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1. Introduction

More than three decades has passed since the first proposal of the photonic crystal (PC) concept [1, 2]. The unique charac-teristic of these crystals, which has made them an unrivaled structure in photonic technology, is their frequency-dependent response. Technically, those frequencies of the electro-magnetic waves which are allowed to propagate through a PC are called modes and the groups of the permitted modes are called bands. Similarly, the prohibited frequencies form bands which are well known as photonic band gaps (PBGs) [3, 4]. It has been also proven that adding any intentional defect in the structure of a PC results in generation of localized and very sharp modes inside its PBG [5–8]. Taking advantage of

the PBGs provided by PCs, great efforts have been devoted to better understanding complicated physical phenomena such as spontaneous emissions in quantum systems and photon/atom localizations [1, 2, 9]. Many suggestions, not limited to the pure science, have also been given in recent decades on the fabrication of new PC-based optoelectronic devices. Different sensors, transducers, omnidirectional highly reflec-tive mirrors, and all-optical switches are some examples in this regard [3–8]. Furthermore, during the last few years, special attention has been paid to the design and fabrication of tunable PC-based devices which might be very coopera-tive in the development of integrated optics technology [6–8, 10–13]. The use of materials with specific characteristics such as temperature-dependent refractive indices, electro/magneto optical features, or significant response to the mechanical ten-sions has been proposed in this regard.

Laser Physics

Terahertz tunable photonic crystal optical filter containing graphene and nonlinear electro-optic polymer

Fatemeh Ghasemi1,3, Samad Roshan Entezar1 and Sepehr Razi2

1 Faculty of Physics, University of Tabriz, Tabriz, Iran2 Faculty of Electrical Engineering, Urmia University of Technology (UUT), Urmia, Iran

E-mail: fatemeh.ghasemi1@gmail.com, s-roshan@tabrizu.ac.ir, sepehr.razi@gmail.com and s.razi@uut.ac.ir

Received 9 September 2018Accepted for publication 23 January 2019Published 20 March 2019

AbstractIn this paper, new photonic crystal optical filters with very unique optical characteristics are presented. The designed structures are composed of stacked SiO2 isotropic dielectric slabs separated by monolayer graphene sheets. Defect layers of nonlinear electro-optical polymer materials are also inserted to behave as the resonant areas. Results obtained by using the transfer matrix method approach clearly reflect the possibility of the wide photonic band gap generation in the absorption spectra of the proposed crystals in the terahertz frequency range. Furthermore, a very strong relation is seen between the defect layer(s) induced resonant modes and the location/number of the defect(s) inside the crystal. To widen the scope of our findings, in addition to studying the dependencies on the structural characteristics, the tunability of the optical features using the external parameters is investigated systematically. It is shown that the number and central frequencies of the resonant modes are controllable in the far-infrared range by adjusting the temperature and wave incident angle. Thus, the proposed crystals might have remarkable applications in the fabrication of multi-channel tunable filters used in optical integrated circuits.

Keywords: photonics crystal, nonlinear electro-optic polymer, photonic band gap

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For instance, Hung et al [14] proposed a semiconductor-dielectric PC structure whose transmittance is thermally tun-able due to the strong, temperature-dependent permittivity of its InSb defect layer. Němec et  al [15] also experimentally proved the successful thermal tunability of a PC formed of alternating layers of quartz and a high-permittivity ceramic. A defect layer of SrTiO3 slab was suggested by them to be inserted into the crystal. In other cases, taking advantage of the electro-optical (EO) features of materials such as Li:NbO3 or liquid crystals, the electrical tuning of the optical features of PCs is suggested [13, 16]. Magnetic tuning is the other pro-posed method, in which an external magnetic field is utilized to tune the optical characteristics of a PC [17, 18]. A few other approaches, such as the use of mechanical strains, have been also proposed [19].

