Surface Plasmon (SP)—collective oscillation of free electrons along a metal surface, is mainly divided into two categories: Surface Plasmon Polaritons (SPPs) and Localized Surface Plasmons (LSPs)^[1-2]. They are widely utilized in surface-enhanced spectroscopy, biochemical sensors^[3-4], nonlinear optics^[5], lasers^[6], optical lenses^[7] and so on. Due to the wave vector mismatch between SPPs and light in free space, SPPs can be excited in structures such as the Kretschmann configuration^[8-10]. While LSPs can be directly excited by light in free space and their resonance frequencies are able to be tuned by the shapes of metallic nanoparticles^[11-14]. However, radiation loss of LSPs results in a large linewidth that significantly limits the potential applications of LSPs ^[15].

According to the research, the relative phase difference generated by oblique incidence and retardation effect affects the interaction between LSPs. And it is demonstrated in suspended metallic nano-sandwich arrays that the coherence condition of LSPs is the same as Wood-Rayleigh anomaly^[16-18]. Fano resonance occurs when a discrete state couples to a continuum state and forms a typical asymmetric line shape in the spectrum^[19-21]. Thus, the coherence of LSPs at specific resonance wavelength and incident angle can be observed via Fano resonance. This phenomenon is also known as plasmonic lattice resonance or Surface Lattice Resonance (SLR)^[22-24]. Because SLR has narrower linewidth and higher Q factor than LSPs of individual particles, it provides a number of applications in areas such as refractive index sensing^[25-27], color printing^{[28, 29]}, optical filter^[30] and fluorescence enhancement^[31]. Commonly, SLR is investigated in the wavelength-dependent spectra when the incident angle is fixed. SHEN Yet al.^[32], for example, proposed a metallic mushroom arrays structure composed of gold caps, photoresist pillars and gold holes, and got hybridized mode of Wood-Rayleigh anomaly and localized surface plasmon. The sensitivity and FOM of that sensor can reach 1015 nm/RIU and 108/RIU, respectively.

In this paper, we investigate the coherence of LSPs in one-dimensional metallic nano-slit arrays built on a sapphire substrate and derive coherence equations for different dielectric-metal interfaces. Due to magnetic surface plasmon resonance, the ring current driven by external electromagnetic field can induce magnetic field enhancement in the slits. A vertical Metal-Insulator-Metal (MIM) sandwich structure is made by metal and dielectric slit, which can be regarded as a Magnetic Resonator (MR). It also can be understood as the combination of capacitors and inductors in LC circuit^[33-35]. Just like X-ray diffraction, we rotate the incident angle to achieve two directional coherent states corresponding to the interface between metal and upper medium, as seen by two dips in angle-resolved spectrum. As the refractive index of the upper medium changes, the two coherence dips shift in opposite directions due to the retardation effect. Compared with using a single dip for sensing, two oppositely moving dips can effectively improve sensitivity.

Fig. 1(a) (color online) shows the proposed structure which is periodic in x-direction and infinite in z-direction. The width and thickness of Au slabs are a=990 nm and b=170 nm, respectively. The period of the structure isP=1000 nm and the slit width is d=10 nm. The refractive index of the sapphire substrate is $n_{\rm{s}} = 1.77$, and the surrounding medium is water and ${n_{\rm{w}}} = 1.33$. The proposed structure can be fabricated by atomic layer lithography^[36-37]. TM polarized light, whose magnetic field component is along the zdirection, illuminates in xy plane at an incident angle of $\theta $.

The simulation tool in this paper is commercial software COMSOL Multiphysics 5.6 based on finite element method. To simplify, we use two-dimensional model and set periodic condition inx-direction. The wavelength-dependent complex dielectric constants of Au are taken from experimental data in ref [38].

