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Abstract
We present a tunable ultra-wideband band-stop filter utilizing graphene with a straightforward staggered structure. The transmission spectrum has been meticulously analyzed using the effective-index-based transfer matrix method (EIB-TMM). The results demonstrate that the filtering properties can be precisely tailored by manipulating the Fermi energy level of graphene. Importantly, we have successfully achieved a remarkable ultra-wideband stopband by optimizing the staggered parameters. Our exploration of redefining the staggered structure through adjustments to three critical parameters has revealed a crucial role in expanding bandwidth. This investigation deepens our understanding of how nonperiodic structures can effectively broaden bandwidth and holds great promise for the prospective design of ultra-wideband band-stop devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Graphene Surface Plasmons (GSPs) have emerged as a fascinating platform for various photonic applications due to their remarkable properties [1–3], such as strong light confinement [4], low losses [5] and tunability [6–8]. The versatile applications of GSPs span a wide spectrum, including optical switches [9–11], splitters [12–14], filters [15–19], perfect absorbers [20–25], intersection [26], sensors [27,28], Bragg reflectors [29], optical modulators [30,31], photodetectors [32] and plasmonic antennas [33,34]. Among this array of applications, the development of tunable band-stop filters [35–38] using graphene-based structures has emerged as particularly intriguing. These filters play a pivotal role in selectively blocking specific wavelengths while allowing the transmission of others, making them essential components in advanced optical communication systems and signal processing.
The precise control and tailoring of the filtering properties of these devices are of paramount importance. Conventional periodic structures [39–41] face limitations in providing the desired bandwidth coverage, impeding their efficacy in ultra-wideband applications. To surmount these limitations, researchers have explored nonperiodic structures. In our previous research [42], we introduced a novel tunable multichannel plasmonic filter comprising a single graphene sheet deposited onto a Fibonacci quasiperiodic structure, While multiple bandgaps were demonstrated, achieving continuous broadening remained a challenge. Similarly, various quasiperiodic structures, such as Thus-morse, Octonacci, and Ternary Fibonacci, have been employed to design filter devices. Although these devices can achieve multi-channel band-stop filtering, their bandwidth remained insufficient for ultra-wideband requirements, and filter bandwidth tunability was limited. Therefore, this paper presents a novel approach to overcome these limitations. we propose a tunable ultra-wideband band-stop filter based on graphene with a simple staggered structure. Through numerical calculations and EIB-TMM, we systematically investigate the filtering characteristics of the proposed structure. To optimize the filter performance, we theoretically analyze the relationship between staggered parameters and filter bandwidth. Our results demonstrate that an ultra-wideband stopband can be achieved and the transmission characteristics can be tuned through the manipulation of graphene's Fermi energy level. This study not only advances the design of tunable filtering devices but also opens up new possibilities in optical communication, radar systems, and nano-plasmonic integrated circuits.
2. Structures and theory
The schematic representation of our proposed structure is illustrated in Fig. 1(a). It consists of a monolayer graphene attached to one-dimensional dielectric and air staggered grating - a nonperiodic structure. In this diagram, dAi represents the width of the dielectric layer, and dBi represents the width of the air layer. we define the staggered structure as
where H is average width of dielectric and air layer. The distribution functions ai and bi are defined asHere, kh and kl are staggered parameters set to values less than 1 to ensure ai and bi remain positive, and m is the total number of cells. These staggered parameters follow arithmetic sequences with average values set to 1. Obviously, the structure degenerate into a periodic structure when kh,l = 0. Thousands of cases of staggered structures with the same average width can be easily obtained, since the average widths are fixed for any kh,l and m.
In the initial research, the staggered parameters are set as kh = 0.4 and kl = 0.1, total number of cells m = 24, and average width H = 30 nm. Figure 1(b) illustrates the normalized width of staggered structure, showing that although the width of dielectric layers and air layers are gradually increasing with the number of cells, the average width of both dielectric layers and air layers remains constant.