Discovery of the allotrope of carbon known as graphene caused a remarkable revolution in optoelectronics and many other sciences/technologies [3, 10–12, 20]. Within a very short time after its discovery, it was proven that inserting graphene defect layers, or choosing graphene as one of the alternating layers of a PC results in very interesting optical features [5–7]. Considering the chemical potential of gra-phene sheets, which are dependent on the external gate volt-age [20, 21], the fabrication of new tunable photonic devices absolutely seems possible. This was the starting point of the suggestion of many valuable graphene-based PC structures in last few years. In almost all the research conducted, the elec-trical tuning of the crystal characteristics by controlling the surface conductivity of the graphene layer(s) was discussed [6–8, 10–13]. Other research has focused on controlling the optical features of graphene-based PCs by an external magn-etic field. Rashidi et al suggested an external magnetic field for controlling the absorbance of a graphene-based PC [17]. Ardakani et  al studied a structure consisting of a graphene layer placed in between two magneto-optical layers that are themselves located between two PCs [18]. They theoretically explored tunability of the optical bistability behavior of the structure using an applied external magnetic field. For further information, one can refer to other reports [3, 4, 21] and also the publications of our research team in [10, 11, 22–25].

In this paper, taking into consideration the favorable char-acteristics of graphene and the recently proposed nonlinear EO polymers, new tunable PC filters are proposed. This

manuscript is organized as follows: theoretical methods and mathematical approaches are presented at first. Optical fea-tures of an ideal one-dimensional (1D) crystal composed of periodic layers of graphene/SiO2 slabs are then presented. Due to the importance of the structural characteristics to the optical response of the crystal, the dependency of the defect modes on the defect location inside the crystal is studied sys-tematically at the next step. Finally, the tunability of the defect layer(s)-induced resonant modes is discussed extensively.

2. Theory

2.1. Materials and physical structures

A crystal composed of alternating layers of graphene and SiO2 is assumed as the ideal PC. Structures including defect layers of nonlinear electro-optical polymer materials are also con-sidered in this study (figure 1). The interfaces of the layers are assumed to be in the x–y plane, and each layer is supposed to be isotropic in this plane. The Z-direction of the Cartesian coordinate system is considered to be normal to the interfaces. Dimensions of the layers are supposed to be 8 μm in x and infinite in y directions, but in contrast, the widths of these lay-ers are assumed to be 0.335 nm, 40 μm, and 8 μm for gra-phene, SiO2, and polymer layers, respectively. Surrounding atmosphere is air, and the incident angle is measured with respect to the z axis.

Before further explanation of the simulation methods, it is worth presenting a short review on the most important optical features of graphene and the polymer because of their impor-tance in nano-electronics/photonics.

2.1.1. Graphene. The optical conductivity of graphene depends strictly on frequency, temperature, as well as its chemical potential and relaxation time. These in turn depend on other parameters such as the material purity [12, 20]. In the case of monolayer graphene sheets, the optical conduc-tivity can be calculated by Kubo formula [26, 27] that takes into consideration both inter and intra-band transitions: σ (ω, T) = σintra + σinter ,

σintra = je2kBT

π� (ω + jΓ)

ŵc

kBT+ 2ln

Äe−

µckBT + 1

äã (1)

Figure 1. Schematics of the double-defect photonic crystal structure.

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σinter = je2

4π�lnÅ

2µc − (ω + jΓ)2µc + (ω + jΓ)

ã. (2)

Here, e,Γ and µc are respectively the charge of an elec-tron, a phenomenological scattering rate, and the chemical potential of the graphene. In addition, T , kB, and � present the Kelvin temperature, Boltzmann’s constant, and reduced Planck’s constant. Among the parameters that the conductiv-ity is dependent on, chemical potential is very important for its tuning. It can be easily changed by adjusting the gating

voltage, µc ≈ �νf (πεrε0V/ed)1/2 [4, 27]. In this equation, νf is the Fermi velocity, εr defines the relative permittivity of the dielectric slab, ε0 is the permittivity of the free space, and d presents the thickness of the dielectric layer.

For a uniaxial graphene stripe in the x–y plane, the main diag-onal elements of the permittivity tensor are non-zero and given by εG = (εG, t, εG, t, εG, ⊥) [21]. By taking into consideration the fact that the normal electric field cannot excite any current in the 2D graphene sheet, the normal component of the permit-tivity can be regarded as εG,⊥ = 1 [12, 25]. But in contrast, the tangential permittivity is more complicated and depends on many parameters:εG,t = εG,⊥ + j (σ (ω, T) /ε0ωdG) [28]. Finally, it is worth noting that the dispersion relation for the TE polarized waves is described by k2

x + k2z = εG,tk2

0 [12, 28].