Firstly, the reflection spectrum of the structure under normal incidence ( $\theta = 0^\circ $) with wavelength ranging from 1000 nm to 1700 nm is shown in Fig. 1(b) (color online). It can be seen a broad dip 1 and a sharp dip 2 are located at 1272 nm and 1388 nm, respectively. The normalized magnetic field intensity distributions ( $|H|/|{H_0}|$) at dip 1 and dip 2 are depicted in Figs. 1(c) (color online) and 1(d) (color online), respectively. The red arrows in Figs. 1(c)~1(d) are proportional to the current density. As can be seen in both Figs. 1(c) and 1(d), the ring current excited in metal produces large magnetic field enhancement in the slit. But for dip1, it originates from magnetic surface plasmon resonance. For dip 2, it is typically asymmetric Fano line-shape caused by the coupling between magnetic surface plasmon resonance mode and the coherent state.

In order to analyze the coherence conditions of magnetic surface plasmon resonance in this structure, we consider the interaction of two neighboring magnetic resonators. As shown in Fig. 2(color online), the purple and green balls in the slits represent magnetic resonators MR 2 and MR 1, respectively. And the relative phase difference between them caused by oblique incidence is ${\varphi _0} = - 2{\text{π}} {{P}}{n_{\rm{w}}} \sin \theta /\lambda$, where λ is incident wavelength in vacuum. The contact between two MRs is mediated by the SPP wave since MR2 and MR1 are connected by metal. The retardation phase difference in Figs. 2(a) and 2(b) are ${\varphi _{21}} = \pm 2{\text{π}} P/ \lambda _{{\rm{SPP}}}^{{\rm{water}}}$ and $\varphi {'_{21}} = \pm 2{\text{π}} P/\lambda _{{\rm{SPP}}}^{{\rm{sub}}}$, in which $\lambda _{{\rm{SPP}}}^{{\rm{water}}}$ and $\lambda _{{\rm{SPP}}}^{{\rm{sub}}}$ are the wavelengthes of SPP along the metal surfaces adjacent to water and substrate, respectively (“±” indicates the wave propagates along $ + x$ or $ - x$ direction, shown as black or white arrows in Fig. 2). Thus, the correlated phase difference between MRs in above two cases are $\Delta \varphi = {\varphi _0} + {\varphi _{21}}$ and $ \Delta \varphi ' = {\varphi _0} + \varphi {'_{21}} $. The coherence of magnetic surface plasmon resonance is achieved when the correlated phase difference meets $2m{\text{π}} $ or $2l{\text{π}} $ (m or l is an integer). Then we can get coherence equations written as

$ P\left( { \pm \sqrt {\frac{{n_{\rm{w}}^2{\varepsilon _m}}}{{n_{\rm{w}}^2 + {\varepsilon _m}}}} - {n_{\rm{w}}}\sin \theta } \right) = m\lambda \quad, $ (1)

$ P\left( { \pm \sqrt {\frac{{n_{\rm{s}}^2{\varepsilon _m}}}{{n_{\rm{s}}^2 + {\varepsilon _m}}}} - {n_{\rm{w}}}\sin \theta } \right) = l\lambda \quad,$ (2)

where ${\varepsilon _m}$ is the permittivity of metal. The process of calculability can refer to ref [16]. The coherence conditions of SPP along metal surfaces adjacent to water and substrate are described by Eq. (1) and Eq. (2), respectively. And + (−) means coherence along $ + x$ ( $ - x$) direction, corresponding to $m > 0$, $l > 0$( $m < 0$, $l < 0$). According to the calculation results of Eq. (1), when $\theta = 0^\circ $, coherence of MRs along the $ + x$ and $ - x$ directions exists simultaneously at a specific wavelength and a standing wave is formed adjacent to the interface between metal and water which is shown by Fig. 1(d). The interaction of magnetic surface plasmon resonance with the coherent state results in Fano resonance as dip 2 shown in Fig. 1(b).

To further investigate the coherent phenomenon in this structure, both the wavelength and the incident angle are varied ( $\lambda $ is from 1000 nm to 1700 nm, $\theta$ is from 0° to 60°).

