Tunable Dual- and Multi-Channel Filter Based on Cantor Photonic Crystals Embedded With Graphene

We theoretically investigate the tunable dual- and multi-channel filter in one-dimensional aperiodic photonic crystals (PCs) embedded with graphene. The dielectrics slabs arrayed alternately submit to the Cantor sequence rule and graphene sheets are local at the interfaces of dielectrics layers. Cantor PCs are fractal structures and support a series of discrete resonant channels in transmission spectra, viz. fractal states. Consequently, dual- and multi-filtering channels could be achieved in the compound systems. The central wavelength of each channel and its value of transmission could be modulating by the chemical potential of grapheme and the incident angle as well. It shows that the dual-channel amplitude of transmission can be varied by over 100% and the multiple channels could be switched on and off by the chemical potential of graphene. This study will be very helpful in designing tunable optical filters.


I. INTRODUCTION
Photonic technology has greatly revolutionized the optical communications in the past decades due to its tremendous progresses made. High speed, low loss, high performance, and reliable optical communication network has been realized through the photonic technology. One of the key components in optical communication systems is the optical filter, which is responsible for choosing specific wavelength of optical information to transmit or reflect.
Various schemes have so far been put forward for realizing the optical filters, among which photonic crystals (PCs) are the most popular ones. PCs [1], [2] are composed of periodic dielectric structures and possess a complete photonic band gap (PBG) in a certain frequency range. Once the frequency falls into the PBG, the propagation of light is strongly suppressed [3], [4]. Therefore, the The associate editor coordinating the review of this manuscript and approving it for publication was Sukhdev Roy. filtering process can be easily realized, and the PBG can also be reconfigured to obtain filters with different spectral characteristics. In some applications, dynamic tunable filters with ultra-narrow bandpass are highly desirable for potential applications in high-precision optical sensing [5], dense wavelength division multiplexing [6], and wavelengthmodulation spectroscopy [7]. On the other hand, quasiperiodic PCs have higher transmission efficiency, smaller band-width, and higher quality factor. Quasi-periodic PCs can support more defect cavities than the conventional periodic PCs [8], [9]. The defect cavities could induce the optical fractal effect in the PBG region. The optical fractal states are transmission modes of light waves. The transmission mode has a peak value of transmittance and a very low reflectivity in quasi-periodic PCs, which well meets the design requirements of the narrowband filters [10], [11], [12]. The properties of quasi-periodic PCs can be changed by certain deterministic rules, such as Fibonacci [13], Octonacci [14], Thue-Morse [15], Rudin-Shapiro [16], and VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Cantor [17], [18], [19]. In addition, other studies conducted in the literature are also briefly discussed to show the importance of various quasi-periodic crystals [20], [21], [22]. However, once the photonic filter is constituted, the amplitude and center frequency of the channel may be permanently fixed, and cannot be conveniently and flexibly adjusted in some specific application scenarios. Graphene is a two-dimensional lattice with perfect honeycomb structure, which has excellent conductivity and many extraordinary optical properties [23], [24], particularly the property of exciting surface polariton [25], [26]. The photonic crystals based on graphene have been extensively studied so far, such as oscillations [27], [28], supermodels [29], optical bistabilities [30], and optical modulators [31], [32]. In addition, the optical characteristic quantity is governed by the surface conductivity of graphene. The graphene surface conductivity is a function of the chemical potential. Generally, the chemical potential of graphene can be changed by loading an external grid voltage on it or by chemical doping [33]. This provides great convenience for graphene to modulate the optical properties induced by the graphenebased photonic crystals. Therefore, here we construct a complex system consisting of a one-dimensional quasiperiodic photonic crystal with graphene to realize a tunable dual-channel optical filter.
Tunable dual-and multi-channel optical filters are studied and numerically investigated in Cantor lattices embedded with graphene. The Cantor lattices are formed by two alternating dielectric slabs with different refractive indices and graphene is imbedded at the interfaces of these dielectric slabs. The discrete transmission channels and extension of channels by increasing the sequence number are demonstrated firstly. Subsequently, the tunability of filtering channels of optical waves is explored by changing the chemical potential of graphene and the incident angle of lights. In order to explore the channel regulation mechanism, the surface conductivity of graphene and the distribution of electric field are given as well. This work could be utilized for optical filters and optical modulators.
