axis beams that have been π-phase shifted

with respect to one another to produce a dark (ornulled) output. Off-axis light is not nulled, hencethe utility of nullers for exoplanet imaging. Interfero-metric coronagraphs have demonstrated, excludingAiry fall-off, a 4 × 106 null for laser light and a 106

null for a 15% bandpass centered on approximately670nm [5].

The primary challenge in producing deep inter-ferometric nulls lies in reducing wavefront error be-tween interfering wavefronts, largely attributed toimperfect optics, and in the case of ground-based ob-servations, to a much greater extent, atmosphericturbulence. Adaptive/active optics (AO) systems re-duce these errors. The performance and capabilitiesof AO are governed by the stroke resolution andrange and the number of elements in the correctivedevice, normally a deformable mirror (DM), as wellas the ability to implement sensitive and accuratewavefront sensing. While AO technology is continu-ally improving and currently becoming practical forground-based observations in the visible and/or near-infrared [6–13], it is unclear whether it will be able toachieve the performance required to produce the∼106 visible nulls for Jupiter-like planets, let alonethe ∼108 null required for terrestrial planets (note

0003-6935/09/264963-15$15.00/0© 2009 Optical Society of America

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that these contrasts assume that the point spreadfunction of the central source is ∼100 less intenseat the locations of the off-axis objects of interest).This reasoning, in addition to considerations con-cerning crucial performance metrics discussed below,has served to motivate the development of robuststellar coronagraphs for space-based extrasolar pla-net detecting and observing.Selecting an output contrast requirement for a

usable instrument is always somewhat arbitrary.Many factors, beyond the scope of any instrument,will influence the level of contrast needed to makea given observation. Of interest for nulling corona-graphy, different planetary systems will have differ-ent intrinsic contrast ratios, as well as a variety ofangular separations, causing them to be observedat different positions in the Airy disk.

If we assume that we can observe for sufficienttime with sufficient sensitivity, the limit to the con-trast that can be detected is the stability (at least inthe ensemble average sense) of the observatory’spoint spread function (PSF). The PSF stability is af-fected by both the stability of the instrument itself,the stability of the telescope, and the stability ofthe detector. While these latter two are beyond thecontrol of an optical instrument designer, we can es-timate the achievable stability by reference to otherinstruments.

It has been shown in [14] that optimal processingof multiple images can suppress the PSF at a level ofabout 1%. Similarly, in [15] it is found that PSF sub-traction reduces the residual PSF of the HubbleSpace Telescope Advanced Camera for Surveys byabout a factor of 100. Less germane, but still illumi-nating, is a study which quantifies the Space Tele-scope Imaging Spectrograph fixed pattern noise asa few percent of the continuum [16]. These resultsindicate that observational and data analysis techni-ques exist to enable observations of exoplanets at asignal-to-background ratio of a few percent. We thuschoose a post coronagraph contrast of 100∶1 asthe limit for detection in Fig. 1 and throughout thispaper.

The crucial performance metrics of a stellar coro-nagraph include its angular and chromatic response.Angular response is frequently proxied by innerworking angle (IWA), the minimum angle at whicha transition from null to transmission occurs. TheIWA is the most important characteristic of a stellarcoronagraph as it determines the cutoff at which anobject can and cannot be detected. As separation an-gles between exoplanets and their host stars aresmall, it is desirable for a stellar coronagraph to havea small IWA. In the case where separation angles arenot much greater than the angular extent of the ob-ject to be nulled, leakage from the object to be nulledcan be of significant concern, and it is therefore alsopreferable for the transition from null to transmis-sion to have as steep a gradient as possible. IWAis usually given in units of λ=D, where λ is the wave-length of operation and D is the diameter of the en-trance aperture for single-aperture systems; formultiaperture systems, D is commonly replaced with2B, where B is the baseline between apertures. Onemight consider taking advantage of the orders ofmagnitude in contrast reduction that can be achievedby observing at infrared wavelengths where thermalemission from extrasolar planets is strong. Increas-ing the observation wavelength, however, requires aproportionate aperture size increase in order fornearby objects of interest to remain outside of theIWA, which, in turn, drastically increases systemcost and design complexity both for ground-basedand space-based applications.

Secondary to IWA, the chromatic response of astellar coronagraph contributes to the integrationtime required to image and measure the spectra ofextrasolar planets. The 106 null bandwidth at a

Fig. 1. (Color online) The interferometric coronagraph proposedin this paper will reduce star–exoplanet contrasts, I�=Ip, to levelsthat will enable direct imaging of exoplanets. Here we present anestimate of I�=Ip for known exoplanets at visible wavelengths onthe sky (crosses) and after interferometric nulling with ideal 0:5m(squares) and 2:4m (diamonds) telescope aperture diameters (in-cluding Airy fall-off at the corresponding field location) observingat a central wavelength of 500nm. The contrasts are calculatedusing data from [51] assuming planets without rings and zero exo-zodiacal background or illumination. The solar system’s planetsobserved at 10parsecs (1parsecðpcÞ ¼ 3:086 × 1016 m) are plottedfor comparison and are denoted by the first one or two letters of theplanet’s name, followed by the telescope aperture diameter. For theon-sky contrasts, only planets with separation angles μ > 10mas(1mas ¼ 10−3 arcsec ¼ 4:848 × 10−9 rad) are plotted. For the nulleddata, only the detectable planets (contrasts below ∼100,μ > 0:4λ=D) are plotted, except Earth, which is included for illus-trative purposes. The long- and short-dashed lines correspond tothe IWA (∼0:4λ=D) and small separation peak intensity angle(0:73λ=D, see Section 2) for the 0:5m and 2:4m designs, respec-tively. It is estimated that nulling with 0:5m or 2:4m telescopewould enable direct detection of 25 or 103 of the known exoplanets,respectively. The right most data point is Fomalhaut b.

