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Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays
Accepted Manuscript Title: Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays Authors: Fatemeh Tabatabaei, Mohammad Biabanifard, Mohammad Sadegh Abrishamian PII: DOI: Reference:
S0030-4026(18)31859-X https://doi.org/10.1016/j.ijleo.2018.11.103 IJLEO 61951
To appear in: Received date: Accepted date:
22 October 2018 23 November 2018
Please cite this article as: Tabatabaei F, Biabanifard M, Abrishamian MS, Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.11.103 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays Fatemeh Tabatabaei1, Mohammad Biabanifard1,* and Mohammad Sadegh Abrishamian1 1
School of Electrical Engineering, K.N. Toosi University of Technology, Tehran, 16315-1355, Iran
*
Corresponding author: [email protected], Tel: +989337201191
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Declarations of interest: none.
ABSTRACT
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In this paper, we propose and investigate a novel terahertz all-optical tunable filter comprised of arrays of graphene
microdisks deposited on a dielectric substrate. The proposed filter exhibits a wide-angle response and also due to the symmetric structure is polarization-insensitive. The transmission line theory (TLT) using the circuit model of graphene disks arrays consists of parallel series R-L-C is exploited to investigate the structure for various lattice
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fill-factor and chemical potential. The results obtained via this approach show excellent agreement with the results of a second different method using the Full-wave numerical modeling. The novel proposed TLT method using an
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easy to implement MATLAB code takes advantage of extremely short computational time (less than a few
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seconds) along with a significant reduction in the required memory in comparison to the Full-wave simulations. In addition, the Kerr nonlinearity is studied through the harmonic balance method in steady-state regime. Our
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investigation reveals that a linear Red-shift occurs in the transmission spectra as the intensity of incident light increases. The proposed filter has potential applications in spectral imaging, communication systems and all-
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optical switching in the terahertz band.
Keywords: Graphene, Terahertz filter, Kerr effect, Harmonic balance method, Circuit model, Disks arrays.
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1. Introduction
In recent years, it has been determined for the research community that the Terahertz region has a high potential
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to respond to numerous engineering impasses in designing high-efficiency modern Opto-electronics devices [1-2]. This region which is called “terahertz gap” (roughly from 0.1 THz to 30 THz), is less studied and many of its capabilities are emerging [3]. One of the main obstacles in this regard was the lack of standard materials to respond to the terahertz radiation appropriately which made the manipulation of lightwave in this region a
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challenging problem. The other difficulty was the fabrication challenges since the dimensions of THz devices are in Nano or few Micrometer scales. Nowadays, owing to the prosperous growth of nanotechnology in various scientific and industrial areas besides the advent of new materials like graphene, a significant attention of researchers has been drawn to the THz technology and its applications [1-2,4]. Graphene, a 2D layer of carbon atoms in which the atoms form a honeycomb lattice, with an atomic thickness presents high thermal conductivity and optical damage threshold besides a large third-order optical nonlinearity [5]. This semi-metal material with zero bandgap possesses high absorption in the infrared and visible range which is 2.3% [6]. Extraordinary properties of graphene have paved
the way for significant improvements in many fields specifically in optics [5]. For instance, this material has been widely used in various devices including modulators, antennas, absorbers, polarizers, and optical fiber sensors [713]. By exploiting the well-known chemical vapor deposition technique (CVD) in the past years, it has been shown that the large-area graphene continues sheets with dimensions up to 30 cm can be efficiently growth and fabricated [14]. The number of grown graphene layers besides its characteristics are determined by the aid of atomic force microscopy and Raman spectroscopy [15-16]. In optical devices, the ability to control light with light is of great importance since it is a fundamental requirement in all optical signal processing in integrated circuits, optical
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communications, and optical computing [17]. In this regard, graphene with an exceptionally strong nonlinear optical response and its corresponding large value of nonlinear optical susceptibilities has caught the attention of
the scientific community [18-21]. Terahertz technology which has been continuously progressing in both sources
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and detectors have fueled the proliferation of designing and fabricating optical devices to control terahertz
electromagnetic lightwaves [22]. Moreover, wide applications of terahertz waves in communication, imaging systems, security, astronomy, biological and medical science make the rapidly growing demand for terahertz devices crystal clear [2,22]. The main blind spot regarding this issue is the absence of materials dealing with the
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terahertz waves. Although the metamaterials have been introduced as an alternative to this demand, there are still
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difficulties in designing and implementing terahertz devices [23-25].
Optical properties of graphene which can be tuned to lie in the terahertz spectrum along with the simple design,
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fabrication and integration have extended the feasibility of producing terahertz devices [26-28]. Considering the
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importance of terahertz filters in the communication systems and spectral imaging [29] we propose a graphenebased all-optical tunable filter. This filter takes advantage of the gate tunability along with the large third-order nonlinearity of graphene to control the transmission frequency. This paper is organized as follows. In section 2,
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graphene and its surface conductivity are introduced. In section 3, the proposed filter and its equivalent circuit model are presented. We investigate the structure for different incident wave polarizations and angles, fill-factors, and chemical potentials in section 4 where all the results yielded by the circuit model approach are validated
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against Full-wave numerical simulations. Furthermore, the nonlinear response of the structure and harmonic balance method is studied in this section. Finally, the paper is concluded in section 5.
