Generalized causality constraint based on duality symmetry reveals untapped potential of sound absorption

Domain: physics
Published: 2025-11-28
Editorial outcome: major_revision

Abstract
Causality constraints are known to bind sound absorption to a limit that can only be achieved by optimizing the system bandwidth for a specific material thickness. This limit is defined on the assumption of a one-port system, generally causing duality symmetry to be overlooked. Here, we define a generalized causality constraint of sound absorption by investigating reflection and transmission of a two-port hybrid monopole-dipole resonator. With our theory, we show that the absorption limit is approached by relying on the well-established critical coupling as well as by matching effective compressibility and density. We experimentally show that the designed resonator absorbance follows the duality symmetry condition embodied in the large bandwidth reported, confirming an intrinsic connection between duality symmetry and scattering causality. A comparison with a traditional foam liner and other competitive works further validates our findings. Our results reveal previously untapped absorption potential in broadband acoustic metamaterials.

Introduction
Causality, a fundamental principle underpinning the asymmetry of unidirectional time flow, governs the upper bound of absorption integral in passive, linear, and time-invariant materials1,2. Former investigations have revealed that achieving perfect absorption at discretized frequencies requires critically coupled non-Hermitian resonances at exceptional points3,4. The manipulation of zeros and poles on the complex plane of the scattering coefficients follows the balance between the mode density and quality factors of the integrated resonator units, thus achieving broadband impedance matching and high absorption. Subsequent key developments have neared this absorption bound, demonstrated by the broadband absorption observed in multiple resonance-based metamaterials across acoustic5, elastic6 and electromagnetic domains7. Studying the absorption bound of causality constraint will guide the design wave functional devices8–11 with optimal broadband performance in the laboratory and pave the way for large-scale industrial applications12.

However, current methodologies neglect a critical question: how do fundamental symmetries dictate absorption bound throughout the entire spectrum? We find clues by considering causality constraints together with duality symmetry13, a fundamental principle embedded in numerous physics equations. For example, the supersymmetric Yang–Mills theory reveals that the strong coupling limit is remarkably equivalent to its weak coupling counterpart, with an interchange of roles between electric and magnetic charges as delineated by Montonen–Olive duality14,15. Maxwell equations are also invariant when swapping the electric and magnetic fields16. Also, duality symmetry is related to the conservation of the helicity of electromagnetic waves, attributed to the invariant degeneracy of permittivity and permeability across different geometries. In acoustics, the implications of duality symmetry are unclear for this context, due to the longitudinal wave nature, which intrinsically prohibits the helicity17. Although the analogy of acoustic duality symmetry has been proposed and investigated in the seminal work17 on the subject, the relationship between acoustic duality and scattering causality–specifically, its impact on the broadband absorption bandwidth bound of two-port systems—remains unexplored.

Results
A generalized causality constraint for a two-port absorber
According to refs. 2, 5, 31 the original causality constraint for a one-port acoustic absorber is expressed as follows:

\int_{0}^{\infty} \ln \left| \frac{1}{S_{m}(\omega)} \right|^{2} \frac{d\omega}{\omega^{2}} \leq \frac{2\pi D}{c_{0} K_{eff}(0)}, (1)

where D represents the sample thickness, c0, K0 are the sound speed and bulk modulus of the background medium (e.g., air or water), and Keff(0) denotes the static effective compressibility (the inverse of the normalized bulk modulus). The variable of integration, d\omega/\omega^{2}, is proportional to d\lambda where \lambda denotes the wavelength. According to energy conservation, the absorption A(\omega) is defined as 1-|Sm(\omega)|2 where Sm(\omega) is the reflection coefficient. The derivation of Eq. (1) assumes a Neumann backing condition, which implies either \partial x p=0 or v=0, corresponding to the scattering process depicted in the left plane of Fig. 1a. Due to mirror symmetry (or anti-symmetry), this boundary condition allows for mapping the pressure field p (or velocity field v) from the left plane to the right plane. Consequently, Eq. (1) is applicable to acoustic problems involving symmetrical pressure excitation and Sm(\omega) becomes the amplitude of monopole scattering, also referred as symmetrical coherent perfect absorption32. How to reduce Keff(0) to relax the stringent constraint? This has attracted great research interests with substantial advances in recent publications9,33–35.

The 1D acoustic equations for harmonic waves can be expressed in a matrix form as: \partial x [p; v] = i\omega [0, \rho; K^{-1}, 0] [p; v], where \rho and K represent the local mass density and bulk modulus, respectively. In this context, we introduce a duality transformation that mixes the pressure and velocity fields:

[p_{\theta}; v_{\theta}] = D(\theta) [p; v] = [\cos\theta, iZ_{0}\sin\theta; i\sin\theta/Z_{0}, \cos\theta] [p; v],

where Z0 = \sqrt{\rho_{0} K_{0}} (see the detailed derivation on acoustic duality transformation with arbitrary \theta in Supplementary Note 1). If we apply a duality transformation with \theta=\pi/2, we specifically consider the exchange p \to v, \rho \to K, and \rho^{-1} \to K^{-1} between Fig. 1a, b. Consequently, the specific impedance Z=p/v is transformed into specific admittance Y=Z^{-1}=v/p. Moreover, in Eq. (1), Sm = (Z-Z0)/(Z+Z0), which can be transformed into Sm \to -Sd because (Z-Z0)/(Z+Z0) \to -(Y-Y0)/(Y+Y0) = -(Z-Z0)/(Z+Z0) = -Sm. Also, the backing condition (\partial x p=0) in Fig. 1a is also replaced by the Dirichlet one (p=0) in Fig. 1b. Therefore, the causality constraint for monopole scattering becomes the complimentary version for dipole scattering22:

\int_{0}^{\infty} \ln \left| \frac{1}{S_{d}(\omega)} \right|^{2} \frac{d\omega}{\omega^{2}} \leq \frac{2\pi D \rho_{eff}(0)}{c_{0} \rho_{0}}, (2)

where Sd(\omega) is the dipole scattering coefficient. In particular, in this case, one can afford near unity absorption over a reasonably large frequency range for a thin absorber, owing to the generally large ratio of \rho_{eff}(0)/\rho_{0}, i.e., the large imaginary part of \rho_{eff}(\omega) around the dynamic absorbing frequency leads to large static and real-valued \rho_{eff}(0), allowed by Kramer–Kronig relations.

