Clarifying the energy storage capabilities of virtual critical coupling under realistic excitation constraints

J\'er\^ome Sokoloff; Olivier Pascal; Th\'eo Delage; Valentin Mazi\`eres

Virtual critical coupling, long touted as a way to pack more energy into a resonator, actually stores less energy than plain continuous-wave excitation once the source is limited to a realistic maximum amplitude; its true merit is reflectio

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2026-08-01 04:10 UTC pith:6HFXX37T

load-bearing objection A clean, narrowly-scoped correction: amplitude-constrained VCC stores less energy than CW, with the constraint choice as the main caveat.

arxiv 2607.22897 v1 pith:6HFXX37T submitted 2026-07-24 physics.optics

Clarifying the energy storage capabilities of virtual critical coupling under realistic excitation constraints

Valentin Mazi\`eres , Th\'eo Delage , J\'er\^ome Sokoloff , Olivier Pascal This is my paper

classification physics.optics

keywords virtual critical couplingenergy storagetemporal coupled-mode theoryreflectionless excitationcontinuous-wave excitationmicrowave resonatorsamplitude constraintplasma ignition

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper re-examines the energy storage claims made for virtual critical coupling (VCC), a technique that uses an exponentially growing incident waveform to excite a resonant cavity without reflections. The authors show analytically, using temporal coupled-mode theory, that the conventional 'ideal' VCC waveform stores more energy than continuous-wave (CW) excitation only because its amplitude is allowed to grow without bound. When both waveforms are constrained to the same maximum available source amplitude — the constrained VCC (CVCC) case — the stored energy is always lower than CW throughout the excitation, by up to a factor of four in a lossless overcoupled cavity. The practical conclusion is that VCC should be understood as an efficiency-improving and waveform-shaping method, not as a way to increase absolute stored energy. This matters for applications like microwave plasma ignition, where the peak intracavity field, not just transfer efficiency, is the quantity of interest.

Core claim

Using temporal coupled-mode theory, the paper derives closed-form expressions for stored energy under CW, ideal VCC (IVCC), and a newly introduced constrained VCC (CVCC) defined by rescaling the IVCC waveform so its maximum amplitude equals the CW peak. The central result is the inequality U_CVCC(t) < U_CW(t) for 0 < t ≤ tf in overcoupled cavities, proved in Appendix B, with the maximum CVCC energy factor at tf equal to (1/4)(γ_ext + γ_int)^2/γ_ext^2, which is 1/4 in the lossless limit. The paper also reinterprets a previously reported 'factor of four' as a consequence of comparing different coupling conditions, not an intrinsic VCC enhancement. The authors validate the theory against a loss

What carries the argument

The central object is the incident waveform and its amplitude normalization. The IVCC signal s_+(t) = s_max e^{−jω0t} e^{ω''_VCC t} grows exponentially with rate ω''_VCC = γ_ext − γ_int; CVCC divides this by e^{ω''_VCC tf} so that its peak at t = tf equals s_max. The energy factor F(t) = U_VCC(t)/U_CW(t), derived from the coupled-mode solution for the modal amplitude, separates the two cases: F_IVCC > 1 because the waveform's unbounded growth delivers more incident energy, while F_CVCC < 1 because the rescaling removes that extra energy. The proof of F_CVCC < 1 rests on the monotonicity of h(x) = (1 − e^{−xt})/x, which orders the decay rates 2γ_ext > γ for overcoupled cavities.

Load-bearing premise

The conclusion holds under the assumption that the practical source limit is the maximum instantaneous amplitude of the incident waveform, and that the CVCC waveform is defined by rescaling the ideal VCC waveform to end at that maximum at tf; under a different constraint, such as equal incident energy or equal average power, the energy ranking could change.

