Compact and Efficient Ring Resonator–Based Plasmonic Lens with Multiple Functionalities

A novel plasmonic lens based on a metal–insulator–metal bus waveguide coupled to a ring resonator is proposed and numerically investigated. The inner edge of the ring resonator was perforated with nanogrooves so that light emitted by these nanogrooves was sharply focused at the center of the ring. The structure relies on interference and resonance theory. The principle of the lens can be extended to a device 1 × N demultiplexer as well as a 1 × N power splitter. These functionalities were validated through finite element method simulations. Results show that the lens had a sharp focus, without any smearing, and a full width at half maximum (FWHM) intensity of around 240 nm. For the 1 × N demultiplexer, the selectivity was high with an FWHM bandwidth of less than 20 nm and crosstalk of less than − 10 dB, whereas the 1 × N power splitter had a narrow bandwidth and was able to split power into N equal parts with negligible imbalance. The proposed focusing structure is compact, and the simulation results show that the structure performs the various functionalities with high efficiency. Due to this, these structures will be of utmost utility in future all-optical signal processing systems.


Introduction
Surface plasmon polaritons (SPPs) are propagating electromagnetic waves coupled to free electron oscillations confined to the metal-dielectric interface [1,2]. A few unique properties of SPPs such as the ability to go beyond the classical diffraction limit, propagation at the nanoscale, and extraordinary transmission make it possible to manipulate photons at the nanoscale [3][4][5]. Faster and more efficient miniature photonic devices can be realized using these properties which can open up new possibilities in the fields of optoelectronics, material sciences, biophotonics, and nanoscience [6][7][8].
Many plasmonic devices have been proposed in the literature to realize various functions and have shown outstanding performance in both simulations and fabricated prototypes [9]. Among such devices, many researchers have shown interest in focusing light at the subwavelength scale due to its importance in nanophotonic circuits and nanofabrication [10,11]. It is well known that conventionally optical focusing is achieved through an appropriate shaping of the wavefront of the emitted light [12]. This same principle of wave shaping is adopted in a variety of SPP-based nanofocusing devices [13][14][15][16][17][18][19][20][21][22][23]. Nano-antenna arrays were also suggested to efficiently focus light [6,12]. A side-illuminated plasmonic planar lens was proposed wherein a set of grooves was used to couple SPP mode propagating in a bus waveguide to free space. The position of the grooves was chosen such that the desired wave shaping could be achieved which results in the focusing of emitted light [24]. Structured grating for out-ofplane focusing was demonstrated in the study of Kumar et al. [25] along with its extension to obtain advanced functionalities such as a power splitter and wavelength demultiplexer. In most of the proposed lens designs in the literature, there may be a few limitations such as large size, low power efficiency, wider full width at half maximum (FWHM) intensity of the focal spot, and deviation in the simulated and the designed focal spot due to Fresnel diffraction. Moreover, generally, the focal spot is not circular in shape but smeared.
Plasmonic wavelength demultiplexers are used to filter the selected wavelength from a bunch of wavelengths and will play a crucial role in all-optical signal processing circuits. These wavelength demultiplexers use a variety of 1 3 resonators as their key component for the wavelength selection with a few exceptions without resonators [9,[26][27][28][29][30][31][32][33][34]. The resonant wavelength of these resonators can be tuned by the proper selection of the geometric parameters and the refractive index of the materials chosen. It can be observed in most of these investigations, two-to three-wavelength selections are considered. A six-wavelength selection with split ring resonators is proposed by Rafiee et al. [34] with an investigation on the effect of the geometrical parameters on the performance of the structure.
Plasmonic power splitters were considered one of the essential components in multichannel photonic integrated circuits. An ultra-wideband Y-splitter consisting of a straight waveguide with a composite H-shaped structure is proposed in the study of Gao et al. [7]. Experimental validation of the proposed structure is carried out at the THz operating frequency range. The splitter proposed in the study of Ye et al. [35] is constructed with a hexagonal resonator with a rotatable silver elliptical core embedded into it. There are four output channels in this structure. Variable transmissions, with selectable output channels, are realized in this structure by appropriately rotating the elliptical core to a different angle. A 1 × 2 metal-insulator-silicon-insulator-metal waveguide switch-cum-splitter with an embedded phase change material such as vanadium oxide and chalcogenide-based Ge 2 Sb 2 Te 5 (GST) is theoretically studied and simulated in the study of Singh [36]. Both 1 × 2 switching and power splitting in the C-band are achieved via tuning the phases of the phase change material through localized heating on the selected output sections.