The materials of A and B are represented as Si and SiO 2 , respectively. The corresponding refractive indices are n A = 3.53 and n B = 1.46. The quasi-periodic system is placed in air and the refractive index of substrate surrounding is set as n 0 = 1. The thicknesses of dielectric slabs are one quarter of an optical wavelength, viz. d A = λ 0 /4/n A = 0.1098 µm, d B = λ 0 /4/n B = 0.2654 µm, where λ 0 = 1.55 µm is the central wavelength. The graphene, denoted by layer C, is embedded arbitrary two dielectrics. Graphene is considered as a metal element in the near-infrared band, and is usually regarded as an ultra-thin two-dimensional dielectric with unique conductivity and equivalent thickness of d g .
The equivalent dielectric constant of graphene is expressed as Here, the symbol i denotes the imaginary unit, k 0 represents the wave vector in vacuum, η 0 is the vacuum resistivity parameter in vacuum. The refractive index of graphene can be characterized by its conductivity as n g = √ ε g . The conductivity of graphene can be obtained from the Kubo formula [33], [34], which could be written as where is the Fermi-Dirac distribution function. ε, µ c , ω represent the particle energy, the chemical potential of graphene, the angular frequency, respectively. e, K B , ℏ denote the electron element charge, and the Boltzmann's constant, the reduced Plank's constant, respectively. The environment temperature is set as T b = 300 K and the relaxation time is set as τ = 0.5 ps.
The first term and the second term of Equation (2) can be simplified to and respectively. Both interband and intraband transitions involve µ c and ω. In other words, the surface conductivity of graphene is a function of chemical potential and light incidence frequency. In the THz band, the interband transition is dominant because the excitation photon energy ℏω is less than 2µ c , and the optical property of graphene is analogue to metals. The one-dimensional photonic multi players embedded by graphene arranges along the horizontal direction. Considering a light impinges on the structure from the left and transmits from the right and ignoring the nonlinear effect of light, we use the transmission matrix method, the transfer matrix of dielectric sheet can be expressed as where ς j = 2πn j d j cosθ j /λ. n j , d j and θ j are the refractive index, the dielectric thickness, and the incident angle respectively. The modified admittance for a transverse magnetic (TM) wave is given by The chemical potential of graphene is a Femi level, so the chemical potential can be controlled by chemical doping or the external grid voltage on graphene. As the voltage load on graphene is positive, the chemical potential of graphene increases, while the voltage load on graphene is negative, the chemical potential decreases. Therefore, the surface conductivity of graphene can be flexibly adjusted by the bias voltage on graphene. Graphene is treated as a thin film with an equivalent thickness. Equivalent refractive index of dielectric is a function of the chemical potential of graphene. Therefore, the refractive index of the equivalent dielectric can be flexibly controlled by the chemical potential of graphene. If graphene is embedded into the photonic crystal, the existence of graphene will affect the overall transmittance of the structure, and the surface conductivity of graphene is a chemical potential function, which can change the transmission spectrum of the entire composite structure through the chemical potential of graphene, and finally regulate the central frequency of resonant modes in the structure.
The relation between the chemical potential µ c and the applied gate voltage V g on graphene can be expressed as where the parameter a 0 is empirical constant and is given by a 0 = 9 × 10 −16 m, v F ≈ 10 6 m/s is the Fermi velocity of electrons in graphene, and V D is the offset bias voltage and it is generally set as V D = 0 for intrinsic graphene.

III. NUMERICAL RESULTS AND DISCUSSIONS
For a TM wave, as it normally impinges on the quasi-periodic photonic crystal multilayers abiding by Cantor sequence and graphene from the left. Figure 2(a-d) respectively give the light transmission spectra of Cantor photonic multilayers with or without graphene for the generation number N = 1, 2, 3 and 4. The graphene chemical potential is given by µ c = 0.5 eV. The horizontal coordinate is the normalized frequency of the incidence (ω − ω 0 )/ω gap .
