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Abstract
A novel, to the best of our knowledge, nanosensor based on a metal-insulator-metal waveguide coupled to a double ring resonator is proposed. The spectral characteristics are studied by finite element method, and a Fano resonance (FR) formed by the interference of the narrowband mode and the broadband mode is discovered. After analyzing the effects of structural parameters on the transmission characteristics, the structure is further optimized by adding a rectangular cavity inside the ring cavity. The maximum sensitivity reaches 1885 nm/RIU with the figure of merit (FOM) of 77. Additionally, a tunable multiple FR system is realized through the derivative structure, which leads to the splitting of the resonance mode and produces two new narrowband modes. Their formation mechanism and performance are studied through a normalized magnetic field distribution and transmission spectrum. The designed structure with excellent performance can discover significant applications in the future nanosensing domain.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Surface plasmon polarons (SPPs) are the collective oscillations of free electrons on the metal surface and incident waves with the same resonance frequency, which mainly exist at the interface between metal and insulator [1,2]. It can effectively enhance the electromagnetic field energy in the nanoscale and overcome the limitation of traditional diffraction [3,4]. Therefore, SPPs can realize the manipulation of light in the subwavelength scale, thus showing broad prospects in the application of subwavelength optical devices. Currently, various plasmonic devices based on SPPs have been proposed, such as optical switches [5,6], plasmonic filters [7–9], slow light devices [10,11], demultiplexers [12,13], couplers [14,15], and nanosensors [16–19].
As an intriguing physical phenomenon, Fano resonance (FR), occurs in some optical systems resulting from the constructive and destructive interferences between a broadband mode and a narrowband mode [20]. Due to its sharp and asymmetrical characteristics [21], FR is extremely sensitive to structural parameters and environmental refractive index. This feature makes FR attract wide concern, and it can be observed in various structures. Du et al. [22] used the gap structure to generate tunable double FRs on the metal surface with large electric field enhancement. Nikoghosyan et al. [23] proposed a theoretical model for observing FRs in a double-well structure in an electric field directed along the growth axis of the structure. Among them, the metal-insulator-metal (MIM) waveguides based on SPPs have the characteristics of strong mode combination, long transmission distance, low loss, and easy manufacturing [24,25], attracting widespread attention in the nanosensing domain. In recent years, there has been increasing research on the refractive index sensors of distinctive structures based on MIM waveguides. Liu et al. [26] proposed a MIM waveguide coupled to a D-shaped cavity with air barriers, whose sensitivity reached 1510 nm/RIU. The structure could identify the blood group. Qi et al. [27] designed a multi-FR structure incorporating a single baffle MIM waveguide coupled with a semicircular resonator and a cruciform resonator, which was discussed by combining the multimode interference coupled mode theory. Rahmatiyar et al. [28] proposed a refractive index sensor containing a ring cavity with circular defects coupled to a MIM waveguide with tapered defects, whose sensitivity reached 1295 nm/RIU.
Herein, we propose a novel refractive index sensor based on a double ring resonator (DRR). The basic structure designed consists of the DRR and a MIM waveguide connected to a rectangular stub. Then the transmission spectrum and normalized magnetic field distribution are calculated numerically by the finite element method, and the formation mechanism of FR is studied accordingly. For improving the sensing performance while maintaining the compactness of the structure, a rectangular cavity is added inside the ring cavity. The influence of key parameters on the transmission characteristics is discussed. After optimization, the sensitivity and figure of merit (FOM) of this structure are as high as 1885 nm/RIU and 77, respectively, far exceeding the sensing performance of the structures mentioned above. In addition, a derivative structure is proposed by changing the position of the rectangular cavity to form three FRs, making the structure obtain more sensing points and reliable sensing results. We can control each FR by adjusting the position of the stub, the size of C1, and the refractive index of the resonant cavity. Compared with other refractive index sensors of the same type, the proposed structure has better compactness, sensitivity, and FOM, which provides a promising candidate for high-performance integrated plasmonic devices.
