Subwavelength aperture light diffraction controlled by resonant metagrating with local symmetry-breaking

Abstract

We present a novel approach to control light diffraction through a single subwavelength aperture (i.e., metallic slit) by designing and exploring a resonant metagrating with compound lattice containing two metaatoms. Such resonant metagrating supports two orthogonal local eigenmodes of opposite symmetry, and a lateral offset between inner metaatoms is introduced to break the lattice symmetry to reduce the orthogonality of the two local eigenmodes. We show that the lateral offset offers additional freedom for controlling aperture diffraction, producing both extraordinarily enhanced and extremely suppressed transmissions. The mechanism is that the resonant metagrating serves as a unidirectional nanocoupler that can efficiently convert the incident light into a surface plasmon wave, whose energy direction depends on the lateral offset. Our findings provide a platform for advanced nanophotonics, including high-efficiency optical sensors and enhanced nonlinear optical devices.

Graphical abstract

Keywords

surface plasmon polariton / bound state in the continuum / extraordinary optical transmission

1 Introduction

Light diffraction through a single aperture (e.g., metallic aperture and slit) is a fundamental phenomenon in optics [1−5], playing a crucial role in modern advanced nanotechnology, including nano-lithography [6, 7], fluorescence correlation spectroscopy [8], and optical trapping [9, 10]. Especially with the development of nanophotonics in recent years and the entry of nanocomponents used to control light into subwavelength scale, how to control the diffraction of light through nanoapertures or slits is a significant issue, which finds substantial applications in high-resolution imaging [11−13], biosensing [14−17], and optical communications [18]. To achieve this goal, various mechanisms and strategies have been proposed over the past decade, including Fabry−Pérot (FP) resonances [19, 20], surface plasmons coupling [21−23], symmetry-breaking nanostructures [24−27], and phase-gradient metasurfaces [28, 29]. Through these strategies, extraordinary optical transmission (EOT) beyond classical aperture theory can be achieved.

Here, we propose an alternative method to control the diffraction of light through a single metallic subwavelength slit, which is achieved by surrounding it with a compound lattice resonant metagrating [see Fig.1(a)]. Such a metagrating structure supports two local eigenmodes: (i) a propagating SPP mode and (ii) a bound state in the continuum (BIC) mode [30−34]. A lateral offset between inner metaatoms (i.e., air grooves) is introduced to break the lattice symmetry, which makes ideal BIC degenerate to an excitable quasi-BIC mode and reduces the orthogonality of the two local eigenmodes. We will demonstrate that the lateral offset offers an additional degree of freedom for controlling subwavelength aperture diffraction, resulting in both extraordinarily enhanced transmission and extremely suppressed transmission. The mechanism is that the resonant metagrating acts as a unidirectional nanocoupler, which can efficiently convert the incident light into surface plasmon polariton (SPP), and its propagation direction depends on the lateral offset. Notably, our achieved transmission efficiency of 0.48 significantly surpasses the conventional Bethe limit T\propto {\left(w_0/\lambda \right)}^4 for subwavelength single-slit transmission [35]. This breakthrough originates from manipulating the orthogonality of two local eigenmodes, which enables a unidirectional SPP coupling mechanism. This approach fundamentally circumvents traditional diffraction limits, establishing a new paradigm for extreme light confinement in nanoscale apertures. Our findings provide a platform for advanced nanophotonic applications, including high-efficiency optical sensors and enhanced nonlinear optical devices, paving the way for next-generation integrated photonic systems.

2 Model and results

To illustrate our idea, Fig.1(a) presents the schematic of the considered geometry, where a single subwavelength air slit in a silver plate is surrounded by two metagratings at the input interface, each covered by SiO₂ cladding. The slit width is w_0, the thickness of the SiO₂ layer/the silver plate is t₁ = 250 nm / t₂ = 500 nm. The metagrating is a periodic array of supercells [Fig.1(b)] containing two grooves (i.e., two metaatoms) with identical widths of w, but different depths denoted as d₁ and d₂, respectively. For geometric parameters, w = 50 nm, and d_2=d_1+\mathrm{\Delta }d with \mathrm{\Delta }d = 40 nm and d₁ being a variable. The permittivity of SiO₂ is {\epsilon }_d=2.25, and the permittivity of silver is {\epsilon }_m={\epsilon }_{\mathrm{\infty }}-{\omega }_p^2/\left[\omega (\omega +\mathrm{i}\tau )\right] [36]. A Gaussian beam of transverse-magnetic (TM) polarization (its magnetic field solely along the y-axis) is normally incident to the entire structure, and the working frequency is f = 550 THz (corresponding to the wavelength of \lambda =545\mathrm{n}\mathrm{m}). To meet the momentum matching condition, i.e., {\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}=k_0\sin {\theta }_{in}+mG, the period of the supercell is set as p=328\mathrm{n}\mathrm{m}, where k_0=2\pi f/c, G=2\pi /p, {\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}=k_0\sqrt{{\epsilon }_d{\epsilon }_m/\left({\epsilon }_d+{\epsilon }_m\right)}=1.68k_0 (being the wave vector of SPPs) and m=\pm 1 is the diffraction order.