What would settle it

Drive an overcoupled resonator with the CVCC waveform of Eq. (8) and with a CW tone whose amplitude is s_max, and compare the time-resolved stored energy |a(t)|^2, measured e.g. via the transmitted or reflected power. The paper predicts U_CVCC(t) < U_CW(t) for all 0 < t ≤ tf and U_CVCC(tf)/U_CW(tf) = (γ_ext + γ_int)^2 / (4 γ_ext^2). Any instant where the CVCC stored energy meets or exceeds the CW value, or a final ratio above that bound, would falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Reported enhancements of VCC stored energy, including an eightfold intensity claim and a factor-of-four prediction, are not intrinsic to the VCC mechanism; they stem from comparing different coupling regimes or from unnormalized waveforms.
For plasma ignition with a fixed peak-power generator, CW excitation yields a higher intracavity field than the amplitude-constrained VCC waveform, so VCC's practical value there is reflectionless operation and tailored transients, not higher peak field.
Efficiency metrics (fraction of incident energy transferred) and absolute stored energy must be reported separately; normalized efficiency gains should not be quoted as stored-energy gains.
Under a peak-amplitude constraint, the best CVCC can do at steady state is match the stored energy of an ideally critically coupled resonator, which is four times below the overcoupled CW steady state in the lossless case.

Where Pith is reading between the lines

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

The ordering is sensitive to the chosen constraint: if one instead fixed total incident energy or average power, the ranking of CW versus VCC could differ; the paper's conclusion is a statement about peak-amplitude-limited sources, which is the common practical case for waveform generators but not the only one.
The same rescaling logic could be applied to other complex-frequency excitation schemes, such as coherent virtual absorption in multi-port systems, to test whether their apparent energy advantages survive a peak-amplitude constraint.
A direct experimental test in the microwave cavity of Ref. [9] — measuring stored energy for CW and CVCC with matched peak amplitude — would confirm the predicted ratio (about one third at tf) and would strengthen the paper's reinterpretation of plasma-ignition VCC results.
If the goal is to maximize stored energy under a peak limit, one might design non-monochromatic waveforms that spend more time near the peak than the CVCC exponential ramp; the paper's framework provides the tool to search for such waveforms.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A clean, narrowly-scoped correction: amplitude-constrained VCC stores less energy than CW, with the constraint choice as the main caveat.

read the letter

The core result: under a fixed maximum input amplitude, ideal VCC's energy advantage over CW is an artifact of its unbounded exponential growth. Rescale the VCC input so its maximum equals the CW amplitude (their CVCC), and the stored energy is strictly below CW for the whole excitation, for overcoupled cavities. The proof is simple coupled-mode algebra, cleanly presented in the appendices; the two inequalities hold, and the validation against the lossless and lossy cavities matches the analytic expressions. The reinterpretation of the 'factor of four' as a comparison between overcoupled and critically coupled resonators, not a VCC enhancement, is also correct and worth having on the record.

What's genuinely new: no one had explicitly done the amplitude-constrained comparison. The paper is a useful corrective to over-readings in the VCC literature. It doesn't reshape the field, but it sharpens what claims you can make.

Soft spots: the conclusion is tied to the maximum-amplitude constraint. That's a reasonable practical constraint, and the authors say so openly, but it's not the only one. If you compare equal incident energy or equal average power, the ordering can reverse. The paper doesn't explore that, and it doesn't ask whether a different waveform respecting the same amplitude bound would beat CW. Those are open questions, not errors. Also, the CVCC waveform starts from essentially zero for long durations, so the comparison depends on the chosen tf, though the mathematical inequality holds for all t ≤ tf.

This is a solid, honest paper. I'd send it to a competent referee. It deserves publication in a good applied physics or optics journal. I'd cite it when discussing VCC energy claims.

Referee Report

0 major / 5 minor

Summary. The paper uses temporal coupled-mode theory to revisit the energy-storage capabilities of virtual critical coupling (VCC) under a practical maximum-amplitude source constraint. It distinguishes the conventional ideal VCC waveform (IVCC), whose amplitude grows exponentially from an initial value smax, from a new constrained VCC waveform (CVCC), obtained by rescaling the IVCC waveform so that its endpoint equals smax. For overcoupled resonators, analytic expressions are derived for the stored energy under CW, IVCC, and CVCC excitation. The paper proves that IVCC always stores more energy than CW (F_IVCC>1), while CVCC always stores less energy than CW throughout the excitation duration (F_CVCC<1), even though CVCC suppresses reflection. The results are validated against a lossless theoretical cavity and a lossy experimental microwave cavity, and the interpretation of previous VCC claims is revisited in light of this amplitude constraint.

