High-Resolution Plasmonic Filter and Refractive Index Sensor Based on Perturbed Square Cavity with Slits and Orthogonal Feeding Scheme

We present a plasmonic bandpass filter and refractive index sensor based on perturbed square cavity resonator with slits, which is fed by orthogonally oriented feeding waveguides. The slits provide size reduction in comparison to conventional square cavities and better coupling to the waveguides, while the perturbation and orthogonal feeding scheme provide a pair of transmission zeros in the response of the structure. In that manner, a bandpass filter with high transmission, narrow bandwidth, and excellent selectivity can be readily designed. Moreover, with a slight modification of the structure, a very sharp Fano-like response can be obtained and employed for refractive index sensing with high sensitivity and figure of merit.

When it comes to filters, which are of crucial importance in optical systems, there have been proposed different plasmonic configurations to achieve filtering operation, such as circular [12], square [8,11], and rectangular cavities [13], as well as stub [5], ring [6,9,10], spiral [7], and complementary split ring resonators [14]. However, most filters cannot simultaneously meet the requirements for a very accurate wavelength selection, low loss, and compactness. Moreover, rarely do filters exhibit multiple transmission zeros in their response which further limit the possibility for high-filtering resolution.
In this paper, we present a plasmonic bandpass filter based on a perturbed square cavity with slits fed by orthogonally oriented waveguides. In that manner, significant compactness is achieved, together with high-transmission level, narrow bandwidth, and excellent selectivity. With a slight modification of the structure, a very sharp fano-like response can be obtained and employed for refractive index sensing with high sensitivity and figure of merit.
The layout of the structure and the corresponding geometrical parameters are shown in Fig. 1. It consists of a square cavity resonator with symmetrically positioned rectangular slits, and small square perturbation positioned at the upper right corner, while the feeding MIM waveguides are mutually orthogonally oriented. The white area represents air whose refractive index is equal to 1, while the blue area represents silver which is characterized using Drude model [27]: where the permittivity at the infinite frequency ε ∞ is equal to 3.7, plasma frequency ω p is 1.38 10 16 rad/s, and the damping frequency γ equals 2.73 10 13 Hz. The width of the MIM waveguides w has been chosen to be 50 nm to provide that only the fundamental transverse magnetic (TM0) mode is supported, while the length of the square cavity L has been set to 400 nm and width of the slits w s 90 nm. The role of the rectangular slits is twofold-they provide size reduction in comparison to the conventional square cavity resonator and they can accommodate the waveguides so better coupling between the waveguides and resonator can be achieved as well as higher transmission level. Fig. 2a shows how the spectral position of the two fundamental degenerative modes, TM 10 and TM 01 , changes with the length of the slits. To that end, Comsol Multiphysics solver and eigenmode analysis have been used. It can be seen that the increase of the l s can significantly reduce the fundamental resonant wavelength. However, the length of the slits should not be excessively increased, since the cavity nature of the resonator and consequently good quality factor would be lost. Therefore, l s has been set to 80 nm.
To achieve a high-performance, narrowband, and highly selective filtering operation, it is desirable to have more than one pole in the response, i.e., to include more than one resonance. At the same time, the poles should be closely positioned in the spectrum and transmission zeros that are closely positioned to the passband should be provided.
To that end, in the proposed configuration, the feeding waveguides are orthogonally oriented, and a small square perturbation is positioned at the upper right corner of the square cavity. As stated previously, the two fundamental modes TM 10 and TM 01 are degenerative modes and have orthogonal field distribution and the same resonant frequency, Fig. 2. To employ them as two closely positioned poles in the filter, the square perturbation is introduced in the structure. Fig. 3a shows how the resonant wavelengths of the modes are changed with the parameter a sq , while Fig. 3b, c shows how the field distribution in the square cavity is changed once the perturbation is introduced.  Higher values of the parameter a sq causes the higher difference in the resonant wavelength of the fundamental modes, i.e., more pronounced blueshift of one of the fundamental modes, Fig. 3a. At the same time, the other fundamental mode remains unchanged, due to the fact that the perturbation is positioned at the corner at which the magnetic field of one mode exhibits peak, and the magnetic field of the other mode exhibits zero and thus it is not influenced by the perturbation.
