Advancing frequency fine-tuning: a theoretical approach to a novel metamaterial-inspired Bi-layer resonator

This study presents a novel adjustable device designed for precise frequency tuning within the S-band of the microwave spectra. In addition to the geometrical design and dielectric behavior of the resonator, this study identifies an influential governing factor that affects the resonant frequency. The proposed method utilizes a bi-layer split ring resonator configuration implemented on a 4×4cm FR4 epoxy substrate with a dielectric constant of 4.4. The substrate is coated with a 35 μm- thick layer of copper and patterned as split ring resonator. Frequency tuning was achieved by spatially separating the two parallel split ring resonators in increments of 800 μm. This innovative approach allows for a shift in the resonant frequency range from 2.36 GHz to 2.61 GHz, covering the desired frequencies in the S-band for applications such as biomedical and wireless communications. This study demonstrates that the alteration in the frequency domain is dependent on the distance between the two layers of split ring resonators. Compared to existing frequency tuning mechanisms, this adjustable bi-layer split ring resonator offers numerous advantages including simplicity, cost-effectiveness, and high sensitivity. The research employs a combination of finite-element simulations and theoretical analysis to validate the findings.


Introduction
Metamaterials, identified first by John Pendry in 1999-2000 [1,2], are engineered materials designed to possess electromagnetic properties that do not occur naturally.They have been widely utilized in various fields such as electromagnetic sensing, biomedical research, optical sensing, ultrasonic sensors, acoustic and seismic sensing, due to their exceptional efficiency and proficiency in research and sensing applications.One of the key advantages of metamaterials is their ability to provide high sensitivity detection mechanisms by exploiting the interaction between electromagnetic waves and substances being tested.In the realm of biomedical research, the radio frequency and microwave regime has proven to be highly beneficial for several decades, facilitating a broad range of diagnostic procedures, from conventional methods like x-rays and MRI to advanced techniques such as infra-red blood glucose sensing, operating at the atomic level.Among the different types of metamaterials, split ring resonators (SRR) represent the simplest form.A split ring resonator consists of a metal structure in the form of a ring or square, with one or multiple splits.When stimulated by electromagnetic waves, the metal portion behaves as an inductor while the split acts as a capacitor.This configuration allows SRR models to be characterized as LC equivalents.These highly sensitive metamaterials rely significantly on the dielectric constant of the split.By varying the dielectric constant, the resonant frequency of the SRR can be dramatically altered.As a result, these resonators are preferred and applied with precision in various sensing mechanisms.
Split ring resonators (SRRs) have garnered significant attention across numerous applications, ranging from biosensing and frequency tuning to dielectric sensing, antenna enhancement, microfluidic works, and microwaves and terahertz applications.Over the years, substantial progress has been made in advancing the efficiency and expanding the scope of applications for split ring resonators.
Split ring resonators exhibit significant promise across a diverse array of applications, encompassing antenna design and wave propagation [3][4][5][6][7][8], the development of microwave and terahertz filters [9][10][11][12], antenna miniaturization [6,[13][14][15], high-resolution microscopy and imaging [16][17][18], terahertz wave generation [19], the advancement of plasmonic devices [20,21], the creation of non-linear optical devices [22][23][24], as well as their application in the biomedical [25][26][27][28] and various sensing domains.For the microfluidic sensing applications, low-loss split ring resonators (SRRs) have demonstrated exceptional efficiency for determining the complex permittivity of common solvents.Silver-coated copper wire at microwave frequencies to showcase the effectiveness of SRRs in microfluidic applications is employed in [29].Additionally, research on biosensing applications utilizing antenna-coupled split ring resonators is conducted in [30].Another noteworthy study focused on enhancing the gain of microstrip patch antennas by incorporating complimentary split ring resonators (CSRRs) in the patch antenna's ground plane is presented in [31].The use of complimentary split ring resonators for sensing ethanol in microfluidic systems is demonstrated employing the sensing mechanism based on permittivity measurements.One of the remarkable features of metamaterials is their application for microfluidic chemical sensing.Furthermore, the benefits of split ring resonators for highly sensitive detection of electrolyte concentrations through permittivity measurements, showcasing the advantages over traditional sensing procedures is demonstrated in [32].The advancements in dielectric sensing using split ring resonators have also been significant over time.Low-cost contactless measurement of the dielectric constant of fluids using multiple CSRRs in the 2.45 GHz frequency range, demonstrating high sensitivity with errors of less than 5 percent is presented in [33].The resonant frequency of a split ring resonator is influenced by its geometry, and changes in the observed resonant frequency compared to the simulated values may occur due to various factors.While these variations might be inconsequential in specific use cases, they assume paramount significance in biosensing and metrological applications, where exacting control over resonant frequencies is imperative.Consequently, it becomes of utmost importance within this context to design devices and systems that enable effective and precise adjustment of resonant frequencies to meet specific needs.Over the past few years, significant strides have been made to address and overcome these challenges.A left-handed material (LHM) is created by combining electromagnetic band-gap (EBG) structures with square split-ring resonators (SSRR).This innovative material finds applications in filters, multiplexers, and couplers.The SSRR array is meticulously fabricated alongside a central conductive strip line to enable coupling.While the device exhibits a spectrum of resonant frequencies, achieving precise frequency control remains a notable challenge [4].An alternative method for obtaining frequency variations, as elucidated in [5], revolves around manipulating the geometry of resonators.This particular approach places a strong emphasis on the design and setup procedures of resonators.The investigation encompasses electric-field coupled (ELC) and complementary