Study of Fano resonance and its application in MIM waveguide using a k-shaped resonator

In this paper, a plasmonic MIM waveguide consisting of a k-shaped resonant cavity is designed, which can support three Fano resonances. The physical mechanisms behind the multiple Fano resonances are studied with the help of field distributions and the changes of geometric parameters. Further, by varying the refractive index of the dielectric medium filled in the air part of the waveguide, property of refractive index sensing is studied. The maximum sensitivity is up to 1250 nm/RIU and the maximum figure of merit is more than 4000. Eventually the slow-light effect is also investigated, and the results show that the maximum optical delay and the group index are about 0.05 ps and 9.23, respectively. All the results may provide some fundamental references for the design of plasmonic optical waveguide devices, and it will also have potential applications in areas such as nanoscale refractive index sensing, slow-light effects, photonic device integration, and so on.


Introduction
Surface plasmon polaritons (SPPs) are electromagnetic waves in the form of collective oscillations, which are initiated by the interaction between incident light and free electrons on the metal surface and can propagate along the metal-dielectric interface with exponential decay in the vertical direction (Yunping et al. 2021;Rabiul et al. 2021). SPPs are able to break through the conventional optical diffraction limit and achieve local field enhancement in the subwavelength range, resulting in miniaturization and high integration of optical devices (Barnes William 2003;Gramotnev Dmitri and Bozhevolnyi 2010). Therefore, waveguide devices based on SPPs have attracted more extensive and in-depth research, for example, refractive index sensors (Shubin et al. 2020;Ben salah et al. 2019;, filters (Yunping et al. 2019;Vishwanath and Habibulla 2019), optical logic devices cavity and the arc type cavity, which will form the proposed asymmetric k-shaped cavity, will contributes to the generation of multiple Fano resonances. At fourth, the novelty can also be found from its better performances shown in Table 2. The transmission properties of the waveguide structure are investigated using the finite element method (FEM). Multiple Fano resonance are formed due to the mutual coupling and interference of broadband continuous state generated by the single stub and the multi-narrowband discrete state generated by the k-shaped resonant cavity. Then, the effect of structural parameters on the transmission characteristics was investigated. Owing to the strong sensitivity of the optical waveguide structure to the refractive index of the medium embedded in the air cavity of the overall waveguide structure, we also investigated the sensing application of the structure. The slow-light properties of the proposed structure have been verified by delay time and group index due to the steep phase distribution of multiple Fano resonances in the spectrum. Figure 1a shows the 3D schematic diagram of the proposed MIM-type plasmonic waveguide system. It consists of a k-shaped resonator coupled to a bus waveguide with a stub. Since the width w of the waveguide and the cavity is much smaller than the wavelength of the incident light, only the fundamental transverse magnetic (TM) modes can be excited (Veronis and Fan 2005), and in this paper, it is set to 50 nm. Figure 1b shows the top view of the structure, where the stub directly connected to the bus waveguide is called C1 (cavity 1), of which the height is L 1 = 140nm . The slot on the left side of the k-type resonant cavity is called C2 (cavity 2), whose height is L 2 with initial value of 490 nm. The right side of C2 is an arc shaped cavity described by a semi-elliptical ring (C3, cavity 3) with long axis along the horizontal direction, of which the relevant geometric parameters are set as b 1 = 220 nm , b 2 = 170 nm , a 1 = 140 nm and a 2 = 90 nm . The coupling space between C2 and C1 is g = 15 nm . The distance between the bottom of the semielliptical ring and the bus waveguide is h 0 = 155 nm.

Structure design and theory
The blue part in Fig. 1 represents air with a relative permittivity of i = 1.0 , while the green part is silver (Ag). Choosing Ag here is based on two reasons. On the one hand, it has the smallest imaginary part of the relative permittivity in the near infrared range and Fig. 1 a 3D schematic diagram of the proposed MIM-type plasmonic waveguide system. b The top view of the structure with the geometric parameters thus its power consumption is lower compared to that of gold and copper (Butt et al. 2019). On the other hand, if using gold (Au) in fabricating the waveguide, its cost is high. Meanwhile, it will agglomerates and forms island when it is thin, and therefore lacks uniformity. In addition, if copper (Cu) and aluminum (Al) are used, it will result in the chemical instability, which is too easy to be oxidized to affect the resonant characteristics (Xiao Gongli et al. 2021;Mishra et al. 2018). The structure can be fabricated by focused ion beam etching (Hindmarch et al. 2012), wet chemical etching or vapour deposition techniques . The special electromagnetic response of silver is commonly described by frequency-dependent complex permittivity using the Drude model (Ying et al. 2021) where ∞ = 3.7 is the relative permittivity at infinity frequency, p = 1.38 × 10 16 rad/s is the inherent oscillation frequency of the bulk plasmon, = 2.73 × 10 13 rad/s is the damping rate used to describe the absorption loss characteristics, and is the angular frequency of the incident light.