According to the energy conservation and Eq. (3), 1-A(\omega) = |R(\omega)|2+|T(\omega)|2 = 1/2(|Sm(\omega)|2+|Sd(\omega)|2) \geq |Sm(\omega)||Sd(\omega)|. The last relation is derived from the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality. By converting this into the causality constraint, we have:

\int_{0}^{\infty} \ln(1-A(\omega)) \frac{d\omega}{\omega^{2}} \leq \frac{1}{2} \int_{0}^{\infty} \left( \ln \left| S_{m}(\omega) \right|^{2} + \ln \left| S_{d}(\omega) \right|^{2} \right) \frac{d\omega}{\omega^{2}} \leq \frac{\pi}{c_{0} L} \left( \frac{1}{K_{eff}(0)} + \frac{\rho_{eff}(0)}{\rho_{0}} \right) = \Gamma, (4)

where L=2D (the defined thickness is effectively doubled due to mirror symmetric boundaries) and \Gamma is the established bound for the two-port setup, reflecting the static material properties. Optimal broadband absorption is defined by the proposed two-port causality constraint. Importantly, the fundamental principle lies on the fact that at long wavelength limit, all 3D scatterings of two-port problems are dominated by long wavelength scatterings as per the weighting factor (1/\omega^{2}). So, the mathematical form of causality constraint inherently embeds the role of duality for two-port 1D systems.

Methods
The simulation was carried out using the COMSOL Multiphysics commercial software, specifically employing the FEM facilitated through the “pressure acoustics, frequency domain” module. The simulation setup included monitoring of incident and transmitted waves via two ports located at the boundaries of the model. The lossy components of the CHR structures, specifically the resonators’ necks and the coupling neck between cavities, were calculated by using the narrow region model. In cases where the cavities were filled with a highly absorptive porous foam69, the Johnson–Champoux–Allard model was utilized, characterized by the following parameters (also adopted from ref. 69): porosity \phi=0.94, fluid resistance Rf=32000 [Pa·s/m2], tortuosity factor \alpha=1.06, viscous characteristic length Lv=56 [\mu m], thermal characteristic length Lth=110 [\mu m]. These parameters were consistently applied across different simulations (including the foam liner in Fig. 5) to maintain a benchmark for comparison. For simulations in the time domain, the “pressure acoustics, time explicit” interface was employed.

The geometrical specifications of the single CHR are as follows (see the marked dimensions in Supplementary Fig. 2): diameter D=1.5cm, length L=2.5cm, l=0.4L, coupling channel diameter d=0.625mm, width w=1.5mm, thickness t=0.95mm, cavity height H=6.25mm, radius R=10.67mm, \tau1=3mm, \tau2=2mm. The perforated rate (or ventilation ratio) \phi=S/S0 = (\pi H^{2})/(\pi D^{2})=0.25, where S represents the cross-sectional area at the narrowest part of the main duct, and S0 denotes the wave-front area or duct cross-sectional area. The thickness of other partition walls is standardized at \Delta w=1mm. A 1/32 reduced model features a sector angle of 360/32=11.25.

The scaling of the sample in Fig. 5 adheres to the following rules: Vc \to \alpha Vc, L \to \alpha^{1/2} L, \phi \to \alpha^{-1/2} \phi, where \alpha is the scaling factor and Vc is the cavity volume occupied by porous foam. The total thickness of the integrated sample is calculated as L = (\sqrt{0.5} + \sqrt{1.5} + \sqrt{2.5} + \sqrt{4}) L0 + 3\Delta w = 14.1cm.

For the acoustic-flow interaction simulations (see Supplementary Fig. 8), the flow resistance was calculated using the “linearized Navier-Stokes, frequency domain” interface. The background pressure drop \Delta P = P2 - P1 was determined by the difference between the output and input ports. The background flow velocity (vf) was varied from -20 to 20m/s, where negative values indicate flow in the opposite direction to the propagation of the acoustic wave. The flow resistance of the integrated CHR structures was calculated as Rf = \Delta P / (vf L). The absorption spectra for five selected flow velocities were computed using the “compressible potential flow” interface, in conjunction with the “linearized potential flow, frequency domain” interface.

The simulated reflection and transmission coefficients [R(\omega) and T(\omega)] were defined at the front and back surfaces of the sample, respectively. These coefficients facilitated the extraction of effective bulk modulus Keff(\omega) and density \rhoeff(\omega). The relationships are expressed as R(\omega) = S11 e^{-2ik0Lwg} and T(\omega) = S21 e^{-2ik0Lwg} [Lwg is the distance between the sample surface to the input or output port, see Supplementary Fig. 2]. For the extraction of effective properties in the experiments, the data were processed using the same algorithm as applied, except that Lwg was substituted with the distance from the sample surface to the nearest microphone.

