Non-Redundant Mask (NRM) Mode

• NRM Mode is an interferometric imaging technique that uses a non-redundant pupil mask, ensuring each hole pair provides a unique baseline for enhanced calibration.
• It leverages precise closure-phase measurements and Fourier analysis to model the PSF, facilitating robust high-angular-resolution imaging and error diagnostics.
• Implemented on instruments like GPI and JWST–NIRISS, NRM enables exoplanet detection and circumstellar imaging through advanced wavefront sensing and calibration.

The Non-Redundant Mask (NRM) Mode is an interferometric imaging technique that employs a carefully designed pupil mask with holes arranged so that each pair of holes defines a unique interferometric baseline. This configuration is implemented on major astronomical instruments such as the Gemini Planet Imager (GPI) and the JWST–NIRISS, enabling high-angular-resolution observations, robust calibration via closure phases, and sensitive diagnostics of instrumental wavefront errors. NRM converts a single-aperture telescope into an interferometric array, optimizing spatial frequency coverage and leveraging the redundancy-free fringe pattern for both scientific imaging and instrumental calibration.

1. Interferometric Principles and Mask Design

NRM is realized by inserting a mask in the pupil plane with $N$ holes, ensuring that each hole pair $(i, j)$ forms a unique baseline vector. For the GPI, a 10-hole mask yields 45 non-redundant baselines; JWST–NIRISS uses a 7-hole mask producing 21 baselines. The mask's geometry enforces non-redundancy, which is pivotal for interferometric self-calibration and for assigning a unique spatial frequency to each baseline.

Mathematically, the image-plane point spread function (PSF) is modeled as the sum of interference terms modulated by the envelope of the individual subaperture:

$$\text{PSF}(k) = P(k) \left\{ N + \sum_{i < j} 2 [\cos(k \cdot (x_i - x_j)) \cos(\Delta\phi_{i, j}) - \sin(k \cdot (x_i - x_j)) \sin(\Delta\phi_{i,j})] \right\}$$

where $P(k)$ is the envelope (primary beam, often the Airy function), $x_i$ are hole positions, and $\Delta\phi_{i,j}$ are piston phase differences. For a mask with hexagonal holes, as in JWST–NIRISS, the model incorporates per-hole amplitude spread functions and precise alignment of projected coordinates.

2. Observing Modes and Instrument Integration

NRM operates as a selectable mode within instruments such as GPI and JWST–NIRISS. On GPI, the NRM mask lies on the apodizer wheel after the deformable mirror, enabling both spectroscopic (integral field spectrograph, IFS) and polarimetric observations. JWST–NIRISS implements the mask within its pupil wheel, supporting imaging at wavelengths between $3$–$5~\mu$m in the AMI mode.

NRM is suited to imaging with an inner working angle down to $\sim$ half the diffraction limit ($\lambda/2B$ for the longest baseline $B$). Through spectroscopic discretization ($\sim$37 channels in GPI IFS mode), NRM enables near-monochromatic fringe extraction and multi-wavelength diagnostics. The mode also functions as a plate scale calibrator and as an attenuator for bright standard stars.

3. Calibration, Data Reduction, and Error Analysis

NRM’s scientific utility depends on precise calibration of instrumental and atmospheric errors using closure phases—sums of baseline phases in closed triangles:

$$\text{Closure Phase}_{123} = \Delta\phi_{12} + \Delta\phi_{23} + \Delta\phi_{31}$$

For point sources, closure phases are theoretically zero, rendering them immune to uniform piston errors across the instrument. Instrumental calibration uses raw closure and kernel phase measurements from science and calibrator stars, employing polynomial time calibration (polycal) and optimized weight schemes to minimize systematic bias.

Error sources include photon noise (with closure phase standard deviation scaling as $$\sigma_\mathrm{CP} \propto N_\mathrm{holes}/\sqrt{N_\mathrm{phot}}$$), flat-field variation, and intra-pixel sensitivity variation. For JWST–NIRISS, high Strehl ratios (90–95%) across $2.77$–$4.8~\mu$m support reliable fringe measurement. Sub-Nyquist sampling (e.g., in F277W) necessitates additional calibration to mitigate IPS-induced phase errors and maintain contrasts required for exoplanet detection ($\sigma_\mathrm{CP} < 10^{-4}$ radians).

