Optical Verification Experiments of Sub-scale Starshades
Anthony Harness, Stuart Shaklan, Phillip Willems, N. Jeremy Kasdin, K. Balasubramanian, Philip Dumont, Victor White, Karl Yee, Rich Muller, Michael Galvin
OOptical Verification Experiments of Sub-scale Starshades
Anthony Harness a,* , Stuart Shaklan b , Phillip Willems b , N. Jeremy Kasdin a,c , K.Balasubramanian b , Philip Dumont b , Victor White b , Karl Yee b , Rich Muller b , MichaelGalvin a a Princeton University, Mechanical & Aerospace Engineering Department, Princeton, New Jersey b Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California c University of San Francisco, College of Arts and Sciences, San Francisco, California
Abstract.
Starshades are a leading technology to enable the detection and spectroscopic characterization of Earth-like exoplanets. In this paper we report on optical experiments of sub-scale starshades that advance critical starlightsuppression technologies in preparation for the next generation of space telescopes. These experiments were conductedat the Princeton starshade testbed, an 80 m long enclosure testing 1/1000 th scale starshades at a flight-like Fresnelnumber. We demonstrate 10 − contrast at the starshade’s geometric inner working angle across 10% of the visiblespectrum, with an average contrast at the inner working angle of . × − and contrast floor of × − . Inaddition to these high contrast demonstrations, we validate diffraction models to better than 35% accuracy throughtests of intentionally flawed starshades. Overall, this suite of experiments reveals a deviation from scalar diffractiontheory due to light propagating through narrow gaps between the starshade petals. We provide a model that accuratelycaptures this effect at contrast levels below − . The results of these experiments demonstrate that there are nooptical impediments to building a starshade that provides sufficient contrast to detect Earth-like exoplanets. This workalso sets an upper limit on the effect of unknowns in the diffraction model used to predict starshade performance andset tolerances on the starshade manufacture. Keywords:
Starshades, optical model validation, high contrast imaging, exoplanet detection, vector diffraction. * Send email correspondence to: Anthony Harness, [email protected]
Starshades have the potential to discover and characterize the atmospheres of Earth-like exoplanetsin the habitable zone of nearby stars.
Their ability to achieve high contrast while maintaininghigh optical throughput and broad wavelength coverage make them the most promising technol-ogy to produce the first spectrum of an exo-Earth atmosphere.
4, 5
In recent years there have beensignificant technological advances that demonstrate the feasibility of building and deploying a star-shade, which have led to a significant increase in the community’s interest in a future starshademission. There is interest in a starshade to rendezvous with NASA’s next flagship mission, theNancy Grace Roman Space Telescope, and a starshade is baselined for the proposed flagshipmission, the Habitable Exoplanet Observatory (HabEx). a r X i v : . [ a s t r o - ph . I M ] N ov hough significant progress has been made, starshades are still an unproven technology, par-ticularly with respect to their optical performance. The distributed architecture’s size (10’s ofmeters diameter over 10,000’s of kilometers) and sensitivity (10 relative change in intensity) areunprecedented—when constructed, the starshade will be the largest visible-light optic ever made.Consequently, we need a reliable means to experimentally validate the design tools that rely on anaccurate prediction of the optical performance. The state of the art optical models operate underthe assumptions of: a scalar theory of diffraction; scale invariance of the Fresnel approximation;and a scalar application of Babinet’s principle. The work presented here focuses on demonstratingthe validity of the first assumption.The diffraction equations employed in the starshade context are invariant with Fresnel number,so while it is impossible to test a full-scale starshade on the ground due to its size, we can validatethe optical models with sub-scale experiments if conducted at a flight-like Fresnel number. Previ-ous experiments in Refs. 13–19 were conducted at Fresnel numbers much larger than that expectedin flight; the experiment in Ref. 20 was done at a flight-like Fresnel number, but its contrast waslimited by the atmosphere. With the Princeton starshade testbed presented here, we are for the firsttime able to achieve 10 − contrast with a starshade at a flight-like Fresnel number.The Starshade to TRL 5 (S5) project was established by the NASA Exoplanet ExplorationProgram to advance starshade technology to TRL 5 in a time frame compatible with a starshaderendezvous with the Roman mission.
6, 8
The work presented here was conducted under S5 toadvance the seminal optical technology of starshades—starlight suppression. We group the ex-periments into two categories, “optical verification” and “model validation”, which reflect the twomilestones set as the criteria needed to reach TRL 5. Optical verification experiments show wecan design an apodization function, which specifies the starshade’s shape, that provides sufficient2ontrast to achieve our stated science goals. These experiments validate the fundamental operationof the starshade and demonstrate that the aforementioned assumptions are valid. Model validationexperiments show that the optical models correctly capture the performance sensitivity to pertur-bations in the starshade shape and set an upper limit to the model uncertainty used in design toolsto derive the starshade shape error budget and tolerances for future missions.To briefly summarize our main results, we have demonstrated 10 − contrast at the geometricinner working angle (IWA) of a starshade with a flight-like Fresnel number across a 10% bandpasswhile reaching a contrast floor of × − beyond the starshade tips. From this we conclude thatwe can predict the nominal starshade performance to at least 10 − contrast. The main limitation tothe contrast at the laboratory scale comes from a polarization-dependent effect of light propagatingthrough the narrow gap between starshade petals. We provide an explanation of this effect inSec. 3.1.1. The model validation experiments demonstrate better than 35% (with an average of20%) model accuracy for a number of different shape perturbations and at multiple wavelengths.This result means we must only carry a contrast margin of 1.35 × in the design error budget. Theresults from these experiments build confidence in our ability to successfully design a starshadethat will provide the contrast needed to detect Earth-like exoplanets.In Sec. 2 we describe the layout and individual components of the starshade testbed and outlinethe experiments performed. Section 3 presents the results of the optical verification experiments,Sec. 4 presents the results of the model validation experiments, and we discuss the implications ofthese results in Sec. 5. We summarize and conclude in Sec. 6. Additional details and results fromthe optical verification experiments can be found in the S5 milestone final reports,
21, 22 which havebeen accepted by the Exoplanet Exploration Program Technical Advisory Committee. Experiment Design
The experiments presented here were conducted at the Princeton starshade testbed, a dedicatedfacility in the Frick chemistry building on the Princeton campus. The testbed was designed toreplicate the flight configuration at 1/1000 th scale as closely as is possible given the differences insize and environment. The scale of the experiment is ultimately limited by the longest separationavailable in an indoor facility on campus. We use the Starshade Rendezvous Probe Mission (SRM)and HabEx mission as the reference flight configurations; parameters for the flight and laboratoryconfigurations are presented in Table 1. The table lists the operational range of the Fresnel number, N , defined as N = R λZ eff , (1)where R is the starshade radius, λ is the wavelength of light, and Z eff is the effective starshade-telescope separation: Z eff = Z src Z tel Z src + Z tel , (2)where Z tel is the distance between telescope and starshade and Z src is the distance between star-shade and light source. Z eff accounts for the finite distance to the diverging beam light source inthe laboratory configuration and is derived in Eq. (4). For the laboratory experiments, Z eff = 17 . m; for the flight configuration, the source (target star) is effectively infinitely far away, making Z eff ≈ Z tel .In this work we use the geometric IWA ( = R/Z tel ) as the point of reference, though futuredesign studies should follow Ref. 24 and set requirements relative to the effective IWA, whichaccounts for the width of the telescope’s point spread function (PSF).4 able 1
Optical parameters for the laboratory experiments and the SRM and HabEx starshade architectures. The rangeof Fresnel numbers is set by the wavelength range. † For apodization design C12/C16 in Table 3.
Laboratory SRM HabExTelescope diameter ( D ) (2 R ) †
26 m 52 mWavelength ( λ ) range 641 - 725 nm †
616 - 800 nm 300 - 1000 nmTelescope - starshade sep. ( Z tel ) ( Z src ) > parsec > parsec Fresnel number ( N ) − † − − For the range of N under study, the Fresnel approximation of diffraction is sufficiently accu-rate to compute the diffraction pattern to contrast levels better than − , an assertion borneout by the successful demonstration of a dark shadow in these experiments. We use the Fresnel-Kirchhoff diffraction equation to describe the diffraction in both the laboratory and flight config-urations. We invoke the standard paraxial and Fresnel approximations and assume circular sym-metry (with radial coordinate at the starshade r ). The electric field incident on the starshade ( U ) is due to a spherical wave of amplitude u emanating at a distance Z src : U ( r ) = u Z src exp (cid:26) iπr λZ src (cid:27) . (3)We assume the starshade’s shape is an approximation of a smooth radial apodization function, A ( r ) , a valid approximation given a sufficient number of petals. The Fresnel-Kirchhoff diffrac-tion integral to compute the on-axis electric field U at the telescope pupil plane (dropping the5eading phase factor) is given by U = 2 πiλZ tel (cid:90) R A ( r ) U ( r ) exp (cid:26) iπr λZ tel (cid:27) r dr = 2 πu iλZ tel Z src (cid:90) R A ( r ) exp (cid:26) iπr λZ src (cid:27) exp (cid:26) iπr λZ tel (cid:27) r dr = 2 πu iλZ tel Z src (cid:90) R A ( r ) exp (cid:26) iπr λ (cid:18) Z src + 1 Z tel (cid:19)(cid:27) r dr . (4)The distance terms in the exponential can be combined into an effective separation parameter, Z eff , given by Eq. (2). We set the amplitude of the incident wave to be unity at the starshade, togive u = Z src . Equation (4) can be rewritten as U = 2 πiλZ tel (cid:90) R A ( r ) exp (cid:26) iπr λZ eff (cid:27) r dr = πi Z eff Z tel (cid:90) N A ( n ) exp { iπn } dn , (5)with dimensionless quantity n = r /λZ eff spanning the range of Fresnel numbers up to N .Written in the dimensionless form of Eq. (5) and neglecting the constant amplitude scale factor ≈ , the integral is independent of R , λ , and Z eff , and depends solely on the Fresnel number.This enables experimental validation via more practical sub-scale experiments: by showing in thelaboratory that high contrast is achieved at flight-like values of N , we show that the full-scale,properly shaped starshade will also produce the same level of contrast. This statement is trueunder the assumption that scalar diffraction theory holds at both scales. The experiments describedhere show that scalar diffraction theory almost holds at small scales, as long as it is corrected withan additive non-scalar component at the edges. Extending the same modeling approach to flightscale shows that the additive terms makes a negligible contribution (see Appendix B), validating6he extension of the scalar diffraction theory result to flight scale.Due to size constraints of the testbed, the laboratory Fresnel number does not extend as low ‡ as those in the full scale configurations, but still falls within the range of Fresnel numbers set bythe span in wavelength. Additionally, the Fresnel number ( N ) quoted in Table 1 is in referenceto the maximum radius of the starshade, but the diffraction integral starts at the base of the petalswith n ∼ and integrates to N . This means our experiments will demonstrate that we accuratelycapture the behavior of the apodization function over the range of Fresnel numbers seen in theflight configurations. Testing at N < has been determined to be sufficiently adequate for thepurposes of model validation. Altogether, by utilizing a 80 m long facility, we are able to test at a flight-like Fresnel numbera starshade that is large enough to be accurately manufactured with existing technology.
