Four Quadrant Phase Mask (4QPM) in Coronagraphy

• 4QPM is a focal-plane coronagraph that divides the image into four quadrants with pairwise π phase differences to cancel on-axis starlight.
• It employs spatial phase transitions and multi-stage designs to achieve broadband starlight suppression in instruments like VLT/SPHERE and JWST.
• Advanced implementations integrate adaptive optics, apodization, and wavefront control to mitigate chromatic and manufacturing limitations for enhanced exoplanet detection.

The Four Quadrant Phase Mask (4QPM) is a focal-plane coronagraphic device engineered to suppress the overwhelming starlight from an on-axis source, enabling high-contrast direct imaging of exoplanets and circumstellar environments. The core innovation of the 4QPM is its use of spatially discrete phase transitions in the focal plane, designed to induce destructive interference of starlight when re-imaged onto the pupil plane. By segmenting the focal plane into four spatial quadrants with pairwise π phase differences, the 4QPM redirects the coherent flux from a centered star in such a manner that, after propagation, it is effectively nulled within the geometric pupil and blocked by a subsequent Lyot stop. This approach has been integral to both ground-based and space-borne coronagraphic instruments and has evolved with advances such as multi-stage implementations, adaptive designs, and integration with wavefront control and differential imaging methodologies.

1. Physical Principle and Mathematical Formalism

The canonical 4QPM operates by imposing a phase function on the focal-plane electric field. For an incoming wavefront E(x, y), the mask's transmission is defined as:

$$M_\lambda(x, y) = \exp\left\{i \phi(\lambda)\frac{1 - \text{sgn}(x)\text{sgn}(y)}{2}\right\}$$

where sgn(x) is the sign function and φ(λ) = π λ₁/λ, with λ₁ as the wavelength for which the phase difference between quadrants is exactly π (Galicher et al., 2011). This configuration is optimal for a single wavelength; for an on-axis source, the phase discontinuity diffracts the star's light such that the field cancels within the pupil after the Lyot stop.

In Fourier terms, the azimuthal phase function φ′(θ′) is:

$\phi'(\theta') = \pi\,\text{Int}(2\theta'/\pi)$

yielding two conditions for perfect starlight nulling (Henault, 2018):

• Zero average: $$a_0 = (1/2\pi) \int_{-\pi}^{\pi} e^{i\phi'(\theta')} d\theta' = 0$$
• Even harmonics only: All odd Fourier coefficients vanish, $a_{2p+1} = 0$

These ensure that the phase mask function yields full extinction of the on-axis starlight in the ideal case.

2. Achromatization: Multi-Stage 4QPM

The limiting factor for monochromatic 4QPM performance is chromaticity; the exact π phase shift is only realized at a single wavelength, causing degradation off-band and leaking stellar light in the Lyot plane. The multi-stage four-quadrant phase mask (MFQPM) strategy overcomes this limitation by cascading n discrete 4QPM stages, each optimized at a designated wavelength λ_i (Galicher et al., 2011). After each stage, the leaked starlight is coherently uniform within the pupil, permitting its elimination in downstream stages.

The residual transmission τ through n stages is:

$$\tau = \int_{\lambda_{\text{min}}}^{\lambda_{\text{max}}} \prod_{i=1}^{n} \cos^2\left(\frac{\pi}{2}\frac{\lambda_i}{\lambda}\right) d\lambda$$

Laboratory results confirm that a three-stage MFQPM achieves broadband attenuation, with on-axis transmission as low as $1.4 \times 10^{-5}$ over a 20% spectral bandwidth and throughput up to 86% even for obstructed pupils. Consequently, MFQPM designs are promising for space and ground observatories requiring large spectral coverage and high contrast.

3. Adaptive and Apodized Designs

Recent advances incorporate adaptive elements to remedy static design limitations, such as tip-tilt sensitivity and manufacturing imperfections (Bourget et al., 2012). Utilization of polarization interferometry, specifically Pancharatnam–Berry phase mechanisms via liquid crystals, generates an achromatic π phase shift by rotating the polarization vector (rather than relying on optical path differences). Ring electrodes allow dynamic adjustment of the mask diameter, phase, and amplitude in real time.