Such compound metagratings are designed to support two local eigenmodes of opposite symmetry [see Fig.1(b)]. One eigenmode is an even-symmetric SPP mode, while the other is a BIC mode with theoretically infinite quality (Q) factor [37]. The BIC mode is an odd symmetric, and cannot be excited by external incidence source. These outcomes are verified by the photonic band diagrams of the compound metagrating, as shown in Fig.2(c) below. With the period (p) unchanged, a lateral offset {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} is introduced to break the mirror symmetry of the supercell, thereby transforming the ideal BIC into a quasi-BIC (QBIC). The lateral offset {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} represents the displacement of the second groove along the x-direction, as depicted in Fig.1(c), where the L/R subscripts represent the left/right metagrating; {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} > 0 defines the shift in the +x direction, and {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} < 0 is for the shift in the –x direction. Due to symmetry breaking, these two local eigenmodes, which are originally completely orthogonal, are non-orthogonal at all [see Fig.1(c)], which can be coupled to incident light at the same time, providing a physics for the realization of unidirectional SPPs excitation.

To illustrate the diffraction transmittance of a single slit encircling by the designed structure, we first examine the case of the metagrating with {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}=0, where the m=\pm 1 order diffraction occurs for normal incidence. It implies that bidirectional SPP excitation happens in each metagrating on both sides of the single slit. Fig.2(a) displays the diffraction transmittance with two parameters of the slit width and the groove depth at the operating frequency of f=550\mathrm{T}\mathrm{H}\mathrm{z}. Two distinct resonance regions are observed, corresponding to two different resonance mechanisms. The first is the SPP resonance in the metagrating (indicated by the yellow dashed line), which can be adjusted by varying the groove depth d₁. The second is the FP resonance occurring within the single slit (marked by the white dashed lines), whose characteristics depend on both the width and length of the slit. The EOT effect occurs around the intersection points of these two resonances, as marked by A (d_1=88\mathrm{n}\mathrm{m} & w_0=22\mathrm{n}\mathrm{m}) and B (d_1=88\mathrm{n}\mathrm{m} & w_0=68\mathrm{n}\mathrm{m}), respectively. Fig.2(b) plots the single-slit transmittance for different slit width for d_1=88\mathrm{n}\mathrm{m}. The resonance at point A exhibits a higher Q factor with a peak transmission of T = 0.26, whereas the resonance at point B exhibits a lower Q factor with a peak value of T = 0.32.

When {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}\ne 0, the symmetry of the metagrating is broken and the BIC mode transits into QBIC mode, thereby modifying the coupling process of incident light to SPPs. Fig.2(c) illustrates the evolution of two eigenmodes (the solid curves) and their Q factors (the dashed curves) with the lateral offset {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}. As |{\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}| increases, the eigenfrequencies of two modes separate gradually, accompanied by an increase in the Q_R of the SPP mode and a significant decrease in the Q_B of the BIC mode. As previously mentioned, the two eigenmodes are no longer perfectly orthogonal. Consequently, the anti-symmetric QBIC mode can interfere with the symmetric SPP modes, producing asymmetric excitations of SPPs. In these calculations, the Ohmic loss of metal is ignored. Fig.2(d) shows the numerically calculated SPPs excitation efficiency vs the lateral offset {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}, which is the ratio of the SPPs energy (at a distance of 5 μm from the center) to the incident energy. Obviously, as the lateral offset |{\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}| increases, the efficiency of SPPs exhibits an asymmetric feature. When {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}>0, the excited SPPs in both metagratings predominantly propagate in the +x direction. Conversely, when , the excited SPPs primarily propagate in the –x direction. The maximum efficiency of one-way SPPs reaches 52% when |{\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}|=9\mathrm{n}\mathrm{m}. This asymmetric SPPs propagation behavior highlights the pivotal role of the lateral offset {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} as a critical parameter for controlling the diffraction of light through a single slit, not just EOT effect.

