Near-field probing of the local density of optical states enhanced by bound states in the continuum in nonlocal metasurfaces

Abstract

Bound states in the continuum (BICs) are optical modes that remain decoupled from free-space radiation. Symmetry-protected BICs in metasurfaces offer powerful means to control light-matter interactions. A key property governing these interactions is the partial local density of optical states (PLDOS), which describes the number of electromagnetic modes available for a photon to occupy at a specific position, frequency and polarization. Here, we employ a terahertz near-field microscope with dual local probes to directly excite and detect quasi-BICs in finite metasurfaces that possess inversion symmetry, corresponding to a symmetry-protected BIC in their infinite counterpart. We observe a strong enhancement of the PLDOS associated with these modes. As the metasurface size increases, the quasi-BIC evolves into a true BIC, with the quality factor diverge while the PLDOS saturates. This establishes an upper limit for enhanced light-matter interaction by BICs. Our findings pave the way for on-chip metasurfaces with maximum light-matter interaction strengths.

Introduction

Bound states in the continuum (BICs) are optical modes that exhibit exceptional spatial localization without any radiation leakage to the far-field, despite their existence within the light cone^1,2. In the absence of material losses, BICs have infinite quality factors (Q) while being tightly confined, which provide an ideal platform for enhancing light-matter interactions^3,4, and improving the efficiency of photonic devices^5,6,7. To quantify the light-matter interaction in photonic systems, the local density of optical states (LDOS) serves as the most direct indicator. A high LDOS corresponds to regions where optical processes, such as energy exchange or spontaneous emission, are more likely to occur^8,9, with the concomitant increase in the decay rate of dipoles located in these regions^10,11. Thus, engineering the LDOS by resonant structures through tuning the Q-factor or mode volume (V) significantly influences fundamental processes in these artificial photonic systems^{12,13,14,15,16}. In a resonant system, such as nonlocal metasurfaces, the LDOS scales with Q/V, suggesting that the infinite Q-factor of BICs would lead to a divergent LDOS in metasurfaces and exceptional light-matter interactions^17,18. Therefore, the experimental study of BIC-related LDOS is not only fundamentally interesting but also of critical relevance for the development of potential on-chip applications.

The LDOS, ρ(r₀, ω), is defined by the number of optical states available for a point dipole to decay. Importantly, the dipole orientation strongly determines the spatial variations in its decay rate, especially in resonant subwavelength systems. Dipoles with different orientations interact with distinct combinations of polarized electromagnetic fields. Therefore, the partial (or projected) local density of optical states (PLDOS) becomes the more relevant quantity to describe light–matter interactions¹⁸. For instance, a dipole p aligned along a unit vector e_d interacts with modes whose electric fields are polarized in the same direction. At this level of detail, the PLDOS, denoted as ρ_p(e_d, r₀, ω), weighs each mode by the electric field projected along the dipole direction e_d. The PLDOS captures the directional dependence of the LDOS, enabling the decomposition into contributions from specific polarizations and spatial orientations. Owing to the metasurface formed by an array of resonating scatterers, the electric field generated by a point dipole can be split into two terms: E(r₀, ω) = E₀(r₀, ω) + E_s(r₀, ω), where E₀(r₀, ω) is the direct emission from the dipole, and E_s(r₀, ω) is the scattered field from the resonators¹⁹. The PLDOS enhancement, which is equal to the enhanced energy dissipation rate of the dipole, is determined by the ratio of the imaginary components of the scattered and intrinsic electric field amplitudes at the position of the dipole and along the dipole’s orientation^8,20:

$$\frac{P}{{P}_{0}}=1+\frac{{{{\rm{Im}}}}\{{{{{\bf{p}}}}}^{*}({{{{\bf{r}}}}}_{0},\omega )\cdot {{{{\bf{E}}}}}_{{{{\rm{s}}}}}({{{{\bf{r}}}}}_{0},\omega )\}}{{{{\rm{Im}}}}\{{{{{\bf{p}}}}}^{*}({{{{\bf{r}}}}}_{0},\omega )\cdot {{{{\bf{E}}}}}_{0}({{{{\bf{r}}}}}_{0},\omega )\}}.$$

(1)