Ring resonators are one of the important blocks in optical integrated circuits and are widely used in applications such as filters, wavelength demultiplexers, switches, logic gates, and sensors in various configurations [34,35,[37][38][39][40][41][42][43]. Electromagnetic energy whose wavelength matches the resonant wavelength is trapped in the ring and can be filtered to a waveguide coupled to it. The resonant frequency of the ring is highly tunable by adjusting its geometrical parameters and through a choice of proper materials. Rings with different radii were designed to have different resonant wavelengths in wavelength-selective devices such as filters and wavelength demultiplexers [34,35]. Split square ring resonators were used to obtain switching functionality in the study of Negahdari et al. [37]. A ring filled with nonlinear material, the refractive index of which is controlled by an external pumping light, is adopted in the realization of logic gates in [38,39]. A single silicon ring resonator-based reversible swap gate is presented in the study of Choure et al. [40]. Ring resonators have also been used as filters in refractive index sensors [41][42][43].
In this paper, we propose a novel compact and efficient plasmonic lens that, contrary to most of the existing plasmonic lenses, can provide a sharp, well-defined focal spot with comparatively smaller FWHM intensity at the focal spot. The simulated focal spot is almost circular in shape with no deviation between the designed and the simulated focal spot. In the proposed structure, the resonating ring is used to focus the light by carving nanogrooves on the ring. Though ring resonators have been used in many applications as mentioned in the previous paragraph, to the best of our knowledge, the use of ring resonators for focusing light has not been proposed.
SPP waves from a metal-insulator-metal (MIM) bus waveguide are evanescently coupled to a circular ring resonator which has nanogrooves carved on it to focus light at the center of the resonator. The proposed design was extended to obtain such important functionalities as a 1 × N demultiplexer and 1 × N power splitter. As mentioned above, most of the demultiplexers available in the literature will be separating two to three wavelengths [26][27][28][29][30][31][32][33][34]. In the proposed work, an effort is made to separate eight different wavelengths (N = 8) with a channel spacing as small as 20 nm with FWHM bandwidth less than 20 nm. The designed power splitter is able to distribute the incident power into eight (N = 8) equal focal spots. Simulations were carried out the using finite element method (FEM). Results demonstrated the functioning of the proposed structures as per the design objectives and specifications. The proposed structures have the potential to play key roles in the development of ultra-compact photonic integrated circuits.

Design and Simulation Results
A 2D schematic structure of the MIM bus waveguide coupled to a ring with nanogrooves is shown in Fig. 1a. Gold is used as metal, and SiO 2 with a refractive index of 1.2 is used as an insulator in the bus and ring. Air occupies the central region of the ring. The frequency-dependent complex relative permittivity of the gold is calculated using the Drude model [44].
To ensure that only the fundamental transverse magnetic (TM) mode propagates in the structure, the width of the insulator in both the bus and ring is set as w = 100 nm, which is much smaller than the operating wavelength. The complex propagation constant (β) of the fundamental TM mode is obtained by Shi et al. [13], and the effective refractive index (n eff ) can be calculated as n ef f = k 0 where k 0 = 2 is the free space wavenumber and λ is the operating wavelength.
The propagation characteristics of the SPP in the ring can be manipulated by varying the width (w), radius (r), and effective refractive index (n eff ) of the ring. For the ring to resonate at the desired wavelength (λ), the circumference of the ring (L) must be an integer multiple of the effective wavelength (λ/n eff ) of SPP. This condition dictates that the radius (r) of the ring has to be as follows [45]: where m is a positive integer indicating the mth resonating mode of the ring. The metal barrier thickness between the bus and ring is chosen as t m .