The experiments presented here can be sorted into two loose categories that track the two S5milestones that this work is tasked to complete: optical verification
21, 22 (presented in Sec. 3)and model validation (presented in Sec. 4). Table 2 provides a summary of six experiments andthe name (production number) of the starshade tested. Improvements to the testbed were madebetween various experiments, the most dramatic of which was the installation of a linear polarizerafter the fiber launcher (see Sec. 2.3) and a linear polarizer as analyzer on a rotation stage in frontof the camera. This change occurred after completing experiments 1, 2, and 6 of Table 2. ‡ The lowest Fresnel number tested is N = 11 . in Sec. 4.4. able 2 Summary of experiments and production number of starshades tested (see Table 4 for details on specificstarshades). Those with OV in the Goal column are optical verification experiments and are presented in Sec. 3. Thosewith MV in the Goal column are model validation experiments and are presented in Sec. 4. < − contrast at the innerworking angle across a 10% bandpass. OV DW213 Exposedpetal tips 3.3 Demonstrate high contrast with a starshadewith realistic petal tips. OV M12P34 Shapeperturbations 4.2 Validate optical models against starshadeswith perturbations built into their shape. MV M12P2 /M12P35 Polarizationstudy 4.3 Improve non-scalar diffraction model bystudying polarization-dependent effects. MV DW9 /DW216 VaryingFresnel The design of the experiment is simple: image a light source from within the deep shadow createdby a starshade and measure the efficiency with which the starshade suppresses the on-axis light.The testbed (shown schematically in Fig. 1) consists of three stations containing a laser, starshade,and camera. The main driver in the testbed design was to maximize the starshade size while main-taining a flight-like Fresnel number, which translates to maximizing the separation. The testbeddesign is set by the longest, straight-line facility to be found on campus, which gives a total testbedlength of 80 m. Since a fraction of the length is needed for propagation of the diverging beam, theeffective separation between starshade and telescope is 17.7 m, which sets the starshade size to 25mm diameter.The beam line is contained in 1 m diameter steel tubing (not a vacuum) to seal the testbed8 ig 1
Layout of testbed showing distances between camera, starshade mask, and laser stations. from stray light and dust and to help stabilize the atmosphere. The tube is wrapped in fiberglassinsulation to minimize the effect of external thermal changes. All equipment is built to be remotelyoperated to minimize how often the testbed is opened, which generates atmospheric turbulence andstirs up dust. Additional details of the testbed design can be found in Refs. 21, 26, 27.
The light source serving as the artificial star is a multi-channel laser diode operating at: 405 nm,638 nm, 641 nm, 660 nm, 699 nm, and 725 nm. The 405 nm light is outside the starshade’soperating bandpass and is used for alignment. The laser is located outside the enclosure andfed in via a polarization-maintaining single-mode fiber optic. The polarization out of the fiberdepends on external environmental conditions, which vary with time. The fiber terminates witha collimator and the output gaussian beam is focused by an objective lens through a pinhole tospatially filter high-order aberrations. Experiments ig 2
Cartoon diagram of laser launching system. The optical fiber enters on the right from the laser outside thetestbed. Light is launched from the fiber, collimated, and passes through a linear polarizer before reaching a beam-splitter. 10% of the light is reflected to a photometer to record the throughput. The other 90% continues to an objectivelens which sends a diverging beam to the starshade. A pinhole at the focus spatially filters high-order aberrations. the transmitted power during observations and allows for the contrast calibration to be adjustedaccordingly. A cartoon diagram of the laser launching system is shown in Fig. 2.
The starshades tested are roughly 25 mm in diameter and are etched into a 100 mm silicon wafer.They are positioned in the middle testbed station and are held by a mask changer (shown in Fig. 3)with a motorized planetary gear that allows us to switch between starshade and calibration masksand to image the mask at different rotation angles.
The starshade mask (shown in Fig. 4) consists of an inner starshade, representative of a free floatingocculter, that is supported in a silicon wafer via radial struts. The outer diameter of the support10 ig 3
Left: starshade station located 50 m from the camera. A wall blocks all light expect that passing through thestarshade. Right: mask holder with a motorized planetary gear that holds the starshade and calibration masks. wafer is also apodized to minimize the diffraction that would occur from the truncation of the beamby the outer diameter. This design results in the starshade mask consisting of N p ( = number ofpetals) transmission regions bounded by the petals of the inner starshade, the radial struts, andpetals of the outer diameter.Both apodization profiles (inner and outer) are designed independent of each other using thenumerical optimization scheme outlined in Ref. 10. Table 3 details the designs of a number ofapodization functions; the minimum radius ( R ) is the radius at which the inner petals start and themaximum radius ( R ) is the radius at which the struts start (where the tips of a free floating occulterwould be). Design A of Table 3 is an earlier design with smaller gaps and a larger maximum radius.Design B was specifically designed for these experiments. We impose a constraint on the radiusof the inner starshade to have Fresnel number < and constrain the gaps between the starshadepetals to have widths > µ m to minimize non-scalar diffraction. We found this to be the largestgap width that provides a valid solution to the optimization problem. Design C16 is the same11 ig 4 Starshade pattern etched into a SOI wafer, manufactured at Microdevices Lab at JPL. Interior to the inner redcircle is the inner starshade representing a free floating occulter. The inner starshade is supported in the wafer by radialstruts. The outer red circle marks the start of the apodization function of the outer diameter. apodization profile as Design B, but is made 3% larger to shift the operating bandpass to cover thelaser’s available wavelength channels. Design C12 is the same as Design C16, but with 12 petalsinstead of 16, which was done to minimize the number of inner gaps between petals, which serveas sources of non-scalar diffraction.
Table 3
Design of apodization functions including the number of petals, operating bandpass, minimum radius (startof the petals), maximum radius (start of the struts), and gap width between petals.
Design Number Operating Minimum Maximum Gap Widthof Petals Bandpass Radius ( R ) Radius ( R ) A 16 600 nm - 670 nm 8.41 mm 17.35 mm 7.5 µ mB 16 600 nm - 690 nm 8.02 mm 12.15 mm 16.2 µ mC16 16 640 nm - 730 nm 8.26 mm 12.53 mm 16.2 µ mC12 12 640 nm - 730 nm 8.26 mm 12.53 mm 21.6 µ m12fter a solution to the optimization problem is found, the apodization profile is multiplied by0.9 to provide width to the radial struts and is then petalized to become the starshade shape shownin Fig. 4. Since the radial struts consist of a constant multiplication applied to the apodizationprofile, they do not diffract into the shadow. The starshade pattern is etched into the device layer of a silicon-on-insulator (SOI) 100 mm wafervia a deep reactive ion etching process. The allowed tolerances on the shape are very small, ∼
100 nm, which is achievable with a direct write electron beam lithography process. The SOIdevice layer is made as thin as is practical (1 µ m - 7 µ m) to minimize non-scalar diffraction as lightpropagates past the optical edge. The 350 µ m thick support wafer is etched from the backside torecess it 50 µ m from the device layer’s optical edge. The final step in the process is to coat the topof the device layer with a thin layer of metal to maintain opacity. Either 0.4 µ m of gold or 0.25 µ mof aluminum is used, both of which have thicknesses more than 50 times greater than their skindepth, so we expect the metal layer to be completely opaque. Details on the manufactured masksare found in Table 4. We refer the reader to Refs. 28, 29 for more details on the manufacturingprocess. The optics system in the camera station has pupil plane and focal plane imaging modes that aretoggled by remotely flipping a lens in/out of the optical path. Contrast measurements are made inthe focal plane imaging mode with the camera focused to the plane of the light source, simulatingan exoplanet observation. We use an f/100 system with a 5 mm diameter aperture, which provides13 able 4
Descriptions of manufactured masks including the apodization design (detailed in Table 3), the thickness ofthe device layer (optical edge), the thickness and type of metal coating, and the perturbations built into the shape.
Name Apodization Edge Thickness Metal Shape PerturbationsDesign ( ± µ m ) CoatingDW9 A 7 µ m Au - 0.4 µ m NoneDW17 B 2 µ m Au - 0.4 µ m NoneDW21 C16 3 µ m Au - 0.4 µ m NoneM12P2 C12 1 µ m Al - 0.25 µ m Displaced edgesM12P3 C12 2 µ m Al - 0.25 µ m Sine waves & exposed tipsthe same number of resolution elements across the starshade as in the flight design. As will beshown in Sec. 3.2, the contrast improves as light at the geometric IWA rolls off with the telescope’sPSF. A telescope that highly resolves the geometric IWA gets an added boost in contrast. As such,to test in a flight-like configuration, we scale the aperture to conserve the number of resolutionelements across the geometric IWA: n resolved = ( R/Z tel ) / ( λ/D ) .In pupil imaging mode, we observe the out-of-band diffraction pattern incident on the entrancepupil and use the bright spot of Arago to align the camera with the starshade, precisely what isdone in the formation flying scheme to maintain starshade alignment.
