1. Introduction

The Terrestrial Planet Finder (TPF) project has been funded by NASA to develop technology to enable the detection of biosignatures in the spectra of Earth-like exoplanets. Targeted small rocky planets would lie in the habitable zone around a star, where the temperatures are neither too cold nor too hot to preclude life, and where liquid water could exist over geological time scales. However, to observe a statistically significant number of stars, an observatory would potentially need to survey stars out to distances of about $20\mathrm{pc}$. Since such planets would subtend angles as small as $40\mathrm{mas}$ from their parent stars, the diffracted starlight would overwhelm the signal from a planet observed with a typical telescope. The ability to measure spectra of Earth-like exoplanets thus requires not only high angular resolution but also high starlight suppression over a broad bandwidth. One wavelength range in which to observe Earth-like planets is the mid-infrared ($\approx 6–20\mathrm{\mu m}$), where not only are there good biosignatures but also the contrast between stars and planets is more favorable than in the optical regime. However, to provide angular resolution sufficient to separate a planet from its star at mid-infrared wavelengths, a single telescope would require a primary mirror diameter greater than $40\text{\hspace{0.17em} m}$, and so interferometry, which combines the light from two or more separated telescope apertures, has been adopted as the baseline design approach of both NASA’s Terrestrial Planet Finder Interferometer (TPF-I) [1] and the European Space Agency’s Darwin [2] projects.

One of the key technology goals of the TPF-I project is thus to demonstrate broadband nulling to the level of $1\times {10}^{-5}$ or better at mid-infrared wavelengths. If the residual starlight is reduced to that amount, then the dominant sources of noise would be scattered light from our own solar system’s zodiacal cloud and light emitted from an equivalent exozodiacal cloud surrounding a typical target star. It is worth mentioning that the Earth–Sun flux ratio is $\sim {10}^{-7}$ at a wavelength of $10\mathrm{\mu m}$, and so another factor of $\sim 100$ in background noise rejection is necessary in order to detect Earth-like planets at a reasonable signal-to-noise ratio. This additional rejection, which requires further modulation steps, is not addressed here, but is the subject of ongoing theoretical and laboratory research [3, 4].

Interference fringes occur when electromagnetic waves propagating from a source follow different paths and are subsequently recombined. The interference upon recombination produces a spatially varying irradiance distribution, or fringe pattern, which is a function of the relative phase of the two beams. For a broadband source, the fringe pattern is modulated by a coherence envelope. In the absence of dispersion, the maximum in the broadband interference pattern occurs at the center of the envelope, because at that location the fringe phase at each wavelength is identical and a simultaneous constructive coherent addition of the electric field vectors across the whole waveband occurs. This can be achieved in a normal interferometer when the OPD at all wavelengths is reduced to zero; then the phase difference at all wavelengths is identically zero. On the other hand, a nulling interferometer produces a minimum in fringe intensity at the center of the coherence envelope [5] because the electric fields in this case cancel each other at all wavelengths across the waveband. For a two-beam interferometer, a central achromatic dark fringe can be achieved if there is an achromatic half-wave, or π, phase shift between incoming beams that is constant across the waveband. The goal of this paper is to evaluate different methods of producing such an achromatic phase shift. Note that some other proposed nulling interferometer arrangements, primarily those with an odd number of telescopes [6], call for achromatic phase shifts other than $\pi \mathrm{rad}$. This more general case is not under consideration here.

2. Interferometric Nulling

The null depth, N, is defined [7] as the ratio of intensities at adjacent dark, $I_{\text{min}}$, and bright, $I_{\text{max}}$, fringes: $N=I_{\text{min}}/I_{\text{max}}$. This is admittedly a nonideal metric, since only one of the two fringe extrema can in general be achromatic. Nevertheless, with the dark fringe achromatic, there is little error incurred in the ratio by some uncertainty in the constructive peak, and furthermore, this quantity is easily measurable, and so is a very serviceable metric. There are a number of factors that contribute to a degradation of the null depth for a nulling interferometer. Serabyn [7] has described the principal contributors that degrade the null, and following that treatment, the instantaneous null depth can be written as

Experiments in nulling interferometry have been ongoing for about the past 10 years and have been undertaken at visible, near-infrared, and mid- infrared wavelengths. Several approaches to achromatic phase shifting have been considered [8] and experimentally tested. The most common variants include (1) using glasses of slightly different thicknesses to introduce a wavelength-dependent dispersive phase delay [9, 10], (2) using a through-focus field flip of the light in one arm of the interferometer [11] (the Gouy phase), and (3) using relative field reversals on reflection in an antisymmetric flat-mirror periscope arrangement [5]. In the absence of aberrations, the third is inherently achromatic, the second is only slightly chromatic as a result of reflections off of mirrors of differing radii of curvature, and the first can be made relatively achromatic by tuning. Other techniques are summarized by Serabyn [8], including field reversal by diffraction and polarization methods such as the Pancharatnam phase. Three other approaches to achromatic phase shifting, which are more recent and beyond the scope of this work, are worth drawing to the reader’s attention: (1) using a deformable mirror to adjust the intensity and phase in a reimaged spectrum, i.e., adaptive nulling [12], (2) the use of total internal reflections with Fresnel rhombs [13], and (3) the use of a checkerboard spatial distribution [14].