2. GEOMETRY MODEL AND ANALYSIS METHOD
The basic structure we proposed consists of a DRR and a MIM waveguide with a rectangular stub as shown in Fig. 1(a). We denote the top ring as C1 and the bottom ring as C2. The outer radii of C1 and C2 are represented by $r$ and $R$, respectively. Furthermore, the height of the rectangular stub on the waveguide is represented by $t$, and the coupling distance between DRR and the rectangular stub is denoted by $g$. In order to guarantee that only the transverse-magnetic fundamental mode (${{\rm TM}_0}$) propagation is performed, the width of the waveguide, the ring cavity, and the rectangular stub are set to $w = {50}\;{\rm nm}$. The white area in Fig. 1(b) represents air, and its relative dielectric constant is 1. The gray area represents Ag, whose relative dielectric constant is represented by the Debye–Drude model as [29]
Silver is used as the filler metal because its low power consumption makes it more conducive to exciting SPPs on its surface. The proposed structure selects quartz as the substrate, and the focused ion beam is sputtered on the silver film on the substrate. Since the size of the silver film in the $z$ direction is 100 nm, which is much smaller than the wavelength of light, the magnetic field characteristics of the 3D model and the 2D model will not be significantly different. In order to adapt the computer’s hardware configuration, we used COMSOL Multiphysics 5.4a to construct a 2D model instead of a 3D model. The normalized magnetic field distribution and transmission spectrum are numerically analyzed by the finite element method. In addition, the ultra-fine meshing is adopted to ensure the accuracy of the simulation, and the perfect matched layer is selected as absorption boundary condition of the system. P1 and P2 in Fig. 1(b) represent the input and output ports of the structure, respectively. The incident light is coupled to the single-mode fiber and the grating to excite SPPs at P1. During the propagation of SPPs along the waveguide, part of the energy is coupled into the DRR through the rectangular cavity. The energy will be transmitted clockwise and counterclockwise, causing interference inside the DRR, and finally output from P2. The propagation equation of SPPs in the waveguide is expressed as follows [30]:
where $q$ represents the wave vector in the waveguide, $k = {\varepsilon _d}/{\varepsilon _m}$ (${\varepsilon _d}$ and ${\varepsilon _m}$ represent the permittivity of dielectric and metal), ${\alpha _c} = {[{{q_0}^2({{\varepsilon _d} - {\varepsilon _m}}) + q}]^{1/2}}$(${q_0}$ denotes the wave vector in the free space).The resonance wavelength of the resonator can be obtained theoretically by [31]
Additionally, the sensitivity and FOM are significant features extensively utilized for measuring the sensor performance. The sensitivity and FOM are expressed as follows [32]:
where $\Delta \lambda$ is the alteration in resonant wavelength, $\Delta n$ represents the variation of refractive index, and FWHM refers to the full width at half-maxima.3. RESULTS AND DISCUSSION
In order to investigate the formation mechanism of FR, Fig. 2 depicts the transmission spectrum of the basic structure, a straight waveguide coupled DRR, and a waveguide with a rectangular stub resonator. The default values of the structural parameters are set as follows: $R = {170}\;{\rm nm}$, $r = {100}\;{\rm nm}$, $t = {50}\;{\rm nm}$, $g = {10}\;{\rm nm}$. The blue curve presents a sharp and obviously asymmetrical line shape, which is the characteristic of the standard Fano line shape. The prerequisite for the formation of FR is to produce continuous broadband mode and discrete narrowband mode. The continuous broadband mode is usually generated by the superradiative mode (bright mode) directly excited by incident light. In our proposed structure, SPPs can excite this mode through the waveguide connected to a rectangular stub resonator. Its spectral shape (red curve) has a flat profile and a wide line width, which is consistent with the characteristics of a continuous broadband mode. The discrete narrowband mode is derived from the subradiation mode (dark mode) excited indirectly, which can maintain the SPPs for a long time and result in a narrower linewidth. In our structure, this mode can be formed by indirect coupling of SPPs to DRR. Therefore, the FR of the basic structure is derived from the interaction of the discrete narrowband modes excited by DRR and the continuous broadband modes formed by the waveguide with a rectangular stub resonator.
In addition, for discussing the role of the rectangular stub resonator, we simulated the transmission spectrum of the straight waveguide coupled DRR (black curve). It can be found that the structure forms a FR dip at $\lambda = {1484}$ nm. After calculation, its FWHM is 40.6 nm, the sensitivity is 1460 nm/RIU, and the FOM is 35.9, whereas the FWHM of the FR dip of the basic structure is 24.8 nm, the sensitivity is 1460 nm/RIU, and the FOM is 58.9. As a result, the addition of the rectangular stub resonator increases the FOM by approximately 64% while maintaining the structural sensitivity. Although the transmittance has increased, the FOM growth caused by the greatly reduced FWHM brings better sensing function to the structure.