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central wavelength of 543nm is the primary figure ofmerit considered in this paper. Throughout this pa-per, the terms “null bandwidth” and “null bandpass”refer to the spectral range and/or region over which astated null depth is achieved. Given sufficient timeand stability for an observation, a coronagraph mustminimally reduce contrasts to a detectable level(∼100∶1) over the spectral range of a narrowband fil-ter. In practice, observing-time constraints make itpreferable for stellar coronagraphs to be capable ofproducing deep nulls over as broad a spectral rangeas possible. Achieving tolerable chromatic perfor-mance for asymmetric designs presents a consid-erable challenge in terms of compensating for ampli-tude and phase mismatch attributed to unbalancedreflections or coating traversals between interferingbeams. In principle, these performance-compromis-ing effects can be minimized by incorporating addi-tional bandwidth-limiting or balancing elements,such as filters or dispersion plates at the cost of ad-ditional design complexity and/or reduced through-put. Fully symmetric nulling interferometers areclearly preferable to their asymmetric counterpartsin that chromatic performance is only limited byalignment, stability, and coating uniformity.We present the design of the monolithic achro-

matic nulling interference coronagraph (MANIC),which nulls by means of geometric field flips andis based on the fully-symmetric rotational shearinginterferometer (RSI) layout presented in [17,18],which itself is essentially a variation on the achro-matic interfero-coronagraph (AIC) [19–22]. Inter-ferometric nullers can be designed to producedeeper achromatic nulls at smaller IWAs than otherstellar coronagraphs. Nullers, however, are subject toalignment precision and stability constraints thatprovide a formidable engineering challenge whendealing with complicated optical layouts consistingof multiple discrete elements. Furthermore, designsmust incorporate an effective means of implementingwavefront sensing and control to correct for system-level wavefront errors due to misalignment andelement figure errors. By incorporating a monolithicoptical design, MANIC simplifies the in-field preci-sion alignment effort, and its layout enables the in-corporation of a well-explored coherent wavefrontsensing technique [23–25].The MANIC nulling optic consists solely of bonded

fused-silica prisms and a custom symmetric beamsplitter (BS) cube. Once assembled, it is virtually im-possible to misalign and considerably more compactthan conventional nullers. Furthermore, MANIC’ssymmetry enables the potential to achieve perfectachromatic performance. These benefits yield a ro-bust nulling optic that simplifies overall system de-sign and integration. The importance of suchcharacteristics cannot be overstated in space-basedinterferometry, where the ability to compensate forwavefront errors due to system-wide static and dy-namic misalignment and surface figure errors is

limited by the performance of the wavefront correc-tive optic.

2. The MANIC Optic

The concept of using a RSI for astronomical researchis suggested in [26–28], and their specific use forspace-based detection of exoplanets by way of nullingat the special case of 90° shear-angle is described in[29]. These papers describe RSIs consisting of tworight-angle prisms and a cubic BS, where [27,28]use phase-compensating plates to match polarizationstates between the two arms of the interferometer atany given rotational shear angle. A nonbulk RSIusing roof mirrors and an extra fold mirror to balancereflections for each polarization component issuggested in [30].

The design path of MANIC reprises that of spa-tially heterodyned spectrometers (SHS) [31]. Earlysounding rocket-based implementations of these in-struments used multioptic designs, including a large,common-path, all-reflective system operating at121:6nm [32] and a wide-field system operating at155nm [33], which were quite capable but sufferedfrom the inherent alignment and stability issues as-sociated with using a large number of optics. Thenext generation of SHS, which used a symmetric de-sign and operated at 308nm, reduced the optical sys-tem to a highly robust compact monolithic prism [34].

The mathematical description of MANIC’s opera-tion is similar to that of the multi-element AIC de-signs presented in [18,21]. The IWA of an AIC issmaller than that of other interferometric corona-graph designs [4], which means that an AIC can ac-cess the same number of targets as other nullers witha smaller aperture. It is shown in [20] that the off-axis throughput of an AIC reaches ∼10% (∼1=3 themaximum throughput at a single image plane posi-tion) as near in as 0:3λ=D. However, for all angles≤0:73λ=D, the same intensity distribution occurs cen-tered on the same location (at 0:73λ=D). This meansthat, while an AIC can detect companion planets at afraction of the first Airy radius, their true positionand brightness must be recovered by additionalmeans. An additional benefit of an AIC design is itscentro-symmetric null, which means that there is nopreferred position angle for detection, unlike the nul-lers with “preferential” directions compared in [4].

A fundamental limitation to a nuller’s performancearises due to off-axis stellar leakage, which, in addi-tion to the (on-sky) angular extent of the object andtelescope resolution, depends on the “nulling order”of the system. MANIC is a single nuller, and, as such,its off-axis leakage is proportional to the square ofthe off-axis angle. For comparison, in a doublenuller, a system in which the dark output of one nul-ler is fed into a second nuller, the off-axis leakage isproportional to the angle to the fourth power.Throughput tends to be higher and IWA is smallerfor single nullers than double nullers. Given anextreme pairing of high contrast and a star withgreat enough extent, however, stellar leakage will


cause double nullers to outperform single nullers.For instance, a double nuller is necessary to reducestellar leakage enough to image Earth-like exopla-nets. The trade in complexity between double andsingle nullers is clear, and ideas presented in this pa-per could be extended to a double-nuller design.Continuing the evolution of RSI/AIC nullers,

MANIC is designed to overcome the problem of pathlength stability. The trade choices made with the de-sign of MANIC are the same as those encountered inmulti-element nullers with the exception of the wa-vefront control approach. It is common to use a DM inone arm of a nuller for wavefront control. To do thiswith MANIC would require putting a DM in eacharm, or a DM in one arm and a fold mirror in theother arm to maintain dispersion and amplitudesymmetry. Such an approach would present a signif-icant complication and limit stability and, therefore,defeat the purpose of a monolithic design. Instead,we choose to use the DM before the nuller. Becauseof the pupil-rotating nature of the instrument, cor-recting before the nuller means only half the pupilcan be corrected. This may be understood by consid-ering a somewhat idealized system where two inputpupil locations (A) and (B) each map to two outputpupil locations (C) and (D), and none of the four pathstraversed, zAC, zAD, zBC, and zBD, are equal. Only inthe special case of zAC − zBC ¼ zAD − zBD could bothoutput locations be corrected simultaneously. Thisconstraint necessitates the use of a mask to blockhalf the output pupil, thereby reducing the through-put by 50%. Finally, due to differing dispersion be-tween the media involved, namely the substrate,optical epoxy used for the BS, and the ambient med-ium (air/vacuum), the wavefront correction can onlybe optimized at a single wavelength at a time.An additional trade encountered by a space-based

monolithic nuller design is the potential risk of per-formance loss due to radiation-induced degradationto the substrate and epoxy. Laboratory-based simu-lations of the space radiation environment [35–38]and the 68-month long NASA Long-Duration Expo-sure Facility satellite [39] have been used to studythese effects on a number of optical materials.Exoplanet imaging missions could conceivably be de-signed for low earth orbit (LEO) or at Sun–Earth L2.Measurements of the LEO radiation environmentare summarized in [40]. Calculations of the expecteddosage for the James Webb Space Telescope locatedat L2 are presented in [41]. In a moderately thick(∼10mm) aluminum enclosure, the optic may be sub-jected to dosages of ∼100 and ∼1000 rad=yr in LEOand at L2, respectively. Taking the radiation-inducedloss of fused silica at 600nm to be 0:1dB=km=rad[42] and the path through MANIC as 270mm, aftera reasonable mission lifetime of five years, it can beexpected that these dosage rates may reducethroughput by 0.25% and 2.5%. While this may bea small effect in terms of signal loss, variance atthe 2.5% level across the pupil could greatly limit

null depth. Such effects are discussed in more detailin Subsection 3.C.5.