electric-field coupled (CELC) resonators, each integrated with a monopole antenna.Diverse configurations are employed to yield a multitude of electrically small antennas (ESA) with extremely low profiles.Moreover, the study delves into feeding mechanisms hinging on both bent monopole excitation and conventional monopole excitation.It is worth highlighting that distinct arrangements and configurations are necessitated to attain varying resonant frequencies.Consequently, while the study offers the potential for creating highly compact antennas with improved performance, it is imperative to recognize that this technique does not lend itself to straightforward frequency tuning approaches.A novel concept for ultraminiature antennas is introduced in [34], involving the integration of a high-permittivity substrate and sub-wavelength resonators.This proposal offers valuable insights into the development of compact, tunable antennas capable of operating across multiple frequency bands.The article delves into the utilization of split-ring resonators fed through coaxial cables, arranged in a split-loop configuration, to underscore the substantial impact of substrate permittivity on resonant frequencies.Notably, the scaling of these resonant frequencies is attained through the careful selection of substrates, necessitating the adjustment of substrate permittivity while keeping the geometrical parameters constant to achieve desired frequency variation.Furthermore, the study underscores the imperative requirement for a substrate permittivity exceeding 25 when targeting device profiles smaller than λ/20.In [35], a novel approach utilizing Surface Integrated Waveguide and Complementary Split Ring Resonator configuration (SIW-CSRR) is presented for the design of a three-port diplexer operating at resonant frequencies below the cutoff frequency of the waveguide.The research investigates the underlying filtering mechanism of this innovative model and explores an advanced cascaded configuration.Additionally, the study introduces a frequency tuning capability that leverages adjustments in the separation between the CSRR loops etched onto the waveguide surface, allowing for the manipulation of the coupling coefficient.It is noteworthy that while this study provides valuable insights into diplexer design, a notable limitation pertains to the labour-intensive process involved in precisely adjusting the CSRR loops to achieve the desired frequency tuning.Introduction of liquid metal to highly coupled split ring resonators, achieving resonant frequency variation through the controlled amount of liquid metal introduced to the system is presented in [36].Frequency tuning of a patch-controlled antenna by varying the split ring resonator pattern was achieved in [8].Other approaches for frequency tuning such as graphene based tuning approaches [37,38], Optical tuning of metamaterials for frequency tuning [39,40] and micro electro mechanical systems (MEMS) [41,42].The work presented in [43] presented an approach of multi-band metamaterial absorber using periodically arranged structures in THz regime.Bi-layer split ring resonators modelling is another trending approach to effectively utilize split ring resonators strength and capabilities.U-Shaped chiral structure based on split ring resonator is employed for polarization rotation realization of linearized polarized waves is discussed in [44].The polarization rotation is dependent upon the polarization angle incident waves and also a relation of T matrix dependency with respect to the propagation direction of the incident wave direction is discussed in this study.The utilization of a chiral bilayered metamaterial, as introduced in reference [45], enables the asymmetric transmission of linearly polarized electromagnetic waves in opposite directions.This presents a valuable avenue for achieving a high-efficiency cross-polarization converter through the incorporation of an array of resonators, thereby shedding light on potential advancements in this field.Bi-layer split ring resonator based circular polarizer is discussed in [46] with a new approach of FTEM (Fission Transmission of Electromagnetic waves) applied for better realization of polarization conversion.Circularly polarized wideband antenna array based on the bi-layer double split ring resonators is discussed in [47].The model presented in the study is utilized to convert linearly polarized waved into circularly polarized form.Similar study of polarization is performed in [48] discussing the circular polarization achievement using bi-layer split ring resonators.Studies incorporating bi-layer split ring resonators collectively offer insights into the exceptional efficiency, straightforward fabrication processes, and compact profiles of metamaterial structures.These attributes render them exceptionally effective for research applications.
In this study, we present a bi-layer model based on split ring resonator (SRR) that offers fine-tuning of frequency by modifying the spatial placement of the resonators.By altering the spatial configuration, the net capacitance changes, leading to variations in the resonant frequency.Achieving precise and accurate frequencies is crucial in applications such as biological sample analysis, military purposes, and metrological aspects.Compared to existing approaches, this proposed model is less complex, highly accurate, and cost-effective, utilizing easily available materials.Furthermore, this approach enables the attainment of very specific frequency ranges within a desired frequency domain.One of the most common challenges in designing resonator models is the mismatch between simulated and experimental results.Redesigning the entire model to address this discrepancy can be arduous and time-consuming.However, our approach offers a highly efficient solution by providing desired frequency responses that align closely with simulation results.By maintaining proper spatial separation between resonators, the desired resonant frequency obtained through simulations can be achieved experimentally, eliminating the need to repeat the entire design, simulation, and fabrication process.
The research is structured into seven sections.The introduction provides an overview of the study.The mathematical formulation of split ring resonators is explained in the second section.The third section focuses on the design of the SRR sensor and analyses the behaviour of multiple layers and their spatial relationship with frequency response.Building upon the second section, the fourth section provides a detailed analysis of the sensor with a specific resonant frequency of 2.45 GHz, including considerations of geometric aspects and separation.The sensitivity of the proposed model is examined in the fifth section.The sixth section presents the results, and the study concludes with the seventh section, offering key insights and conclusions derived from the research.