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2. Graphene surface conductivity In this section, some basic formulas about graphene’s complex surface conductivity are reviewed. Thereinafter, the equivalent circuit model of the graphene disks arrays is discussed in detail. In the absence of a magnetostatic
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field and spatial dispersion the graphene conductivity which is derived from the well-known Kubo formula is defined as follows (by assumption of e σ ω,μc ,,T
j t
as time-harmonic dependency) [30]:
f f ε f f ε f d ε f d je 2 ω j 2 Γ 1 dε ε dε 2 0 0 ω j 2 Γ 2 4ε 2 ε π 2 ε ω j 2 Γ
(1)
where ω is the angular frequency, μc is the chemical potential, Г is a phenomenological scattering rate which is inversely related to the electron relaxation time 1 (2) [31], T is the temperature, -e is the electron charge, ħ
1 is the reduced Plank’s constant, f d e c / k BT 1 is the Fermi-Dirac distribution [30] where ε is the
energy and kB is the Boltzmann constant. More in detail, the electron relaxation time and chemical potential are dependent on each other via m c e F2 , where μm is the carrier mobility and νF ≈ 106 m/s is the Fermi velocity [32]. In this work, the electron relaxation time is considered to be constant as τ = 3 ps and T = 300 K is assumed as room temperature [33]. The first and the second term in Eq. (1) account for the intraband and interband conductivity of graphene
ns
2
2 F2
0 f d f d 2c d
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respectively. The Chemical potential, μc, is influenced by carrier density ns that is defined as [30]: (2)
Therefore, changing carrier density by means of applying a gate voltage or chemical doping leads to variation in
0 sVg
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the chemical potential value [30]. Carrier density can be substituted with ns
de
,where Vg is the bias
voltage, d is the thickness of the substrate, ε0 is the permittivity of the free space, and εs is that of the substrate
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[34]. Afterward, the chemical potential can be derived via numerical techniques.
c 2 ln ec / kBT 1 j 2 k BT e2 K BT
2
(3)
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intra , c , , T j
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After some mathematical calculations, the intraband conductivity of graphene is derived as [30]:
be approximated as follows [30]:
je2 2 c j 2 ln 4 2 c j 2
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inter , c , ,0
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and in the terahertz region with low energy photons, where kBT << |μc| is satisfied, the interband conductivity can
(4)
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Fig. 1(a, b) respectively depict the absolute value of the interband and intraband conductivity of graphene with the chemical potential variations in the THz spectrum. As seen, the intraband conductivity is the dominant term within this frequency range since it is of the order of 10-3 while the interband term is of the order of 10-6.
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Consequently, the interband conductivity can be ignored in the graphene modeling at the terahertz regime. Taking into account the chemical potential μc of graphene and contribution of interband or intraband conductivity, this material is capable of presenting metallic or dielectric behaviour [5]. According to Pauli blocking principle, if photon’s energy ħω is less than 2EF where Fermi energy is EF ≈ μc, it is not absorbed and there is no interband
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transition. In this regard, |μc| > ħω/2 leads to the contribution of intraband transition which lies in the terahertz range, while for |μc| < ħω/2 there is no blocking effect and both the interband and intraband transitions must be taken into account [5]. In order to investigate the nonlinear behaviour of bulk materials, the third-order susceptibility χ(3) is used. However, due to the 2D intrinsic nature of graphene, the current sheet approach is exploited to characterize the nonlinear response. Therefore, the nonlinear conductivity coefficient of graphene is introduced as [18]:
3 j
3 e 2 e F 2 . 32 2 3
(5)
(b)
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(a)
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Fig. 1. The absolute value of the (a) interband and (b) intraband conductivity of graphene with chemical potential and frequency.
Fig. 2 illustrates the variation of the third-order conductivity of graphene in terms of chemical potential and
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frequency.
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Fig. 2. The absolute value of the nonlinear graphene conductivity with chemical potential and frequency.
In low frequency ranges where the intraband transition plays the main role, the nonlinear conductivity coefficient is proportional to λ3. On the contrary, when the frequency increases and both the interband and intraband conductivities reach comparable values, quantum mechanical studies show that the nonlinear coefficient is proportional to λ4 [18,20]. It is noteworthy that the frequency range in this work lies within the THz spectrum which keeps the nonlinear conductivity proportional to λ3. Furthermore, the other nonlinear phenomena such as two-photon absorption and third-harmonic generation are ignored throughout this study [33].
3. Proposed structure and its equivalent circuit model The proposed filter, as schematically illustrated in Fig. 3 (a), is comprised of arrays of graphene microdisks with radius a and periodicity L which are deposited on a dielectric substrate. A unit cell of periodic array of graphene disks is demonstrated in Fig. 3 (b). (b)
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(a)
Dielectric
y
x
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k(z)
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Graphene
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z
Fig. 3. (a) Schematic 3-D view of the proposed tunable filter using graphene disks arrays and (b) a unit
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cell of periodic array of graphene disks.
A TM-polarized plane wave with normal incidence travelling in the z-direction is employed to illuminate the
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structure, as shown in Fig. 3 . In what follows, the radius of the disks a = 1 μm and the fill-factor 2a/L = 0.9 are considered. The distribution of the electric field in terms of frequency and x-coordinates at constant y and z is
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demonstrated in Fig. 4. This figure depicts how the electric fields in x-direction distribute as the frequency changes.
In literature, Full-wave modeling and numerical methods are widely used to simulate the performance of the
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optical devices [33,36]. One of the main drawbacks of this technique is its long runtime which depending on the dimensions of the device and meshing resolution might vary from several hours to several days. In contrast, the
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transmission line method which is an efficient and accurate approach that can be used to simulate complex optical devices specifically the graphene-based ones along with significant reduction in the runtime and minimum of the required hardware [12,35,37-38]. In this regard, analytical circuit model using the quasi-static approximation and perturbation theory has been proposed for arrays of graphene disks and is verified with the Full-wave numerical modeling [12,39]. Therefore, the required allocated time of the circuit modeling approach is just less than a few
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seconds, regardless of the dimensions of the structure [12,35]. As a result and in this work, the simulations are conducted through two different and independent methods: (i) Full-wave modeling simulation and (ii) the circuit modeling approach while using an easy to implement MATLAB code. The induced current which specifies the linear and nonlinear dispersive attributes of graphene is considered as: 2 J J L J NL ˆE ( g 3 E ) E .