Therefore, a direct measurement of the PLDOS enhancement requires determining the complex electric field at the exact position where the dipole source is located, highlighting the significant complexity of these measurements. Due to these limitations, LDOS enhancements are commonly indirectly obtained from measurements of the lifetime or decay rate of excited states^21,22,23. However, this approach is not applicable to BICs, as they do not couple to the far-field continuum. Instead, near-field techniques with specific excitation or detection methods, are required to investigate these dark modes and visualize their spatial field distribution^24,25,26. For example, Dong et al. used electron energy loss spectroscopy to probe the near-field response of optically inaccessible BIC modes in silicon nano-antenna arrays, revealing strong spatial confinement of the associated electromagnetic energy at nanometer resolution²⁶. Hoof et al. used a double THz near-field probe microscope to measure the near field amplitude and phase of BICs and reveal the symmetry protection that suppresses radiation losses²⁵. With the advancement of near-field techniques spanning from the visible to the terahertz (THz) regime, these methods have become widely utilized for controlling and probing electromagnetic waves in engineered photonic and metamaterial platforms^27,28,29,30. Among them, THz near-field microscopy integrated with dual microprobes offers a powerful approach for investigating diverse photonic phenomena, including spatial coherence and Fourier components of resonant modes^25,31. It should be noted that the experimental observation of a pure BIC is not possible, since the required symmetry protection is inevitably broken in real systems by finite size, fabrication imperfections, or probing.

In this work, we present the direct experimental evidence of the enhanced PLDOS associated with quasi-BICs in finite metasurfaces supporting pure BICs in their infinite counterparts. Combined with the theoretical calculation, we find a significant enhancement of PLDOS at the BIC frequency resulting from the collective response of the resonant array, and observe its saturation with increasing array size. The PLDOS enhancement of quasi-BICs in finite systems has thus an upper bound despite the divergence of the Q-factor due to the increase of the mode volume. Our measurements are validated by numerical simulations. Additionally, we reveal the spatial distribution of the PLDOS for different frequencies, highlighting the strong field enhancement and extreme confinement of the this mode to the surface. These results will guide future designs of finite-size metasurfaces with a maximum light-matter interaction strength.

Results

Symmetry-protected BICs are supported by structures that have inversion symmetry^2,32. The odd field symmetry of BICs mismatches with the even field of propagating modes in the continuum, leading to a full decoupling of BICs from the continuum^33,34. Here, we investigate a metasurface formed by a periodic array of Au rods with a periodicity of p = 300 μm, where one unit cell consists of two identical rods with a separation of d = 80 μm along the x direction, as shown in Fig. 1a. The length and width of each rod are 200 μm and 40 μm, respectively. The metasurface has been fabricated using standard lithography (details on the fabrication can be found in the “Methods” section). Given the designed dimensions, we expect a BIC mode emerging at a frequency of  ~0.395 THz as demonstrated in ref. ³⁵. This is supported by a theoretical analysis using a coupled detuned-dipole model, which shows that the imaginary component of the system’s eigenvalue vanishes at this specific frequency.

Fig. 1: Sample configuration and far-field transmission.

a Illustration of a unit cell of the metasurface and transmission measurements with changing incidence angles. The inset shows an optical microscope image of a unit cell in the fabricated metasurface, where the separation between the rods is d = 80 μm. b The left panel displays the THz transmission amplitude of the metasurface for incidence angles of 0^∘, 20^∘, 30^∘, and 40^∘. The right panel shows the corresponding relative transmission (Rt) spectra, normalized to the transmission at normal incidence. These normalized spectra highlight the emergence of a quasi-BIC mode, which becomes prominent under off-normal incidence. In these measurements, the THz electric field is polarized along the y-direction. c The calculated transmission spectra and extracted quality factor (Q-factor) of the quasi-BIC mode as a function of the incidence angle. The symbols represent the extracted quality factor (Q-factor) of the quasi-BIC mode, given by Q = f₀/Δf, where f₀ is the resonant frequency of the quasi-BIC mode, and Δf is the linewidth (half-width at half-maximum) of the quasi-BIC mode in the Rt spectra retrieved from Fano fitting. The solid lines represent fits to the extracted Q-factors using the relation Q ∝ 1/θ².

THz far-field measurements

To investigate the far-field properties of the metasurface and validate the presence of the BIC, we measured the amplitude transmission spectra T(ω, θ) through the sample at various angles θ of incidence with a far-field THz-TDS system (Menlo K15), and normalized them to the transmission of the bare substrate. The THz electric field was oriented parallel to the long axis of the rods (y direction), as depicted in Fig. 1a. At normal incidence, T(ω, 0) does not show any signature of the BIC (Fig. 1b). The broad resonance in the transmission spectrum at 0.495 THz corresponds to the even mode in the array resulting from the in-phase half-wavelength (λ/2) resonances along the long axis of the rods³⁵. A narrow feature appears in T(ω, θ) around 0.365 THz for off-normal incidence (θ = 40°). These findings align with the expectation that incident THz radiation cannot couple to the BIC at normal incidence, rendering it invisible to the far-field incident plane wave. However, the inversion symmetry is broken under oblique illumination and the BIC evolves into a quasi-BIC that can couple to the continuum. The right panel of Fig. 1b shows the transmission at θ = 20°, 30°, 40° normalized by the transmission at normal incidence, highlighting the leaky quasi-BIC resonance. Detailed measurements and simulations of the angle-dependent transmission and the quasi-BIC can be found in Supplementary Information (Fig. S1).

