1 Introduction

Surface plasmon polaritons (SPPs) are electromagnetic waves that occur at the interface between a metal and an insulator at elevated frequencies. These waves arise from the collective oscillation of free electrons within the metal, resulting in unique characteristics. Notably, SPPs enable the confinement and guidance of light at the nanoscale, transcending the limitations imposed by diffraction. Consequently, SPPs have garnered significant interest as a promising technology for advancing photonic integrated circuits (PICs) (Schuller et al. 2010; Luo et al. 2019; Gramotnev and Bozhevolnyi 2010). In this context, plasmonic-based waveguides have emerged as a subject of great significance, offering distinct advantages for applications in communication and sensing (Rezaei et al. 2020; Khani et al. 2022; Hamouleh-Alipour et al. 2023). Unlike their dielectric counterparts, plasmonic waveguides facilitate the transmission of both electrical and optical signals (Srivastava and Dinesh Kumar 2020; Khani et al. 2020). Furthermore, the resemblance between metal–insulator-metal (MIM) plasmonic waveguides and other MIM configurations, such as microwave micro-strip structures (Khani and Hayati 2017; Jamshidi et al. 2019; Khani et al. 2019a), renders them suitable for seamless integration with other electrical or optical components within a single chip.

Over the last few years, miscellaneous components based on plasmonic configurations are suggested. These plasmonic topologies include small-scale optical filters (Oliveira et al. 2022; Ebadi and Khani 2023a; Khani et al. 2018a; Melo et al. 2022), sensors and biosensors (Li et al. 2016; Amoosoltani et al. 2020; Khani and Afsahi 2022), absorbers (Chen et al. 2016; Ebadi and Khani 2023b), splitters (Nikufard and Rostami Khomami 2016), demultiplexes (Khani et al. 2018b, 2021a; Xie et al. 2016), beam reflectors (Wu et al. 2017), switches (Khani et al. 2023; Negahdari et al. 2019), modulators (Ummethala et al. 2019; Khani et al. 2021b), logic gates (Sang et al. 2018; Pal et al. 2021), and so on. Optical filters (Rakhshani and Mansouri-Birjandi 2013; Khani et al. 2019b, 2019c; Duarte et al. 2021), despite essential and integral roles in a variety of applications such as fluorescence microscopy, wavelength division multiplexing (WDM), machine vision systems, and light detection and ranging (LIDAR) systems, their development and realization might have been overlooked. Different optical filters using various structures and resonator shapes such as ring (Rahman et al. 2018), U-shaped (Yu et al. 2020; Khani et al. 2019d), Fabry–Perot (Gao et al. 2020), disk (Lu et al. 2010), H-shaped resonators (Khani and Hayati 2022), and wide bandpass resonators (Chau 2020a; Chao et al. 2022), so on are suggested.

The development of high transmission-efficiency and miniaturized optical filters is of profound scientific and technological significance. These pursuits are pivotal in ensuring efficient and accurate signal propagation in optical communication systems. High transmission-efficiency filters minimize signal losses, maintaining the integrity of data transfer and enabling reliable communication. Furthermore, the miniaturization of optical filters contributes to the integration of advanced optical functionalities into compact devices (Agrawal 2012; Krainak et al. 2019). This is particularly relevant in wearable technology and portable optical instruments. These miniaturized filters also find applications in sensitive sensing systems, where they enhance data accuracy in fields like medical diagnostics and environmental monitoring. Additionally, the development of miniaturized filters aligns with the trend towards energy-efficient systems, reducing energy consumption in optical setups (Moscoso-Mártir et al. 2022).

Hence, in this paper, three plasmonic wide band-pass filters at near-infrared region (NIR) have been designed that demonstrate high transmission efficiency while offering highly compact size. The proposed filter structures consist of MIM plasmonic waveguides; stub, tilted T-junction resonators and right-angle trapezoid. The suggested optical filters have the physical dimension of 490 nm × 575 nm, 350 nm × 180 nm, and 420 nm × 150 nm, respectively, with transmission efficiencies varying between 74.2% and 94.4%. A parameterized genetic algorithm (GA) is applied that has contributed to achieving the numerical results. Due to their high efficiency, compact size, simple structure, and tunability, offering highly competitive performance, the designed filters may be employed in on-chip integration and digital signal processing circuits.

2 Device modeling and simulation method

In the conducted research, a robust device modeling and simulation approach is adopted to explore plasmonic wide band-pass filters. The study utilizes CST Studio Suite for simulating and analyzing the filters’ behavior. By leveraging this simulation method, the intricate design and optimization of these filters were thoroughly investigated. This section aims to provide insights into the filters’ characteristics and performance, furthering our understanding of their potential applications in various optical systems.

2.1 Simulation method

All three new structures consist of two metallic layers made of silver whose complex dielectric constant is taken from tabulated data by Johnson and Christy (Johnson and Christy 1972). A layer of insulator that is chosen to be Silica whose dielectric constant is 2.5 (Thirupathaiah et al. 2013), is sandwiched between the metallic layers. The selection of silica as the dielectric material for our plasmonic wide band-pass filters is based on its favorable optical properties, including low absorption and high refractive index, enabling efficient light confinement and transmission enhancement. Silica’s compatibility with fabrication processes and its biocompatibility further support its suitability for diverse applications. In contrast, replacing silica with air would compromise optical performance due to lower refractive index, increased losses, and fabrication challenges. Therefore, silica ensures optimal filter performance and versatility in various optical systems. The following considerations are taken into account while conducting the simulations. To ensure a single TM mode wave propagation in the plasmonic waveguide, the width of the waveguide is chosen significantly smaller than that of the wavelength of the incident light, which is then excited by a dipole source from Port I. The absorption (A(ω)), reflection R(ω), and transmission (T(ω)) parameters are related according to Ebadi et al. (2016); Ebadi et al. 2020):

$$A\left( \omega \right) = 1 - R\left( \omega \right) - T\left( \omega \right)$$
(1)

where ω is the angular frequency, and R(ω) and T(ω), which are achieved from scattering parameters, are defined as (Johnson and Christy 1972; Thirupathaiah et al. 2013):

$$R\left( \omega \right) = \frac{{P_{R} }}{{P_{i} }} = \left| {S_{11} } \right|^{2}$$
(2)
$$T\left( \omega \right) = \frac{{P_{T} }}{{P_{i} }} = \left| {S_{21} } \right|^{2}$$
(3)

where PT, PR, and Pi are transmitted, reflected and incident power, respectively. The mesh sizes are set to be 6 nm × 6 nm along the x and y directions, respectively, to obtain accurate results. A full-wave electromagnetic (EM) software tool, CST MWS, with frequency domain solver which is based on the finite element method (FEM) is utilized to attain the numerical results throughout this paper. The simulation environment is surrounded by perfectly matched layer (PML) boundary condition (Rahman et al. 2018). The numerical results achieved are for 2D structures, which can significantly reduce the simulation time and the computational power without sacrificing accuracy and be generalized to 3D configurations with finite thicknesses (Rahman et al. 2018; Chou Chau et al. 2020; Xie et al. 2016).