The monolayer graphene sheet is modeled by random-phase approximation (RPA), the optical conductivity with a Drude-like expression in the middle-infrared region [43,14]:
where $\hbar $ is the reduced Plank’s constant, e is the elementary charge, ω is the angular frequency of the incident light, and τ is the carrier relaxation time, which satisfies the relationship $\tau = {{\mu {E_f}} / {\textrm{(}e}}{\nu _f}^2)$ where the carrier mobility $\mu = 10000c{m^2}/(V.s)$, the Fermi velocity ${\nu _f} = {10^6}m/s$, and ${E_f}$ is the chemical potential of graphene which can be tuned by gate voltage and chemical doping, etc.The dispersion relation of Surface Plasmon Polaritons (SPPs) propagating in the graphene/substrate waveguide is described as [43,44]
The EIB-TMM is applied to investigate the transmission and optimize the parameters of the proposed structure. As an analogy of one-dimensional photonic crystal, the transfer matrix of the staggered structure is expressed as
3. Results and analysis
3.1 Dispersion relations of the graphene waveguide
We investigate the dispersion properties of the graphene waveguide through numerical methods. In our calculations, the permittivity of substrate is assumed as 3.9. The real part of effective mode index in air/graphene/dielectric waveguide and air/graphene/air waveguide under different Fermi energies are illustrated in Fig. 2(a) and Fig. 2(b), respectively. It is evident that the effective index of the surface wave propagating along the graphene sheet on a dielectric substrate is higher than that along the graphene sheet on an air substrate. This difference provides a valuable effective index contrast for one-dimensional photonic crystals. As the Fermi energy level decreases, there is an associated increase in the real part of the effective index. Moreover, the effective index contrast becomes more pronounced at higher Fermi energy levels, which presents an opportunity for tunable devices. Consequently, we can achieve a tunable plasmonic filter by alternately stacking graphene-dielectric waveguides and graphene-air waveguides.
Fig. 2. (a) The real part of effective mode index in air/graphene/dielectric waveguide under varying Fermi energies. (b) The real part of effective mode index in air/graphene/air waveguide under varying Fermi energies.
3.2 Crucial factors affecting the filter performance
Figure 3 displays the transmission spectra of the staggered structure with a fixed chemical potential of Ef = 0.9 eV, while keeping other parameters consistent with the previously mentioned ones. Additionally, in periodic structure, the width of each cell equals the average width H. By comparison, one can see that the stop band width of staggered structure surpasses that of the periodic structure.
Fig. 3. The transmission spectra of the staggered structure when Ef = 0.9 eV. (a) The transmission spectra of periodic structure and non-periodic structure. (b) The transmission spectra of staggered structure with different number of cells.
Despite the inherent challenges in establishing a rigorous theoretical framework for staggered structure, we can qualitatively explain the broadening effect using the established theory of short-range order and long-range disorder, commonly employed in the realm of solid-state physics to describe nonperiodic structures. In the context of staggered structures, wherein the difference between neighboring cells is relatively small, a series of continuous cells can effectively be considered as identical, resulting in a periodic arrangement that satisfies the Bragg condition. Consequently, the staggered structure can be conceptually divided into several periodic substructures, each characterized by distinct valleys exhibiting low transmission. The position of these valleys is governed by the cell width, which gradually varies across the structure. Due to the continuous variation in cell width, the corresponding valleys undergo a shifting effect. This shifting distance is directly related to the disparity between the minimum and maximum cell width, a parameter commonly referred to as the “staggered parameter”. As depicted in Fig. 3(b), the discrepancy between the minimum and maximum cell width has increased as the number of cells increases, resulting in broader transmission spectrum ranges, and the low transmission peak around 3.7µm gradually approaches zero. Thus, the staggered parameter plays a pivotal role in determining the width of the broadening ranges of transmission spectrum. Furthermore, when the staggered parameter remains fixed, the disparity between two adjacent cells diminishes as the number of cells increases. This can result in valleys overlapping, giving rise to a series of broad and low-transmission ranges. Therefore, the staggered parameter governs the width of these extensive ranges, while the number of cells influences the quantity of valleys within each range and their interconnection pattern.
To gain a deeper understanding of the mechanisms behind graphene surface plasmon propagation in the staggered structure, the finite-difference time-domain (FDTD) method is employed to calculate the distribution of the electromagnetic field. Figure 4 presents the magnetic field profiles of graphene surface plasmon propagation across the staggered structure. In Fig. 4(b), it is evident that the incident wave experiences reflection at a precise wavelength of 3.2µm, while effectively transmitting through the structure at 2.4µm, a region situated outside the stopband region visualized in Fig. 4(a). This observation aligns with the spectral analysis depicted in Fig. 3(a).
Fig. 4. Simulated magnetic field intensity profiles of the staggered structure at the incident wavelength at 2.4µm and 3.2µm, with the structure parameters are the same as Fig. 3(a).