For discussing the effects of the key parameters of the structure on the transmission characteristics, the outer radii of the two rings were adjusted. First, we adjusted the outer radius $R$ of C2 from 150 to 190 nm at 10 nm intervals with other parameters set to default values. Similarly, the outer radius $r$ of C1 was altered from 80 to 160 nm at 20 nm intervals with other parameters set to default values. The transmission spectra of the two cases are shown in Fig. 3. It is clear that the FR dips both have a significant redshift with the increase of $R$ and $r$. According to Eq. (3), the increase of $R$ and $r$ means that the effective length $L$ of the DRR becomes larger, and the resonant wavelength of FR is proportional to $L$. As a result, the resonant wavelength will be increased significantly. Moreover, the wavelength shift and the refractive index variation were used for linear fitting, and the sensitivity of the structure with different parameters was calculated. According to the results, the sensitivity reaches the highest when $R = {190}\;{\rm nm}$ and $r = {160}\;{\rm nm}$, which are 1660 nm/RIU and 1580 nm/RIU, respectively.
Fig. 3. (a) Transmission spectra of the structure with different $R$. (b) Transmission spectra of the structure with different $r$.
What follows is the discussion of the influence of $t$, the height of the rectangular stub on the waveguide, on the transmission characteristics. We adjusted $t$ from 30 to 70 nm at 10 nm intervals, and other parameters were set to the default values. It can be found from the transmission spectrum in Fig. 4(a) that the transmittance decreases slightly as $t$ increases. However, the wavelength position of the FR dip is almost unchanged. Moreover, the FWHM gradually increases, resulting in the decrease of FOM. The calculated FOM is 62.13, 61.9, 60.8, 58, and 54.3. Through the above research, we found that the sensitivity will be significantly affected by $R$ and $r$, and $t$ only affects the FWHM of the spectrum. Hence the following deduction is derived: the FR resonance wavelength is mainly affected by the narrowband mode, whereas the broadband mode only changes the spectral waveform. This can be explained by the normalized magnetic field distribution. Since the energy at the FR dip is almost limited to the ring cavity, the resonance wavelength is extremely sensitive to changes in the parameters of the ring cavity. Afterwards, we analyzed the effects of coupling distance on the transmission characteristics. Figure 4(b) shows the transmission spectrum of the structure with different coupling distances in the range of 5–20 nm, and other parameters are set as default values. The simulation results reveal that a blueshift occurs in the transmission spectrum with the increase of $g$, while the transmittance at the FR dip is improved and the spectral width narrows sharply. The phenomenon results from the weaker coupling strength between DRR and the rectangular stub on the waveguide with the increase of $g$. In order to maintain a narrow half-width and low transmittance, we chose 10 nm as the most suitable coupling distance.
Fig. 4. (a) Transmission spectra of the structure with different $t$. (b) Transmission spectra of the structure with different $g$.
For improving the sensing performance while maintaining the compactness of the structure, we optimized the structure without expanding the outer dimensions. According to the above research, the improvement of sensitivity mainly depends on the narrowband mode, which is formed by DRR. We tried to add a rectangular cavity at the bottom of C2 as shown in Fig. 5. The width and height of the rectangular cavity are set to $w = {50}\;{\rm nm}$ and $H = {120}\;{\rm nm}$, respectively. The other parameters were set to the default values. The transmission spectrum in Fig. 6(a) reveals that the optimized structure produces a new FR at 1667 nm. Compared with the FR of the basic structure, it has a narrower FWHM and higher transmittance at the dip. After calculation, the sensitivity achieves 1671 nm/RIU with the FOM of 80.3, whereas the sensitivity and FOM of the basic structure under the default parameters are 1460 nm/RIU and 60.8. Consequently, the sensing performance of the new structure has been significantly improved compared with that of the basic structure.
According to the normalized magnetic field distribution in Figs. 6(b) and 6(c), the SPPs in the two structures are coupled to the DRR through the bus waveguide. The field energy of the basic structure at the dip is mainly concentrated in the interior of C1 and the bottom of C2, and the SPPs rarely propagate to P2. The field energy of the new structure at the dip is mainly concentrated in C1 and in the rectangular cavity, and some SPPs propagate to P2. Similar to the basic structure, the internal magnetic field of C1 and the bottom magnetic field of C2 are still in antiphase, and the stub on the waveguide is in phase with the bottom of C2. It will cause constructive interference between the two excitation paths and enhance the transmission. This explains the increase in the transmittance of the new structure.
