Machine learning analysis of a Fano resonance based plasmonic refractive index sensor using U shaped resonators

Figure 1 plots the designed surface plasmon-based RIS structure. The suggested structure comprises a USR coupled to an IUSR, both coupled to an MIM WG, and it is symmetrical about the centerline. The values of the defined geometrical dimensions equal l₁ = 75, l₂ = 40, l₃ = 50, l₄ = 100, l₅ = 240, s = 100, g = 25, and w = 100 (all in nm). The insulator and metal materials utilized for the presented sensor are air and silver, respectively. The air material has the relative permittivity of ε_d = 1. Furthermore, the Drude model defines the relative permittivity of silver⁴⁴.

In Eq. 1, ε_∞ = 3.7 denotes the medium dielectric constant for the infinite frequency. Also, \({\omega }_{p}=1.38\times {10}^{16}\) Hz is the bulk plasma frequency, ω shows the angular frequency of incident light, and \(\gamma =2.73\times {10}^{13}\) Hz demonstrates the electron collision frequency.

Figure 2 illustrates the transmission curve of the proposed RIS (Fig. 1), which is obtained using the FDTD method. It is found that two FR modes with sharp edges appear in a wide wavelength range. In Fig. 2, the FR peaks and valleys are labeled with λ_p1 = 659 nm, λ_p2 = 871 nm, and λ_v1 = 598 nm, λ_v2 = 891 nm, respectively. The transmittance values of λ_p1 and λ_p2 are 59.9% and 70.8%, respectively. Furthermore, the transmittance values of λ_v1 and λ_v2 are almost near zero.

In the next step, the field profiles of \(|{H}_{z} |\) for the RIS at peaks and valleys of the FR modes are represented in Fig. 3. As observed, two valley wavelengths of 598 and 891 nm do not achieve the right (output) port, while the peak resonance wavelengths of 659 and 871 nm transmit to this port. As observed in Fig. 3b, the energy at λ_p1 = 659 nm is mainly concentrated within both USR and IUSR. Furthermore, Fig. 3c shows that for λ_p2 = 871 nm, the most energy is only focused on the USR. It means that both resonators play a key role in the formation of the first peak, while the USR has a more significant influence on the formation of the second peak. In the following, this issue can be seen more clearly by calculating the transmission spectra of each resonator and waveguide separately.

To more accurately investigate the two generated FR modes, the two building block structures of the MIM WG coupled to IUSR (Structure 1) and USR (Structure 2) are also simulated separately, and their transmission spectra are compared to the transmission curve of the proposed RIS in a single figure. Figure 4a,b show the schematic topologies of Structure 1 and Structure 2, respectively. Also, the transmission curves of the building blocks and the main RIS are presented in Fig. 4c.

The transmission curve of Structure 1 (IUSR coupled to the MIM WG) is presented by a green dotted line. It can be observed that Structure 1 creates a wide notch at a wavelength of 620 nm. Also, the transmission curve of Structure 2 (USR coupled to the MIM WG) is displayed by a black dot-dash line. As seen, two narrow notches at wavelengths of 634.2 and 879.4 nm are generated by Structure 2.

The suggested RIS is designed by combining both basic structures (Structure 1 and Structure 2), and its transmission spectrum is illustrated by an orange solid line. By the interaction between the broadband mode of Structure 1 and the narrowband modes of Structure 2, the two FR modes mentioned above are produced.

After designing the sensor structure, its RI sensing performance is examined. The transmission spectra of various filling media with the RI from 1 to 1.05 for Δn = 0.01 is presented in Fig. 5a. It indicates that when the RI of the analyte (insulator media) is increased, the transmittance curve shifts to higher wavelengths, regularly. In order to quantitatively evaluate optical sensors, various parameters have been defined. One of the most important parameters is sensitivity which can be given by⁴⁵:

In Eq. 2, Δλ and Δn demonstrate the shift of the resonance wavelength and the variation of the RI, respectively. Also, RIU is the RI unit. To calculate the sensitivity of the RIS, the data points of λ_v1 and λ_v2 are fitted by linear functions (Fig. 5b,c). The obtained linear functions for λ_v1 and λ_v2 are given in Eqs. 3 and 4, respectively.