4. Imaging Algorithms and Reconstruction

NRM data reduction pipelines typically include dark/flat/background subtraction, bad pixel correction, and Fourier transformation of interferograms. Algorithms for fringe fitting include direct image-plane modeling (e.g., the Lacour–Greenbaum method, as in ImPlaneIA) and Fourier-space extraction. Observational strategies involve dithering for flat-field calibration and near-contemporaneous calibrator observations for instrumental drift tracking.

Image reconstruction from incomplete Fourier coverage uses deterministic algorithms (e.g., BSMEM with maximum entropy regularization), and stochastic methods (e.g., SQUEEZE with simulated annealing and MCMC sampling). Kernel phase projection via SVD is implemented to obtain phase observables robust to correlated errors.

5. Scientific Applications and Performance Metrics

NRM is optimized for high angular resolution applications inaccessible to direct imaging or coronagraphy: detection of close-in exoplanet companions, gap-structure mapping in transition disks, and resolved binary astrometry. Typical performance on GPI indicates $5\sigma$ contrast sensitivities of $2-3 \times 10^{-3}$ near $\lambda/D$, correlated with residual AO-measured wavefront error.

Closure phase stability in laboratory settings can reach median standard deviations of 0.0046 radians in H band; on sky, atmospheric turbulence degrades this to 0.016–0.055 radians, yet binary companions at $88.4 \pm 0.5$ mas and contrast ratios $5.94 \pm 0.09$ have been detected within tens of minutes. Polarimetric NRM achieves differential visibility precisions of $\sim 0.4\%$.

In JWST–NIRISS AMI, the non-redundant mask achieves an inner working angle of $\sim 70$ mas, with contrasts up to $8$–$9$ magnitudes. AMI has been applied successfully to exoplanet and circumstellar disk imaging, AGN studies, and zodiacal light detection.

6. Wavefront Sensing and Advanced Techniques

NRM sampling introduces non-redundant pupil diversity, which can resolve phase ambiguities in in-focus wavefront sensing algorithms such as Gerchberg–Saxton. NRM-derived fringe phases provide coarse piston measurements over mask holes, which, when enforced in initial GS iterations, break the sign-flip ambiguity in phase retrieval:

$$\phi_\mathrm{updated}(x) = \begin{cases} \phi_\mathrm{NRM}(x) & x \in \text{NRM holes} \ \text{GS-updated } \phi(x) & \text{elsewhere} \end{cases}$$

This technique has proven effective on segmented mirrors (e.g., JWST primary), enabling in-focus sensing and backup wavefront diagnostics without hardware-based phase diversity, with measured performance robust against moderate jitter and finite bandwidth.

7. Comparative Methods and Future Directions

NRM contrasts with filled-aperture kernel phase imaging, which preserves throughput and is most applicable at high Strehl ratios, but is limited by systematic errors at lower Strehl. Simulated and observational contrast curves show that NRM can outperform kernel phase at $\lesssim \lambda/D$ for bright targets, while kernel phase is more sensitive for faint objects and spectral applications.

Next-generation facilities (e.g., GMT, ELTs) will benefit from more uniform aperture geometry, enabling higher-order masks and improved ($u,v$) coverage. NRM principles extend to photonic applications, as in the Costas-array-based optical phased arrays, achieving $N^2$ scaling of resolvable points and mitigating the control complexity seen in traditional OPAs (Fukui et al., 2021).

Table: Comparison of NRM Implementations

Instrument	Mask Holes	Baselines	Inner Working Angle	Typical Contrast	Spectral Bands
GPI	10	45	$\sim \lambda/2B$	$2-3 \times 10^{-3}$	$Y, J, H, K$
JWST–NIRISS	7	21	$\sim 70$ mas	$8$–$9$ mag	$2.77$–$5~\mu$m

NRM continues to advance high-contrast astrophysical imaging by delivering robust interferometric and calibration capabilities, and its principles now extend into photonic device engineering and machine learning contexts where elimination of redundancy leads to optimal efficiency. The mode’s ongoing development—in data reduction, calibration strategy, and algorithmic innovation—is integral to future scientific performance in both ground-based and space-based observatories.