1. Introduction

The interaction between nanostructures and light fields is a highly consequential research topic in contemporary science, with vast potential for application in diverse fields such as optical imaging [1], nano-holography [2], and optical sensing [3–5]. Precise manipulation of light absorption, transmission, and scattering properties at the nanoscale [6–9], particularly in terms of scattering directionality [10–13], has emerged as a key research direction in these fields. Currently, there are diverse strategies for achieving unidirectional scattering, ranging from structural design to material property manipulation, with proposals based on metal [14–16], dielectric [17–23], and hybrid nanoantennas [24]. Resonance modes in metal antennas are typically dominated by electric dipoles [25], necessitating more complex designs for unidirectional scattering and being limited by significant Ohmic losses. In contrast, high-refractive-index dielectric materials, such as silicon (Si), germanium (Ge), and gallium arsenide (GaAs), while maintaining low optical losses [26], exhibit a rich set of electromagnetic response modes [27,28]. Relying solely on a single silicon particle can achieve control over the directionality of scattering by satisfying the Kerker conditions (the phase and amplitude relationship between electromagnetic multipole moments) [29]. Nevertheless, the predominant approach for achieving unidirectional scattering relies on the interference between electric multipoles [15,16] or between electric and magnetic multipoles [17–24], while investigations concerning magnetic multipoles interference remain scarce.

In this context, the use of cylindrical vector beams (CVBs) has opened up a new dimension in the interaction between nanostructures and light [30]. CVBs, characterized by spatially varying polarization states [31–34], are capable of producing pronounced longitudinal electromagnetic components upon being subjected to tight focusing [35–37]. They can flexibly excite various resonance modes of nanostructures, including electromagnetic dipole modes oriented along the optical axis [38], toroidal modes [39], anapole modes [40], and Fano resonances [41–44]. Notably, the tightly focused azimuthally polarized beams (APBs) can selectively excite magnetic resonances in gold split-ring structures, inducing Fano resonances dominated by magnetic dipoles [42,43]. This flexible modulation of resonance modes paves the way for various innovative applications [45–48], particularly in the field of displacement sensing. Transverse unidirectional scattering based on the interaction between CVBs and nanostructures has rapidly become a focal point of research [49–57], particularly for on-chip integrated transverse micro-displacement measurement. However, studies on longitudinal displacement sensing based on this mechanism are underreported. Although traditional optical-based longitudinal displacement measurements are relatively mature, their complex structures and low on-chip integration limit further development [58].

To address these challenges, our research delves into the interaction between APBs and a silicon nanodimer comprising two non-concentric nanorings. We propose a scheme to selectively excite magnetic dipoles and quadrupoles within the silicon structure, leading to Fano resonances and transverse unidirectional scattering dominated by magnetic multipoles. Additionally, we further explored the potential for sensing longitudinal displacements. This investigation focused on monitoring alterations in the directionality of this unidirectional scattering, which we found to be highly correlated with the longitudinal position of the nanodimer. Our work not only provides innovative solutions for magnetic multipole-dominated unidirectional scattering but also offers new perspectives for resolving the existing challenges of on-chip integrable longitudinal displacement sensing.