The extracted effective properties from the CHR structures are fitted with the Lorentz model with the locked broadband duality symmetry. In addition, the dissipation factors (\beta m=d = \beta m = \beta d) are fixed according to closed-form solutions from Supplementary Note 2. The target of our fitting is to minimize the difference between the extracted \rhoeff(\omega) and Keff(\omega) and the Lorentz model, obtained via the optimization toolbox (Levenberg–Marquardt algorithm for nonlinear fitting) provided by the commercial software of MATLAB.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This is an interesting work which examines critical coupling and perfect absorption of acoustic waves in 1D two-channel scattering systems. The central idea of the work that the optimal broadband absorption conditions are realized in “duality-symmetric” resonators where the monopole- and dipole-mode coefficients (essentially, the density and compressibility) have similar resonant frequency dispersions with similar parameters. The theoretical results are supported by experimental realization of such “duality-symmetric” samples based on coupled Helmholtz resonators.\nIn my opinion, the results are novel, important, and, in principle, deserve publication in Nature Communications.\nI have several remarks to be considered by the authors.\n1. It seems to me that the duality symmetry is not the fundamental principle underlying the optimal broadband absorption but rather an artefact of the 1D two-channel scattering considered here.\nIndeed, in the general 3D case, there is no duality symmetry between the scalar pressure field and vector velocity field (in contrast to the electromagnetic duality symmetry between the vector electric and magnetic fields). An acoustic analogy of the dual-symmetric approach to electromagnetism has been examined in\nL. Burns et al., New J. Phys. 22, 053050 (2020).\nFurthermore, the critical coupling and total absorption (zero scattering) of waves can be achieved in a single-channel non-Hermitian scattering (either reflection or transmission) without duality. See, e.g., examples in\nK. Y. Bliokh et al., Phys. Rev. Lett. 97, 243904 (2006); Rev. Mod. Phys. 80, 1201 (2008).\nM. Asano et al., Nat. Commun. 7, 13488 (2016).\nTherefore, only in the specific case of 1D acoustics (with the scalar velocity field) and two-channel scattering (requiring two conditions for the total absorption), the effective duality starts to play the key role.\n2. In the general 3D case, the specific “duality” interference between the electric and magnetic dipoles (in electromagnetism) or the dipole and monopole modes (in acoustics) are well studied in the context of wave scattering by small particles.\nIndeed, the duality-symmetric electromagnetic case (orthogonal electric and magnetic dipoles with appropriate phases and amplitudes) is known as the Kerker conditions, or Huygens source, which results in the vanishing backscattering. Acoustic analogues of the Kerker conditions and Huygens sources have been recently considered in\nL. Wei and F. J. Rodríguez-Fortuño, New J. Phys. 22, 083016 (2020).\nY. Long et al., Natl. Sci. Rev. 7, 1024 (2020).\nMoreover, the electromagnetic and acoustic Kerker effects can be generalized to the interference of higher-order multipoles.\nSee, e.g., recent review\nI. Toftul et al., arXiv:2410.23670 (2024).\nI think, in these cases, considering absorbing small particles, the duality symmetry leads to specific scattering regimes but does not provide total absorption (because of 3D geometry and multiple scattering channels).\n3. Finally, the two similar resonant dependences of the density and compressibility parameters (or monopole and dipole scattering coefficient) in the problem under consideration can be considered in the context of two overlapping resonances, which can lead to “super-absorption”:\nK. Ladutenko et al., Nanoscale 7, 18897 (2015).\nor “super-scattering”:\nZ. Ruan and S. Fan, Phys. Rev. Lett. 105, 013901 (2010).\nThis is probably related to Refs. [47-49] in the present paper.\nThus, I suggest the authors to consider these remarks, and discuss relations of the present work to other cases and works: 1D vs. 2D/3D cases, two-channel vs. single- and multiple-channel scatterings, duality in the context of Kerker conditions and Huygens sources, and overlapping of resonances for super-absorption.", "reviewer_2": "This manuscript makes a clear and original contribution. By combining a duality transformation with causality-based absorption bounds, the authors derive a generalized two-port absorption constraint and identify broadband duality symmetry as the missing ingredient that unlocks additional absorption potential. The theoretical development is well reasoned and supported by a Lorentzian surrogate model, while the experimental implementation using CHRs demonstrates an impressively broad absorption band that convincingly outperforms a foam liner of comparable thickness. This strong agreement between theory, simulation, and experiment substantially reinforces the central claim. The authors also acknowledge current limitations (high-order modes, end corrections) and suggest possible mitigations. Overall, the work is important and well executed; however, before final acceptance in Nature Communications, I recommend minor revisions to address the following points.\n(a) The step combining Eqs. (1) and (2) with AM–GM implicitly relies on the mirror/symmetric boundary mapping and the way monopole/dipole fields are superposed. Please make these symmetry/backing assumptions explicit and clarify whether Eq. (4) remains valid for asymmetric two-port boundary impedances.\n(b) Eq. (7) correctly places the zeros on the real axis for single-mode Lorentzian resonators. However, in realistic multi-mode, lossy systems this condition is necessary but not sufficient. The authors should justify this and discuss the resulting limitations.\n(c) The physical interpretation and impact of the higher-order correction terms δₘ(ω) and δd(ω) in the modified surrogate model remain ambiguous. Treating these higher-order contributions as frequency-independent may bias the absorption integral and the FOM at higher frequencies. The authors could provide either sensitivity curves showing how dispersive δ(ω) alters γ/Γ or a short justification beyond the 1/ω low-frequency weighting argument.\n(d) The FOM depends heavily on the estimated bound Γ from simplified static calculations that neglect end corrections. As the authors note, this could underestimate Γ and thereby inflate the reported FOM values. Please quantify this possible bias.\n(e)The authors neglect many recently reported high-performance, unique metamaterial absorbers (e.g., doi.org/10.1016/j.mattod.2025.06.029; Nat. Commun. 2023:14, 4028) in the state-of-the-art discussions and performance comparisons.\n(f) The core advantage of ventilative design is highlighted, but no quantitative data such like pressure drop and flow resistance versus absorption performance is provided.