Monochromatic null depths between roughly $1\times {10}^{-7}$ and $1\times {10}^{-6}$ have been demonstrated using laser light at visible [4, 15, 17] and mid-infrared [16] wavelengths. At near-infrared wavelengths nulls have been achieved between $1\times {10}^{-6}$ and $1\times {10}^{-5}$ using diode lasers with bandwidths of up to about 5% [18, 19]. Wider bandwidth nulling interferometers have attained null depths between $1\times {10}^{-5}$ and $1\times {10}^{-3}$; see, for example, Refs. [16, 20, 21, 22, 23]. Recently, Samuele et al. [15] demonstrated an almost $1\times {10}^{-6}$ null at visible wavelengths with a bandwidth of 15%. The goals of the work described in this paper were to test and compare the three techniques listed above for achromatic phase shifters at mid-infrared wavelengths, including both performance and ease of setup, alignment, and use, aiming at achieving broadband null depths at roughly the $1\times {10}^{-5}$ level currently envisioned as necessary for TPF-I.

3. Achromatic Nulling Testbed

The Achromatic Nulling Testbed (ANT) was designed to explore the three different methods of achieving an achromatic π phase shift described above in a single laboratory space. The dispersive glass plates and through-focus methods were the subject of experimental work at JPL prior to 2006, and the dispersive plates method is the technique used in the Keck nuller [4, 16, 24, 25, 26]. The results of that research are summarized only briefly below because much of that work was published earlier. Subsequent work has been devoted to the periscope approach, which is described more fully in the section that follows. A comparison of the results from the different achromatic phase shifting techniques is given in Table 1.

3A. Testbed Overview

The three methods were tested in the same laboratory but on different breadboard optical tables that shared the same broadband thermal source. Two key developments that supported all three methods in the suite of testbed work were the installation of a new argon plasma source and the procurement of novel mid-infrared single-mode fibers made from chalcogenide glass. The source layout for the later experiments is shown in Fig. 1.

The null signal for the interferometer is a chopped signal generated with a mechanical chopper wheel located in a focal plane near the infrared source (Fig. 1). The detector is a HgCdTe single pixel detector connected to a preamplifier and a lock-in amplifier. The source is chopped at $100\mathrm{Hz}$, which reduces the $1/f$ noise of the detector. In addition, chopping allows us to electronically subtract the thermal background, which would otherwise overwhelm the small interferometric null signal. The chopper wheel driver supplies the lock-in frequency to the electronics.

Because our nulling experiments are performed with room temperature optics, the large thermal background limits the dynamic range of the measurements. Prior mid-infrared nulling experiments at JPL used a ceramic filament as a source. The need for a brighter source led to the development of an argon arc source modeled after the work of Bridges and Migdall [20, 27]. This arc source is roughly eight times brighter in the mid-infrared than a $1500\mathrm{K}$ ceramic filament, thereby increasing the system dynamic range by the same factor. The resultant (random error) dynamic range is set by the ratio of the chopped source signal within a single mode to the background noise reaching the detector and was of the order of ${10}^6$ at $100\mathrm{Hz}$. Systematic errors (drifts) turned out to be more of a limitation (see below).

In addition to the broadband source, a ${\mathrm{CO}}_2$ laser was used for alignment purposes and for verifying the monochromatic performance of the interferometer. The laser is an indispensable tool in isolating different sources of null degradation, in particular distinguishing between achromatic and chromatic effects. Prior mid-infrared laser experiments at JPL had already demonstrated laser nulls better than $1\times {10}^{-6}$ [16].

Single-mode mid-infrared fibers were manufactured for this work because nulling interferometers under consideration for exoplanet detection would benefit from the spatial filtering and suppression of higher-order optical aberrations provided by the fibers [28]. Use of single-mode fibers allows the tolerances on the optics in the beam train to be lowered and become tractable [29]. Higher-order wavefront aberrations that would otherwise reduce the visibility of the fringes (depth of the null) are rejected by the spatial filter. Moreover, errors in tilt in each arm of the interferometer are translated into small errors in received intensity, which are relatively straightforward to correct. The experiments described in this paper make use of single-mode chalcogenide fibers developed for the TPF-I project and tested by Ksendzov et al. [29]. For all three versions of the testbed, the output light from the single-mode fiber is collimated by an off-axis parabolic mirror to provide a $25\mathrm{mm}$ diameter input source beam for the nuller (Fig. 1).