The off-normal excitation breaks the inversion symmetry of the system, causing the BIC to evolve into a quasi-BIC that becomes accessible from the far-field. As the angle of incidence increases, the resonance redshifts and broadens, indicating a reduction in Q-factor. These spectral changes are more clearly observed in the relative transmission spectra, normalized by the transmission at normal incidence, as shown in the right panel of Fig. 1b. The observed redshift and Q-factor reduction are qualitatively consistent with simulation results in Fig. 1c, obtained using the scuff-transmission method in SCUFF-EM (see Methods). In these simulations, the BIC in an infinite array of rod dimers appears at 0.395 THz and exhibits an infinite Q-factor. We note a slight mismatch between the experimental and simulated resonance frequencies. This mismatch arises from idealized assumptions in the simulations with a lossless, infinite, and homogeneous environment, which differ from real experimental conditions (a more detailed discussion is provided in Fig. S1 of Supplementary Information). Despite this discrepancy, the BIC framework remains an effective tool for the design of nonlocal metasurfaces.

THz near-field observation of the quasi-bound states in the continuum

The inversion symmetry of the metasurface prevents BICs from coupling to the continuum of diffraction modes^26,36, rendering them invisible in the far-field. Here, we employ two THz microprobes to locally excite and probe the THz near-field response of the metasurface, where the microprobes function as sub-wavelength photoconductive antennas, as demonstrated in ref. ²⁰. The probes locally perturb the metasurface symmetry, transforming the symmetry-protected BIC into a quasi-BIC with a finite radiative linewidth, thereby enabling coupling to and measurement of the otherwise inaccessible BIC. We stress that this local perturbation of the inversion symmetry takes place only at the excitation and detection spots, in contrast to most works in the literature on quasi-BICs, where the symmetry is broken in every unit cell. This makes our system sufficiently large arrays, where the finite size effects can be neglected, the closest possible experimental realization and characterization of a pure BIC. Due to the much smaller size of the photoconductive antennas compared to the wavelength of THz radiation (~λ/38 at the BIC frequency), the microprobe acting as a THz emitter can be approximated as a radiating point dipole. The other photoconductive antenna serving as the detector is placed at a fixed distance of 10 μm from the emitter, as shown in Fig. 2a. The details on alignment of microprobes can be found in Supplementary Information (Figs. S2–S4). This close proximity ensures that the emission and detection occur at the “same position” within a sub-wavelength distance as shown in (i) and (ii) of Fig. 2a. In the experiment, both THz antennas are oriented along the y-direction or the x-direction (along the long or short axis of the rods), fixing the orientation of the dipole moment of the THz emitter and the detected field component of the near-field. Therefore, the detected signal is proportional to the PLDOS, G_yy(r₀, r₀, ω) or G_xx(r₀, r₀, ω) (see next section), which represents the components of the dyadic Green function along the y- or x-directions at the position of the emitter, respectively. Based on this unique setup, we have previously investigated the PLDOS enhancement above gold, InSb, and quartz interfaces, achieving excellent agreement with theoretical calculations for a point dipole as a function of height and demonstrating that the microprobes do not alter significantly the PLDOS²⁰. In what follows, we simplify the notation by not expressing the spatial and frequency dependencies of the Green function.

Fig. 2: Experimental near-field setup and measured electric field enhancement of a quasi-bound state in the continuum (BIC).

a Illustration of the THz near-field setup, in which: (i) is a photograph of the sample during the measurement taken by a microscope camera, and (ii) is an illustration of the image to highlight the functional elements, the gold electrodes of the photo-conductive antennas on top of the low-temperature-grown GaAs substrate. The red circles (and the corresponding thick red lines in the main figure) represent the laser beams used for the photo-excitation of the LT-GaAs (see also Methods section). b Normalized peak value of the THz transients (E_peak) measured in this configuration and mapped across one unit cell of the metasurface. The dashed black boxes outline the two gold rods. c THz transients measured at the three positions marked in (b), corresponding to the edge of a rod (circle symbol), corner (diamond symbol), and edge (star symbol) of the unit cell. d THz intensity spectra ∣E(r₀, ω)∣² at the selected positions, obtained from the Fourier transformation of the THz transients in (c). The inset shows the normalized THz intensity spectrum measured on bare quartz ranging from 0.2 THz to 1.4 THz with a frequency resolution of 20 GHz.