\n(g) The conclusion mentions AI/optimization but gives no concrete route to extend duality symmetry to high-order modes; can you add a brief roadmap?", "reviewer_3": "This manuscript investigates the reflection and transmission properties of acoustic monopole-dipole resonators in a two-port configuration, establishing a sound absorption theory under generalized causality constraints and revealing the intrinsic connection between duality symmetry and scattering causality. The research indicates that achieving optimal absorption approaching the absorption bound not only depends on critical coupling but also requires satisfying broadband duality symmetry (equivalent to matched effective compressibility and density). Although some theoretical results have been mentioned in Ref. 23, the elaboration in this manuscript is more clear and thorough. Moreover, the designed coupled Helmholtz resonator (CHR) has experimentally observed broadband duality-symmetric sound absorption characteristics, and the further designed integrated CHR exhibits better sound absorption performance than traditional foam liners and previously reported competitive studies, with a high degree of consistency between experimental data and theoretical models. Based on this, I am willing to recommend its publication once the following issues are properly resolved.\n1. The structure constructed in the manuscript is intrusive, while the foam liner used for comparison is non-intrusive, and the rigor of such a comparison is questionable. It is recommended to supplement comparative data with foam liners that have the same ventilation ratio to enhance the rigor of the research.\n2. The calculation method of the experimental data in Figure 3(h) and Figure 4(e) needs to be explained. Since there is a lack of experimental data in the low-frequency region, and according to Formula 12, the integral needs to start from 0Hz, it is necessary to clarify the specific calculation process of the experimental values and the handling method for the missing low-frequency data.\n3. It is necessary to clarify whether filling porous materials to change the dissipation factor β will have an impact on the volume of the air cavity that controls the modulus. In addition, listing the detailed geometric parameters of the structure in the manuscript will help readers reproduce the experimental results and conduct further verification.\n4. The layout of the figures needs further optimization, such as the arrangement of images in Figure 2 and the order of subfigures in Figure 3, to improve the readability of the figures.\n5. Attention should be paid to distinguishing symbols in the formulas, such as the sample thickness d and the differential symbol d, the sound pressure symbol p and the zero position p, to avoid confusion."}, "author_response": "Responses to Reviewers’ Comments\nRe: Manuscript ID NCOMMS-25-57293\n[The reviewer’s comments are listed with the italic text. The authors’ responses are shown by the normal text. The revised/added texts in the manuscript are highlighted.]\nReviewer #1 (Remarks to the Author):\nThis is an interesting work which examines critical coupling and perfect absorption of acoustic\nwaves in 1D two-channel scattering systems. The central idea of the work that the optimal\nbroadband absorption conditions are realized in “duality-symmetric” resonators where the\nmonopole- and dipole-mode coefficients (essentially, the density and compressibility) have\nsimilar resonant frequency dispersions with similar parameters. The theoretical results are\nsupported by experimental realization of such “duality-symmetric” samples based on coupled\nHelmholtz resonators. In my opinion, the results are novel, important, and, in principle, deserve\npublication in Nature Communications.\nWe sincerely thank you for the constructive comments, based on the meaningful comparison of\nexisting literature with our work. We appreciate the recognition of the novelty and importance\nof our work, and have carefully considered all your suggestions. Below we provide point-by-\npoint responses.\nI have several remarks to be considered by the authors.\n1. It seems to me that the duality symmetry is not the fundamental principle underlying the\noptimal broadband absorption but rather an artefact of the 1D two-channel scattering\nconsidered here. Indeed, in the general 3D case, there is no duality symmetry between the scalar\npressure field and vector velocity field (in contrast to the electromagnetic duality symmetry\nbetween the vector electric and magnetic fields).\nIndeed, acoustic wave is scalar in nature and a direct one-to-one correspondence between scalar\npressure (𝑝) and 3D vectorial velocity (𝒗) does not exist. But for plane waves, we can always\nbuild up the duality relation by pairing 𝑝 with the scalar 𝑣 along the direction of pressure\ngradient (∇𝑝). For example, a single anisotropic scatterer in 3D free space, with proper design,\nshall exhibit protection through duality symmetry for a specific direction of plane wave\nincidence (a reflectionless state); however, this protection may not hold if the orientation of\nexcitation is altered (e.g., significant backscattering occurs). If the input and output modes are\nfrom higher-orders in the general 3D, we will agree that the direct application of duality\nprinciple of overlapping monopole and dipole modes may be not proper for extending\nabsorption bandwidth. However, this does not affect the core point of this article. When\nconsidering absorption problems, we usually focus on the most challenging part: plane waves\n(low frequency, broadband). We added the following to the subsequent discussion of Eq. (4):\n Optimal broadband absorption is defined by the proposed 2-port causality constraint.\nImportantly, the fundamental principle lies on the fact that at long wavelength limit, all 3D\nscatterings of two-port problems, become specular reflection and plane-wave transmission,\ndefining the effective compressibility and density. Also, the absorption integral is\ndominated by long wavelength scatterings as per the weighting factor (1/𝜔²). So, the\nmathematical form of causality constraint inherently embeds the role of duality for two-\nport 1D systems.\nIn the concluding remarks, we have added a sentence:\n The focus of our research is on one-dimensional two-port systems or their equivalent\nscattering problems characterized by reduced dimensions (see the comparison of\nwaveguide and free space scattering setups in Supplementary Figure S7). In contrast, the\nmore intricate landscape of higher-order scatterings in higher dimensions may present\nchallenges that the straightforward pairing and cancellation between a monopole and a\ndipole—enforced by the duality symmetry—may not adequately address.\nand to clearly define the scope of our work, we added a Supplementary Figure S7.