Background theory and design of split ring resonators
Split ring resonators are represented as an LC equivalent circuit, with the metal acting as an inductor and the gap between the rings acting as a capacitor (as illustrated in figure 1).When an external magnetic field varying over time is applied along the perpendicular axis of the resonator, the resulting excitation forces the electric current to flow through the splits between the rings.This constriction of the current produces a strong displacement current in the splits.As a result, the gap between the rings behaves as a capacitor, which gives rise to a distributed capacitance.
The inductance of single circular loop of radius R is given by (1), where w is the width of the loops as shown in figure 1 [49][50][51].
where r o is w 4 [50].When dealing with two conducting loops whose radii are similar in size, the mutual inductances between loops is estimated as The small factor x is given by x = r r 2r The capacitance C 1 for the inner loop of a resonator with a radius of r 2 is given by (4) [50] Cs 1 and Cs 2 represent the capacitances that arise in the gaps between resonators with radii of r 1 and r 2 , respectively.
The capacitance per unit length (C pul ) of charged metal strips, which are separated by a distance d and have a width w, is calculated using (5) [50] If a charge of +Q is distributed over half of a reference resonator loop, a corresponding charge of -Q is induced on the corresponding half of the second loop, due to the effect of the primary charge.The length along which this charge distribution is referred as the effective length (l eff ).
The mutual capacitance between the two loops is expressed in terms of the effective length and capacitance per unit length using (6) [50].
In order to make an approximation, the curved loops can be substituted with parallel metal strips that have a separation distance of 'w' between them.The length of this coplanar charged line is expressed as (7) [50] ( ) If the charges (+Q and −Q) are distributed uniformly along the metal strips, the effective length can be calculated as p = l a.
eff However, the non-uniform distribution of charges along the curved loops must be taken into account as it significantly affects the results.In order to determine the effective length of loop resonators, we utilize King's Theory of loop antennas [50,51].
Using this concept, the effective length of loop resonators is found to be = l a 2 .
eff Therefore, the equivalent capacitance (C mut ) of the split loops of SRR is approximated as (8) [50] In the case where the length of the split in the resonator is significantly greater than the separation distance, the capacitance resulting from the gap, Cs The value of C mut is given by (10).
Therefore, the net capacitance C net is expressed as (11).
Resonant frequency for this oscillator is hence formulated as (12).
As an extension of the previously discussed numerical results, this study introduces a new resonator model.Figure 3 shows the schematic of the resonator model along with its equivalent circuit diagram.The presented model has been optimized through the incorporation of a microstrip line and two double-split ring resonators, enabling the attainment of the desired high-frequency range while maintaining a low-profile resonator design.