Simulations were carried out using FEM. The resonating wavelength of the ring is obtained by simulating the basic structure of a bus waveguide coupled to a ring (see the inset of Fig. 1b). The geometrical parameters considered for the simulation are m = 7, t m = 15 nm, w = 100 nm, and r = 460 nm for the excitation of the seventh resonating mode in the ring. Figure 1b shows the plot of the normalized x component of the time-averaged power (P x ) at the output port (P 2 ) of the bus waveguide with respect to wavelength (λ). Figure 1c and d shows the z component of the magnetic field intensity (|H z | 2 ) plotted for 650 nm and 670 nm wavelengths, respectively. It is visible from the figure that at the resonating wavelength (Fig. 1c), almost all the power is coupled to the ring with minimum power exiting the bus at port P 2 . While computing the radius of the ring, it must be noted that n eff varies slightly along the coupling length between the ring and the bus [46]. However, in the calculations, we have taken n eff to be constant in Eq. (1). This will result in the ring radius (r) not exactly matching the one given by Eq. (1).
To focus the light in the resonator, nanogrooves were introduced on the inner metal wall of the ring. Each of these grooves has width (w g ) and thickness (t g ). If the size of the grooves is much smaller than the operating wavelength, they can be modeled as point sources [24]. The power in the ring is partly coupled to the groove. Each groove will emit light toward the center of the ring. Power coupling efficiency from ring to groove depends on the thickness of the metal barrier between the ring and the nanogroove (t i ). The smaller this thickness, the more power will be coupled to the grooves. A proper choice of the thickness of the metal barrier should be found so that the intensity of the light emitted by each of the grooves is almost the same. Nanogrooves must be positioned on the inner metal wall of the ring such that light emitted from each of the grooves must collectively generate a circular wavefront that converges at the center of the ring. As the distance between the center of the circular ring and each of the nanogrooves is equal for all the grooves, for constructive interference at the center, the distance between two consecutive nanogrooves must be an integer multiple of the effective wavelength (λ/n eff ) of the SPP in the ring. This implies that with the mth resonating mode in the ring, a maximum of m nanogrooves can be placed equidistantly (see d in Fig. 1a) in the ring. Note that there must be a sufficient number of nanogrooves present in the structure to generate a circular wavefront that can focus at the center. For 650 nm wavelength, the structure parameters for the simulation were chosen as m = 7, w = 100 nm, t m = 15 nm, r = 467 nm, w g = 50 nm, t g = 50 nm, and t i = 5 nm. These values were found to be close to optimal for 650 nm wavelength as reported in the Appendix. Nanogrooves were Fig. 1 a Schematic sketch of the proposed plasmonic lens consisting of circular ring resonator with nanogrooves b normalized power (P x ) measured at the output of bus (at P 2 ) versus wavelength (λ); the inset shows a schematic of structure considered. Intensity (|H z | 2 ) distribution for wavelength 650 nm in (c) and 670 nm in (d) for the ring resonator as in the inset positioned equidistantly as shown in Fig. 1a and were oriented such that, near the coupling region of SPP from bus to ring, grooves were located symmetrically on both sides of the y-axis.
The magnetic field (H z ) and its intensity (|H z | 2 ) are shown in Fig. 2. With m = 7, the ring is resonating at the seventh mode, and seven complete cycles can be seen in the ring in Fig. 2a. Almost each of the nanogrooves was emitting light with almost the same phase which resulted in the generation of a circular wavefront as can be seen. The circular wavefront resulted in the focusing of the wave at the center of the ring as shown in Fig. 2b. The cross section of the focal spot is considerably small as shown in Fig. 2c where intensity (|H z | 2 ) is plotted along the x-axis as well as the y-axis through the center of the ring (focal spot). It can be seen from this figure that the FWHM intensity is around 240 nm along both axes demonstrating a considerably sharp focal spot. It can be observed from Fig. 2b that, with the ring resonating, almost all the power in the bus is coupled to the ring. In turn, nanogrooves on the ring emit almost all the power toward the center resulting in an intense focal spot. From these discussions, it can be confirmed that the proposed plasmonic lens structure is efficient. Also, due to the use of a single-ring resonator for focusing and with the light focusing at the center of the ring, the proposed structure is considerably compact with no deviation between the designed and the simulated focal spot.
