1. Introduction

To enable the direct detection of exoplanets around nearby stars, the much brighter starlight must first be greatly suppressed. Stellar coronagraphy [1,2] is effective at stellar suppression beyond separation angles of a few diffraction beam widths (i.e., a few λ/D, where λ is the observation wavelength and D the telescope diameter), but exoplanets closer to their host stars will remain beyond the reach of typical coronagraphs, thus limiting the number of exoplanets observable coronagraphically [3,4]. However, smaller separations can be accessed with nulling interferometry [5–9] (hereafter “nulling”), which uses destructive interference to reject starlight. Originally aimed at detecting mid-infrared (∼ 10 µm) thermal exoplanet emission by nulling the starlight arriving at separate space-based telescopes [5], shorter-wavelength (near-infrared and visible) nulling can provide access to reflected-light exoplanet spectra with interferometric baselines short enough to fit within a large single-aperture telescope (hence the nomenclature “cross-aperture nulling”). This technique has recently been demonstrated by the Palomar Fiber Nuller [8,9], the Guided Light Interferometric Nulling Technology [10] and the Vortex Fiber Nuller [11–13] (VFN) instruments.

Early cross-aperture nullers [6] were based on free-space beamsplitters combining beams from different telescope subapertures. However, a simpler photonic approach is possible. Specifically, a pupil-plane phase mask that produces any type of anti-symmetric stellar focal-plane field distribution will prevent the centered starlight pattern from coupling to the symmetric mode of a single-mode (SM) fiber [8,9,14,15] (Fig. 1), while allowing off-axis exoplanet light through. As any anti-symmetric focal-plane field distribution will have a central value of zero, the Fourier transform relationship between focal- and pupil-plane fields then implies a zero-average pupil-plane (i.e., mask output plane) field [16]. Many zero-average field distributions within a round pupil are possible, but the simplest pupil-plane phase patterns that lead to an asymmetric focal plane field distribution are, in Cartesian and polar coordinates, respectively, the “phase knife” mask with a π phase step across a pupil bisector [16,17] (Fig. 1), and the optical vortex phase mask with an azimuthal phase wrap around the center of the mask [11,12]. The phase knife is the limiting case of a dual-subaperture nuller wherein each subaperture is large enough to encompass half the pupil, and is the SM nuller configuration with the largest peak coupling efficiency to off-axis exoplanets [16]. Moreover, in contrast to the vortex fiber nuller, the phase-knife nuller provides azimuthal sensitivity to off-axis exoplanets [16], which can be used to disambiguate multiple exoplanets within the central fiber mode. To date, dual-subaperture fiber nulling has reached a calibrated on-sky null depth (i.e., a stellar rejection level) of 1.6 × 10⁻⁴ (root mean square) [9], and has detected stellar-diameter signals down to depths of 4.8 × 10⁻⁴ [8,9], while the VFN has detected companions to ∼ 2.5 × 10⁻³ [13], neither yet sufficient to detect exoplanets. A deeper nulling capability is thus essential to being able to use SM nulling to probe the centralmost regions around nearby stars.

 

Fig. 1. SM fiber-nuller layout, consisting of a pupil-plane phase mask (here a phase-knife mask) in a collimated beam, a tip-tilt mirror, and a lens that focuses the beam onto a SM fiber (Thorlabs P1-630A-FC-2). Top center: our phase-knife mask’s optical axis structure is indicated by the two sets of parallel lines within the circular mask. The arrows show the effect of the mask on both input linear polarization states. Curve at lower left: cross-cut through the resultant anti-symmetric focal-plane field. Images at bottom right: Coupling maps to the point spread function measured through our SM fiber, with the mask out (top) and mask in (bottom), obtained by scanning the tip-tilt mirror in 2 dimensions. For direct comparison, both maps are on the same intensity scale (i.e., with the same color bar), both being normalized to the central peak of the mask-out case. The lateral angles are in common arbitrary units.

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The phase-knife mask has a very simple structure, consisting of two uniform halves with a π radian phase shift between them. Here we present a phase-knife mask structure that is based on the use of geometric phase, and we describe its implementation using a thin liquid-crystal polymer (LCP) layer. We also present laboratory nulling results obtained using this mask and discuss potential improvements.