Significance. If accepted, the paper provides a valuable clarification of a recurring ambiguity in the VCC literature: VCC enhances the efficiency of energy transfer into a resonator, but not the absolute stored energy when the source is subject to a maximum instantaneous amplitude. The main technical strength is the explicit, self-contained TCMT derivation: Equations (9)–(16) follow cleanly from the model, and Appendices A and B give elementary proofs of the two inequalities. The numerical validations reproduce the expected trends and are consistent with the analytic formulas. The authors are also careful to state the central caveat—the choice of maximum amplitude as the practical constraint—in Section II-C and again in Sections V and VI. The paper is therefore a useful contribution to the practical interpretation of VCC, with relevance to photonics and microwave plasma applications.

minor comments (5)

[Section III.B] The notation 't ≫ 1/τ' appears before any definition of τ. Presumably τ=1/γ (or the condition should read t ≫ τ, with τ=1/γ). Please define τ and correct the dimensional inconsistency.
[Section III.B, Eq. (16) and Table II] The quantity F_max_CVCC is called the 'maximum' CVCC energy factor, but no monotonicity proof is given to show that F_CVCC(t) is increasing up to t=tf. If this is meant only as the quasi-steady value at the endpoint, say so explicitly and provide a brief argument (e.g., monotonicity of the product in Eq. (B1)) or soften the word 'maximum.'
[Section IV.A, Fig. 3(d)] The text states that the CW efficiency first increases to about 0.8 and then decreases to a value close to 0.2, and then says that the efficiency 'tends to zero.' These statements are compatible only on different time horizons; please clarify that the 0.2 value is the efficiency at t=tf, while the asymptotic value is zero.
[Section V.B, Eq. (20)] The notation U_ss_CVCC,OC(t) combines 'quasi-steady-state' with a time argument t that can be much smaller than the settling time. This is confusing; consider denoting it as the quasi-steady CVCC envelope or explicitly stating that the quasi-steady approximation is used for the waveform shape and that t represents the time within the excitation window.
[Figures 3 and 5] Several panels use very different vertical scales and small insets. The figures are readable, but adding explicit axis labels and legends to each panel would improve clarity, especially in Fig. 5(b).

Circularity Check

0 steps flagged

No significant circularity; the central inequality is derived from TCMT and explicit waveform definitions, not assumed.

full rationale

The paper's central claims (Eqs. 9, 10, 13–15 and Appendices A–B) follow algebraically from the temporal coupled-mode equations (Eqs. 1–5) and from the explicit definitions of CW, IVCC, and CVCC waveforms (Eqs. 6–8). No fitted parameter is renamed as a prediction: the VCC growth rate ω''_VCC = γ_ext − γ_int is imported from prior VCC theory as background, and the experimental cavity parameters in Section IV.B come from a previously published experiment [9], used as an external benchmark rather than as an input to the proof. The CVCC normalization (Eq. 8) is an openly stated modeling choice for comparing under a fixed maximum excitation amplitude; the paper repeatedly qualifies its conclusions as applying under that constraint. Appendix B proves F_CVCC(t) < 1 by monotonicity of h(x) = (1−e^{−xt})/x and by e^{2ω''(t−t_f)} ≤ 1 for t ≤ t_f, so the inequality is a derived consequence, not an assumed one. The only caveat is scope—the ordering can change under a different source constraint—but this is acknowledged explicitly and is a question of framing, not circularity. Self-citations to the authors' prior work are contextual and do not carry the derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No parameters are fitted to data; all system parameters come from cited experiments. The only hand-chosen design parameter is the pulse duration tf. The constraint that the peak amplitude is the practical limit is an interpretive assumption rather than a mathematical derivation.