In order to get a better insight into the obtained results, a comparison of the proposed lens structure with others available in the literature is presented in Table 1. The parameters used for comparison are operating wavelength, the topology of the structure, simulated focal length, deviation in the simulated focal length compared with the designed one, and FWHM intensity of the focal spot. For most of the lenses considered in Table 1, there is a deviation in the designed and simulated focal spot, whereas for the lens proposed  in this paper, the light is focused at the center of the ring resonator with no deviation in the simulated and designed focal spot. The FWHM intensity for the proposed lens is comparatively low among all other designs considered here. These comparisons help in concluding that the structure proposed in this paper generates a sharp focal spot.

The 1 × N Demultiplexer
The above-discussed focusing structure can be readily extended to perform as a 1 × N demultiplexer to separate N constituent wavelengths from an SPP wave propagating in the MIM bus waveguide. The schematic of this extended  structure is shown in Fig. 3 where N is taken as 8. There are a total of N ring resonators, placed on either side of the bus, with their geometrical parameters such that each one resonates at a particular wavelength. As explained earlier, nanogrooves on the inner metal layer of the rings emit light at the resonant wavelength of the ring. This emitted light, of a particular wavelength, is focused at the center of the ring which results in the structure functioning as a 1 × N demultiplexer. Wavelengths selected for the simulation of the 1 × 8 demultiplexer and the corresponding radius of the ring resonators calculated using Eq. (1) are given in Table 2. The other parameters were fixed as m = 7, w = 100 nm, t m = 25 nm, w g = 50 nm, t g = 70 nm, and t i = 15 nm. The simulation results are shown in Fig. 4 where intensity (|H z | 2 ) is plotted for different wavelengths. It can be observed that the rings in association with the nanogrooves can spatially separate the wavelengths and focus them as desired. A sharp focal spot, without any smears, with a considerably small FWHM intensity which is less than 300 nm can be observed for all the wavelengths. Simulation results reveal that there is a possibility of two or more rings resonating at the same wavelength. This is because a ring designed to resonate at a particular wavelength at a mode m can also resonate at a higher wavelength for a mode < m. As an example, a ring designed for 650 nm is also seen to be resonating at 770 nm  in Fig. 4g. We invite the reader's attention to the fact that the ring designed for 770 nm is resonating at m = 7 but the ring designed for 650 nm is resonating at 770 nm with m = 6.
This, however, does not affect the functioning of the demultiplexer because focusing on the desired wavelength at the center of the corresponding ring is controlled by the positioning of the nanogrooves. Thus, even though the ring designed for 650 nm resonates at 770 nm, that ring does not focus on the 770 nm wavelength because the position of the nanogrooves in that ring does not support the constructive interference of 770 nm. It is worth mentioning here that in most of the cavity-based demultiplexers, due to the abovementioned fact wherein a resonator resonates at more than one wavelength, it is sometimes not possible to differentiate the wavelengths. One of the distinct features of the structure proposed here is that it is immune from this problem.
Field intensity (|H z | 2 ) at the focus of all the eight ring resonators versus wavelength is plotted as shown in Fig. 5. Frequency selectivity, a figure of merit for a demultiplexer, is measured, and FWHM bandwidth was found to be less than 20 nm for all the selected wavelengths. Crosstalk, another figure of merit of a demultiplexer, was found to be less than − 10 dB for all the wavelengths. This indicates an acceptably high suppression of the unwanted wavelengths at the focal spot.
