1. Introduction

Recently, the crossing point of bands has been achieved at the Brillouin zone center in two-dimension (2D) dielectric photonic crystals (PhCs) by accidental degeneracy of modes [1–3]. At the crossing point, the threefold degeneracy of modes generates a Dirac cone and an additional ñat band intersecting the Dirac cone. This is different from that of the crossing point (called a Dirac point) at the six corners of the hexagonal Brillouin zone in which only Dirac cones exist [4–7]. As a result, the crossing point at the Brillouin zone center is called a Dirac-like point [1–3]. If the threefold degeneracy occurs by a monopole mode and two dipolar modes, the PhCs can be taken as an effective zero-index materials (ZIMs) in which the permittivity ($\epsilon$) and the permeability ($\mu$) are zero at the Dirac-like point [1,2,8]. From Maxwell equations, if $\epsilon$and $\mu$ are zero, the optical longitudinal modes could exist, in additional to the well-known transverse modes [2]. The additional flat band at the Dirac-like point is closely connected with the longitudinal modes. For a homogeneous ZIM, the flat band is dispersionless and the longitudinal mode is a “dark mode” which cannot couple with external light. But for a PhC mimicking a ZIM, there is always some spatial dispersion, so that the band is not perfectly dispersionless away from the zone center. Therefore, if a Gaussian beam with non-zero k-parallel components is incident on the PhC with a finite size, the longitudinal mode could be excited [2]. In a ZIM, the effective wavelength is extremely large, which leads to many usual scattering properties [2,9–12]. However, in a PhC mimicking a ZIM, the longitudinal mode may influence the wave propagations in the PhC. In Ref [2], Huang et al. showed that slightly above the Dirac-like point, one can find a regime of transport where the effective index remains close to zero and the longitudinal mode is not excited. Nevertheless, if the longitudinal mode is excited, say, precisely at the Dirac-like point, how will a wave propagate? Up to now the longitudinal mode is rarely studied and people know little about its properties. In this paper, by introducing random positional shift in the PhCs, we study the dependence of the longitudinal mode on the disorder. We find that, at the Dirac-like point where the longitudinal mode is excited, the transmittance decreases as the random degree increases. However, at a frequency slightly above the Dirac-like point, in which the longitudinal mode is absent and at the same time the PhC still mimics a near-zero medium, the transmission is insensitive to the disorder because the effective wavelength in the PhC is very large.

2. Transport properties of disordered PhCs

The band structure of PhCs is calculated by means of the plane-wave expansion method [13]. The square-lattice PhCs are composed of alumina rods. The radius (r), height (h) and dielectric constant ($\epsilon$) of the rods are 3 mm, 10 mm, and 8.35, respectively. The lattice constant ($a$) of the PhC is 13.36 mm, as is indicated in the inset of Fig. 1(a). Under these parameters, in Fig. 1(a) we give the band structure of the transverse electric (TE) polarization, in which the electric field is along rod axis and denoted by E_Z. From Fig. 1(a), a triply degenerate state is formed at the $\Gamma$ point, which is the Dirac-like point, at the frequency of 13.24 GHz. Moreover, the dispersion as a function of wave vectors in the x and y directions are plotted in Fig. 1(b). From Fig. 1(b), we see Dirac cones around the $\Gamma$ point and a flat band intersecting the Dirac-like point. The field patterns of the three degenerated states at the Dirac-like point are shown in Fig. 1(c). We see that the accidental degeneracy occurs by a monopole mode and two dipolar modes. Therefore, at the Dirac-like point, the PhC can mimic an effective ZIM.

 

Fig. 1 (a) The band structure of a 2D PhC for the TE polarization. The rods have a relative permittivity 8.35, $r=\text{3}\text{mm}$ and $a=\text{13}\text{.36}\text{mm}$. At the $\Gamma$ point (see the inset), a Dirac-like point is formed at 13.24 GHz. (b) Three-dimensional surface plot of the band structure shown in (a) as a function of wave vectors in x and y directions. (c) The electric field patterns of the threefold degenerate modes near the Dirac-like point with a very small k along $\Gamma -\text{X}$ direction.

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Now we make positional shift of rods to introduce disorder in the PhC sample as is shown in Fig. 2(a). The positional disorder is introduced as follows. Starting from the perfect square lattice position of the rods $\overrightarrow{r_{i,0}}$, we create a new position $\overrightarrow{r_{i,\Delta }}=\overrightarrow{r_{i,0}}+\Delta (\cos {\theta }_i\overrightarrow{e_x}+\sin {\theta }_i\overrightarrow{e_y})$, where ${\theta }_i$ is randomly chosen from {-π/4, 0, π/2, π, 3π/2} giving the direction of the shift, $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are the unit vectors in x and y directions, $\Delta$ gives the length of the shift, see Fig. 2(b), and $\delta =\Delta /a$ is the factor describing the random degree. The larger $\delta$ is, the bigger the random degree is. $\Delta$ is selected to be 0, 0.3, 0.6, 0.9 and 1.2 mm, respectively. In this paper, we care about the straight transmission in which the transmitted wave has the same direction as that of the incident wave [14,15], i.e., in the $\Gamma -\text{X}$ direction in the first Brillouin zone. To realize this purpose in the experiment, we put wave-absorbing materials behind the PhC sample in the metallic waveguide (see Fig. 2(a)), which will be explained in detail in the next paragraph.