The slops of the linear functions (Δλ⁄Δn) are the RI sensitivity values at valley wavelengths. It can be concluded that the RI sensitivity for λ_v1 and λ_v2 are 571.4, and 872.9 nm/RIU, respectively. As observed, sensitivity is a parameter that specifies the value of wavelength shift for a determined RI variation, and it cannot show the resolution of a sensor³⁴. As a result, another more comprehensive parameter to evaluate the sensors’ performance is the FoM parameter. This factor is defined as Eq. 5 for sensors with FR spectra⁴⁵.

In Eq. 5, ΔT, Δn, and T are the change of the transmittance, the change of the RI, and the transmittance of the system, respectively. Based on Eq. 5, for the RIS designed in this paper, the FoM distribution of the wavelength is obtained (Fig. 6). Figure 6 shows that the maximum FoM value is obtained for λ_v1 = 598 nm which equals 14,993 RIU^-1. It is worth mentioning that the FoM value obtained for λ_v2 = 891 nm is 2178 RIU^-1. Therefore, the first valley wavelength with the highest FoM is considered for the sensing application of the designed RIS. Considering the remarkable sensitivity and FoM values obtained for the presented topology, it can be a suitable option for RI sensing.

Since the geometrical parameters of the suggested RIS can affect its transmission properties, the transmission performance for numerous values of some parameters (l₁, l₂, l₃, and g) is investigated here. The first column of Fig. 7a,d,g,j displays the transmission curves of the RIS for various values of l₁, l₂, l₃, and g, respectively. Furthermore, the relationship between the FoM and various values of mentioned parameters for λ_v1 and λ_v2 are shown in the second column of Fig. 7b,e,h,k and the third column of Fig. 7c,f, i,l, respectively. Figure 7a displays that when the value of l₁ is increased, the locations of both FR modes are relatively constant, while Fig. 7b,c demonstrate that increasing l₁ increases the FoM of λ_v1 initially and then decreases, and decreases the FoM of λ_v2. As discussed, the wavelength of λ_v1 is chosen for sensing performance, and on the other hand, although the FoM of λ_v2 also changes, the range of these variations is much smaller than the FoM of λ_v1. Consequently, the value of 75 nm for l₁, which causes the highest FoM for λ_v1 is chosen.

Figure 7d shows that increasing l₂ shifts the first valley wavelength to the lower wavelengths, while the location of the second valley wavelength is constant. Also, by increasing l₂, the highest FoM of λ_v1 is obtained at l₂ = 40 nm and the FoM of λ_v2 increases (Fig. 7e,f). Therefore, for the same reasons as in the previous case (variations of l₁), l₂ = 40 nm is chosen. Variations of l₃ are demonstrated in Fig. 7g–i. Figure 7g demonstrates that changing the l₃ value varies the locations of both FR modes. In other words, variations of l₃ result in a plasmonic structure with tunable resonance mods. The variations of the FoM of λ_v1 and λ_v2 are similar to the previous case (l₂). Figure 7h,i show these cases. As a result, l₃ = 50 nm is chosen. The gap of g is the last parameter in which its variation is studied (Fig. 7j–l). Figure 7j indicates that increasing the g value shifts the FR modes to the higher wavelengths. Furthermore, the FoM values of both FR modes increase initially and then decrease when the g value is increased (Fig. 7k,l). The highest FoM values for both modes occur at g = 25 nm which is selected.

Machine learning analysis

As known, using FDTD and FEM simulation methods for optical systems is time-consuming and needs extra memory, computational power, and processing time. Therefore, utilizing a method that can reduce the resources and time required to simulate is necessary. ML techniques such as the ERT RM can solve these challenges by predicting essential parameters and specifying missing values. Therefore, this section discusses the RMs briefly and shows how these techniques can reduce the needed resources for the proposed sensor structure by 90%. Regression analysis specifies dependent parameter values (transmittance) according to independent parameter values (wavelength), and uses three steps as follows:

Step 1: Simulating the RIS utilizing a larger wavelength’s step size.