free parameters (1)

excitation duration tf = 600 T0 (lossless example), 70 ns (lossy example)
Free design parameter defining the endpoint of CVCC; the inequality F_cvcc<1 holds for any tf, but the maximum value at t=tf depends on it.

axioms (5)

standard math Single-resonator temporal coupled-mode equations (1)-(2) with e^{-jωt} convention
Standard model from Haus and Suh et al. [10,11]; used for all derivations.
domain assumption Incident fields of the form A_+ e^{-jω0t} e^{ω''t} (Eq. 3)
Complex-frequency excitation family used in VCC; prior literature [1,2].
domain assumption Overcoupled regime γext > γint
Needed for ω''_VCC=γext−γint>0 and both appendix proofs; paper states this assumption (§II-B).
domain assumption VCC growth rate ω''_VCC = γext − γint
Taken from prior VCC theory [1,2], not derived in this paper.
ad hoc to paper Practical source constraint is maximum instantaneous amplitude
Defines the CVCC comparison; the central conclusion changes if another constraint is imposed.

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read the original abstract

Virtual critical coupling (VCC) has emerged as a promising approach for achieving reflectionless excitation of resonant systems through tailored incident waveforms. However, the energy storage enhancement often attributed to VCC is generally assessed without considering practical constraints imposed by the excitation source. In this work, we revisit the energy storage capabilities of VCC under a realistic maximum-amplitude constraint. Using temporal coupled-mode theory, we derive analytical expressions for the stored energy of continuous wave (CW), ideal VCC (IVCC), and constrained VCC (CVCC) excitations. While the conventional IVCC excitation leads to higher stored energy than CW excitation due to its exponentially increasing incident amplitude, we show that this enhancement originates from the larger incident energy delivered by the unconstrained waveform. When the maximum excitation amplitude is fixed, the proposed CVCC excitation stores less energy than CW excitation throughout the excitation duration, while preserving the high energy transfer efficiency and reflectionless excitation enabled by VCC. The analytical expressions are used to analyze both an ideal lossless resonator and an experimentally lossy microwave cavity previously used for plasma ignition by VCC. The results clarify that VCC should primarily be regarded as a method for improving energy transfer efficiency and controlling transient excitation, rather than as an intrinsic mechanism for increasing the absolute stored energy under realistic source limitations.

Figures

Figures reproduced from arXiv: 2607.22897 by J\'er\^ome Sokoloff, Olivier Pascal, Th\'eo Delage, Valentin Mazi\`eres.

**Figure 2.** Figure 2: FIG. 2: Incident signals used for CW, IVCC and CVCC [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparison between CW, IVCC and CVCC excitations of the t [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Incident signals used for CW, IVCC, and [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Comparison between CW, IVCC and CVCC excitations of the e [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 linked inside Pith

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Baranov, Alex Krasnok, and Andrea Al` u

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and to interpret the behavior of the proposed CVCC excitation in the lossless case. The second experimental case enables us to interpret the behavior of the proposed CVCC excitation in the lossy case and provides insight into the advantages and limitations of using VCC instead of CW for plasma ignition. To this end, as in reference [2], two additional qua...
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Q. Zhong, L. Simonson, T. Kottos, and R. El-Ganainy. Coherent virtual absorption of light in microring res- onators. Physical Review Research , 2(1):013362, March 2020

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Th´ eo Delage, Olivier Pascal, J´ erˆ ome Sokoloﬀ, and Valentin Mazi` eres. Experimental demonstration of vir- tual critical coupling to a single-mode microwave cav- ity. Journal of Applied Physics , 132(15):153105, October 2022

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Th´ eo Delage, J´ erˆ ome Sokoloﬀ, Olivier Pascal, Valentin Mazi` eres, Alex Krasnok, and Thierry Callegari. Plasma Ignition via High-Power Virtual Perfect Absorption. ACS Photonics, 10(10):3781–3788, October 2023

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Jakob Hinney, Seunghwi Kim, Graydon J. K. Flatt, Ip- shita Datta, Andrea Al` u, and Michal Lipson. Eﬃcient excitation and control of integrated photonic circuits with virtual critical coupling. Nature Communications , 15(1):2741, March 2024