30, 31
To perform calibration measurements, a neutral density filter (optical density > − ) is tog-gled into the optical train by a motorized stage. A linear polarizer on a motorized rotation stageserves as a polarization analyzer. The detector is an Andor iXon Ultra 888 EMCCD with 13 µ mpixels. For low noise performance, the detector is operated with its conventional amplifier, i.e., notelectron-multiplying, and is liquid cooled down to -90 ◦ C.14 .6 Calibrations
A circular aperture mask is used to calibrate the throughput of the system and convert measure-ments of the occulted light source to a contrast value (see Appendix A for a definition of contrast).The calibration mask is a 50 mm diameter circle etched through a silicon wafer and switches posi-tion with the starshade mask via the motorized mask changer. For each set of observations, two setsof images are taken: one with the starshade mask in the beam and one with the calibration maskin the beam. When observing with the calibration mask, a neutral density filter is placed in theoptical path. The measured count rate for both sets of images are used in Eq. (11) of Appendix Ato calculate contrast. We refer the reader to Ref. 21 for additional details on the calibration process.
The first category of experiments are meant to verify the fundamental concept of a starshade bydemonstrating that we can design the starshade’s shape to provide the optical performance neededto image exoplanets. These experiments validate most of the assumptions (e.g., binary approxi-mation to a smooth function) made in the equations used to design the apodization function thatdefines the starshade’s shape. Verification is achieved by demonstrating better than 10 − contrastacross a wide bandpass with a starshade in a flight-like optical configuration. The results presentedin this section represent the completion of Milestones 1A and 1B of the S5 Project, whichsatisfy the first of two main requirements in the Starlight Suppression technology developmentplan. .1 Monochromatic contrast In this first experiment, we demonstrate the best contrast achieved with the highest quality mask(DW17) at a single wavelength ( λ = 638 nm). In the contrast image shown in Fig. 5, the brightestfeatures are two lobes that are aligned with the polarization vector of the incident light (the intrinsicpolarization of the fiber is slightly elliptical at a 40 ◦ angle) and which remain fixed as the maskis imaged at different rotation angles. The bright lobes are due to polarization-dependent changesin the electric field as light propagates through the narrow gaps between petals. We call this the“thick screen effect” and describe it in detail in Sec. 3.1.1. While these lobes are relatively brightat their peak, they are confined to two lobes at the inner gaps between petals and the contrastsignificantly improves in regions of the image away from the lobes. This means that despite thelobes, 10 − contrast is achieved over a significant fraction of the image at the geometric IWA.This is a key feature of the starshade: any light leaking around the starshade is confined to the edgeof the starshade in the image and rolls off with the telescope’s PSF at image locations away fromthe edge.The primary impact of the lobes is to slightly reduce the image area over which − contrastis achieved. The contrast is better than − over 44 % of a λ/D wide annulus centered at thegeometric IWA and quickly rises to 100 % at 1.05 × the IWA. Figure 8 shows the contrast averagedover a λ/D wide annulus as a function of angular separation and simulates the effect of rotatingthe starshade during the exposure to smear out the diffracted light. The average contrast at theIWA is . × − and quickly falls with angular separation to the contrast floor of × − .Figure 22 shows that the floor at large angles is set by the non-scalar diffraction lobes and Rayleighscattering by the atmosphere. ig 5 Contrast map for monochromatic experiment at λ = 638 nm with mask DW17. The starshade pattern is overlaidand the dashed circle marks the geometric IWA. The bright lobes in Fig. 5 are aligned with the input polarization vector, remain stationary as thestarshade rotates, and have a brightness that is an order of magnitude above that predicted byscalar diffraction theory. This was a new discovery that only appeared once high contrast levelswere achieved at a flight-like Fresnel number. We have since developed a theory, deemed the thickscreen effect, which readily explains their origin.The Fresnel-Kirchhoff diffraction formula, which is used to derive the apodization function17epresented by the starshade’s shape, makes the assumption that the electromagnetic field can berepresented by a single scalar wave function that satisfies the scalar wave equation, a valid as-sumption for most optical systems with features larger than the wavelength of light. More specif-ically, F-K diffraction assumes an infinitely thin, perfectly conducting diffraction screen and thatthe field in the plane of the screen takes Kirchhoff’s boundary conditions, where the field is zeroon the screen and is unchanged in the aperture.The starshades in the lab configuration are small enough that these assumptions begin to breakdown. The gap between two petals is ∼
20 wavelengths across and the screen (optical edge) isup to half as thick as the gaps are wide, so the gaps resemble waveguides more than thin screens.As light propagates past the thick edge of the starshade mask (through the waveguide), energyis lost due to the finite conductivity of the walls . The interaction with materials of the edge ispolarization-dependent and induces a slight change in the complex field (in both amplitude andphase) and Kirchhoff’s boundary conditions are no longer valid – the propagation of light can nolonger be described by scalar diffraction theory alone.The result of the interaction with the edge is an attenuation of the transmission coefficientthrough the gap. This can be approximated by a ∼ wavelength wide boundary layer aroundthe starshade’s edge that blocks light and changes the effective apodization profile of the petal,negating some of the light suppression. In effect, the width of each petal increases by ∼ λ on eachside. The strength of the edge effect is roughly the same moving radially outwards; however, thetransmission of the apodization profile rapidly increases radially, so the edge effect is strongest inthe narrow gaps between petals where it occupies an appreciable fraction of the transmission area.This explains why the lobes are only seen in the innermost regions. The interaction of light withthe edge is also dependent on the polarization of the incident light and thus the morphology of the18obes depend on the input polarization state.Our theory of the edge effect is supported by a newly developed optical model and its com-parison to experimental results is shown in Sec. 4.3. Under this theory, these effects should be neg-ligible at flight scales. The relevant parameter is the width of the thick screen-induced boundarylayer relative to the transmission area. The optical edges for a flight starshade will induce roughlythe same wavelength wide boundary layer, but the transmission area is 1000 × larger, meaningthe contribution from non-scalar diffraction is × lower and can be considered negligible (seeAppendix B for a derivation of this argument). Full vector models of flight-scale starshades arebeyond the scope of this paper, but will be addressed in the future. Testing over a wider wavelength range increases the applicability of the experiment to a moreflight-like configuration and demonstrates that a starshade can maintain its high contrast perfor-mance over a scientifically interesting bandpass. In this experiment we tested mask DW21 at fourdiscrete wavelengths that span a 10% (85 nm) bandpass. The design of DW21 is identical to, but3% larger than, DW17 in order to shift the starshade’s operating bandpass to cover the availablelaser wavelengths. Figure 6 shows again that better than 10 − contrast is achieved at the IWA andthat the performance is dominated by the thick screen effect. The contrast is relatively constantacross the bandpass, while the peak contrast is higher than in the monochromatic experiment dueto DW21 being slightly thicker than DW17, which produces a larger thick screen effect. The mor-phology of the polarization lobes in the λ = 660 nm data is most likely due to a misalignmentbetween the camera and starshade – Fig. 7 shows a model image where the camera is shifted off-axis by 1 mm and the lobes are distorted to one side of the starshade. Figure 8 shows the contrast19veraged over a λ/D wide annulus, where the contrast is slightly worse at longer wavelengths asthe PSF broadens and more light from the polarization lobes are leaked into the IWA. The averagecontrast at the IWA across the wavelengths is . × − . This experiment shows the starshadedoes not suffer any fundamental degradation in performance by operating across a wider bandpass,as is expected by theory. Fig 6
Contrast images of mask DW21 at four discrete wavelengths spanning a 10% bandpass. ig 7 Experiment (left) and model (right) images of mask DW21 at λ = 660 nm. In the model the camera is shiftedoff-axis by 1 mm, which distorts the polarization lobes, suggesting this can account for the morphology of the lobesseen in the data. In order to suspend the fragile silicon starshade in the testbed, the starshade design incorporatesradial struts that keep the starshade attached to the outer supporting wafer. This means the end ofthe petal never terminates at a tip. The diffraction equations show that the critical features of thestarshade are the inner valleys between the petals and the outer tips of the petals, i.e., where thepetal shape has a large azimuthal component. To verify that critical features such as tips behavein an expected manner, and to make the test article reflect a more flight-like configuration, wetested a mask built with exposed tips. Figure 9 shows the design of mask M12P3 where the innerstarshade is slightly rotated relative to the struts and outer apodization function to expose tipsat the end of the petals. As the petalized starshade is an approximation to a radial, azimuthally-21 ig 8
Contrast averaged over a λ/D wide annulus as a function of angular separation for different wavelengthobservations of mask DW21. Also included is the monochromatic observation of mask DW17. The solid black linedenotes the starshade’s geometric IWA. symmetric apodization profile, rotating the inner starshade does not change the approximated radialapodization profile and therefore should achieve the same contrast.Figure 10 shows the residuals between experiment and model for a stacked image of maskM12P3 at λ = 725 nm. We imaged the starshade rotated by 0 ◦ , 120 ◦ , and 240 ◦ , de-rotated theimages to align on the perturbations, and then median combined. Imaging at different orientationslessens the impact of the central polarization lobes. The residuals ( | experiment − model | ) ofthe stacked images are shown in Fig. 10. In addition to the exposed tips, M12P3 has sine wave22 ig 9 A closeup of one petal in the design of mask M12P3. The inner starshade is rotated relative to the outer, creatingexposed tips at all petals (seen at 12.5 mm). The blue hatched area is transparent; the white area is the silicon wafer. perturbations built into its shape for model validation (see Sec. 4.2.2) and a defect leftover fromthe manufacturing process, which dominate the contrast in the residual image.Taking a λ/D wide annulus at the angular separation of the tips, and excluding the portionthat lies on the sine wave perturbation (at 6:00 in the image), we find the average residual contrastin the annulus is × − . Inspection of Fig. 10 shows there is not significant residual light atthe location of the tips and that most of the residual is due to leakage of unmodeled non-scalardiffraction from the inner gaps of the starshade and from the perturbations. From this we concludethat the exposed tips behave as expected (to the 10 − contrast level) and the starshade still providessufficient contrast outside the IWA. 23 ig 10 Residuals between experiment and model of stacked [log-scale contrast] images of mask M12P3 at λ = 725 nm. Lab