2. Geometry and theoretical model

The structural scheme proposed in this paper is demonstrated in Fig. 1(a), in which an azimuthally polarized beams (APBs) is tightly focused by a focusing lens with high numerical aperture (NA = 0.9), subsequently interacting with a silicon nanodimer situated at the focal plane (z = 0 nm). The nanodimer is precisely aligned such that its internal nanoring's center coincides with the center of the focal point (origin), where the polarization singularity of the APBs is situated. Figure 1(b) displays the nanodimer's structural representation in both three-dimensional and two-dimensional Cartesian coordinates, detailing the associated geometrical parameters. The dimer features a pair of non-concentric nanorings, each with a uniform height of H = 180 nm. Specifically, the internal nanoring's radii are R₁ = 70 nm (inner) and R₂ = 170 nm (outer), while the external nanoring's radii are R₃ = 290 nm (inner) and R₄= 365 nm (outer). The centers O₁ and O₂ of the inner and outer rings, respectively, exhibit an offset in the x-direction, which is denoted as D = 45 nm. Figure 1(c) illustrates the electromagnetic field distribution at the focal plane of the APBs, offering an intuitive visualization of the silicon nanodimer's position within the optical fields. The spatial distribution of the APBs is calculated using the vector diffraction theory [35,36], with white arrows in the figure indicating the spatial polarization direction. Results reveal that the tightly focused APBs generates a transverse electric field E_t with azimuthal polarization, a transverse magnetic field H_t with radial polarization, and a pronounced longitudinal magnetic field H_z polarized along the z-axis. The precise locations of the nanodimer relative to the optical fields are indicated by black dashed lines. Notably, the intensity distribution indicates that the amplitudes of the transverse electromagnetic fields are minimal in the central region, whereas the longitudinal magnetic field forms a distinct focusing spot. Consequently, in the vicinity of the dimer's central region, H_z is the predominant component. Furthermore, all full-wave simulations are performed in the COMSOL Multiphysics, utilizing dielectric constant data for the silicon material from Palik's Handbook of Optical Constants [59]. Our investigation commences with the scenario in which the dimer is embedded in a homogeneous medium with a dielectric constant of ${\varepsilon _d} = 1$.

 

Fig. 1. Interaction between tightly focused azimuthally polarized beams (APBs) and a silicon nanodimer. (a) Schematic of the APBs’ z-axis propagation, focused by an objective lens with a numerical aperture (NA) of 0.9. (b) Diagram illustrating the nanodimer structure, comprising two non-concentric nanorings with a uniform height of H = 180 nm. The inner ring features radii of R₁ = 70 nm (inner) and R₂ = 170 nm (outer), while the outer ring has radii of R₃ = 290 nm (inner) and R₄ = 365 nm (outer). The centers O₁ and O₂ of the inner and outer rings, respectively, exhibit an offset in the x-direction, which is denoted as D = 45 nm. (c) Theoretical field intensity distribution on the focal plane of APBs, with the black dashed line indicating the relative position of the nanodimer within the beams.

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To elucidate the underlying interaction mechanisms of the nanodimer system, we separately analyzed the effects of the interaction of the inner and outer nanorings, as well as the entire nanodimer, with the APBs. Quantitative analyses of the scattering intensity, near-field, and current density distributions under APBs excitation are conducted and are depicted in Fig. 2. The scattering spectra for the individual inner ring, the integrated dimer, and the outer ring are sequentially presented in Fig. 2(a), revealing distinct resonance features. Employing the principles of hybridized models [60], an isolated inner nanoring exhibits a narrow magnetic dipole resonance at approximately 1200 nm ($ {|{{\omega_{ - ,HN1}}} } \rangle $), excited by the longitudinal magnetic field. Conversely, the APBs excites the isolated outer nanoring to a broader resonance at approximately 1500 nm (${ {|{{\omega_{ - ,HN2}}} } \rangle _{(2 )}}$). The offset in outer nanoring's position induces a pronounced magnetic quadrupole resonance near 800 nm ($ {|{{\omega_{ - ,HN2}}} } \rangle_{\textrm{(1)}}$). Interactions between the nanorings yield resonance coupling, evident in the dimer's scattering intensity, which manifests split resonance peaks at 840 nm, 1010 nm, 1090 nm, and 1600 nm (${ {|{\omega_ -^ + } } \rangle _{(1 )}}$, ${ {|{\omega_ -^ + } } \rangle _{(2 )}}$, ${ {|{\omega_ -^ + } } \rangle _{(3 )}}$, $ {|{\omega_ -^ - } } \rangle $, respectively), alongside a distinct spectral dip at 1140 nm, indicative of a magnetic Fano resonance. Figure 2(b) portrays the resonance modes at these wavelengths, highlighting the associated local field H_z and current densities (black arrows). The H_z of Resonance ${ {|{\omega_ -^ + } } \rangle _{(1 )}}$ demonstrates distinct different polarities between the inner and outer regions of the outer ring, indicative of magnetic quadrupole characteristics. The emergence of ${ {|{\omega_ -^ + } } \rangle _{(1 )}}$ results from the magnetic dipole-quadrupole resonant interactions, specifically, the hybridization of modes $ {|{{\omega_{ - ,HN1}}} } \rangle $ and ${ {|{{\omega_{ - ,HN2}}} } \rangle _{(1 )}}$. Then, Resonances ${ {|{\omega_ -^ + } } \rangle _{(2 )}}$ and $ {|{\omega_ -^ - } } \rangle$ correspond to same H_z polarities within each ring, with current densities forming intraring vortices. The emergence of ${ {|{\omega_ -^ + } } \rangle _{(2 )}}$ and $ {|{\omega_ -^ - } } \rangle $ results from the magnetic dipole-dipole resonant interactions, specifically, the hybridization of modes $ {|{{\omega_{ - ,HN1}}}} \rangle $ and ${ {|{{\omega_{ - ,HN2}}} } \rangle _{(2 )}}$. Moreover, the ${ {|{\omega_ -^ + } } \rangle _{(3 )}}$ peak displays opposite H_z polarities across the outer ring and divergent current density vortex directions, signaling a pronounced magnetic quadrupole resonance characteristic. Therefore, the emergence of ${ {|{\omega_ -^ + } } \rangle _{(3 )}}$ results from the magnetic dipole-quadrupole resonant interactions, specifically, the hybridization of modes $ {|{{\omega_{ - ,HN1}}}} \rangle $ and ${ {|{{\omega_{ - ,HN2}}} } \rangle _{(1 )}}$.

