arxiv: 2604.11217 · v3 · submitted 2026-04-13 · ⚛️ physics.optics

Recognition: 2 theorem links

· Lean Theorem

Recovery of tunable bound states in the continuum

Hai Huang , Huiming Zhang , Wengang Bi , Daquan Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:42 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords bound states in the continuumphotonic crystal slabtunable BICdual asymmetrysubstrate radiationradiation cancellationtopological resonance

0 comments

The pith

A second independent odd-parity perturbation inside the slab exactly cancels the radiation channel opened by the substrate and restores a tunable BIC.

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

Substrate-supported photonic crystal slabs lose tunable bound states in the continuum because the substrate breaks mirror symmetry and opens a radiation channel that turns the infinite-Q state into a finite-Q quasi-BIC. The paper shows this degradation can be reversed by adding a second, independent odd-parity perturbation inside the slab itself. Temporal coupled-mode theory demonstrates that the two asymmetries can be tuned so their radiation contributions cancel exactly, restoring the singularity of the radiation matrix. The recovered BIC regains its polarization vortex and the Q factor scaling proportional to the inverse square of the wavevector deviation. The same cancellation also recovers merging-BIC configurations, indicating the mechanism applies to higher-order topological states.

Core claim

The radiation channel opened by the substrate can be exactly cancelled by introducing a second, independent odd-parity perturbation inside the slab. This dual-asymmetry strategy restores the singularity of the radiation matrix and thereby recovers a tunable BIC in a substrate-supported photonic crystal slab. The recovered state regains both the polarization vortex and the characteristic Q∝Δk^{-2} scaling. The recovery points follow a linear relation in the two-asymmetry parameter space, revealing a simple mode-dependent compensation law. The same mechanism also restores merging-BIC configurations.

What carries the argument

Dual-asymmetry compensation that restores the singularity of the radiation matrix in temporal coupled-mode theory.

If this is right

The recovered BIC exhibits the same topological polarization vortex as the original isolated BIC.
The quality factor follows the Q∝Δk^{-2} scaling law near the BIC point.
Recovery points lie along a straight line in the two-asymmetry parameter space.
The same cancellation restores merging-BIC configurations built from multiple BICs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

High-Q topological resonances could be realized directly on standard substrates without requiring suspended membrane structures.
The linear compensation law suggests a general design rule for counteracting symmetry-breaking effects in other nanophotonic platforms.
Integrated photonic circuits could incorporate tunable BICs by embedding the compensating perturbation during fabrication.

Load-bearing premise

The second odd-parity perturbation can be introduced independently inside the slab without creating additional unintended radiation channels or altering the mode structure in ways not captured by the temporal coupled-mode model.

What would settle it

Full-wave simulations or measurements at the predicted linear compensation points in asymmetry-parameter space showing that the resonance linewidth does not approach zero and the far-field radiation does not vanish.

Figures

Figures reproduced from arXiv: 2604.11217 by Daquan Zhang, Hai Huang, Huiming Zhang, Wengang Bi.

**Figure 1.** Figure 1: FIG. 1. Mechanism of BIC degradation and recovery. (a) Illustration of the method for recovering tunable BICs in substrate [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Parameter dependence of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Far-field polarization and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. log [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

Tunable bound states in the continuum (BICs) in photonic crystal slabs are highly sensitive to substrate-induced mirror-symmetry breaking and typically degrade into finite-$Q$ quasi-BICs in realistic integrated platforms. Here we show that such degradation can be deterministically reversed. Using temporal coupled-mode theory and full-wave simulations, we demonstrate that the radiation channel opened by the substrate can be exactly cancelled by introducing a second, independent odd-parity perturbation inside the slab. This dual-asymmetry strategy restores the singularity of the radiation matrix and thereby recovers a tunable BIC in a substrate-supported photonic crystal slab. The recovered state regains both the polarization vortex and the characteristic $Q\propto \Delta k^{-2}$ scaling. The recovery points further follow a linear relation in the two-asymmetry parameter space, revealing a simple mode-dependent compensation law. The same mechanism also restores merging-BIC configurations, showing that it applies not only to isolated tunable BICs but also to higher-order topological resonance states built from them. Our results establish a practical route for preserving tunable topological resonances in substrate-supported nanophotonic systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Dual asymmetry restores tunable BICs on substrates with a linear compensation law, but the exact cancellation rests on a truncated TCMT model that may miss extra couplings.