2. Phase knife mask

Although a phase-knife mask’s π radian phase step could be provided monochromatically by passage through different thicknesses of glass on opposite sides of the mask bisector, or by reflection off a two-segment mirror [18], the ultimate desire for achromatic performance would require multiple glasses or coatings, with a sharp dividing line between them. On the other hand, geometric phase [19] allows for a very simple phase-knife mask structure, as a relative π phase shift can be brought about by rotating the polarization vectors on opposite sides of the bisector into opposition. This can be effected with orthogonally oriented (i.e., “crossed”) half-wave-plates (HWPs) on opposite sides of the pupil bisector. Indeed, as illustrated in Fig. 1 for this case, both input polarization states are rotated into opposition across the bisector, thus providing polarization-independent operation. (Note that the two orientations do not need to be at ± 45° to the bisector as drawn for this to be the case, but merely perpendicular to each other.)

Opposed output fields require a retardance of π radians. This could be provided by a pair of appropriately cut and oriented macroscopic birefringent crystals, but as this is inconsistent with the goal of a single continuous full-pupil phase mask (i.e., with no additional path differences in possible need of compensation), we instead aimed at a single substrate with differing coating structures on its two halves. Specifically, we aimed at a common LCP layer with optical axes orientated orthogonally to each other across the bisector, as in Fig. 1, and with each side oriented at ± 45° to the bisector. Our initial goal here was a simple single-layer (hence, monochromatic) LCP device, as such a device can be used in nulling demonstrations by operating at the wavelength at which its retardance crosses π.

A test phase knife was manufactured by Beam Engineering by spin coating a polymerizable liquid crystal (LC) layer over a substrate having a photoalignment layer that was split into areas of orthogonal alignment directions by the action of a polarized laser beam. The thickness of the LC layer was aimed at a half-wave retardation at the He-Ne laser wavelength (633 nm). Details of the fabrication technology can be found in Refs. [20] and [21]. The optical properties of the resultant device were measured on an Axoscan Muller Matrix Imaging Polarimeter. The optical axis orientations were found to be 44.9°± 0.1° and -44.96° ± 0.1° relative to the bisector, giving a relative angle of 89.9°, and the device retardance was found to cross π at roughly 660 nm (Fig. 2). The retardance is quite uniform (to better than a degree) with the error estimated slightly off the best wavelength to avoid folding the extracted phases across π. The ∼ 15 µm wide bisector region is evident in Fig. 2, as are a few several-micron sized deviations from uniformity. As this mask is intended for use in the pupil plane, neither is cause for concern.

 

Fig. 2. Top – the measured optical-axis orientation map of our phase-knife mask. Center – map of the retardance measured near the center of the mask slightly off the best (i.e., 180°-crossing) wavelength (to avoid ambiguities introduced when crossing 180°). Bottom – the measured average mask retardance vs. wavelength for the region in the white box in the top panel.

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3. Nulling setup and measurements

The mask’s nulling capability was tested in our Infrared Coronagraphic Testbed (IRCT) [22], with its layout modified to include an output SM fiber as in Fig. 1. Not shown is the input fiber that sends the light from a supercontinuum laser to an off-axis paraboloidal mirror to be collimated upstream of the optics shown in Fig. 1. The central wavelength and bandwidth (∼ 2 nm) were selectable at the supercontinuum source. The phase-knife mask was mounted between a circular mask defining the pupil (≈ 15 mm in diameter) and a thin wire that blocks the bisector area. An achromatic doublet focuses the light onto the face of a SM fiber, and an actuated tip-tilt mirror allows scanning the output point spread function (PSF) across the stationary SM fiber tip, with the fiber output providing local samples of the PSF as coupled through the fiber. The resultant measured source coupling efficiency maps for both the mask-in and mask-out cases (with both maps normalized to the central value of the mask-out map), are shown in Fig. 1. As seen in Fig. 1, the measured peak coupling efficiency with the mask in the beam is 43% of the peak without the mask. Normalizing to the theoretically expected peak no-mask coupling efficiency of 81%, this translates to an absolute off-axis peak coupling efficiency of 35%, consistent with predictions [16].

To optimize the on-axis null depth (i.e., the ratio of the central “mask in” and “mask out” fluxes), various experimental settings and parameters were explored, the most significant one being the central wavelength. Figure 3 shows that the null depth varies roughly quadratically with wavelength, as expected for the linear retardance dependance seen (Fig. 2), with the deepest null occurring near 657 nm. As shown in Fig. 1, we used the tip-tilt mirror to map regions of the PSF large enough to determine the central peak flux in the mask-out case, and to reach both of the off-axis peaks in the mask-in case, to allow for proper normalization. Slow short scans were then made along the line connecting the two off-axis mask-in peaks to measure the central minimum repeatedly (Fig. 3). To calibrate the null depth measurements, the scans were dark-subtracted, and the mask-in scans corrected for the measured mask transmission of 93%, before normalizing by the peak of the no-mask image.