Sensor design
The sensor is designed on FR4 epoxy substrate with the dielectric constant of 4.4.To achieve a resonant frequency of 2.45 GHz for the minimal-sized resonator, the dimensions are employed in microns (μm).The substrate is shaped into a square with dimensions of 40000 μm in length and width, and a layer of copper with a thickness of 35 μm is added on top.A microstrip line is designed to facilitate electromagnetic coupling, with dimensions of 5000 μm in width and 4000 μm in length.Two consecutive co-centric split ring resonators were designed on each side of the substrate, with their splits positioned 180°apart.The inner ring has an outer radius (r 1 ) of 3000 μm, while the outer ring has an outer radius (r 2 ) of 5000 μm, with each ring having a width of 1000 μm.A top view of the split ring resonator model is shown in figure 4.This design is achieved through careful material selection and precise dimensioning of the substrate, microstrip line, and split ring resonators to optimize sensor performance.
The centre-to-centre separation (x) between pairs of split-ring resonators is set to 24000 μm.The design is simulated using student version of Ansys Electronic Desktop (2021) High-Frequency Simulation Software (HFSS).In this study, simulations of a single-unit cell device are conducted as an essential initial step to gain insights into the fundamental properties and behaviour of the material in its isolated state.The primary objective is to explore the inherent characteristics of the single-cell device (two concentric split ring resonators with inner and outer radius r 1 and r 2 placed at a distance x, see figure 4), excluding any influence from neighbouring unit  cells.While periodic array simulations can unveil collective effects and extended behaviours in bulk materials, a single-cell device enables through investigation of the material's properties in isolation.This isolation is crucial for a comprehensive understanding of its behaviour and potential applications.