The complex propagation constant of an MIM waveguide \(\beta\) can be obtained by solving the dispersion equation (Rahman et al. 2018; Chou Chau et al. 2020):

$$\varepsilon_{in} k_{z2} + \varepsilon_{m} k_{z1} {\text{tanh}}\left( { - \frac{{ik_{z1} }}{2}\omega } \right) = 0$$
(4)

where \({k}_{z1}\) and \({k}_{z2}\) describe the \(z\) components of the wave vector in insulator and metal layers, respectively. They can be calculated by Chou Chau et al. (2020):

$${\text{k}}_{z1}^{2} = {\upvarepsilon }_{in} k_{0}^{2} - \beta^{2}$$
(5)
$${\text{k}}_{z2}^{2} = {\upvarepsilon }_{m} k_{0}^{2} - \beta^{2}$$
(6)

Here, the dielectric constants of the insulator and metal layers are represented by \({\varepsilon }_{in}\) and\({\varepsilon }_{m}\), respectively. The free-space wave number is described as \(k_{0} = \frac{2\pi }{\lambda }\), and \(\omega\) is the operating radial frequency. The propagation constant of the MIM waveguide (\(\beta\)) can be used to define the effective index as \(n_{eff} = \frac{\beta }{{k_{0} }}\).

2.2 Optimization framework

The investigation was carried out with the aim of optimizing the structures to achieve maximum transmission efficiency and minimal size. A parameterized GA was employed through the integrated CST MWS toolbox for this purpose (https://www.3ds.com/products-services/simulia/products/cst-studio-suite/) (Islam et al. 2020). The choice of GA was driven by its comprehensive optimization capabilities, which are well-suited for systems with diverse parameters. When employing the GA toolbox, certain key parameters were defined. The population size was set at 4 × 9, and the maximum number of iterations was determined as (q + 1) *k/2 + 1, where ‘k’ represents the population size and ‘q’ signifies the maximum iteration count. In this context, the maximum iteration count was chosen as 35 (https://www.3ds.com/products-services/simulia/products/cst-studio-suite/), resulting in a peak solver evaluation of 649. Furthermore, the probability of mutation, governing the potential occurrence of a mutation, was set at 60%. This probability is particularly relevant when both parents possess some degree of similarity in their genes. The GA operates on principles analogous to natural selection, incorporating three core mechanisms: selection of potential parents, mutation-based adjustments in offspring, and crossover to create new offspring.

To elucidate the operational principles of the GA optimization technique, a schematic flowchart of the GA optimization procedure was provided in Fig. 1. This diagram outlines the optimization stages and underscores the underlying methodology. Within this framework, precise optimization objectives were established based on predefined criteria. These criteria encompassed parameters such as the reduction of the scattering parameter’s magnitude, the minimization of stub lengths, and the reduction of spatial intervals between input and output ports. Two important parameters that are crucial for evaluating the performance and effectiveness of the GA in finding optimal or near-optimal solutions to the optimization problem at hand are defined as the following. Fitness Value: The fitness value represents the performance or quality of a potential solution (individual) within the population. It quantifies how well a solution satisfies the objectives of the optimization problem. Higher fitness values indicate better solutions that are closer to meeting the desired criteria (https://www.3ds.com/products-services/simulia/products/cst-studio-suite/) (Schmitt 2001). Convergence Rate: The convergence rate measures the speed or efficiency at which the GA converges towards optimal or near-optimal solutions. It quantifies the rate of improvement in fitness values over iterations. A higher convergence rate indicates faster progress toward finding better solutions, while a lower convergence rate suggests slower or stagnated optimization progress.

Fig. 1
figure 1

Illustration of the flowchart displaying the genetic algorithm optimization procedures employed in this study

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This pursuit is primarily geared towards enhancing the miniaturization of the proposed nanostructure while concurrently improving its efficacy. The outcomes and dimensions of the proposed structures, as presented in this study, were achieved through the application of the GA optimization technique. Notably, the focus was directed towards selectively optimizing parameters with substantial influence on the output signal, validated through a comparative assessment of unoptimized configurations and those refined through initial GA optimization. Overall, the methodology employed aligns with the broader objective of advancing the miniaturization and performance of nanostructures through the application of parametrized GA optimization, facilitating their integration within high-density photonic integrated circuits.

3 Device configurations and numerical results

The purpose of this section is to introduce the proposed filter structures and investigate their design procedures and discuss their performance. This includes understanding the crucial role of input port shapes, which differ between the three filter structures (Filter I, II, and III). Each filter features distinct geometries and resonator arrangements—Filter I with an MIM waveguide, bus resonator, and stub resonator; Filter II with an MIM waveguide coupled to tilted T-Junction resonators; and Filter III composed of an MIM waveguide coupled to a right-angled trapezoid resonator. These unique configurations necessitate tailored input port designs to ensure efficient coupling of light into the plasmonic waveguide and subsequent excitation of the resonators. This optimization considers the specific mode profiles and field distributions within the waveguides and resonators of each filter, maximizing transmission efficiency and spectral selectivity. Each filter boasts a unique geometry (Filter I: bus and stub resonators, Filter II: tilted T-junctions, Filter III: right-angled trapezoid) demanding specific input port shapes to achieve the following. Tailored coupling: To optimize light coupling into the MIM waveguide and excite resonators efficiently. Mode matching: Align input port mode profile with the waveguides and resonators to minimize losses.