 

Fig. 2. Hybridization models for nanorings. (a) Scattering spectra characteristics resulting from the interaction of a single inner nanoring, the nanodimer, and a single outer nanoring with APBs. (b) Distribution of the longitudinal component of the magnetic field for different coupling modes: dipole-quadrupole anti-bonding mode (840 nm), dipole-dipole anti-bonding mode (1010 nm), dipole-quadrupole anti-bonding mode (1090 nm), dipole-quadrupole Fano resonance mode (1140 nm), and dipole-dipole bonding mode (1600 nm). The direction of current density is indicated by black arrows.

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The formation of the Fano dip is scrutinized via the bonding and anti-bonding models inherent to hybridization models. The anti-bonding mode is prominent at ${ {|{\omega_ -^ + } } \rangle _{(1 )}}$ and ${ {|{\omega_ -^ + } } \rangle _{(2 )}}$, with opposing current density vortices in the inner and outer rings. ${ {|{\omega_ -^ + } } \rangle _{(3 )}}$ is characterized by bonding mode on one side and a stronger anti-bonding mode on the other, with the latter prevailing in intensity. These anti-phase multipole moment oscillations within the nanorings culminate in a lower-energy radiation mode. In contrast, the bonding mode at $ {|{\omega_ -^ - } } \rangle $, where vortex directions align, leads to a more intense radiation mode. The interplay of these modes within the Fano resonance bandwidth results in destructive interference and the subsequent pronounced spectral Fano dip.

In the following study, we employed a scattering multipole decomposition approach that beyond the long-wavelength approximation [61,62] to quantitatively analyze the scattering spectrum arising from interactions involving inner ring-APBs, outer ring-APBs, non-concentric nanodimer-APBs, and concentric nanodimer-APBs, respectively. The computational outcomes are depicted in Fig. 3. In the single inner ring system, the scattering spectrum is predominantly governed by magnetic dipole (MD), as illustrated in Fig. 3(a). Within the single outer ring system, the scattering spectrum exhibits a decomposition into MD and magnetic quadrupole (MQ), as depicted in Fig. 3(b). Notably, around 800 nm, MD prevail, whereas around 1600 nm, magnetic dipoles completely dominate. Additionally, within the non-concentric nanodimer system, multipole decomposition analysis delineates the scattering spectrum into contributions from MD and MQ, as illustrated in Fig. 3(c). The contribution of the total electric dipole (TED), represented by the black curve, was so diminutive that it could be disregarded. These results reinforce the understanding that the interactions within the nanodimer system are purely magnetic multipolar in nature.

 

Fig. 3. Multipole decomposition of scattering spectrum. Normalized scattering intensity and contributions from multipole moments: (a) single inner ring system, (b) single outer ring system, (c) non-concentric nanodimer system, and (e) concentric nanodimer system. These include total electric dipole (TED), magnetic dipole (MD), electric quadrupole (EQ), magnetic quadrupole (MQ), and their sum. Dashed line represents the overall value from full-wave simulation. (d) and (f) are specific multipole moments contained in the magnetic dipole and magnetic quadrupole within non-concentric nanodimer system and concentric nanodimer system, respectively.