(6)
Regarding Eq. (6) and values of graphene conductivity shown in Fig. 1 and Fig. 2, a strong excitation is necessary to make the total value of σ3|E|2 comparable with σg, in order to be considered in calculations. (b)
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(a)
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Fig. 4. Distribution of (a) |Ex| and (b) |Ez| in the structure at different frequencies for μc = 0.26 eV.
In order to investigate the linear part, an accurate and analytically developed transmission line method is used
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[35,38]. In this method, the arrays of graphene disks are equivalent to infinite number of series R-L-C circuits in
R1
R2
Rn
L1
L2
... Ln
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circuit model of the proposed filter is shown in Fig. 5.
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which the dual capacitive-inductive attribute of the graphene disks is precisely presented in [39]. The equivalent
C2
Cn
ZS
C1
ZL
Fig. 5. The equivalent circuit model of the proposed graphene-
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based filter.
By considering the incident electric field to be polarized along the x-axis, each of the series R-L-C corresponds
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to a mode of the graphene disks which are defined by [39]:
Rn
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Ln
Cn
L2 K n
2 S n2
Re{ s1}
(7)
L2 K n Im{ s1}
2 S n2
(8)
2 S n2 eff L2 K n q1n
(9)
Where [39]: Sn
a df ( ) 1n 0 d
f1n ( ) d
(10)
Kn
S 1n 1n dS , 1n f1n cos
(11)
The first three eigenfunctions f1n(ρ) and eigenvalues q1n of the graphene disks arrays are given in tables 1 and 2 of [39] respectively. It is noteworthy that by designing the structure near its first mode, the higher order modes of the disk arrays have a minor effect on the accuracy of the circuit model [40]. Therefore, in this study we design the proposed filter near the first mode and use n = 1 as it is considered in [40]. Also, effective medium approximations can be determined using the Maxwell-Garnett formula [41-42]:
2 i ( i m ) i 2 m , 2 m i i ( m i )
(12)
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eff m
where εeff is the effective dielectric constant of the medium surrounding the graphene microdisks. εm, and εi are the permittivity of the free space and the substrate respectively. Also, δi is the volume fraction of the inclusions.
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Assuming a low permittivity substrate, the effective dielectric constant is taken into account as εeff = 1 [33].
4. Results and discussion
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In this section, the proposed structure is investigated for various chemical potentials, fill-factors, and also incident angles. The results obtained via the circuit model approach are discussed and compared with those of the Full-
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wave simulations. Afterward, the Kerr effect using the harmonic balance method is studied.
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Before proceeding, it should be noted that owing to the symmetric structure, identical responses to the TE- and TM-polarized incident waves are expected. This can be considered as an advantage over the other structures
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composed of graphene ribbons [33]. In this regard, the structure is illuminated under different polarizations of the incident source at μc = 0.3 eV. The transmission spectra are rendered in Fig. 6 (a), which imply that the structure
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is polarization-insensitive. Furthermore, perfect agreement between the results calculated by the circuit model approach (CMA) and those ones obtained by the Full-wave modeling method (FMM) is observed. (b)
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(a)
Fig. 6. (a) Normalized value of the transmission spectra of the graphene microdisks arrays with frequency and (b) with angle of incident and frequency at μc = 0.3 eV.
Successively, the structure is illuminated by a TM-polarized plane wave with various incident angles, with respect to the positive z-axis, at μc = 0.3 eV. As depicted in Fig. 6 (b), the structure shows a wide-angle response. The resonant frequency of our structure remains constant as the angle of incident increases, whereas a structure composed of graphene ribbons reveals a shift in the transmission spectrum in accordance with the variation of the incident angle, as reported in [33]. It is worthy of note that in all of the following simulations, the incident wave is considered to be normal with TM-polarization. The effect of coupling strength between disks (which is determined with fill-factor 2a/L) on the transmission
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spectrum is illustrates in Fig. 7. To do so, the period L is assumed to be constant while the radius a is changed.
Fig. 7. Normalized transmission spectra of the tunable graphene-
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based filter versus frequency with different fill-factors at μc = 0.2 eV.
The resonant frequency corresponding to each of the series R-L-C is defined as f r 1 (2 Ln Cn ) . Increasing
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the fill-factor reduces the gap between the microdisks, which leads to a stronger light-graphene interaction. More in detail, according to the table 2 of [39], as the fill-factor becomes larger, the eigenvalue q1n gets smaller. Consequently, regarding Eq. (9), increasing the fill-factor contributes to a Red-shift in the resonant frequency. In
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addition, the quality factor of each series R-L-C is expressed as Q (1 / Rn ) Ln Cn , which is inversely proportional to the width of its corresponding resonance. As seen in Fig. 7, the larger fill-factor accounts for a
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broader resonance which is the result of a reduction in q1n. In all of the following figures, the markers represent the results obtained by the Full-wave modeling simulation and the lines show that of the circuit model approach. As seen in Fig. 7, the CMA and FMM results are verifying each other. The transmission spectrum is also influenced by the chemical potential variations. Regarding Eqs. (7), (8), and
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(9) as the chemical potential is increased, both the real and imaginary parts of σ-1 increase correspondingly. As a result, Ln and Rn increases but Cn remains constant. Therefore, as depicted in Fig. 8, the resonant frequency corresponding to its R-L-C branch decreases which contributes to a Blue-shift in the transmission spectrum.
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Fig. 8. Normalized transmission spectra of the graphene-based terahertz filter for different values of chemical potential.
In what follows, the nonlinear response of the proposed structure is investigated. First off, in order to have an explicit vantage point of the expected effect of the incident lightwave intensity on the transmission spectrum, its
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variation from the graphene effective permittivity point of view is studied. By assumption of graphene as a very
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thin dielectric layer, the nonlinear susceptibility 3( gr ) of graphene in term of its nonlinear conductivity is defined
j 3 , 0c
(13)
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3( gr )
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as [33,43-44]:
2
1 3( gr ) E .