For MIM waveguide structures, the dispersion relations for surface plasmon polaritons (SPPs) supporting the TM mode are (Zhaojian et al. 2018) where w refers to the width of the waveguide, is incident light wavelength in vacuum, i and m are respectively the permittivity of embedded dielectric medium and metal, SPP is propagation constant of SPPs, and k 0 = 2 ∕ is the wave number.
For the proposed structure, when the electromagnetic wave is incident from the left port of the bus waveguide, it will first directly couple with stub, and a standing wave will be formed when C1 meets the resonance condition, which is (Xianshi and Xuguang 2008) In Eq. 5, = SPP = k 0 Re n eff is the propagation constant of the excited SPPs, is the additional phase shift caused by the reflection of SPPs at the top of the stub, and the positive integer m is the resonance order. Further, the SPPs can also be coupled into the k-shaped resonant cavity forming standing wave, which will enhance the local field in the cavity, and the needed resonance condition of C2 and C3 can be written as (Xinyuan et al. 2015;Qin et al. 2009) and here L eff is the effective length of C2 and C3. To collect the input and output power, two power monitors are set at ports A and B, respectively. The transmittance is defined Page 5 of 17 75 by the ratio of the output power P out to the input power P in , i.e., T = P out ∕P in . Satisfying a sufficient thickness in the z-axis direction, the results obtained with the three-dimensional model will be the same as those calculated using two-dimensional case (Qiqiang et al. 2021), and so we will perform two-dimensional simulations to save computational resources and improve the computational speed. Frequency domain solver of FEM based software COMSOL Multiphysics 5.6 is applied in this paper, and in the simulation process, the size of the simulation domain is selected as 1600 nm × 1400 nm . We used extremely fine mesh grid to ensure the convergence and maintain the accuracy of the calculation results. A perfect matching layer of 100 nm is placed at the upper and lower boundaries of the simulation domain. Meanwhile, scattering boundary condition is applied along the top and bottom sides, while numeric ports with input power P in = 1 W∕m along with boundary mode analysis are used at the input and output sides. To be noted is that 2D simulation is adopted in this paper in order to save both computing cost and time since some reported literatures and our earlier researches have shown that the results from 2D simulation agree well with those from 3D simulation if the waveguide device thickness is large enough. In this way, the transmission characteristics of the waveguide structure can be better described.

Simulation and discussion
To investigate the generation principle of Fano resonance, the proposed structure is decoupled into two cases: the first case is that a stub (C1) is directly connected to a bus waveguide, the second one is that the bus waveguide is directly coupled to the k-shaped resonant cavity. For convenience, these two cases and the proposed structure are respectively named as str-1, str-2 and str-3, as shown in Fig. 2a, b and c. The corresponding transmission spectra are given in Fig. 2d, where the parameters are shown in Table 1. For str-1, as shown by the black curve, it is clear that the transmission spectrum is a broadband continuum state, which can be considered as a super-radiant mode. For str-2, as shown by the blue curve, it can be regarded as multiple narrowband discrete states with sub-radiant modes. When the resonance condition in str-3 is satisfied, the overlap of broadband continuum state and narrow discrete states will lead to the generation of sharp asymmetric spectral profile, as shown by red curve in Fig. 2d. It is called Fano resonance and is suggested to be a kind of exotic optical phenomenon formed by the coherent interference of narrow-band discrete and broad-band continuum states. To be noted is that, whether the working wavelength is too large or too small, the radiation of stub based broadband continuum state will be reduced, which will not enough to couple with the narrow discrete states to form Fano resonance. Therefore, only three Fano resonances are observed in the spectral line.