Both microprobes are positioned at a height of 10 μm above the surface for spatial mapping of the PLDOS. The time-dependent THz electric field of the locally excited metasurface is spatially mapped on an x-y grid with a step size of 4 μm. While the line spread function of the near-field probes has been measured to be 20 μm (see Supplementary Information, Fig. S5), we chose a smaller step size for a detailed mapping. Figure 2b displays the distribution of the measured THz electric field normalized to the maximum across a unit cell, obtained from the peak value of time-dependent THz amplitude. As shown in this figure, the normalized THz electric field along the y-direction is enhanced at the top and bottom edges of the rods.

To analyze the near-field measurements in more detail, we extract the THz transients at each position. Fig. 2c displays the time-dependent THz electric-field amplitudes at the positions marked in Fig. 2b, which are located at the top edge of the rod, at the corner, and at the edge of the unit cell, respectively. The THz transient measured at the edge of the rod exhibits a pronounced oscillatory behavior with a long decay time of over 35 ps and an increased peak intensity. These oscillations are also present, although less pronounced, in the measurement at the corner of the unit cell. Fourier transformation of the THz transients provides the frequency-dependent intensity of the THz field ∣E(r₀, ω)∣², plotted in Fig. 2d for the three locations. The near-field measurements provide direct evidence of a greatly enhanced electric field at the edge of the rod, observed as the peak in the THz intensity centered at 0.395 THz. This peak, corresponding to the oscillations in the THz transients, originates from the field enhancement owing to the quasi-BIC mode. The peak has a finite line width, which results from material losses in the Au rods and the limited spectral resolution caused by the short time window of the THz transients measurements^37,38. Additionally, a high but relatively lower field enhancement is observed at the corner of the unit cell shown in Fig. 2d, while it is nearly absent at the edge where the quasi-BIC’s electric field is minimal. The enhancement of the field at the corner of the unit cell illustrates the interaction between neighboring unit cells in the array originating from the THz wave scattering by the Au rods, and highlights the importance of the extended metasurface for the formation of the BIC.

Enhancement of the partial local density of optical states

The dyadic Green’s function, $\overleftrightarrow{{{\bf{G}}}}$, is a second-rank tensor represented by a 3 × 3 matrix for the x-, y-, and z-field components. The trace of the imaginary component $\overleftrightarrow{{{\bf{G}}}}$ at the position of the source accounts for the total LDOS corresponding to the three possible orthogonal dipole orientations. The PLDOS is used to describe the LDOS of a dipole oriented along a principal axis^18,20. The THz microprobes used in the setup generate and detect linearly polarized broadband THz radiation³⁹. Therefore, the setup is particularly suited for measuring the PLDOS.

To gain insight into the PLDOS originating from a quasi-BIC in a metasurface of finite size, we calculated the frequency-dependent PLDOS enhancement with respect to vacuum, ${{{\rm{Im}}}}({G}_{yy}({{{{\bf{r}}}}}_{0},{{{{\bf{r}}}}}_{0},\omega ))/{{{\rm{Im}}}}({G}_{0}({{{{\bf{r}}}}}_{0},{{{{\bf{r}}}}}_{0},\omega ))$, for a point dipole oriented along the y-direction. The calculations were performed using the package SCUFF-EM, as detailed in the Methods section. Figure 3a, b demonstrates an enhancement near the top and bottom edges of the Au rods for both a single dimer and an array of 9 × 9 dimers, at their respective resonance frequencies. Moreover, this enhancement increases significantly with the number of unit cells in the array. Unlike local metasurfaces, where each unit cell or meta-atom responds independently to the incident field, nonlocal metasurfaces support optical modes that arise from coherent interactions among many identical neighboring units. This nonlocality is evident in the behavior of the quasi-BIC mode in the metasurface system studied here. Closer inspection of the frequency-dependent PLDOS enhancement at the position of the rod’s edge (see Fig. 3d) reveals a number of peaks around the frequency of the BIC mode for the infinite array, which increase in number and become sharper as the array size increases (Further details are presented in Fig. S6 of Supplementary Information.). These peaks are collective hybridized modes, formed by coherent coupling of localized resonances across the unit cell array. They manifest as standing wave patterns extending over the entire structure. The inset in panel (ii) of Fig. 3d shows a magnified view of the five distinct modes for the 9 × 9 array. A spatial map of the first three modes is shown in Fig. 3e. (The PLDOS distribution of other modes is presented in Fig. S7 of Supplementary Information.) These modes experience radiative losses due to the finite nature of the array. For a perfect, lossless and infinite system, the collective modes would resonate with infinite Q-factor, defining BICs with zero linewidth^40,41.