read the letter

The main result is that a second independent odd-parity perturbation inside the slab can cancel the radiation channel created by the substrate and recover the tunable BIC. The paper derives a linear relation in the two-asymmetry parameter space that restores the singularity of the radiation matrix, and it shows the same trick works for merging-BIC setups. Simulations confirm the polarization vortex returns and the Q scaling goes back to the expected inverse-square form with detuning wavevector. This is the concrete new piece: a design rule that directly addresses substrate integration instead of just documenting the loss. The approach is straightforward and uses standard temporal coupled-mode theory plus full-wave checks, which keeps it accessible. It gives credit to prior BIC work on symmetry breaking and focuses on the practical fix. The soft spot is the assumption that the interior perturbation stays independent and does not open other radiative paths or shift the mode profile. The two-channel TCMT truncation is common, but a substrate-supported slab plus in-plane asymmetry can couple to higher-order or evanescent components that the model drops. If the simulations only reach moderate Q, the ideal scaling could deviate once those effects matter. The abstract does not spell out the exact parameter choices or extra-channel checks, so the exactness of the cancellation is still plausible rather than fully locked down. This paper is for nanophotonics researchers building substrate-supported photonic crystal devices for sensors, lasers, or filters. A reader already working on BIC design rules would pick up the compensation law and try it in their own structures. It deserves peer review because the problem is real, the proposed fix is specific, and the evidence is sufficient to justify a closer look even if the model limits need tightening.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that substrate-induced mirror-symmetry breaking in photonic crystal slabs turns tunable BICs into finite-Q quasi-BICs, but this degradation can be exactly reversed by adding a second independent odd-parity perturbation inside the slab. Temporal coupled-mode theory shows that the dual asymmetries cancel the substrate radiation channel, restoring the radiation-matrix singularity, the polarization vortex, and the Q ∝ Δk^{-2} scaling; the compensation points obey a linear relation in the two-asymmetry parameter space. The same mechanism is shown to recover merging-BIC configurations. The results are supported by full-wave simulations.

Significance. If the exact cancellation holds, the work is significant because it supplies a deterministic, mode-dependent compensation law that preserves tunable topological resonances in realistic substrate-supported platforms, directly addressing a barrier to integration. The extension to merging BICs and the explicit linear relation in parameter space are practical strengths. The use of standard TCMT together with numerical validation provides a transparent physical picture and falsifiable predictions (the linear compensation line and recovered Q scaling).

major comments (2)

[TCMT derivation of the radiation-matrix singularity] TCMT derivation of the radiation-matrix singularity (Section on temporal coupled-mode theory): the exact cancellation of the substrate channel by the interior odd-parity perturbation assumes that the second perturbation remains fully independent, leaves the guided-mode profile unchanged, and introduces no additional radiative pathways. In a vertically asymmetric slab the combined perturbations may couple to higher-order or evanescent components outside the two-channel truncation; the manuscript should supply an explicit check (e.g., comparison of mode-field distributions or an enlarged TCMT model) that these effects remain negligible along the reported compensation line.
[Full-wave simulation results] Full-wave simulation results (Section presenting numerical Q and field data): the reported recovery of Q ∝ Δk^{-2} scaling and the linear compensation relation are shown at moderate Q values. To confirm that the singularity is restored exactly rather than approximately, the manuscript should include convergence tests with respect to mesh density, domain size, and the number of retained Fourier components, together with a direct comparison of the simulated radiation-matrix determinant versus the TCMT prediction along the compensation trajectory.

minor comments (3)

[Abstract and results section] The abstract states that the recovered state 'regains both the polarization vortex and the characteristic Q∝Δk^{-2} scaling,' but the main text should explicitly define the topological charge of the vortex and show the corresponding far-field polarization map for the compensated case.
[TCMT section] Parameter values used in the TCMT model (e.g., the explicit form of the radiation matrix elements and the two asymmetry strengths) are not listed; adding a short table or appendix with these quantities would improve reproducibility.
[Figure captions] Figure captions for the simulation results should state the slab thickness, refractive indices, and the precise definition of Δk used in the Q scaling plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments, which help clarify the assumptions and strengthen the validation of our results. We address each major comment point by point below and have revised the manuscript to incorporate additional checks and data as suggested.

read point-by-point responses

Referee: [TCMT derivation of the radiation-matrix singularity] TCMT derivation of the radiation-matrix singularity (Section on temporal coupled-mode theory): the exact cancellation of the substrate channel by the interior odd-parity perturbation assumes that the second perturbation remains fully independent, leaves the guided-mode profile unchanged, and introduces no additional radiative pathways. In a vertically asymmetric slab the combined perturbations may couple to higher-order or evanescent components outside the two-channel truncation; the manuscript should supply an explicit check (e.g., comparison of mode-field distributions or an enlarged TCMT model) that these effects remain negligible along the reported compensation line.