 

Fig. 2 (a) A view of the experimental set-up which is used to measure the transmission. The top slab of the rectangular waveguide is not shown. (b) The amplified schematic diagram of the red dotted area in (a), showing how disorder is introduced in the PhCs.

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Our simulations are carried out by computer simulation technology microwave studio software based on a finite integration method. The perfect-electric-conductor and the absorbing boundary conditions are used in the simulation. The square-lattice PhCs are made up of 338 (26 line × 13 column in the x-y plane) cylindrical alumina rods. We simulate the straight transmission spectra after the emitting and receiving waveguide ports are set to have the same size (45 mm × 10 mm). Numerical simulation shows that, after the beam reaches the PhC, the beam is very similar to a Gaussian beam with non-zero k-parallel components. Our experimental layout consists of a network analyzer of Agilent Technologies (Agilent N5222A), a rectangular waveguide and microwave horn antennas whose feeding and detecting probes are connected to the input and output ports of the Vector Network Analyzer. The sample is placed symmetrically with respect to the antennas and oriented such that normal incidence of the emitted waves coincided with the $\Gamma -\text{X}$ direction in the first Brillouin zone. The parameters of rods are the same as above. In our experiments, we put wave-absorbing materials to make the illuminating and detecting waveguide ports be 45 mm × 10 mm. In the propagation direction, near each side face of the waveguide, wave-absorbing materials are used to minimize the disturbance of reflections from the metallic faces. The transmitted wave has the same direction as that of the incident wave, excluding the transmitted wave emerging from other directions. So the straight transmission spectra are measured by the experimental set-up as is shown in Fig. 2(a). Figure 2(b) gives the enlarged view of the red dotted area in Fig. 2(a) and shows how disorder is introduced in the PhCs. In simulations and experiments, the wave with TE mode is launched into the sample. Moreover, averaged spectra over different disordered configurations are used in the simulations and experiments, respectively. The transmission spectra lines are slightly smoothed by Savitzky-Golay method to remove the remaining traces of the spikes.

Figure 3(a) gives the straight transmission spectra by simulation. The transmission spectra are somewhat complex and we firstly discuss the transmission spectrum for a perfect PhC. The incident wave in the simulation is similar to a Gaussian beam with many k-parallel components. When a Gaussian beam is incident on a photonic graphene at the frequency of a Dirac point at the K point in the Brillouin zone, a transmission dip will appear at the Dirac point. This is due to the wave-filtering effect of non-zero k-parallel components by the conical singularity [16,17]. When a Gaussian beam is incident on a PhC at a Dirac-like point at the Brillouin zone center, a transmission peak induced by the longitudinal mode will appear at the middle of the previous transmission dip induced by the wave-filtering effect. Because the peak has a bandwidth, there would be two dips appearing almost symmetrically around the peak. If the PhC size is large enough, the two dips both come from the wave-filtering effect. However, if the PhC size is not large enough, the crossing point of the band structure will be not well closed and a tiny gap will appear below or above the Dirac-like point. In Fig. 3(a), the tiny gap induced by the finite size effect appears around 13 GHz below the Dirac-like point. Numerical calculations show that the transmission at this tiny gap will noticeably increase as we increase the periodic numbers in the propagation direction. Also, because of the finite size effect, the Dirac-like point deviates a little bit from 13.24 GHz for the infinite structure and becomes 13.17 GHz, as is indicated by the black dotted line in Fig. 3(a). Then, we discuss the influence of the disorder on the transmissions. In Fig. 3(a) the transmittances at the tiny gap around 13 GHz boost as $\delta$ increases from 0 to 0.088 with a step of 0.022. In contrast, at the Dirac-like point of 13.17 GHz, the transmittance decreases as $\delta$ increases. This behavior is similar to that at the frequency in pass bands. However, slightly above the dip to the right of the Dirac-like point such as at 13.5 GHz, the transmission is insensitive to thedisorder, which is different from that at the Dirac-like point. To investigate these phenomena in a more explicit manner, we take averages over a small frequency range of 0.05 GHz. The averaging ranges indicated by gray bars are shown in Fig. 3(a). In Fig. 3(b) we show the variances of the transmittances as $\delta$ increases at frequencies of 12.5, 13.17 and 13.5 GHz, respectively. At 12.5 GHz (a frequency in the pass band) and at the Dirac-like point of 13.17 GHz, the larger the value of $\delta$ is, the smaller the transmittance is. In contrast, at 13.5 GHz which is slightly above the dip to the right of the Dirac-like point, the transmittances are insensitive to the change of disorder.