A. Principle of Operation

The MANIC nulling optic, which is depicted in Fig. 2,consists of a symmetric cubic BS (see Fig. 3) andstandard prisms including two right-angle prisms,two rhomboid prisms, and two rectangular prismscontacted in an arrangement that produces a flipin each interferometer arm about orthogonal axes.Upon recombination, there exists a π-phase shiftand pupil flip between the two beams in the dark out-put, producing a centro-symmetric null. The beamsproduced by an off-axis source, when recombined,are tilted with respect to one another and appearat two position angles in the image. The materials,dimensions, and angular tolerances for MANIC arespecified in Table 1.

As illustrated in Fig. 2, light enters the optic afterbeing collimated by a telescope and reflecting off of aDM. Upon entering the optic and either being re-flected or transmitted at the first BS encounter,the beams pass through rectangular prisms that me-chanically stabilize the right-angle prisms and serveto eliminate the optical surfaces that would exist intheir absence. The beams are then flipped aboutorthogonal axes by the right-angle prisms. Next,the beams pass through rhombic prism periscopesthat serve to translate the beams such that they over-lap, balance s- and p-reflection total internal reflec-tion phase shifts, and provide access to the brightoutput. Finally, before exiting the optic at the brightand dark outputs, the beams are recombined at the

Fig. 2. (Color online) MANIC consists of optically contactedprism pairs and a symmetric BS that are arranged to produce sym-metric beam traversals and access to both bright and dark outputs.The right-angle prisms achieve a π-phase shift by orthogonal flipsof the electric field, which produces the centro-symmetric null thatis characteristic of an AIC design. The rhombic prism periscopesbalance s- and p-component reflections and provide access to thebright output, which may be used for AO wavefront sensing.


BS. Note that the beam input and output locationsshown in Figs. 2 and 3 can be interchanged. Thebeam input can be moved to any of these locationsto allow for flexibility in designing how the optic ismounted and integrated with other subsystems.

1. Symmetry-Induced Achromaticity

TheMANIC design is fully symmetric, which enablesthe theoretical potential to achieve perfect achro-matic performance. In an equal-path Michelson in-terferometer, the dark output overlaps the inputand is produced by the π=2 difference in phase shiftbetween transmitted and reflected beams. A deepnull may be produced provided the BS transmission,T, and reflection, R, coefficients are matched suchthat TlTl is identical to RlR0

l at all wavelengths,where 0 denotes reverse traversal of the BS later,and l is the polarization component. By introducinga field inversion π-phase shift (rather than a path de-lay) with orthogonal right-angle prisms, TlR0

l andRlTl must be matched instead. As mentioned above,the rhomboid periscopes, paired with the right-angleprisms, balance the s- and p-component reflections.

To accommodate the limitation that Rl never exactlyequals R0

l, a symmetric BS, which is described inmore detail in Subsection 2.B.2, is incorporated inthe design.

2. Coherent Wavefront Sensing

Achieving the active control needed to produce deepnulls requires wavefront sensing to measure wave-front errors in the nulling pupil. Given the extreme(nanometer to subnanometer) wavefront matchingrequirements (null leakage is proportional to1 − cosð2πdÞ, where d is the wavefront error inwaves), it is undesirable to use a separate wavefrontsensor located in front of the nulling optic or in aseparate beam path because it would introducenon-common path errors in the beam and degradethe wavefront correction. To eliminate such errors,it is preferable to sense the wavefront at the darkoutput where, unfortunately, the wavefront error sig-nal is weak.

Our solution to overcome the sensitivity limita-tions of using the dark output is based on the conceptof coherent calibration, in which one mixes a brightreference wavefront with the nulled output. If the re-ference wavefront is coherent with the leakage, itwill form interference fringes that can be sensed,while the off-axis signal will not be coherent withthe reference, and therefore will not contaminatethe calibration measurement. This method of wave-front sensing has recently been explored in [23–25].Coherent calibration will be performed by passingthe bright output of MANIC through a spatialfilter to remove all wavefront errors and then mixingit with a portion of the nulled output at a BS(see Fig. 4).

B. Fabrication Strategy

Details associated with the technical realization ofthe MANIC nulling optic are discussed briefly below.

1. Prism Bonding

The method used to bond the individual prism pairsthat comprise the monolith is critical to the perfor-mance of MANIC. Optical epoxy is the simplestmeans of coupling prisms, and it is the only meansof bonding surfaces consisting of different materials.However, the drawbacks of using optical epoxy, whichcan include mechanical stresses across the interfacedue to different material properties, ghosting due todiffering refractive indices, and small misalignmentsdue to the finite thickness of the epoxy layer, compro-mise the performance of a nuller. Optical contactingis a more precise method of bonding that can be usedfor prisms of the same material with surfaces thatare sufficiently clean and figure matched. Alterna-tively, hydroxide catalyzed bonding (HCB), the meth-od chosen to fabricate MANIC, produces bonds thatare as precise and transparent, stronger, and morereliable than optical contact bonds. Furthermore,

Fig. 3. BS design for the MANIC enabling a fully symmetriccoating traversal for the bright and dark output beam pairs.

Table 1. MANIC Design Parameters

Materials and coatingsGlass HomosilBS epoxy Epotek 301–2BS coating SiO2 on 1=2AgFold coatings None (TIR)Entrance and exit coatings AntireflectionDimensions (mm)Overall 120 × 120 × 60BS 60 × 60 × 60Angle tolerance {high, low}-precision (arcsec)BS epoxy wedge and 45° {1, 10}RAa90°, Rhomboid parallel, and pyramid errors {1, 10}RA and Rhomboid pitch and yaw {2, 20}RA and Rhomboid roll {6, 60}

aRA ¼ right-angle.


HCB is a process that is less demanding and less ex-pensive than that of optical contacting.

2. Symmetric Beam Splitter

The layout of the symmetric BS is shown in Fig. 3.Symmetry is obtained by simultaneously coatingcomplimentary halves of the hypotenuses of thetwo right-angle prisms comprising the BS. This coat-ing and assembly design make it such that the twodestructively interfering beams, one being firsttransmitted then reflected (TR) and the other beingfirst reflected then transmitted (RT), traverse thesame layer structure of substrate/epoxy/splitter coat-ing/substrate or substrate/splitter coating/epoxy/substrate, ensuring that all Fresnel reflection andtransmission coefficients are balanced for each polar-ization component throughout the optic. A strategyfor fabricating and measuring the precision of aBS cube for white light interferometry is describedin [43].

3. Total Internal Reflection Folding

All reflections other than those in the BS rely on totalinternal reflection (TIR), the benefit of which ismanyfold: TIR is lossless, the surfaces need not becoated, which reduces cost and eliminates issueswith coating nonuniformity, and the surfaces are less

susceptible to environmental degradation, therebymaking the optic more robust.

