1. Introduction
During the past decades, photonic crystals have drawn considerable attention due to the unique properties of photonic band gaps originating from the consequence of Bragg scattering [1-3]. In recent years, the study of photonic crystals has been expanded into the area of artificial metamaterials [4] after the realization of composite medium with simultaneously negative permeability and permittivity [5, 6]. Subsequently, characteristics of photonic band gaps were investigated widely in photonic crystals containing different metamaterials, such as isotropic [7-9] and anisotropic [10] negative index materials, single negative materials [11, 12], as well as hyperbolic metamaterials [13]. In particular, photonic band gaps have been investigated in quasiperiodic photonic crystals containing metamaterials with the Fibonacci sequence [14, 15]. Subsequently, many works have been reported in quasiperiodic Fibonacci photonic crystals containing metamaterials, including band edge states [16], plasmon polaritons [17], Anderson localization, and Brewster anomalies [18], non-Bragg-gap solitons [19], as well as second harmonic generation [20], etc.
Photonic band gaps were also studied in photonic crystals containing NZPIM [21-23]. The most interesting property of NZPIM is optical DP with a double-cone structure, which was realized for the first time in 2009 [24]. Subsequently, the propagation properties of optics in different structures with NZPIM have been studied, for example, the Zitterbewegung effect [25], thermal emission [26], the transmission and Goos-H $\ddot{{\rm{a}}}$ nchen shifts [27-29], NZPIM waveguide [30-32], and nonlinear guide waves [33, 34], etc. These results predict that simple NZPIM structures can be created as an alternative to studying exotic phenomena in graphene, where DP with a double-cone structure is formed at which the conduction and valence touch each other [35-37].
For the photonic band gaps in photonic crystals containing NZPIM, there exist an omnidirectional passing band and a kind of special band, which result from the interaction of the evanescent and propagating waves [21]. Many novel properties such as the defect tunneling effect and DP band gap have been presented [22]. A broadband wave plate based on a one-dimensional photonic crystal containing NZPIM was also demonstrated [23]. However, up to now, quasi-periodic photonic crystals with NZPIM, such as quasi-periodic Fibonacci photonic crystals [14-20], have not been discussed.
In this paper, we study the transmission properties in a quasi-periodic photonic crystal containing NZPIM with different Fibonacci sequences. We obtain some transmission band gaps, including DP gap, zero-$\bar{n}$ gap, Bragg gap, and gaps beside the DP gap induced by total reflection. We also demonstrate the corresponding GH shifts and phase shifts in detail. The characteristics of the band gaps, GH shifts, and phase variations depend on the incident angle and periodic number, as well as layer thickness.
2. Model and basic equations
Consider a quasi-periodic photonic crystal with the structure in each cell following the Fibonacci sequence [15, 16], Sj, by a recurrent relation Sj+1 = Sj, Sj−1, with S0 = B and S1 = A, where j is the generation number of the Fibonacci unit cell, the first few sequences are S2 = AB, S3 = ABA, S4 = ABAAB, S5 = ABAABABA, and so on. For example, as shown in figure 1, we consider in this paper the transmission properties of the following Fibonacci sequences, (AB)N, (ABA)N, and (ABAAB)N, where N is the number of periods. We assume that the layers A with thickness d1 are made of NZPIM with permittivity ε1 and permeability μ1. We also assume that layers B are the conventional dielectric medium (for example, air) with thickness d2, and the permittivity and permeability are ϵ2 = 1 and μ2 = 1, respectively. In this paper, we consider a TE-polarized light beam with frequency ω incident from air at an angle θ onto the Fibonacci quasi-periodic photonic crystals containing NZPIM (TM-mode can be treated in the same way).