A comparison of the performance of the proposed demultiplexer with that available in recent studies is carried out and is presented in Table 3. Demultiplexing wavelengths, the topology of the structure, materials used in the MIM structure, the number of output channels, channel Fig. 6 a Schematic of 1 × 2 power splitter, b intensity (|H z | 2 ), c intensity (|H z | 2 ) along the x-axis through the focal spot of both ring resonators, and d variation of intensity (|H z | 2 ) at the focal spot along with transmittance with respect to metal barrier thickness (t m ) for 650 nm wavelength As seen from Table 3, most of the studies focus on two to three channels except a few [32,35] where six channels are considered. However, in the proposed work, eight channels are considered for simulation and successfully separated spatially. Further, in comparison to the other demultiplexers, the proposed demultiplexer has a comparably minimum FWHM bandwidth with minimum spacing between the channels and reasonable crosstalk. These features make it suitable for wavelength division multiplexing applications.

The 1 × N Power Splitter
By choosing the N ring resonators to resonate at the same wavelength, it is possible to design a 1 × N power splitter. The power splitting ratio (intensity at the focal spot) can be adjusted by varying the metal barrier thickness (t m ) between the bus and each ring. To begin with, the schematic of a 1 × 2, 3-dB power splitter with two rings of identical dimensions is shown in Fig. 6a. The geometrical parameters considered for the simulations are m = 7, w = 100 nm, t m = 15 nm, r = 469 nm, w g = 50 nm, t g = 50 nm, and t i = 5 nm.
The simulated field profile (|H z | 2 ) is shown in Fig. 6b and the intensity through the focal spot of both the ring resonators along the x-axis in Fig. 6c, for 650 nm wavelength. Almost all the power propagating in the bus is coupled to the rings and results in a sharp and intense focal spot similar to the one obtained for the plasmonic lens. It can be seen from the figure that the focal spots are of almost equal intensity. This result demonstrates the equal power-splitting functionality of the proposed structure. Variation in the intensity (|H z | 2 ) at the focal spot as a function of metal barrier thickness (t m ) is shown in Fig. 6d. On the same set of axes, the transmittance of the bus defined as the x component of the time-averaged power (P x ) at port P 2 normalized with respect to that at port P 1 is also plotted. It can be observed that as t m increases, the intensity at the focal spot decreases with an increase in transmittance. Results of Fig. 6d play a key role in the extension of the above-discussed 1 × 2 power splitter to a general 1 × N power splitter.
The design can be extended to obtain a 1 × N power splitter by placing N rings of identical geometrical parameters on either side of the bus. To ensure the required power is coupled to the rings, the metal barrier thickness (t m ) at each power splitting stage is arrived at from the results plotted in Fig. 6d. Note that power coupled to the power splitting stages will drop progressively due to metal loss in the bus as well as the power coupled into the preceding stages. A 1 × 8 power splitter is demonstrated as in Fig. 7a with the geometrical parameters considered for the simulation are as given by m = 7, w = 100 nm, r = 469 nm, w g = 50 nm, t g = 50 nm, and t i = 5 nm, for 650 nm wavelength. From the figure, it can be observed that there are four pairs of ring resonators arranged sequentially along the bus. To ensure equal coupling of power to the rings, t m must be the same for each pair. This t m is arrived at from Fig. 6d and was found to be 44 nm, 37 nm, 30 nm, and 16 nm, respectively, for the pairs from left to right. Figure 7a shows the simulated intensity distribution (|H z | 2 ), and Fig. 7b shows the intensity (|H z | 2 ) along the x-axis through the focal spot for all the 8-ring resonators. It can be observed that all 8 focal spots have almost equal intensity with almost zero power exiting at the output port of the bus. The sharp and intense focal spot along with the fact that there is negligible power exiting the bus shows that the power-splitting functionality is very efficient. Needless to say, the distribution of the rings need not be symmetric as in Fig. 7a, and it is also possible to obtain an unequal power splitting ratio by appropriately adjusting t m .

Conclusion
A compact and efficient plasmonic focusing lens based on a MIM bus waveguide, coupled with a circular ring resonator perforated with nanogrooves, is proposed and numerically demonstrated. The proposed structure uses a ring resonator to focus the light. Appropriate selection of the radius of the ring associated with proper positioning of the nanogrooves ensures that the ring resonates for a particular wavelength and generates an intense and sharp focus at the center of the ring. A suitable choice of the metal barrier thickness between the bus and the ring as well as the ring and the grooves results in maximum power coupling from the bus to the focal spot leading to an efficient lens. The proposed structure is compact with a ring radius of around 500 nm and with the focal spot at the center of the ring. The simulated focal spot has a symmetric FWHM intensity of 240 nm which results in a sharp focus.