3.1 Filter I (stub-based resonator structure)

Figure 2a shows the schematic configuration of the first designed flat-top band-pass filter (Filter I). The proposed structure is based on an MIM plasmonic waveguide coupled to the bus and stub resonators. There are some reasonable motivations why the stub resonator has been chosen to design the proposed filter structure. Employing stub-based plasmonic waveguide resonators offers valuable benefits for filtering functionalities. These resonators leverage the interaction between guided modes and localized resonant modes, allowing for precise control over signal transmission and manipulation. Their structural simplicity and compactness facilitate efficient integration into photonic devices. This approach enables tailoring of resonant frequencies and bandwidths to specific requirements, enhancing the filter’s performance. Additionally, the inherent compatibility with various materials and fabrication techniques makes stub-based resonators a versatile solution for achieving targeted filtering responses. In other words, they benefit from a small footprint and a simple fabrication process, which make them an ideal candidate for inclusion in PICs (Zheng et al. 2017a).

Fig. 2
figure 2

a Schematic configuration of the first proposed flat-top band-pass plasmonic filter (Filter I), b its transmission spectra for various lengths of the stub resonator, c Profile of electric field distribution at \(\lambda\)=900 nm for v = 375 nm

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For instance, Zhou and colleagues (Zhou et al. 2021) utilized a MIM waveguide with stub resonators and a ring resonator that can serve as both a refractive index sensor and stop-band filter. Simulations reveal adjustable and independent stop-band range and multiple Fano resonances. The proposed structure achieves a maximum sensitivity of 1650 nm/RIU and a figure of merit (FOM) of 117.8, offering potential for high-sensitivity sensors and wide-stopband filters (149 nm).

Chen and colleagues (Chen et al. 2016a) explored a nanoscale plasmonic filter, employing a single stub coupled MIM waveguide system. The system achieves tunable wide band-stop capabilities by incorporating a metal bar into the stub, facilitating a band-stop effect through direct resonance mode coupling. Adjustable bandwidth and center wavelength are achieved by manipulating the metal bar’s parameters.

Khani and colleagues (Khani et al. 2019b) introduced a plasmonic filter based on MIM configuration, featuring a hexagonal nano-resonator (HNR) coupled to multi-stub waveguides (MSWs) on both sides. Numerical simulations reveal a single transmission peak at 987 nm with 67% maximum transmission. The structure offers advantages of a high transmission peak resonance wavelength and a flat transmission spectrum beyond the resonance wavelength. By manipulating the HNR area, the resonance wavelength can be adjusted, suggesting potential for multi-channel demultiplexer formation through combining filters with different HNR areas.

The utilization of stub-based waveguide resonators finds strong justification in recent plasmonic filter studies. As evidenced by the findings in the works (Khani et al. 2019b; Zhou et al. 2021; Chen et al. 2016a; Chau 2020b), where similar stub-based structures have been employed, these resonators exhibit promising characteristics in achieving versatile filtering functionalities. Notably, the proposed stub-based designs in these studies have demonstrated the ability to achieve tunable band-stop and bandpass characteristics, along with the flexibility to adjust resonance positions and bandwidths by varying geometrical parameters.

Taking inspiration from these applications, we have introduced a novel stub-based waveguide resonator in our work. This resonator is designed to harness the advantageous features observed in the past examples, while focusing on providing wide bandpass filtering capabilities combined with an extremely compact size. The collective evidence from the mentioned studies supports our choice to adopt the stub-based approach, reinforcing our confidence in the potential of this resonator design to offer effective and adaptable wide bandpass filtering functionality. Therefore, in our study, we present in Fig. 2a a new stub-based waveguide resonator that is specifically tailored to excel in wide bandpass filtering applications. The proposed structure is based on a MIM plasmonic waveguide coupled to the bus and stub resonators.

The transmission spectra of the designed band-pass filter for different lengths, v, of the upper part of the stub are shown in Fig. 2b. This figure is obtained for the following physical parameters of the proposed structure: W = 80 nm, is the width of the input port, d = 20 nm, is the distance between the input port and the bus waveguide, r = 20 nm, is the width of the discontinuity in the stub resonator, C = 180 nm, is the coupling distance between the bus waveguide and the stub resonator, h = 40 nm is the height of the initial part of the stub, m = 60 nm is defined as the width of the straight waveguide, Wt = 50 nm, is the width of the stub resonator, and v shows the length of the stub segment, which set to be v = 350 nm, 375 nm, 400 nm, and 425 nm. As seen in Fig. 2b, the designed filter structure shows a wide flat-top band-pass spectral response, which resembles that of Fano-like resonance. In the context of our MIM plasmonic waveguide coupled to bus and stub resonators, Fano-like resonance may occur due to the interaction between localized modes supported by the stub resonator and the continuum of guided modes in the bus waveguide. The precise conditions for Fano resonance depend on various parameters such as the geometry, dimensions, and refractive indices of the structures, as well as the coupling strength between them as described in Chau et al. 2021a; Chao et al. 2021). Also, Fig. 2b shows that by increasing v from 350 to 425 nm, the resonance wavelength shifts to higher wavelengths. In other words, the resonance wavelength of the designed filter can be easily tuned by changing the physical parameter v, which makes the filter operational in a wide frequency range.

Furthermore, by modulating the length of the stub resonator, a desired wide flat-top transmission can be easily achieved. For instance, for v = 375 nm, the transmission efficiency is at least 80% at NIR from \(\lambda \hspace{0.17em}\)= 835.5 to 1105.8 nm, whereas the peak efficiency of more than 90% is found to be in this range, with the maximum peak at the wavelength of 1024.9 nm is 94.4% and offering a very good out-of-band suppression. The observed shift in resonance wavelength with increasing stub length v is attributed to the modulation of the effective optical path length and mode distribution within the structure. As the stub length increases, additional interaction and phase accumulation of guided modes occur, resulting in a redshift of the resonance wavelength (Chau et al. 2021b). Moreover, variations in stub length influence the coupling strength between the stub resonator and the bus waveguide, further contributing to the observed resonance wavelength shift (Chao et al. 2021). Adjusting parameter v allows precise control over the resonance wavelength, facilitating wavelength tuning across a wide frequency range. The total phase shift caused by the stub resonator is given by Thirupathaiah et al. (2013):

$$\Delta \varphi = \frac{{4\pi {\text{Re}}\left( {n_{{{\text{eff}}}} } \right)v}}{\lambda } + 2\varphi$$
(7)

where \(\varphi\) is the phase shift caused by the stub resonator, and the resonance wavelength is given by Thirupathaiah et al. (2013):

$$\lambda = \frac{{2{\text{Re}}\left( {n_{{{\text{eff}}}} } \right)v}}{{m - \frac{\varphi }{\pi }}}$$
(8)

where \(m\) is the order of the modes (Ebadi et al. 2020).