and model images were created by median combining images taken at three starshade orientations.The brightest spots are due to intentional sine wave perturbations (see Sec. 4.2.2) and an unintentional manufacturingdefect. High contrast is still achieved at the location of the exposed tips (marked by the dashed circle). There will always be slight deformations in the starshade’s shape due to manufacturing errors,deployment errors, and thermal and mechanical stresses. A contrast error budget sets toleranceson the allowed deformations by balancing aspects of the mechanical design deemed to be mostchallenging. One purpose of the model validation experiments is to validate the accuracy atwhich the models used to derive the error budget capture the sensitivity of contrast performance toshape perturbation; by observing starshades with known perturbations built into the shape, we can24alidate how the contrast changes in a known way. The validation accuracy set by the experimentsdetermines the Model Uncertainty Factor (MUF) between contrast and shape in the error budget.The MUF is a multiplicative term applied to the intensity (contrast) of each term in the error budgetthat provides a margin for inaccuracy of the model. Reducing the MUF through experimentationallows us to reduce the contrast margin budgeted to model uncertainty and leads to a more efficientdesign. The shape changes selected for the validation experiments are related to the mechanicalarchitecture of the SRM design and are representative of the shape errors in the error budget. The model we are trying to validate uses scalar diffraction only, which is believed to be suf-ficiently accurate for the flight-scale starshade. However, since the discovery of non-scalar, thickscreen effects in the lab configuration, additional work must be done to include these effects in themodel in order to properly demonstrate that the perturbations are accounted for at flight scales.In this section, we first present results from two experiments testing perturbations that dominatethe error budget; experiments on other perturbations are in progress and will be reported at alater date. For the perturbed shape tests, the perturbations are made large enough so that thepredicted light leakages are dominated by scalar diffraction, with only slight contributions fromthe thick screen effect in the inner parts of the starshade. Later in this section, we present resultsfrom a polarization study demonstrating the non-scalar model accurately captures the thick screeneffect. Estimates of this model applied to flight-scale starshades shows that the effect is negligibleas expected (see Appendix B), but a full analysis is saved for a later date. We end this sectionwith an experiment testing the Fresnel number dependence on contrast by testing a starshade ata Fresnel number outside of its designed Fresnel space. We show the model accurately capturesthis transition, as well as the transition between the scalar and non-scalar diffraction dominatedregimes. The results presented in this section are progress towards Milestone 2 of the S5 Project,25hich will satisfy the final requirement in the Starlight Suppression technology development plan. Several methods are available for solving the scalar diffraction problem, all of which have beenshown to be in agreement. Two methods, a boundary diffraction wave method and a similar an-gular integral method,
9, 12 convert the two-dimensional diffraction equation into a one-dimensionalline integral around the occulter’s edge and are well suited to capture the large dynamic rangeof sizes in the starshade shape. In this work, we use the model of Ref. 36, which uses a two-dimensional Fresnel propagator (shown to be in agreement with the boundary methods ) to in-clude non-scalar diffraction. We provide a brief summary of the optical model in Appendix C andwe show in Sec. 4.3 that our implementation of non-scalar diffraction agrees with the data. Here we present results from testing two masks with different classes of perturbations: displacededges and sine waves. Each mask has two perturbations of different sizes built into its design.The sizes are chosen to produce a signal in the image that is bright enough to overcome contri-butions from the thick screen effect, but faint enough to be informative to model validation. Oneperturbation is located on the inner starshade and is made brighter since it is closer to the centralpolarization lobes; the other defect is located on the outer starshade and is allowed to be fainter.The perturbed masks have 12 petals to minimize the thick screen effect; fewer gaps between petalsmean there are fewer sources of non-scalar diffraction and the lobes are (12 / times as bright.Details of the perturbations are presented in Table 5 and the locations of inner and outer perturba-tions are shown in Fig. 11. 26 ig 11 Location of inner and outer perturbations on the perturbed masks M12P2 and M12P3. The red arrows are aquiver plot of the sine waves on M12P3 with amplitudes greatly exaggerated. The blue hatched area is transparent;the white area is the silicon wafer.
In comparing contrasts between experiment and model, we draw a photometric aperture of ra-dius λ/D around the perturbation and calculate the average contrast in that aperture using Eq. (12)of Appendix A, with error given by Eq. (14) of Appendix A. The expected contrast presented inTable 5 is the average contrast in the photometric aperture, calculated under the assumption ofscalar diffraction only. 27 able 5
Description of shape perturbations, including their expected contrast (photometric average, see Eq. (12)) attwo wavelengths. Each perturbed mask ( a M12P2, b M12P3) has an inner and outer perturbation of different size.
Perturbation Location Description Expected Mean Contrast λ = 641 nm λ = 725 nmDisplaced Edge Inner Petal a µ m tall, 414 µ m long 2.3 × − × − Displaced Edge Outer Petal a µ m tall, 532 µ m long 6.5 × − × − Sine Wave Inner Petal b µ m amp., 4 cycles over 2.9 mm 1.9 × − × − Sine Wave Outer Petal b µ m amp., 5 cycles over 2.3 mm 9.6 × − × − The displaced edge perturbation simulates the effect of a petal edge segment being displaced fromits nominal position during petal assembly on the ground. Figure 12 shows a 3.7 µ m tall dis-placed edge segment built into the design of the manufactured mask M12P2. The locations of theperturbations on the starshade petals are shown in Fig. 11.Figure 13 shows the experimental and model contrast images at four wavelengths across thebandpass. The model includes polarization effects and manufacturing defects found on the maskthat are larger than 30 square microns. The input polarization vector is horizontal in the imageso that the perturbations are away from the inner polarization lobes. Figure 14 plots the averagecontrast in a λ/D radius photometric aperture centered on each perturbation. By design, the innerperturbation is brightest and drops in contrast with wavelength. The model shows good agreementwith the experimental data.Figure 15 plots the difference in the photometric aperture averaged contrast between the ex-perimental and model data for each of the perturbations. The percent difference is calculated as Percent Difference = | Model − Experiment | Model × , (6)28 ig 12 Microscope image of the inner displaced edge perturbation on M12P2. The red line denotes the edge of thenominal starshade shape. The 3.7 µ m step in the edge simulates a displaced edge segment. where the values for Model and Experiment are the photometric aperture averaged contrast. Theerror bars in Figure 15 are the experimental uncertainty propagated to the percent difference, andare calculated as Percent Uncertainty = σ Experiment
Model × , (7)where σ Experiment is calculated from Eq. (13) of Appendix A.The agreement is better than 20% for all perturbations except for the outer perturbation at λ = 699 nm. The difference for the outer perturbation at 699 nm is ∼ ; Fig. 13 shows that it ismuch dimmer than expected and does not follow the trend of the different wavelengths. This effectis seen in three orientations of the mask (rotated by ± ◦ ) and has been repeated several times.We currently do not have a good explanation as to why this happens, but we are continuing torefine the characterization and model of the optical edges in hopes of explaining this observation.29 ig 13 Contrast images of M12P2 − displaced petal edge segment perturbations. Experimental data are in the leftcolumn, model data are in the right column. The rows are for the different wavelengths. The perturbation on the innerpetal is circled in cyan and the perturbation on the outer petal is circled in red. The red ‘x’ marks the center of thestarshade. ig 14 M12P2 − displaced edge perturbations. Contrast for the Inner and Outer perturbations across the four wave-lengths. Solid markers are experimental data; open markers are model data. The contrast is the photometric apertureaverage given by Eq. (12) and the error bars (on experimental data only) are σ , given by Eq. (14). Error bars onexperimental data that are not visible are smaller than the symbols. Sinusoidal changes to the edge shape can occur if individual edge segments are misplaced in sucha way that their envelope creates a sine wave with respect to the nominal edge position. This is aparticularly harmful perturbation if the sine wave is in sync with the Fresnel half-zones and theyconstructively interfere. This also places a strong wavelength dependence on the contrast they31 ig 15
M12P2 − displaced edge perturbations. Percent difference between experimental and model contrast for theInner and Outer perturbations across the four wavelengths. The contrast is the photometric aperture average given byEq. (12) and the error bars are the percent uncertainty given by Eq. (7). The value for the Outer perturbation at 699 nmis off the chart (at 60%). induce. The locations of the sine wave defects are shown in Fig. 11.Figure 16 shows the images taken of this perturbed mask. For M12P3, in addition to theintentional perturbations, a large defect is left over from the manufacturing process and is thebrightest source in the image. SEM images show this is a large defect that extends vertically belowthe wafer’s device layer and has a complicated, protruding structure. We estimate the area to be32 ig 16 Contrast images of M12P3 − sine wave perturbations. Experimental data are in the left column, model dataare in the right column. The rows are for the different wavelengths. The perturbation on the inner petal is circledin cyan, the perturbation on the outer petal is circled in red, and the unintentional manufacturing defect is circled inmagenta. The red ‘x’ marks the center of the starshade. − Fig 17
M12P3 − sine wave perturbations. Contrast for the Inner and Outer perturbations across the four wavelengths.Solid markers are experimental data; open markers are model data. The contrast is the photometric aperture averagegiven by Eq. (12) and the error bars (on experimental data only) are σ , given by Eq. (14). Error bars on experimentaldata that are not visible are smaller than the symbols. Figure 17 plots the average contrast in a photometric aperture centered on each perturbation.The sine wave perturbation has a strong response with wavelength, and the inner and outer per-34urbations occasionally switch which is brightest, behavior that agrees with model predictions.Figure 18 shows the comparison between the experimental and model contrasts, with the percentdifference calculated from Eq. (6) and the uncertainty on that difference calculated from Eq. (7).For this mask, since the inner and outer perturbations are of equal brightness, the inner one gener-ally performs worse as it is closer to and suffers more contamination from the central polarizationlobes. Both the inner and outer perturbations agree with the model to better than 35% at all wave-lengths. This agreement is not as good as that of the displaced edges mask, which we attribute tothe presence of the large manufacturing defect. Due to the complicated, three-dimensional struc-ture of the manufacturing defect, it is difficult to model the interaction between the defect andthe sine wave perturbations. We believe this unmodeled interaction is the source of the largerdiscrepancy.