Here, ω gap is the photonic bandgap expressed as ω gap = 4ω 0 arcsin|[Re(n B )−Re(n A )]/[Re(n A )+Re(n B )]| 2 /π , ω 0 = 2πc/λ 0 is called as the central frequency of incidence. ω gap is a typical important parameter to characterize photonic crystals. The real term of the material refractive index determines the propagation phase of the traveling wave, and ultimately determines the resonant wavelength of the transmission mode. In addition, the photonic band gap ω gap of photonic crystals is also determined by the real term of the material refractive index. Within the dotted line range marked in Figure 2(a-d), one can see that there are 0, 2, 10, 26 resonant states in red curves as the generation number increases from 1 to 4. These resonances are symmetrically distributed around the center and their peak values are all 1. The number of transmission modes increases exponentially with the generation number of dielectric sequence. As graphene sheets are imbedded at the interfaces of slabs, the spectrum on the left side of the center point shifts slightly to the right, while the spectrum on the right side of the center point shifts slightly to the left. In the first three generations of the iterative sequence, the total number of resonance peaks remains unchanged. Since the fourth generation, the total number has changed. As a light wave illuminates normally on the photonic quasicrystals, defects in Cantor PCs can induce optical fractal states. The optical fractal states correspond to a series VOLUME 11, 2023 of transmission modes. The bandgap in the transmission spectrum can naturally be used as photonic filters by analogy with the energy band structure of electrons in semiconductors. Hence, dual-and multi-filtering channels could be achieved for appropriate iteration sequence.   In order to demonstrate how graphene sheets modulate the transmission performance and the center frequency of the filter channel, we sampled the channels where the channels in the middle of the band gap are marked as T 1 , T 2 , and the corresponding central frequencies are ω 1 , ω 2 , respectively. Figure 3(b-e) show the transmittance and the normalized frequency of the channel versus the graphene chemical potential for a center wavelength. One can see that the chemical potential of graphene is less than or equal to 0.3 eV, the transmittance T 1 equals to 0.5, and while it exceeds 0.35 eV, the value rapidly steps up to one as shown in Figure 3(b). In Figure 3(c), as µ c is less than or equal to 0.3 eV, the center frequency of the transmission mode decreases with the increase of the chemical potential of graphene. As µ c exceeds 0.3 eV, the center frequency of the transmission mode increases with the increase of the chemical potential of graphene. The center frequency of the transmission mode is the minimum at the chemical potential of 0.3 eV. Similarly, this trend also exists for the other channel. In Figure 3(d), by increasing the chemical potential of graphene to 0.45 eV, the transmittance T 2 remains 0.5, and while it exceeds 0.5 eV, the value rapidly steps up to one. The center frequency of the transmission mode decreases first and then increases with the increase of graphene value as shown in Figure 3(e). The central frequency of the transmission mode is the minimum for µ c = 0.45 eV. It is not difficult to find that a small change in the graphene value will cause a change in the central frequency of the transmission mode. In addition, the transmittance of the transmission mode changes significantly if and only if the value of graphene increases to a certain value, and the maximum transmittance is close to 1. It shows that the dual-channel amplitude of transmission can be varied by over 100%.  the range of (− 0.418, 0.7). While in the range of (− 0.7, −0.418), the value sharply jumps down to zero. As µ c increases to 0.4 eV and 0.5 eV, the curve shifts to the right overall, the normalized frequency at the jump is (ω − ω 0 )/ω gap = 0.067 and (ω − ω 0 )/ω gap = 0.544 respectively, and the corresponding Re(σ ) in each curve is 0.6 ×10 −5 S. The parameter Re(σ ) is the counterpart of the imaginary part of the graphene equivalent dielectric constant Im(ε g ), this indicates the optical loss in graphene. Larger Re(σ ) means greater loss and weaker transmitted intensity. As graphene is adjusted to a certain range, the optical loss remains unchanged, and the corresponding transmittance value could not decay, close to 1. This approves the phenomenon that the transmittance changes with the normalized frequency of incident light for different chemical potential as shown in Figure 3(a). Therefore, this characteristic can be used for manipulating the channel frequency. Figure 4(b) provides the real part of surface conductivity Re(σ ) of the photonic lattice. The parameters space is composed of µ c and incident frequency ω. One can see that, the value of Re(σ ) remains constant by raising the graphene chemical potential to a certain value, and keeping on increasing the graphene chemical potential or turning up the incident frequency. Since Re(σ ) dominates the loss in graphene of light, changing µ c or modulating ω is an important step to control the propagation of light in complex multilayer. The jump of this turning point, that is, the abrupt increase of Re(σ ) from zero to a certain value, is the result of electron transition from intraband to interband in graphene. Figure 4(c) gives the imaginary part of surface conductivity Im(σ ) varying with µ c . The whole curve of Im(σ ) moves to the right as the chemical potential increases. By increasing the frequency of light, there is an inflection point of each curve, on the left side of the inflection point, Im(σ ) decreases, and then increases smoothly to a constant value as the frequency passes the inflection point. The parameter of Im(σ ) determines the real part of graphene equivalent permittivity and Im(σ ) can be modulated by µ c , so we can flexibly adjust the transmittance through the chemical potential of graphene.