Fig. 3: Enhancement of the partial local density of optical states (PLDOS) by the quasi-bound state in the continuum (BIC) in finite-size systems.

Calculated and measured PLDOS enhancement P_yy/P₀ as a function of frequency and position along the y-direction at half of the width of one rod of the unit cell, indicated by the dotted line in the inset of (a). The calculated PLDOS enhancement for a single dimer (a) and for an array consisting of 9 × 9 dimers (b). c Measured PLDOS enhancement of the metasurface consisting of 84 × 84 unit cells. d The calculated (left y-axis, solid lines) and measured (right y-axis, open symbols with filled area underneath) frequency-dependent PLDOS enhancement at the edge of a rod (dashed horizontal lines marked in (a–c)) in arrays with different numbers of dimers N × N: (i) 84 × 84 (measurement only), (ii) 9 × 9, (iii) 7 × 7, (iv) 5 × 5, (v) 3 × 3, and (vi) 1 × 1. The peaks in the PLDOS enhancement for dimer arrays (N > 1) result from the collective nature of the BIC confined in the finite array. The inset in (ii) highlights the three lowest-frequency sharp peaks in the PLDOS enhancement spectrum of the 9 × 9 array, corresponding to the 1st, 2nd, and 3rd resonant modes. e Spatial distribution of the PLDOS enhancement for the 9 × 9 array at the respective PLDOS peaks, corresponding to the labeled peaks in the inset of panel (ii) in (d). f Calculated quality factor (Q-factor), and g PLDOS enhancement extracted from the lowest-frequency peak as a function of the number of dimers in the array (averaged over 10 positions near the rod edge, i.e., over 10 μm along the y-direction).The associated error bars represent the standard deviation calculated from the same 10 measurement points. The solid lines represent exponential fits to the data.

The THz near-field microscope allows us to experimentally determine the PLDOS enhancement along the same principal y-axis of the metasurface containing 84 × 84 unit cells. A bare quartz substrate, identical to the sample substrate, was used as the reference to normalize the measurements and isolate the optical response of the metasurface. As illustrated in Fig. 3c, we observe a significant PLDOS enhancement at the frequency of the quasi-BIC in the measurements, in agreement with the calculated results. Fig. 3d, panel (i), shows the measured frequency-dependent PLDOS enhancement at the edge of a central rod in the array. Rather than a series of sharp peaks compared to the results from calculations in panels (ii)–(v), we observe a broad peak in the measured results with a very large enhancement. This difference is due to the limited spectral resolution set by the range of the time-delay stage, which is used to acquire the THz transients. In addition, the differences between the measured and calculated values for each sample could be partly attributed to intrinsic material losses within the samples and refractive index mismatches at the interface. Peaks slightly broader than those in the simulations can be observed in the measurements of the PLDOS enhancement of arrays with sizes of 3 × 3 and 5 × 5 dimers, as illustrated in Fig. 3d, panels (iv) and (v). In addition, a very broad resonance with low PLDOS enhancement can be observed in the measurements and calculations of the single dimer (Fig. 3d, panels (vi)). This dependence of the peak width on the array size illustrates the nonlocal resonance of the metasurface due to the collective response of the dimers^16,42.

An intriguing question is why the calculated and the experimental PLDOS do not show a diverging behavior at the BIC frequency. A symmetry-protected BIC mode has an infinite lifetime and exists only in ideal systems that extend infinitely in at least one direction. In practice, the BIC arises from the collective resonances of constituent scatterers- in this case, individual dimers^26,35,43,44. However, the finite size of the arrays used in both calculations and experiments leads to radiation leakage at the boundaries, resulting in a finite lifetime; thus, the ideal BIC manifests as a quasi-BIC whose lifetime depends on the array size⁴¹. Still, the question remains about the relation between the BIC lifetime and the PLDOS. We have calculated the Q-factor, defined as the reciprocal of the mode’s lifetime, as a function of the array size. The Q-factor was extracted from the first peak (fundamental resonance) of the frequency-dependent PLDOS enhancement, which is described in Fig. S8. Figure 3f, g shows the calculated Q-factor and the PLDOS enhancement as a function of the total number of dimers in the array, respectively, indicating that while the Q-factor of the quasi-BIC diverges when the size of the metasurface is increased (as theoretically pointed out for finite sphere arrays in ref. ⁴¹), the PLDOS enhancement remains finite. The measured values can be found in Fig. S9. Intuitively, the PLDOS depends on the Q-factor and the mode volume V through the relation  ~Q/V. While the Q-factor increases with the size of the metasurface, the mode volume V also grows, which explains the saturation (or very slow growth) of the PLDOS enhancement for large metasurfaces despite the diverging Q-factor.