2.1.2. Polymer. Recently new polymer materials having elec-tro-optic (EO) coefficients much higher than that of the Lith-ium niobate (Li:NbO3) or liquid crystals have been proposed [29, 30]. However, limited reports have been presented on the usage of these polymers as a key building block of PCs up to now [16]. There are also reports on the failure of their usage in PC structures, such as their small light confinement ability or causing small band gaps relative to the high index materials [30]. Thus, much more systematic studies are necessary in this regard before they could replace the common materials which are well-known as the standard platforms for fabrications of many photonic elements. Here in this paper, utilizing the valuable experimental data in [31], a nonlinear EO polymer is selected as the defect layer. External tuning voltage is applied along the x direction. The nonlinear polymer is considered isotropic, and its response time is supposed to be instant. It is also assumed to be poled such that the axis of the strongest EO interaction is aligned along the x-direction. Thus, it is reason-able to assume that only the x-components of the optical and modulating fields are relevant. This x-directed electric field is able to change the refractive index of the crystal via the linear EO effects. The change in the refractive index is calculated by n′

e = ne − (1/2) n3eγ33E, γ33 = 150 pm V−1 [16, 31], in

which the associated EO coefficient is represented by γ33.

2.2. Theoretical basis

Propagation of the electromagnetic wave through the crys-tal is studied using the well-known transfer matrix method (TMM) [10–12, 24]. This approach, which is an appropriate method for investigating the structures containing any number of layers, takes into account many important possible physical phenomena. Considering the reflections from interfaces and

taking into account the polarization state of the incident wave are two instances in this regard [21, 22]. Furthermore, TMM can be easily extended to any convoluted multilayer structure with much less computational complexity compared to the other approaches such as finite element or finite difference frequency domain methods [11, 32].

Associating dynamical and propagation matrices for each layer of the stack, its transfer matrix, which describes the propagation of the input field, is extracted as [12, 21]

Mi = MDi MPi =

Çcos (kizdi) − j

pisin (kizdi)

−jpisin (kizdi) cos (kizdi)

å. (3)

Using this matrix and inserting the related parameters of each layer, variations of the wave amplitude in passing a layer (i = graphene, SiO2, or polymer) is determined. Here, pi is

equal to kiz/ωµ0, kiz =»

k20εi − k2

x and kix = kx = k0sinθ0 for

TE polarizations.Taking into consideration the relations for single layers, the

response of the total crystal is simply determined by multiply-ing the matrices of all the involved layers. Therefore, for the ideal crystal with N periods, the total transfer matrix is

M = (MGMSiO2)N=

Çm11 m12

m21 m22

å. (4)

Similarly, total matrices for proposed structures with one or two defect layers could be extracted as

M = (MGMSiO2)N1 (Mpoly) (MGMSiO2)

N2 =

Çm11 m12

m21 m22

å

(5)

M = (MGMSiO2)N1 Mpoly(MGMSiO2)

N2 Mpoly(MGMSiO2)N3 =

Çm11 m12

m21 m22

å.

(6)Reflection (R), transmission (T), and Absorption (A) of the crystals could be calculated by R = |r|2, T = |t|2, and A = 1 − T − R, respectively. Reflectance and transmittance coefficients are described by [21]

r =(m11 + m12pn+1) p0 − m21 − m22pn+1

(m11 + m12pn+1) p0 + m21 + m22pn+1 (7)

t =2p0

(m11 + m12pn+1) p0 + m21 + m22pn+1. (8)

Here, p0 is equal to Ä√

µ0/ε0

äcos (θ0) and pn+1 is √

µn+1/εn+1cos (θn+1). Air is considered in our simulations as the surrounding input and output media, and ε0 and εN+1 are denoting the relative dielectric constants of the input and output planes.