 

Fig. 3. Left – Null depth (i.e., rejection) vs. wavelength measured with our LCP phase-knife mask on the IRCT. Right – Measured null depth vs. displacement along the centerline between the two off-axis peaks in the mask-in case, for ten short scans centered roughly midway between the two peaks. Only the immediate region of the central minimum was scanned here.

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The best measured null depth in Fig. 3 is 2.2 × 10⁻⁵, a level likely limited by a number of factors, including bandwidth (Fig. 2), the slight deviation from orthogonality of the crossed optical axes, other slight mask imperfections and asymmetries, and residual misalignments, as attested by the slightly unequal peak heights seen in the mask-in image in Fig. 1. Similar monochromatic laboratory null depths have been achieved with the vortex fiber nuller [23], which however has a lower off-axis transmission peak and lacks azimuthal sensitivity. Moreover, the demonstrated laboratory rejection of the phase-knife nuller is over an order of magnitude deeper than has been achieved to date in any on-sky nulling observations, and is at a level that could enable the direct detection of Hot Jupiter exoplanets with a sufficiently long baseline.

4. Discussion

Cross-aperture null depths on astronomical telescopes will be limited by many factors, including phase instability (which can be mitigated by an adaptive optics system), starlight leakage due to both finite stellar diameters and pointing errors (which effectively mimic larger stellar diameters), and mask chromaticity. The demonstrated laboratory performance of our mask should already allow reaching the current on-sky performance limit set by these error and stellar leakage terms, which is not particularly deep. However, stellar diameter leakage will be even larger on the next generation of 30 m telescopes, as the stellar leak is proportional to the square of the telescope diameter⁷. Several system limitations will thus need to be addressed before the full potential of nulling phase masks can be reached on sky.

First, there is the mask bandwidth. As is well known, the bandwidth of HWPs can be broadened by using a sequence of three rotated HWPs [24,25]. This approach has already been used to make vector vortex phase masks more broadband [26,27], so as the simple phase-knife structure should be much easier to manufacture than spatially-variant vortex phase masks, the same multi-layer solution should be easy to apply.

Next is the leakage due to finite stellar-diameter, and the related pointing-error leakage. Here we suggest a simple mask modification to address this issue. We take our cue from early space-based nulling interferometer concepts [28–31] that used the interference between more than two telescopes to spatially broaden the central null fringe shape to better cancel the light from a star of finite angular extent (in detail, both the focal-plane fields and their slopes can be made to cross zero on axis, leading to a broader, higher-order null). By analogy, we can similarly generalize the two-region phase-knife mask to a larger number (i.e., 3 or 4) of phase regions (or “stripes”), as seen in the 2^nd and 3^rd images from the top in the central column of Fig. 4, in an attempt to produce wider central nulls that would be less sensitive to stellar-diameter (and pointing-error) leakage. In the symmetric three-stripe case, one particular central stripe width will allow that stripe’s central focal-plane field to cancel the combined focal-plane fields from the outer symmetric pair of stripes, while in the anti-symmetric four-stripe case, the on-axis focal plane fields from both the inner and outer stripe pairs will cancel each other separately, allowing their widths to be used as an extra degree of freedom that can be used to modify the shape of the central null fringe in the focal plane. As illustrated in Fig. 4, such considerations suggest that there is a general topological correspondence between separated-aperture multi-telescope nullers (left column) and single-aperture nulling phase masks (central column): in any horizontal row in the figure, the separate phase areas in either column can be displaced and deformed into each other while keeping the same phase pattern in place. This extends even to the optical vortex case (bottom right), which can be seen to be the limiting case of both a circular nulling array [31] and a staircase-type phase mask, such as, e.g., the four-quadrant phase mask [FQPM] [32] and the eight-octant mask [33].