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where c is the velocity of light in free space. Then, the complex permittivity can be expressed as [33,36,44]: (14)
n2
3 3( gr ) 4n02
0 ,
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The corresponding graphene nonlinear refractive index (in units of m2/W) can be modeled by [33,45]:
(15)
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where 0 0 0 377 is the free space impedance and n0 is the linear refractive index (n = n0 + n2I0). The sign of Re(n2) can change in accordance with the angle of the complex value of n0. Since other nonlinear phenomena such as two-photon absorption are ignored, the imaginary part of nonlinear susceptibility 3( gr ) of
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graphene is zero and when |Re(n2)| = |Im(n2)|, a sign-transition occurs in Re(n2) . On the whole, the sign of Re(n2) related to the parameters of the presented results is positive [33]. The effective permittivity of graphene can influence the excitation frequency of its SPs. Taking into account Eq. (14), variation of the incident light intensity contributes to a change in the effective permittivity, which finally results in an all-optical tunable shift in the transmission spectrum. Increasing the complex permittivity of graphene leads to a Red-shift in the transmission spectra as demonstrated in Fig. 9, for μc = 0.1 eV, which it can be achieved by increasing the intensity of the incident light in Eq. (14). More in detail, the coupling between the microdisks is
affected by the surrounding medium and denser medium results in a Red-shift in the resonant frequency. All of
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the following simulations are carried out for the chemical potential of μc = 0.1 eV.
Fig. 9. Transmission spectra with frequency for different values of
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the permittivity for μc = 0.1 eV.
It is worthy of note that simulating the Kerr effect in a structure and investigating the behaviour of dispersive
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materials such as graphene is not possible with the commercial packages including CST or Lumerical [33].
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However, the harmonic balance method, a hybrid frequency- and time-domain technique, is a common technique to deal with cases which contain transmission lines, nonlinearities and dispersive effects [46]. Moreover, the
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harmonic balance method has been used to analyze the Kerr nonlinearity in microribbons graphene arrays [33]. In this study, the same procedure is exploited in order to analyze the Kerr effect in the microdisks graphene arrays.
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The increment in the light intensity modifies the reflected and transmitted power. This variation cycle carries on until the contribution of the reflected and transmitted power reach to a steady state response and finally, the effective permittivity in Eq. (14) is determined which can be calculated by the harmonic balance method [33].
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At low intensity of the incident light, the response of the structure is similar to that of the linear regime which its equivalent circuit model is presented in previous section. For higher light intensity, the resonant frequency is
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modified by the increased contribution of the nonlinear behaviour of graphene, which is revealed as a Red-shift in the transmission spectrum as depicted in Fig. 10 (a). Variation in the light intensity alters the equivalent capacitance based on Eq. (9), while the equivalent resistance and reactance remain unchanged. Furthermore, the results exhibited in Fig. 10 (a) are in agreement with the expected results based on Fig. 9.
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Fig. 10 (b) exhibits the transmission contrast ratio which is an important characteristic for all-optical devices. It is defined as CR = 10log(T1/T0), where T1 and T0 are respectively the normalized transmission at I0 = 0.43 MW/cm2 and I0 = 0.012 MW/cm2. The frequency shift caused by different light intensities is illustrated in Fig. 11 which implies a linear behaviour in response to the increased light intensity. It is noteworthy that any modification in the structure such as variation in the chemical potential of graphene or the lattice fill-factor leads to a different required irradiance intensity to observe the nonlinearity in the graphene disks arrays and a change in the transmission spectrum along with frequency shift.
(a)
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(b)
Fig. 10. Transmission spectra using the harmonic balance method for (a) the incident source intensity of I0 = 0.012 MW/cm2 and
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I0 = 0.43 MW/cm2, (b) transmission contrast ratio for the two values of the incident light intensity as high and low intensity.
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Fig. 11. Frequency shift with incident light intensity.
Moreover, the irradiance intensities used in this study are sufficiently below the optical damage threshold of graphene which is reported to be 1 MW/cm2 for continuous wave [47] which makes the fabrication of the proposed structure a feasible matter. More in detail, fabrication of graphene-based devices is achieved with fast localization
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of the graphene sheets on desired substrates which results in devices without notable defects, corrugations, or morphological deformations [48]. The gate tunability, low required irradiance intensity in the nonlinear regime, and working in THz region are some of the characteristics of the studied filter. The electric field on the surface of graphene disk is highly confined and enhanced since this is one of the main characteristics of the surface plasmons in graphene, as mentioned previously. As a result, the required incident intensity to observe the nonlinear response is reduced in comparison with noble metals, which is one of the main merits of graphene-based all-optical devices. Other noble metals can be used in the proposed structure, replacing the graphene disks, but at the cost of losing
the gate tunability, higher required irradiance intensity owing to the weaker nonlinear attributes in comparison with graphene, and the resonant frequency which lies in the visible and near-infrared spectrum. Finally, simulating the proposed filter using the Full-wave modeling method takes more than 1 hour with a computer possessing the following characteristics: processor, Intel(R) Core(TM) i7-7700 CPU @ 3.60 GHz: installed memory (RAM), 32.0 GB. While for the same computer by usage of an easy to implement MATLAB code which is based on the circuit model, the simulation time is reduced to a less than 0.5 s.
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5. Conclusion To conclude, we have proposed a novel all-optical tunable filter in the THz spectrum composed of arrays of graphene microdisks settled on a low permittivity substrate. An accurate and analytically developed transmission
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line method has been employed to investigate the structure and all of the results have been verified with the Fullwave numerical modeling. The structure reveals a wide-angle and polarization-insensitive response along with taking advantage of the gate tunability. In addition, the proposed structure can be practically achieved using CVD fabrication method. Exploiting the harmonic balance method, we discussed the Kerr effect in the graphene
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microdisks in steady-state regime. The results exhibited a linear Red-shift in the transmission spectrum as the irradiance intensity increased. The proposed filter can be extensively utilized in spectral imaging, communication
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systems and all-optical switching in the terahertz spectrum.