To minimize vibrations and unwanted path length fluctuations, the achromatic phase shifters and interferometers were vibrationally isolated using multiple levels of isolation. Each interferometer was built on an optical breadboard that sat on passive air-filled isolators supported on an optical table. The optical table was floated on compressed-air filled isolation legs. These table legs rested on an isolation pad built into the floor of the laboratory, which has a separate foundation from the rest of the building. In order to minimize acoustic vibrations and the effects of room air flow, each interferometer was surrounded by a plexiglas housing that was supported from the floor of the laboratory. This housing enclosed each interferometer without contacting it. Acoustic vibrations that might otherwise interfere with the measurements were therefore transmitted to the floor.

Finally, note that deep nulling requires a high degree of symmetry, which is most easily achieved by using a beam combiner based on a reversed pair of beam splitters, in, e.g., a modified Mach–Zehnder configuration [6, 7]. However, such an arrangement does lead to somewhat increased complexity, and to somewhat lower signal levels (individual signals are a factor of 2 lower in the case of an extra $50/50$ beam splitter). On the other hand, since in a laboratory interferometric testbed the source itself first needs to be split in two to provide two equivalent input beams, it is possible to make use of the necessary source beam splitter to provide a simplified laboratory arrangement. In this case the first beam splitter is used to split the source beam, and a reversed beam splitter is used to recombine the beams. This combination of beam splitters of course defines a normal Mach–Zehnder configuration and thus produces exactly the same effect as a modified Mach–Zehnder beam combiner would, i.e., each combining beam sees one beam splitter reflection and one beam splitter transmission. Thus, the desired symmetry is maintained with this simpler system, and so it is possible to make use of a normal Mach–Zehnder configuration in the laboratory with no loss of fidelity.

3A1. Dielectric Phase Shifters

The first implementation of dispersive phase correctors used a single pair of ZnSe plates to introduce a quasi-achromatic dispersive phase delay. This work followed on the previous experience obtained during development of the nulling combiner of the Keck interferometer [20, 21]. The nulling bandwidth for this approach is limited by the wavelength dependence of the phase delay. Experimental results gave rms null depths of $8.8\times {10}^{-5}$, which were within approximately a factor of 3 of the $3.7\times {10}^{-5}$ theoretical limit for a $3\text{-}\mathrm{\mu m}$ bandwidth centered at $10.0\mathrm{\mu m}$ [24].

The successful results obtained using a single-glass dispersive phase delay led to the construction of a dual-glass phase delay architecture [26] as seen in Fig. 2. The overall balancing of chromatic effects obtained using two glasses can extend the π phase shift over a larger bandwidth, or provide a deeper null depth over a narrower bandwidth. The challenge for the dual-glass architecture is the increased complexity involved in optimizing the differential thicknesses of the two glasses. The solution space for two glasses allows very deep nulling over a large range of glass thicknesses but requires fine thickness adjustment in one glass [30]. Thickness adjustments as small as $100\mathrm{nm}$ are needed to optimize the null.

The dual-glass phase shifter used motorized rotation stages to turn ZnSe and ZnS plates (Fig. 3) of approximately $15\mathrm{mm}$ thickness with a $2\mathrm{arcmin}$ wedge. The wedge prevents parasitic fringes due to the Fabry–Perot effect in a plane parallel plate. The optimal thickness differences were calculated with lens design software to give minimum OPD between two beams over the chosen passband. The optimal differential thicknesses for ZnSe and ZnS over our 25% passband centered at $9.5\mathrm{\mu m}$ are 454.03 and $172.48\mathrm{\mu m}$, respectively. The results vary slightly ($\pm 0.5\mathrm{\mu m}$ thickness difference) depending on the optimization method chosen. Thickness measurements of the glass plates were made using a laser metrology system with submicrometer accuracy [31], showing a thickness difference near optimal for ZnSe of $449.0\mathrm{\mu m}$ but a poorer match of $144.9\mathrm{\mu m}$ for ZnS. The plate rotation required to compensate the thickness error in ZnS was over $7°$. This introduced a polarization-dependent intensity imbalance due to Fresnel reflection effects and decreased the thickness adjustment resolution, which is angularly dependent. In addition, deviation from normal incidence introduces beam shear as the glass thickness is adjusted. This beam shear contributes to beam intensity mismatch when coupled into single-mode fiber, thus degrading the null depth.

Theoretical calculations predicted that an ideal dual-glass compensator could produce deeper nulls for a given bandwidth than a single glass. However, this testbed achieved results only comparable to previous single-glass experiments (see Table 1), likely due to the nonideal thickness of the ZnS plates, and the consequent need to rotate the glass plates to nonoptimal angles. The beam shear, glass thickness, and polarization issues can be avoided by using pairs of opposed laterally translatable wedges of glass, as in the case of the Keck interfero meter nuller [32], but as this would double the number of glass elements, it has not yet been employed in our laboratory testbed work. It is also possible to implement the two-glass solution with two opposing wedges of different materials [19], but this has not been implemented here.