Figure 1. Examples of Fibonacci quasi-periodic photonic crystals containing NZPIM, (a) S2: (AB)N, (b) S3: (ABA)N, and (c) S4: (ABAAB)N. |
The permittivity and permeability of the NZPIM, ϵ1 and μ1, can be taken in the Drude model ${\varepsilon }_{1}\left(\omega \right)={\mu }_{1}\left(\omega \right)=1-{\omega }_{D}^{2}/{\omega }^{2}$ [24, 25], with ωD as the frequency of the optical DP. It is clear that the permittivity and permeability of the NZPIM can change from negative to zero and then to positive near DP. NZPIM has linear dispersion near DP [24], which can be written as
$\begin{eqnarray}\kappa \left(\omega \right)=\displaystyle \frac{\omega -{\omega }_{D}}{{\upsilon }_{D}}=\displaystyle \frac{2(\omega -{\omega }_{D})}{c},\end{eqnarray}$
with νD = dω/dκ ≈ c/2 (c is the speed of light in vacuum) is the group velocity of the light near DP (ω = ωD). Generally, the frequency domain to match the linear dispersion of NZPIM is $\tfrac{2}{3}{\omega }_{D}\lt \omega \lt 2{\omega }_{D}$ [24, 27, 30]. In this paper, we suppose the frequency of DP is ωD = 2π × 10 GHz.Via the transfer matrix method [38], the transmission coefficient of the incident beam can be written in the following form
$\begin{eqnarray}M=\left(\begin{array}{cc}\cos {\delta }_{j} & -{\rm{i}}\,\sin \,{\delta }_{{\rm{j}}}/{{\rm{q}}}_{{\rm{j}}}\\ -{\mathrm{iq}}_{{\rm{j}}}\,\sin \,{\delta }_{{\rm{j}}} & \cos \,{\delta }_{j}\end{array}\right),\end{eqnarray}$
with δj = −kjzdj and ${q}_{{\rm{j}}}=\tfrac{{k}_{{jz}}}{{k}_{0}{\mu }_{j}}$, kjz(j = 1, 2) are the z components of the transmitted wave vector in NZPIM and air, in which ${k}_{1z}=\sqrt{{\kappa }^{2}(\omega )-{h}^{2}}$ and ${k}_{2z}=\sqrt{{\left(\omega /c\right)}^{2}{\varepsilon }_{2}{\mu }_{2}-{h}^{2}}$ with $h=(\omega /c)\sqrt{{\varepsilon }_{2}{\mu }_{2}}\sin \theta $ being the propagating constant.Substituting the boundary values of the corresponding electric field and magnetic field into the transfer matrix, the transmission coefficient t(ω) can be represented as
$\begin{eqnarray}t(\omega )=\displaystyle \frac{2\cos \theta }{[{M}_{11}+{M}_{12}\cos \theta ]\cos \theta +[{M}_{21}+{M}_{22}\cos \theta ]},\end{eqnarray}$
where Mij(i, j = 1, 2) represent the elements of the 2 × 2 transfer matrix for the composite medium. The transmittance of the light beam through the quasi-periodic Fibonacci photonic crystals can be written as T(ω) = ∣t(ω)∣2. Subsequently, the transmission properties, for example, the GH shifts [39] of the transmitted beam can be calculated analytically with the stationary phase method $\begin{eqnarray}{\rm{\Delta }}=-{\rm{d}}{\rm{\Phi }}(\omega )/{\rm{d}}h,\end{eqnarray}$
where ${\rm{\Phi }}(\omega )={\tan }^{-1}[\mathrm{Im}t(\omega )/\mathrm{Re}t(\omega )]$ is the phase shift of the transmitted beam [40].3. Transmittance properties of photonic crystals with Fibonacci sequence
3.1. Transmission properties of (AB)N
Firstly, we study the transmission characteristics of the (AB)N structure. We set the thicknesses of NZPIM and conventional medium to d1 = 12 mm and d2 = 6 mm, and the number of periods is N = 16. In figures 2(a) and (b), we display the dependence of the transmittance T(ω) on frequency ω/2π at different incident angles. We can see in figure 2(a) that an asymmetric band gap, DP gap (gap I), appears at a certain range of the frequency near the DP for oblique incidence with θ = 10o. However, the DP gap vanishes for normal incidence with θ = 0o, which is suitable for arbitrary Fibonacci sequences and the number of periods. As shown in figure 2(b), the width of the DP band gap gradually increases with the increase of the incident angles, and the central frequency of the DP band gap also gradually shifts upwards in frequency with the increase of the incident angles.
Figure 2. Transmittance T(ω) of structure (AB)N on the incident frequency with different incident angles θ = 0°, 10° (a) and θ = 20°, 25°, 30° (b), respectively. The thicknesses of the medium are d1 = 12 mm and d2 = 6 mm. The number of periods is N = 16. |
In the frequency range ω < ωD, the refractive index of NZPIM is negative, and there exhibits an omnidirectional zero-$\bar{n}$ band gap (gap III) when the incident angle is above a threshold. The zero-$\bar{n}$ gap occurs around the frequency ω/2π = 8.165 GHz, at which the average refractive index of the periodic structure satisfies the requirement [n1(ω)d1 + n2d2]/(d1 + d2) = 0. In fact, this zero-$\bar{n}$ gap has been studied extensively in a one-dimensional stack of layers with alternating right-handed and left-handed materials [4]. When the incident angle increases, the edge of the zero-$\bar{n}$ gap shifts toward lower frequencies slightly while the width becomes bigger.