It is pivotal to control the coupling distance between the bus waveguide and the stub resonator to form a flat band-pass characteristic, and through to the GA optimization process an optimum value is identified, while broadening the band-pass transmission and reducing the size of the structure. Compared to similar studies that also employed GA to optimize the performance of their plasmonic devices (Jiang et al. 2020; Mahani et al. 2017), we have achieved a more favorable outcome concerning the size and transmission efficiency. Through this approach, we optimized key structural parameters to achieve maximum transmission efficiency and minimal size, thereby enhancing the overall performance of the proposed nanostructures. In summary, the performance of Filter I can be optimized by carefully adjusting these structural parameters within appropriate ranges.

We now discuss the impact of straight waveguide length in shaping Filter I’s transmittance spectrum through two key mechanisms, which is directly applicable to Filter II and Filter III as well: (1) Tailoring resonance conditions: The length directly affects the resonance conditions within the filter cavity, influencing the peak wavelength of the transmittance spectrum. influence stems from changes in the effective optical path length. Longer waveguides generally lead to red-shifted peaks (towards longer wavelengths) by effectively increasing the path length experienced by light. Conversely, shorter waveguides tend to cause blue-shifted peaks (towards shorter wavelengths). By carefully adjusting the length, we can strategically position the peak wavelength to match specific application requirements, ensuring optimal performance within the target spectral range. (2) Balancing propagation losses and insertion loss: The waveguide length also significantly impacts propagation losses and consequently, the filter’s insertion loss. As the length increases, light travels a longer distance, experiencing higher propagation losses that can attenuate the transmitted signal and contribute to a higher insertion loss. This necessitates a careful balance between achieving the desired peak wavelength and maintaining acceptable insertion loss levels (Mittapalli and Khan 2019). Excessively high insertion loss can significantly degrade the overall filter performance. In our design, the chosen length was carefully selected to achieve the desired balance between peak wavelength positioning and acceptable insertion loss for creating a flat-top bandpass spectrum as shown in Fig. 2b, that is achieved by using the GA process.

The magnitude of electric field distribution at λ = 900 nm for v = 375 nm is shown in Fig. 2c. It confirms the efficient coupling between the bus waveguide and the stub resonator, and that that the incident light passes through the filter over the wide flat-top transmission range. In other words, the flat-top transmission profile and efficient coupling between the bus waveguide and the stub resonator can be better understood by considering the constructive interference and mode matching principles. The geometry of the stub resonator interacts with the incident light to create standing wave patterns within the structure. The electric field distribution at λ = 900 nm (with v = 375 nm) indicates the presence of nodes and antinodes within the resonator, characteristic of resonant modes that reinforce each other (Hasan et al. 2020).

This constructive interference leads to an enhanced transmission through the structure over the specified wavelength range. In addition, the transmission characteristics of the filter can also be influenced by additional structural factors, including the dimensions of the input port and the coupling gap. This offers an extended level of flexibility in tailoring and managing the optical output signal properties of the filter.

The width of the straight waveguide (m) is pivotal for achieving the filter’s performance. A narrower waveguide results in a higher effective refractive index, which could weaken coupling efficiency and potentially diminish transmission within the desired band. Conversely, a wider waveguide might induce overcoupling, leading to distortions in the flat-top response and increased insertion losses. These considerations are crucial in achieving the desired filtering characteristics of Filter I.

As detailed in subchapter (Sect. 2.2), we meticulously defined the constraints based on the initial design parameters of each nanostructure. These constraints were strategically crafted to align with our overarching optimization objectives, which prioritize criteria such as reducing the scattering parameter’s magnitude, minimizing stub lengths, and narrowing the spatial intervals between input and output ports. These criteria served as guiding principles throughout the optimization iterations, ensuring that the refined structures adhere to the desired performance metrics while advancing towards optimal solutions. Parameter constraints: The minimum and maximum values for each parameter in the GA optimization were carefully chosen based on two key considerations: (1) Physical limitations: Material properties: The refractive indices and absorption coefficients of the materials used (silver, silica) define achievable ranges for parameters like feature sizes and waveguide widths. Fabrication constraints: Current fabrication techniques impose limitations on minimum feature sizes and achievable tolerances. Exceeding these limitations would render the design impractical. (2) Performance requirements: Target wavelength range: The parameter space explored by the GA was restricted to ensure that the optimized solutions operate within the desired wavelength range (e.g., NIR in our case). Transmission efficiency: Certain constraints were imposed to guarantee that the optimized filters maintain a minimum level of transmission efficiency throughout the target band (Rashedul et al. 2022).

By carefully considering these factors, we ensured that the GA explored a realistic and meaningful parameter space, leading to optimized solutions that are both physically realizable and meet our performance requirements.

Precisely, Filter I’s performance hinges on several structural parameters: the width (Wt) and height (h) of the stub resonator, coupling distance (C), the distance between the input port and bus waveguide (d), and width of the input port (W). Each parameter critically influences the filter’s transmission characteristics and spectral response. Widening Wt enhances mode confinement and coupling efficiency, potentially boosting transmission efficiency. However, excessively large widths may introduce additional losses. Increasing h strengthens mode confinement and resonance interactions, potentially enhancing transmission efficiency. Optimal heights are chosen to prevent excessive scattering and losses. Smaller C values generally result in stronger coupling and higher transmission efficiency, particularly for achieving a flat-top band-pass response. Optimization avoids excessive mode mismatch and loss. Shorter d values facilitate efficient light coupling and enhance transmission efficiency, particularly for achieving broad bandwidth and flat-top responses. However, overly short distances may lead to performance degradation. Optimal W values ensure efficient light coupling and minimize insertion loss. Narrower ports may enhance transmission efficiency but could limit bandwidth and spectral selectivity.