3A2. Through-Focus Phase Shift (Gouy Phase)

In the through-focus or Gouy phase approach, the beam in one arm of the interferometer is sent through a matched pair of off-axis parabolic (OAP) mirrors and passes through a focus, while the other beam travels the same distance but reflects off plane mirrors. This layout is shown in Fig. 4. The passage through the focus causes a field inversion, which is equivalent to introducing a π phase shift [33, 34]. Slight achromaticities may arise due to differing angles on the curved and flat mirrors.

Precise alignment of the OAPs is an extremely important aspect for this approach. The alignment method involved a He–Ne laser boresighted to the IR beam, target irises, and shear plates. Even so, this method only provided pointing alignment of the interferometer beams to approximately $1\mathrm{arcmin}$. In addition, any clocking of the parabolas can lead to imperfect subtraction because of the resultant aberrations and polarization effects. Unfortunately, the use of irises and shear plates during alignment could not effectively determine the relative clocking of the two OAPs to very high accuracy. For example, our lens design software model of the interfero meter predicted that a $1°$ clocking of one OAP produces a combination of astigmatism and coma with a peak-to-valley magnitude of $\lambda /15$ at $\lambda =9.5\mathrm{\mu m}$. However, the alignment method we used was found to be insensitive to this amount of clocking, leaving a small amount of differential aberration between the two beams, which can leave a phase error that limits the null depth. A peak-to-valley error of $\lambda /15$ gives approximately $\lambda /50\mathrm{rms}$ phase error ($0.125\mathrm{rad}\mathrm{rms}$), which limits the null depth to $\approx 1\times {10}^{-3}$.

The best nulling results for the through-focus interferometer yielded a null depth of only $6.7\times {10}^{-4}$ over a 25% bandwidth, which was at least a factor of two worse than the previous dispersive glass nulling results [26]. The primary difficulty was the inability to align the system to sufficient accuracy. For this reason it also proved difficult to obtain repeatable results. Successive alignments yielded null depths ranging from $5\times {10}^{-3}$ to $7\times {10}^{-4}$, without any easily discernible difference in the system alignment. The use of through-focus Gouy phase would likely produce better nulls if an improved alignment method, such as a laser Fizeau interferometer, were implemented. It is also possible that longer focal length OAPs might reduce the sensitivity to misalignment, and thereby improve the achievable null depths.

4. Periscope Phase Shifter and Nuller

The periscope phase shifter [5] has no powered optics in the arms of the interferometer and is therefore easier to align. The phase shift is accomplished through an electric field flip (or pupil inversion) of one pupil relative to the other within an antisymmetric periscopelike arrangement of mirrors, as shown in Fig. 5. The periscope layout is fully antisymmetric, and because of the geometric nature of the field flip, the π phase shift is intrinsically achromatic for matched optical trains. Our field inversion periscope is composed of four mirrored prisms optically bonded to a single glass block, referred to here as the periscope monolith. Because this approach yielded the best results, we now discuss it in more detail.

4A. Input and Output Optics

The periscope nuller used single-mode spatial filters both at the input and the output. Each spatial filter is composed of two OAP mirrors with a single-mode mid-infrared fiber made from chalcogenide glass in between. The single-mode chalcogenide fiber has a core diameter of $23\mathrm{\mu m}$ and a cladding diameter of $170\mathrm{\mu m}$. Characterization of the fibers was done at JPL by Ksendzov et al. [29]. As shown in Fig. 1, an OAP mirror focuses light from the broadband source onto the single-mode fiber and the output of this fiber is collimated using another OAP. The first spatial filter is used at the interferometer input to provide a broadband artificial star that is spatially coherent (i.e., an unresolved point source). This input fiber is not strictly required, although it greatly simplifies the alignment tolerances by reducing sensitivity to beam shear. The beam is then steered into the nuller as shown labeled “From Source” in Figs. 6, 7.

The ${\mathrm{CO}}_2$ laser and arc sources, which were common to all ANT layouts, were effectively co-aligned by means of injection into the same single-mode fiber. This guarantees that the laser and broadband sources are injected into the interferometer identically. The second spatial filter is implemented at the output of the interferometer. This additional chalcogenide fiber filters the wavefronts in the combined beam. Tilt errors between the interferometer arms are converted to an intensity mismatch when the beams are coupled into the single-mode fiber at the output.

4B. Mach–Zehnder Interferometer

As noted earlier, to simplify the layout, our laboratory nulling architecture is essentially a classical Mach–Zehnder interferometer. In the case of the periscope architecture, the periscope field flip optics are then located between the two beam splitters. The source and input optics, including a single-mode fiber, are shown in Fig. 1. The overall layout of the interferometer is shown in Fig. 6, and a close-up of the input beam splitter (Beam splitter 1) and the periscope monolith is shown in Fig. 7. Light from a broadband infrared source is transferred through the single-mode fiber, as illustrated in Fig. 1, and directed upward to Beam splitter 1, shown in Figs. 6, 7. The two resulting beams are reflected down into the periscope monolith where the electric field inversion is performed as illustrated in Fig. 5. One interferometer beam reflects off the piston mirror, which is driven by laser metrology for implementation of path length (phase) control via a picomotor and piezoelectric transducer (PZT) on a translation stage. The two interferometer beams are combined at Beam splitter 2 and the nulled output is then directed to the output single-mode fiber and HgCdTe detector.