References
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Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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S0030-4026(18)31859-X https://doi.org/10.1016/j.ijleo.2018.11.103 IJLEO 61951
To appear in: Received date: Accepted date:
22 October 2018 23 November 2018
Please cite this article as: Tabatabaei F, Biabanifard M, Abrishamian MS, Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.11.103 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Terahertz polarization-insensitive and all-optical tunable filter using Kerr effect in graphene disks arrays Fatemeh Tabatabaei1, Mohammad Biabanifard1,* and Mohammad Sadegh Abrishamian1 1
School of Electrical Engineering, K.N. Toosi University of Technology, Tehran, 16315-1355, Iran
*
Corresponding author: [email protected], Tel: +989337201191
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Declarations of interest: none.
ABSTRACT
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In this paper, we propose and investigate a novel terahertz all-optical tunable filter comprised of arrays of graphene
microdisks deposited on a dielectric substrate. The proposed filter exhibits a wide-angle response and also due to the symmetric structure is polarization-insensitive. The transmission line theory (TLT) using the circuit model of graphene disks arrays consists of parallel series R-L-C is exploited to investigate the structure for various lattice
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fill-factor and chemical potential. The results obtained via this approach show excellent agreement with the results of a second different method using the Full-wave numerical modeling. The novel proposed TLT method using an
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easy to implement MATLAB code takes advantage of extremely short computational time (less than a few
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seconds) along with a significant reduction in the required memory in comparison to the Full-wave simulations. In addition, the Kerr nonlinearity is studied through the harmonic balance method in steady-state regime. Our
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investigation reveals that a linear Red-shift occurs in the transmission spectra as the intensity of incident light increases. The proposed filter has potential applications in spectral imaging, communication systems and all-
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optical switching in the terahertz band.
Keywords: Graphene, Terahertz filter, Kerr effect, Harmonic balance method, Circuit model, Disks arrays.
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1. Introduction
In recent years, it has been determined for the research community that the Terahertz region has a high potential
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to respond to numerous engineering impasses in designing high-efficiency modern Opto-electronics devices [1-2]. This region which is called “terahertz gap” (roughly from 0.1 THz to 30 THz), is less studied and many of its capabilities are emerging [3]. One of the main obstacles in this regard was the lack of standard materials to respond to the terahertz radiation appropriately which made the manipulation of lightwave in this region a
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challenging problem. The other difficulty was the fabrication challenges since the dimensions of THz devices are in Nano or few Micrometer scales. Nowadays, owing to the prosperous growth of nanotechnology in various scientific and industrial areas besides the advent of new materials like graphene, a significant attention of researchers has been drawn to the THz technology and its applications [1-2,4]. Graphene, a 2D layer of carbon atoms in which the atoms form a honeycomb lattice, with an atomic thickness presents high thermal conductivity and optical damage threshold besides a large third-order optical nonlinearity [5]. This semi-metal material with zero bandgap possesses high absorption in the infrared and visible range which is 2.3% [6]. Extraordinary properties of graphene have paved
the way for significant improvements in many fields specifically in optics [5]. For instance, this material has been widely used in various devices including modulators, antennas, absorbers, polarizers, and optical fiber sensors [713]. By exploiting the well-known chemical vapor deposition technique (CVD) in the past years, it has been shown that the large-area graphene continues sheets with dimensions up to 30 cm can be efficiently growth and fabricated [14]. The number of grown graphene layers besides its characteristics are determined by the aid of atomic force microscopy and Raman spectroscopy [15-16]. In optical devices, the ability to control light with light is of great importance since it is a fundamental requirement in all optical signal processing in integrated circuits, optical
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communications, and optical computing [17]. In this regard, graphene with an exceptionally strong nonlinear optical response and its corresponding large value of nonlinear optical susceptibilities has caught the attention of
the scientific community [18-21]. Terahertz technology which has been continuously progressing in both sources
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and detectors have fueled the proliferation of designing and fabricating optical devices to control terahertz
electromagnetic lightwaves [22]. Moreover, wide applications of terahertz waves in communication, imaging systems, security, astronomy, biological and medical science make the rapidly growing demand for terahertz devices crystal clear [2,22]. The main blind spot regarding this issue is the absence of materials dealing with the
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terahertz waves. Although the metamaterials have been introduced as an alternative to this demand, there are still
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difficulties in designing and implementing terahertz devices [23-25].
Optical properties of graphene which can be tuned to lie in the terahertz spectrum along with the simple design,
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fabrication and integration have extended the feasibility of producing terahertz devices [26-28]. Considering the
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importance of terahertz filters in the communication systems and spectral imaging [29] we propose a graphenebased all-optical tunable filter. This filter takes advantage of the gate tunability along with the large third-order nonlinearity of graphene to control the transmission frequency. This paper is organized as follows. In section 2,
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graphene and its surface conductivity are introduced. In section 3, the proposed filter and its equivalent circuit model are presented. We investigate the structure for different incident wave polarizations and angles, fill-factors, and chemical potentials in section 4 where all the results yielded by the circuit model approach are validated
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against Full-wave numerical simulations. Furthermore, the nonlinear response of the structure and harmonic balance method is studied in this section. Finally, the paper is concluded in section 5.