To explain the mechanism underlying of Fano resonance phenomenon, the distribution of magnetic field components Hz at different peaks and dips are plotted in Fig. 3a-l. For str-1, when entering the stub cavity (C1) from the left side of the bus waveguide, the SPPs are reflected at the top of stub and form a standing wave, which suppresses the transmission of SPPs to the output port of the bus waveguide, thus forming a transmission dip near the wavelength of 920 nm. The magnetic field distribution in Fig. 3a shows that the energy at resonance is mainly concentrated in the stub, while there is almost no energy at the output port. For str-2, when the SPPs propagate along the x-direction, the energy will be coupled into the k-shaped resonant cavity through the near field, and the on-resonant standing waves will prevent the SPPs from being transmitted to the output port of the bus waveguide. The magnetic field distribution at four different dips of narrow-band spectrum is shown in Fig. 3b-e, respectively. It can be seen that the energy at different resonant wavelengths is gathered in different parts of the k-shaped resonator. At the wavelength of 750 nm, as shown in Fig. 3b, the magnetic field is mainly distributed in C2. At 955 nm, as shown in Fig. 3c, the magnetic field is mainly concentrated in the two ends of C3 (the semielliptical ring) and the bottom of C2. At 1315 nm, as shown in Fig. 3d, the magnetic field is mainly focused in C2. While at 1490 nm, as shown in Fig. 3e, the magnetic field is almost completely confined in C3. The resonances shown in Fig. 3a-e can be verified by Eqs. (5) and (6). Figure 3f-l depict the distribution of the magnetic field Hz at the wavelengths corresponding to the seven points marked by "a"-"g" on the red spectral line. The magnetic field distributions at the wavelengths corresponding to different Fano peaks and dips are different. At the peaks "a"-"c", most of the magnetic field energy can always be  transmitted to the output port, as shown in Fig. 3g, i and k. Obviously, there are differences in the field intensity, which are in accordance with the peak transmission coefficient shown on the red curve in Fig. 2d. In addition, there is almost no magnetic field energy distribution at the output port, as shown in Fig. 3f, h, j and l, because of the near zero transmission coefficient at the dips marked by "d"-"g". Specifically, when the working wavelength is 740 nm (point "d"), as shown in Fig. 3f, the input SPPs wave entering from the left side will first pass across C1 and then coupled into C2 through the gap. In terms of Eq. 6, one can find that the working wavelength satisfies the 2nd order resonant condition of C2, leading to the nearly perfect of magnetic field confinement in C2. Similarly, as shown in Fig. 3h (point "e"), it is the 2nd order resonance of C3. And in Fig. 3l (point "g"), it is the 1st order resonance of C2. The magnetic field distribution at 990 nm (point "f") shown in Fig. 3j is very similar to that of Fig. 3a, and at this time, due to the existence of weak coupling between C1 and C2, the mode effective refractive index will become smaller than that of str-1, thus the corresponding resonant wavelength of C1 become bigger than that Fig. 3 a Distribution of magnetic field Hz corresponding to the transmission dip of str-1. b-e Distribution of Hz at the four transmission dips of str-2. f-l Distribution of Hz of str-3 at the seven points marked by "a"-"g" in the red curve in Fig. 2d corresponds to str-1. In brief, at the three Fano peaks, the magnetic field distribution in C1 is too weak to be seen, which is due to the destructive interference of SPPs directly entered into C1 and that returned from the k-shaped resonator, and then transmission with different intensities can be achieved. However, at the four Fano dips, the magnetic field will be limited in different parts of C1, C2 and C3, thus the transmission is blocked. Next, we investigated the effects of structural parameters on the transmission properties of the proposed waveguide structure. First, parameter L 1 is changed from 100 to 180 nm with an interval of 20 nm with other parameters been fixed, and the transmission spectra are shown in Fig. 4a. It can be observed that with the gradually increasing of L 1 , the shape and symmetry of the spectral line of the Fano resonance in the middle show obvious changes because the broadband continuum spectrum coming from C1 will have a redshift with the increase of L 1 . The change trends of the position corresponding to the different peaks and dips are shown in Fig. 4d, which clearly demonstrate the effect of L 1 on them. In addition, it is worth noting that the middle Fano spectral line is nearly symmetric when L 1 = 140 nm , and this phenomenon can be approximated by PIT (plasmon induced transparency). The symmetry of a spectral line of Fano resonance can be judged by the asymmetry factor F = Δ high ∕Δ low (Piao Xianji and Sunkyu 2012; Kunhua et al. 2018;Kun et al. 2019), where Δ high and Δ low denote the wavelength offsets between a Fano peak and the dips on both sides of it. When F = 1 , the transmission spectral line are symmetric and the transmission characteristic can be called as PIT. Calculation shows that when L 1 = 100 nm , 120 nm , 140 nm , 160 nm , and 180 nm , the corresponding F corresponding to the middle Fano spectral line is about 0.09 , 0.25 , 1 , 5.25 , and 13 , respectively. Namely, the gradual increase of L 1 will lead to the change of F from less than 1 to equal to 1 and finally greater than 1, which fully indicates the evolution between the Fano effect and the PIT effect. The results are closely related to the redshift of the spectrum due to the increase of stub height.
