Phase-Apodized-Pupil Lyot Coronagraphs for Arbitrary Telescope Pupils

Emiel H. Por

REVIEW 4 major objections 5 minor 44 references

Phase-Apodized-Pupil Lyot Coronagraphs for Arbitrary Telescope Pupils

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Phase-only aperture apodization plus a Lyot stage—the PAPLC—reaches inner working angles of 1.4λ/D at 10⁻¹⁰ contrast with >75% throughput, including on realistic segmented and obstructed telescope pupils up to 30% central obscuration.

desk verdict A promising coronagraph design method with an honest parameter study, but the headline broadband performance rests on a suspect achromatization argument and an unproven phase-only relaxation. read the letter →

arxiv 1908.02585 v1 pith:7MH4VRVQ submitted 2019-08-07 astro-ph.IM astro-ph.EP

Emiel H. Por This is my paper · ORCID

classification astro-ph.IMastro-ph.EP

keywords coronagraphyhigh-contrastimagingexoplanetdirectapodizingphaseplateLyotcoronagraphconvexoptimizationapodizationknife-edgefocal-planemask

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that replacing the amplitude apodizer of an apodized-pupil Lyot coronagraph (APLC) with a phase-only apodizer—the phase-apodized-pupil Lyot coronagraph (PAPLC)—yields a coronagraph that matches the APLC for point-symmetric dark zones and substantially outperforms it for one-sided dark zones. The central result is that with a knife-edge focal-plane mask, the PAPLC reaches inner working angles as small as 1.4λ/D at a raw contrast of 10⁻¹⁰ with post-coronagraphic throughput above 75%, for telescope apertures with central obscurations up to 30%. A sympathetic reader would care because deep-contrast exoplanet imaging is currently limited by the trade-off between inner working angle, throughput, and robustness to real telescope pupils; the PAPLC appears to relax that trade-off while still handling struts, segments, and non-circular edges. The paper also presents concrete designs for the VLT/SPHERE instrument and the LUVOIR-A aperture.

What carries the argument

The load-bearing device is the convex optimization of the complex transmission of the pre-apodizer. The apodizer is written as $A(x)(X(x)+iY(x))$, the phase-only constraint is relaxed to $|X(x)+iY(x)|\le 1$, and the objective is the real part of the on-axis non-coronagraphic field, which removes the piston degeneracy while keeping the problem convex. Because the electric field in the post-coronagraphic focal plane is a linear map of the apodizer transmission (Fourier propagation through the pupil, knife-edge mask, and Lyot stop), the stellar-leakage contrast constraints become convex inequalities; a linearized tip-tilt constraint prevents the optimizer from sneaking a centroid shift into the phase solution. The same machinery yields the point-symmetric designs as a real-valued special case. For one-sided dark zones, a wavelength-dependent phase tilt added to the apodizer centers the growing PSF on the knife-edge over the band, giving theoretical achromatization.

What would settle it

Run the paper's relaxed convex optimization problem for the LUVOIR-A pupil and check whether the optimal complex transmission has any region with $|X+iY|<1$; if it does, the physical phase-only apodizer cannot match the optimized contrast. Independently, simulate the broadband LUVOIR-A knife-edge design at 2.2λ/D over a 10% band with the wavelength-dependent tilt applied and look for any dark-zone pixel above 10⁻¹⁰ raw contrast.

Watch

Extended reading notes

Core claim

The central discovery is that the PAPLC design problem can be made convex by writing the apodizer as a complex transmission $X(x)+iY(x)$ with the phase-only constraint relaxed to $|X(x)+iY(x)|\le 1$, and that in practice the optimum is still phase-only. The post-coronagraphic electric field is a linear function of that transmission, so the $10^{-c}$ stellar-leakage constraints are convex, and the piston ambiguity is removed by maximizing the real part of the on-axis field. Since the APLC's solution space (real nonnegative apodizer) is a subset of this complex-amplitude space, the PAPLC is guaranteed to match or beat the APLC for the same masks, dark zone, and contrast. The one-sided PAPLC uses an offset knife-edge focal-plane mask; the apodizer creates the one-sided dark zone in the focal plane, and the Lyot stop deepens the contrast by several orders of magnitude, reaching 1.4λ/D at 10⁻¹⁰ contrast and up to about 80% throughput for central obscurations up to 30%. Point-symmetric annular-mask designs are found as a real-valued special case and show only marginal gains over the APLC.