4. Optical Path Difference Compensation

Interferometric coronagraphs that utilize common-path geometries, notably the system described in[44], do not suffer from optical path difference(OPD) between interfering beams. Note that, in thispaper, OPD refers to the zeroth-order (piston) wave-front error between arms introduced by themonolith.WhileOPDcannotbeavoided in the initial fabricationof MANIC, it can be removed at a single wave-length to limit dispersion effects and thereby widenthe deep null bandpass. All results and analysis pre-sented in Subsection 3.B assume that OPD betweenthe TR and RT arms of MANIC has been removedat 543nm by contacting parallel compensator plates,i.e., controlled-thickness etalons, to the right-angleprisms (see Fig. 4). Determining the compensatorplate thicknesses can be accomplished by measuringtheaveragepiston required to null a singlewhite lightor narrowband source over all locations in the usablehalf of MANIC’s dark output pupil. The precisionwith which the required thickness can be measured,polished, and assembled is limited, and there willbe, therefore, uncorrected path difference betweenthe two interferometer arms. The effect of thisuncorrected bulk path difference is discussed inSubsection 3.C.6.

5. Birefringence Considerations

Homosil was chosen as the prism material for its lowrefractive index, high Abbe number, and highly iso-tropic and homogeneous qualities. For applicationsthat are less sensitive to small optical path errors,Homosil is considered to be an isotropic material.In the case of deep nulling interferometry (>105 con-trast reduction), however, the scale of MANIC couldlead to significant path difference between extraor-dinary and ordinary waves introduced by Homosil’s3:6nmcm−1 bar−1 residual birefringence. This de-pends on how the randomly oriented crystal axesare assembled with respect to one another, whichis difficult to predict, as well as how the optic ismounted. If necessary, birefringence effects can beovercome by filtering the dark output with a linearpolarizer at a cost of losing an additional half the re-maining signal of the object of interest.

6. Dimensioning

MANIC’s dimensions (see Table 1) were selectedbased on a combination of considerations. Minimally,the prisms that make up the monolith must be largeenough that they can be easily handled and polishedby an optician. In terms of producing high-precisionangles, larger facets act as longer levers for register-ing the relevant angles between prism when they areassembled. Larger optics also mean larger aperturesfor interferometric measurement, at the cost of an

Fig. 4. The layout of MANIC enables the incorporation of a well-explored coherent wavefront sensing technique that may be usedto control the wavefront corrective deformable mirror (DM) at theinput. In this schematic, the bright output is spatially filteredusing a pinhole (PH) and used as a reference to be mixed with aportion of the dark output. Beam splitters (BS) are used to samplethe dark output and recombine it with the bright output reference.The signal at the wavefront sensing camera (WFS) is interpretedby a computer (CPU) that drives the DM. The nulled image isdetected at a science camera (SCI). Also shown are the compensa-tor plates (CPs) that will be contacted to the right-angle prisms tominimize optical path difference between the two halves of themonolith and thereby increase the deep null bandwidth (see Sub-section 2.B.4), and the polarization filter (PF) that may be neededto reduce birefringence-related leakage (see Subsection 2.B.5).


increase in polishing time and, of course, addedmass. Assuming no limit to the ability to handlethe ensemble optic or measure the surface figuresof all the relevant individual facets, a different upperlimit is reached. Homogeneity and isotropy require-ments set an upper limit due to the limited size ofhigh quality ingot from which the prisms can bemade. Ultimately, some balance must be struckbetween these trade-offs, as a beam compressor orexpander can be used with a range of nuller sizes.

3. Modeling

The description of MANIC’s theoretical performanceis similar to that of its multi-element counterpartspresented in [18,21], however, propagation throughnonunity index gives rise to additional considera-tions. While the ideal MANIC is fully symmetric,and therefore perfectly achromatic, manufacturingerrors contribute wavefront errors across the pupilthat produce wavelength-dependent leakage, pri-marily due to dispersion. The leakage can be attrib-uted to OPD produced by a wedged epoxy layer, TIRphase shift imbalance due to prism facet angle er-rors, and to a lesser extent, refraction at the input.The relative contributions of these leakage sourcesis considered in Subsection 3.C.We have used analysis andmodeling to ensure that

the fabrication precision will not compromise the in-strument performance. Using wave and Fourier op-tics to analytically determine the levels of tolerablepolarization rotation, amplitude mismatch, and OPDfor deep nulling interferometric coronagraphs is rela-tively straightforward andwell understood. A deriva-tion of such symmetry requirements for nullers ispresented in [45] for a monochromatic point source,a polychromatic source of finite extent, and the latterin the presence of fluctuations. Determining how in-dividual random fabrication errors compound one an-other to produce asymmetries in multisurface opticalsystems with complex geometries, however, does notlend itself to an analytical approach.

A. Model Description

To appreciate the cumulative effects of high- and low-precision fabrication errors (see Table 1), MANICwas modeled using ZEMAX ray tracing software innonsequential mode. The model incorporates prismfacet tilt errors, wedge in the epoxy layer of theBS, dimensional mismatches between prism pairs,as well as translational and rotational alignment er-rors. Refractive index data were modeled usingSellmeier and Schott coefficients for Homosil andEpotek 301–2, respectively, and linear interpolationof complex refractive index data was used for silver.An ideal MANIC assumes homogeneous and iso-

tropic materials, uniform and identical BS coatings,perfect surface quality of all facets, and no scattering.Deviations from these idealizations in the fabricatedoptic are intrinsically spatially varying. Such effectsare better suited to modeling through numerical im-plementation of the equations presented by [21],

using appropriate spatial resolutions and erroramplitudes. A minimum intensity threshold forray splitting was set to limit rays to a single passthrough each arm of the system. All of the aforemen-tioned assumptions serve to minimize computationalcomplexity while also enabling a method of ray-pair-based system optimization, whose descriptionfollows below.

The various prism facet and alignment angleswere randomly perturbed to simulate fabrication er-rors using two sets of user-defined limits, which arelisted in Table 1. The two sets of angles were selectedbased on a combination of considerations. The pri-mary challenge lies in trading the relative detrimentto the null depth posed by specific errors with theirfabricability. While individual prisms with 10 arcsecangle tolerances are readily available through anumber of optics suppliers, polishing and measuringtechniques are capable of producing 1 arcsec toler-ances or better, depending on the scale of the optic.In assembly, the facet angle errors of individualprisms add to produce alignment errors, e.g., wedgeand pyramid errors in the BS and rhombic prismsadd to produce tip and tilt alignment errors in theright-angle prisms. Because these alignment errorsare constrained in two directions by polished planeinterfaces when joined by optical contacts, they areeasier to control than the rotational alignment, e.g.,the rotation of a right-angle prism about an axis nor-mal to its large face, and these tolerances are scaledaccordingly.