 

Fig. 4. Comparison of space-based multi-aperture nulling interferometer configurations (left column) with single-aperture nulling phase masks (central column). Phases of 0 and π are shown as white and black, respectively. Intermediate phases in the bottom row are shown in various shades of gray. Each small circle in the left column represents a separate telescope aperture. Central column – single-aperture “linear” nulling phase masks (in which all mask phase transitions are along straight lines) of increasing complexity (top to bottom), arranged to show their topological correspondence to the separated-aperture nulling cases to their left. Right column: “Round” single-aperture phase masks (in which at least one of the phase transitions is circular) that also topologically correspond to the leftmost entries in their rows. (The dashed vertical line in the central phase dimple case is not a real structure; it is there simply to make the phase correspondence to the leftmost image in its row more obvious.)

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To test the hypothesis that additional phase stripes can broaden the central cross-aperture null fringe of a SM phase-knife fiber nuller, we calculated coupling efficiencies through a SM fiber for upstream 2, 3, and 4 phase-stripe masks. In Fig. 5, the computed coupling efficiencies for the phase-knife nuller, the symmetric 3-stripe nuller, and two different anti-symmetric 4-stripe nuller cases (with “narrow” and “wide” outer stripes) are compared. As can be seen in the figure, adding phase stripes does indeed broaden the bottom of the central null fringe, while leaving the peak level of the coupling efficiency relatively unaffected in all cases. The central null region of the sole symmetric 3-stripe nuller solution is seen to be significantly broader and flatter than the phase-knife’s narrow null, while in the anti-symmetric 4-stripe case, varying the outer to inner stripe-width ratio allows the central null shape to be smoothly altered, from the pure phase-knife’s relatively narrow central fringe bottom (for minimal outer-stripe widths), to a much broader and flatter central null fringe shape when the outer stripe width is a larger fraction of the pupil diameter. The pair of 4-stripe examples shown in Fig. 5 thus highlight the control over the null fringe shape and the flexibility that such mask modifications enable. (In contrast, large multi-telescope nulling configurations are much more difficult to alter at will.)

 

Fig. 5. Top left panel: Calculated coupling efficiency profiles along the direction perpendicular to the phase transitions for the cases of the phase-knife mask, the 3-stripe mask (for which the phase transition is at at 33% of the radius) and two different 4-stripe masks (with the outer stripes starting at 75% and 56% of the aperture radius, called the “narrow” and “wide” stripe cases, respectively). Middle left panel: Blowup of the central null region. Central column (color): the corresponding pupil amplitudes and phases. The potential mask optical axis configurations are shown at the very top left – each successive mask stripe produces an output field flip relative to its neighboring stripes. Rightmost column: Corresponding fiber coupling efficiency maps out to ± 3λ/D. Bottom left panel: Coupling efficiency azimuthal response curves through each coupling map’s pair of peaks.

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However, in exchange for a broader null fringe, the off-axis transmission peaks to be used for exoplanet detection move outward. Even so, as the inner half power points of the curves in Fig. 5 (i.e., their “inner working angles”, or IWAs) are 1.22 λ/D for the wide 4-stripe nuller case and 0.91 λ/D for the 3-stripe nuller case, all of their IWAs are still well inside the normal coronagraphic IWA of a few λ/D considered possible for the high contrast coronagraphs on the Roman Space Telescope [34], and the Habitable Exoplanet Observatory (HabEx) [35], Large UV/Optical/IR Surveyor (LUVOIR) [36], and Habitable Worlds Observatory (HWO) [37] exoplanet mission concepts. Of course, as adding even more phase stripes would eventually lead to a phase grating with a spacing a that is << D, for which the two innermost grating response lobes would be at ∼ ± λ/a >> λ/D, one cannot add too many phase stripes without the response peaks moving too far off-axis, i.e., into the regime already accessible by means of coronagraphy.

One can also make use of both lateral directions, as was already the case for separated-aperture nulling concepts. In particular, by superposing two orthogonal phase knives, one arrives at an FQPM [32]. As can be seen in the 4^th row of Fig. 4, the FQPM (center column) has a phase pattern topologically equivalent to the 2d separated-telescope “Angel-cross” nulling configuration [38] (left column), which is known to provide a broader central null than a single-baseline nuller. We therefore also calculated coupling efficiencies for a FQPM nuller, and found that it also provides a broader central cross-aperture null, but this curve is not plotted in Fig. 5 to keep from cluttering the figure. Thus, the general correspondence illustrated in Fig. 4 holds true – not only do the static multi-telescope nulling configurations have topologically-equivalent pupil-plane phase masks that can be used for cross-aperture nulling interferometry, but increasing the number of phase regions in both cases broadens and flattens the central nulls.