\nFigure S7. The scattering setups of previous research, our current focus, and the prospective causality-\nconstraint generalization to higher-order dimensions and multi-port systems. Notably, free-space scattering\ncan be effectively equivalent to waveguide transport of waves, as both share the same mathematical formulation of\nthe scattering matrix, provided that the dimensions (𝑁×𝑁) are identical.\nAn acoustic analogy of the dual-symmetric approach to electromagnetism has been examined\nin L. Burns et al., New J. Phys. 22, 053050 (2020). Furthermore, the critical coupling and total\nabsorption (zero scattering) of waves can be achieved in a single-channel non-Hermitian\nscattering (either reflection or transmission) without duality. See, e.g., examples in K. Y. Bliokh\net al., Phys. Rev. Lett. 97, 243904 (2006); Rev. Mod. Phys. 80, 1201 (2008). M. Asano et al.,\nNat. Commun. 7, 13488 (2016).\nWe agree that the analogy of acoustic duality symmetry has been discovered, inspired by the\nstate-of-the-art concept developed by optics community. In the introduction part, we added the\nmissing reference with proper discussion:\n Although the analogy of acoustic duality symmetry has been proposed and investigated in\nthe seminal work [New J. Phys. 22, 053050 (2020)] on the subject, the relationship\nbetween acoustic duality and scattering causality—specifically, its impact on the\nbroadband absorption bandwidth bound of two-port systems—remains unexplored.\nWe emphasize that the exploration of duality symmetry in broadband absorption system is rare\nin past works. More importantly, it is a blank area if this duality symmetry is related to the\nbandwidth limit set by causality constraints [as per Eq. (4)]. Our work filled this gap.\nThe shared seminal works are excellent examples of the critical coupling and total absorption\n(zero scattering) for waves. However, existing studies primarily focus on single frequencies or\na limited number of discretized frequencies, which restricts bandwidth. The objective of our\nresearch is to elucidate the fundamental relationship between scattering causality, absorption\nbandwidth—quantified through a spectral integral—and duality symmetry. While achieving\nnarrowband perfect absorption is nothing new, maximizing absorption band holds significant\nvalue for practical applications.\nTherefore, only in the specific case of 1D acoustics (with the scalar velocity field) and two-channel scattering (requiring two conditions for the total absorption), the effective duality starts to play the key role.\nThanks for confirming that for 1D case, the duality plays a key role. For 2D/3D general cases,\nwe argue that\n Certain examples in both 2D and 3D scenarios can be simplified to a 1D case, provided\nthat the dimensions of the scattering matrix are equivalent, specifically 2×2 (see\nSupplementary Figure S7). For instance, a meta-surface arranged in a periodic layout may\nabsorb perfectly at one incident angle, but only undergo total retro-reflection at another\nangle, referred as extremely asymmetrical absorption [Physical Review Letters 123.21\n(2019): 214302]. Although the radiation channels are based on plane waves (equivalent to\nscalar velocity field), there exist vectorial velocity fields in the near-field region, with non-\nradiative local modes of the scatterer.\nAlso, it is well-documented that many absorbers experience diminished performance under\noblique incidence. Additionally, in the context of oblique incidence within the long-wavelength\nregime—where the primary contribution to the absorption integral arises—only specular\nreflection with possible plane-wave transmission will be observed, with no involvement of\ndiffraction or higher-order scattering channels.\n2. In the general 3D case, the specific “duality” interference between the electric and magnetic\ndipoles (in electromagnetism) or the dipole and monopole modes (in acoustics) are well studied\nin the context of wave scattering by small particles. Indeed, the duality-symmetric\nelectromagnetic case (orthogonal electric and magnetic dipoles with appropriate phases and\namplitudes) is known as the Kerker conditions, or Huygens source, which results in the\nvanishing backscattering. Acoustic analogues of the Kerker conditions and Huygens sources\nhave been recently considered in L. Wei and F. J. Rodríguez-Fortuño, New J. Phys. 22, 083016\n(2020). Y. Long et al., Natl. Sci. Rev. 7, 1024 (2020). Moreover, the electromagnetic and\nacoustic Kerker effects can be generalized to the interference of higher-order multipoles. See,\ne.g., recent review I. Toftul et al., arXiv:2410.23670 (2024). I think, in these cases, considering\nabsorbing small particles, the duality symmetry leads to specific scattering regimes but does\nnot provide total absorption (because of 3D geometry and multiple scattering channels).\nThank you for the insightful comment and for pointing out the excellent references. Our key\ncontribution here is to demonstrate the critical role of duality in achieving the fundamental\nbroadband limit for the low-frequency absorption. As for the Kerker conditions you mentioned,\nindeed it shares the fundamental physics with the derived critical coupling conditions. After\nderiving Eqs. (8-9), we have added the following comments:\n This condition, which dictates a specific amplitude and phase relationship between the\nmonopole and dipole scattering coefficients, is the acoustic equivalent [Wei & Rodríguez-\nFortuño (2020) and Long et al. (2020)] of the Kerker condition known in electromagnetism\nand leads to a Huygens-source-like radiating or scattering profile.\nThe unexplored link between the maximized absorption integral and duality symmetry with\ncritical coupling condition, is what we want to highlight. Acoustic analogue of the Kerker\nconditions is not the main innovative point in our work.\nWe agree that the duality symmetry is not the sufficient condition to provide total absorption.\nSmall particle indeed supports multiple scattering channels at higher frequencies. As we\ndiscussed, what we focus on is the low-frequency and broadband two-port scattering problems.\nSo, this is not contradictory to our theory for subwavelength scatters in two-port system. Even\nwith duality symmetry, imperfect absorption may be caused by the deviation from the critical\ncoupling required loss [see our experimental results in Supplementary Figure S3], as well as\nthe multiple 3D scattering channels you mentioned. The conclusion is that for two-port\nscattering problems, sufficient and necessary conditions for maximized absorption integral are\nthe simultaneous compliance of duality symmetry and critical coupling.