The staggered parameter holds significant influence over the width of the transmission spectrum's broadening ranges. In Fig. 5, the stop band ranges, defined by transmission values below 10% for all incident waves, are visually presented. As per Formula 2, it is evident that when kh and kl equal zero, the staggered structure degenerates into a periodic structure, resulting in a stop band range of 1.09µm. Remarkably, the stopband range reaches its maximum value of 5.09µm when kh = kl = 1, representing a notable enhancement of 4.67 times compared to the periodic structure.
While the analysis suggests that the stop band range should increase with an increase in the staggered parameter, the results do not entirely align with this expectation. This discrepancy arises due to the presence of low transmission peaks within the bandgap when m assumes finite values, as illustrated in Fig. 3(b). Additionally, in Fig. 5, the stop band range is not displayed for cases when kh = 0.2, kl = 0.4, and kh = 0.4, kl = 0.8. This is attributed to the absence of effective transmissions greater than 10% in the short wavelength range for these specific staggered parameters.
3.3 Dynamically tunable ultra-wideband filter
As can be observed from the Fig. 6(a), with the increase in the average width of the staggered structure, the bandgap shifts towards longer wavelengths, and the bandwidth of the bandgap broadens. This behavior can be attributed to the formation of short-range periodic structures as the average width increases. According to the Bragg condition, these structures create a bandgap that shifts toward longer wavelengths, resulting in a slight enlargement of the bandgap range. Consequently, the long-range disordered structures also experience an increase in bandwidth due to this broadening effect.
Fig. 6. (a) The transmission spectra of staggered structure with different average widths for a Fermi energy level of 0.9 eV. (b) The transmission spectra of staggered structure with different Fermi energy levels of graphene for H = 30 nm. Other structure parameters are consistent with those in Fig. 1(b).
In Fig. 6(b), as the Fermi energy level of graphene increases, the bandwidth gradually decreases. This phenomenon arises because an increase in the Fermi energy level of graphene leads to a decrease in the effective refractive index. Following the Bragg condition, this reduction in the bandgap caused by the short-range ordered structures narrows the bandgap in the long-range disordered non-periodic structures
The results presented in the Fig. 6 highlight the intricate interplay between the average width of the stacked structure and the Fermi energy level of graphene in modulating bandgap characteristics. This comprehensive understanding of the bandgap behavior in such complex structures has significant implications for the design and optimization of novel optoelectronic devices, where precise control over the bandgap properties is crucial. Moreover, these findings provide valuable insights for researchers exploring the potential of graphene-based photonic and electronic devices.
Furthermore, we investigated the impact of the graphene Fermi energy level on the transmission spectra. As shown in Fig. 7, the bandgap shifts toward shorter wavelengths, as the Fermi energy level increases. This shift is due to the decrease in the graphene structure's effective refractive index, following the Bragg condition for short-range ordered structures. Simultaneously, this shift also causes the bandgap in the long-range disordered structures to move towards shorter wavelengths.
Fig. 7. The transmission spectra of staggered structure with different Fermi energy levels for m = 30. Other structure parameters are consistent with those in Fig. 1(b).
The tunability of the bandgap through manipulation of the Fermi energy level holds promising implications for tailored optical responses in various devices. This discovery opens exciting possibilities for designing advanced photonic and optoelectronic devices, such as tunable ultra-wideband filters, ultra-wideband absorber and modulators.
Finally, we have provided a comprehensive comparison between our presented structures and previous research. This comparison is based on critical filter indicators, including bandwidth, width, and overall structure. A detailed parametric comparison can be found in Table 1. It is evident that the presented structure has been effectively optimized in terms of bandwidth, width, and overall structure. These results can be attributed to the implementation of a staggered structure.
Table 1. Comparison of final generated broadband filter with previous work
4. Conclusion
In conclusion, we have presented a novel tunable ultra-wideband band-stop filter based on graphene with a simple staggered structure. Our thorough analysis using the EIB-TMM revealed that the filtering properties can be finely adjusted by varying the Fermi energy level of graphene. Optimizing the staggered parameters allowed us to achieve a remarkable 4.67-fold enhancement in bandwidth when compared to the periodic structure, successfully realizing an ultra-wideband stopband. This investigation deepens our understanding of how nonperiodic structures can effectively extend bandwidth, thereby advancing the design of future ultra-wideband tunable band-stop devices. The tunable nature of graphene-based filters offers potential for transformative advancements in building active integrated photonic circuits
Funding
Natural Science Foundation of Shandong Province (ZR2019YQ01, ZR2021MA037); National Natural Science Foundation of China (11505100, 11975132).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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