Besides being usable for cross-aperture nulling, each of the phase mask types seen in Fig. 4 could in principle also be used for full-aperture coronagraphy, with an appropriate change of scale for any masks possessing radial phase variations. Specifically, in the focal-plane case, any radial phase steps must lie within the core of the stellar point spread function, and so must be implemented on a much smaller spatial scale than in the pupil-mask case, for which the phase steps would occur at a significant fraction of the pupil diameter. Indeed, several of the phase masks depicted in Fig. 4 are already familiar from coronagraphy, specifically the phase knife [17,39], the FQPM [32] and the round Roddier phase “dimple” [40,41] (top element of the rightmost column), but there have evidently been gaps in the known coronagraphic phase mask zoo, which can be filled in here by analogy with the separated aperture phase patterns shown in the left-hand column of Fig. 4. This allows the definition of the linear “3-stripe phase mask” and the “4-stripe phase mask,” as discussed for the case of nulling, and also the round “dimpled phase-knife”, all of which should provide improved performance over the simplest phase-knife coronagraph, just as they do for the phase-knife cross-aperture nuller.

The duality between pupil-plane nullers and focal-plane coronagraphs is easy to demonstrate, especially for the case of purely azimuthal phase masks, which require no radial change of scale between their use in the pupil and focal planes. We therefore moved the phase-knife mask that was used for the nulling experiment described herein to the IRCT’s internal focal plane to carry out a simple coronagraphic demonstration. Specifically, imaging the pupil plane downstream of the focal-plane phase-knife mask (i.e., the “Lyot” plane), we find a Lyot plane light distribution (Fig. 6) that is remarkably similar to the theoretical prediction for the appearance of the Lyot plane in a phase knife coronagraph [17]. Further phase mask development should therefore benefit both applications.

 

Fig. 6. Top - the configuration of the IRCT used to demonstrate that our phase-knife mask can also operate as a coronagraphic mask. Bottom – the resultant measured Lyot-plane image.

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Finally, additional steps for even further potential post-nulling stellar suppression are possible, including both phase mask (and concomitant null fringe) rotation relative to the observed field, and spectral discrimination [42] between stars and exoplanets. Mask rotation should allow significant reduction of residual post-nulling stellar leakage for round, uniform stars and other background signals, while enabling exoplanet azimuths to be determined. Synchronous angular detection via mask rotation, as originally suggested for the separated telescope case [5,29,30], can be considered, but only for sufficiently bright companions (i.e., for large enough telescopes), as the averages of the azimuthal coupling curves in Fig. 5 are 0.15 (phase knife), 0.094 (3-stripe), 0.074 (4-stripe wide) and 0.077 (4-stripe narrow). Dual-azimuth (i.e., on-off) mask rotational chopping would be more efficient for companions with known azimuths, by better maximizing time spent integrating on a companion’s location, while also providing the necessary off-source calibration observation.

5. Summary and prospects

A planar phase-knife mask based on a single LCP layer has been used to demonstrate a simple SM-fiber-based photonic nuller that can reach angles smaller than those reachable by typical coronagraphs. A point-source rejection of 2.2 × 10⁻⁵ has been demonstrated here for a phase mask that has a measured peak off-axis coupling efficiency of 35%. This particular phase-mask structure is shown to be part of a family of phase masks with a common (deformable) topology shared by separated-aperture nullers, cross-aperture nullers and full-aperture coronagraphs. Indeed, the same phase-knife mask has allowed a demonstration of both nulling and coronagraphic operation.

Comparing cross-aperture SM nulling with coronagraphy, it is worth noting their spatial complementarity, with SM nulling operating in a small on-axis dark hole (or dark fringe) inside the much larger regions typical of coronagraphic dark holes that span many spatial modes. While it is even more difficult to suppress starlight in the brightest, centralmost region of a star’s Airy pattern, compensating this are three factors: closer to stars, planets are both brighter and presumably more numerous, while at long wavelengths, small IWA nullers can also compensate for larger diffractions beams.

A number of promising paths to improved null depths (i.e., stellar rejection) have been identified, using masks modified both with additional LCP layers to make them more broadband, and with a larger number of phase regions to broaden the nulled region spatially, so as to better suppress finite-sized stars. Such phase masks used in SM cross-aperture nulling may thus provide a unique photonic method of unveiling exoplanets hiding in the innermost regions around nearby stars.