Metallic nanostructures supporting plasmonic resonances are widely studied as they can lead to subwavelength field confinement with a strong enhancement of field intensity and electron-matter interaction [1, 2]. However, compared to dielectric systems, they are affected by dissipative losses that limit the achievable Q-factors of plasmonic resonances known as surface plasmon polaritons (SPPs) [2–4]. Thus, metallic nanostructures are subject to a fundamental trade-off: the very properties that make plasmonics attractive—the tight spatial localization of plasmon modes—are intrinsically linked to mechanisms that cause significant energy dissipation.

Bound States in the Continuum [5–8] are solutions of the wave equation that are resonant with the continuous spectrum; nevertheless, they are not radiatively coupled to the continuum, due to symmetry mismatch, resonance anticrossing, or other mechanisms. They can be transformed into quasi-bound states in the continuum (q-BICs) that couple to far-field radiation in a controlled manner, by employing design criteria based e.g. on symmetry reduction [9, 10], which can also yield topological singularities with full circular polarization [11–14]. BICs are a vibrant topic in nanophotonics, leading to significant new research pathways on, e.g. generalized Kerker effect [15], exceptional BICs [16], asymmetric or Janus BICs [17], singularities in phase spectra [18], enhancement of nonlinear effects [19, 20].

In metallic systems, q-BICs are also subject to nonradiative losses due to the metal’s finite conductivity. However, compared to SPPs, the radiative loss can be controlled by proper design. After a theoretical proposal on hybrid plasmonic-photonic structures [21], experimental research focused on increasing the Q-factor of q-BICs [22, 23] and on tailoring light-matter coupling via the critical coupling condition [24, 25], while further theoretical research concerned the role of surface-lattice resonances [26] and robustness against weak spatial coherence and polarization [27]. BICs in plasmonics systems promise to yield improved solutions to the fundamental trade-off between increasing field localization and dissipation, however, reaching this goal requires physically inspired designs for controlling field profiles together with radiative and nonradiative losses.

Chirality, namely the absence of mirror symmetries, is a fundamental property of many physical and biochemical systems, including most biomolecules [28, 29]. A common manifestation of chirality is circular dichroism (CD) in optical properties like absorption and emission. Recently, chiral BICs in plasmonic systems have attracted significant interest [30–33] because they yield narrow resonances that can be exploited for applications requiring strong electric-field enhancement and resonant high-Q chiral behavior.

In a recent work [34], we predicted the existence of chiral q-BIC resonances in a metal nanohole array with symmetry-broken holes. Such resonances arise as absorption peaks (or transmission dips) in the regime of extraordinary optical transmission by the nanohole array, via coupling to SPPs [35–37]. The concept of symmetry breaking leads to a robust q-BIC design that yields nearly maximum CD at small angles of incidence. Such chiral q-BICs have , field intensity enhancement ∼160, CD, with a simple 2D metasurface based on the deformation of circular holes into oval holes. Moreover, BICs can couple to an active medium to form hybridized BIC polaritons, which become chiral polaritons upon symmetry breaking [38].

For SPP resonances in the quasi-static approximation, the Q-factor tends to peak at a fixed wavelength independent of structure geometry [3, 4]. Even beyond the quasi-static approximation, increasing the structure size does reduce the ohmic losses, but it can decrease the total Q-factor due to increase of radiative scattering [39]. In this work, we consider the chiral plasmonic q-BIC in a wide spectral range, focusing on the interplay between ohmic and radiative losses. We demonstrate that high CD and Q-factor, as well as strong field enhancement, can be achieved at the same time at longer resonance wavelengths. Such trends for the symmetry-broken q-BIC are very different from those expected for SPP resonances, and they effectively break the trade-off between Q-factor and field enhancement in plasmonic systems. To elucidate the mechanisms that underlie such improvement, we introduce a novel procedure that allows us to separate the radiative and non-radiative contributions to the Q-factor. We find that on increasing the lattice constant, both dissipative and radiative losses are reduced, which we explain in terms of field localization at the air/metal interfaces at the bottom of the hole sidewalls. The same mechanism leads to an increase of the CD for the chiral plasmonic q-BIC at a finite angle of incidence. Thus, for the present q-BIC resonance based on symmetry breaking, field localization is found to reduce (rather than increase) the losses, which challenges the common paradigm in plasmonic systems. The present analysis and results go much beyond our previous work [34], and they may be generally useful in the field of nanoplasmonics as a new way to actively tailor the loss mechanisms. The present results can be exploited in view of the realization of chiral plasmonic resonances with high Q-factor and enhanced light–matter interaction, which may find application in linear and nonlinear nano-plasmonic devices.