Load-bearing premise

The relaxed optimization problem allows the apodizer amplitude to dip below unity, and the paper's results rely on the assertion—not a proof—that every optimal solution is still phase-only, plus the assumption that the monochromatic one-sided designs hold their 10⁻¹⁰ contrast after the proposed achromatization.

Editorial extensions

If this is right

Because the PAPLC design space contains the APLC design space, any PAPLC design is guaranteed to match or beat the best APLC for the same pupil, focal-plane mask, Lyot stop, dark zone, and design contrast.
For one-sided dark zones, the knife-edge PAPLC reaches 1.4λ/D inner working angle at 10⁻¹⁰ contrast with throughput above 75% on pupils with up to 30% central obscuration, a combination not available from the corresponding APLC at these parameters.
The LUVOIR-A knife-edge PAPLC design (IWA 2.2λ/D, maximum throughput 78%) roughly triples the maximum throughput of the APLC comparison design on the same aperture, while cutting the inner working angle by 1.5λ/D.
Alignment of residual atmospheric dispersion along the knife edge makes the coronagraph effectively insensitive to that dispersion, relaxing the atmospheric dispersion corrector requirement to about 1λ/D instead of a few tenths or hundredths of λ/D.
The design method is not restricted to circular or unobstructed pupils: the VLT/SPHERE and LUVOIR-A case studies show it handles struts, segments, dead deformable-mirror actuators, and non-circular outer edges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference (not in the paper): the one-sided dark zone covers about half the field of view, so survey-mode mapping of a full circumstellar disk would require observing at several roll angles, and the throughput penalty of that mode is not included in the headline >75% figure.
Inference (not in the paper): the proposed achromatization assumes the wavelength-dependent tilt is implemented exactly; a laboratory test across a 10–20% band could confirm whether the 10⁻¹⁰ contrast floor survives real dispersion, and would be the natural next experiment.
Inference (not in the paper): because the same convex machinery was used to harden APLCs against low-order aberrations, a similar inclusion of aberration modes should produce robust PAPLC designs; the paper explicitly leaves that to future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Emiel Por's PAPLC paper is a serious piece of design work. The architecture is new: combine an APP-style phase apodizer with a Lyot stop and optimize the phase with a convex program. The one-sided knife-edge case is the real contribution, with parameter studies showing ~1.4λ/D IWA at 10^-10 contrast on obstructed pupils, and the VLT/SPHERE and LUVOIR-A case studies are genuinely impressive. The point-symmetric case is basically a phase-only APLC, and the paper says so; that negative result is honest and useful. The optimization problem is clearly posed, the linearization of contrast constraints is standard, and the use of HCIPy means the designs are reproducible. The author also states plainly that the designs are not robust to aberrations or Lyot-stop misalignment.

Two soft spots matter, and one is load-bearing. First, the headline 1.4λ/D at 10^-10 comes from a parameter study that is explicitly monochromatic (Section 4.1). The achromatization in Section 4.2 argues that adding a tilt to the phase plate makes the design achromatic, but that ignores the physical plate's chromaticity: a plate made for λ0 has phase (λ0/λ)φ0 at wavelength λ. A tilt then produces a physical shift that is independent of λ, not one that scales with the PSF size. So the mask edge moves relative to the rescaled PSF, and 10^-10 contrast is not established over band. Figure 7's broadband panel is qualitative, not a contrast curve. If the author can demonstrate broadband 10^-10 with the actual chromatic phase, fine; as written, the abstract overstates the result.