In the frequency range ω > ωD, the refractive index of NZPIM is positive. For all incident angles, there exhibits another band gap (gap II), the Bragg band gap, which is analogous to those studied in conventional one-dimensional photonic crystals due to Bragg scattering. From figure 2, we can see the center of the Bragg band gap always shifts towards higher frequency when the incident angle increases. Thus the Bragg band gap will disappear in the range $\tfrac{2}{3}{\omega }_{D}\lt \omega \lt 2{\omega }_{D}$ when the incident angle is large enough, with which the periodic structure of (AB)N can be used as a broadband wave plate [23].
Figure 3(a) illustrates the dependence of the GH shifts of the transmitted light on the incident frequency ω/2π. Obviously, the GH shifts are always negative when ω < ωD and are always positive when ω > ωD, due to the fact that the refractive index of NZPIM has different signs on both sides of DP. At DP, the refractive index of NZPIM is zero, and the GH shifts are zero for all incident angles. Thus the GH shifts change from negative to positive at DP. It is further demonstrated that the GH shifts are giant near both the low- and high-frequency edges of all three band gaps, especially the DP gap. For example, when θ = 30°, the GH shifts are about −254.2λ and 200.7λ near the low- and high-frequency edges of the DP gap, respectively.
Figure 3. (a) The dependence of GH shifts in the structure (AB)N on incident frequency with different incident angles θ = 20° (dash-dotted line), θ = 25° (solid line), and θ = 30° (dashed line), respectively. (b) The phase shift Φ of structure (AB)N on incident frequency with incident angle θ = 20°. The thickness of the medium are d1 = 12 mm and d2 = 6 mm. The number of period is N = 16. |
In order to indicate the variations of GH shifts clearly, we show in figure 3(b) the dependence of the phase shift Φ(ω) on incident frequency ω/2π with the incident angle θ = 20°. It is clear that the phase shift is subject to rapid variations at the edges of each gap, leading to a great shift of the outgoing beam just like the case of total internal reflection.
The number of periods N of the photonic crystal also affects the transmission properties of the light beam. For comparison, when the incident angle is θ = 25°, we display in figure 4 the transmittances of the beam when the numbers of periods are N = 8, N = 16 and N = 48, respectively. It is clear that the DP gap is not sensitive to the number of periods with its width and position almost remaining unchangeable. However, the zero-$\bar{n}$ gap and Bragg gap exist only when the number N is above a threshold (N = 16, 48), below which, the band gaps disappear at N = 8. Interestingly, by increasing the number N = 48, another special complete transmission band gap (gap IV) emerges alongside the DP gap around the frequency ω/2π = 14.27 GHz when the incident angle is larger than θc = 20°. This band gap results from the total reflection when the critical incident angle is exceeded.
Figure 4. Transmittance T(ω) of structure (AB)N with incident angle θ = 25° when the number of periods are N = 8 (a), N = 16 (b), and N = 48 (c), respectively. The thickness of the medium are d1 = 12 mm and d2 = 6 mm. |
We demonstrate in figures 5(a) and (b) the effect of the thickness of layers A and B on the transmittances and the GH shifts with the incident angle θ = 25° and the number N = 48. When the thickness d2 of the conventional medium increases, all of the four band gaps move toward lower frequencies. When the thickness d1 of NZPIM increases, the zero-$\bar{n}$ gap moves toward a higher frequency, whereas, the Bragg gap and gap IV move toward lower frequencies. For the DP band gap, the width will become larger with the left edge of the gap moving toward a lower frequency and the right edge of the gap moving toward a higher frequency, as shown in figure 6. The peak positions of the GH shifts near the edges of each gap display the same dynamics with the gaps, as shown in figure 5(b). The corresponding variation of the phase shift is shown in figure 5(c) when d1 = 12 mm and d2 = 6 mm.