Material tuning approach
In the active tuning mechanism, external control or manipulation is leveraged to actively modify parameters within a system or device, enabling the attainment of desired performance characteristics.This process frequently employs feedback systems or external components to dynamically alter the system's behaviour in real-time.On the other hand, passive tuning involves configuring a system or component to instinctively react to specific frequencies or conditions without necessitating active intervention or external control.Passive tuning relies on inherent properties, such as resonance or impedance matching, to optimize the system's performance.In the context of our specific scenario, the boundary box housing wave ports is stimulated via an active tuning approach, wherein parameters are actively adjusted through external control or manipulation to achieve the desired performance outcomes.In contrast, the resonator positioned within the enclosure adopts a passive tuning mechanism, relying on its intrinsic properties to naturally respond to specific frequencies or conditions, thereby obviating the need for active intervention.
The metamaterial unit cell model is enclosed by walls made of perfect electric conductors (PEC) and perfect magnetic conductors (PMC), respectively oriented in parallel and perpendicular directions with respect to the Zaxis.Excitation of the unit cell model is achieved through the utilization of two waveports serving as input and output ports, as shown in figure 5.
The microstrip line functions as a transmission medium for the propagation of electromagnetic waves, offering the capability to guide and manipulate electromagnetic signals through careful control of its physical dimensions and electrical properties.In the context of this configuration, it serves as the essential means for both transmitting signals to the split ring resonator (SRR) and receiving signals from it.In essence, the microstrip line plays a pivotal role in ensuring the controlled and efficient conveyance of waves between distinct points, thus forming an integral component for reliable signal transmission within the designated path.In a two-port network, the resonant frequency is indicated by the dip in the S 21 parameter.To investigate the effect of adding another layer of the designed SRR, a bi-layer metamaterial model is simulated.The simulation involves analysing the S 21 scattering parameter over the same frequency sweep range of 0.5 GHz to 4.0 GHz used for the single-layer model.The resonant frequency characteristics of the resonator align with the theoretical and numerical aspects presented in section 2. The results show that the resonant frequency for the bi-layer approach (with separation of 1600 μm between layers) is observed to be 2.55 GHz, which is lower than the resonant frequency of 2.764 GHz observed in the single-layer model.
The schematic representation of the procedure is shown in figure 6.Each layer of simulation setup includes precisely two double-split-ring resonators (DSRRs).This configuration is opted for a simplified exploration of resonance phenomena.By employing two DSRRs per layer, the individual interactions and inherent properties are thoroughly analysed.To investigate the effect of layer separation on resonant frequencies in a two-layer metamaterial, simulations were performed at successive intervals of 800 μm between the layers (see figure 7).The S 21 parameter was analysed to determine the resonant frequencies for each separation distance.
The resonant frequencies are determined for different separations between two layers of the SRRs.The resonant frequencies for 2400 μm separation, 3200 μm separation, and 4000 μm separation are observed as 2.60 GHz, 2.63 GHz, and 2.65 GHz respectively.The separation between layers is then increased in increments of 800 μm, ranging from 1600 μm to 7200 μm, and the resonant frequency is observed for each increment of separation.Figure 8 shows the comparison between the resonant frequency of a single layer and the variation in resonant frequencies with spatial separation of resonators.
Figure 9 shows the relationship between frequencies and their corresponding distances.The tuning of resonant frequency in the designed model is achieved by increasing the separation between two consecutive layers.By varying the spatial distance between two resonators, a frequency range of 2.55 GHz to 2.76 GHz is obtained.

Optimization for 2.45 GHz
To comply with the allowed ISM frequency of 2.45 GHz for wireless research and applications, several geometrical modifications are made to the original design depicted in figure 4.These modifications involve variations in dimensions such as the length and width of the substrate, outer radius of the inner ring and outer radius of the outer ring, width of the rings, and microstrip line.These dimensions are altered multiple times until the desired resonant frequency of 2.45 GHz is achieved.The optimized model with dimensions for the 2.45 GHz resonant frequency is presented in figure 10.
Figure 11 shows the plot of S 21 parameter for the modified dimensions, where the resonant frequency is identified as the dip in the S 21 parameter curve.
Table 1 presents a comparison of the design parameters for the original model shown in figure 4 and the modified model shown in figure 10.
The surface current density and electric field of the resonator model presented in figure 10 is simulated and shown in figure 12.  Simulations were conducted to evaluate the distribution of surface current density across the proposed model and to validate its magnetic resonance characteristics at specific frequencies.In figure 12(a), the profile of surface current density along the metallic rings is presented.Notably, at its resonant frequency of 2.45 GHz, the surface current vectors exhibit a uniform distribution relative to the symmetry axis of the model.This behaviour is in line with the expectations since the resonator sustains a circulating current along the metallic rings.The surface current density (J) spans from 0.03 A m −1 to 28.947 A/m.Consequently, at this specific frequency, the effective magnetic field is oriented perpendicular to the device.Furthermore, for reference, we have included the accompanying electric field vectors in the figure 12(b).The E-field varies from a minimum of 6.047 V m −1 to a maximum of 39235.858V/m. Figure 12 provides valuable insights into the electromagnetic characteristics of the model as revealed by surface current density and E-field simulations conducted in HFSS.The simulations showcase a wide spectrum of electromagnetic field strengths and current densities within the structure.These results, thoughtfully presented on a logarithmic scale, offer significant insights into the resonant behaviour and electromagnetic interactions inherent to the model under examination.