Second, the convex relaxation to |X+iY|≤1 is asserted to yield phase-only solutions, but no proof or amplitude maps are shown. The VLT/SPHERE design's statement that dead-actuator light is 'blocked at the apodizer' sounds like an amplitude operation. If the optimized solutions are not phase-only, the comparison to APLCs is not apples-to-apples and the 'always beats APLC' argument collapses. This should be easy to check and needs to be in the paper.

The circularity note about design contrast being a constraint is fair but minor. The citation pattern looks fine, and HCIPy use is reproducible.

Recommendation: send to peer review. The method is novel, the parameter study useful, the case studies valuable, but the paper needs a revision that presents broadband contrast verification with a real chromatic phase plate (or weakens the abstract) and proves or demonstrates phase-only convergence, e.g., by showing amplitude maps and the performance of a projected phase-only solution.

Referee Report

4 major / 5 minor

Summary. This paper introduces a convex optimization framework for designing phase-apodized-pupil Lyot coronagraphs (PAPLCs), in which a phase-only pre-apodizer is paired with a focal-plane mask and a Lyot stop. The method is applied to two configurations: annular focal-plane masks producing point-symmetric dark zones, and knife-edge masks producing one-sided dark zones. Parameter studies with simplified circular obstructed pupils show that point-symmetric PAPLCs perform comparably to APLCs, while one-sided designs achieve smaller inner working angles (down to 1.4 lambda/D) and higher throughput (up to about 75-80%) at design contrasts down to 10^-10, based on monochromatic optimizations. Case studies for VLT/SPHERE and LUVOIR-A demonstrate the approach on segmented, strutted, and non-circular pupils. Section 4.2 argues that the one-sided designs can be made achromatic by centering the knife-edge mask and adding a phase tilt, and Section 5.3 evaluates tip-tilt robustness. The paper also states that the relaxed optimization problem from phase-only to sub-unity complex amplitude in practice returns phase-only solutions, and that PAPLCs always perform as well as or better than APLCs.

Significance. If the headline performance holds under broadband illumination and with physically realizable phase-only apodizers, the knife-edge PAPLC would be an important new coronagraph option for high-contrast imaging on obscured and segmented apertures, offering substantially better throughput and smaller inner working angles than APLCs. The convex optimization methodology is clearly presented, builds on prior work by the same author, and the use of the open-source HCIPy package supports reproducibility. The point-symmetric PAPLC analysis and the VLT/SPHERE and LUVOIR-A case studies are valuable contributions. However, the central claims currently rest on two unverified assumptions: that the relaxed optimization solutions are exactly phase-only, and that the tilt-based achromatization actually delivers the advertised broadband contrast. These gaps make the significance contingent on additional verification.

major comments (4)