\n3. Finally, the two similar resonant dependences of the density and compressibility parameters\n(or monopole and dipole scattering coefficient) in the problem under consideration can be\nconsidered in the context of two overlapping resonances, which can lead to “super-absorption”:\nK. Ladutenko et al., Nanoscale 7, 18897 (2015). or “super-scattering”: Z. Ruan and S. Fan,\nPhys. Rev. Lett. 105, 013901 (2010). This is probably related to Refs. [47-49] in the present\npaper. Thus, I suggest the authors to consider these remarks, and discuss relations of the present\nwork to other cases and works: 1D vs. 2D/3D cases, two-channel vs. single- and multiple-\nchannel scatterings, duality in the context of Kerker conditions and Huygens sources, and\noverlapping of resonances for super-absorption.\nThank you for your insightful observation. We have added:\n Electromagnetic superabsorption reported by K. Ladutenko et al. [Nanoscale 7,\n18897(2015)] indeed serves as an optical counterpart to our study. However,\nsuperscattering represents a distinct phenomenon that does not necessitate critical\ncoupling; rather, it maximizes backscattering as opposed to minimizing it. This effect can\nbe viewed as an anti-effect to super-absorption, although both phenomena share a common\nfeature: the alignment of two fundamental resonances, albeit with their phases reversed.\nAdditionally, in the concluding remarks, we have emphasized the following:\n The strategy of overlapping resonances to enhance wave-matter interaction is a well-\nestablished principle, leading to phenomena such as super-absorption [Phys. Rev. Lett.\n105, 013901 (2010)] and super-scattering [Nanoscale 7, 18897 (2015)]. Our work builds\non this principle by applying it to the specific case of acoustic monopole-dipole degeneracy.\nReviewer #2 (Remarks to the Author):\nThis manuscript makes a clear and original contribution. By combining a duality\ntransformation with causality-based absorption bounds, the authors derive a generalized two-\nport absorption constraint and identify broadband duality symmetry as the missing ingredient\nthat unlocks additional absorption potential. The theoretical development is well reasoned and\nsupported by a Lorentzian surrogate model, while the experimental implementation using CHRs\ndemonstrates an impressively broad absorption band that convincingly outperforms a foam\nliner of comparable thickness. This strong agreement between theory, simulation, and\nexperiment substantially reinforces the central claim. The authors also acknowledge current\nlimitations (high-order modes, end corrections) and suggest possible mitigations. Overall, the\nwork is important and well executed; however, before final acceptance in Nature\nCommunications, I recommend minor revisions to address the following points.\nWe sincerely appreciate your confirmation of the originality of our contributions, as well as the\nstrong alignment observed between theory, simulation, and experiment. We believe that your\ninsights demonstrate a comprehensive understanding of our work, and we have taken the liberty\nof addressing them for your consideration.\n(a) The step combining Eqs. (1) and (2) with AM–GM implicitly relies on the mirror/symmetric\nboundary mapping and the way monopole/dipole fields are superposed. Please make these\nsymmetry/backing assumptions explicit and clarify whether Eq. (4) remains valid for\nasymmetric two-port boundary impedances.\nThank you for raising the question regarding the key step associated with the AM–GM\ninequality. To clarify, we focus on mapping the acoustic fields resulting from the mirror or\nsymmetric boundaries, rather than the boundaries themselves. Due to the equal weighting ratio\n(scattering coefficients 𝑆ₘ = −𝑆d), Eq. (4) is specifically compatible with impedance\nmatching backing. Symmetrical impedance setup is the most common case for reflection-\ntransmission problems of waveguide scattering or wave modulation by flat meta-surfaces.\nRegarding the asymmetric configuration of two-port boundary impedances, we think this is a\nsmart idea. It is possible to incorporate scattering fields with varying weighting ratios to\ngenerate a field distribution that accommodates the impedance of asymmetric ports. However,\nthe mathematical expression for the integral of −ln (1−𝐴) becomes increasingly complex\nand may lack closed-form solutions (AM-GM does not directly apply), unlike the elegant\nsolutions of Eq. (8) and Eq. (9). Nevertheless, this presents a valuable opportunity and a\nsignificant avenue for future generalization and numerical application, although it falls outside\nthe scope of the current study.\n(b) Eq. (7) correctly places the zeros on the real axis for single-mode Lorentzian resonators.\nHowever, in realistic multi-mode, lossy systems this condition is necessary but not sufficient.\nThe authors should justify this and discuss the resulting limitations.\nThank you for this insightful question. The single-mode assumption is employed to gain\ntheoretical insights and to capture the primary features of the sample for experimental\nrealization. In the context of multi-mode, lossy systems, the critical coupling condition has been\nextensively investigated in many prior studies through numerical methods. For the theoretical\nmodeling of impedance conditions involving multiple resonances, an effective approach is to\nutilize the continuum model of the resonance frequency distribution (as demonstrated in our\nprevious work, as per Figure 2 of Physical Review Applied 17.4 (2022): 047001). The\nconclusion drawn from this analysis is that optimal absorption bandwidth necessitates a\nreduction in system loss to accommodate the increased mode density via multiple resonances.\nIn the main text [after Eq. (7)], we added the discussion on limitations and generalization:\n For multi-resonant systems, Eq. (7) is not applicable; consequently, the optimal loss must\nbe determined by considering the mode density distribution [National Science Review 12,\nnwaf199 (2025)]. A theoretical model based on continuum resonance distribution has been\ndeveloped for this purpose [Physical Review Applied 17.4 (2022): 047001]. In this study,\nwe prefer to utilize the single-mode version of Eq. (5) to gain insights within a more\nconcise framework.\n(c) The physical interpretation and impact of the higher-order correction terms δₘ(ω) and δd(ω)\nin the modified surrogate model remain ambiguous. Treating these higher-order contributions\nas frequency-independent may bias the absorption integral and the FOM at higher frequencies.\nThe authors could provide either sensitivity curves showing how dispersive δ(ω) alters γ/Γ or\na short justification beyond the 1/ω low-frequency weighting argument.