Geometrical aspects
To analyse the impact of geometrical parameters on the resonant frequency of the design, several variations are made.The resonators are designed with the following geometrical modifications: i. Concentric circular rings

Bi-layer approach for 2.45 GHz model
A bi-layer approach is employed for the model depicted in figure 10 with a resonant frequency of 2.45 GHz.Two consecutive layers are placed at a separation of 800 μm between them.In addition to the geometrical variations, the effect of the position of the microstrip line on the resonant frequency is also investigated by varying the placement of the microstrip line on the upper or bottom surface of the substrate in both layers simultaneously.Various geometrical variations are analysed and simulated for 800 μm separation between consecutive layers, including: i. Concentric rings with microstrip on the upper surface of the Substrate.
ii. Concentric rings with microstrip on the bottom surface of the Substrate.
iii.Only outer rings with microstrip on the upper surface of the substrate.vi.Only inner rings with microstrip on the bottom surface of the substrate.
Figure 15 illustrates the geometrical variations for the bi-layer metamaterial model with a separation of 800 μm between two consecutive layers.
The designs depicted in figure 15 underwent resonance frequency analysis with consideration given to all the geometrical variations mentioned.The S 21 parameter plot for all of these geometrical variations is presented in figure 16.The dashed lines on this figure corresponds to the plots obtained when the microstrip is located at the bottom of the substrate.
The consistent resonant frequency regardless of whether the microstrip line is positioned on the top or bottom face of the substrate, and in the absence of a ground plane, is ascribed to several underlying factors.The inherent symmetry of the structure, arising from the lack of a ground plane and the location of the SRRs on the substrate's top surface, creates a comparable electromagnetic environment in both scenarios.Furthermore, the uniform electrical characteristics of the substrate and the dominance of a specific mode of excitation attenuates the influence of microstrip line placement on the resonant frequency response.
In the case of the model depicted in figure 15(a), which features a microstrip at the top with concentric rings, the separation between the two consecutive layers is gradually increased.This is done in increments of 800 μm, starting from an initial separation of 800 μm between layers and going up to 17600 μm.At each individual increment of the 800 μm step, the frequency shift is analysed.Figure 16 displays the S 21 parameter versus resonant frequency plot for the single-layer model, along with the resonant frequency at each step increment for the bilayer design.