Second, as shown in Fig. 4b, the effect of parameter L 2 on the transmission spectrum is explored. When L 2 changes from 470 to 530 nm with an interval of 10 nm, both left and right Fano spectral lines show significantly redshift, while the middle one nearly unchanged. This is because, as shown in Fig. 3, the middle Fano resonance is mainly determined by C3, while the other two are mainly determined by C2. The increase of L 2 leads to increase of the effective length of C2, which in turn causes an increase in the resonant wavelength and eventually shows a trend of redshift. Contour plot shows in Fig. 4c depicts more in detail the dependence of the transmission spectrum on the parameter L 2 and the incident wavelength. Figure 4e presents more clearly the changes of Fano peaks and dips with L 2 .
Next, we study the dependence of the transmission spectrum on parameters b 1 and a 1 . Figure 5a shows the simulation result when a 1 changes from 130 to 170 nm with a step of 10 nm, and clearly, the spectrum shows a trend of redshift. The right two Fano profiles possess faster shift speed than the left one, which shows that parameter a 1 will have dramatic effect on these two Fano profiles. In fact, with the increase of a 1 , together with increase of the effective cavity length of C3 and the change in its shape, the coupling position between C2 and C3 will change accordingly. Therefore, the field coupled in these two cavities will be redistributed and the resonance characteristic will also change, leading to the redshift of the spectral lines. Figure 5c demonstrates this more obviously. Furthermore, Fig. 5b indicates the transmission spectra with parameter b 1 varying from 220 to 300 nm. Under this circumstance, although the effective cavity length and the shape of C3 also changes, its effect on the right two Fano profiles are not very significant, which can also be found from Fig. 5d. However, the change of leftmost one is complex, which also comes from the field re-distribution and resonant characteristic due to the coupling between C2 and C3.
Finally, we investigated the effect of parameter g , namely, the coupling distance on the transmission spectrum. To be noted is that under this circumstance, the entire k-shaped cavity (C2&C3) will move upward, which is different from the cases discussed above. The simulation results when parameter g increases from 10 to 40 nm with a step of 5 nm are given in Fig. 6a. It is obvious that all the Fano resonances will become weaker with the increase of g, which is easy to be understood that the reduced coupling strength between the stub (C1) and the k-shaped cavity (C2&C3) will inevitably destroy the conditions for the generation of Fano resonances gradually. And if the value of g is too large, Fano resonances will disappear and the spectral line will approximate to the case of str-1, which can be seen in Fig. 6b and c more obviously.