Our proposition that the bright central lobes are due to the thick screen effect can be validated byexamining the polarization induced as light propagates past the mask. In the explanation providedin Sec. 3.1.1, the change in the electric field is dependent on the polarization direction relative tothe sidewall of the mask. The effect is dominant in the inner gaps between petals, which can beapproximated by parallel plates aligned with the clocking angle of the petal. Assuming horizontallinearly polarized light, petals at the 3:00 and 9:00 positions have all s -polarization (electric fieldparallel to wall), petals at 12:00 and 6:00 have all p -polarization (electric field perpendicular towall), and petals in between have a mix of both. The s -polarization is subject to a greater changein the electric field (the field goes to zero at the walls of a perfect conductor ), so the polarizationlobes are aligned with the input polarization direction. Figure 19 confirms this as we view hori-35 ig 18 M12P3 − sine wave perturbations. Percent difference between experimental and model contrast for the Innerand Outer perturbations and the unintentional manufacturing defect across the four wavelengths. The contrast is thephotometric aperture average given by Eq. (12) and the error bars are the percent uncertainty given by Eq. (7). zontally polarized input light with an analyzing polarizer in front of the camera rotated to differentangles. When the analyzer is exactly orthogonal to the input polarizer, scalar diffraction theorypredicts no light should be visible. However, we see four lobes that are the result of unbalancedvertical polarization induced by the petals at ◦ , ◦ , ◦ , ◦ .Figure 20 shows experimental and model data at λ = 641 nm for the 7 µ m thick mask DW936 ig 19 Experimental (top) and model (bottom) images of mask DW9 imaged with the camera analyzer rotating relativeto the input polarizer direction (horizontal in the image). The angle given is that between the analyzer and polarizer,where 0 ◦ means they are aligned, 90 ◦ means they are crossed. Data are taken with λ = 641 nm. with the analyzer aligned and crossed with the input polarization. The model agrees well with theexperiment and is able to replicate the major polarization features, even at contrast levels below − . Figure 21 shows experimental and model data at λ = 641 nm for the 3 µ m thick maskDW21 (these data are taken with linear polarizer installed and thus differ from the DW21 datataken in Sec. 3.2). Again, the model agrees well and the peak contrast is ∼ × fainter thanthose of DW9, which is consistent with the thick screen effect scaling linearly with edge thickness,as should be expected. This demonstration shows we understand the source of the non-scalardiffraction, are able to replicate it’s dependency on edge thickness, and that it is due solely to thesmall scale of the experiment and will not be an issue for flight. Appendix B provides an estimateof the strength of this effect at flight scales. The sub-scale experiments presented here are informative for a starshade mission because of the37 ig 20
Experimental (left) and model (right) images of mask DW9 at λ = 641 nm. In the top row, the camera analyzeris aligned with the input polarizer (with a slight 15 ◦ misalignment). In the bottom row, the camera analyzer is crossedwith the input polarizer. Note the difference in colorbar scales. scale invariance of the diffraction equations under the Fresnel approximation. Therefore it is crit-ical to validate this assumption via an exploration of the Fresnel parameter space, starting in re-gions of high contrast and watching the contrast degrade as the experiment moves beyond thedesign space. Future testing should deconstruct the Fresnel number into its three constituent parts ( R, λ, Z ) and explore the contrast dependence of each around the flight-like Fresnel number, ig 21 Experimental (left) and model (right) images of mask DW21 at λ = 641 nm with polarized light. In the toprow, the camera analyzer is aligned with the input polarizer. The bright spot at 10:00 is dust on the mask. In the bottomrow, the camera analyzer is crossed with the input polarizer. Note the difference in colorbar scales. however, the size of this testbed puts a practical constraint on how much of the Fresnel parameterspace can be explored. Section 3.2 explored the wavelength dependence within the designed Fres-nel space; in this section, we explore the size dependence and push beyond the designed Fresnelspace to where the contrast begins to degrade. This experiment is a first step in an exploration ofFresnel space and should be followed by more extensive tests in the future.39igure 22 shows the results of testing a starshade outside of its designed Fresnel space, wherethe change in Fresnel number is achieved by changing the size of the starshade. Masks DW17 andDW21 have identical apodization functions, but DW21 is 3% larger, which means λ = 725 nmlight is in the designed Fresnel space of DW21 ( N = 12 . , but outside that of DW17 ( N = 11 . .Figure 22 shows the results of both masks at 725 nm; the contrast rises steeply with Fresnel num-ber and the model captures this transition well, with even better model-experiment agreement atthe brighter contrast. At these brighter contrast levels, the diffraction is completely dominated byscalar diffraction and both the scalar and non-scalar models converge. As the model agreement im-proves as the experiment moves beyond the regime where non-scalar diffraction is present and intoa scalar-only regime, we build confidence that the scalar model will accurately predict performancein configurations where non-scalar diffraction is less prominent. The starshade enables the detection of exoplanets because it provides high contrast at a smallIWA, which means it operates at a moderately low Fresnel number where the diffraction is gov-erned by complicated, near field equations. The apodization function that provides sufficientlyhigh contrast is found by solving an optimization problem that minimizes the electric field in theFresnel-Kirchhoff diffraction equation. This radial apodization function is then approximated bya petalized binary occulter that is the starshade’s edge. Until now, it had not been shown that wehave the tools capable of designing an apodization function that sufficiently suppresses diffractionat low Fresnel numbers and that the petalized approximation is valid. As the Fresnel number isdecreased, diffraction becomes harder to control as the apodization function spans fewer Fresnelzones over which to average out imperfections. As such, it was necessary to demonstrate that high40 ig 22
Azimuthally averaged contrast for masks DW17 and DW21 at λ = 725 nm, and the corresponding vector andscalar propagation models. The vertical lines mark the tip of the inner and outer apodization profiles. Also shown areestimates of Rayleigh scattering generated from the models of Ref. 32. contrast was achievable at low Fresnel numbers. The experiments presented here show for the firsttime that the design tools used to solve for the apodization function are capable of achieving − contrast at a flight-like Fresnel number. The experiment contrast floor was ∼ − at the IWA and ∼ − near the outer starshade, allowing us to conclude that approximations in the model are noworse than − for the nominal starshade design.The observed thick screen effect represents a worse case scenario in which our assumption of apurely scalar theory of diffraction is no longer valid and the optimized apodization is no longer ap-plied to the appropriate problem. However, even at the small scales of the laboratory configuration,the contribution from non-scalar diffraction is below the target contrast level. Additionally, we ex-41ect deviations from scalar diffraction theory to go away at larger scales as the sizes of features getmuch larger than the wavelength; scaling to a larger starshade for flight should work in our favor.If our theory of the thick screen is correct, the edges of the full-sized starshade will induce thesame wavelength wide boundary layer near the edge, but the size of the starshade will be 1000 × larger, meaning the impact of the non-scalar diffraction will be 10 × smaller in intensity and willbe reduced to a negligible amount (see Appendix B). Additional work may be needed to completethe validation of the non-scalar diffraction model, but the experiments completed thus far havebuilt confidence that we can achieve the same, if not better, contrast at larger scales. Given historicunderstanding of how light behaves around features with sizes comparable to the wavelength, itwas known non-scalar diffraction could be present, but we previously did not have adequate toolsto quantify the effect. The experiments in this work have helped to develop those tools and willallow us to apply them to future configurations.The model validation experiments aim to determine the accuracy to which the model can pre-dict the contrast sensitivity to known effects. Some of the experimental noise is due to unknownmanufacturing errors, misalignment, turbulence, and stray reflections, and is not attributable to themodel. Without detailed measurements of these error sources, we can only set an upper limit to themodel inaccuracy. The results presented in Sec. 4.2 show the model remains accurate to at least the35% level, with an average difference of 20%. The model agreement is even better ( < ) for thedisplaced edge perturbation data on mask M12P2, as that mask did not have a large manufacturingdefect as mask M12P3 did. This leads us to believe that the dominant source of model disagree-ment is due to uncharacterized perturbations in the manufactured mask, where due to the thickscreen effect, the three-dimensional structure of the mask becomes relevant. The large manufac-turing defect in M12P3 has a complex vertical structure that is difficult to adequately characterize42nd thus difficult to include in the model. Global defects, such as a variable overetching, are alsodifficult to characterize and can contribute to the model disagreement.Figure 23 plots the measured contrast against the model prediction for all perturbation data,along with the absolute difference between the two and the measurement error. The measurementstrack well with the model, even down to the lowest contrast levels. The measurement-model differ-ence and the measurement error follow the same trend as the measurements, meaning the fractionalerror is not increasing. These results show that the model inaccuracy scales with the size of theperturbation and thus justifies the use of a multiplicative MUF. To provide sufficient margin in theerror budget due to model uncertainty, the contrast from each perturbation must by increased by afactor of 1.35 × . While this conclusion certainly holds for perturbations down to 10 − contrast, dueto the non-scalar diffraction present in the laboratory configuration, comment on the behavior ofperturbations at lower contrast levels requires a cautious extrapolation of the trend seen in Fig. 23.The experiments presented here are only the first of a suite of model validation experiments,so these conclusions on model accuracy are not complete. Future experiments will include globaland mixed perturbations and will help us further investigate coherent effects. We have presented results from a number of experiments demonstrating the best contrast achievedwith a starshade at a flight-like Fresnel number. We achieved better than − contrast, the levelneeded to detect Earth-like exoplanets, at the geometric IWA and across a scientifically interestingbandpass. The starshades tested were not perfectly manufactured, their small size introduced non-scalar diffraction, and the experiments were conducted in air. That we were able to achieve suchhigh contrast even in the presence of these factors shows the efficiency and robustness in which43 ig 23 Model vs. measured data for all perturbations on the displaced edge and sine wave perturbations and fourwavelengths. The absolute difference between model and measured data and the measurement error are also shown. starshades operate and should garner confidence in the community for a successful starshade mis-sion in the future.In future work, we will continue model validation experiments on test masks with more pertur-bations representative of those in flight. These include: displacing a single petal radially; displac-ing all petals radially; and combining an edge segment displacement with a petal displacement.We will also continue to refine the thick screen model through tests of masks with different thick-44esses and formally apply that model to the flight design to show these effects are negligible atflight scales. Completion of these experiments will advance the starlight suppression technologyfor starshades to TRL 5 and starshades will be ready for selection for the next exoplanet mission.