Result
The resonant frequency behaviour of resonators is highly dependent on the design parameters.It is observed that resonators with multiple rings are more effective in providing low-frequency response at smaller dimensions.Additionally, significant variations in frequency responses can be observed for vertical in-plane displacements between split ring resonators, which is applied for achieving frequency tuning applications.
The graph presented in figure 17 showcases the relationship between separation (in μm) and frequency (in GHz) with a gradual increment of 800 μm.Each specific separation value is accompanied by its corresponding frequency.Starting with a separation of 800 μm, the frequency is measured at 2.36 GHz.As the separation increases, the frequencies demonstrate a gradual increment as follows: at 1600 μm, the frequency is 2.4 GHz; at 2400 μm, the frequency is 2.43 GHz; at 3200 μm, the frequency is 2.45 GHz; at 4000 μm, the frequency is 2.47 GHz; at 4800 μm, the frequency is 2.49 GHz; at 5600 μm, the frequency is 2.51 GHz.Continuing this pattern, at 6400 μm, the frequency is 2.52 GHz; at 7200 μm, the frequency is 2.54 GHz; at 8000 μm, the frequency is 2.55 GHz; at 8800 μm, the frequency is 2.56 GHz; at 9600 μm, the frequency is 2.57 GHz; at 10400 μm and 11200 μm, the frequency is 2.58 GHz.Further increases in separation lead to frequencies of 2.59 GHz at 12000 μm and 12800 μm.At 13600 μm, the frequency reaches 2.6 GHz, and the same value persists at 14400 μm, 15200 μm, and 18400 μm.The frequencies return to 2.6 GHz at 16000 μm and 16800 μm, and finally, at 17600 μm, the  frequency increases to 2.61 GHz.Notably, after a separation of 14400 μm, the frequencies appear to saturate around the value of 2.6 GHz.The precise attainment of a frequency shift from 2.36 GHz to 2.61 GHz is realized through the meticulous implementation of the technique of spatial separation variation of split ring resonators (SRRs), ensuring seamless transformation without necessitating any modifications to the underlying design parameters.The observed shift in resonant frequency with varying separation between the two layers is attributed to electromagnetic coupling, which leads to changes in mutual capacitance and inductance.Once the influence of coupling diminishes significantly, saturation occurs resulting in the stabilization of resonant behaviour.The gap between the split ring resonators (SRRs) plays a significant role in redistributing and affecting electromagnetic fields throughout this process.This approach's broadband feature provides multiple frequency ranges based on the step separation between the layers.While the resonant frequency may reach saturation beyond a certain separation between layers, multiple and precise frequency ranges can still be obtained.This simplifies the complex fabrication process of resonators for each required frequency, and Table 2. Comparison of traditional frequency tuning approaches along with their respective limitations.

Tuning Method Description Limitations References Circuit element-based technique
Involves altering the electrical properties of the metamaterial using active circuit components such as varactor diodes, semiconductors, tunable-capacitors or circuits.
Intricate circuitry, energy consumption, the risk of signal attenuation, restricted tuning range, and issues related to scalability. [

52-54]
Active Tuning Approach Dynamic adjustments of material properties, including but not limited to permittivity, permeability, conductivity, and doping concentrations, are achieved through external sources.
External power sources are necessary, often requiring the implementation of intricate circuitry.Achieving precise frequency ranges can be challenging.
Narrow tunability range due to inherent material properties.Demands significant effort, and achieving fine-tuning can be challenging.
[ 4,5,35,57] Bi-Layer chiral concept Resonator design using two layers of chiral materials to shift polarization and frequency of electromagnetic waves.
Constraints include limited frequency range adjustability, the necessity for precise fabrication, and susceptibility to environmental variations [44,45] Bi-Layer step separation approach for frequency tuning Employing a dual-layer metamaterial configuration to achieve tunability through the exploitation of interlayer interactions.Material parameter adjustments primarily concentrate on the inserts/pads positioned within the resonators.
Requires precise fabrication, sensitive to environmental factors.provides a cost-effective and efficient method for achieving fine tuning of frequency.It is also observed that the resonant frequency behaviour does not significantly change with respect to the microstrip position.Whether the microstrip is placed on the top layer of the substrate or at the bottom of the substrate, the resonant frequency does not change drastically.However, the number and dimensions of the rings of resonators play a significant role in deciding the resonant frequency behaviour of the resonator (See figure 15). Figure 18 shows the resonant frequency as a function of step separation.The results clearly indicate that the resonant frequency increases with increasing separation and reaches a constant value of 2.61 GHz after the layer separation reaches 14,400 μm.Table 2 presents a compilation of conventional methodologies, as well as those employed in this study, for the purpose of tuning metamaterials, accompanied with their respective limitations.
To ensure precise and reliable outcomes, the utilization of the spatial separation approach is employed to attain the desired frequency, accounting for inevitable variations between simulated and observed results arising from factors such as environmental conditions, instrumentation limitations, design accuracy, and methodology.This cost-effective and straightforward design of split ring resonators not only enables fine-tuning of frequencies but also proven invaluable in applications such as biosensing and imaging, where the utmost precision and accuracy of frequencies are indispensable.Consequently, the approach elucidated in this study presents a cost-effective and uncomplicated solution for achieving frequency fine-tuning using split ring resonators.