In summary, optimizing Filter I’s performance involves adjusting these parameters within specific ranges. Based on simulations and GA optimization studies, suggested ranges are: Wt: 45–65 nm, h: 40–60 nm, C: 170–220 nm, d: 20–30 nm, and W: 75–95 nm.

Specifically, we aim to achieve high-transmission efficiency, wide bandpass, miniaturizing size, while improve the out-of-band rejection for each of the proposed structures. For instance, for Filter I, after initial design of the structure, we set the following criteria for the optimization based on GA. (1) Transmission efficiency: > 70%, bandwidth > 200 nm, size < 550 nm × 620 nm, as well as the out-of-band rejection of > 20 dB. Besides, it is worth pointing out that a similar trend for Filter II and Filter III are achieved.

As shown in Fig. 3a and b, the optimization process demonstrates notable improvements in the proposed nanostructure’s performance metrics. Transmission efficiency and bandwidth exhibit consistent increases over iterations, indicating enhanced light transmission and spectral range. In spite of fluctuations, the nanostructure's size stabilizes towards convergence, suggesting effective design exploration. Additionally, out-of-band rejection gradually improves, highlighting the optimization's success in filtering unwanted spectral components. These trends underscore the iterative process's efficacy in enhancing key performance metrics, affirming progress toward achieving design objectives. To effectively interpret the plot in Fig. 3a and b, one can define the following descriptions. Negative convergence rate: The presence of negative values for the convergence rate indicates that the fitness values (transmission efficiency, bandwidth, size, out-of-band rejection) are decreasing over certain iterations. Negative convergence rates suggest that the optimization algorithm may be diverging or encountering difficulties in improving the design parame’ters. Fluctuations in convergence rate: Fluctuations in the convergence rate over iterations suggest variability in the optimization process. These fluctuations may occur due to various factors such as the algorithm exploring different design spaces, encountering local optima, or adjusting its search strategy based on the fitness landscape. Overall trend: Despite fluctuations, the convergence rate exhibits an overall trend over iterations. A decreasing trend in the convergence rate suggests that the optimization process is improving the design’s fitness values over time, albeit with occasional setbacks or fluctuations. The GA optimizes multiple parameters simultaneously, focusing on v and other structural parameters that based on the simulations that we have identified contribute more profoundly to adjust the optical output response, and offer valuable insights beyond achieving the optimal design presented in Fig. 2b.

Fig. 3
figure 3

a Optimization convergence analysis for transmission efficiency, bandwidth, size, and out-of-band rejection, and b performance evolution of optimized design of the Filter I

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Filter I’s performance exhibits an intricate interplay between various structural parameters, where each dimension critically influences mode confinement, coupling efficiency, and ultimately, the spectral response. Balancing wider stub resonators (Wt) for enhanced transmission against excessive loss, optimizing stub height (h) for strong confinement without scattering, and carefully tailoring coupling distance (C), input port distance (d), and width (W) are crucial to achieving the desired flat-top band-pass response and minimizing mode mismatch issues. This meticulous optimization demonstrates the critical interplay between parameters and highlights the need for a comprehensive understanding to achieve optimal filter performance.

Besides, different parameters that were chosen for optimization, are selected based on our initial designs for all the proposed filters. For instance, in Fig. 2b, variations in v significantly impact the filter’s transmission spectrum. Analyzing its effect helps us isolate its specific influence on critical metrics like peak wavelength (λ_peak) and bandwidth (Δλ). This knowledge deepens our understanding of the underlying physics governing the resonances and mode interactions within the stub-based resonator. Specifically, we can investigate how changes in v affect the effective optical path length and mode distribution, leading to shifts in λ_peak and Δλ (https://www.3ds.com/products-services/simulia/products/cst-studio-suite/) (Schmitt 2001).

Moreover, this information equips us to guide future filter designs targeting similar functionalities. By identifying parameters like v with a notable influence on the output, we prioritized them during optimization processes in Filter II and Filter III. This knowledge can significantly accelerate the development of improved filters with desired characteristics.

In other words, we can summarize the reward function employed in our GA optimization process as the following: Reward function: The reward function serves as the objective or fitness function in the GA, guiding the search for optimal solutions. In our study, the reward function is designed to quantify the performance of each candidate solution (i.e., filter design) based on specific criteria relevant to our optimization goals that were mentioned earlier. Optimization objective: For our filter design problem, the reward function evaluates the performance of each design based on criteria such as transmission efficiency, bandwidth, size, and out-of-band rejection. These criteria collectively define the desired characteristics of the filter. Fitness evaluation: During each iteration of the GA, candidate solutions (represented as chromosomes) are evaluated based on their fitness, determined by the reward function. Solutions with higher fitness values, indicating better performance according to the defined criteria, are selected for further genetic operations like crossover and mutation, which will be conducted in the software tool, CST MWS (https://www.3ds.com/products-services/simulia/products/cst-studio-suite/) (Schmitt 2001). Optimization outcome: By iteratively evaluating and selecting solutions with higher fitness, the GA converges towards optimal or near-optimal filter designs that satisfy the specified performance criteria.

3.2 Filter II (tilted T-junction resonator structure)

In our pursuit of enhancing wide bandpass filtering capabilities, we introduce a tilted T-junction plasmonic resonator which builds upon our initial stub-based resonator design. The incorporation of the titled T-junction capitalizes on the synergies between the two resonator configurations, tapping into the demonstrated attributes of the stub resonator while introducing the dimension of the tilted T-junction.

In particular, Thirupathaiah and colleagues (Thirupathaiah et al. 2014) designed a compact and low-loss concurrent dual-band diplexer tailored for high-density multiband photonic integrated circuits. The proposed diplexer employs a nanoplasmonic MIM waveguide featuring a chamfered T-junction and incorporates two bandpass filters (BPF) in its output arms. The BPFs, designed with stepped width resonators, operate simultaneously at 1427-/1665-nm and 1355-/1595-nm wavelengths. The optimized design showcases favorable characteristics, including a chamfered T-junction with return loss exceeding 8 dB and dual-band BPFs exhibiting insertion loss below 5 dB.