To further optimize the performance, we adjusted $H$ from 110 to 150 nm at 10 nm intervals. The transmission spectrum in Fig. 7(a) shows that the FR has a significant redshift and the transmittance increases. Through the linear fitting in Fig. 7(b), the sensitivity is calculated to be 1637, 1671, 1722, 1777, and 1885 nm/RIU, and the corresponding FOM is 79.1, 80.3, 81.2, 79.3, and 77. Obviously, a larger $H$ will result in a greater sensitivity. Nevertheless, it is unwise to improve the sensitivity by increasing $H$ further. As $H$ becomes larger, the significant increase of FWHM will lead to a sharp decrease of FOM, and the greatly increased transmittance is not conducive to the sensing performance of the structure. Furthermore, when the rectangular cavity is further close to C1, strong interference will occur between them, thereby destroying the existing resonance mode. Figure 7(d) depicts the transmission spectrum when $H = {164}\;{\rm nm}$. The original FR disappears and is replaced by a new FR at 1099 nm. According to the normalized magnetic field distribution at the dip of the new FR, the top of the rectangular cavity is in phase with C1 and opposite to C2. Additionally, the bottom of the DRR is anti-phase with the waveguide, causing destructive interference and reducing the transmittance. Obviously, the formation mechanism of NFR is different from FR. It leads to the conclusion that when the distance between the rectangular cavity and C1 is too close, a new FR will be formed. After calculation, the NFR has a very narrow FWHM of 9.5 nm, but its sensitivity is not high enough. We finally set $H$ to 150 nm to obtain the best sensing performance.
Fig. 6. (a) Transmission spectra of the basic structure and the optimized structure. (b) HZ field distribution at the FR dip ($\lambda = {1476}\;{\rm nm}$) of the basic structure. (c) HZ field distribution at the FR dip ($\lambda = {1667}\;{\rm nm}$) of the optimized structure.
Fig. 7. (a) Transmission spectra of the optimized structure with different $H$. (b) Fitting line of sensitivity at the FR dip. (c) Variation trend of FOM with the increase of $H$. (d) Transmission spectra of the optimized structure when $H = {164}$ nm, and the HZ field distribution at the FR dip.
Fig. 8. (a) 2D schematic of the derivative structure. (b) Transmission spectra of the derivative structure.
In addition, we proposed a derivative structure plotted in Fig. 8(a). In this structure, the rectangular cavity in C2 was placed horizontally. The distance between the bottom edge of the rectangular cavity and the bottom of C2 is $D = {150}\;{\rm nm}$, and other parameters were set to the default values. The transmission spectrum in Fig. 8(b) shows that three FRs are formed, which are labeled F1, F2, and F3, respectively. For studying the formation mechanisms of the FRs in the derivative structure, the magnetic field distribution at $\lambda = {680}$, 973, and 1477 nm are illustrated in Figs. 9(a)–9(c). Obviously, most SPPs is concentrated at the top and bottom of C1 to excite a narrowband mode, which interferes with the broadband mode generated by the stub on the waveguide to form F1. It demonstrates that F1 is mainly affected by C1. Similarly, the SPPs at F2 dip are mainly concentrated in the rectangular cavity and the bottom of C2. It implies that the rectangular cavity is coupled with the bottom of C2 to form a new narrowband mode from Fig. 9(b). Furthermore, the energy at the dip of F3 is mainly limited to C1 and the bottom of C2. This is similar to the magnetic field distribution of the basic structure at the FR dip. Previous studies have revealed that the derivative structure could form three narrowband modes, which are derived from different formation mechanisms. Compared with the basic structure, the addition of the rectangular cavity causes the splitting of the resonance mode, thereby generating two new high-order narrowband modes [33]. They are coupled with the broadband mode caused from the stub on the waveguide to generate new FRs (F1, F2). This makes it possible for the structure to have multiple sensing points. Since the change of the narrowband mode is the main reason that affects the resonance wavelength, the positions of the three FR dips can be adjusted by altering the structural parameters.