Appendix A: Contrast Definition
Contrast is defined as the amount of light within a resolution element of a telescope (at the imageplane), divided by the peak brightness of the main light source as measured by that telescope whenthere is no starshade in place. The following equations define the contrast in terms of quantitiesmeasured in the lab; Table 6 provides descriptions of the variables used in the definition.
Table 6
List of variables used in contrast definition.
Symbol Description C i Contrast at pixel i Γ i Transfer function mapping to pixel iγ Ratio of PSF peak after propagation through free spaceto that of the calibration mask A Peak value of apodization function (= 0 . s i Counts collected in pixel i during exposure of length t [ct] P Laser power [W] ν Neutral density filter transmission ε Transmission through atmosphere τ Throughput of camera optics Q Quantum efficiency of detector [ e − /ph] E Photon energy [J/ph] G Camera inverse-gain [ e − /ct] t Exposure time [s] m Denotes mask measurement u Denotes unocculted / calibration measurementFree space propagation is not possible in the confinements of the lab, thus we use a circu-45ar calibration mask to measure the unocculted brightness and convert to a free space brightnessthrough modeling. In the following definitions, the subscript of a symbol will denote the obser-vation mode, with m denoting measurements made when the starshade mask is in place and u denoting unocculted measurements when the calibration mask is in place.We define the contrast at pixel i as C i = Γ im Γ free space = Γ im γA Γ u , (8)which is a theoretical construct specifying the reduction in brightness the starshade mask provides,relative to the on-axis unocculted source.The peak value of the apodization function ( A ) in the denominator accounts for the fractionof light that is blocked by the radial struts supporting the inner starshade. The value γ is the ratioof the peak of the PSF after propagation through free space to that after propagation through thecircular calibration mask, and relates the contrast measured in the lab to that expected from a free-floating starshade. More details on this conversion can be found in Ref. 21. The transfer functionis tied to a measurement through the equation s ix = (cid:18) P νεQτ t Γ i EG (cid:19) x , (9)where x ∈ { m, u } . We drop the superscript on s u and assume it is on-axis. We assume there isno ND filter during starshade measurements ( ν m = 1 , ν u ≡ ν ) and that the photon energy, cameragain, and camera throughput do not change between observation modes. Substituting Eq. (9) into46q. (8), we rewrite the contrast as C i = (cid:18) νs im t u P u γA s u t m P m (cid:19) (cid:18) ε u Q u ε m Q m (cid:19) . (10)We assume that values in the right parentheses have the same mean between observation modes,but whose true value during a given observation is distributed normally around the mean withvariance σ . In other words, Q u = Q m ≡ Q , σ Q u = σ Q m ≡ σ Q , and similarly for ε . Thissimplifies the contrast definition to C i = νs im t u P u γA s u t m P m . (11)For model validation, we calculate the average contrast over n pixels that lie in a photometricaperture of radius λ/D centered at that pixel as C = 1 n n (cid:88) i C i = 1 n νt u P u γA s u t m P m n (cid:88) i s im ≡ α n (cid:88) i s im , (12)where we have wrapped values independent of pixel into the constant variable α . The uncertaintyin α is given by σ α α = σ s u s u + σ γ γ + σ ν ν + σ P u P u + σ P m P m + 2 σ ε ε + 2 σ Q Q . (13)The variance in the counts of the unocculted image, σ s u , is given by Eq. (15). We estimate thevalues of the rest of the uncertainties of α in Ref. 21 to find σ α ∼ . . Assuming independent47easurement errors, the uncertainty in the average contrast is propagated to σ C C = σ α α + (cid:80) ni σ s im ( (cid:80) ni s im ) , (14)where σ s im is the variance of pixel i in the mask image.The dominant contributions to the uncertainty in counts collected during each exposure ( s u , s m ) are: photon noise from the source, background light, and detector dark noise and read noise. Wecan ignore noise from clock induced charge in the detector electronics, as this is estimated tobe < × − events/pixel. Read noise is estimated from the variance of 2 bias frames of 10 µ sexposure time to be σ R = 3.20 e − /pixel/frame. We combine the number of counts from backgroundlight and detector dark noise into the variable d , which is estimated from dark exposures taken withan equal exposure time. The uncertainty in the measurement of s counts in a single image j is givenby σ s j = s j G + 2 σ d + σ R . (15)For each observation mode, a number ( n frames ) of frames are taken and median-combined intoa master image. The variance in s counts obtained from n frames frames is given by σ s = σ s j n frames . (16)Additional details on noise sources and calibrations can be found in Ref. 21.48 ppendix B: The thick screen effect at flight scales In this appendix, we estimate how the thick screen effect scales with starshade size and argue theeffect is negligible at flight scales. To start, we derive an expression for the change in intensity atthe telescope due to the thick screen effect (assuming it induces a local change in amplitude only)and show that the expression is consistent with experimental results. We then apply the expressionto the flight-scale starshade and show the effect is negligible.
B.1 Derivation of intensity estimation
We begin with the Fresnel-Kirchhoff diffraction equation and assume an incident plane wave with z = Z eff . The on-axis electric field U is given by U = 2 πiλz (cid:90) R A ( r ) exp (cid:26) iπr λz (cid:27) r dr , (17)where A ( r ) is the circularly symmetric apodization function. We characterize the thick screeneffect as a change in the apodization function ( α ) relative to the nominal shape ( A ) such that A ( r ) = A ( r ) + α ( r ) . Equation (17) then becomes U = 2 πiλz (cid:90) R [ A ( r ) + α ( r )] exp (cid:26) iπr λz (cid:27) r dr = 2 πiλz (cid:90) R A ( r ) exp (cid:26) iπr λz (cid:27) r dr + 2 πiλz (cid:90) R α ( r ) exp (cid:26) iπr λz (cid:27) r dr = U nominal + ∆ U . (18) U nominal is the nominal electric field in the scalar diffraction limit and ∆ U is the change inelectric field as a result of the thick screen effect. In Sec. 3.1.1 we posited, and the models con-49rm, that the presence of the thick screen induces a change in the electric field in a narrow ( ∼ λ )boundary layer around the edge. For now, we restrict ourselves to the case where the presence ofthe screen induces a change in amplitude only, resulting in a ∼ λ wide boundary layer around thestarshade edge with zero transmission. Since the width of the boundary layer, which we defineas δ , is roughly constant for each edge (we neglect differences between polarization states), thechange in the apodization function is related to the boundary layer width by α ( r ) = , r < R δN p πr , R ≤ r ≤ R , (19)where N p is the number of starshade petals, R is the minimum radius at which the petals start,and we’ve acknowledged that there is no change in the apodization before the petals starts. Thethick screen effect can now be written as ∆ U = 2 πiλz (cid:90) RR (cid:18) δN p πr (cid:19) exp (cid:26) iπr λz (cid:27) r dr = δN p iλz (cid:90) RR exp (cid:26) iπr λz (cid:27) dr . (20)The integral in Eq. (20) can readily be evaluated in terms of Fresnel integrals. We define thecomplex Fresnel integral as F ( u ) ≡ C ( u ) + iS ( u ) ≡ (cid:90) u e iπt / dt , (21)50nd make the appropriate substitutions to write the thick screen effect as ∆ U = (cid:18) δN p iλz (cid:19) (cid:114) λz (cid:104) F (cid:16) √ N (cid:17) − F (cid:16)(cid:112) N (cid:17)(cid:105) , (22)where N is the Fresnel number at the starshade radius and N is the Fresnel number at the start ofthe petals. The change in (on-axis) intensity due to the thick screen effect is then | ∆ U | = δ λz N p (cid:12)(cid:12)(cid:12) F (cid:16) √ N (cid:17) − F (cid:16)(cid:112) N (cid:17)(cid:12)(cid:12)(cid:12) . (23)We note that the leading factor looks like a Fresnel number across the width of the boundary layer. B.2 Comparison to experimental results
We will now use Eq. (23) to estimate the change in intensity for the laboratory configuration andcompare to experimental results of mask DW9. Equation (23) is the change in on-axis intensity atthe telescope plane, so we will compare our estimates to the suppression values calculated frompupil plane images. Suppression is the total intensity incident on the telescope’s aperture when thestarshade is occulting the star, relative to the total intensity of the unblocked star.At a wavelength of λ = 638 nm, finite-difference time-domain (FDTD) simulations of themask DW9 geometry yield an average change in amplitude consistent with a boundary layer widthof δ = 0 . µ m (averaging over polarization states). We input the parameters from Table 3 intoEq. (23) to estimate the change in intensity to be | ∆ U | = 1 . × − . Figure 24 shows thesuppression plot (image of the pupil plane) for DW9 at λ = 638 nm. These data were takenwithout any polarizing elements, so represent a rough average over polarization states. The peaksuppression is . × − and the average over the aperture is . × − , which is greater than,51ut within an order of magnitude of, that predicted by Eq. (23). These results show our estimate ofthe thick screen effect is within reason and we note that Eq. (23) was a lower limit as it assumedan amplitude-only change in field due to the thick screen. Fig 24
Suppression (pupil plane) image of DW9 at λ = 638 nm, without any polarizing elements. B.3 Scaling up to flight
By examining Eq. (23), we can see why the non-scalar diffraction should be negligible at flightscales. Scaling from the lab configuration to that of flight, the Fresnel numbers are the same and δ will be roughly the same (the optical edges of the flight design are of similar thickness to those inthe lab), so the thick screen effect intensity goes as | ∆ U | ∼ z − . The effective starshade-telescope52eparation for flight is 10 × that in the lab (the starshades are 1000 th scale and the separation scalesquadratically with size for a given Fresnel number), so we can expect the non-scalar diffraction tobe 10 × lower in intensity.Replacing λz in the denominator of Eq. (23) with R N gives | ∆ U | ∼ δ /R , which is thesame argument given in Sec. 3.1.1 that the thick screen effect goes as the area of the boundary layerrelative to the transmission area. A final observation: the leading factor in Eq. (23) is similar to aFresnel number across the boundary layer, which becomes increasingly small at large separations(because the width remains constant), meaning there is little phase variance across the width. Appendix C: Optical Model Description
The optical model summarized here is presented in full in Ref. 36. The scalar diffraction prob-lem we are solving uses the standard Fresnel and Kirchhoff assumptions that allow us to expressthe diffraction propagation as a Fourier transform. Because the inner starshade is connected tothe outer supporting wafer via the radial struts, the starshade pattern can be treated as individualapertures and standard Fourier techniques sufficiently resolve the starshade. If the coordinatesof the screen and observation planes are given as ( ξ, η ) and ( x, y ) , respectively, we can write theFresnel-Kirchhoff diffraction equation as U ( x, y ) ∝ F (cid:104) U ( ξ, η ) · A ( ξ, η ) · e ik z ( ξ + η ) (cid:105) ( x,y ) , (24)where U is the initial field incident on the screen, A represents the aperture function of the screen,and F [ · ] ( x,y ) is the Fourier transform. Using the Kirchhoff boundary values, A is a binary functionthat is 0 on the screen and 1 in the aperture. Our method implements non-scalar diffraction into this53odel by replacing the boundary values of A with a complex field that arises from local diffractionat the edge of the screen. The presence of the screen only affects its immediate surrounding, so weonly change the values of A in a narrow ( ∼ λ wide) seam around the edge of the screen, similarto the method proposed by Braunbek. The field in the seam around the edge is solved for via an FDTD simulation of light propa-gating past the edge of a metal-coated silicon wafer. This simulation solves Maxwell’s equationsnear the edge and allows us to simulate the geometry and material properties of the manufacturedstarshade masks. In order to get the model to match the results of Sec. 4.3, we had to account forscalloping and tapering of the mask sidewalls left by the etching process. The edge properties thatproduced the best fit to the data (1 ◦ taper angle and scallops 0.8 µ m tall and 0.2 µ m deep) wereconfirmed to be those of the real mask from scanning electron microscope images of the edge of amanufactured mask. Figure 25 shows the simulated geometry of the edge along with SEM imagesof a manufactured mask, confirming we are properly simulating the complex edge structures. Disclosures
Some of the work presented here is included in the S5 Milestone Reports of Refs. 21, 22 and inRef. 40. Anthony Harness is a guest editor for this special section.