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Subsequently, we focus on analyzing the benefits of the proposed non-concentric nanodimer. In the concentric nanodimer system, the scattering spectrum can be decomposed into MD and MQ contributions, as illustrated in Fig. 3(e). Significant disparities can be observed in the MD between the concentric and single ring systems (Fig. 3(a) and 3(b)), whereas the MQ closely resembles that of the outer ring system (Fig. 3(b)). This underscores the predominant influence of the concentric system on MD, contrasting sharply with the non-concentric configuration, which affects both MD and MQ (Fig. 3(c)). Further examination of the specific multipole moments involved in the concentric system is presented in Fig. 3(f), where the moments are resolved into MD (0, 0, m_z) and MQ (M_xx, M_yy, M_zz) components. The fundamental disparity between concentric and non-concentric systems lies in the effective excitation of the crucial component $i{k_d}/6{v_d}\ast {\textrm{M}_{xz}}\; $ (Fig. 3(d) and 3(f)). Notably, among these multipoles, only the scattering patterns of ${\textrm{m}_z}/{v_d}$ and $i{k_d}/6{v_d}\ast {\textrm{M}_{xz}}\; $ have components along the x-axis [61]. Thus, to achieve transverse unidirectional scattering along the x-axis, employing a non-concentric structure is particularly crucial.

Hence, we have meticulously elucidated the theoretical model for the non-concentric dimer system, discussing the origins of the split-ting in the scattering spectrum and the occurrence of pure magnetic Fano resonances. This provides a solid foundation for further exploration of the practical applications of this model.

3. Results and discussion

We present a detailed discussion of the nanodimer in two distinct scenarios: embedded within a homogeneous medium and placed upon a silica substrate, exploring its applications in transverse unidirectional scattering and longitudinal displacement sensing.

3.1 Homogeneous medium

Initiating with the homogeneous medium (${\varepsilon _d} = 1$), this section delineates the occurrence of transverse unidirectional scattering driven by magnetic multipoles, resulting from non-concentric nanodimer-APBs interactions (refer to Fig. 4). The far-field radiation patterns of ${\textrm{m}_z}/{v_d}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$, which theoretical models suggest to dominate, exhibit odd and even symmetries along the x-axis, respectively (Fig. 4(a)). Consequently, coherent extinction (amplification) occurs in the negative x-direction, and coherent amplification (extinction) in the positive x-direction when their intensities are equal and in-phase (out-of-phase). In this sense, achieving unidirectional scattering along the x-axis is feasible. For a comprehensive validation, we encompass all electromagnetic multipole moments, establishing their linkage with far-field radiation as expressed in Eq. (1) [61,62]:

Here, n is the unit vector along r, p and m are the electric and magnetic dipoles, Q and M are the electric and magnetic quadrupoles, and ${v_d} = c/\sqrt {{\varepsilon _d}} $ and ${k_d} = k\sqrt {{\varepsilon _d}} $ are the speed of light and the wavenumber in a medium with dielectric constant ${\varepsilon _d}$, respectively. In this model, only

 

Fig. 4. Transverse unidirectional scattering driven by magnetic multipole moments. (a) Theoretical schematic illustrating transverse unidirectional scattering induced by magnetic quadrupole moment ($\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$) and magnetic dipole moment (${\textrm{m}_z}/{v_d}$). (b) The black curve depicts the phase difference ratio to π between ${\textrm{m}_z}/{v_d}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$, while the red curve represents the amplitude ratio of $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ to ${\textrm{m}_z}/{v_d}$. Both ratios equal 1 at a wavelength of 1130 nm. (c) A 3D plot and planar projection demonstrate transverse far-field scattering at a wavelength of 1130 nm. The black curve corresponds to the theoretical calculation, while the red dashed line represents the simulated result.