After perturbing all of the relevant angles, a singleray at normal incidence to the input face, k̂in;⊥, wastraced through the two arms of the optic. This pro-duces two rays in the output with directions k̂RTand k̂TR. The null axis of the optic, k̂in;null lies inthe direction that produces the result k̂RT∥k̂TR. Thisdirection was found by iteratively applying the rulek̂in;j ¼ k̂in;j−1 − ðk̂RT;j−1 − k̂RT;i−1Þ=2 for j ¼ 1; 2; 3…and k̂in;0 ¼ k̂in;⊥.

To consider the effects of the fabrication errors ofMANIC alone, a perfectly collimating telescope wasassumed. A perfectly collimated beam was modeledusing a symmetric grid of monochromatic and iden-tically polarized rays (i.e., wavefront normals). Thebeam was injected into the perturbed optic at an an-gle k̂in;null. We anticipate using a DM with MANIC(see Fig. 2) that is 10mm on a side, a readily obtain-able size for commercially available DMs, and there-fore model the system with rays spanning a single10mm diameter pupil. After passing though the op-tic, the sum of electric fields of each ray pair, i, ori-ginating from (Ai) and (Bi) input pupil locationsthat map to the output pupil location (Ci is mea-sured. Since the full pupil cannot be corrected (seeSection 2), rays emerging in the other half of the out-put pupil (Di) are masked.

An optimization routine is run to minimize thecomplex amplitude sum of the electric field at allpoints in the useful portion of the output. The opti-mization is accomplished by first compensating with


one of the right-angle prisms to simultaneously mini-mize OPD between the two arms and the combinedelectric field, Eð0Þ ¼ ETRð0Þ þ ERTð0Þ, where (0) isthe center of the unmasked portion of the pupil.Secondly, the starting points of all other ray pairszoðAiÞ and zoðBiÞ are adjusted by dzðAiÞ ¼ −dzðBiÞto adjust path lengths through the optic, zðAiÞ andzðBiÞ, thereby adjusting the phase to minimizeEðCÞi ¼ ERTðCiÞ þ ETRðCiÞ. The first step is tanta-mount to optically contacting controlled-thickness

compensator plates on the right-angle prisms afterthe initial fabrication to minimize the OPD betweenthe two arms (see Subsection 2.B.4). Adjusting the zoffset in the second step is equivalent to using a DMat the input to remove higher-order wavefront errors.

B. Model Results

After perturbing the optic and then optimizing theinput angle and compensators at the 543nm greenhelium–neon center wavelength, the chromatic leak-age of a point source and the off-axis leakage of amonochromatic extended source were modeled byscanning in wavelength and input angle, respec-tively. Other than proximity to the center of the visi-ble band and its availability as a source, the choice ofoptimization wavelength is arbitrary. The chromaticand angular leakage may be treated independentlysince the effects of input refraction, which is consid-ered in Subsection 3.C.3, is small. The total leakagewas calculated by taking the quotient of the sum ofcoherent intensities divided by the sum of incoherentintensities for all ray pairs in the useful portion of thepupil described above:

Ltotal ¼Xni¼1

jEðCiÞj2jERTðCiÞj2 þ jETRðCiÞj2

; ð1Þ

where n is the number of ray pairs and the null isgiven by

null ¼ 1=Ltotal: ð2Þ

1. Chromatic Performance

The chromatic leakage based on 1000 Monte Carloperturbations for each set of angle tolerances opti-mized at 543nm was modeled from 0.5 to 0:6 μm,and the median, minimum, maximum, and 90% con-fidence levels are presented in Fig. 5. The data showthat significant gains in 106 null bandpass can bemade by increasing fabrication precision, particu-larly in that of the BS epoxy layer wedge, which isdiscussed in more detail in Subsection 3.C.1.

The likelihood of producing 106 nulls over a rangeof bandwidths for 1000 low-precision and 1000 high-precision perturbations of MANIC optimized at543nm is plotted in Fig. 6. The low-precision toler-ances yielded a minimum bandwidth of 1nm and a50% likelihood of achieving a bandwidth of at least5 to 6nm. The high-precision tolerances yielded aminimum bandwidth of 10nm and a 50% likelihoodof achieving a bandwidth of at least 50 to 60nm.

2. Off-Axis Null Response

Angular response data for an optic optimized at543nm and tilted relative to the null axis at anglesranging from 10−5 to 102λ=D are plotted in Fig. 7.Included with these data is a plot of the theoreticalresponse of the AIC design determined by [21]. The

Fig. 5. Fabrication errors in the MANIC will produce chromaticphase errors, which limit the deep-nulling bandwidth. Here wepresent the modeled chromatic leakage from a point source for500 high-precision (top) and 500 low-precision (bottom) perturba-tions optimized at 543nm. The minimum and maximum leakageare shown as dotted curves. The dashed curves represent the sam-ple median. The solid curves show the upper and lower limits ofthe 90% confidence interval. While the low-precision modelachieves a 106 null for laser light, the high-precision modelachieves the same performance over a ∼4–18% bandpass.


data show that for angles greater than ∼10−3λ=D theoff-axis performance for the different precision levelsis practically identical. At small angles, the low-precision design has more leakage due to OPD aris-ing from the epoxy layer wedge and the total internalreflection phase shift error. This figure also illus-trates the effects of stellar leakage, Ls, due to the fi-nite (off-axis) extent of the star. In the examplepresented here, the Sun at 10 pc leaks at a level of∼5 × 10−4, but the added PSF (∼Airy) suppressionfactor of ∼100 makes Jupiter detectable at a nulledcontrast of ∼100 relative to the background. It isimportant to note that for angles greater than1λ=D, coherence is greatly reduced, and the methodof computing the leakage becomes somewhatunphysical.

3. Deformable Mirror Stroke

The ultimate limit to the achievable null depth usinginterferometric coronagraphs is, in part, a function ofhow well the recombined wavefronts are matched.This is, in turn, a function of the “flatness” of theincoming wavefront. The order of magnitude to con-sider for a 106 null is approximately 10−3 wave-lengths. Because a DM has limited stroke to

correct end-to-end system errors, it is important toreserve as much stroke for difficult to manage toler-ances and elements whose alignment or surfacesmay vary dynamically in the presence of thermalgradients or other environmental perturbations. Aplot of DM stroke versus input angle required to

Fig. 6. The 106 null bandwidth of MANIC scales approximatelylinearly with fabrication precision. Here we plot the likelihood ofachieving 106 null bandwidths centered on 543nm with low-precision specifications (bold, lower abscissa) and high-precisionspecifications (fine, upper abscissa), each based on 1000 MonteCarlo perturbations. The plot shows that 99% and 98% of the mod-eled high- and low-precision modeled systems achieve a bandpassof ∼20nm and 3nm, respectively, while 50% achieve ∼50–60 and5–6nm, respectively. Furthermore, a MANIC fabricated with thehigh-precision specifications would have a ∼20% chance of nullingat the 106 level over the entire V-band, which has a∼100nm band-width centered on ∼550nm.