3. Results and discussions

3.1. Resonant modes

In a PC composed of 22 periods of graphene/SiO2 layers, defect layer(s) can be inserted in any position in between (i.e. with any number of periods on its top/bottom). Taking

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into consideration how a defect layer is able to change the optical features of a PC, it is very critical in many cases to have enough knowledge of its exact position. Hence, to widen our insight, dependency of the crystal absorption spectrum on the position of the defect layer is investigated at the first step. Furthermore, the dependency of the optical characteris-tics on the number of defects is studied and their results are compared. Two defective crystals (photonic crystals including single (SDPC) and double (DDPC) defects) are considered in this regard. Values of the geometrical, electrical, thermal, and optical parameters in our simulations are considered as fol-lows: dgraphene = 0.335 nm, dSiO2 = 40 μm, dpolymer = 8 μm, T = 300 K, μ = 0.9 eV, and Γ = 1 THz. Moreover, it is worth noting that only waves with TE (E = (0, Ey, 0), H = (Hx, 0, Hz)) polarizations are considered in the study.

The inset in figure 2(a) presents the absorption spectrum of the ideal perfect crystal in the frequency range of 4.5 to 8 THz. Two PBGs are observed in this interval, but no signs of resonant modes are observed inside the PBGs of the ideal crystal. Insertion of a defect layer inside the crystal results in generation of the resonant modes in the gaps. Figure 2 illus-trates the localized first and second resonant modes in the 5.03 to 5.5 THz and 7.05 to 7.14 THz frequency ranges for SDPC in terms of the defect layer positions. The defect breaks the symmetry (structural and translation symmetries) and a reso-nance takes place around it. As a result, a mode is generated inside the PBG.

In order to better explore the resonant modes of the SDPC, intervals of the PBG are illustrated separately in figures 2(a) and (b). In these graphs, N1 denotes the number of periods on top of the defect layer and modes 1 and 2 represent the reso-nant modes in the first and second PBGs of the spectrum. It is clear that the position of the defect layer can significantly affect both the first and second modes, but their dependency is to some extent different. In the case in which the defect layer is the first interface that the light interacts with, no resonant mode is observed in the first gap. Additionally, no mode is generated in the second PBG. Moving the defect layer from the top layer towards the middle of the crystal, a peak starts appearing in the first and second gap ranges. By

increasing the N1 value, the central frequency of the resonant modes in both the PBGs shift gradually to the higher fre-quencies. Furthermore, their width reduces and the heights (intensities) of the peaks increase. Finally, in the case of N1 = 6 periods, the first mode gains its maximum value of 98.6% absorption while the absorption of the second mode reaches 81%. However, the second mode obtains its maxi-mum absorption value of 90% at N1 = 7. In this case, the first mode reaches 90% of absorbance. Adding even one more period of layers, the absorption value of the first mode starts to decrease rapidly. However, for N1 = 8, the absorption value of the second mode remains almost unchanged com-pared to N1 = 7. Since then, the height of the second mode starts to decrease and its width increases. A very small shift towards the lower frequencies is also observed in both the modes after their maximum absorbance. Similarly, further raising the N1 value, the height of the first mode decreases rapidly and finally vanishes when the defect layer is placed at the last layer of the crystal (similar to when it was in the initial extremity of the PC).

In the case of DDPC, two resonant modes are observed in each PBG. The dependency of these modes on the posi-tion of the defect layers is presented separately in figure  3. In these graphs, N1 denotes the number of periods before the first defect layer, and N2 is the number of the periods in between the defects. The modes are numbered from low to high frequencies. Optimum period numbers that are neces-sary for generation of the well-separated defect modes in the gaps is investigated at the first step. We start with the case in which the defect layers are sticking together and 10 peri-ods of layers exist symmetrically on both sides. No resonant mode is observed in the frequency range from 5.2 to 5.45 THz, figure  3(a), or in the second selected range from 6.8 to 7.2 THz, figure 3(b). Placing a single period of layers in between the defect slabs, one mode appears at ~5.26 THz in the first selected interval. In this case, a mode at 6.81 THz is also observed in the second PBG range. By adding one more period, two well-separated modes are formed in both the gaps. In the first stop band, the first mode has much lower height than the second one. However, in the second gap, the intensity

Figure 2. Absorbance spectra of 1D PCs with a single polymer defect layer. Resonant mode (a) in the first and (b) second PBG intervals. Inset illustrates the absorption spectrum of the ideal perfect crystal in the frequency range of 4.5 to 8 THz. N1 changes from 0 to 21 and denotes the number of periods of graphene/SiO2 layers that are placed before the single defect of the crystal. Both graphs are for normal incidence at T = 300 K and μ = 0.9 eV.