The 2D reflectance spectrum is shown in Fig. 3(a) (color online). The calculated coherence conditions of the upper metal interface are drawn with dashed lines named $C_{m = - 1}^{{\rm{water}}}$$C_{m = 1}^{{\rm{water}}}$and $C_{m = - 2}^{{\rm{water}}}$ , respectively, and those of the lower metal interface calculated by Eq. (2) are drawn with solid lines named $C_{l = - 2}^{{\rm{sub}}}$ and $C_{l = 1}^{{\rm{sub}}}$, respectively. A dark broad band (approximately from 1215 nm to 1400 nm) between two vertical green dashed lines in Fig. 3(a) originates from magnetic surface plasmon resonance. Other narrow bands result from the coherence of surface plasmons. As it can be seen, the calculated coherence lines match well with the simulation results, except for the region between two vertical green lines due to the strong interaction between coherence and surface plasmon resonance.

We take $\lambda = 1\;150\;{\rm{nm}}$ (position marked by vertical dashed line in Fig. 3(b) (color online)) and change the incident angle to explore several coherent states. In Fig. 4(a) (color online), four coherent states named A, B, C and D are shown by the reflection dips in angle-resolved spectrum. Moreover, the normalized magnetic field intensity distributions of these states are drawn in Fig. 4(b) (color online). The black arrows are proportional to Poynting vectors. From A and B, it can be seen that the near fields are mainly localized above the upper surface of metal due to the coherent states of SPP adjacent to water. Based on Eq. (1), A( $m = 1$) and B ( $m = - 2$) are the coherent states along $ + x$ and $ - x$ directions, which is confirmed by the energy flow. Similarly, C ( $l = - 2$) and D ( $l = 1$) are the coherent states of SPP adjacent to the substrate along $ - x$ and $ + x$ directions.

According to the two variables $\lambda $ and $\theta $ in Eq. (1), there are two ways to realize refractive index sensing: wavelength interrogation and incident angle interrogation. And there are two sensitivity definitions ${S_\lambda } = \Delta \lambda /\Delta n$ and ${S_\theta } = \Delta \theta /\Delta n$, in which the latter is used in this work. Because wavelength interrogation is conventionally used in previous work^{[25-27, 32]}, it is necessary to derive the relation between ${S_\lambda }$ and ${S_\theta }$. In the wavelength range of $1000 \sim 1700\;{\rm{nm}}$, $\sqrt {{\varepsilon _m}/(n_{\rm{w}}^2 + {\varepsilon _m})} $ can be approximated to 1, because $|{\varepsilon _m}| > > n_{\rm{w}}^2$ (the value of $\sqrt {{\varepsilon _m}/(n_{\rm{w}}^2 + {\varepsilon _m})} $ varies from 1.006 to 1.023 in this range when ${n_{\rm{w}}} = 1.33$). So, Eq. (1) can be simplified as:

$ P( \pm {n_{\rm{w}}} - {n_{\rm{w}}}\sin \theta ) = m\lambda \quad.$ (3)

From Eq. (3), we can deduce the relation between the two sensitivity definitions:

$ {S_\theta } = \frac{m}{{{n_{\rm{w}}}P\cos \theta }}{S_\lambda } \quad.$ (4)

Next, we change the refractive index of the upper medium to study the sensing performance of this structure. As shown in Fig. 5(a) (color online), with the increase of the refractive index ${n_{\rm{w}}}$, dip A and dip B move in opposite directions. We take ${\theta _A}$ and ${\theta _B}$ as the angles corresponding to dip A and dip B, respectively. The reverse movement can be seen intuitively in Fig. 5(b) (color online). When λis fixed and ${n_{\rm{w}}}$ becomes larger, the coherent states of A and B take positive and negative signs in Eq. (1), resulting in larger ${\theta _A}$ and smaller ${\theta _B}$, respectively. While in Eq. (2), " $ \pm $" sign related term remains unchanged, and the increase of ${n_{\rm{w}}}$ will inevitably lead to decrease of $\theta $. As a result, the angles of both dip Cand D( ${\theta _C}$, ${\theta _D}$) are slightly shifted to small values as ${n_{\rm{w}}}$ increases. However, as demonstrated in Fig. 5(c) (color online), wavelength interrogation cannot lead to reverse movement of double dips. In this case, the incident angle is $\theta = 5^\circ $ (position marked by horizontal dashed line in Fig. 3(b)) and the two dips are caused by coherence of $m = 1$ and $m = - 1$ corresponding to interface between metal and upper medium.