[Section 2.2, Eqs. (10)-(11)] The relaxation from the phase-only constraint X^2+Y^2=1 to the inequality X^2+Y^2<=1 is asserted to produce phase-only solutions in practice, but no evidence is provided: there are no amplitude maps, histograms of |X+iY|, or statistics across the parameter studies. This is load-bearing because the physical coronagraph uses a phase-only apodizer; if any optimized solution has regions with |X+iY|<1, the phase-only projection is not guaranteed to meet the contrast and throughput claims. Moreover, the VLT/SPHERE design in Section 5.1 explicitly blocks light at dead deformable-mirror actuator locations 'at the apodizer,' which is an amplitude operation and directly contradicts the phase-only assertion. The authors should demonstrate that all final designs (including the VLT/SPHERE case) saturate the norm constraint, or they should treat amplitude as part of the physical model and re-derive the affected performance numbers.
[Section 4.1 and Abstract] The headline values in the abstract - inner working angle as close as 1.4 lambda/D at contrasts of 10^-10 and maximum post-coronagraphic throughput of >75% - come from the parameter study in Section 4.1, which states explicitly that 'All masks were calculated for a single wavelength only: we presume monochromatic light.' The abstract and conclusions present these numbers without the monochromatic qualifier. Section 4.2's achromatization argument is qualitative and does not quantitatively account for the chromatic scaling of the phase plate's non-tilt phase components; no evaluation of Cdesign as defined in Eq. (3) over a finite band is presented for any one-sided design. The authors should either run a broadband end-to-end simulation of the 1.4 lambda/D design over the intended band (e.g., 10% or 20% bandwidth) and report the achieved Cdesign, or clearly state in the abstract that the demonstrated 10^-10 contrast is monochromatic and that broadband achromatization is a proposal that remains to be verified.
[Section 2.2, paragraph after Eq. (11)] The statement that 'a PAPLC will always perform the same or better than an APLC for a given telescope pupil, dark zone geometry and design contrast' is not justified as a universal claim. The APLC solution space is indeed a subspace of the relaxed complex-amplitude problem, but the actual PAPLC apodizer is constrained to be phase-only. If the relaxed optimum has |X+iY|<1, the phase-only projection may have lower throughput or worse contrast. The empirical comparisons in Sections 3 and 4 are reasonable evidence for the specific cases studied, but the unconditional 'always' wording should be qualified by the tightness of the relaxation, or replaced by a statement about the specific designs presented.
[Table 1 and Section 5.3] The comparison between PAPLC and APLC throughput in Table 1 is a central demonstration of the PAPLC's advantage, but the APLC designs are described only as a 'preliminary solution' for VLT/SPHERE and 'a part of a coronagraph design study' for LUVOIR-A, courtesy of other groups. These APLC designs may not have been optimized with the same hyperparameter search (focal-plane mask size, Lyot stop geometry) used for the PAPLCs. To make the factor-of-two-to-three throughput comparison fair, the authors should either optimize the APLC baselines with the same procedure described in Section 3, or provide the design parameters and optimization effort for both families so that the reader can judge the comparison.

minor comments (5)

[Eq. (8)] In Eq. (8), 'X(x) + iY(y)' should presumably read 'X(x) + iY(x)'.
[Section 2.2] The phrase 'the solutions space for APLCs' contains a typo; it should be 'the solution space for APLCs'.
[Figure 6] In the rendered figure, the y-axis label 'Throughput' is missing from the CO=20% and CO=30% panels; please ensure all panels have axis labels.
[Section 4.2] The sentence 'This makes the one-sided PAPLC completely achromatic in theory (barring experimental eﬀects)' is too strong given that the chromaticity of the phase plate's non-tilt phase components is not analyzed. A more cautious wording such as 'approximately achromatic to first order, subject to verification' would better match the evidence presented.
[Appendix A and Section 2.1] For reproducibility, the authors should report numerical details such as the pupil grid size, the number of optimization variables, and the solver settings (e.g., Gurobi version and optimality tolerances).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the PAPLC performance numbers are outputs of a constrained optimization, not fits or definitions in disguise.

full rationale

The paper's derivation chain is a design optimization, not a prediction from fitted inputs. In Eq. (11), the contrast requirement is imposed as a constraint (|E_coro|^2 < 10^{-c}|E_noncoro|^2) and the throughput and inner working angle are optimized outputs; reporting that the resulting design reaches the design contrast is a statement that the constraint is satisfied, not a prediction of an independently measured quantity. The 'PAPLC always performs at least as well as APLC' argument rests on solution-space inclusion: APLC amplitude masks are feasible for the complex-amplitude relaxed problem, and if the relaxed optimum is phase-only, it is feasible for the PAPLC. This is a valid mathematical argument conditional on the paper's explicit, unproved observation that 'in practice all solutions turn out to be phase only' (Section 2.2). That assertion is an evidence gap, not circularity. Similarly, the monochromatic limitation is admitted directly: 'All masks were calculated for a single wavelength only' (Section 4.1), and the broadband achromatization in Section 4.2 is presented as a theoretical construction rather than a hidden reuse of the result it claims to explain. The self-citation to Por (2017) supplies a convex-optimization method and the analogous APP/SPC comparison, but the central PAPLC derivation and parameter studies are performed in this paper with explicit equations and quoted hyperparameters. The honesty of the paper's own caveats (phase-only not guaranteed, no robustness to Lyot-stop misalignment, monochromatic parameter study) further confirms that the weak points are unverified assumptions rather than definitional circularity. No load-bearing step reduces to its own input, so the appropriate finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central design rests on standard optical propagation assumptions plus several numerical-heuristic assumptions that are stated but not proven. Hyperparameters of the focal-plane mask and Lyot stop are searched, not derived. No new physical entities are introduced.