4C. Dispersion Compensation

The only transmissive components required in the interferometer are two ZnSe beam splitters. These beam splitters have, potentially, a difference of up to $4\mathrm{\mu m}$ in their respective optical thicknesses because of manufacturing tolerances and thus introduce different amounts of chromatic dispersion into each beam. Since the beam splitters have a $2\mathrm{arcmin}$ wedge, differences in beam height at each beam splitter will add to the effective differential glass thickness. Left uncorrected, a $4\mathrm{\mu m}$ difference would limit the null depth to $2.8\times {10}^{-5}$ for a 25% passband centered at $9.5\mathrm{\mu m}$ (Fig. 8). A discussion of the beam splitter effects and design can be found in Martin et al. [16]. The actual beam splitter thickness differences were not measured, but residual dispersion in the interferometer indicates a glass thickness difference of as much as $8\mathrm{\mu m}$. Manufacturing tolerances alone do not account for this difference, and thus the effect is likely due to the glass wedge angle. Beam centration on the beam splitters was not a major driver during alignment, and observations indicate up to $10\mathrm{mm}$ difference between the position on the first and second beam splitters. To compensate for the resultant dispersion, wedged ZnSe compensator plates were included in each interferometer beam, which can be rotated with respect to each other to match the beam splitter differential optical thickness. Equalizing the amount of ZnSe in each beam path to better than $1\mathrm{\mu m}$, in theory allows a null depth of $1.7\times {10}^{-6}$. The dispersion compensating plates are shown in the interferometer layout of Fig. 6.

4D. Intensity and Phase Control

One of the most important requirements for a deep null is equal intensities in the two beams, so a precise means of adjusting the actual intensity is needed. The degree of required control is discussed below. Intensity balancing is accomplished by the insertion of a pair of crossed wires in each beam, which are moved laterally by a picomotor translation stage to differentially adjust the intensity. (Note, this adjustment does not provide control of wavelength- dependent amplitude differences, which may be present due to different beam splitter coatings, diffraction effects, or other sources within the interferometer.)

Phase control is split up into several contributions. The OPD between the arms of the interferometer is affected by vibration (high frequency), thermal drifts (low frequency), and dispersion (static). While nulling, the phase difference between the interferometer arms is maintained using an active control loop to drive the piston mirror in Beam 1. Although the passive vibration isolation effectively minimizes the higher frequency OPD effects, a laser-based heterodyne metrology control system operating at $633\mathrm{nm}$ is used to actively control the OPD for frequencies below $10\mathrm{Hz}$. The metrology control loop operates at a $100\mathrm{kHz}$ sampling rate and a $10\mathrm{kHz}$ control bandwidth. A $1\mathrm{mm}$ diameter metrology beam is injected through the back side of Beam splitter 1 decentered by approximately $5\mathrm{mm}$ from the infrared optical axis as shown in Fig. 7. This allows the metrology signal to measure the OPD along virtually the same path as the broadband infrared beam. The metrology beam is combined with a local oscillator, and the resultant signals are detected from the back side of Beam splitter 2.

5. Error Budget for the Periscope Nuller

As introduced earlier, Eq. (1) expresses the primary factors affecting nulling performance. For the single-polarization case discussed here, we can neglect the polarization term, $\mathrm{\Delta }{\mathrm{\Phi }}_{s-p}$. We used two infrared polarizers with a rejection ratio of ${10}^{-4}$ each to achieve a total linear polarization purity of $\sim {10}^{-8}$. Table 2 summarizes the factors affecting null depth on the periscope nuller, the measured level of control, and the estimated contribution to null depth for each factor. These factors are discussed in detail in the following sections.

5A. Polarization Rotation

An alignment error within the periscope monolith would cause a polarization rotation error. This rotation can also be understood as a pupil rotation. To achieve a null of $1\times {10}^{-5}$ for the system, the pupil rotation must be within $3\mathrm{mrad}$ of π. This turns out to be relatively easy to accomplish during construction of the periscope monolith. An autocollimator was used to monitor alignment of each mirror surface of the monolith. The mirrors were optically contacted to a glass base plate to form the monolith. Using a wet optical contacting method allowed each mirror to be adjusted to $15\mathrm{arcsec}$ tolerance [35], at least a factor of 4 better than the pupil rotation error requirement.