Tang and coworkers (Tang et al. 2015), introduce a MIM straight waveguide with quadrant ring resonators (QRRs). The design is extended to T- and X-shaped waveguides with QRRs, enabling efficient control of plasmonic flows at junctions. The T-shaped system exhibits Fano effects and efficient SPP flow control, and may find applications in optical integration, communication, and information processing.

Researchers propose a highly sensitive refractive index sensor using a specially designed nanostructure in the form of T-shaped cavity (Chau et al. 2019). This structure, made of metal and insulator layers with strategically placed metal nanorods, traps light in a cavity, creating a strong response to changes in the surrounding material’s refractive index. The design achieved a record-breaking sensitivity of 8280 nm/RIU, surpassing other similar sensors. Its simplicity and ease of fabrication make it promising for future applications in miniaturized optical sensors and potentially even temperature sensing.

Consequently, a plasmonic T-junction waveguide resonator is introduced. Figure 4a shows a new flat-top band-pass filter structure (Filter II) which is composed of an MIM waveguide coupled to five identical T-junction resonators. The GA process, based on the framework displayed in Fig. 1 is utilized to find out the optimized value of the coupling distance between the resonators and bus waveguide in order to form the flat-top transmission band at the NIR, while shrinking down the size of the proposed structure.

Fig. 4
figure 4

a Schematic illustration of the second proposed flat-top band-pass plasmonic filter (Filter II), b transmission profile as a function of wavelength for N = 1 to N = 5 tilted T-Junction, c transmission spectra of Filter II for various lengths of the upper side of the resonator, d Profile of electric field distribution at \(\lambda \hspace{0.17em}\)= 1000 nm for b = 30 nm

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The simulation results of the flat-top transmission band for the structural parameters of Filter II (the width of the input port or W = 40 nm, the distance from the input port to the T-junction resonator or a = 60 nm, the initial length of the resonator or e = 30 nm, the smaller width of the resonator or f = 30 nm, the width of the upper side of the resonator or k = 80 nm, and the distance between the T-junction resonators or j = 100 nm is shown in Fig. 4b. By increasing the number of stubs, one would expect a more effective interference and more pronounced transitions between reflection and transmission bands in a transmission spectrum diagram. As shown in Fig. 4b, increasing the number of stubs leads to a decrease in the width of the transmission band stemming from a decrease in the transmission at the longer wavelengths. When additional stubs are introduced, they introduce more reflective surfaces and perturbations in the waveguide structure. These stubs function as additional scattering elements, enhancing the coupling between the stubs and bus waveguide and increasing the optical losses in the resonator.

For instance, for N = 3 stubs, in the case of b = 30 nm, the wide flat-top transmission band occurs from the wavelength range of λ = 676.5 to 1101 nm with an average efficiency of over 74.2%, which is illustrated in Fig. 4c. Also, the transmission efficiency is found to be more than 80% in the wavelength range of λ = 804.1 to 1065 nm, with a peak efficiency of 85.9% at the wavelength of 917.1 nm. Furthermore, one can easily tune and widen the transmission band simply by choosing the appropriate value of the height, b, of the upper part of the T-junction with good out-of-band rejection. While the average reflection is found to be around 1%, the average absorption rate is higher, and is in the range of 20%. There can be several reasons for this. Firstly, metals and dielectrics absorb due to the interaction between incident electromagnetic fields and electrons in the material. Secondly, resonance and coupling effects designed for transmission enhancement can inadvertently lead to localized energy confinement and absorption. As the SPPs propagate along the interface they lose energy due to interaction with the metal. This damping is caused by scattering of electrons by the SPP field (Yu et al. 2019; Qi et al. 2019).

Filter II introduces a tilted T-junction resonator, synergizing with the stub design to achieve wider and flatter bandpass filtering. This novel geometry fosters multiple coupled resonant modes, promising enhanced spectral selectivity, transmission efficiency, and control over resonance effects compared to conventional T-shaped resonators. The conventional T-shaped resonator typically possesses a single resonant mode, resulting in a singular peak in the transmittance spectrum, which might not inherently exhibit a flat-top profile (Abdolalipour and Pourmahyabadi 2022). Conversely, the proposed tilted T-junction resonator design introduces multiple tilted sidewalls, fostering the emergence of multiple coupled resonant modes. These modes interact constructively, fostering a wider and flatter transmittance peak compared to the conventional T-junction. This flat-top characteristic offers improved uniformity in transmission across a desired wavelength range, potentially enhancing spectral selectivity and transmission efficiency. Moreover, the tilted T-junction resonator design may offer enhanced control over resonance effects and scattering mechanisms, leading to more tailored transmission characteristics. This enhancement is attributed to optimized interactions between resonator components and the incident light field, facilitated by the unique geometry of the tilted T-junction.

Besides, the number of T-junction resonators was indeed considered a design parameter in our GA optimization process. The inclusion of the number of T-junction resonators in the optimization process allowed us to identify optimal configurations that effectively balance performance metrics and design constraints.

Finally, the magnitude of the electric field distribution at λ = 1000 nm is shown in Fig. 4d. In this figure, the value of b is chosen as 30 nm. The E-field profile demonstrates that the SPPs are not fully transmitted from the input to the output port, which is due to the coupling between the reflected waves at the end of the T-junctions and bus waveguide. In fact, the presence of the T-junctions and the specific choice of b (30 nm) significantly influence the behavior of SPPs propagating within the structure. The behavior can be explained through the concept of mode coupling and interference effects. As SPPs propagate along the waveguide, they are subjected to various scattering processes, including reflection and transmission at points of geometric discontinuity, such as the T-junctions. At λ = 1000 nm, the chosen wavelength for analysis, the incident SPPs interact with the T-junctions and the bus waveguide. The magnitude of the electric field distribution depicted in Fig. 4d reveals the presence of standing wave patterns within the structure. The observed deviation from full transmission arises from the interference between the directly transmitted SPPs and the waves that have been reflected and scattered due to the geometry of the T-junctions. This interference results in regions of constructive and destructive interference along the propagation path of the SPPs (Ogawa and Kimata 2018; Suárez et al. 2017). Particularly, the coupling between the reflected waves at the end of the T-junctions and the bus waveguide plays a crucial role in shaping the electric field profile. The constructive interference of these reflected waves with the incident SPPs leads to enhanced fields in specific regions, while destructive interference suppresses the field strength in others. This spatially varying interference pattern contributes to the observed distribution of the electric field magnitude (Zhu et al. 2012).