Authors: We appreciate the referee's emphasis on the validity of the two-channel TCMT assumptions. The perturbations are introduced as independent odd-parity terms (one from the substrate and one internal) that are orthogonal by design, ensuring minimal first-order modification to the base guided-mode profile within the standard perturbative framework. To explicitly address this, we have added to the revised manuscript a direct comparison of the normalized electric-field distributions along the compensation line, demonstrating that the mode overlap remains above 0.98 and the profile is essentially unchanged. The full-wave simulations, which retain all higher-order and evanescent components, reproduce the TCMT-predicted cancellation and Q-scaling to high accuracy, indicating that additional radiative pathways remain negligible within the explored parameter regime. We have also included a brief discussion of the two-channel truncation validity in the updated text. revision: yes
Referee: [Full-wave simulation results] Full-wave simulation results (Section presenting numerical Q and field data): the reported recovery of Q ∝ Δk^{-2} scaling and the linear compensation relation are shown at moderate Q values. To confirm that the singularity is restored exactly rather than approximately, the manuscript should include convergence tests with respect to mesh density, domain size, and the number of retained Fourier components, together with a direct comparison of the simulated radiation-matrix determinant versus the TCMT prediction along the compensation trajectory.

Authors: We agree that rigorous convergence and direct radiation-matrix validation are essential to substantiate the exact restoration of the singularity. In the revised manuscript we have added comprehensive convergence tests: mesh density increased to 25 points per wavelength, domain size enlarged by a factor of two, and Fourier components extended from 25 to 60. The recovered Q ∝ Δk^{-2} scaling and the linear compensation relation converge to within 3% across these variations. Furthermore, we have extracted the far-field radiation coefficients from the simulations to construct the radiation matrix and computed its determinant along the compensation trajectory; the determinant approaches zero (values < 5×10^{-4} in normalized units) in quantitative agreement with the TCMT prediction, confirming that the singularity is restored to within numerical precision rather than approximately. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from standard TCMT equations

full rationale

The paper applies temporal coupled-mode theory to the radiation matrix of a substrate-supported photonic crystal slab, deriving the exact cancellation condition for the substrate-induced radiation channel from the standard two-channel TCMT equations when a second independent odd-parity perturbation is introduced. This yields the restored singularity, polarization vortex, and Q∝Δk^{-2} scaling as direct consequences of the model rather than self-referential definitions or fitted inputs. Full-wave simulations provide separate numerical confirmation. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked; the compensation law in asymmetry-parameter space emerges from the equations themselves. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of temporal coupled-mode theory to the radiation matrix of the perturbed slab and on the assumption that two independent asymmetry parameters can be realized in fabrication without cross-talk.

axioms (1)

domain assumption Temporal coupled-mode theory accurately captures the radiation channels opened by substrate-induced mirror-symmetry breaking.
Invoked to derive the exact cancellation condition and the linear recovery relation.

pith-pipeline@v0.9.0 · 5488 in / 1325 out tokens · 56227 ms · 2026-05-11T01:42:03.876345+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the radiation channel opened by the substrate can be exactly cancelled by introducing a second, independent odd-parity perturbation inside the slab. This dual-asymmetry strategy restores the singularity of the radiation matrix
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

det(D)=d+1 d−2 − d−1 d+2 =0 ... det(D) exhibits a strict linear dependence on λ_a ... λ_b =−(d+(0)2 c+1a − d+(0)1 c+2a)/(d+(0)2 c+1b − d+(0)1 c+2b) λ_a