Alignment of the incidence angle at the beam splitters is a much more difficult process. Due to the three-dimensional layout of the periscope nuller, errors in alignment angle result in shear and path length errors through the monolith. Also, differences in reflectance angle on the two beam splitters may cause wavelength-dependent intensity differences in the beams. An autocollimator, a coordinate measurement arm, and mechanical alignment targets were used to align the interferometer such that the incidence angles on the two beam splitters matched to within $2\mathrm{arcmin}$. Broadband beam splitter coatings are generally insensitive to incident angle changes of this magnitude. Based on this modeling, the wavelength-dependent intensity effects should be negligible. In addition, the alignment method ensured that the beams exiting the monolith were parallel to better than $1\mathrm{arcmin}$, and each mirror surface within the monolith was correct to within $1\mathrm{arcmin}$ of the nominal $45°$ angle. For a statistically likely case, in which all four monolith components have opposite $30\mathrm{arcsec}$ angular errors, the resultant pupil rotation difference between interferometer arms is $2\mathrm{arcmin}$ ($1.2\mathrm{mrad}$). We can therefore allocate a tolerance of pupil rotation of ${\alpha }_{\text{rot}}=1.2\times {10}^{-3}\mathrm{rad}$. This allocation alone would limit the null depth to a negligible contribution of ($1/4$) ${{\alpha }_{\text{rot}}}^2=3.6\times {10}^{-7}$.

5B. Intensity and Phase Balance

As mentioned previously, beam intensity balance is controlled with thin wires in each beam. Measurement of the individual beam power for the inter ferometer arms shows that adjustment of the wires allows the average intensity difference to be controlled to better than 0.25%. If we set $\delta I=2.5\times {10}^{-3}$, this residual intensity difference alone should limit the achievable null depth to $N_{\delta I}=(1/4)(\delta I{)}^2=1.6\times {10}^{-6}$.

The heterodyne metrology system provides data on both the passive path length control through vibration isolation and the active path control when the control loop is on. Calculation of the cumulative rms phase error from open loop metrology data indicates that the passive vibration isolation provides residual optical path stability of $2–3\mathrm{nm}\mathrm{rms}$ for frequencies above $10\mathrm{Hz}$, as shown in Fig. 9. Performing the same calculation on closed loop metrology data shows that OPD control maintains the same $2–3\mathrm{nm}\mathrm{rms}$ path stability for frequencies below $10\mathrm{Hz}$. If we set $x=3\mathrm{nm}$ in $\mathrm{\Delta }\mathrm{\Phi }=(2\pi /\lambda )x$, this level of path fluctuation alone should limit the achievable null depth to $N_{\mathrm{\Delta }\mathrm{\Phi }}=(1/4)(\mathrm{\Delta }\mathrm{\Phi }{)}^2=1.0\times {10}^{-6}$.

The rotation, intensity, and phase terms considered above are all monochromatic contributions to the null. Given that the best measured monochromatic (laser) null measured on the periscope nuller is $N_{\text{laser}}=3.3\times {10}^{-6}$, the agreement with the sum of the three estimated monochromatic contributions, $3.0\times {10}^{-6}$, is excellent. This monochromatic laser null is a factor of a few worse than earlier ${\mathrm{CO}}_2$ laser work [16], because the goals of the ANT were not deep laser nulls per se. Instead, the laser only needed to be nulled to a level below that required by the broadband experiments to follow.

5C. Chromatic Dispersion

In addition to monochromatic terms, there are two wavelength-dependent terms that contribute to degradation of the null. The first is chromatic dispersion due primarily to thickness differences in the beam splitter and recombiner that must be accounted for. As described previously, ZnSe compensator plates are used to balance this dispersion. A differential glass thickness will cause the broadband fringe envelope to be asymmetric about the central null fringe. The asymmetry can be adjusted by rotating the compensator plates to balance the intensity of the fringes immediately adjacent to the central null. (Alternatively, a long-scan Fourier transform can be used, but this technique, when implemented, proved no more sensitive than the balancing of fringe minima.) A corrected fringe asymmetry of approximately 2% was routinely achievable, which corresponds to a differential glass thickness calculated to be approximately $1\mathrm{\mu m}$. The index of refraction of ZnSe at $\lambda =10\mathrm{\mu m}$ is 2.4, and therefore the null fringe would be found when $1\mathrm{\mu m}$ of ZnSe is matched with $2.4\mathrm{\mu m}$ of air path. This glass mismatch introduces a phase slope across the passband, and if we assume that the phase difference is identically zero at the center of the band (${\lambda }_{\text{center}}=9.5\mathrm{\mu m}$, 25% bandwidth), then there is an rms phase of $\mathrm{\Delta }{\mathrm{\Phi }}_{\lambda }=2.6\times {10}^{-3}$. If all other effects were perfectly compensated, this would limit the null depth to $(1/4)(\mathrm{\Delta }{\mathrm{\Phi }}_{\lambda }{)}^2=1.7\times {10}^{-6}$.

The second wavelength-dependent term is the chromatic intensity difference between beams. This difference may be a result of beam splitter or mirror reflectivity differences. We did not have a means by which to measure the chromatic intensity difference between the interferometer beams, but the effect can be modeled in software. Since the monochromatic intensity difference is effectively an average across the waveband, we expect any chromatic difference to be equal or less in magnitude than the monochromatic term. Allowing a 0.25% linear slope in intensity across a 25% bandpass, centered at $9.45\mathrm{\mu m}$, produces a null depth limitation of only $3.3\times {10}^{-8}$.