We define the total sensitivity of two dips as:

$ S = \Delta ({\theta _A} - {\theta _B})/\Delta {n_{\rm{w}}} \quad.$ (5)

According to the analysis above, we can infer that $ \Delta {\theta }_{A}/\Delta {n}_{{\rm{w}}}＞0 $ and $ \Delta {\theta }_{B}/\Delta {n}_{{\rm{w}}}＜0 $. The sensitivities of dip A and dip B are $ {S_A} = |\Delta {\theta _A}/\Delta {n_{\rm{w}}}| = \Delta {\theta _A}/\Delta {n_{\rm{w}}} $ and $ {S_{{B}}} = |\Delta {\theta _{{B}}}/\Delta {n_{\rm{w}}}| = - \Delta {\theta _{{B}}}/\Delta {n_{\rm{w}}} $, respectively. As a result, the total sensitivity $ S = {S_A} + {S_B} $ is better than the sensitivity of any single dip. Since dip A and dip B have different sensitivity and FWHM, we calculate their FOMs separately. The definitions of FOMs are FOM_A=S_A/FWHM_A and FOM_B=S_B/FWHM_B. Sensitivities and FOMs of different refractive index are listed in Table 1. The maximum sensitivities of two single dips are 39.2°/RIU and 102.4°/RIU and the corresponding FOMs are 115.2/RIU and 214.1/RIU, respectively. Due to the reverse movement, the total sensitivity of two dips can reach 141.6°/RIU. We compare the proposed device with previously reported devices in Table 2. The comparison result shows that our structure has a better performance.

Because of the narrow width and poor molecular mobility in the slit area, we consider two conditions in Fig. 6 (color online). In condition 1, the refractive index of the slit region is consistent with sensing medium. While in condition 2, the refractive index of the slit region is fixed at 1.33. The comparison result on the right of Fig. 6 shows that in these two cases, the positions of the four dips are nearly identical. Because the variation of the slit medium only affects the resonant wavelength of magnetic surface plasmon which is independent of incident angle. And the positions of four coherence dips mainly depend on the refractive index of the upper medium. Therefore, the inconsistency between the refractive index of slit region and upper medium has few influences on the sensing performance of this type of sensor.

Comparing coherence of surface plasmons in this structure (described by Eq. (1)) with lattice diffraction, such as X-ray diffraction (described by Bragg's theorem, $2d\sin \theta = m\lambda $), there are some differences between them. First, only relative phase difference is considered in lattice diffraction. Besides, the retardation effect related to the interaction of surface plasmon must be considered in the coherence of surface plasmons. As a result, the coherence dips manifest opposite movement in refractive index sensing.

In summary, we study the coherence of magnetic surface plasmon resonance in periodic metal nano-slit arrays and propose a convenient and high-accuracy sensing method. The coherence equations of surface plasmons in the structure are obtained by analyzing the correlated phase difference, and they are verified in 2D reflection spectrum. Due to the retardation effect, the two coherence dips in angle-resolved spectrum can move oppositely as the refractive index of the sensing medium changes. For sensing applications, compared with one dip used for sensing, two inversely moving dips can improve the sensitivity much more efficiently. In addition, the inconsistency between the refractive index of slit medium and upper medium has few influences on the sensing performance, which can lead to wide practical applications.