 

Fig. 3 (a) The simulated transmission spectra for different samples as the random degree $\delta$ increases from 0 to 0.088 with a step of 0.022. The dotted vertical line indicates the Dirac-like point. The gray bars indicate the regions of averaging used in (b). (b) The averaged transmissions versus $\delta$ when the frequencies are at 12.5 GHz (black squares), 13.17 GHz (red triangles) and 13.5 GHz (blue circles), respectively The error bars indicate the tolerance. The lines with different styles are guide to the eyes.

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Here we try to explain the different transmission behaviors in Fig. 3. In our simulations, because the incident wave is similar to a Gaussian beam, the flat band of longitudinal mode might be excited. To explain this, in Fig. 4 we give the electric field distribution when the wave goes through the perfect PhC at the frequencies of 12.5, 13.17 and 13.5 GHz, respectively. At 12.5 GHz, the beam is not a plane wave after it goes through the PhC and the fields inside the PhC are normal transverse modes. At 13.17 and 13.5 GHz, the PhC can mimic a ZIM and a near-zero medium, respectively, and the beam would become a plane wave after it goes through the PhC. However, only at 13.5 GHz we see this wavefront transformation effect. This demonstrates that the longitudinal mode is excited at the Dirac-like point and strongly disturbs the beam propagations, as is shown by Fig. 4(b). We also see that the field pattern in the PhC in Fig. 4(b) is quite different from those in Figs. 4(a) and 4(c). The longitudinal mode at the Dirac-like point makes the disorder dependence similar to that at the frequency in the pass band, as is shown in Fig. 3. However, at 13.5 GHz, the longitudinal mode is absent and the beam becomes a nearly plane wave after it goes through the PhC, owing to the wavefront transformation effect of near-zero medium. Because the effective wavelength in the PhC at 13.5 GHz is very large and there is no disturbance of the longitudinal mode, the transmittances are relatively insensitive to the disorder.

 

Fig. 4 The electric field distribution after a beam goes through the perfect PhC at different frequencies. The entrance face is marked by a purple short line.

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In Fig. 5(a) we show the measured straight transmission spectra when $\delta$ increases from 0 to 0.088 with a step of 0.022. Owing to the tolerance of the dielectric constant and rod size, the Dirac-like point in the experiment (indicated by the dot-vertical line) is 13.21 GHz which deviates a little bit from the simulated one (13.17 GHz). Overall, the experimental results agree well with the simulated results in Fig. 3(a). Because of the finite size effect, a very small gap around 12.9 GHz also exists below the Dirac-like point, as is shown in Fig. 5(a). Figure 5(b) shows the variances of the averaged transmissions as $\delta$ increases at the frequencies of 12.5, 13.21 and 13.5 GHz, respectively. The small frequency range for averaging are indicated in Fig. 5(a). Both the transmissions at 12.5 GHz and 13.21 GHz decrease as $\delta$ increase. In contrast, the transmissions at 13.5 GHz are relatively insensitive to the random degree. It should be pointed out that, in the case $\delta$ = 0.088, the right band of the transmission peak decreases noticeably, and it looks that there is a small gap around 13.3 GHz above the Dirac-like point. Different from the tiny gap below the Dirac-like point, this small gap is not induced by the finite size effect. When we increase the column number of a perfect PhC from 13 to 17, the numerical transmissions at the tiny gap below the Dirac-like point increase while those at the dip to the right of the Dirac-like point are hardly changed. So far we have experimentally observed the effects of disorders on the longitudinal mode at the Dirac-like point and on the near-zero materials slightly above the Dirac-like point.

 

Fig. 5 (a) The measured transmission spectra for different samples as the random degree $\delta$ increases from 0 to 0.088 with a step of 0.022. (b) The averaged transmissions versus $\delta$ when the frequencies are at 12.5, 13.21 (the Dirac-like point) and 13.5 GHz, respectively. The lines with different styles are guide to the eyes.

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3. Conclusion

In conclusion, we study the effects of disorder on the wave propagations of PhCs around the Dirac-like point. At the Dirac-like point, the transmission peak induced by the longitudinal mode decreases as the random degree increases. However, at a frequency slightly above the Dirac-like point, in which the longitudinal mode vanishes and the effective index is still near zero, the transmission is insensitive to the disorder. Our results are helpful to understand the optical longitudinal mode and the influence of disorder on the near-zero materials.