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of the first mode is larger than the second one. Adding more periods of graphene/SiO2 layers in between the defect layers, two peaks in the first stop band get closer and their intensity increases. Finally, at N2 = 10, they merge together com-pletely and at N2 = 12, the single merged mode has its maxi-mum amount of absorption value. From here, the peaks do not separate anymore and adding more periods reduces the absorption value of the crystal. Instead, the single peak gets broader and shifts slightly towards the lower frequencies. In the case of N2 = 20, which presents the crystal with defect layers at its two extremities, there exist no perfect peak at the first PBG. Similarly, in the second frequency range, increasing N2, the separation and height differences of the peaks reduces, but despite the first PBG interval, they merge at N2 = 11. At N2 = 19, the merged mode is eliminated almost completely. Using the above-presented results, in the following, the value of N2 is fixed as 3 and the dependency of the modes on the number of periods placed before the first defect is studied. Figures 3(c) and (d) illustrate the relation between the absorp-tion spectrum and the N1 value. It is worth noting that the number of periods existing after the second defect (N3) can be calculated by subtracting the values of N1 and N2 from the total number of 22.

Taking into consideration N1 = 0, i.e. light faces the defect layer at first, just one broad peak is generated in both stop bands. Adding one period on top of the crystal, two different wide peaks start to form in both bands. Inserting one more period (N1 = 2), two resonant modes are formed more clearly in both PBGs. Increasing the N1 value, the absorption amount of both modes of the first gap increases until they reach their maximum value of 93% and 78% at N1 = 5. Separation of the peaks is not affected by N1 until the N1 = 3 period and later. The intensities of the peaks are reduced with further increase in N1, and after N1 = 12, they are almost totally eliminated. The changes of the modes in the second selected frequency inter-val are to some extent similar to the first gap. Increasing the N1 value up to 6, absorption intensity of the first mode increases,

but its central frequency does not change significantly. Further increasing the N1 value, the height of the mode decreases and finally is eliminated completely. The second resonant mode of this interval behaves similarly: it first increases to a maximum value at N1 = 6 periods, and then decreases and finally van-ishes after N1 = 12.

3.2. Incident angle dependency

In the following two subsections, the tunability of the defect layer(s)-induced resonant modes is studied. Considering the perfect crystal as the one for which very sharp, intense, and well-separated resonant peaks are observed in its PBGs, in the following the number of layers is fixed as N1 = 6 and N2 = 15 for SDPC and N1 = 6, N2 = 3, and N3 = 11 in the case of DDPC. Figures 4(a) and (c) illustrate the dependency of the absorption spectrum to the wave incident angle. The color maps of the absorbance spectra versus frequency and incident angle are also presented in figures  4(b) and (d) to better reflect the changes. It is clearly observed that in both crystals, including single and double defect layers, both the resonant modes and the total PBG intervals of the spectra are significantly affected by the incident angle. A serious blue-shift is observed for all the resonant modes with incident angle increase. However, the variations in the intensity of the modes are not so severe, and in most cases is less than ~10%. The shift in absorbance spectrum might be related to the changes of the phase difference (ϕ = kzldl). When the incident angle increases, then the wavelength λ has to decrease accordingly to obtain the same phase change ϕ.

3.3. Temperature dependency

In this section, the dependency of the crystal absorbance spectrum on the temperature is explored. For both SDPC and DDPC, it is seen in figure  5 that by raising the crystal temper ature from 100 up to 300 K, the central frequency of

Figure 3. Absorbance spectra of 1D PCs with two polymer defect layers. (a) and (c) show the two resonant modes in the first PBG, and (b) and (d) show the two resonant modes in the second PBG intervals. N2 denotes the number of graphene/SiO2 layers placed in between the defect slabs of the crystal. N1 denotes the number of graphene/SiO2 layers on top of the first defect layer. Both graphs are for normal incidence at T = 300 K and μ = 0.9 eV.