Step 2: Training the RM with the simulation data.

Step 3: Predicting the transmittance of middle wavelengths utilizing the trained model.

Two vital parameters for the regression analysis used in this paper (ERT model) are the minimum needed sample size for node splitting (n_min) and the number of randomly selected properties at each node (K). The parameter of K = 1……p is used to determine the property strength applied to compute the goal. Here p shows the independent variables utilized to anticipate the goal parameter. To have more precision, the K value should be larger⁴⁶.

In this paper, to forecast the transmittance, six parameters are used. They are l₁, l₂, l₃, g, n, and wavelength (λ). It is worth mentioning that the value of n_min is considered 3. This can be explained below. First, all values between 2 and 10 are considered for n_min, and no significant difference is observed in the outputs. Since different figures for various values of five selected parameters (l₁, l₂, l₃, g, and n) are given in the following, to reduce the number of similar figures and reduce the number of pages, only one case (n_min = 3) is reported. It is worth mentioning that the scope of K is (1 ≤ K ≤ total number of attributes). Since the number of input features of the problem equals one (we only have one input that is wavelength (λ)), k = 1 is considered here. In the ERT RM, a set of “m” unpruned regression trees is produced. They are RT₁, RT₂, RT₃, …, RT_m. By calculating the total of each tree’s forecasts and taking the arithmetic mean of the outcomes, the final prediction is attained. Equation 6 demonstrates it⁴⁷.

In Eq. 6, m is the number of trees, and x demonstrates the value of the independent parameter. Some performance indices may be utilized to calculate the accuracy of the trained ERT RM. These indices are R-Square Score (R²S), Adjusted R Square Score (Adj-R²S), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE) which are expressed by Eqs. 7–10⁴⁷

In these Equations, N illustrates the total number of samples that were used in the validation of the RM. As mentioned, the ERT RM is used to predict the transmission value in this work. This model is applied for the test case of 10% (TC-10). Two distinct subsets of the FDTD-generated data are utilized in TC-10. In the first subset, 10% of the simulation data are selected for training the ERT RM. It is worth mentioning that these data are chosen with equal row spacing. The other 90% of the simulation data are considered for the accuracy of the model’s predictive (the second subset).

Figure 8 displays the heat map of the Adj-R²S of the ERT RM for n_min = 3. It is worth noting that in this figure, all values are rounded to four decimal places, and the exact values ​​are shown in parentheses. Adj-R²S for different values of l₁, l₂, l₃, g, and n are depicted in Fig. 8a–e, respectively. As observed, Adj-R²S values are close to 1. As a result, it can be said that high prediction accuracy is obtained. In other words, a tight link exists between simulated (actual) and predicted values.

Figure 9 illustrates the predicted values by the ERT RM versus the actual values of the transmission curves for various values of l₁ (Fig. 9a), l₂ (Fig. 9b), l₃ (Fig. 9c), g (Fig. 9d), and the RI of n (Fig. 9e). The obtained results for all parameters indicate that predicted and actual values are matched. Therefore, the high accuracy of the prediction is also concluded from this figure.

The RMSE generated for various values of all aforementioned parameters (l₁, l₂, l₃, g, and n) using comparative bar charts are demonstrated in Fig. 10. Figure 10a,d,e displays that the RMSE value less than 0.0018 is obtained for all various values of l₁, g, and n. Furthermore, for all different values of l₂ and l₃, the RMSE value less than 0.002 is attained. In fact, these figures show the low error prediction error.

Finally, the predicted versus the simulated transmittances of the suggested RIS are shown in Fig. 11. This figure shows that both curves are in good agreement. Consequently, using the ML-based ERT RM can reduce the simulation time for designing the plasmonic RIS by 90%.