The sketches of figure 1 present schematic views of the metasurface featuring a nanohole array investigated in this work. A gold layer of thickness t = 100 nm is embedded between SiO2 and air, as shown in the 3D view (a). A square lattice with period a, composed of oval-shaped air holes, is etched into the gold layer. The nanostructure is excited from the air side with right- or left-circularly polarized light (CPL) at either normal incidence or at an angle θ from the normal with light incident along the xz orientation. Figure 1(b) shows the 2D view of the unit cell in the xy plane, just above the glass. Each nanohole consists of two half-ellipses sharing a common major axis of length while having differing semi-axes r2, r3. We define the hole deformation as

and D is taken to be the symmetry-breaking parameter in this work.

Figure 1. Scheme of the metasurface studied in this work. (a) 3D view: a layer of gold (thickness t = 100 nm) on a glass substrate contains a square lattice of air holes with period a. (b) 2D view of the unit cell: each hole has oval shape made of two half-ellipses with a common axis of length and differing semi-axes .

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To analyze the optical response—specifically, absorption spectra and CD - we utilize the Bloch-mode scattering matrix approach implemented in the open-source software EMUstack. This method employs a finite element approach to compute Bloch modes in each 2D layer of the structure, while electromagnetic fields are expanded in plane waves in the outer layers and propagated through the layers using a scattering matrix formalism [40, 41].

For field profiles calculations, we employ EMUstack and 3D finite difference time domain (FDTD) method using Lumerical-Ansys commercial simulator. In Lumerical, we set perfectly matched layers in the z-direction. The two computational methods generally yield closely matching resonance energies, typically agreeing within  eV. Further discussion and examples of convergence tests are given in appendix A. The dielectric function of Au is taken from Lumerical database from [42]. This dielectric function is also used in EMUstack. For SiO2, we used , where n = 1.45.

Both methods enable the calculation of transmittance (T), reflectance (R), absorbance (A), and field profiles. We concentrate on the CD in absorption, which is defined in terms of absorbance for left- or right circular polarization (lcp, rcp) as

Each oval hole is invariant under mirror reflection in the yz plane, which is also a symmetry plane of the square lattice. Thus, the metasurface is invariant under mirror reflection in the yz plane and it does not show CD at normal incidence (no intrinsic chirality) or along the yz orientation. For oblique incidence along any orientation different from yz, the metasurface with the incident beam are in a configuration without mirror planes and a finite CD occurs (extrinsic chirality [43–45]). This effect is expected to be maximum for incidence along the xz orientation.

The Q-factor can be enhanced through proper optimization, especially when operating the plasmonic metasurface that sustains q-BIC modes in the infrared range of the spectrum at telecom wavelengths. For this reason, in this section, we explore the properties of the plasmonic q-BIC in the IR range. To achieve this, we investigate the behavior of the q-BIC resonance as a function of lattice constant a. We present in figure 2 the absorption spectra calculated at normal incidence (), for lattice constants ranging from 500 nm to 1500 nm. The radii are scaled as , , and , with deformation D = 0.2. We clearly see that as the lattice constant increases, the q-BIC resonance becomes narrower and sharper. Instead, the SPP peak gets weaker. At the largest lattice constant a = 1500 nm, the q-BIC dominates the absorption spectrum while the SPP is almost vanishing.

Figure 2. Absorption spectra of the metasurface studied in this work, for left circular polarization at normal incidence, for different lattice constants. Parameters: , , (deformation D = 0.2). The curves are vertically shifted by multiples of 0.1 for clarity, as indicated in the spectra for a = 600 and 1500 nm.

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We present in figure 3 the distribution of the electric field intensity in the xz-plane for three different lattice constants. The upper panels display the field on a linear scale, which reveals a progressive enhancement of the electric field near the edges of the metallic film as the lattice constant increases. To more clearly examine the field distribution within and around the metal, the lower panels show the same data plotted on a logarithmic scale. This log-scale visualization allows us to identify subtle variations in field intensity inside the metal. We observe that as the lattice constant increases, the electric field becomes increasingly localized at the corners and edges of the metallic layer, with minimal penetration into the bulk of the metal. This behavior suggests a trend toward stronger confinement at the surface, and a reduction in field intensity within the metal itself. Physically, this follows from the fact that the real part of the metal’s dielectric function becomes more negative with a reduction of the resonant frequency, approaching the behavior of an ideal metal. As a result of reduced penetration in the metal, dissipation losses will be reduced, since less electromagnetic energy is absorbed within the material. This is an important trend, and it will be studied in detail in section 4.