The perturbation somewhat rotates the original fields, and the field poles and zeros are located at the corners of the square cavity. Consequently, the proposed feeding scheme allows that the two modes are simultaneously excited, which would not be possible in the configuration without perturbation. Moreover, each waveguide feeds the cavity in the manner that it is simultaneously coupled to the points of strong and weak magnetic fields of the modes. Intuitively, this can be understood that the modes are simultaneously strongly and weakly excited, and consequently transmission peaks and transmission dips can occur very closely in the spectrum.
Based on the previous analysis and the optimization of the parameters, a bandpass filter has been designed and the final values of the geometrical parameters are: L = 400 nm, l s = 80 nm, w s = 90 nm, w = 50 nm, g = 20 nm, and a sq = 20 nm.
The transmission response obtained in Comsol Multiphysics is shown in Fig. 4. To find the scattering matrix for the proposed structure four planes were defined (A, B, C, D) as it is shown in Fig. 1. The distance between A and B and C and D is equal to 50 nm, while the distance between B and the filter and between C and the filter is 400 nm. In these four planes, the magnetic field H z in complex form was determined, and using those values the reflected and transmitted wave components can be separated, and the elements of the scattering matrix calculated [28]. Since the system is a symmetrical and reciprocal, S 11 = S 22 and S 12 = S 21 . The transmission coefficient T and the reflection coefficient R can be calculated as T = |S 21 | 2 and R = |S 11 | 2 , respectively.
The central wavelength of the passband is 1228 nm, and the passband is characterized by good transmission level. Excellent selectivity and narrow bandwidth are provided by two transmission zeros that are symmetrically positioned around the passband at 1171 and 1285 nm. Full width at half maximum (FWHM) is only 23 nm, i.e., 1.87%, which confirms very narrow bandwidth not readily obtained in other plasmonic filters.
The proposed configuration can be also analyzed using temporal-coupled mode theory (CMT) using the following expressions [29]: where a 1 and a 2 represent the resonant modes, ω 1 and ω 2 their resonant frequencies, and τ 10 and τ 20 decay times of their internal loss. The parameters τ 11 , τ 12 , τ 21 , and τ 22 , are decay times of the coupling between the resonant modes and waveguides, and θ 11 , θ 12 , θ 21 , and θ 22 , are the corresponding coupling phases, while s i± are the field amplitudes in each waveguide (i = 1, 2, for outgoing (−) or incoming (+) from the resonator). The coefficient κ w stands for the coupling between the waveguides, while the coupling coefficient between the two cavity modes is denoted by μ. Namely, according to the field distribution, it can be considered that both modes are coupled to both feeding waveguides, which is expressed through the second and third terms in Eqs. (4) and (5). The fourth term in the same expressions indicates that the two waveguides are mutually coupled, and although the coupling is very weak, it contributes to the existence of the pair of symmetrically positioned transmission zeros [30]. Since s 2+ can be considered zero, the transmission response can be expressed as follows: M ¼ 1 The resonant frequencies ω 1 and ω 2 and the internal decay times have been determined using eigenmode analysis in Comsol Multiphysics solver and the following expression τ n0 = 2Re(ω i )/(2ω i Im(ω i )), where i = 1, 2. The coupling factor μ has been approximated as μ = ω 2 /(2(ω 2 -ω 1 )) [31]. The decay times between the resonant modes and waveguides have been approximated as ω 2 /δω, where δω is the width of the frequency range in which the S 11 phase changes the phase from -π/2 to π/2. After the optimization and fitting of the CMT model response to the numerical one, the following values of the parameters have been obtained: ω 1 = 2π 2.44 10 15 rad/s, ω 2 = 2π 2.47 10 15 rad/s, τ 10 = 356 fs, τ 20 = 363 fs, τ 11 = 120 fs, τ 12 = 650 fs, τ 21 = 650 fs, τ 22 = 120 fs, τ 22 = 120 fs, τ w = 700 s, θ 11 = 0.1π, θ 12 = −0.55π, θ 21 = 0.25π, θ 22 = 0.6π, θ w = 1.1π, and μ = 1. 15 10 13 . Figure 4 shows the comparison between the numerically obtained response and the fitted one and an excellent agreement can be observed.