Spatial distribution of partial local density of optical states

Given that the PLDOS enhancement at the frequency of the quasi-BIC mode is supported by collective resonances in the dimers coupled across the array, it is crucial to investigate the spatial distribution of the PLDOS in the metasurface. This distribution defines the relevant mode volume of the quasi-BIC and the extent to which light-matter interactions can be modified by the metasurface^45,46. The calculated PLDOS enhancements are normalized to the corresponding values in vacuum at each frequency, whereas the experimental results are normalized to the PLDOS measured on the bare quartz substrate at the same frequency. From the calculations of the 9 × 9 dimer array, we obtained the spatial distribution of the PLDOS enhancement in the central unit cell and in the xy-plane at a height of 10μm above the metasurface, as plotted in Fig. 4a–c for 0.35 THz, 0.395 THz, and 0.45 THz, respectively. The enhancement of the PLDOS is strongly localized at the edges of the rods and at the frequency of the BIC for the infinite array (see Fig. 1c), 0.395 THz, rapidly decreasing beyond these areas and frequency. The greatest enhancement occurs in an area of 40 × 20 μm², which is almost three orders of magnitude smaller than λ². This result illustrates the extremely high degree of spatial confinement of the electric field with a small volume resulting from the origin of the BIC on the λ/2 resonances in the rods²⁵. This strong confinement may even be surprising, as the BIC mode is supported by collective resonances in the dimers coupled across the infinite array. For frequencies slightly away from the BIC frequencies, the small PLDOS enhancement also rapidly decreases with the spatial extent.

Fig. 4: Spatial maps of the enhanced partial local density of optical states (PLDOS).

a–c Calculated spatial distribution of y-component of the PLDOS enhancement, P_yy/P₀ in the central unit cell of a 9 × 9 array at the frequencies 0.35 THz, 0.395 THz, and 0.45 THz, respectively. d–f Measured spatial distribution of with PLDOS enhancement, P_yy/P₀, with the y-polarized near-field excitation and detection at 0.35 THz, 0.395 THz, and 0.45 THz, respectively. The calculated and measured spatial distributions are obtained at a height of 10 μm over the metasurface with an array size of 84 × 84 dimers.

Using the near-field microscope, we experimentally determined the spatial extent of the PLDOS by scanning the THz microprobes with a resolution of 4 μm across the xy-plane above the metasurface. We also observe a PLDOS enhancement that is highly confined to the edges of the rods, as shown in Fig. 4d–f. The measured PLDOS enhancement appears more pronounced at the upper ends of the rods than at the lower ones, which is caused by a small intentional misalignment between the THz emitter and receiver, as shown in Fig. 2a. This misalignment prevents cross-talk between emitter and receiver that would be caused by specular reflection of the laser radiation used to gate the photoconductive switches. In addition, the phase information between the two bars within a dimer has been demonstrated and discussed in Fig. S10 of the  Supplementary Information. For x-polarized excitation and detection, which determines P_xx/P₀, both calculations and experiments indicate that the PLDOS enhancement occurs along the long sides of the rods (cf. Fig. S11 of Supplementary Information).

Out-of-plane confinement of the enhanced partial local density of optical states

In addition to the in-plane spatial distribution of the PLDOS discussed so far, the high confinement of the quasi-BIC should also extend to the out-of-plane direction, confirmed in Fig. S12 of  Supplementary Information, which reflects the evanescent decay of the electric field enhancement toward the continuum⁴⁷. Experimentally, we determined this PLDOS confinement to the surface by positioning the microprobes at (90 μm, 250 μm) for y-polarization and (180 μm, 210 μm) for x-polarization, corresponding to the position in Fig. 2b, and gradually moving them in the vertical (z) direction away from the metasurface with a step size of 1 μm.

Figure 5a shows the THz transients for y- and x-polarized THz excitation and detection, respectively, in (i) and (ii). At 5 μm above the surface, the THz transients exhibit a strong under-damped oscillation, corresponding to the long lifetime of the quasi-BIC. As the distance from the surface increases, this oscillation rapidly weakens and becomes damped. At 35 μm above the surface, the oscillation is barely visible, illustrating the strong field confinement of the quasi-BIC to the surface⁴⁷.

Fig. 5: Out-of-plane confinement of the enhanced partial local density of optical states (PLDOS).

a THz transients and b THz amplitude spectra measured by the near-field microprobes E_y and E_x, respectively. The measured points are (90 μm, 250 μm) for y-polarization (y-pol) and (180 μm, 210 μm) for x-polarization (x-pol), corresponding to the position in Fig. 2b. c, d Measured PLDOS enhancements with near-field probing of y- and x-polarizations, P_yy/P₀ and P_xx/P₀, as a function of frequency and height. e, f Calculated P_yy/P₀ and P_xx/P₀ at positions (90 μm, 250 μm) and (180 μm, 210 μm), respectively, as a function of frequency and height. g Decay process of the PLDOS enhancement at the quasi-BIC frequency over h/λ₀, for (i) y-polarization, and (ii) x-polarization, where h is the height above the surface of the metasurface and λ₀ is the wavelength of the quasi-BIC mode. The calculated (lines) and measured (symbols) PLDOS enhancement are normalized by the maximum value.