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2. Graphene surface conductivity In this section, some basic formulas about graphene’s complex surface conductivity are reviewed. Thereinafter, the equivalent circuit model of the graphene disks arrays is discussed in detail. In the absence of a magnetostatic
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field and spatial dispersion the graphene conductivity which is derived from the well-known Kubo formula is defined as follows (by assumption of e σ ω,μc ,,T
j t
as time-harmonic dependency) [30]:
f f ε f f ε f d ε f d je 2 ω j 2 Γ 1 dε ε dε 2 0 0 ω j 2 Γ 2 4ε 2 ε π 2 ε ω j 2 Γ
(1)
where ω is the angular frequency, μc is the chemical potential, Г is a phenomenological scattering rate which is inversely related to the electron relaxation time 1 (2) [31], T is the temperature, -e is the electron charge, ħ
1 is the reduced Plank’s constant, f d e c / k BT 1 is the Fermi-Dirac distribution [30] where ε is the
energy and kB is the Boltzmann constant. More in detail, the electron relaxation time and chemical potential are dependent on each other via m c e F2 , where μm is the carrier mobility and νF ≈ 106 m/s is the Fermi velocity [32]. In this work, the electron relaxation time is considered to be constant as τ = 3 ps and T = 300 K is assumed as room temperature [33]. The first and the second term in Eq. (1) account for the intraband and interband conductivity of graphene
ns
2
2 F2
0 f d f d 2c d
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respectively. The Chemical potential, μc, is influenced by carrier density ns that is defined as [30]: (2)
Therefore, changing carrier density by means of applying a gate voltage or chemical doping leads to variation in
0 sVg
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the chemical potential value [30]. Carrier density can be substituted with ns
de
,where Vg is the bias
voltage, d is the thickness of the substrate, ε0 is the permittivity of the free space, and εs is that of the substrate
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[34]. Afterward, the chemical potential can be derived via numerical techniques.
c 2 ln ec / kBT 1 j 2 k BT e2 K BT
2
(3)
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intra , c , , T j
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After some mathematical calculations, the intraband conductivity of graphene is derived as [30]:
be approximated as follows [30]:
je2 2 c j 2 ln 4 2 c j 2
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inter , c , ,0
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and in the terahertz region with low energy photons, where kBT << |μc| is satisfied, the interband conductivity can
(4)
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Fig. 1(a, b) respectively depict the absolute value of the interband and intraband conductivity of graphene with the chemical potential variations in the THz spectrum. As seen, the intraband conductivity is the dominant term within this frequency range since it is of the order of 10-3 while the interband term is of the order of 10-6.
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Consequently, the interband conductivity can be ignored in the graphene modeling at the terahertz regime. Taking into account the chemical potential μc of graphene and contribution of interband or intraband conductivity, this material is capable of presenting metallic or dielectric behaviour [5]. According to Pauli blocking principle, if photon’s energy ħω is less than 2EF where Fermi energy is EF ≈ μc, it is not absorbed and there is no interband
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transition. In this regard, |μc| > ħω/2 leads to the contribution of intraband transition which lies in the terahertz range, while for |μc| < ħω/2 there is no blocking effect and both the interband and intraband transitions must be taken into account [5]. In order to investigate the nonlinear behaviour of bulk materials, the third-order susceptibility χ(3) is used. However, due to the 2D intrinsic nature of graphene, the current sheet approach is exploited to characterize the nonlinear response. Therefore, the nonlinear conductivity coefficient of graphene is introduced as [18]:
3 j
3 e 2 e F 2 . 32 2 3
(5)
(b)
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(a)
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Fig. 1. The absolute value of the (a) interband and (b) intraband conductivity of graphene with chemical potential and frequency.
Fig. 2 illustrates the variation of the third-order conductivity of graphene in terms of chemical potential and
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frequency.
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Fig. 2. The absolute value of the nonlinear graphene conductivity with chemical potential and frequency.
In low frequency ranges where the intraband transition plays the main role, the nonlinear conductivity coefficient is proportional to λ3. On the contrary, when the frequency increases and both the interband and intraband conductivities reach comparable values, quantum mechanical studies show that the nonlinear coefficient is proportional to λ4 [18,20]. It is noteworthy that the frequency range in this work lies within the THz spectrum which keeps the nonlinear conductivity proportional to λ3. Furthermore, the other nonlinear phenomena such as two-photon absorption and third-harmonic generation are ignored throughout this study [33].
3. Proposed structure and its equivalent circuit model The proposed filter, as schematically illustrated in Fig. 3 (a), is comprised of arrays of graphene microdisks with radius a and periodicity L which are deposited on a dielectric substrate. A unit cell of periodic array of graphene disks is demonstrated in Fig. 3 (b). (b)
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(a)
Dielectric
y
x
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k(z)
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Graphene
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z
Fig. 3. (a) Schematic 3-D view of the proposed tunable filter using graphene disks arrays and (b) a unit
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cell of periodic array of graphene disks.
A TM-polarized plane wave with normal incidence travelling in the z-direction is employed to illuminate the
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structure, as shown in Fig. 3 . In what follows, the radius of the disks a = 1 μm and the fill-factor 2a/L = 0.9 are considered. The distribution of the electric field in terms of frequency and x-coordinates at constant y and z is
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demonstrated in Fig. 4. This figure depicts how the electric fields in x-direction distribute as the frequency changes.
In literature, Full-wave modeling and numerical methods are widely used to simulate the performance of the
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optical devices [33,36]. One of the main drawbacks of this technique is its long runtime which depending on the dimensions of the device and meshing resolution might vary from several hours to several days. In contrast, the
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transmission line method which is an efficient and accurate approach that can be used to simulate complex optical devices specifically the graphene-based ones along with significant reduction in the runtime and minimum of the required hardware [12,35,37-38]. In this regard, analytical circuit model using the quasi-static approximation and perturbation theory has been proposed for arrays of graphene disks and is verified with the Full-wave numerical modeling [12,39]. Therefore, the required allocated time of the circuit modeling approach is just less than a few
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seconds, regardless of the dimensions of the structure [12,35]. As a result and in this work, the simulations are conducted through two different and independent methods: (i) Full-wave modeling simulation and (ii) the circuit modeling approach while using an easy to implement MATLAB code. The induced current which specifies the linear and nonlinear dispersive attributes of graphene is considered as: 2 J J L J NL ˆE ( g 3 E ) E .