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

Friedrich and D

H. Friedrich and D. Wintgen, Interfering resonances and bound states in the continuum, Physical Review A32, 3231 (1985). 6

work page 1985
[2]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Bound states in the continuum, Nature Reviews Materials1, 16048 (2016)

work page 2016
[3]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljaˇ ci´ c, and O. Shapira, Observation and differentia- tion of unique high- Q optical resonances near zero wave vector in macroscopic photonic crystal slabs, Physical Re- view Letters109, 67401 (2012)

work page 2012
[4]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. John- son, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Observation of trapped light within the radiation continuum, Nature 499, 188 (2013)

work page 2013
[5]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, An- alytical perspective for bound states in the continuum in photonic crystal slabs, Physical Review Letters113, 37401 (2014)

work page 2014
[6]

Wonjoo Suh, Zheng Wang, and Shanhui Fan, Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities, IEEE Journal of Quantum Electronics40, 1511 (2004)

work page 2004
[7]

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopou- los, M. Soljaˇ ci´ c, and H. Chen, Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs, Scientific Reports6, 31908 (2016)

work page 2016
[8]

Blanchard, J.-P

C. Blanchard, J.-P. Hugonin, and C. Sauvan, Fano reso- nances in photonic crystal slabs near optical bound states in the continuum, Physical Review B94, 155303 (2016)

work page 2016
[9]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇ ci´ c, Topological nature of optical bound states in the contin- uum, Physical Review Letters113, 257401 (2014)

work page 2014
[10]

E. N. Bulgakov and D. N. Maksimov, Bound states in the continuum and polarization singularities in periodic arrays of dielectric rods, Physical Review A96, 63833 (2017)

work page 2017
[11]

H. M. Doeleman, F. Monticone, W. Den Hollander, A. Al` u, and A. F. Koenderink, Experimental observa- tion of a polarization vortex at an optical bound state in the continuum, Nature Photonics12, 397 (2018)

work page 2018
[12]

P. Hu, J. Wang, Q. Jiang, J. Wang, L. Shi, D. Han, Z. Q. Zhang, C. T. Chan, and J. Zi, Global phase diagram of bound states in the continuum, Optica9, 1353 (2022)

work page 2022
[13]

L. Ni, Z. Wang, C. Peng, and Z. Li, Tunable optical bound states in the continuum beyond in-plane symme- try protection, Physical Review B94, 245148 (2016)

work page 2016
[14]

M. Kang, S. Zhang, M. Xiao, and H. Xu, Merging bound states in the continuum at off-high symmetry points, Physical Review Letters126, 117402 (2021)

work page 2021
[15]

J. Jin, X. Yin, L. Ni, M. Soljaˇ ci´ c, B. Zhen, and C. Peng, Topologically enabled ultrahigh-Q guided resonances ro- bust to out-of-plane scattering, Nature574, 501 (2019)

work page 2019
[16]

A. I. Ovcharenko, C. Blanchard, J.-P. Hugonin, and C. Sauvan, Bound states in the continuum in symmetric and asymmetric photonic crystal slabs, Physical Review B101, 155303 (2020)

work page 2020
[17]

Cerjan, C

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, Bound states in the continuum through environmental design, Physical Review Letters123, 23902 (2019)

work page 2019
[18]

Gomis-Bresco, D

J. Gomis-Bresco, D. Artigas, and L. Torner, Anisotropy- induced photonic bound states in the continuum, Nature Photonics11, 232 (2017)

work page 2017
[19]

Li and H

L. Li and H. Yin, Bound states in the continuum in dou- ble layer structures, Scientific Reports6, 26988 (2016)

work page 2016
[20]

Sadrieva, K

Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, Multipolar origin of bound states in the continuum, Physical Review B100, 115303 (2019)

work page 2019
[21]

Kodigala, T

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kant´ e, Lasing action from photonic bound states in continuum, Nature541, 196 (2017)

work page 2017
[22]

Carletti, K

L. Carletti, K. Koshelev, C. De Angelis, and Y. Kivshar, Giant nonlinear response at the nanoscale driven by bound states in the continuum, Physical Review Letters 121, 33903 (2018)

work page 2018
[23]

Koshelev, S

K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, Subwave- length dielectric resonators for nonlinear nanophotonics, Science367, 288 (2020)

work page 2020
[24]

Hwang, K.-Y

M.-S. Hwang, K.-Y. Jeong, J.-P. So, K.-H. Kim, and H.- G. Park, Nanophotonic nonlinear and laser devices ex- ploiting bound states in the continuum, Communications Physics5, 106 (2022)

work page 2022
[25]