We continuously adjust the graphene chemical potential and depict the imaginary part of surface conductivity Im(σ ) varying with the incident frequency in the parameter space as shown in Figure 4(d). It demonstrates that Im(σ ) produces an effect after the graphene chemical potential increases to a certain value, and shows a trend of decreasing rapidly and then increasing to a certain value in the incident frequency. Im(σ ) determines the real part of graphene equivalent permittivity, and graphene chemical potential can change the value of Im(σ ), so we can flexibly adjust the transmittance through the chemical potential of graphene.   Figure 2). One can see that the transmittances of channel are different for different µ c . In Figure 5(a-b), the generation number N is set as 3, as µ c ≤ 0.35 eV, the transmittance T 3 is equal to 0.0007. While µ c = 0.4 eV, the transmittance jumps to 0.639. The transmittance reaches the maximum 0.8093 at µ c = 0.6 eV. The transmittance decreases with the increase of the chemical potential as µ c > 0.6 eV. For the channel of T 4 , the transmittance T 4 is close to 0.0007 as µ c ≤ 0.4 eV. In the range of (0.45 eV, 0.8 eV), the transmission value of light waves increases with the chemical potential of graphene. The maximum transmittance is T m = 0.8428 for T 4 around µ c = 0.8 eV, and then the transmittance T 4 decreases with the increase in µ c .
The transmittance of T 5 changing with the chemical potential could be observed in Figure 5(c). In the chemical potential range of (0.2 eV, 0.35 eV), the maximum transmittance of T 5 approximates to zero. There is abrupt step of channel transmittance of T 5 = 0.049 at µ c = 0.4 eV and the value of transmission rises to 0.4413. The maximum transmittance of T 5 is T m = 0.685 at µ c = 0.6 eV. Compared with T 3 , the value of T 5 and its maximum are all decrease. In Figure 5(d), obviously, the transmittance T 6 is zero as µ c ≤ 0.4 eV and rapidly increases as µ c is in the range of (0.4 eV, 0.75 eV) and the maximum transmittance rises to the value of 0.7233 at µ c = 0.75 eV. Therefore, the transmittance can be tuned by the chemical potential in the application of photonic filters. But if we want to get a dual-channel filter, and apply photonic crystals with different sequence, the graphene should be modulated in a compromise range. It shows that the multiple channels could be switched on and off by the chemical potential of graphene.  The parameter space is composed of the incident angle and the normalized frequency (ω − ω 0 )/ω gap . One can see that there are two defect modes in the parameter space. With the increase of frequency, no new defect modes appear. With the change of the incident angle, the defect mode moves along the dotted line in the parameter space. For each defect mode, the transmittance increases slightly with the increase of the incident angle, and the profile of the defect mode moves to a higher central frequency. Figure 6(b) presents the transmission spectra for four given incident angles. One can see that the spectrum profile moves rightwards overall by increasing the incident angle. The normalized frequency of the resonant defect modes is higher for a larger incident angle and the transmission values of the peaks are higher relatively. Therefore, using the composite structure as a dual-channel photonic filter, the transmittance and center frequency of the filter channel can be flexibly adjusted through the chemical potential of graphene and the incident angle of light waves.