5D. Source Size and Single-Mode Fibers

The spatial coherence of the broadband source is another limiting factor for the null depth. The source size term in Eq. (1) indicates that both the source spatial extent and the interferometer spatial resolution contribute to this effect. With ideal single-mode fibers on the input and output of the interferometer there would be no contribution from this term—the Mach–Zehnder experiment configuration has an effective interferometer baseline b of zero (both arms sample the same piece of wavefront), and an ideal input fiber has a source diameter of ${\theta }_{\text{dia}}=0$ [28].

Furthermore the use of an ideal single-mode fiber as an output spatial filter translates the phase distribution in the pupil plane into its single average phase value within the fiber. With appropriate amplitude (beam intensity) control, an ideal output spatial filter therefore creates an opportunity for perfect nulling. However, in practice, insufficient higher- order mode suppression of the fibers may be a limiting factor. For the chalcogenide fibers used here, the fiber cladding modes are suppressed by a rejection factor of 1000 [29]. Since stray light from the cladding may add incoherently to the null signal at the detector, this suppression factor is important for realizing deep, low-noise nulls.

There are two potential sources of light in the fiber cladding: first, light from the source image on the input fiber tip that falls outside of the fiber core area, and second, aberrated light that arrives at the second fiber tip outside of the core of the ideal point-spread function (due to aberrations in the interferometer optics). In the first case, light from our broadband argon arc source is focused onto a $60\mathrm{\mu m}$ pinhole, which is then imaged onto the input fiber tip with a magnification of 0.6. If all light from the $36\mathrm{\mu m}$ pinhole image that does not couple into the $23\mathrm{\mu m}$ core is instead coupled into the cladding, then the irradiance in the cladding is approximately equal to that in the core. Assuming no net loss of cladding modes in the interferometer, the cladding light is suppressed by a factor of $r=1000$ per fiber. The result is a null contribution of $N_{\text{clad}}\approx r^{-2}\approx {10}^{-6}$. In the second case, only one fiber is traversed by the aberrated light generated within the interferometer. Based on the surface quality of the ANT optics, the rms wavefront error is likely to be about $0.03\mathrm{rad}\mathrm{rms}$ at $10\mathrm{\mu m}$, so that approximately 0.1% of the light would couple into the output fiber cladding. The resulting null depth contribution would thus again be $\sim {10}^{-6}$. Thus, while more uncertain than some of the other terms discussed, the cladding leakage likely does not provide a major limitation.

5E. Polarization Delay

Due to the orientation of beam splitters in the interferometer there is a built-in orientation for s and p polarization. From the symmetry of the design, in the ideal case, one would expect no net phase difference between the two polarizations. In practice we found an estimated $10–15\mathrm{nm}$ difference in path length for the s and p polarizations. This measurement was performed using a gold wire grid infrared polarizer. The metrology system was used to hold the interferometer at the null fringe while the polarizer was rotated through $90°$. Metrology piston control was then moved in $10\mathrm{nm}$ steps to again minimize the null fringe. This polarization dependent path length difference is reflected in the nulling results shown in Table 1. Allocating $x_{s-p}=10\mathrm{nm}$, we have $\mathrm{\Delta }{\mathrm{\Phi }}_{s-p}=(2\pi /\lambda )x_{s-p}=1.1\times {10}^{-5}$. We consistently observed deeper nulls with a polarizer in place than without. While there are methods for introducing polarization rotation, such as using the Pancharatnam phase [36], no method of adjusting the polarization phase difference was included in this testbed. In order to achieve the deepest nulls, the broadband experiments were limited to single-polarization nulling.

5F. Error Budget Summary

Table 2 summarizes the expected error budget for single-polarization nulling measurements taken at 20% bandwidth. The parameters, alignment tolerances, and calculations are as expressed in Eq. (1). Several of the resultant error budget terms are similar, $\sim 1–2\times {10}^{-6}$, so no single term dominates the resultant error budget. The net error budget yields an estimate for the best possible null depth of $\sim 6\times {10}^{-6}$, although it must be remembered that several of the terms entering this sum are rather rough estimates.

6. Periscope Nuller Results

The periscope nuller consistently yielded $10\mathrm{\mu m}$ laser single-polarization null depths of the order of $5\times {10}^{-6}$, with best average values of $3.3\times {10}^{-6}$. The laser nulls were used primarily as a diagnostic tool to confirm that monochromatic effects were controlled to the desired levels. The best single- polarization broadband nulls achieved to date with the periscope nuller are displayed in Figs. 10, 11. Both the 20% and the 25% bandwidth data were obtained using infrared bandpass filters with a center wavelength of $9.45\mathrm{\mu m}$. The only change to the interferometer between the two data sets was the bandpass filter used. The data show average nulls of $2\times {10}^{-5}$ for 20% bandwidth and $4\times {10}^{-5}$ for 25% bandwidth, with a few best nulls of the order of $9\times {10}^{-6}$ for the 20% case. The null clearly drifts slowly and regularly between $\sim 9\times {10}^{-6}$ and $3–4\times {10}^{-5}$. This slow drift is also present in “dark” data, shown in Fig. 12, suggesting that a low level electronic drift or instability is limiting the long-term average null depths attainable with white light to $\sim {10}^{-5}$. This drift is not as much a limiting factor for laser measurements, because of the higher signal levels. Nevertheless, even in the presence of this drift, the best short-term nulls seen, $\sim 9\times {10}^{-6}$, are very close to the prediction of the error budget, $\sim 6\times {10}^{-6}$.