To clearly show the decay of the quasi-BIC from the surface, the amplitude spectra are obtained by performing a Fourier transform of the THz transient signal, shown in Fig. 5a, after subtracting the contribution from direct dipole interaction. A clear peak at 0.395 THz, which corresponds to the quasi-BIC, is observed in the frequency spectra for both y- and x-polarized THz excitation and detection, respectively, shown in Fig. 5b, and the intensity of the peak rapidly decreases with increasing probe height above the surface.

The tight confinement of the quasi-BIC to the surface is even more clearly visible in the frequency-dependent PLDOS enhancement, which is experimentally obtained by Fourier transforming the THz transients measured on the metasurface and dividing them by the THz transients measured on a quartz substrate, as shown in Fig. 5c, d for both polarizations. The PLDOS enhancement reaches a maximum at the BIC frequency of 0.395 THz for both polarizations and is confined within a height of 20μm from the surface. The equivalent calculated PLDOS enhancements, P_yy/P₀ and P_xx/P₀, are shown in Fig. 5e, f and confirm the experimental results.

The spatial extent of the quasi-BIC mode in the xy-plane can be further quantified by examining the normalized PLDOS at the frequency of the quasi-BIC mode as a function of the relative height, h/λ₀, as displayed in Fig. 5g. The enhancement of the PLDOS in such a metasurface occurs only within a distance of λ₀/20, demonstrating strong out-of-plane confinement of the quasi-BIC. This high spatial confinement, evidenced by the PLDOS enhancements (shown in Figs. 4 and 5), provides strong evidence of the potential of symmetry-protected BICs in metasurfaces to enhance light-matter interaction. This can be leveraged for highly integrated on-chip applications such as sensing, nonlinear optics, and lasing.

Discussion

BICs are a unique class of optical modes in photonic systems with theoretically infinite lifetimes in idealized systems, arising from destructive interference or symmetry-protection mechanisms. These dark modes remain decoupled from radiative channels while being highly localized to the photonic system, resulting in an exceptionally high PLDOS. The spatial distribution of the PLDOS, often confined to subwavelength regions, underscores the ability of BICs to concentrate optical energy in small mode volumes. Furthermore, the collective nature of quasi-BICs in finite metasurfaces highlights their sensitivity to size and boundary effects, where the interplay of mode confinement plays a crucial role in determining the achievable PLDOS enhancement. While our study focuses on the influence of array size on PLDOS enhancement with fixed structural parameters, we note that other parameters, such as the gold film thickness, can also significantly affect the PLDOS. This occurs through its impact on material losses, field confinement, and collective mode formation. These characteristics make BICs and quasi-BICs not only a fundamental topic in the study of optical modes, but also a powerful platform for designing photonic devices with tunable emission, absorption, and light-matter interaction strengths.

In this work, we have used two THz microprobes positioned at a subwavelength distance from each other to locally excite and probe the complex electric field distribution of a quasi-BIC in a metasurface with inversion symmetry formed by an array of Au dimer rods. The infinite-size counterpart of these metasurfaces supports symmetry-protected BICs. The near-field probes allow direct measurements of the partial local density of optical states of quasi-BICs associated with finite metasurfaces, which was found to be significantly enhanced at the edges of the rods, extending over the unit cell due to the collective resonant field enhancement. Numerical calculations showed the saturation of the PLDOS with the size of the array at moderate sizes of a few unit cells, despite the increase in the Q-factor for larger arrays. The increase in Q-factor in symmetry-protected quasi-BICs is unavoidably linked to the increase in the mode volume due to the array size and the nonlocal character of these modes, which leads to the saturation of the ratio Q/V that defines the PLDOS enhancement. Furthermore, we theoretically and experimentally investigated the spatial confinement to the surface of the PLDOS at the BIC frequency for the infinite array, revealing a strong confinement accompanied by a large enhancement.

Owing to the scale invariance of Maxwell’s equations, the design principles and insights developed here for terahertz metasurfaces can be naturally extended to the infrared and visible regimes. In particular, the ability to engineer quasi-BIC modes and enhance the PLDOS opens up promising avenues for controlling light–matter interactions in nanophotonic systems. This has important implications for photonic and quantum technologies, such as the enhancement of the emission efficiency of single-photon sources, the tailoring of spontaneous emission rates in cavity quantum electrodynamics, or the improvement of nonlinear optical responses in integrated photonic platforms.