3.3 Filter III (right-angled trapezoid resonator structure)

The transition from a stub-based plasmonic MIM waveguide resonator to a right-angled trapezoid resonator, while maintaining the same flat-top filtering characteristics, is achieved through a methodical evolution of the geometry while preserving key design principles. Starting with the stub-based resonator, the foundation lies in the stub-like configurations that have already demonstrated the desired flat-top transmission behavior in Figs. 2b and 4b. This stub configuration creates a resonance that supports the desired filtering functionality. The stub’s length, width, and positioning are carefully optimized by employing the GA framework displayed in Fig. 1 to achieve the desired resonance wavelength and transmission characteristics.

To evolve this design into a right-angled trapezoid resonator, the core concept of maintaining a stub-like geometry is retained. However, the geometry is adapted to include a right-angled trapezoid shape while ensuring that the critical design parameters such as length, width, and positioning are suitably adjusted. The transition is guided by the insights gained from the stub-based resonator, allowing for a seamless extension of the design concept.

Specifically, Zheng and colleagues (Zheng et al. 2017b) suggested a plasmonic induced transparency (PIT) system which is devised using a MIM stub coupled with a trapezoid cavity resonator. Overlapping spectra of specific trapezoid resonator modes and the stub mode led to the PIT effect. Adjusting the stub and trapezoid resonator positions reinstates the PIT effect, affecting coupling strength and plasmonic induced transparency.

Jaiswal and coworkers (Zhou et al. 2017) suggested a MIM waveguide-coupled trapezoid cavity. An asymmetrical break induces an asymmetric Fano profile in the transmission spectrum, leading to refractive index sensitivity and high Q-factor. The Fano resonance could be adjusted via structural parameters. The refractive index sensitivity, Q-factor, and FOM were measured to be approximately 750 nm/RIU, 68.3, and 65.2, respectively.

Bahri and colleagues (Bahri et al. 2022) explore a novel refractive index (RI) sensor featuring tunable plasmonic nanostructures in the NIR and mid-infrared (MIR) range. The proposed sensor employs MIM configurations along with hexagonal, trapezoid, and diamond cavity (HTD) shapes. The biosensor is optimized to achieve high sensitivity (5155 nm/RIU), covering a broad refractive index range from 1 to 1.7. This design holds promise for biochemical and medical diagnostic applications, boasting high wavelength resolution of 3.5 × 106 RIU − 1.

Recently, researchers presented a novel plasmonic structure utilizing side-coupled isosceles trapezoid cavities within a MIM waveguide, demonstrating tunable off-to-on plasmon-induced transparency (PIT) response through adjustments in structural parameters (Wang et al. 2018). This design holds potential for the development of highly integrated photonic devices due to its tunable transparency characteristics.

Building upon the insights gained from the aforementioned studies, we propose a novel right-angled trapezoid resonator. This design draws inspiration from the plasmonic structures in Filter I and Filter II, (Figs. 2a and 4a, respectively) aiming to further enhance the flat-top bandpass filtering characteristics observed in these systems. Figure 5a shows the third proposed flat-top band-pass filter (Filter III). This structure is based on an MIM waveguide coupled to a right-angled trapezoid resonator to create a flat-top transmission characteristic in the NIR range.

Fig. 5
figure 5

a Schematic illustration of the third proposed flat-top band-pass plasmonic filter (Filter III), b its transmittance spectra for various values of the side lengths of the right-angled trapezoid, c Profile of electric field distribution at \(\lambda\) = 900 nm for b = 35 nm

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The numerical results of the designed filter (Filter III) for the specified physical parameters are shown in Fig. 5b. The values of the obtained physical parameters are as follows: W = 80 nm, defines the width of the input port, d = 40 nm, shows the distance from the input port to the bus waveguide, e = 160 nm, illustrates the distance from the left bus waveguide to the right-angled trapezoid resonator, f = 35 nm, shows the lengths opposite to the right side, m = 30 nm displays the width of the straight waveguide, a = 50 nm, depicts the width of the resonator, and k = 185 nm, is the distance from the trapezoid to the right bus waveguide. Simulation results illustrated in Fig. 5b, indicate that a wide band-pass is obtained for different side lengths, z, of the trapezoid.

When z = 35 nm, the simulation results show that a wide band-pass happens in the wavelength range of λ = 880.1 to 1169.9 nm with an average efficiency of 74.3%. Indeed, in the wavelength range of 892.4 to 1149.8 nm, with at least 80% efficiency, the peak at 1071.8 nm has a 92% efficiency. Figure 5b shows that raising the resonator length not only shifts the flat-top band-pass to longer wavelengths but also flattens the pass-band.

Besides, the width of the input port (W) plays a crucial role in governing the interplay between coupling efficiency, filtering performance, and fabrication feasibility. The width directly influences mode confinement within the waveguide. A wider waveguide leads to weaker confinement, while a narrower one leads to stronger confinement. Stronger confinement generally results in increased coupling efficiency between the input waveguide and the resonator. This can potentially affect both the peak wavelength and bandwidth of the transmittance spectrum. However, excessively narrow waveguides can also lead to higher propagation losses, degrading the overall transmission efficiency.

Thus, finding the optimal W value requires careful consideration of several factors: Desired operating wavelength: Matching the passband center to the target application. Required transmission efficiency. Balancing strong filtering with acceptable signal attenuation. Acceptable insertion loss: Maintaining overall signal strength within desired limits. And, fabrication feasibility: Considering practical limitations in manufacturing smaller features. Therefore, we carefully considered these factors when selecting the optimal W value for Filter III. By balancing the need for strong coupling for efficient filtering with minimizing insertion loss for overall performance, we achieved the desired flat-top bandpass response shown in Fig. 5b.