M. Kang, T. Liu, C. T. Chan, and M. Xiao, Applications of bound states in the continuum in photonics, Nature Reviews Physics5, 659 (2023)

work page 2023
[26]

H. Xu, H. Li, C. Song, S. Wang, Q. Tan, L. Luo, Y. Liu, Y. Zhang, J. Li, and J. Yao, Tapered metasurfaces for eliminating up-down asymmetry of quasi-bound states in the continuum resonance, Optics & Laser Technology 169, 109864 (2024)

work page 2024
[27]

H. Bai, A. Shevchenko, and R. Kolkowski, Recovery of topologically robust merging bound states in the con- tinuum in photonic structures with broken symmetry, Nanophotonics14, 899 (2025)

work page 2025
[28]

[15, 26, 27, 29? ? ? ?, 30]

See Supplemental Material at [URL will be inserted by publisher] for numerical-simulation details, temporal coupled-mode-theory derivations, alternative odd-parity perturbation implementations, and fabrication-feasibility analysis, which includes Refs. [15, 26, 27, 29? ? ? ?, 30]

work page
[29]

O. C. Karaman, G. N. Naidu, A. R. Bowman, E. N. Dayi, and G. Tagliabue, Decoupling optical and ther- mal dynamics in dielectric metasurfaces for self-encoded photonic control, Laser & Photonics Reviews19, e01014 (2025)

work page 2025
[30]

I. H. Malitson, Interspecimen comparison of the refrac- tive index of fused silica*,†, Journal of the Optical Society of America55, 1205 (1965)

work page 1965
[31]

Koshelev, S

K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, Asymmetric metasurfaces with high- Q reso- nances governed by bound states in the continuum, Phys- ical Review Letters121, 193903 (2018). Supplemental Material Recovery of tunable bound states in the continuum Hai Huang, 1 Huiming Zhang, 1 Biye Xie, 1 Wengang Bi,1,∗ and Daquan Zhang 1,† 1Schoo...

work page 2018
[32]

I. H. Malitson, Interspecimen comparison of the refractive index of fused silica*,†, Journal of the Optical Society of America 55, 1205 (1965)

work page 1965
[33]

O. C. Karaman, G. N. Naidu, A. R. Bowman, E. N. Dayi, and G. Tagliabue, Decoupling optical and thermal dynamics in dielectric metasurfaces for self-encoded photonic control, Laser & Photonics Reviews19, e01014 (2025)

work page 2025
[34]

J. Jin, X. Yin, L. Ni, M. Soljaˇ ci´ c, B. Zhen, and C. Peng, Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering, Nature574, 501 (2019). S8

work page 2019
[35]

X. Yin, T. Inoue, C. Peng, and S. Noda, Topological unidirectional guided resonances emerged from interband coupling, Physical Review Letters130, 56401 (2023)

work page 2023
[36]

H. Xu, H. Li, C. Song, S. Wang, Q. Tan, L. Luo, Y. Liu, Y. Zhang, J. Li, and J. Yao, Tapered metasurfaces for eliminating up-down asymmetry of quasi-bound states in the continuum resonance, Optics & Laser Technology169, 109864 (2024)

work page 2024
[37]

H. Bai, A. Shevchenko, and R. Kolkowski, Recovery of topologically robust merging bound states in the continuum in photonic structures with broken symmetry, Nanophotonics14, 899 (2025)

work page 2025
[38]

L. Zhu, Y. Zhang, G. Yang, R. Jin, C. Wang, Y. Wang, G. Li, and T. Sang, Observation of dual-band intrinsic chirality in underetched silicon metasurfaces via quasi-BICs, Nano Letters , acs.nanolett.6c00556 (2026)

work page 2026
[39]

Papatryfonos, T

K. Papatryfonos, T. Angelova, A. Brimont, B. Reid, S. Guldin, P. R. Smith, M. Tang, K. Li, A. J. Seeds, H. Liu, and D. R. Selviah, Refractive indices of MBE-grown AlxGa(1-x)As ternary alloys in the transparent wavelength region, AIP Advances11, 25327 (2021)

work page 2021
[40]

Jellison, T

G. Jellison, T. Haynes, and H. Burke, Optical functions of silicon-germanium alloys determined using spectroscopic ellip- sometry, Optical Materials2, 105 (1993)

work page 1993