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MD (m_x, 0, m_z) and MQ (M_xx, M_yy, M_zz, M_xz) components are operative, simplifying Eq. (1) upon their insertion:

Here r, θ, φ are the spherical coordinates. Considering only the far-field radiation in the -x (θ = π/2, φ = 0) and + x (θ = π/2, φ = π) directions simplifies the expression further:

Equations (3) and (4) corroborate that the radiation is solely contingent on ${\textrm{m}_z}/{v_d}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$. The ratio of their amplitudes (red curve) and the difference in their phases (black curve) are shown in Fig. 4(b). At the wavelength of 1130 nm, the simultaneous satisfaction of the phase-matching and amplitude-matching criteria enables the occurrence of unidirectional scattering in the negative x-direction. Further elucidated in Fig. 4(c) through three-dimensional and planar projections (with black lines representing theoretical calculations and red dashed lines indicating simulation results), the nanodimer-APBs interactions at 1130 nm culminates in superior transverse unidirectional scattering driven by magnetic multipoles (magnetic dipole-quadrupole), an effect rarely reported in related works. In the x-y plane, theoretical calculations incorporating only dipoles and quadrupoles show good concordance with simulation outcomes; however, deviations emerge in the x-z plane as displacement from the x-axis increases. These discrepancies are likely due to influences from higher-order multipoles.

Further analysis is undertaken to delineate the distribution of current density within nanodimer during unidirectional scattering, as depicted in Fig. 5. Theoretical models predict distinct distributions associated with m_z and M_xz moments (Fig. 5(a)). Here, the m_z distribution forms a single vortex-like pattern on the x-y plane, whereas the M_xz configuration consists of two vortices with opposite circulations. Horizontal cross-sections of the nanodimer at -60 nm, 0 nm, and +60 nm in the x-y plane (Fig. 5(b)), marked by red dashed lines, delineate the directionality of vortex rotation. Notably, at -60 nm, a conspicuous m_z pattern emerges, presenting a singular vortex in both the inner and outer rings. Conversely, at 0 nm and +60 nm, the current density distinctly embodies M_xz features, with the currents in the inner and outer nanorings jointly constituting two counter-rotating vortices. These observations affirm that the offset of the external ring engenders asymmetric coupling across the nanodimer, invigorating the typically magnetic quadrupole mode.

 

Fig. 5. Distribution of current density within the nanodimer at the wavelength of 1130 nm. (a) Theoretical distribution of current density for magnetic dipole moment (m_z) and quadrupole moment (M_xz). (b) Distribution of current density at three cross-sections (-60 nm, 0 nm, and 60 nm) within the nanodimer. Black arrows indicate the direction of current, and red dashed lines represent the overall rotation direction of current density.

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In the nanodimer system, the optical response is significantly influenced when the lateral coordinate is fixed while the longitudinal coordinate varies. Specifically, the nanodimer is invariably localized at the transverse central region of APBs with a dominant longitudinal component H_z. Longitudinal shifts of the nanodimer alter the spatial distribution of H_z, thereby modulating the asymmetric coupling effect between the internal and external nanorings mentioned earlier. We quantified the effect of longitudinal displacement on magnetic multipole moments by tracking the intensity of the ${\textrm{m}_z}/{v_d}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ across a displacement range from z = 0 nm to z = 2500 nm at the wavelength of 1130 nm. As depicted in Fig. 6(a), the magnitudes of ${\textrm{m}_z}/{v_d}$ (red curve) and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ (green curve) decline with increasing distance from the focal plane, yet they exhibit differing rates of decay. The analysis of phase and amplitude between $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ and ${\textrm{m}_z}/{v_d}$ (Fig. 6(b)) reveals a stable phase-matching, with an approximate phase difference of π. Furthermore, the amplitude ratio gradually rises from 1 within the displacement range of 0 nm to 2000nm, indicating a transition from amplitude-matching to mismatched states. These variations manifest in the far-field radiation patterns depicted in Fig. 6(c), where the solid line represents theoretical calculations and the dashed line corresponds to simulation results. A marked reduction in radiation intensity occurs in the negative x-direction correlating with increased longitudinal displacement, attributed to diminishing H_z strength. Conversely, far-field radiation intensification in the positive x-direction arises due to reduced coherent cancellation from changing amplitude-matching. However, this increase is moderated by the overall decrease in H_z intensity. In accordance with previous works [49], [50], [54–56], we adopt the scattering directionality, Dx, to precisely express the correlation between scattering power along the x-axis and longitudinal displacement. The specific definition of Dx is the quotient obtained by dividing the difference in scattering power between the negative and positive directions of the x-axis by the sum of the two and the theoretical formulation for the logarithmic representation of Dx is presented as follows:

 

Fig. 6. Longitudinal displacement sensing at a wavelength of 1130 nm. (a) Variation of the magnetic dipole moment ${\textrm{m}_z}/{v_d}$ and magnetic quadrupole moment $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ with respect to the longitudinal position z of the nanodimer. (b) Ratio of the phase difference between $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ to pi, as well as the ratio of their magnitudes, as a function of the longitudinal position z. (c) Scattering field changes in the x-y plane at longitudinal positions of 0 nm/500 nm/1000 nm/1500 nm. Solid lines represent theoretical calculations, while dashed lines represent simulation results. (d) Variation of scattering directionality Dx, which related to scattered powers on the x-axis, with respect to the longitudinal position z. The red dotted line represents simulation values, while the black curve represents theoretical calculations.

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Figure 6(d) reveals the relationship between 10*logDx and the longitudinal displacement z, with simulation values indicated by red dotted lines and theoretical values by a black curve. In the displacement range from 0 nm to 2000nm, both theoretical and simulated values demonstrate a decreasing trend of 10*logDx with increasing z. Thus, 10*logDx can serves as a reliable indicator of the nanodimer's longitudinal positioning within (0 nm, 2000nm) based on the variation in scattering powers along the x-axis. Specifically, the scattering field powers along the x-axis is initially measured, followed by the calculation of Dx. By analyzing the functional relationship between Dx and the longitudinal position z, the precise value of z can be deduced, thereby enabling accurate longitudinal position sensing through optical intensity detection.

3.2 Silica substrate

To evaluate the potential applicability of our scheme in the fields of transverse unidirectional scattering and longitudinal displacement sensing, we examined its performance on a silica substrate, featuring a refractive index of n_s = 1.45. The system is illuminated from above the substrate, and light propagates from the + z to -z direction.

The presence of the substrate makes the scattering process more complex [63,64]. Three components may be identified in the scattered light of the nanodimer: light propagating directly in the upper medium, light propagating in the upper medium via interface reflection, and light transmitted into the substrate. Therefore, we divide the scattering space into two regions - above (air) and below (substrate) - and examine them separately. Notably, the silica substrate has a high transmittance and the scatterer-substrate ensemble achieves superior impedance matching relative to the air, facilitating increased energy penetration into the substrate. Moreover, the presence of a substrate introduces changes in the simulation. The interaction between the APBs and the substrate results in the appearance of transmitted and reflected light. Consequently, the background field cannot be considered exclusively as the APBs. To address this issue, a two-step simulation approach can be implemented [65]. Initially, the electromagnetic field distribution in the entire space was computed, considering the interaction between the APBs and the substrate, but excluding the nanodimer. Then, the electromagnetic field obtained in the previous step was utilized as the background field to interact with both the nanodimer and substrate, leading to the generation of the scattering field.

Obtaining an analytical solution for this problem poses a formidable challenge due to the continuous generation of transmitted and reflected light resulting from the re-excitation of scatterer by reflected light. We approximate the transverse unidirectional scattering condition in the case with the silica substrate as $|\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}|/ |{\textrm{m}_z}/{v_d}|= 1$; $\textrm{arg}(\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}) - \textrm{arg}({{\textrm{m}_z}/{v_d}} )= 0\mathrm{\,or\,\pi }.$ The simulation outcomes regarding the phase difference and amplitude ratio are illustrated in Fig. 7. Based on the original parameters (R4 = 365 nm), we observed a redshift in the wavelength that facilitated phase-matching but did not achieve amplitude-matching, as demonstrated in Fig. 7. In order to verify the effectiveness of this scheme even with the substrate, adjustments were made to the structural parameters. Specifically, the outer radius of the external nanoring (R₄) was reduced from 365 nm to 335 nm. As a result of this modification, we successfully achieved simultaneous phase and amplitude matching at wavelength of 1085 nm.

 

Fig. 7. Influence of geometric parameters on the magnetic multipole performance of a nano dimer in the presence of silica substrate. The figure investigates the specific impact of varying the outer radius (R4) of the external nanoring when the nanodimer is placed on a silica substrate on the phase difference and amplitude ratio between $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ and ${\textrm{m}_z}/{v_d}$.