Conclusion
In conclusion, this research presents an innovative method that utilizes a bi-layer of split ring resonators for achieving precise frequency tuning in the S-band of microwave spectra.The resonators are designed on FR4 epoxy substrate, resulting in a compact device with dimensions of 4cm x 4cm.By adjusting the separation between the resonators within a range of 800 μm to 17600 μm, the desired resonant frequency range of 2.36 GHz to 2.61 GHz is achieved.The findings emphasize the critical role of the inter-resonator distance in influencing alterations within the frequency domain.Furthermore, the proposed model reveals that the resonant frequency is influenced not only by the design and dielectric properties of the substrate but also by the separation between the layers of split ring resonators.This innovative approach simplifies the frequency tuning process compared to conventional methods, offering exceptional effectiveness and versatility in finely adjusting resonant frequencies.Moreover, the model demonstrates a remarkable sensitivity to changes in permittivity, enabling frequency shifts through variations in the dielectric material between the resonator layers.These findings underscore the considerable potential of this approach in a wide range of applications, including biomedical applications, communication systems, military technologies, and research purposes.Additionally, this study presents a method that greatly streamlines the process of correlating simulated data with physical results, ensuring precise frequencies tailored to specific requirements.
The capacitance C 2 of the outer loop in a resonator, which has a radius of r 2 , is expressed as (3)[50]

Figure 2 .
Figure 2. Equivalent Circuit of dual ring concentric split ring resonator.

Figure 3 .
Figure 3. (a) Schematic of dual ring concentric split ring resonator coupled to a microstrip line (b) Circuit model of the SRR.L & C mut represent the inductance and capacitance of the SRR, respectively.Ls is the per-unit-length inductance of the microstrip transmission line.

Figure 4 .
Figure 4. Schematic top view of resonator model.

Figure 5 .
Figure 5. Schematic representations of boundary conditions for simulated single-unit cell device.

Figure 6 .
Figure 6.Separation of 1600 μm between the two layers.

Figure 8 .
Figure 8. S 21 versus resonant frequency as a function of step separations.

Figure 9 .
Figure 9. Resonant frequency comparison with respect to separation between bi-layers of resonator.
ii.Only outer rings iii.Only inner rings The designed models are illustrated in figure 13.The research conducted in this study explores the influence of various geometrical variations on the resonant frequency and S 21 parameter values of a model.When the model is configured with concentric double rings on (figure 12(a)), the resonant is measured at 2.4 GHz, accompanied by an S 21 parameter value of −35.3 dB.Alternatively, when the resonator is designed with only outer rings (figure 12(b)), a shift in the resonant frequency occurs, resulting in a value of 2.57 GHz, while the S 21 parameter is observed at −29.8 dB.Similarly, for the resonator featuring solely inner rings (figure 12(c)), the S 21 parameter is determined to be −33.2dB, alongside a resonant frequency value of 2.45 GHz.The frequency comparison for these distinct geometrical variations is illustrated in figure 14.

Figure 12 .
Figure 12.Visualization of vector field line plots representing (a) Surface Current Density and (b) Electric Field for the unit-cell device.

Figure 13 .
Figure 13.Resonator model with (a) Concentric circular rings (b) Only outer rings (c) Only inner rings.

Figure 14 .
Figure 14.Resonant frequency comparison for design parameters mentioned in figure 9.

Figure 15 .
Figure 15.Geometrical variations in resonator model including (a) Concentric rings with microstrip on top (b) Concentric rings with microstrip at bottom (c) Outer rings with microstrip on top (d) Outer rings with microstrip at bottom (e) Inner rings with microstrip on top (f) Inner rings with microstrip at bottom.

Figure 16 .
Figure 16.(a) S 21 versus Frequency plot with reference to model designs shown in figure 14(b) 800 μm separation between bi-layers of the resonator.

Figure 17 .
Figure 17.S 21 versus frequency plot with gradual increase in step separation between two layers.

Figure 18 .
Figure 18.Resonant frequency plot as a function of step separation.

Table 1 .
Design comparison for 2.45 and 2.76 GHz resonator models.