Figure 3. xz cut at y = 0 of for a unit incident field, for three different lattice constants, in linear scale (upper panels) and log scale (lower panels), for LCP at normal incidence. Parameters: , , , (deformation D = 0.2). Thin dashed lines indicate the metal regions.

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In figure 4 we show absorption maps for both right- and left-CPL, as well as the CD maps, again for  nm. These plots provide a comprehensive view of how the optical response of the metasurface evolves with increasing lattice constant. The angle of incidence θ is varied symmetrically between and 6∘ to capture the angular dependence. The prominent absorption peak that shifts to lower energy for finite angles () becomes sharper and more defined with increased lattice constants, indicating enhanced resonant behavior and improved coupling to the q-BIC mode. The CD maps, shown in the lower row of figure 4, reveal the asymmetry in the metasurface’s response as CD() = −CD(θ). This behavior is a direct consequence of the metasurface possessing a mirror symmetry in the yz plane and it highlights the effect of extrinsic chirality. The maps also show that for each lattice constant, the maximum CD occurs at small, non-zero θ, which increases as the lattice constant a increases. Quantitatively, the peak CD value rises from at a = 500 nm to at a = 1500 nm. This trend indicates a strengthening of the chiral response in the long-wavelength regime for the q-BIC resonance. Taken together with the reduction of electric field in the metal shown in figure 3, this suggests that the maximum CD values are limited by dissipative losses in the metal. Furthermore, these results demonstrate the robustness of the chiral response with respect to variations in both lattice constant and the angle of incidence.

Figure 4. Absorption for RCP (upper panels), LCP (middle panels) and CD maps (lower panels) as a function of incidence angle and energy, for three different lattice constants. Parameters: , , .

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We performed an in-depth study of the characteristics of q-BIC spatial profile for different lattice constants by analyzing the local enhancement of the electric field. Our focus was on left-handed CPL, considering two key excitation scenarios: normal incidence, and oblique incidence at the angle that maximizes the CD. So, figure 5 shows the squared modulus of the electric field within the xy cross-section near the bottom surface of Au film (nm). These fields are taken at the energies of maximum absorption of the q-BIC resonances: for normal incidence (panels (a) to (c)), and for oblique incidence (panels (d) to (f)), again for  nm. At normal incidence, some degree of field enhancement is observed for all three lattice constants, reflecting the weak but non-negligible coupling of the incoming light to the q-BIC mode. In particular, this coupling is weakest for a = 500 nm and becomes progressively more substantial as the lattice constant increases. This trend suggests that a larger periodicity empowers more effective coupling between the incident light and the q-BIC mode.

Figure 5. xy cut at z = 5 nm of for three different lattice constants, for a unit incident field with LCP at normal incidence (upper panels) and oblique incidence (lower panels), calculated at the energies of maximum absorption. Parameters: , , .

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Under oblique incidence, the trend is significantly more pronounced. As shown in figures 5(d)–(f), the electric field enhancement becomes appreciably stronger and the field is more localized at the interface with the metal, with the most considerable effect observed for a = 1500 nm. In particular, oblique incidence lifts the symmetry protection that usually prevents coupling to the q-BIC mode at , effectively unlocking the q-BIC resonance and allowing for enhanced absorption and near-field intensity. This enhanced coupling under oblique incidence aligns with the trends observed in the absorption and CD maps, showing that local field enhancement and resonance sharpening occur at the same time.

In figure 6, we present a comprehensive analysis of how key optical properties of the q-BIC evolve with increasing lattice constant: the Q-factor, the maximum local field intensity enhancement , and the maximum CD. The Q-factor and maximum intensity enhancement are evaluated under both normal and oblique incidence, while the CD is at the finite angles θ that maximize the peak CD value and which are also shown in panel (c), right scale.