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We further investigated the variation of ${\textrm{m}_z}/{v_d}$ and $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ with longitudinal displacement z at wavelength of 1085 nm. The simulation results, depicted in Fig. 8(b), demonstrate that the phase difference remains constant at π as the vertical displacement z ranges from 0 to 2000nm, confirming phase matching. However, the ratio of amplitudes gradually increases from 1, leading to a progressive mismatch in amplitude, consistent with the conclusions obtained in the absence of a substrate. Figure 8(c) delineates the simulated far-field radiation profiles. When the nanodimer is positioned at z = 0 nm, it exhibits favorable transverse unidirectional scattering characteristics. In the x-y plane, the variation of the scattering field along the x-axis with the increase of longitudinal displacement z recapitulates the similar trends observed in the absence of a substrate. In the x-z plane, the scattering field within the substrate demonstrates a similar behavior to the observations made along the x-axis when observed within the angular range of 15 degrees. Meanwhile, the 10*logDx is also employed to quantitatively analyze the correlation between scattering power along the x-axis and longitudinal displacement z. As shown in Fig. 8(d), a consistent decrease in 10*logDx is observed as z increases within the range of 0 to 1200 nm. In this context, the power of the scattering field along the x-axis is first measured, followed by the calculation of Dx. Subsequently, leveraging the functional relationship between Dx and the longitudinal displacement z, as illustrated in Fig. 8(d), the value of z can be deduced. This result reaffirms the efficacy of our proposed method, which utilizes light intensity measurement for longitudinal displacement detection, even in the presence of a silica substrate.

 

Fig. 8. Longitudinal displacement sensing in the presence of silica substrate at a wavelength of 1085 nm. (a) Schematic representation of the interaction between the APBs and the nano-dimer positioned on a silica substrate. The system is illuminated from above the substrate, and light propagates from the + z to -z direction. (b) Ratio of the phase difference between $\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$ and ${\textrm{m}_z}/{v_d}$, corresponding to the longitudinal displacement z, to $\mathrm{\ \pi }$, along with the ratio of their magnitudes. (c) Simulated results illustrating the variation of the scattering field in the x-y and x-z planes at different longitudinal positions (0 nm/500 nm/1000 nm/1500 nm). (d) Simulated results demonstrating the variation of the numerical quantity 10*logDx, which describes the scattering power along the x-axis, with longitudinal displacement z.

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4. Conclusion

In summary, our investigation has successfully achieved transverse unidirectional scattering and longitudinal displacement sensing by engaging the interaction between tightly focused azimuthally polarized beams (APBs) and a silicon nanodimer (comprising two non-concentric nanorings). The strategic positioning of the nanodimer within the APBs, with the center of the inner ring aligned with the singularity of the light field, enables selective excitation of magnetic dipole and quadrupole resonances. Through simulation analysis, we have validated the feasibility of this scheme and provided detailed insights into the generation mechanism of Fano resonance dominated by magnetic multipoles, utilizing hybridization models. At a wavelength of 1130 nm, the nanodimer exhibits distinct transverse unidirectional scattering properties, resulting from the interaction between the magnetic dipole moment (${\textrm{m}_z}/{v_d}$) and quadrupole moment ($\textrm{i}{k_d}/6{v_d}\mathrm{\ast }{\textrm{M}_{xz}}$). Moreover, by longitudinally varying the position of the nanodimer within the range of z = 0 nm to z = 2000nm, we observed a decreasing trend in scattering directionality Dx, which is related to the scattering power along the x-axis. This finding highlights the potential of our proposed scheme for longitudinal displacement sensing. For the configuration with a silica substrate, we also achieved conditions for unidirectional scattering at a wavelength of 1085 nm by adjusting structural parameters of the nanodimer. Within the range of z = 0 nm to z = 1200 nm, a similar trend is observed, with scattering directionality Dx decreasing as z increases. In summary, our study advances the control of optical scattering in nanostructures and provides valuable insights for designing on-chip integrated longitudinal displacement sensors. The demonstrated transverse unidirectional scattering and the potential for longitudinal displacement sensing have promising applications in various fields, such as nanophotonics and integrated sensing devices.