(6)
Regarding Eq. (6) and values of graphene conductivity shown in Fig. 1 and Fig. 2, a strong excitation is necessary to make the total value of σ3|E|2 comparable with σg, in order to be considered in calculations. (b)
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(a)
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Fig. 4. Distribution of (a) |Ex| and (b) |Ez| in the structure at different frequencies for μc = 0.26 eV.
In order to investigate the linear part, an accurate and analytically developed transmission line method is used
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[35,38]. In this method, the arrays of graphene disks are equivalent to infinite number of series R-L-C circuits in
R1
R2
Rn
L1
L2
... Ln
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circuit model of the proposed filter is shown in Fig. 5.
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which the dual capacitive-inductive attribute of the graphene disks is precisely presented in [39]. The equivalent
C2
Cn
ZS
C1
ZL
Fig. 5. The equivalent circuit model of the proposed graphene-
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based filter.
By considering the incident electric field to be polarized along the x-axis, each of the series R-L-C corresponds
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to a mode of the graphene disks which are defined by [39]:
Rn
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Ln
Cn
L2 K n
2 S n2
Re{ s1}
(7)
L2 K n Im{ s1}
2 S n2
(8)
2 S n2 eff L2 K n q1n
(9)
Where [39]: Sn
a df ( ) 1n 0 d
f1n ( ) d
(10)
Kn
S 1n 1n dS , 1n f1n cos
(11)
The first three eigenfunctions f1n(ρ) and eigenvalues q1n of the graphene disks arrays are given in tables 1 and 2 of [39] respectively. It is noteworthy that by designing the structure near its first mode, the higher order modes of the disk arrays have a minor effect on the accuracy of the circuit model [40]. Therefore, in this study we design the proposed filter near the first mode and use n = 1 as it is considered in [40]. Also, effective medium approximations can be determined using the Maxwell-Garnett formula [41-42]:
2 i ( i m ) i 2 m , 2 m i i ( m i )
(12)
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eff m
where εeff is the effective dielectric constant of the medium surrounding the graphene microdisks. εm, and εi are the permittivity of the free space and the substrate respectively. Also, δi is the volume fraction of the inclusions.
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Assuming a low permittivity substrate, the effective dielectric constant is taken into account as εeff = 1 [33].
4. Results and discussion
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In this section, the proposed structure is investigated for various chemical potentials, fill-factors, and also incident angles. The results obtained via the circuit model approach are discussed and compared with those of the Full-
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wave simulations. Afterward, the Kerr effect using the harmonic balance method is studied.
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Before proceeding, it should be noted that owing to the symmetric structure, identical responses to the TE- and TM-polarized incident waves are expected. This can be considered as an advantage over the other structures
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composed of graphene ribbons [33]. In this regard, the structure is illuminated under different polarizations of the incident source at μc = 0.3 eV. The transmission spectra are rendered in Fig. 6 (a), which imply that the structure
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is polarization-insensitive. Furthermore, perfect agreement between the results calculated by the circuit model approach (CMA) and those ones obtained by the Full-wave modeling method (FMM) is observed. (b)
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(a)
Fig. 6. (a) Normalized value of the transmission spectra of the graphene microdisks arrays with frequency and (b) with angle of incident and frequency at μc = 0.3 eV.
Successively, the structure is illuminated by a TM-polarized plane wave with various incident angles, with respect to the positive z-axis, at μc = 0.3 eV. As depicted in Fig. 6 (b), the structure shows a wide-angle response. The resonant frequency of our structure remains constant as the angle of incident increases, whereas a structure composed of graphene ribbons reveals a shift in the transmission spectrum in accordance with the variation of the incident angle, as reported in [33]. It is worthy of note that in all of the following simulations, the incident wave is considered to be normal with TM-polarization. The effect of coupling strength between disks (which is determined with fill-factor 2a/L) on the transmission
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spectrum is illustrates in Fig. 7. To do so, the period L is assumed to be constant while the radius a is changed.
Fig. 7. Normalized transmission spectra of the tunable graphene-
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based filter versus frequency with different fill-factors at μc = 0.2 eV.
The resonant frequency corresponding to each of the series R-L-C is defined as f r 1 (2 Ln Cn ) . Increasing
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the fill-factor reduces the gap between the microdisks, which leads to a stronger light-graphene interaction. More in detail, according to the table 2 of [39], as the fill-factor becomes larger, the eigenvalue q1n gets smaller. Consequently, regarding Eq. (9), increasing the fill-factor contributes to a Red-shift in the resonant frequency. In
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addition, the quality factor of each series R-L-C is expressed as Q (1 / Rn ) Ln Cn , which is inversely proportional to the width of its corresponding resonance. As seen in Fig. 7, the larger fill-factor accounts for a
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broader resonance which is the result of a reduction in q1n. In all of the following figures, the markers represent the results obtained by the Full-wave modeling simulation and the lines show that of the circuit model approach. As seen in Fig. 7, the CMA and FMM results are verifying each other. The transmission spectrum is also influenced by the chemical potential variations. Regarding Eqs. (7), (8), and
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(9) as the chemical potential is increased, both the real and imaginary parts of σ-1 increase correspondingly. As a result, Ln and Rn increases but Cn remains constant. Therefore, as depicted in Fig. 8, the resonant frequency corresponding to its R-L-C branch decreases which contributes to a Blue-shift in the transmission spectrum.
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Fig. 8. Normalized transmission spectra of the graphene-based terahertz filter for different values of chemical potential.
In what follows, the nonlinear response of the proposed structure is investigated. First off, in order to have an explicit vantage point of the expected effect of the incident lightwave intensity on the transmission spectrum, its
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variation from the graphene effective permittivity point of view is studied. By assumption of graphene as a very
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thin dielectric layer, the nonlinear susceptibility 3( gr ) of graphene in term of its nonlinear conductivity is defined
j 3 , 0c
(13)
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3( gr )
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as [33,43-44]:
2
1 3( gr ) E .