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Pith/arXiv arXiv 2026

[1] [1]

Baranov, Alex Krasnok, and Andrea Al` u

Denis G. Baranov, Alex Krasnok, and Andrea Al` u. Co- herent virtual absorption based on complex zero excita- tion for ideal light capturing. Optica, 4(12):1457–1461, December 2017

2017

[2] [2]

and to interpret the behavior of the proposed CVCC excitation in the lossless case. The second experimental case enables us to interpret the behavior of the proposed CVCC excitation in the lossy case and provides insight into the advantages and limitations of using VCC instead of CW for plasma ignition. To this end, as in reference [2], two additional qua...

[3] [3]

Virtual Critical Coupling

Younes Ra’di, Alex Krasnok, and Andrea Al` u. Virtual Critical Coupling. ACS Photonics, 7(6):1468–1475, June 2020

2020

[4] [4]

Trainiti, Y

G. Trainiti, Y. Ra’di, M. Ruzzene, and A. Al` u. Coher- ent virtual absorption of elastodynamic waves. Science Advances, 5(8):eaaw3255, August 2019

2019

[5] [5]

Zhong, L

Q. Zhong, L. Simonson, T. Kottos, and R. El-Ganainy. Coherent virtual absorption of light in microring res- onators. Physical Review Research , 2(1):013362, March 2020

2020

[6] [6]

Experimental demonstration of vir- tual critical coupling to a single-mode microwave cav- ity

Th´ eo Delage, Olivier Pascal, J´ erˆ ome Sokoloﬀ, and Valentin Mazi` eres. Experimental demonstration of vir- tual critical coupling to a single-mode microwave cav- ity. Journal of Applied Physics , 132(15):153105, October 2022

2022

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Araujo-Martinez, Alex Krasnok, Sergey V

Aurora C. Araujo-Martinez, Alex Krasnok, Sergey V. Kutsaev, Andrew H. Seltzman, and Alexander Yu. Smirnov. Virtual Critical Coupling in High-Power Res- onant Systems. IEEE Transactions on Plasma Science , 53(9):2410–2418, September 2025

2025

[8] [8]

Anoma- lies in light scattering

Alex Krasnok, Denis Baranov, Huanan Li, Mohammad- Ali Miri, Francesco Monticone, and Andrea Al´ u. Anoma- lies in light scattering. Advances in Optics and Photonics, Vol. 11, Issue 4, pp. 892-951 , December 2019

2019

[9] [9]

Complex-frequency excitations in photonics and wave physics

Seunghwi Kim, Alex Krasnok, and Andrea Al` u. Complex-frequency excitations in photonics and wave physics. Science, 387(6741):eado4128, March 2025

2025

[10] [10]

Plasma Ignition via High-Power Virtual Perfect Absorption

Th´ eo Delage, J´ erˆ ome Sokoloﬀ, Olivier Pascal, Valentin Mazi` eres, Alex Krasnok, and Thierry Callegari. Plasma Ignition via High-Power Virtual Perfect Absorption. ACS Photonics, 10(10):3781–3788, October 2023

2023

[11] [11]

Hermann A. Haus. Waves and Fields in Optoelectronics . Prentice-Hall, 1984

1984

[12] [12]

Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities

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 Electronics, 40(10):1511–1518, October 2004

2004

[13] [13]

Jakob Hinney, Seunghwi Kim, Graydon J. K. Flatt, Ip- shita Datta, Andrea Al` u, and Michal Lipson. Eﬃcient excitation and control of integrated photonic circuits with virtual critical coupling. Nature Communications , 15(1):2741, March 2024

2024

[14] [14]

Transiently Driven Reﬂectionless Resonant Microwave Plasmas via Virtual Critical Coupling, July 2026

Muhammad Rizwan Akram and Abbas Semnani. Transiently Driven Reﬂectionless Resonant Microwave Plasmas via Virtual Critical Coupling, July 2026. arXiv:2607.02323 [physics.app-ph]

Pith/arXiv arXiv 2026