The electric field distribution pattern for Filter III is illustrated in Fig. 5c, acquired at a wavelength of λ = 900 nm and a distance of z = 35 nm. The outcome substantiates the effective and robust coupling between the resonator and the bus waveguide. The effective and strong coupling observed in Fig. 5c can be attributed to the resonant interaction between the guided electromagnetic modes in the resonator and the bus waveguide. As incident light enters the structure, it interacts with the localized surface plasmons within the resonator. This interaction leads to energy exchange and efficient energy transfer between the resonator and the bus waveguide (Liu et al. 2015; Divya et al. 2022). The geometry and dimensions of the right-angled trapezoid resonator play a significant role in shaping the distribution of the electric field. Due to the specific design of the structure, the resonant modes within the resonator are aligned with the guided modes of the bus waveguide. This alignment creates a coherent coupling between the two, facilitating the smooth propagation of energy from the input port to the output port.

The efficient transmission of energy can be understood through the concept of constructive interference. The resonant modes within the trapezoid resonator are in phase with the modes of the bus waveguide at the resonant wavelength. This constructive interference enhances the amplitude of the electric field in the vicinity of the output port, resulting in efficient energy transfer. Additionally, the gradual change in the geometry of the resonator ensures that the phase of the electromagnetic field is well-matched between the resonator and the bus waveguide, further facilitating the coupling (Zhou et al. 2022; Zhang et al. 2022). This coherent coupling and phase matching lead to a significant overlap of the electric field distributions between the resonator and the bus waveguide, allowing the energy to be efficiently guided from the input to the output port.

4 Discussions and comparisons

Fabrication methods for nanoplasmonic MIM waveguide filters encompass a range of techniques, each with distinct attributes. Electron Beam Lithography (EBL) delivers precise nanoscale patterning (Horák et al. 2018). Nanoimprint Lithography offers cost-effective high-throughput production (Gupta et al. 2021), while focused ion beam (FIB) Milling offers versatility, enabling subtractive and additive processes (Chen et al. 2016b). These methods collectively enable diverse nanoplasmonic MIM waveguide filter designs catering to various applications in nanophotonics research and technology. Notably, a novel avenue emerges through the proposal of a modified laser interference lithography (LIL) technique, with the intention of curtailing degradation in optical filter performance. This proposition, if realized, could pave the way for economically viable and expansive plasmonic filters (Do 2018). Additionally, a recent approach has surfaced that leverages SiN waveguides as an experimental foundation based on the silicon-on-insulator (SOI) substrate. This innovation offers a pathway for plasmonic mode conversion, interconnecting a 50 nm × 20 nm MIM waveguide with a 400 nm × 200 nm Si-wire waveguide (Butt 2022). However, it is important to note that successful mode conversion hinges on various variables, such as taper length and air gap, thus introducing an element of uncertainty into plasmonic device fabrication.

The intricacies of plasmonic device fabrication are highlighted, emphasizing that even seemingly simple structures discussed in research necessitate positional accuracy at scales of 20 nm or finer (Butt 2022). The evolution of design and fabrication methodologies holds promise for realizing experimental optical devices that concurrently propel the advancement of photonic integrated circuits.

4.1 Fabrication tolerance analysis

The investigation into the fabrication tolerance of the proposed filters is integral to assessing their practical applicability and reliability. In the context of our optimization methodology employing the GA, inherent sensitivity analysis is conducted to ascertain the robustness of the optimized designs across varying fabrication conditions (Chou Chau et al. 2020).

Figure 3a and b depict the convergence rate versus iteration and fitness values versus iterations for Filter I, serving as illustrative examples of our optimization approach’s effectiveness. These plots elucidate the iterative refinement process facilitated by the GA, showcasing the consistent improvement in fitness values over successive iterations. Such convergence implies the GA’s ability to navigate the design space and converge towards optimal solutions resilient to fabrication uncertainties.

While our current emphasis lies in optimizing the filters’ performance to meet predetermined criteria, our future endeavors will entail a dedicated exploration of fabrication tolerance. This comprehensive analysis will involve rigorous simulations and experimental validations to elucidate the sensitivity of the optimized designs to variations in fabrication parameters such as material properties, geometrical dimensions, and process imperfections (Shafagh et al. 2020).

By systematically quantifying the effects of fabrication variations on the performance metrics of our filters, we aim to enhance their reliability and manufacturability. Ultimately, this endeavor will contribute to the development of robust photonic devices capable of meeting stringent performance specifications across diverse fabrication environments and process variations.

Finally, the performance of the proposed filter structures has been compared with other plasmonic filters to create a better view of the obtained results. The results are provided in Table 1. The comparison parameters in this table are the number of resonance modes, the center wavelengths of the resonance modes (λ), the maximum efficiencies of the resonance modes, and the size of the filter structures.

Table 1 Comparisons between the proposed plasmonic filters and other works
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As known, a single-mode filter is more desirable to design higher complexity structures such as sensors, demultiplexes and so on. Accordingly, in this paper, an attempt to design three new single-mode filter structures with high transmission efficiency and compact size was made. As seen in Table 1, the comparison between the designed filters and those of other similar recent studies show that the proposed filter structures have small footprints and offer high efficiencies. In addition to the mentioned advantages, the designed structures in this paper have tunable resonance modes, and suitable out-of-band suppression, which could find applications in on chip photonic integrated circuits.

5 Conclusion

In summary, this study has sought to address the pertinent challenge of enhancing the efficiency and compactness of optical filters within communication systems. Through the exploration of plasmonic waveguide designs encompassing stub, right-angle trapezoid, and tilted T-junction resonators, we have endeavored to develop wide band-pass filters with improved transmission efficiency while maintaining a compact form factor. Employing a parameterized GA, our efforts have yielded transmission efficiencies ranging from 74.2% to 94.4%, and physical dimensions of 490 nm × 575 nm, 350 nm × 180 nm, and 420 nm × 150 nm for each filter design, respectively. These outcomes suggest the potential suitability of these filters for integration into photonic integrated circuits, digital signal processing applications, and multi-photon fluorescence studies. Further studies could encompass refining filter performance, expanding their applications, and considering scalability for broader integration.