The focusing structure's design was extended to obtain a 1 × N demultiplexer as well as a 1 × N power splitter with N taking a value as high as 8 without compromising any of the advantages of the focusing structure. The demultiplexer has a channel spacing of 20 nm with an FWHM bandwidth of less than 20 nm. Light is focused at the center of the ring corresponding to the demultiplexed wavelength with comparatively minimum light leaking out of the other rings. The measured crosstalk is − 10 dB which is highly acceptable as per the standards in the literature. Eight different rings resonating at the same wavelength generate eight focal spots with equal intensity at their center. This confirms the functioning of a 1 × 8 equal power splitter. Due to its efficiency and compact size, the proposed structure will be a versatile component in future all-optical signal processing integrated circuits.
The effect of geometrical parameters on the performance of the proposed plasmonic lens is discussed in this Appendix. In the results discussed, the parameters m, w, and w g are fixed as 7, 100 nm, and 50 nm respectively, whereas t i , t g , t m , r, and the orientation of the nanogrooves (θ) are varied. Figure 8a depicts the effect of variation of t i on the intensity of the focus. The intensity is seen to be falling with the increase in t i due to a decrease in power coupled to the nanogrooves. The impact of variation of the groove thickness (t g ) on the intensity of the focal spot is shown in Fig. 8b. As t g changes, n eff varies at the groove positions resulting in variation in the resonant mode distribution in the ring. Consequently, the power coupled to the nanogrooves varies. This is visible if Fig. 9a, b is compared with Fig. 2a, b where field (H z ) and intensity (|H z | 2 ) distributions were plotted for t g = 90 nm and 50 nm, respectively. Also, with the increase in t g , due to the reduced free space area enclosed by the ring structure, light emitted by the nanogrooves may not be able to converge at the center. These effects lead to a decrease in intensity at the focal spot. Fig. 8 Variation of intensity (|H z | 2 ) at focal spot as a function of a metal barrier thickness (t i ), b groove thickness (t g ), c metal barrier thickness (t m ), d radius of the ring (r) from the designed value, and e angular orientation (θ) Fig. 9 Simulated magnetic field (H z ) in (a) and its intensity (|H z | 2 ) in (b) for the proposed plasmonic lens with t g = 90 nm at 650 nm wavelength Intensity at the focal spot can be seen to be decreasing with variation in t m on either side of an optimum value as shown in Fig. 8c. This is due to the blue shift of the resonant wavelength of the ring [28]. The effect of variation in the radius (r) of the ring from its designed value is plotted in Fig. 8d. To focus the light emitted from the nanogrooves, it is essential that the spacing between them must be an integer multiple of the effective SPP wavelength (λ/n eff ) in the ring. Deviation in r from its designed value will alter the spacing between the nanogrooves, which disturbs the condition for constructive interference. This leads to a decrease in the intensity at the focal spot. Finally, the impact of the angular orientation of the nanogrooves (θ) on the intensity of the focal spot is shown in Fig. 8e. The rotation angle (θ) is defined as in the inset of Fig. 8e. There is an optimum value of θ which is periodic with a period of 2π/m. The optimum value of θ arises because the orientation of the nanogrooves has to be such that they are positioned at the peak intensity points of the resonant mode within the ring.
The above discussion reveals that variation in any of the parameters impacts the n eff resulting in a variation of the resonant mode in the ring. Light gets focused with the high intensity only if the parameters are such that light emitted by each nanogroove is maximum and results in a circular wavefront.
Author Contribution Theoretical ideas were developed by both authors. Simulations were carried out by Dr. Aparna U and Dr. M. Sathish Kumar prepared the manuscript.
Funding Open access funding provided by Manipal Academy of Higher Education, Manipal.
Data Availability Data underlying the results presented in this paper are available with the authors and may be obtained from the authors upon reasonable request.

Competing Interests
The authors declare no competing interests.
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