Figure 5. Transmittance T(ω) (a) and corresponding GH shifts (b) of structure (AB)N on incident frequency with different thicknesses d1 = 12 mm, d2 = 6 mm (dash-dotted lines), d1 = 14 mm, d2 = 6 mm (solid lines), and d1 = 12 mm, d2 = 7 mm (dashed lines), respectively. (c) The phase shift Φ of structure (AB)N on incident frequency with the thickness of the medium are d1 = 12 mm and d2 = 6 mm. The number of periods is N = 48 and the incident angle is θ = 25° for all the figures. |
Figure 6. Transmittance T(ω) around DP gap of structure (AB)N on incident frequency with different thicknesses d1 = 12 mm, d2 = 6 mm (dash-dotted lines), d1 = 14 mm, d2 = 6 mm (solid lines), and d1 = 12 mm, d2 = 7 mm (dashed lines), respectively. The number of periods is N = 48 and the incident angle is θ = 25°. |
3.2. Transmission properties of (ABA)N
Next, we calculate the transmittance of the beam in figure 7(a) in the quasi-periodic (ABA)N structure (N = 8). When the incident angle increases, the widths, edges, and centers of the DP gap and zero-$\bar{n}$ gap are similar to the case of (AB)N structure. The zero-$\bar{n}$ band gap occurs around the frequency ω/2π = 8.944 GHz. Similar to figure 4, we can see that the Bragg band gap of the (ABA)N structure disappears when the number N is small (N = 8). However, gap IV appears around the frequency ω/2π = 12.61 GHz with a small number N = 8, whereas it requires a large enough value in (AB)N structure (N = 48). This gap also shifts upwards in frequency when the incident angle increases. Figure 7(b) demonstrates the dependence of GH shifts on the frequency in (ABA)8 structure with different incident angles. Similarly, the values of shifts are always large at the edges of the gaps and change from negative to positive at DP.
Figure 7. Transmittance T(ω) (a) and corresponding GH shifts (b) of structure (ABA)N on incident frequency with different incident angles θ = 20° (dash-dotted lines), 22° (solid lines), and 24° (dashed lines), respectively. The periodic number is N = 8 and the thicknesses of the medium are d1 = 12 mm and d2 = 6 mm. |
Increase the number of periods of the (ABA)N structure (N = 16), similar to figure (4), the Bragg band gap will appear again for all incident angles. As shown in figures 8(a) and (b), we display four transmission band gaps and phase shifts when the incident angle is θ = 30°. We also checked (not shown) the effect of the thickness of the layers on the transmission properties, and found the results are similar to figures 5(a) and (b).
Figure 8. Transmittance T(ω) (a) and corresponding phase shift Φ (b) of structure (ABA)N on incident frequency. The initial parameters are d1 = 12 mm, d2 = 6 mm, N = 16, and θ = 30°. |
3.3. Transmission properties of (ABAAB)N
For higher-order Fibonacci levels, the distribution of transmission band gaps is like a Cantor-like set. For example, in figure 9(a), we show the transmittance of the Fibonacci sequence S4 [(ABAAB)10]. Compared with the first lower-order Fibonacci levels S2 and S3, it is obvious that the properties of the band gaps become complicated. Besides the DP gap, Bragg gap, zero-$\bar{n}$ gap, and gap IV, we can see another Bragg gap (gap V) emerges when ω > ωD. Below DP (ω < ωD), there exists another negative refractive gap (gap VI) besides the DP gap induced by total reflection. The corresponding GH shift and the phase shift are also displayed in figures 9(b) and (c).
Figure 9. Transmittance T(ω) (a), GH shift (b), and phase shift Φ (c) of structure (ABAAB)N on incident frequency. The initial parameters are d1 = 12 mm, d2 = 6 mm, N = 10, and θ = 24°. |
4. Conclusion
It has been shown that many exotic phenomena in graphene could be simulated with the relatively simple optical method [21, 24, 27, 30] after the realization of optical DP. Motivated by the experimental realization of a graphene superlattice [41], electronic band gap structures and transport properties have been extensively investigated. For instance, DP appears in the graphene superlattice, and it is exactly located at the energy with the zero-$\bar{k}$ gap [26, 42]. This gap is analogous to the photonic zero-$\bar{n}$ gap in the photonic crystals containing negative-index and positive-index materials [4]. The phenomena indicate that one can simulate the corresponding properties of the electronic band gap in graphene using optical metamaterials, such as left-handed materials and NZPIM.
In this work, we have studied the transmission properties in a Fibonacci quasi-periodic photonic crystal containing NZPIM via the transfer matrix method. Specifically, we considered three Fibonacci sequences, (AB)N, (ABA)N, and (ABAAB)N. We should emphasize here that the method can be applied to quasi-periodic photonic crystals with higher order Fibonacci levels. More band gaps will appear irregularly. Because of linear dispersion, an asymmetric gap appears near the DP only at an oblique incidence. Zero-$\bar{n}$ gaps and Bragg gaps exist when the refractive indexes of NZPIM are negative (ω < ωD) and positive (ω > ωD), respectively. We also obtained gaps induced by total reflection on both sides of the DP gap. We demonstrated the corresponding GH shifts and phase shifts in detail. The characteristics of the band gaps, GH shifts, and phase variations depend on the incident angle, number of periods, and layer thickness. Our results may have some potential applications in integrated optics and optical devices, and also suggest analogous phenomena of valence electrons in graphene Fibonacci superlattices [43].