free parameters (5)

Focal-plane mask geometry (fID, fOD for annular; fedge for knife edge) = fID = 2*DZ_min - 5 lambda/D to 2*DZ_min; fOD = 2*DZ_max; fedge = DZ_min to DZ_min - 1.0 lambda/D
These mask parameters are brute-force hyperparameters; final IWA and throughput depend on them.
Lyot stop inner and outer diameters (LID, LOD) = LID = CO to CO + 0.4; LOD = 0.85 to 1 (parameter study); case studies use fixed or thickened pupils
Search ranges are scanned and optimized; the paper notes shrinking the Lyot stop had no positive effect for one-sided designs.
Dark zone inner radius (DZ_min) and outer radius (DZ_max) = Point-symmetric DZ_min=2.0-3.5 lambda/D, DZ_max=13.25 lambda/D; one-sided DZ_min=0.4-2.0 lambda/D, DZ_max=8 lambda/D
Dark zone geometry is chosen to target an IWA; the actual IWA is measured after optimization.
Design contrast exponent c (10^-c) = 10^-5 to 10^-10
This is a design requirement rather than a fitted quantity, but the headline performance is only claimed at these chosen contrasts.
Expected Strehl Sexpected = Not reported; updated iteratively until convergence
Used to linearize the contrast constraints in Appendix A; if the iteration converges to a value different from the achieved Strehl, the constraints may be mis-specified.

assumptions (5)

standard math Fraunhofer propagation between pupil and focal plane is accurately modeled by Fourier transforms and their inverses.
Used throughout Section 2.1 and Eq. 7 for the propagation operators P and P^-1.
domain assumption The telescope pupil can be represented by a scalar complex-amplitude transmission A(x) with negligible polarization, chromatic, and non-scalar effects.
Eq. 7c builds the pupil field as A(x) exp(i phi(x)); this is standard for coronagraph design studies.
ad hoc to paper The relaxed complex-amplitude problem |X+iY|<=1 yields solutions with |X+iY|=1 in all cases studied.
Section 2.2 states 'in practice all solutions turn out to be phase only', but no proof is given; this is required for the PAPLC to be a pure phase apodizer.
domain assumption The linearized contrast constraints using expected Strehl Sexpected and iterative updating converge to the true contrast constraint.
Appendix A, Eqs. A1c-A1j, replaces the quadratic contrast ratio with linear constraints; no convergence proof is provided.
domain assumption Achromatization by adding a wavelength-dependent phase tilt renders the one-sided PAPLC completely achromatic in theory.
Section 4.2 asserts this and illustrates one design, but does not optimize or verify broadband performance at design contrast.

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Cite this review

Pith. "Pith review of Phase-Apodized-Pupil Lyot Coronagraphs for Arbitrary Telescope Pupils." pith.science (2026). https://pith.science/paper/7MH4VRVQ

@misc{pith2026190802585,
  author       = {Pith},
  title        = {Pith review of: Phase-Apodized-Pupil Lyot Coronagraphs for Arbitrary Telescope Pupils},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7MH4VRVQ}},
  note         = {Machine review of arXiv:1908.02585}
}