Methods

Sample fabrication

The gold metasurfaces were fabricated on a 2.5 × 2.5 cm² quartz substrate using a standard lithography process. The AZ 5214 UV photoresist was spin-coated on the substrate with a thickness of 2 μm. The exposure was done with a hard mask by UV lithography (MA6 Mask Aligner). Subsequently, the exposed sample was developed in an AZ 726 MIF solution for 1 min. The metal film was prepared by the electron beam evaporation of a 5-nm-thick titanium film as the adhesion layer and an 80-nm-thick gold film. The lift-off process of the samples was done with an acetone solution and the samples were cleaned with ultrapure water.

PLDOS calculations

Numerical calculations of the PLDOS were carried out with the Surface Current/Field Formulation of Electromagnetism (SCUFF-EM), which is a free, open-source software implementation of the boundary-element method (or the “method of moments") of electromagnetic scattering⁴⁸. In particular, we make use of the tool SCUFF-LDOS for computing the electromagnetic LDOS at points inside or outside compact or extended material bodies. Actually, it computes the scattering part of the dyadic Green’s functions of the geometry in question. For nonperiodic geometries, SCUFF-LDOS does six scattering calculations for each (x, ω) point–specifically, in which the incident field is the field of an electric or magnetic point source oriented in each of the three Cartesian directions. The same geometrical parameters as in the experimental measurements were used, but gold rods were considered as perfectly planar rectangles embedded in a uniform medium with n = 1.55. This refractive index is the average of those of air and the supporting quartz substrate, n = 2.1. This approximation is known to yield consistent results except for very close to the interface⁴⁹.

THz near-field emission and detection setup

The THz near-field setup consists of two commercially available THz near-field microprobes (TeraSpike TD-800-X-HR-WT, Protemics GmbH). The tip of the microprobes is made of low-temperature-grown GaAs (LT-GaAs), with two microstructured electrodes that define a photo-conductive antenna. A short laser pulse with a duration of 100 fs and a central wavelength of 780 nm is used to generate and probe the THz amplitude with a delay stage as a function of time, i.e., the THz transient pulse, by exciting the electrically biased emitter and the unbiased detector, respectively, as shown in Fig. 2a. Its working principle is the same as the standard far-field THz-TDS, which is a phase-sensitive technique that retrieves the transient THz electric field E(t) as a function of time t⁵⁰. By applying a Fourier transform to E(t), the complex THz electric field E(ω) can be obtained across a broad frequency range. The frequency resolution of the terahertz near-field time-domain spectroscopy (THz-TDS) system is approximately 20 GHz, determined by the 50 ps time window from the delay stage. The size of the THz source, defined by the emitter microprobe, is  ~20 μm, which is much smaller than the period of the metasurface (300 μm) and the wavelength at the frequency of the quasi-BIC (λ₀ = 760 μm). Therefore, we approximate the source as a point dipole.

The discriminating advantage of polarization-sensitive microprobes over a standard far-field setup lies in their ability to measure the field component E(t) along the orientation of the detector microprobe at the position r₀ where it was emitted (within a subwavelength distance given by the separation between the emitter and detector microprobes), thus allowing the determination of the fully vectorial complex field E(r₀, ω)³⁹. Such a near-field THz-TDS system is ideally suited to experimentally determine the PLDOS, which corresponds to the principal components of ${{{\rm{Im}}}}(\overleftrightarrow{{{\bf{G}}}}({{{{\bf{r}}}}}_{0},{{{{\bf{r}}}}}_{0},\omega ))$, defined by the orientation of the dipole moment of the emitter microprobe²⁰.

The PLDOS of the metasurface is spatio-temporally resolved at every point via a raster scan (step size: 4 μm) with sub-THz-cycle temporal resolution. Fast Fourier transforms of the THz transients at each position provide the complex spectral information with the real and imaginary components of the THz field. Owing to the limited scan range of the time-delay stage, zero padding is applied to the time-domain data prior to the Fourier transformation to enhance the frequency spectral resolution in the frequency domain. The measured PLDOS enhancements P_yy/P₀ and P_xx/P₀ are retrieved from the imaginary field component of the measured THz electric near field on the array, ${{{\rm{Im}}}}(E({{{{\bf{r}}}}}_{0},\omega ))$, under y- or x-polarization of the near-field emitter and detector, normalized by the measurements on the underlying quartz substrate, ${{{\rm{Im}}}}({E}_{0}({{{{\bf{r}}}}}_{0},\omega ))$.