(a)

(b)

Fig. 7. The ability of a coronagraph to null an object dependsstrongly on the object’s angular extent. The angular response data(crosses) plotted here comprise 1000 random high-precision (top)and low-precision (bottom) perturbations and were generated bytilting the optic relative to the nominal null axis. The solid linesare the AIC theoretical angular response derived by [21]. Theshort- and long-dashed vertical lines represent the extent of theSun (βSun) and maximum separation of Jupiter (μJup) at 10pcfor λ=D ¼ 0:2arcsec. For large angles, the off-axis performancefor the different precision levels is practically identical. At smallangles (<10−3λ=D), TIR phase rotation mismatches become signif-icant, and the low-precision design exhibits greater leakage. Thisleakage is inconsequential compared to that coming from the finiteextent of the Sun at 10pc (Ls).


correct the perturbed 1000 high- and low-precisionmodels is presented in Fig. 8. The stroke maximaof the entire sample were found to be 0.513 and4:845 μm for the high- and low-precision models, re-spectively. For a DM with a moderate 10 μm of stroke(DMs with ∼1 to 50 μm are commercially available)this corresponds to a difference between using ap-proximately 5% and half of the total available stroketo correct the nulling optic. In the latter case, signif-icantly less margin is left for correcting the rest of thesystem and potential dynamic errors.

C. Analytic Verification

The angular and chromatic responses of MANICpresented above arise due to a combination of differ-ent sources of dispersion and amplitude imbalancebetween the two arms of the interferometer. Therelative contribution of these error sources are sum-marized in Table 2 and considered independentlybelow.

1. Beam Splitter Epoxy Wedge

A considerable amount of leakage can be attributedto wedge in the BS epoxy layer, which introduceschromatic path error between the TR and RT arms.The OPD between the RT and TR arms due to thewedge may be approximated by

dλo;BSw ¼ nλo;elBSw=λo ¼ nλo;eðρ tan αÞ=λo ≈ nλo;eρα=λo;ð3Þ

where lBSw is the physical difference in epoxy layerthicknesses and ρ is the distance between the split-ting and recombination beam centers in the BSplane, α is the BS wedge angle error along the direc-tion connecting these points, λo is the central com-pensator optimization wavelength, and nλo;e is theindex of refraction of the epoxy at this wavelength.The optical thickness traversed through the compen-

sator plates at the optimization wavelength, dλo;CP, isideally made to equal dλo;BSw, which sets forth the re-quirement lBSw ¼ nλo;elBSw=nλo;s where nλo;s is the in-dex of refraction of the compensator plate substrateat the central compensator optimization wavelength.The wavelength-dependent optical path difference,

dλ;BSw ¼ lBSwλ

��nλ;e − nλ;snλo;enλo;s

��; ð4Þ

may then be used to determine the chromaticleakage due to BS epoxy layer wedge,

Fig. 8. The strokes required to null the pupil after minimizingOPD between the two arms of the monolith are readily obtainablein commercially available DMs. Stroke is plotted here versus theinput angle at which theMANICmodels must be tilted to align thenull axes in each half of the optic at 543nm. Crosses and trianglescorrespond to the high- and low-precision designs, respectively.

Table 2. MANIC Null Leakage Sources

Source Symbol Figure(s) Section(s) Contribution

Stellar a Ls 7 1, 2, and 3.B.2 5 × 10−5, 10−3

BS epoxy wedge b Lλ;BSw 5, 6, and 9 3.B.1 and 3.C.1 <10−5, 10−3

Radiation degradation c LA;rad - 3.C.5 ≲6 × 10−6, 6 × 10−4

Uncorrected OPD d Ldλ ;cp - 3.C.6 ≲2 × 10−6, 3 × 10−4

TIR phase shift e Lϕ;TIR 7, 8, and 10 3.C.2 ≲10−8, 10−6

Field rotation f Lrot - 3.C.4 ≲10−9, 10−7

BS incidence g LA;θi - 3.C.5 ≲10−8

Input refraction h Ldλ ;r 8 3.C.3 <10−11

Pupil overlap LA;Φ - 3.C.5 ≈0

aAssuming a Sun-like star at 10pc, observing at λ ¼ 543nm with D ¼ 0:5, 2:4m.bFor α ¼ 1, 10arcsec over 500–600nm.cAssuming maximum imbalance of 0.25% and 2.5% due to radiation-induced transmission loss after 5 years at LEO, L2 with no

amplitude correctiondFor 4% and 20% bandpasses at a central wavelength of 543nm.eValues extrapolated from the high-, low-precision model results.fFor σrot ¼ 6, 60arcsec.gFor θi ¼ 10arcsec.hMaximum value over 500–600nm for θi ¼ 10arcsec.


Lλ;BSw ¼ 1 − cosð2πdλ;BSwÞ; ð5Þ

which is plotted in Fig. 9 for Epotek 301–2 withρ ¼ ffiffiffi

3p ð30Þ ≈ 52:0mm, and α ¼ 5 and 50 μrad (1 and

10 arcsec) over the entire range of available indexdata. This figure suggests that the 106 null bandpassdepends strongly on Lλ;BSw; therefore, in addition tochoosing an epoxy with a dispersion gradient thatbest matches that of the compensator plate sub-strate, significant fabrication effort should be de-voted to minimizing wedge in the BS epoxy layer.

2. TIR Folding Phase Errors

Another source of leakage arises due to unbalancedphase shifts accumulated over the five reflections en-countered by each beam. MANIC uses TIR to accom-plish all reflections aside from those encountered atthe BS. Each internal reflection can give rise to phaseshift mismatch due to reflections at unequal anglesof incidence. TIR of s- and p-polarized waves is ac-companied by a phase shift of

ϕs;λ ¼ 2tan−1

��cos θc;λcos θi

�2− 1

�12�; ð6aÞ

ϕp;λ ¼ 2 tan−1

�−

1

sin2 θc;λ

��cos θc;λcos θi

�2− 1

�12�; ð6bÞ

respectively, where θc;λ ¼ sin−1ð1=nλÞ is the criticalangle for TIR in a unity index medium, nλ is the re-fractive index of the glass, and θi is the angle of in-cidence. All reflections in theMANIC optic nominallyoccur at 45° incidence. The actual angles of incidencevary on the scale of the angular precision at whichthe prisms are polished and assembled. The totalTIR phase shift difference between the two arms con-sisting of an equal number, N, of s and p reflections,is given by

δϕTIR ¼XNq¼1

ðϕTR;s;q − ϕRT;s;qÞ þ ðϕTR;p;q − ϕRT;p;qÞ; ð7Þ

where the λ subscripts used in Eq. (6) have beenomitted. For MANIC, N ¼ 2. The null leakage fromTIR phase shift errors, Lϕ;TIR, is given by