Fig.3(a) shows the phase diagram of single-slit transmission as {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}} changes, where the horizontal axis is {\mathrm{\Delta }}_{\mathrm{L}}, while the vertical axis is {\mathrm{\Delta }}_{\mathrm{R}}. According to their signs, the phase diagram is divided into four distinct regions. Region IV represents the enhancement region, whereas Region I, conversely, corresponds to the suppression region. In Region I, at {\mathrm{\Delta }}_{\mathrm{L}}=-9\mathrm{n}\mathrm{m} and {\mathrm{\Delta }}_{\mathrm{R}}=9\mathrm{n}\mathrm{m} (i.e., point A), the diffraction transmission reaches its minimum value, i.e., T_min = 0.02, which is caused by the fact that the excited SPPs in both metagratings propagates outward from the single slit. This result is further confirmed by the magnetic field energy flow distribution in Fig.3(b), where the black arrows indicate the propagation direction of the energy flow. The Regions II and III are moderate regions, featured by the two excited SPPs propagating in the same direction. For a fixed {\mathrm{\Delta }}_{\mathrm{R}}({\mathrm{\Delta }}_{\mathrm{L}}), increasing the lateral offset {\mathrm{\Delta }}_{\mathrm{L}}({\mathrm{\Delta }}_{\mathrm{R}}) initially enhances the single-slit transmittance, followed by a subsequent decrease. This trend is consistent with the SPPs excitation efficiency as a function of lateral offset {\mathrm{\Delta }}_{\mathrm{L}}({\mathrm{\Delta }}_{\mathrm{R}}) shown in Fig.2(d). Fig.3(c) and (d) display the magnetic field distributions corresponding to point B (i.e., {\mathrm{\Delta }}_{\mathrm{L}} = 9 nm and {\mathrm{\Delta }}_{\mathrm{R}} = 9 nm) in Region II and point C (i.e., {\mathrm{\Delta }}_{\mathrm{L}} = –9 nm and {\mathrm{\Delta }}_{\mathrm{R}} = –9 nm) in Region III, respectively. In both cases, all excited SPPs propagate either to the right or the left (as indicated by white arrows), with only a portion of the energy flowing into the single slit. In Region IV, where {\mathrm{\Delta }}_{\mathrm{L}} > 0 and {\mathrm{\Delta }}_{\mathrm{R}} < 0, SPPs from both sides of metagratings propagate towards the single slit, thereby significantly enhancing the transmission of the single slit. In particular, at point D (i.e., {\mathrm{\Delta }}_{\mathrm{L}} = 9 nm and {\mathrm{\Delta }}_{\mathrm{R}} = – 9 nm), the transmission reaches its maximum value, i.e., T_max = 0.48. Fig.3(e) shows the corresponding magnetic field distribution, in which the energy flow, indicated by the black arrows, propagates towards the center, resulting in enhanced transmission.

The frequency response of the unidirectional SPPs and associated EOT effect has been explored, with obtained results as shown in Fig.4. The red curve in Fig.4 is for the case of the D point in the phase diagram. For comparison, the transmission spectra of both the single silt with symmetric metagrating (i.e., {\mathrm{\Delta }}_{\mathrm{L}/\mathrm{R}}=0) (the blue curve) and the single silt only (the gray curve) are presented. Compared with the case of the single slit alone, the bidirectional SPP excitations results in a narrower resonance with Q = 26, and the transmission efficiency is enhanced from 0.05 to 0.33 through SPPs channel coupling. Additionally, the symmetry breaking of the metagrating leads to asymmetric SPP excitations and one-way SPPs propagation, which further improves the transmission efficiency, with the maximum value reaching 0.5.

3 Conclusion

In conclusion, we have demonstrated that the compound metagrating can provide a way to control the diffraction transmission of light through a single subwavelength aperture. The compound metagrating consists of a supercell with two metaatoms, which supports two local eigenmodes of opposite symmetry, i.e., an even-symmetric SPP mode and an odd-symmetric BIC mode. We have shown that the light diffraction of single subwavelength slit can be freely controlled by altering the inner lateral offset between the two metaatoms, producing the transmission changing from the near zero to the significantly enhanced value of about 0.5. The mechanism is that the compound metagrating can act as an efficient coupler that transforms the incident light to unidirectional propagating SPPs, and this process can be controlled by local symmetry breaking of the metagrating. The designed structure can extend to three-dimensional structures such as gradient refractive index circular plates [38]. However, due to the polarization dependence of SPPs, their propagation direction may become more diffuse, which inevitably degrades the transmission efficiency. These results are promising in many applications, such as imaging [39, 40], planar lenses [41], and sensing [42, 43].

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