The proposed structure with multiple sharp asymmetric Fano line shapes is expected to be sensitive to the change of the refractive index, which can be used in the field of refractive index sensing. For the refractive index sensing effect of Fano resonances, it is generally measured by sensitivity (S) and figure of merit (FOM). Sensitivity is defined as the shift of the resonance wavelength caused by the change of refractive index. FOM can be used to describe the rate of change of relative transmittance intensity. They are denoted respectively as (Ying et al. 2018;Hongxue et al. 2018): where T , n 0 and T( , n) are respectively the transmission before and after the refractive index is changed. Δn = n − n 0 is the difference of the refractive index. Meanwhile, the sensing resolution is also a criterion often used to characterize the sensing property (Rabiul et al. 2021;Yiyuan et al. 2015;Xuewei et al. 2018), which is usually used to determine the limiting value of the refractive index change detected by the sensor, and it can be defined as: where Δ denotes the wavelength detection resolution, which is usually available for highresolution spectrum analyzers (Alireza and Nosrat 2013). For simplicity, we studied the sensing characteristics by filling all the air parts with different dielectrics of different refractive indices. By setting that the refractive index varies within the range from 1.00 to 1.10 with a step of 0.02, the simulation results are shown in Fig. 7a. It is clear that with the increase of the refractive index, the spectral lines undergo a trend of redshift. It can also be found from Fig. 7b that the change is nearly linear for the different Fano peaks and dips. Calculation result can also prove that the sensitivity corresponding to Fano peaks a, b, and c is about 750 nm∕RIU , 1000 nm∕RIU and 1250 nm∕RIU , respectively. Figure 7c shows the calculated FOM, and one can find that the maximum value of FOM is more than 4000. Figure 7d gives the dependence of the wavelength offset of the three Fano peaks on the change of refractive index, from which one can find that the larger the refractive index deviation, the larger the wavelength offset, which also means the larger the sensing resolution. The calculated sensing resolution corresponding to the three peaks is as high as about 1.33 × 10 −5 RIU , 1.0 × 10 −5 RIU and 0.8 × 10 −5 RIU with wavelength detection resolution being selected as Δ = 0.01 nm (Yiyuan et al. 2015). Finally, the overall performances, especially the sensing property of the proposed waveguide were compared with other works reported in the literature. Table 2 shows the comparison results, including structure design, maximum of FOM, Sensitivity and so on. It is worth mentioning that all these works are based on the results of simulation. We can see that the structure proposed in this paper has the advantages of more Fano peaks, high FOM and simple structure, which provides ideas for designing refractive index sensors with better performance.
It is well known that slow-light effect is important in optical information storage and optical switching. Therefore, it has been a great research hotspot in the field of nano-integrated optics. Because the sharp change between the transmission dip and peak of the spectra line of the proposed waveguide is most likely to produce an abrupt phase shift, which will lead to a change in the group index and a certain time delay. The group index is usually used to describe the slow light effect, which is defined as follows  where c represents speed of light in vacuum, D = 1.6 m denotes the transverse length of the waveguide structure, g = d ( )∕d is the delay time, while ( ) is the transmission phase shift and v g is the group velocity.
The transmission spectrum is plotted again in Fig. 8a and the corresponding transmission phase shift is depicted in Fig. 8c. It is shown that abrupt phase shift occurs at the position where Fano resonance is formed, which means the generation of slow-light effects. One can find that the slope of the phase shift change is negative near the peak of Fano resonance, which implies normal dispersion. As shown in Fig. 8b, the positive delay time rightly indicates this slow-light effect. In contrast, if the slope of the phase shift is positive, it means the anomalous dispersion and the calculated delay time should be negative,  which represents a fast light effect. We find that the maximum optical delay time of about 0.05 ps can be obtained near the Fano Peak b, and the delay time corresponding the other two peaks is about 0.037 ps and 0.032 ps , respectively. The calculated group index is given in Fig. 8d, where a group index of about 9.23 is produced near Peak b, which means that the group velocity can be reduced by a factor of nearly 9.23 . The above study fully demonstrates the potential application of the proposed waveguide in the field of slow light.

Conclusion
In the paper, we designed a plasmonic MIM waveguide structure consisting of a k-shaped resonator coupled to a bus waveguide with a stub, and based on which, we studied the generation mechanism of three Fano resonances and their dependencies on separate geometric parameters. For the aim of sensing application, we studied the change of the transmission spectrum when the air parts of the waveguide is replaced by dielectrics with refractive index increasing from 1.00 to 1.10 with a step of 0.02. Simulation result shows that the transmission spectra will redshift linearly with the increase of refractive index. The maximum sensitivity corresponding to one of the three Fano resonances is up to 1250 nm∕RIU , and the maximum FOM is more than 4000. At last, the slow-light effect is investigated. The results show that a maximum value of optical delay is about 0.05 ps and the corresponding group index is about 9.23. The research results obtained is this paper can provide some references for the design of plasmonic waveguide devices and are expected to achieve potential applications in the fields of nanoscale refractive index sensors, slow-light devices, and so on.