Furthermore, apodized 4QPM systems employ pupil-plane amplitude masks (apodizers) optimized using convex linear programming to redirect leaking starlight outside the obstructed telescope aperture (Carlotti, 2012). The optimization objective is maximization of apodizer throughput subject to electric field amplitude constraints within the Lyot stop:

$$\text{maximize} \sum_{i,j}A(x_i, y_j)\Delta x\Delta y$$

with

$$-10^{-c/2} \leq P(\tilde{x}_k, \tilde{y}_l) \leq 10^{-c/2}$$

Simulation results for VLT-like apertures show achievable contrast of a few $10^{-10}$ at $1\lambda/D$ and up to 64% maximum throughput.

4. Wavefront Control and Low-Order Sensing

Wavefront aberrations, particularly low-order errors (tip, tilt, focus), may introduce starlight leaks that degrade coronagraphic contrast at small inner working angles. To mitigate these, Lyot-based Pointing Control Systems (LPCS) and Lyot-based Low Order Wavefront Sensors (LLOWFS) have been engineered for phase mask coronagraphs (Singh et al., 2013, Singh et al., 2014). Both systems exploit starlight diffracted by the phase mask and reflected by a modified Lyot stop, employing linear relationships such as:

$$I_R(\alpha_x, \alpha_y) - I_0 = \alpha_x S_x + \alpha_y S_y$$

where $I_0$ is the reference image and $S_x, S_y$ are calibration frames for unit tip and tilt, allowing extraction of pointing errors with accuracies between 2–12 nm at 1.6 μm. This supports sub-milli-arcsecond alignment crucial for small IWA imaging.

5. Practical Implementation in Instrumentation

4QPM-based coronagraphs have been deployed and tested in major instruments such as VLT/SPHERE and JWST. Dark hole techniques—focal-plane wavefront control using pairwise probing and electric field conjugation—are increasingly integrated, actively suppressing residual speckles in a defined region (the “dark hole”; e.g., 183–625 mas from a star) (Galicher et al., 27 Mar 2024). Combined with reference differential imaging (RDI), which subtracts the quasi-static PSF of a reference star, the technique yields a threefold improvement in H-band detection limits, an order-of-magnitude reduction in speckle noise after calibration, and robust independence from the astrophysical signal.

A representative performance metric in differential calibration is:

$$v(i,p) = \mathrm{STD}\left(\frac{I_i}{\langle I_i \rangle} - \frac{I_{R,p}}{\langle I_{R,p} \rangle}\right)$$

Such approaches facilitate fast “star hopping,” rapid dark hole creation, and effective combination of dark hole and RDI methods without compromising extended or point-like astrophysical targets.

6. Limitations and Manufacturing Constraints

While theoretically the 4QPM provides perfect nulling, manufacturing tolerances are severe: residual phase errors δφ′ scale the starlight leakage as (Henault, 2018):

$$I_2(0'', 0'') \approx \frac{(\sigma_{\delta\phi'})^2}{4\pi^2}$$

with leakage shooting rapidly as the fourth power of RMS phase error. Physical construction must realize sharp transitions ($<1\,\mu m$) and minimal surface errors ($<2\,nm$ rms per lens). Environmental cleanliness and transition thickness remain active development areas.

7. Context Within Broader Phase Mask Technologies

The 4QPM is situated among a family of coronagraphic concepts (Roddier, vortex, dual-zone) (Dohlen, 2018), sharing the principle of manipulating the focal-plane phase pattern to redirect starlight. Compared to other designs, the 4QPM offers a simple symmetric nulling geometry and is robust to pupil apodization and integration with active wavefront control, but less so to chromatic and alignment errors. Hybrid systems (such as those leveraging Mach–Zehnder interferometry) demonstrate the theoretical equivalence between spatial phase manipulation and path interferometry in nulling unwanted starlight.

8. Prospects for Future High-Contrast Imaging

MFQPM architectures, adaptive phase masks, and advanced calibration strategies are under ongoing development to reach contrasts required for Earth-analog detection ($10^{10}$), multi-band spectral characterization, and integration with deformable mirrors and self-coherent cameras. The simplicity and maturity of 4QPM technology and its compatibility with both chopped Lyot stops and spectrometers position it as a mainstay in instrument architectures targeting direct exoplanet imaging for ELTs, JWST successor missions, and fast AO-assisted survey platforms. Further advancements center on minimizing phase transition thickness, enhancing environmental stability, and refining multi-modal calibration pipelines.