Figure 6. Main properties of the q-BIC as a function of lattice constant: (a) Q-factor and (b) maximum local field enhancement at normal incidence and at the angle θ ≠ 0 of maximum CD, (c) maximum absorption CD and the corresponding angle θ, all evaluated at the q-BIC resonance energy. Parameters: , , .

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As shown in figure 6(a), the Q-factor of the q-BIC mode increases with the lattice constant at both normal and oblique incidence. This trend reflects the enhanced optical confinement and reduced radiative losses associated with larger periodicities, which better support the formation of high-Q factors in the IR. However, the Q-factors for oblique incidence are consistently lower than those at normal incidence. This reduction stems from the fact that oblique incidence breaks the symmetry protection that stabilizes the ideal BIC, allowing the mode to couple more efficiently with free-space radiation, thereby lowering the Q factor compared to normal incidence.

Figure 6(b) shows the corresponding maximum local field intensity enhancement of the q-BIC modes. The intensity enhancement increases with lattice constant at both normal and oblique incidence, showing the same trend of the Q-factor. This is an important finding, as the simultaneous increase of the two quantities effectively breaks the trade-off between field localization and losses. Interestingly, the field intensity under oblique incidence shows an even more dramatic increase with lattice constant, compared to normal incidence. This can be attributed to the efficient coupling to the q-BIC radiative mode, allowing more energy to be concentrated in localized hot spots, particularly near metallic edges and corners. A more quantitative explanation will be given later in section 4.

The peak CD values shown in figure 6(c) are achieved at the optimal angle of incidence for each lattice constant. These CD values are less than one because of dissipation losses, and they are reduced at longer resonance wavelengths. Indeed, the peak CD values increase with the lattice constant and the maximum value occurs for a = 1500 nm and equals 0.94, still limited by the residual dissipation losses at this lattice constant. Moreover, in appendix B we compare the maximum CD for q-BIC and SPP resonances, showing that the chiral properties of the q-BIC are much improved over those of the SPP, in a wide range of lattice constants.

To further investigate the trends in field enhancement and chiral properties, in figure 7(a) we show 1−CD, while in figure 7(b) we show the maximum and average field intensity at normal and oblique incidence, both as a function of the Q-factor, using logarithmic scale. It can be seen that 1−CD, confirming that the maximum CD of the chiral BIC is limited by losses. The scaling properties in figure 7(b) are more complicated, as the maximum field intensity increases as , while the average field intensity increases as . This nontrivial behavior requires a physical explanation. To this extent, in figure 7(c) we show cuts of the intensity distribution along the x direction for y = 0 and z = 5 nm, i.e. just above the interface with glass. It can be seen that, with increasing the lattice constant, the electric field intensity becomes more localized near the interface with the metal. Therefore, we can propose the following explanation for the scaling properties in figure 7(b). The average field intensity increases proportional to Q, just like in a Fabry–Perot resonator, because of the formation of a transmission resonance, i.e. due to a spectral effect. The maximum field intensity increases like Q2 because of the combination of a spectral and a spatial effect: beyond the formation of a narrow transmission resonance that makes an energy enter the resonator, the redistribution of the electric field intensity close to the metal/air sidewalls gives an additional factor growing when the resonance wavelength is increased towards the infrared.

Figure 7. (a) Scaling of 1-CD as a function of Q-factor: the points are calculated by EMUstack, while the solid line represents a fitting curve . (b) Scaling of maximum and average local field enhancement, at normal incidence and at oblique incidence at the angle of maximum CD: the points are calculated by EMUstack, while the solid lines represent fitting curves (for ) and (for ). (c) as a function of for y = 0 and , at normal and oblique incidence, for three different lattice constants (the curves are vertically shifted for clarity). Parameters: , , .

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We notice another puzzling result in figure 7(b): while the field intensity increases as a function of Q-factor both at θ = 0 and at θ ≠ 0, when going from θ = 0 to θ ≠ 0 the Q-factor decreases but the field intensity increases. Again, this behavior will be explained below in section 4.

In this section, we analyze the loss mechanisms of the q-BIC by separating radiative and nonradiative contributions to the losses. The procedure for extracting the radiative part of the loss consists in multiplying the imaginary part of the dielectric function of the metal by a reduction factor f and extrapolating to , as explained in appendix C.

In figure 8 we consider the simplest case, namely the q-BIC as a function of deformation, for a fixed lattice constant a = 500 nm at normal incidence. Figure 8(a) displays the total, radiative, nonradiative linewidths and it shows that t