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where c is the velocity of light in free space. Then, the complex permittivity can be expressed as [33,36,44]: (14)
n2
3 3( gr ) 4n02
0 ,
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The corresponding graphene nonlinear refractive index (in units of m2/W) can be modeled by [33,45]:
(15)
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where 0 0 0 377 is the free space impedance and n0 is the linear refractive index (n = n0 + n2I0). The sign of Re(n2) can change in accordance with the angle of the complex value of n0. Since other nonlinear phenomena such as two-photon absorption are ignored, the imaginary part of nonlinear susceptibility 3( gr ) of
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graphene is zero and when |Re(n2)| = |Im(n2)|, a sign-transition occurs in Re(n2) . On the whole, the sign of Re(n2) related to the parameters of the presented results is positive [33]. The effective permittivity of graphene can influence the excitation frequency of its SPs. Taking into account Eq. (14), variation of the incident light intensity contributes to a change in the effective permittivity, which finally results in an all-optical tunable shift in the transmission spectrum. Increasing the complex permittivity of graphene leads to a Red-shift in the transmission spectra as demonstrated in Fig. 9, for μc = 0.1 eV, which it can be achieved by increasing the intensity of the incident light in Eq. (14). More in detail, the coupling between the microdisks is
affected by the surrounding medium and denser medium results in a Red-shift in the resonant frequency. All of
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the following simulations are carried out for the chemical potential of μc = 0.1 eV.
Fig. 9. Transmission spectra with frequency for different values of
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the permittivity for μc = 0.1 eV.
It is worthy of note that simulating the Kerr effect in a structure and investigating the behaviour of dispersive
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materials such as graphene is not possible with the commercial packages including CST or Lumerical [33].
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However, the harmonic balance method, a hybrid frequency- and time-domain technique, is a common technique to deal with cases which contain transmission lines, nonlinearities and dispersive effects [46]. Moreover, the
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harmonic balance method has been used to analyze the Kerr nonlinearity in microribbons graphene arrays [33]. In this study, the same procedure is exploited in order to analyze the Kerr effect in the microdisks graphene arrays.
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The increment in the light intensity modifies the reflected and transmitted power. This variation cycle carries on until the contribution of the reflected and transmitted power reach to a steady state response and finally, the effective permittivity in Eq. (14) is determined which can be calculated by the harmonic balance method [33].
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At low intensity of the incident light, the response of the structure is similar to that of the linear regime which its equivalent circuit model is presented in previous section. For higher light intensity, the resonant frequency is
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modified by the increased contribution of the nonlinear behaviour of graphene, which is revealed as a Red-shift in the transmission spectrum as depicted in Fig. 10 (a). Variation in the light intensity alters the equivalent capacitance based on Eq. (9), while the equivalent resistance and reactance remain unchanged. Furthermore, the results exhibited in Fig. 10 (a) are in agreement with the expected results based on Fig. 9.
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Fig. 10 (b) exhibits the transmission contrast ratio which is an important characteristic for all-optical devices. It is defined as CR = 10log(T1/T0), where T1 and T0 are respectively the normalized transmission at I0 = 0.43 MW/cm2 and I0 = 0.012 MW/cm2. The frequency shift caused by different light intensities is illustrated in Fig. 11 which implies a linear behaviour in response to the increased light intensity. It is noteworthy that any modification in the structure such as variation in the chemical potential of graphene or the lattice fill-factor leads to a different required irradiance intensity to observe the nonlinearity in the graphene disks arrays and a change in the transmission spectrum along with frequency shift.
(a)
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(b)
Fig. 10. Transmission spectra using the harmonic balance method for (a) the incident source intensity of I0 = 0.012 MW/cm2 and
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I0 = 0.43 MW/cm2, (b) transmission contrast ratio for the two values of the incident light intensity as high and low intensity.
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Fig. 11. Frequency shift with incident light intensity.
Moreover, the irradiance intensities used in this study are sufficiently below the optical damage threshold of graphene which is reported to be 1 MW/cm2 for continuous wave [47] which makes the fabrication of the proposed structure a feasible matter. More in detail, fabrication of graphene-based devices is achieved with fast localization
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of the graphene sheets on desired substrates which results in devices without notable defects, corrugations, or morphological deformations [48]. The gate tunability, low required irradiance intensity in the nonlinear regime, and working in THz region are some of the characteristics of the studied filter. The electric field on the surface of graphene disk is highly confined and enhanced since this is one of the main characteristics of the surface plasmons in graphene, as mentioned previously. As a result, the required incident intensity to observe the nonlinear response is reduced in comparison with noble metals, which is one of the main merits of graphene-based all-optical devices. Other noble metals can be used in the proposed structure, replacing the graphene disks, but at the cost of losing
the gate tunability, higher required irradiance intensity owing to the weaker nonlinear attributes in comparison with graphene, and the resonant frequency which lies in the visible and near-infrared spectrum. Finally, simulating the proposed filter using the Full-wave modeling method takes more than 1 hour with a computer possessing the following characteristics: processor, Intel(R) Core(TM) i7-7700 CPU @ 3.60 GHz: installed memory (RAM), 32.0 GB. While for the same computer by usage of an easy to implement MATLAB code which is based on the circuit model, the simulation time is reduced to a less than 0.5 s.
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5. Conclusion To conclude, we have proposed a novel all-optical tunable filter in the THz spectrum composed of arrays of graphene microdisks settled on a low permittivity substrate. An accurate and analytically developed transmission
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line method has been employed to investigate the structure and all of the results have been verified with the Fullwave numerical modeling. The structure reveals a wide-angle and polarization-insensitive response along with taking advantage of the gate tunability. In addition, the proposed structure can be practically achieved using CVD fabrication method. Exploiting the harmonic balance method, we discussed the Kerr effect in the graphene
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microdisks in steady-state regime. The results exhibited a linear Red-shift in the transmission spectrum as the irradiance intensity increased. The proposed filter can be extensively utilized in spectral imaging, communication
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systems and all-optical switching in the terahertz spectrum.
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Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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