As mentioned above, there are two transmission peaks (denoted by T 1 andT 2 ) for N = 2. The corresponding wavelengths are the values of λ 1 = 1.999 and λ 2 = 1.273 µm, respectively. Here, we use the inverse transmission matrix method (in blue curve) and the finite difference time domain (in color bar) [33] method to simulate the electric field distribution, and the results of the two methods are consistent. For the light beams normally incident from the left, the Z-component of electric field intensity of T 1 for the Cantor lattices is given in Figure 7(a). One can see that the electric field intensity is mainly localized at the center of the defect B and attenuates from the center to the left and right. The localization of electric field manifests that the transmission peak is a resonant state. The electric field distribution at the peak of T 2 is given in Figure 7(b). The two largest distributions of electric field appear in defect layers BBB. This phenomenon also supports the characteristics of electric field localization, which indicates that the transmission peak is also a resonant mode. The above characteristic is necessary to realize dual-filtering channels. The distributions of electric fields are simulated using the software COMSOL and MATLAB as shown in Figure 7. The other results are calculated by MATLAB programs. The software of CorelDRAW is used to edit the figures.
Hyperbolic metamaterials composed by graphene and dielectric multilayers can support a Fabry-Perot resonant mode and the central wavelength of resonance could be modulated by the chemical potential of graphene and the thickness of dielectric [35]. Metamaterial terahertz filters have been achieved by periodic metallic rings with gaps. Graphene stripes are local at these gaps and the resonance frequency of the system can be altered by 40% by tuning the conductivity of graphene [36]. Graphene can also be utilized for tuning the reflection of lightwaves and it shows that the dual-band amplitude of reflection can be varied by over 15% and resonant positions can be shifted by over 90 cm −1 [37]. Furthermore, the composite structure of photonic crystal and graphene can realize the optical regulation of terahertz band [38], [39], [40]. We here investigate the dual-and multi-channel filter of transmission based on optical fractal resonances in quasi-periodic crystals, of which graphene is embedded at the position of the strongest local electric field. Therefore, the transmittance and the central wavelength of resonance can be greatly modulated by the chemical potential of graphene. It manifests that the dual-channel amplitude of transmission can be modulated by over 100% and the 8438 VOLUME 11, 2023 multiple channels of filters could be switched on and off by the chemical potential of graphene in our systems.

IV. CONCLUSION
In summary, tunable dual-and multi-channel filter are investigated in one-dimensional aperiodic photonic crystals (PCs) embedded with graphene. The aperiodic PCs composed of two different dielectric slabs submit to the Cantor sequence law. The Cantor PCs can support a series of discrete resonant channels in transmission spectra, which provides the condition to realize dual-and multi-filtering channels. The central wavelength of each channel and its value of transmission could be modulated by changing the graphene chemical potential. The dual-channel amplitude of transmission can be changed by over 100% and the multiple channels could be turned on and off by the chemical potential of graphene. Moreover, the value of transmission increases and the central wavelength of the filter channel moves toward the short-wave direction as the incident angle is turned up. The study could provide a feasible scheme for tunable optical filters.
XIAOLING CHEN received the M.S. degree in control theory and application from East China Jiaotong University, in 2015. She is currently a Teacher with the School of Electrical and Information Engineering, Hubei University of Science and Technology. She has published several international journals. Her research interests include surface plasmon photonics of graphene and research on optical artificial supernormal materials.
PU ZHANG received the M.S. degree in applied mathematics from the Huazhong University of Science and Technology, in 2007. She is currently an Associate Professor with the School of Education, Hubei University of Science and Technology. Her research interest includes micro nano photonics. In particular, she is very interested in the study of non-Hermitian optical properties of surface plasmon polaritons.
DONG ZHONG (Member, IEEE) received the Ph.D. degree in communication and information system from the Wuhan University of Technology, in 2015. He is currently a Professor with the School of Electrical and Information Engineering, Hubei University of Science and Technology. He has published more than 70 scientific research papers and has presided more than 40 projects. He holds more than ten national invention patents. His research interest includes micro nano photonics.
JUNJIE DONG received the M.S. degree in signal information processing from Chang'an University, in 2014. He is currently a Researcher with the Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology. His research interest includes micro nano photonics.