7. Conclusion

Of the three nulling approaches that were examined, the periscope nuller yielded the best broadband single-polarization nulls. This may have been the result of the inherently achromatic architecture of this phase shifter, however this nuller was implemented within an interferometer that used potentially dispersive elements: a matched beam splitter/ recombiner pair with an inevitable thickness difference. The best nulling results obtained with the periscope nuller are quite close to the null depth predicted by a detailed error budget, and the attainable long-term average null is limited by a slight electronic instability or drift at approximately the ${10}^{-5}$ level. These broadband nulls are, to our knowledge, the best broadband mid-infrared nulling results achieved to date without the use of active optics. The periscope nuller has thus been able to demonstrate performance levels close to those demanded by mid-infrared terrestrial exoplanet observations. Of course as a result, the performance already exceeds that required for nulling observations of Hot Jupiters [37].

The performance of the other two approaches tested might also have yielded improved results had the testbeds been equipped with higher- resolution alignment techniques. The limitations in the dual-glass approach were attributed to insufficient resolution in the adjustment of differential glass thickness. In particular, opposed pairs of wedged glass elements [32] can provide improved thickness resolution and would also remove many other issues from the table (such as beam shear and polarization effects). The limitations in the through-focus approach, attributed to insufficient resolution in pointing and clocking alignment, can of course also be improved upon. Accurate alignment and balancing of the relevant parameters is of course critical to the success of any nuller, and as pointed out earlier, symmetry and stability are paramount [6, 7]. It should also be noted that both these architectures were implemented in a dual polarization arrangement. Given the polarization limitations in the periscope nuller, it is possible that single- polarization measurements for the dual-glass and through focus methods could have yielded comparable results.

In parallel with this research, work in adaptive nulling has also been conducted at JPL [12], which has yielded null depths of $1.1\times {10}^{-5}$ at 32% bandwidth, slightly exceeding the performance reported here. In fact, the static and active methods of compensation are complementary, in that the stroke of active nulling components is limited, especially in relation to long mid-infrared wavelengths. Thus, a flight nulling system would likely rely on both types of nuller, with a classical broadband nuller to get within the stroke range of the final active nulling stage. Thus, assuming such a hybrid approach, many of the necessary phase shifting capabilities needed for a TPF-I-like flight system have been demonstrated to approximately the levels required for exoplanet observations in the mid-infrared.

The work described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA).

Table 1. Experimental Results for Various Achromatic Nulling Testbed Architectures

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Table 2. Contributing Factors for Single-Polarization Null Depth on Periscope Nuller

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Fig. 1 Source and input spatial filter layout. ${\mathrm{CO}}_2$ laser and broadband source are co-aligned into the pinhole and single-mode fiber. Output of the source module is the input to the nuller.

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Fig. 2 Phase plate nuller layout. One phase plate in each beam is rotated to optimize the differential glass thickness to generate quasi-achromatic phase shift.

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Fig. 3 Dual-glass phase plate adjustment via rotation stages. One plate of each glass type is rotated.

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Fig. 4 Through-focus nuller layout. Matched focal length parabolic mirrors provide the mechanism for the Gouy phase shift. The microbolometer array is used to view the infrared beam during alignment.

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Fig. 5 Electric field (pupil) inversion in the periscope nulling architecture.

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Fig. 6 Periscope nuller layout. Note, the metrology beam enters from the top (back side of Beam splitter 1) and exits to a separate detector (not shown) from the back side of Beam splitter 2.

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Fig. 7 Photograph of the periscope nulling interferometer including Beam splitter 1 and the field flip mirrors in the periscope monolith.

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Fig. 8 Null depth versus differential glass thickness. Bandwidth for calculation is 25% centered at $9.45\mathrm{\mu m}$.

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Fig. 9 Metrology data showing path length stability. The upper trace shows the raw, unprocessed metrology signal, indicating the real-time path length stability. The lower trace shows the cumulative rms path difference as a function of vibration frequency.

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Fig. 10 Null fringe measurement using a bandpass filter with a full width at half-maximum width of 20%, centered at $9.45\mathrm{\mu m}$. The fringe signal is normalized to the constructive peak signal.

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Fig. 11 Null fringe measurement for 25% bandwidth, centered at $9.45\mathrm{\mu m}$.

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Fig. 12 Electronic noise on the periscope nuller. The vertical scale is the same as the null measurements. This is the signal measured by the detector and electronics with the infrared source blocked.

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