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Figure 4. Dependency of the absorption spectra of the 1D PCs with single and double defect layers to the wave incidence angle. Graphs of the upper row (a) and (b) belong to the SDPCs, and the lower ones (c) and (d) are results for DDPCs. Figures (b) and (d) are color maps of absorbance versus frequency and incident angle. All graphs are for normal incidence at T = 300 K and μ = 0.9 eV.

Figure 5. Tunability of the absorbance spectra of the (a) SDPC and (b) DDPC by the crystal temperature alterations. Results are for normal incidence at μ = 0.9 eV, N1 = 6, N2 = 15 for SDPC and N1 = 6, N2 = 3, N3 = 11 for DDPC.

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all resonant peaks shifts towards the lower values (redshift happens). However, the height (intensity) of the modes is not affected by temperature so much. In comparison to the incident angle dependency, changes of the absorption spectra due to the temperature alterations are not so severe. In other words, the shift in frequency as a result of temperature increase from 100 to 300 K is just ~3%. Excluding the PBG regions, the absor-bance of the structures in other frequency intervals is almost the same for the three selected temper atures. The other impor-tant point worth noting is the independ ence of the PBG width from both the incident angle and temperature alterations.

In order to analyze the results more quantitatively, the quality factor (Q-factor), defined as the ratio of the peak cen-tral frequency to its FWHM, is calculated for all the modes as a figure of merit (figure 6). It is clear that in the case of SDPC, figures 6(a) and (c), the Q-factor of the second mode (the mode in the second gap) is much more than the first one mode. The Q-factors of both modes are almost independent of the temperature, and they remain nearly constant in the temper ature interval of 100–350 K. However, the Q-factors of both modes are seriously dependent on the wave incidence angle, such that a very sharp increase is observed by increas-ing the incident angle. The amount of change is much more severe for the defect mode in the second PBG (defect mode 2). The dependency of the Q-factors of the modes in the PBGs of the DDPC to temperature and incident angle are similar

to the behaviors observed in SDPC. The first mode of each PBG has lower Q-factor than the second one, but by raising the temper ature, their difference decreases to some extent. The maximum value of the Q-factor is observed in the case of the defect mode 4 (the second mode of the second PBG). Increasing the wave incident angle, the Q-factors of all the modes increase. However, the amount of changes in the third and fourth modes are much more remarkable compared to the first and second modes.

4. Conclusion

Absorbance of TE polarized electromagnetic waves of 4.5–8.5 THz propagating through 1D-PCs composed of graphene/SiO2 alternative layers and defect layers of nonlinear EO polymers was studied in detail. Results show that the defect layer(s)-induced resonant modes inside the PBGs are seriously dependent on the location of the defect(s). In other words, the central frequency, full width at half maximum, and number of resonant modes are extremely varied with the number of alter-nating layers existing before/after/between the defect layer(s). Dependencies of the resonant modes in the first and second PBGs are almost identical.

Tunability of the resonant modes was also studied by tak-ing into consideration the dependency of the optical features

Figure 6. Q-factors of the defect modes versus the crystal temperature and wave incident angle; (a) and (c) are for SDPC, (b) and (d) are for DDPC. Modes are numbered from left to right in the frequency range (i.e. lower frequencies to higher ones).

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on the wave incident angle and crystal temperature. It was found that when the incident angle is increased, the PBGs of crystals with both single and dual defect layers shift to higher frequencies. The resonant modes also show the same behavior in compliance. In contrast, a redshift in central frequencies of the resonant modes is observed when increasing the crystal temperature from 100 to 300 K. However, the PBG width is independent of the wave incident angle and crystal temper-ature. Furthermore, the Q-factor of the modes is almost inde-pendent of the temperature changes but depends seriously on the incident angle. Thus, it is reasonable to conclude that the proposed structures have great potential to be used in fabrica-tion of new generations of tunable optical filters.

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Laser Phys. 29 (2019) 056201