Acknowledgments
The authors thank Jonathan Arenberg for his helpful and constructive comments. This work wasperformed in part at the Jet Propulsion Laboratory, California Institute of Technology under acontract with the National Aeronautics and Space Administration. Starshade masks were manu-factured using the facilities at the Microdevices Lab at JPL. This project made use of the resources54 ig 25
Left: SEM image of the scalloped vertical profile of the wafer edge for a manufactured mask. Right: electricpermittivity map of the
Meep FDTD simulation cell displaying the material geometry. from the Princeton Institute for Computational Science and Engineering (PICSciE) and the Officeof Information Technology’s High Performance Computing Center and Visualization Laboratoryat Princeton University.
References
Nature , 51 – 53 (2006). [doi:10.1038/nature04930].2 S. Seager, N. J. Kasdin, and S. R. P. Team, “Starshade Rendezvous Mission probe con-cept,” in
American Astronomical Society Meeting Abstracts , American AstronomicalSociety Meeting Abstracts (2018). https://smd-prod.s3.amazonaws.com/science-red/s3fs-public/atoms/files/Starshade2.pdf .3 S. Gaudi, S. Seager, B. Mennesson, et al. , “The Habitable Exoplanet Observatory HabEx55ission concept study final report,” arXiv e-prints (2020). https://arxiv.org/abs/2001.06683 .4 D. Spergel and THEIA Collaboration, “THEIA: Telescope for Habitable Exoplanets andInterstellar/Intergalactic Astronomy,” in
American Astronomical Society Meeting Abstracts , American Astronomical Society Meeting Abstracts (2010).5 M. Turnbull, T. Glassman, A. Roberge, et al. , “The Search for Habitable Worlds. 1. TheViability of a Starshade Mission,”
Publications of the Astronomical Society of the Pacific , 418 (2012). [doi:10.1086/666325].6 P. Willems, “Starshade to TRL5 (S5) technology development plan,”
Jet PropulsionLaboratory Publications (2018). https://exoplanets.nasa.gov/internal_resources/1033 .7 B. Crill and N. Siegler, “Exoplanet Exploration Program: 2019 Technology Plan Appendix,”
Jet Propulsion Laboratory Publications
D-102506 (2018). https://exoplanets.nasa.gov/internal_resources/1123/ .8 P. Willems, “NASA’s starshade technology development activity,”
Journal of AstronomicalTelescopes, Instruments, and Systems (this issue) (2020).9 W. Cash, “Analytic modeling of starshades,”
The Astrophysical Journal , 76 (2011).[doi:10.1088/0004-637X/738/1/76].10 R. J. Vanderbei, E. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolarplanets,”
ApJ , 794 – 798 (2007). [doi:10.1086/519452].11 E. Cady, “Boundary diffraction wave integrals for diffraction modeling of external occulters,”
Optics Express , 15196 (2012). [doi:10.1364/OE.20.015196].562 A. Harness, S. Shaklan, W. Cash, et al. , “Advances in edge diffraction algorithms,” J. Opt.Soc. Am. A , 275 – 285 (2018). [doi:10.1364/JOSAA.35.000275].13 D. Leviton, W. Cash, B. Gleason, et al. , “White-light demonstration of one hundred partsper billion irradiance suppression in air by new starshade occulters,” in UV/Optical/IR SpaceTelescopes: Innovative Technologies and Concepts III , Proc. SPIE , 465 – 476 (2007).[doi:10.1117/12.742927].14 R. Samuele, T. Glassman, A. Johnson, et al. , “Starlight suppression from the starshade testbedat NGAS,” in
Techniques and Instrumentation for Detection of Exoplanets IV , Proc. SPIE , 21 – 29 (2009). [doi:10.1117/12.824889].15 E. Cady, K. Balasubramanian, M. Carr, et al. , “Progress on the occulter experiment at Prince-ton,” in
Techniques and Instrumentation for Detection of Exoplanets IV , Proc. SPIE , 38– 47 (2009). [doi:10.1117/12.826359].16 R. Samuele, R. Varshneya, T. Johnson, et al. , “Progress at the starshade testbed at NorthropGrumman Aerospace Systems: comparisons with computer simulations,” in
Space Tele-scopes and Instrumentation 2010: Optical, Infrared, and Millimeter Wave , Proc. SPIE ,1733 – 1744 (2010). [doi:10.1117/12.856334].17 D. Sirbu, E. Cady, N. J. Kasdin, et al. , “Optical verification of occulter-based high contrastimaging,” in
Techniques and Instrumentation for Detection of Exoplanets V , Proc. SPIE ,416 – 427 (2011). [doi:10.1117/12.894213].18 T. Glassman, S. Casement, S. Warwick, et al. , “Measurements of high-contrast starshade per-formance,” in
Space Telescopes and Instrumentation 2014: Optical, Infrared, and MillimeterWave , Proc. SPIE , 805 – 817 (2014). [doi:10.1117/12.2054680].579 T. Glassman, M. Novicki, M. Richards, et al. , “2012 TDEM: Demonstration of Star-shade Starlight-Suppression Performance in the Field. Final Report,”
Jet Propulsion Labora-tory Publications (2012). https://exoplanets.nasa.gov/exep/files/exep/GlassmanTDEM2012_FinalReport.pdf .20 A. Harness, W. Cash, and S. Warwick, “High contrast observations of bright stars with astarshade,”
Experimental Astronomy , 209 – 237 (2017). [doi:10.1007/s10686-017-9562-1].21 A. Harness, S. Shaklan, N. J. Kasdin, et al. , “Starshade technology development activityMilestone 1A: Demonstration of high contrast in monochromatic light at a flight-like Fres-nel number,” Jet Propulsion Laboratory Publications (2019). https://exoplanets.nasa.gov/internal_resources/1210 .22 A. Harness, S. Shaklan, N. J. Kasdin, et al. , “Starshade technology development activityMilestone 1B: Demonstration of high contrast in broadband light at a flight-like Fresnel num-ber,”
Jet Propulsion Laboratory Publications (2019). https://exoplanets.nasa.gov/internal_resources/1211 .23 A. Boss, R. Oppenheimer, J. Pitman, et al. , “ExoTAC Report on Starshade S5 Mile-stones
Jet Propulsion Laboratory Publications (2019). https://exoplanets.nasa.gov/internal_resources/1214/ .24 J. W. Arenberg, A. S. Lo, T. M. Glassman, et al. , “Optical performance of the New WorldsOcculter,”
Comptes Rendus Physique , 438–447 (2007). [doi:10.1016/j.crhy.2007.03.005].25 M. Born and E. Wolf, Principles of Optics , Cambridge University, 7 ed. (1999).26 M. Galvin, Y. Kim, N. J. Kasdin, et al. , “Design and construction of a 76m long-travel58aser enclosure for a space occulter testbed,” in
Advances in Optical and Mechanical Tech-nologies for Telescopes and Instrumentation II , Proc. SPIE , 2031 – 2048 (2016).[doi:10.1117/12.2231093].27 Y. Kim, D. Sirbu, M. Galvin, et al. , “Experimental study of starshade at flight Fresnel num-bers in the laboratory,” in
Space Telescopes and Instrumentation 2016: Optical, Infrared, andMillimeter Wave , Proc. SPIE , 1143 – 1153 (2016). [doi:10.1117/12.2231112].28 K. Balasubramanian, D. Wilson, V. White, et al. , “High contrast internal and external coro-nagraph masks produced by various techniques,” in
Techniques and Instrumentation for De-tection of Exoplanets VI , Proc. SPIE , 644 – 652 (2013). [doi:10.1117/12.2231112].29 K. Balasubramanian, A. J. E. Riggs, E. Cady, et al. , “Fabrication of coronagraph masks andlaboratory scale star-shade masks: characteristics, defects, and performance,” in
Techniquesand Instrumentation for Detection of Exoplanets VIII , Proc. SPIE , 74 – 89 (2017).[doi:10.1117/12.2274059].30 M. Bottom, S. Martin, E. Cady, et al. , “Starshade formation flying I: optical sens-ing,”
Journal of Astronomical Telescopes, Instruments, and Systems (1), 1 – 17 (2020).[doi:10.1117/1.JATIS.6.1.015003].31 L. Palacios, A. Harness, and N. J. Kasdin, “Hardware-in-the-loop testing of formation flyingcontrol and sensing algorithms for starshade missions,” Acta Astronautica , 97 – 105(2020). [doi:10.1016/j.actaastro.2020.02.011].32 P. Willems and A. Harness, “Rayleigh scattering in the Princeton starshade testbed,” in
Tech-niques and Instrumentation for Detection of Exoplanets IX , Proc. SPIE , 193 – 201(2019). [doi:10.1117/12.2528077]. 593 J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account forthick mask effects in photolithography,” in
Optical Microlithography XVI , Proc. SPIE ,1611 – 1619 (2003). [doi:10.1117/12.488803].34 J. D. Jackson,
Classical Electrodynamics , John Wiley & Sons, 3 ed. (1998).35 W. Braunbek, “Neue N¨aherungsmethode f¨ur die Beugung am ebenen Schirm,”
Zeitschrift f¨urPhysik , 381 – 390 (1950). [doi:10.1007/BF01329835].36 A. Harness, “Implementing non-scalar diffraction in Fourier optics via the Braunbekmethod,”
Optics Express , 34290–34308 (2020).37 S. Shaklan, L. Marchen, E. Cady, et al. , “Error budgets for the Exoplanet Starshade (Exo-S)probe-class mission study,” in Techniques and Instrumentation for Detection of ExoplanetsVII , Proc. SPIE , 314 – 327 (2015). [doi:10.1117/12.2190384].38 G. Blackwood et al. , “Starshade Readiness Working Group Recommendation to As-trophysics Division Director,”
Jet Propulsion Laboratory Publications
CL (2016). https://exoplanets.nasa.gov/system/internal_resources/details/original/339_SSWG_APD_briefing_final.pdf .39 A. F. Oskooi, D. Roundy, M. Ibanescu, et al. , “Meep: A flexible free-software packagefor electromagnetic simulations by the FDTD method,”
Computer Physics Communications (3), 687 – 702 (2010). [doi:10.1016/j.cpc.2009.11.008].40 A. Harness, S. Shaklan, N. J. Kasdin, et al. , “Demonstration of 1e-10 contrast at the innerworking angle of a starshade in broadband light and at a flight-like Fresnel number,” in
Tech-niques and Instrumentation for Detection of Exoplanets IX , Proc. SPIE , 185 – 192(2019). [doi:10.1117/12.2528445]. 60 nthony Harness is an Associate Research Scholar in the Mechanical and Aerospace EngineeringDepartment at Princeton University. He received his Ph.D. in Astrophysics in 2016 from the Uni-versity of Colorado Boulder. He currently leads the experiments at Princeton validating starshadeoptical technologies.