Lϕ;TIR ¼ 1 − cos δϕTIR: ð8Þ

For illustrative purposes, Lϕ;TIR for N ¼ 1 is plottedin Fig. 10 for errors of �22:5 arcsec around 45° at543nm for cases of either the first or second quantityon the right-hand side of Eq. (7) being equal to zero.The plot shows that Lϕ;TIR becomes significant fordeviations exceeding 0:1mrad (20 arcsec) and0:18mrad (36 arcsec) for p- and s-component reflec-tions, respectively. Neglecting TIR at spatial fre-quencies greater than unity, this phase shift maybe considered a source of OPD similar to but distin-guishable from OPD attributed to unequal physicalpaths through the two arms of the interferometer.In this sense, TIR phase shift is measured along withthe physical path error by performing the white lightnull described in Subsection 2.B.4, and their asso-ciated leakages are minimized with the compensatorplates described in the same section.

3. Input Refraction

Facet errors between the two arms of the nuller movethe “on-axis” angle relative to the input surface awayfrom normal. As such, chromatic null leakage attrib-uted to chromatic path error due to input refractionshould be considered. This error can be approxi-mated by treating the optic as a two-dimensionalair/glass/air parallel interface with no BS epoxylayer. Using simple geometry and Snell’s Law itcan be shown that

lλ;r ≈ nλzf1 − cos½θrðnλÞ�g

¼ nλzf1 − cos�sin−1

�sin θinλ

��; ð9Þ

where lλ;r is the wavelength-dependent refraction-induced path error, θr is the refraction angle, nλ isthe wavelength-dependent refractive index of theglass, and z is the thickness of the glass slab.Assuming zero OPD can be achieved at an arbi-trary wavelength, λo, using compensator plates, the

Fig. 9. The greatest detriment to the 106 null bandwidth is pro-duced by dispersion mismatch between the BS epoxy and the cor-rective compensator plates, which must be made from the samematerial as the prisms to be optically contacted. This plot showsthe analytically calculated null leakage due to path error intro-duced by a wedge in the BS Epotek 301–2 epoxy layer for wedgeangles of α ¼ 1 (solid curve) and 10arcsec (dashed curve), with thedistance between the splitting and recombination beam centers inthe BS plane, ρ ¼ 52mm. Note that the curves in this plot closelyresemble the corresponding median chromatic leakage curves inFig. 5.


wavelength-dependent path error may be ex-pressed as

dλ;r ¼ jlλ;r − lλo;rj=λ

¼ nλzλ

��1 −

sin2θin2λ

�1=2

−

�1 −

sin2θin2λo

�1=2

��; ð10Þ

where we make use of the trigonometric identitycos½sin−1ðxÞ� ¼ ð1 − x2Þ1=2. Chromatic null leakagedue to refraction-induced dispersive OPD is given by

Ldλ;r ¼ 1 − cosð2πdλ;rÞ: ð11Þ

Figure 8 shows that the maximum input angle en-countered in the perturbation study is approximately10 arcsec. Using this input angle with λo ¼ 543nmand z ¼ 270mm in Eqs. (10) and (11) yields amaximum leakage of Ldλ;r ∼ 10−11 over a bandpassof 500 to 600nm, and it can therefore be concludedthat input refraction has an insignificant effecton MANIC fabricated with the aforementionedtolerances.

4. Field Rotation

A final source of leakage included in themodel is thatof field rotation. The field rotation between interfer-

ing beams is produced by the right-angle prisms.While the right-angle prisms are constrained in“tip/tilt” relative to the rest of the monolith by com-parably well-controlled pyramid errors in the otherindividual prisms, the nominal “roll” angle of theright-angle prisms is more difficult to locate, andis accordingly afforded a larger tolerance. The conse-quent (geometric) leakage due to field rotation isdominated by this error and given by

Lrot ¼ 2½1 − cosðωÞ�; ð12Þ

where ω is the deviation from π, having a maximumvalue of 2σrot, and σrot is the roll tolerance. For thelow- and high-precision models, the maximum ex-pected leakage is ∼10−8 and ∼10−10, respectively.This amount of leakage, while not negligible as inthe case of that due to input refraction, remains alesser contribution to the degradation to the overallnull.

5. Amplitude Balance

Given a system with no scattering and perfect align-ment, i.e., no phase errors due to OPD or tilt errors,the only potential source of null leakage would bethat due to imbalance in the electric field amplitudes,Lamp, which, for a uniform error across the pupil, isequal to the square of the fractional error betweenamplitudes. In fringe contrast terms, the balance be-tween combined electric field amplitudes determinesthe modulation between constructive and destruc-tive interference. In general, coating nonuniformityin themultiple fore-optics and nuller BSwill give riseto spatially varying leakage. It is assumed that thisspatial variation in beam amplitude will be domi-nated by larger fore-optics, namely the telescope pri-mary and secondary, which are more difficult to coatand introduce significant scattering. In the final sys-tem, as is the case with other nullers, such effects candominate the leakage, and depending on the nulldepth requirement, this may ultimately require am-plitude control in the AO. Systems capable of achiev-ing such control are explored in [46–48]. (Note thatsuch a subsystem is not depicted in Fig. 4.) Here weconsider other nonspatially varying amplitude leak-age sources introduced only by the MANIC nul-ling optic.

The model described above assumes a perfectlyuniform BS coating due to the complication ofintroducing spatial asymmetry. This assumption isreasonable to make if the coatings are applied simul-taneously (see Subsection 2.B.2). The angles of inci-dence at the BS coating vary between the two beams,however, introducing a small uniform error in theamplitude between TlRl and RlTl. The largest inputangle of∼10 arcsec needed to locate the null axis (seeFig. 8) was used to estimate the maximum LA;θi dueto incidence angle error at 500 and 600nm for eachpolarization component, the greatest value of whichwas found to be∼10−8 for the s-component product at

Fig. 10. Unbalanced TIR phase shifts due to small errors inincidence angles encountered by interfering beams in the twohalves of the optic contribute significant leakage. Here we plot theanalytical null leakage of a monochromatic point source due to p-component (solid line) and s-component (dashed line) TIR phaseshift mismatch, Lϕ;TIR, from a single deviation in incidence anglearound 45° (0:7854 rad) at 543nm for Homosil in a unity indexenvironment. Lϕ;TIR becomes significant at the 10−6 level for devia-tions exceeding 0:1mrad (20arcsec) and 0:18mrad (36arcsec) forp- and s-component reflections, respectively. In the low-precisiondesign, where input angles can approach 10arcsec (see Fig. 8),it is easy to appreciate the magnitude of this effect.