abstract

The phase-apodized-pupil Lyot coronagraph (PAPLC) is a pairing of the apodized-pupil Lyot coronagraph (APLC) and the apodizing phase plate (APP) coronagraph. We describe a numerical optimization method to obtain globally-optimal solutions for the phase apodizers for arbitrary telescope pupils, based on the linear map between complex-amplitude transmission of the apodizer and the electric field in the post-coronagraphic focal plane. PAPLCs with annular focal-plane masks and point-symmetric dark zones perform analogous to their corresponding APLCs. However with a knife-edge focal-plane mask and one-sided dark zones, the PAPLC yields inner working angles as close as $1.4\lambda/D$ at contrasts of $10^{-10}$ and maximum post-coronagraphic throughput of >75% for telescope apertures with central obscurations of up to 30%. We present knife-edge PAPLC designs optimized for the VLT/SPHERE instrument and the LUVOIR-A aperture. These designs show that the knife-edge PAPLC retains its performance, even for realistic telescope pupils with struts, segments and non-circular outer edges.

Figures

Figures reproduced from arXiv: 1908.02585 by the authors.

**Figure 1.** The optical layout of the PAPLC with a) pointsymmetric dark zones, and b) one-sided dark zones follows a standard Lyot-style optical setup. The focal-plane mask for point-symmetric dark zones is annular, while it is a knife edge for the one-sided dark zone. In this study we optimize the pre-apodizer (in orange), viewing the parameters of the focalplane mask and the Lyot stop (in green) as hyperparameters. restrict… view at source ↗

**Figure 2.** The definition of all masks used in this work. These masks are used for the parameter study in Sections 3 and 4. Centered masks are used for both pointsymmetric and one-sided dark zones. The left-justified masks are for two-sided dark zones, while the right-justified masks are used for one-sided dark zones. In general though, arbitrary telescope pupils, Lyot masks, focal-plane masks and dark zone geometries can b… view at source ↗

**Figure 3.** Some examples of PAPLC designs with point-symmetric dark zones. For two sets of parameters, we show both the APLC design and the PAPLC design. The phase patterns for the PAPLC consist of regions of 0 or π radians in phase, while the APLC designs consist of regions of 0 and 1 transmission. We show a 10% broadband image just in front of the focal-plane mask in log-scale from 10−5 to 100 , and the post-coronagraphic im… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Throughput vs inner working angle for various contrasts for an annular dark zone. Solid lines and solid points are APLC designs, dashed lines and open points are PAPLC designs. The design contrast ranges from 10−5 to 10−10. Each point is a coronagraph design for which …

**Figure 5.** Figure 5: Some examples of PAPLC designs with one-sided dark zones. The color scale for phase is from −1 rad to 1 rad but typically the phase pattern rms is ∼ 0.4 rad. We show the image at the focal-plane mask with a translucent focal-plane mask to show the positioning of the fo…

**Figure 6.** Figure 6: Throughput vs inner working angle for various contrasts for a one-sided dark zone. All designs are PAPLC designs. The design contrast ranges from 10−5 to 10−10. Each point is a coronagraph design for which all hyperparameters (focal-plane mask offset, and Lyot stop inn…

**Figure 7.** Figure 7: Raw post-coronagraphic images for a one-sided dark zone with an inner working angle of 1.6λ/D with increasing imperfections. Top left: Only tip-tilt jitter with 0.003λ/D rms. Top right: tip-tilt jitter and 20% broadband light. Bottom left: tip-tilt jitter, broadband …

**Figure 8.** Figure 8: The case study design for VLT/SPHERE. We show the apodizer phase pattern, focal-plane mask and Lyot stop. Additionally, we show the light in each of the coronagraphic planes: before and after the focal-plane mask (on a logarithmic scale), and before and after the Lyot …

**Figure 9.** Figure 9: The case study design for the LUVOIR-A telescope. We show the apodizer phase pattern, focal-plane mask and Lyot stop. Additionally, we show the light in each of the coronagraphic planes: before and after the focal-plane mask (on a logarithmic scale), and before and aft…

**Figure 11.** Figure 11: Slices of the normalized irradiance for varying values of the RMS tip-tilt error on the star for both the VLT/SPHERE and LUVOIR-A design. The different RMS values were chosen to show the transition from no effect to a significant effect on the normalized irradiance. A…

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Pith tools

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