\nThanks for raising this question. It is difficult to measure the dispersion of a function using a\nsingle variable. But we can justify that it is a good approximation in our case to adopt non-\ndispersive higher-order terms because they do not bias the absorption integral. In the discussion\nafter Eq. (12), we highlighted the quantified data summary:\n The absorption contributed by higher order terms is ignorable. Otherwise, the calculated\n𝛾 via simulation data (considering higher-order contribution) will significantly deviate\nfrom the lowest-order theoretical value Γ₁. However, this is not true because we have 𝛾 =\n48.9%Γ₁, which is very close to Γ₁ = 47.5%Γ₁. It is important to emphasize that this\napproximation is applicable solely to our specific case and does not yield a universal\nconclusion for other resonator structures.\n(d) The FOM depends heavily on the estimated bound Γ from simplified static calculations that\nneglect end corrections. As the authors note, this could underestimate Γ and thereby inflate the\nreported FOM values. Please quantify this possible bias.\nYes. To facilitate the calculation of the estimated bound Γ for the works being compared, we\nomitted end corrections, as it is not feasible to evaluate Γ without comprehensive information\nregarding the reference works. Nonetheless, we established a unified evaluation framework\nbased on the Figure of Merit (FOM) for all cases to ensure a fair comparison. After the\ndiscussion of Eq. (13), we added:\n To quantify the resulting bias, we can compare the calculated Γ of single CHR and\nintegrated CHRs: the respective accurate values are Γ = 6.1×10⁻⁴ [s] and 4.8×\n10⁻⁴ [s], while the FOM-based values are Γ = 4.6×10⁻⁴ [s] and 4.0×10⁻³ [s]. This\nobserved bias is both expected and justifiable, as near-field interactions contribute to end\ncorrection effects. Furthermore, rather than integrating over the entire frequency band, the\nsimplified FOM places greater emphasis on high absorption and low-frequency bands\n(absorption greater than 50%), resulting in higher scores for broadband absorbers.\n(e) The authors neglect many recently reported high-performance, unique metamaterial\nabsorbers (e.g., doi.org/10.1016/j.mattod.2025.06.029; Nat. Commun. 2023:14, 4028) in the\nstate-of-the-art discussions and performance comparisons.\nThanks for suggesting the latest absorbers. We have added more discussion and citations:\n Many recently reported high-performance and innovative metamaterial absorbers [Nature\nCommunications 14.1 (2023): 4028.; Materials Today (2025.06.029)] may also be re-\nevaluated and optimized using our new figure of merit (FOM) evaluation framework. We\nFigure S8. The interaction between the background flow and the acoustic absorption structures\n(integrated CHRs). (a) The relationship between the pressure drop (Pa) and the background flow velocity\n(m/s). A positive flow velocity indicates that the flow is in the same direction as sound propagation, while a\nnegative flow velocity indicates the flow is in the opposite direction. (b) The corresponding fluid resistance as\na function of flow velocity. (c) The absorption spectra at selected flow velocities. Simulation details are\nprovided in the Methods section of the main text.\nforesee a promising future for acoustic-mechanical metamaterials that transcends the\nlimitations of conventional materials.\n(f) The core advantage of ventilative design is highlighted, but no quantitative data such like\npressure drop and flow resistance versus absorption performance is provided.\nWe have carried out additional acoustic-flow interaction simulation (see Supplementary Figure\nS8). Therefore, quantitative data of pressure drop and flow resistance versus absorption\nperformance are provided for highlighting the ventilative design. Also, we have updated\nMethods to describe how we did this simulation:\n For the acoustic-flow interaction simulations (see Supplementary Figure S8), the flow\nresistance was calculated using the 'Linearized Navier-Stokes, Frequency Domain'\ninterface. The background pressure drop (Δ𝑃 = 𝑃₂−𝑃₁) was determined by the\ndifference between the output and input ports. The background flow velocity (𝑣_f) was\nvaried from -20 m/s to 20 m/s, where negative values indicate flow in the opposite\ndirection to the propagation of the acoustic wave. The flow resistance of the integrated\nCHR structures was calculated as 𝑅_f = Δ𝑃/(𝑣_f 𝐿) . The absorption spectra for five\nselected flow velocities were computed using the 'Compressible Potential Flow' interface,\nin conjunction with the 'Linearized Potential Flow, Frequency Domain' interface.\n(g) The conclusion mentions AI/optimization but gives no concrete route to extend duality\nsymmetry to high-order modes; can you add a brief roadmap?\nA concrete approach to enhancing duality symmetry can be pursued through advanced\noptimization and artificial intelligence technologies. The proposed roadmap involves the\ndevelopment of an efficient optimization framework that leverages feature engineering of\ndatasets. So, we have added heuristic discussion in the concluding remarks:\n For example, the impedance network could be effectively established with the multi-\nobjective and coupling-driven optimization, while considering the additional requirements\non the mechanical strength and lightweight design [ACS Applied Materials & Interfaces\n15, 9940-9952 (2023); Virtual and Physical Prototyping 18, e2166851 (2023); Advanced\nFunctional Materials 35, 2420207 (2025)].\nBy integrating mechanical data into the model, we can incorporate additional application\nrequirements. Furthermore, the inclusion of modes enriched by mechanical and acoustic\ndomains is advantageous for the exploration of higher-order modes.\nReviewer #3 (Remarks to the Author):\nThis manuscript investigates the reflection and transmission properties of acoustic monopole-\ndipole resonators in a two-port configuration, establishing a sound absorption theory under\ngeneralized causality constraints and revealing the intrinsic connection between duality\nsymmetry and scattering causality. The research indicates that achieving optimal absorption\napproaching the absorption bound not only depends on critical coupling but also requires\nsatisfying broadband duality symmetry (equivalent to matched effective compressibility and\ndensity). Although some theoretical results have been mentioned in Ref. 23, the elaboration in\nthis manuscript is clearer and more thorough. Moreover, the designed coupled Helmholtz\nresonator (CHR) has experimentally observed broadband duality-symmetric sound absorption\ncharacteristics, and the further designed integrated CHR exhibits better sound absorption\nperformance than traditional foam liners and previously reported competitive studies, with a\nhigh degree of consistency between experimental data and theoretical models. Based on this, I\nam willing to recommend its publication once the following issues are properly resolved.\nWe would like to express our sincere gratitude for confirming that (1) our theoretical innovation\nthat establishes a connection between duality symmetry and scattering causality; (2) the\nsignificant consistency observed between competitive experimental data and refined theoretical\nmodels. The issues raised have indeed contributed to the enhancement of our manuscript, and\nwe present our point-to-point responses below.