600nm, and is, therefore, considered a minor contri-butor to the overall null leakage.Pupil overlap leakage, LðA;ΦÞ, which is caused by

translational shear due to translational alignmentof the right-angle and rhomboid prisms as well astranslational misalignment at the input, is assumedto be masked perfectly. The trade-off between pupilshear error and the masking thereof is throughput.It is important to maximize throughput to achieve asufficient signal from an exoplanet over as short atime as possible. Because of the π pupil-rotating nat-ure of the MANIC, any features in the pupil that donot have π-rotation symmetry must be masked tolimit LA;Φ, which is proportional to the fractionalarea of the beam overlap asymmetry. This is a sys-tem-level consideration that is assumed to be easilyavoidable with a minimal amount of alignment effort(translating the nuller relative to the input beam toget the output beams to overlap) and using a mask toblock areas of rotational nonoverlap.As discussed in Section 2, after long-duration expo-

sure to space radiation, environment transmissionloss in the substrate and epoxy could give rise to sig-nificant amplitude imbalance. If exposed uniformly,it would be expected that the transmission would de-crease uniformly, and a negligible imbalance wouldbe expected. It is conceivable, however, that nonuni-form shielding or an average preferred orientationwith respect to the Sun could produce a gradientin the exposure. The radiation-induced throughputreduction values of 0.25% and 2.5% quoted earlierin this paper for a five year mission in LEO and atL2, respectively, may be used to estimate this effect.A conservative estimation would assume that thedegradation takes place purely in one arm. Suchan approximation would predict nuller-induced am-plitude leakages, LA;rad, of 6 × 10−6 and 6 × 10−4, re-spectively, and would thus require extra shieldingof the optic for a long-duration mission or requiresome level of amplitude control as mentioned above.

6. Uncorrected Bulk OPD

An additional leakage source that was not modeledbut can readily be estimated analytically is that ofbulk (piston) OPD between the two arms of the opticfollowing correction with the compensator plates de-scribed in Subsection 2.B.4. Determining the com-pensator plate thicknesses can be accomplished bymeasuring the average piston required to null a sin-gle white light or narrowband source over all loca-tions in the usable half of MANIC’s dark outputpupil. Such a measurement can be made with ap-proximately λ=20 accuracy. Because they are to be op-tically contacted, it is the surface figure of thecompensator plates and the right-angle prisms, allof which can be polished to λ=20, in addition to theerror in the controlled thickness of ≲10nm [49] thatlimit the accuracy of the OPD correction. Assumingthat the measurement uncertainty and the figure er-rors in the four contacted surface pairs add randomly,

we can expect to control the overall OPD to an accu-racy of �50nm.

While the remaining net OPD at any pupil locationcan be reduced for any given wavelength using a DM,bulk dispersion effects will contribute chromaticleakage, Ldλ;cp, which can be approximately equalto the mean square dispersion across a given pass-band [45]. For a path error of 50nm in Homosil,Ldλ;cp reaches a level of 10−6 level for a 4% bandpasscentered on 543nm, which is comparable to the over-all performance of the low-precision model, butwould yield a narrower bandpass than that whichwas found for the high-precision model.

4. Conclusions

We have described a monolithic nulling interferom-eter consisting of bonded standard prisms and a sym-metric cubic BS fabricated to a precision that couldnull a Sun-like star at 10pc (2 × 10−3λ=D ¼ 0:2arcsecextent for D ¼ 0:5m at λ ¼ 543nm) to reach the Airyfall-off aided ∼100∶1 star-to-planet contrast for aJupiter-like planet (located at 2:2λ=D) over a ∼4%bandpass. While lower contrast systems could be de-tected at separations as small as 0:38λ=D, the radialposition and intensity of objects within 0:73λ=D can-not be determined with certainty.

The various leakage sources that determine theperformance of the nuller are summarized in Table 2.The most significant leakage is stellar leakage,which is a well-known drawback of RSI/AIC designs.On the other hand, the small IWA is a major benefitof such a design, one which significantly reduces theaperture size requirement. The distinct advantage ofa monolithic nuller is improved path length stability,an issue that challenges all nuller designs. Once as-sembled, such an optic would be virtually impossibleto misalign and considerably more compact than con-ventional nulling interferometers. These benefitsyield a robust optic that simplifies overall system de-sign and integration and reduces the overall systemcost, as well as the AO budget needed to correct dy-namic instrument misalignment. The major trade-offs, however, seem to be (1) a reduction of bandpassdue to a limited ability to remove in-bulk OPD-related dispersion and (2) a 50% reduction inthroughput from having to mask half of the outputdue from having the DM before the nuller ratherthan in one of its arms.

As expected, fabrication-induced leakage levelswere shown to depend on the level of fabrication er-ror built into the optic, predominantly that of the BSepoxy layer wedge. The off-axis performance of thedesigns modeled in this study performed at the the-oretical limit down to ∼10−3λ=D, at which point TIRphase error and pupil rotation leakages become sig-nificant. Presumably, with the ability to polish andassemble the prisms and beam splitter to a higherdegree of precision than assumed in this work, thecloser one could get to producing an optic that per-forms near the fully achromatic theoretical limit.


The theoretical performance of MANIC, as pre-sented in this paper, is achromatic in the sense thatit can be “tuned” to produce zero OPD between beamsat any give wavelength within the ability of thewavefront corrective device. Assuming the use of aDM correcting in unity index material, monochro-matic nulls can be produced with a depth that is afunction of the overall stability of the system. Thebandwidth of the null, however, is dependent onthe ability to carefully match dispersion. While itwould introduce a small complication to the design,one could balance dispersion over different band-passes using carefully engineered dispersion platessuch as those discussed in [5,47].The issue of off-axis leakage and AIC designs has

been discussed and revisited numerous times start-ing with its inception [21]. Nonetheless, an AIC iscapable of detecting nearby solar system-like gasgiants. Although it would greatly complicate overallsystem design, the authors are keenly interested inthe prospects of a concept proposed by [50] to im-prove the off-axis rejection of extended objects usingmodified AIC designs in tandem. Such a technique,or even simply a standard coronagraph could be usedwith the next implementation of the MANIC to ex-tend its performance toward enabling direct detec-tion of exo-Earths.

The authors of this paper thank IanMiller and JeffWimperis (LightMachinery, Inc.) for their assistancewith the design of the MANIC. The authors also ac-knowledge the many useful discussions on nullinginterferometry held with Bruce Martin Levine, Mi-chael Shao, and James Kent Wallace (Jet Propul-sion Laboratory) during the ongoing development ofthe Planetary Imaging Concept Testbed Using aRocket Experiment (PICTURE) project funded underNational Aeronautics and Space Administration(NASA) grant NNG05WC17G. Development of theMANIC is funded under NASA grant NNX07AH89Gand a Boston University Photonics Center SeniorPhotonics Fellowship.

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monolithic achromatic nulling interference coronagraph: design and performance

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