Since the proposed structure exhibits the narrowband response and transmission zeros, it can be intuitively understood that with slight modifications in the structure, a sharp fano-like response can be obtained, which can be used for the sensing purposes. To that end, the structure without the square perturbation has been investigated. Figure 5 shows that in this case a response with one transmission pole and a pair of asymmetrically positioned transmission zeros can be obtained, where one of the zeros forms a very steep peak-to-dip slope. Namely, Fig. 5 shows how the response is changed when the symmetry axis of the first waveguide (with A and B planes) and the corresponding slit is moved away from the axis shown in the gray dotted line in Fig. 1, while the other parameters remain the same. For the deviation from the gray axis d = 20 nm a good trade-off between the transmission level and peak-to-dip slope is obtained, 0.61 and 30 nm, respectively,  and this structure has been further investigated for the sensing purposes. We note here that in the sensing structure the parameter l s is equal to 70 nm, which causes a slight blueshift if the response, whilst the other parameters remain the same as in the case of the filter.
Sensing potential of the structure is demonstrated through its response for different refractive indices of the dielectric marked in white in Fig. 1. The refractive index has been varied from 1 to 1.08 with the step of 0.02 and the corresponding responses are shown in Fig. 6. The sensitivity of a sensor (nm/ RIU) is usually defined as the shift in the resonance wavelength per unit variations of the refractive index [24]. According to the responses, the sensitivity is equal to 1150 nm/RIU.
Another important and more illustrative parameter of a sensor performance is figure of merit (FOM), which is defined as FOM = ΔT/TΔn [24]. T denotes the transmittance in the proposed structure and ΔT/Δn denotes the transmission change at the fixed wavelength induced by a refractive index change. Fig. 7 shows FOMs calculated for the refractive index 1.02, 1.04, 1.06, and 1.08 as FOM = (T n = 1.02/1.04/1.06/ 1.08 − T n = 1 )/(T n = 1 Δn). Although the curves differ in the maximal values, it can be considered that FOM can reach the values of the order of 10 3 -10 4 .
In Table 1, the sensitivity and FOM of different structures are compared to those of the proposed structure. As it can be seen, the proposed structure has comparable or higher FOM than the other structures, except for [21,22]. As for the sensitivity, only the structures in [22,23] exhibit significantly better sensitivity. However, high FOM and sensitivity in the structure in [22] is achieved at the expense of the low-transmission level, while the structure in [23] exhibits very low FOM. Moreover, we note again that the proposed structure has a twofold purpose which represents its particular strength. Besides operating as a filter, the proposed structure can also be readily transformed to a sensor with excellent performance, which is not the case with other sensors.
To conclude, we proposed a novel plasmonic structure based on perturbed square cavity resonator with slits, which has a great potential for applications in filtering operation and sensing. A small square perturbation introduced in the structure and the orthogonal feeding scheme provides twopole, narrowband, and highly selective bandpass response which makes the proposed structure one of the most compact filter with excellent performance and FWHM of only 1.87%. A slight modification of the proposed structure allows for fano-like response with a very steep peak-to-dip slope, which can be readily applied in sensing. Sensing potential has been demonstrated through the response for different refractive indices of the dielectric. Sensitivity of 1150 nm/RIU and FOM of the order 10 3 -10 4 have been achieved, which is comparable to the state-of-the-art solutions, and thus the proposed structure can be considered as a good candidate for sensing applications.