Stuart Shaklan is the supervisor of the High Contrast Imaging Group in the Optics Section of theJet Propulsion Laboratory. He received his Ph.D. in Optics at the University of Arizona in 1989and has been with JPL since 1991.
Phil Willems is an Optical Engineer at the Jet Propulsion Laboratory, where he is the manager ofthe S5 Starshade Technology Development Activity. He received his BS degree in physics fromthe University of Wisconsin-Madison in 1988, and his PhD degree in physics from the CaliforniaInstitute of Technology in 1997.
N. Jeremy Kasdin is the Assistant Dean for Engineering at the University of San Francisco andthe Eugene Higgins professor of Mechanical and Aerospace Engineering, emeritus, at PrincetonUniversity. He received his Ph.D. in 1991 from Stanford University. After being the chief systemsengineer for NASA’s Gravity Probe B spacecraft, he joined the Princeton faculty in 1999 where heresearched high-contrast imaging technology for exoplanet imaging. From 2014 to 2016 he wasVice Dean of the School of Engineering and Applied Science at Princeton. He is currently the Ad-jutant Scientist for the coronagraph instrument on NASA’s Wide Field Infrared Survey Telescope.61 ist of Figures λ = 638 nm with mask DW17.The starshade pattern is overlaid and the dashed circle marks the geometric IWA.6 Contrast images of mask DW21 at four discrete wavelengths spanning a 10% band-pass. 62 Experiment (left) and model (right) images of mask DW21 at λ = 660 nm. In themodel the camera is shifted off-axis by 1 mm, which distorts the polarization lobes,suggesting this can account for the morphology of the lobes seen in the data.8 Contrast averaged over a λ/D wide annulus as a function of angular separation fordifferent wavelength observations of mask DW21. Also included is the monochro-matic observation of mask DW17. The solid black line denotes the starshade’sgeometric IWA.9 A closeup of one petal in the design of mask M12P3. The inner starshade is rotatedrelative to the outer, creating exposed tips at all petals (seen at 12.5 mm). The bluehatched area is transparent; the white area is the silicon wafer.10 Residuals between experiment and model of stacked [log-scale contrast] images ofmask M12P3 at λ = 725 nm. Lab and model images were created by mediancombining images taken at three starshade orientations. The brightest spots aredue to intentional sine wave perturbations (see Sec. 4.2.2) and an unintentionalmanufacturing defect. High contrast is still achieved at the location of the exposedtips (marked by the dashed circle).11 Location of inner and outer perturbations on the perturbed masks M12P2 andM12P3. The red arrows are a quiver plot of the sine waves on M12P3 with ampli-tudes greatly exaggerated. The blue hatched area is transparent; the white area isthe silicon wafer.12 Microscope image of the inner displaced edge perturbation on M12P2. The redline denotes the edge of the nominal starshade shape. The 3.7 µ m step in the edgesimulates a displaced edge segment. 633 Contrast images of M12P2 − displaced petal edge segment perturbations. Exper-imental data are in the left column, model data are in the right column. The rowsare for the different wavelengths. The perturbation on the inner petal is circled incyan and the perturbation on the outer petal is circled in red. The red ‘x’ marks thecenter of the starshade.14 M12P2 − displaced edge perturbations. Contrast for the Inner and Outer pertur-bations across the four wavelengths. Solid markers are experimental data; openmarkers are model data. The contrast is the photometric aperture average given byEq. (12) and the error bars (on experimental data only) are σ , given by Eq. (14).Error bars on experimental data that are not visible are smaller than the symbols.15 M12P2 − displaced edge perturbations. Percent difference between experimentaland model contrast for the Inner and Outer perturbations across the four wave-lengths. The contrast is the photometric aperture average given by Eq. (12) andthe error bars are the percent uncertainty given by Eq. (7). The value for the Outerperturbation at 699 nm is off the chart (at 60%).16 Contrast images of M12P3 − sine wave perturbations. Experimental data are inthe left column, model data are in the right column. The rows are for the differentwavelengths. The perturbation on the inner petal is circled in cyan, the perturbationon the outer petal is circled in red, and the unintentional manufacturing defect iscircled in magenta. The red ‘x’ marks the center of the starshade.647 M12P3 − sine wave perturbations. Contrast for the Inner and Outer perturbationsacross the four wavelengths. Solid markers are experimental data; open markersare model data. The contrast is the photometric aperture average given by Eq. (12)and the error bars (on experimental data only) are σ , given by Eq. (14). Error barson experimental data that are not visible are smaller than the symbols.18 M12P3 − sine wave perturbations. Percent difference between experimental andmodel contrast for the Inner and Outer perturbations and the unintentional man-ufacturing defect across the four wavelengths. The contrast is the photometricaperture average given by Eq. (12) and the error bars are the percent uncertaintygiven by Eq. (7).19 Experimental (top) and model (bottom) images of mask DW9 imaged with thecamera analyzer rotating relative to the input polarizer direction (horizontal in theimage). The angle given is that between the analyzer and polarizer, where 0 ◦ meansthey are aligned, 90 ◦ means they are crossed. Data are taken with λ = 641 nm.20 Experimental (left) and model (right) images of mask DW9 at λ = 641 nm. In thetop row, the camera analyzer is aligned with the input polarizer (with a slight 15 ◦ misalignment). In the bottom row, the camera analyzer is crossed with the inputpolarizer. Note the difference in colorbar scales.21 Experimental (left) and model (right) images of mask DW21 at λ = 641 nm withpolarized light. In the top row, the camera analyzer is aligned with the input polar-izer. The bright spot at 10:00 is dust on the mask. In the bottom row, the cameraanalyzer is crossed with the input polarizer. Note the difference in colorbar scales.652 Azimuthally averaged contrast for masks DW17 and DW21 at λ = 725 nm, andthe corresponding vector and scalar propagation models. The vertical lines markthe tip of the inner and outer apodization profiles. Also shown are estimates ofRayleigh scattering generated from the models of Ref. 32.23 Model vs. measured data for all perturbations on the displaced edge and sine waveperturbations and four wavelengths. The absolute difference between model andmeasured data and the measurement error are also shown.24 Suppression (pupil plane) image of DW9 at λ = 638 nm, without any polarizingelements.25 Left: SEM image of the scalloped vertical profile of the wafer edge for a manufac-tured mask. Right: electric permittivity map of the Meep FDTD simulation celldisplaying the material geometry.
List of Tables † For apodization design C12/C16 in Table 3.2 Summary of experiments and production number of starshades tested (see Table 4for details on specific starshades). Those with OV in the Goal column are opticalverification experiments and are presented in Sec. 3. Those with MV in the Goalcolumn are model validation experiments and are presented in Sec. 4.66 Design of apodization functions including the number of petals, operating band-pass, minimum radius (start of the petals), maximum radius (start of the struts),and gap width between petals.4 Descriptions of manufactured masks including the apodization design (detailed inTable 3), the thickness of the device layer (optical edge), the thickness and type ofmetal coating, and the perturbations built into the shape.5 Description of shape perturbations, including their expected contrast (photomet-ric average, see Eq. (12)) at two wavelengths. Each perturbed mask ( a M12P2, bb