To verify the accuracy of the above analysis, we conducted a study of the influence of $D$ and $r$ on the FRs. We first increased $D$ from 140 nm to 155 nm at 5 nm intervals, and kept other parameters the same as the default values. As shown in Fig. 10(a), F1 has a redshift and the transmittance is reduced, whereas F2 has a significant redshift and the transmittance is slightly increased. Figure 10(c) clearly depicts the variation trend of each FR’s wavelength. The increase of $D$ could significantly influence the resonance wavelength of F2, and the offset of F2 is 61 nm. The results confirm the previous analysis that the narrowband mode of F2 is mainly formed by the SPPs in the rectangular cavity and the bottom of C2. The position of the rectangular cavity is likely to influence the SPPs confined inside it, thereby affecting the narrowband mode. In addition, F1 is slightly influenced by $D$, and the offset of F1 is 26 nm. Although the narrowband mode of F1 is mainly formed by SPPs at the top and bottom of C2, there still exists a small amount of energy confined in the rectangular cavity. Furthermore, F3 is almost unaffected by $D$, and the offset of F3 is only 6 nm. This result verifies the previous speculation that the formation mechanism of F3 in the derivative structure is consistent with that of the FR in the basic structure. The effects of the outer radius of C1, which was set to 97, 100, 103, and 106 nm, were investigated below. The transmission spectrum is shown in Fig. 10(b), and the resonant wavelengths of F1, F2, and F3 are displayed in Fig. 10(d). It is manifest that F1 and F3 both have evident linear redshifts with the increase of $r$. F1 primarily originates from the narrowband mode formed by C1, and the narrowband mode of F3 is formed by C1 and C2 together. Hence, both F1 and F3 will be significantly affected by the outer radius of C1, and their offsets are 84 and 45 nm, respectively. Furthermore, F2 has a relatively small shift of 40 nm, because the energy limited in C1 is less than that of the other two modes. The above results are identical to the results of previous analysis.
Fig. 10. (a) Transmission spectra of the derivative structure with different $D$. (b) Transmission spectra of the derivative structure with different $r$. (c) Variation trend of each FR’s wavelength with different $D$. (d) Variation trend of each FR’s wavelength with different $r$.
Finally, in order to discuss the influence of the refractive index on these three FRs, we adjusted $n$ from 1 to 1.05 at intervals of 0.01 and set the structure parameters as follows: $D = {155}\;{\rm nm}$, $r = {100}\;{\rm nm}$, $R = {170}\;{\rm nm}$, $t = {50}\;{\rm nm}$. As shown in Fig. 11(a), all three FRs have linear redshifts. This can also be explained by Eq. (3); a larger refractive index will lead to a larger ${\mathop{\rm Re}\nolimits} ({{n_{\text{eff}}}})$, which is proportional to the resonance wavelength. The offsets of F1, F2, and F3 are 32, 48, and 72 nm, respectively. Hence, the wavelength of the FRs can be adjusted by changing the refractive index in a certain range. In addition, Fig. 11(b) exhibits the excellent linearity performance of the three FRs. According to the fitting consequence, the linear correlation coefficient of F1, F2, and F3 is 0.99977, 0.9983, and 0.9999, respectively. We also calculated the linear correlation coefficients of the basic structure and optimized structure, respectively, which are 0.99995 and 0.99996. Compared with the basic structure and the optimized structure, the linear correlation coefficient of the three FRs formed by this derivative structure all have a slight decrease. The optimized structure has the highest correlation coefficient of 0.99996, followed by the basic structure, whose correlation coefficient is 0.99995. But their numerical difference is very small, and they are all extraordinary close to 1. It indicates they can all be applied to refractive index sensing. Furthermore, Among the three FRs of the derivative structure, F1 has the largest FOM of 71.8, and F3 has the highest sensitivity of 1448.5 nm/RIU. It is worth noting that although the derivative structure realizes a tunable multiple FRs system, its sensitivity and FOM are not improved compared to the basic structure and optimized structure.
Fig. 11. (a) Transmission spectra of the derivative structure with different $n$. (b) Fitting line of sensitivity at the three FR dips.
4. CONCLUSION
In summary, a high-performance nanosensor based on a MIM waveguide coupled to a DRR was proposed. As the incident waves enter the waveguide, SPPs are generated and a broadband mode will be formed. Meanwhile, part of the energy is coupled to DRR to form a narrowband mode. The two modes interfere with each other to form the FR In the near field. By analyzing the influence of geometric parameters on the transmission characteristics, we conclude that the sensitivity is mainly affected by the outer size of the DRR. For improving the sensing performance while maintaining compactness, we placed a rectangular cavity vertically inside the DRR. When the rectangular cavity was placed horizontally, the mode splitting resulted in two high-order narrowband modes, providing multiple sensing points for the structure. But the sensitivity and FOM are not improved compared to the basic structure and optimized structure. The maximum sensitivity of the structure is 1885 nm/RIU with the FOM of 77. The structure can measure some physical quantities that have good linear relationships with the refractive index. For example, it can detect the hemoglobin concentration when the resonant cavity is filled with blood samples, or measure temperature when it is filled with ethanol. The structure proposed has the characteristics of high performance, compactness, and flexibility, which have many potential applications in the field of nanosensing.
Funding
National Natural Science Foundation of China (61875250, 61975189); Natural Science Foundation of Zhejiang Province (LD21F050001, LY21F040001); Key Research and Development Project of Zhejiang Province (2021C03019).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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