\n1. The structure constructed in the manuscript is intrusive, while the foam liner used for\ncomparison is non-intrusive, and the rigor of such a comparison is questionable. It is\nrecommended to supplement comparative data with foam liners that have the same ventilation\nratio to enhance the rigor of the research.\nWe appreciate your insightful observation. Our primary comparison is grounded in the principle\nof equal volume, which relates to effective compressibility, alongside the standard\nconfiguration of a conventional foam liner. Your recommendation to include data on an\nintrusive foam is also valuable and adopted, as it reflects the equivalent condition of effective\ndensity.\nConsequently, in the revised manuscript, we have incorporated the simulated absorption\nspectrum (displayed in dB scale for better illustration) of the intrusive foam and compared it\nwith those of the existing foam liner and our integrated CHR structures (see the added\nSupplementary Figure S9). The results are comprehensible and contribute significantly to the\nenhancement of our theoretical insights. Although the intrusive foam exhibits equivalent static\neffective density and compressibility compared to our engineered structures, its performance\nremains inferior to that of our designs (𝛾/Γ ≤ 33.1% ). Nonetheless, it surpasses that of\nFigure S9. Additional simulation data for the intrusive foam. (a) The geometric layout of the intrusive foam,\nwhich shares the same duct shape and volume as our integrated CHRs. The porous resonator domain was calculated\nusing the same JCA parameters. (b) Absorption spectra are displayed on a dB scale: 10×log₁₀(1−𝐴) for the\nintrusive foam, foam liner, and integrated CHRs. The legend shows the percentage between the full band integration\nof Eq. (12) (𝜔→∞) and Γ.\nconventional foam liners (𝛾/Γ ≤ 18.5%), because static duality symmetric material properties\nare met.\n2. The calculation method of the experimental data in Figure 3(h) and Figure 4(e) needs to be\nexplained. Since there is a lack of experimental data in the low-frequency region, and according\nto Formula 12, the integral needs to start from 0Hz, it is necessary to clarify the specific\ncalculation process of the experimental values and the handling method for the missing low-\nfrequency data.\nWe appreciate your nice question regarding the data processing methods employed in both the\nexperimental and simulation contexts. Indeed, while the simulation and theoretical models can\ngenerate absorption data across a continuous range, from near-zero frequency to a high cutoff\nfrequency, it remains difficult to obtain full band accurate data through experimental means\n(especially for low frequencies). If the experimental data of the absorption coefficient are\nslightly amplified by experimental errors at low frequencies, then we will obtain an\noverestimated value of the integral. Therefore, our strategy is to discard low-frequency data\nbelow 300 Hz and focus on the frequency range we are more concerned about. We have clarified\nmore details in the Methods:\n To compare the experiment-based data with simulation predictions, an offset value\nΔ𝛾 from simulation data was utilized. For comparisons illustrated in Fig. 4(f) and Fig.\n5(e), the adopted formula (data presented by circles) was that 𝛾(𝜔) = ∫_{2π×300}^{ω} −ln(1−\n𝐴_{exp}(𝜔)) + Δ𝛾 , where 𝐴_{sim}, 𝐴_{exp} denote absorption spectra from simulation\nand experiments respectively and Δ𝛾 = ∫_{0}^{2π×300} −ln(1−𝐴_{sim}(𝜔)) .\n3. It is necessary to clarify whether filling porous materials to change the dissipation factor β\nwill have an impact on the volume of the air cavity that controls the modulus. In addition, listing\nthe detailed geometric parameters of the structure in the manuscript will help readers\nreproduce the experimental results and conduct further verification.\nYes, porous materials do have an impact on the volume of the air cavity. To make this point\nclear, we added more words in the main text:\n The remaining variable 𝑉_{air} should be evaluated in different ways for pure air domain\nand porous domain (𝑉_{air} = 𝑉₀ + 𝑉_{porous}). In pure air domain (e.g., the narrowed\nchannel), it is the same as the actual volume 𝑉₀ occupied by air. In porous domain, due\nto the comparable scale of micro-structures and viscous boundary layer thickness, the\nporous material absorption should be treated as isothermal process, according to Physical\nReview Letters 134.23 (2025): 237001, and then, 𝑉_{porous} = 𝛾_{air} 𝑉₀ 𝜑 where 𝑉₀ is the\nresonator’s cavity volume without porous materials, the porosity 𝜑 excludes the porous\nskeleton volume ratio and 𝛾_{air} is the adiabatic index of air.\nAs for the detailed geometric parameters of our sample, we have highlighted it in the Methods\nand Supplementary Figure S2:\n The geometrical specifications of the single CHR are as follows (see the marked\ndimensions in Supplementary Figure S2): diameter 𝐷₀ = 1.5 cm, length 𝐿 = 2.5 cm,\nℓ = 0.4𝐿, coupling channel diameter 𝑑₁ = 0.625 mm, width 𝑤 = 1.5 mm, thickness\n𝑡 = 0.95 mm, cavity height 𝐻 = 6.25 mm, radius 𝑅 = 10.67 mm, 𝜏₁ = 3 mm,\n𝜏₂ = 2 mm. The perforated rate (or ventilation ratio) 𝜙 = 𝑆/𝑆₀ = (𝜋𝐻²)/(𝜋𝐷₀²) =\n0.25, where 𝑆 represents the cross-sectional area at the narrowest part of the main duct,\nand 𝑆₀ denotes the wave-front area or duct cross-sectional area. The thickness of other\npartition walls is standardized at Δ_s = 1mm. A 1/32 reduced model features a sector\nangle of 360∘/32 = 11.25∘.\n4. The layout of the figures needs further optimization, such as the arrangement of images in\nFigure 2 and the order of subfigures in Figure 3, to improve the readability of the figures.\nThanks for this suggestion. We have split original Fig. 2 into two figures (current Fig. 2 and\nFig. 3). Also, we optimized the subfigure’s order (of original Fig. 3, and current Fig. 4) in the\nnew version.\n5. Attention should be paid to distinguishing symbols in the formulas, such as the sample\nthickness d and the differential symbol d, the sound pressure symbol p and the zero-position p,\nto avoid confusion.\nWe appreciate your suggestion on the formula symbols. We have replaced the sample thickness\n𝑑 by 𝐷 throughout the manuscript. The zero-position 𝑝 in Fig. 1 is to explain the zero\npressure boundary condition, not to define position. We have fixed this problem by adding a\ntext near the confusing 𝑝 in Fig. 1."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have convincingly responded to my remarks and improved the manuscript. I recommend it for publication in Nature Communications.", "reviewer_3": "I have checked the response provided by the authors to all review comments and the revised manuscript. I'm satisfied with the additional information offered in this revised version. Therefore I recommend the publication."}, "author_response": ""}]

Decision letter