1 Introduction

Since the existence of photonic band gaps (PBGs) (Joannopoulos et al. 1995), the photonic crystals (PCs) become a attractive research focus after firstly proposed in 1987 (Yablonovitch 1987; John 1987), in which the electromagnetic (EM) waves can be forbidden because of the interference of multiple Bragg scattering (Joannopoulos et al. 1995). It means that the researchers can manipulate the EM waves to propagate in a certain direction or frequency only by arranging the different dielectric materials in space periodically. Thus, many important and valuable practical devices can be realized by the PCs (Mekis et al. 1996; Russell 2003; Biancalana 2008). If two different polarized EM waves (TE and TM waves) are forbidden at same time, the complete PBGs (CPBGs) can be obtained. To the PCs, if the CPBGs can be achieved in any direction, the omnidirectional band gap (OBG) will be obtained. Compared to the PBGs, the OBGs have more practical values, which can be used to design many applications (Xiang et al. 2007; Hart et al. 2002; Winn et al. 1998). However, the OBG is difficult to obtain in the conventional dielectric PCs, whose PBGs also cannot be manipulated and strongly depend on the topology of PCs. To overcome those shortcomings, the dispersive materials or metamaterials are introduced into the conventional PCs to enlarge or obtain the tunable PBGs, such as metal (Chern et al. 2006), semiconductor (Tian and Zi 2005), plasma (Hojo and Mase 2004) and superconductor (Aly et al. 2014). In the nature, the physical features of plasma can be manipulated by many external factors (Sakai and Tachibana 2012; Ginzburg 1970). Thus, the plasma photonic crystals (PPCs) become an good candidate to achieve the tunable CPBGs or OBGs since it was firstly proposed by Hojo et al. (Hojo and Mase 2004) in 2004. Especially, more complicated EM modes and interesting phenomena can be observed in the magnetized PPCs (Qi 2012; King et al. 2013; Zhang et al. 2015c). Thus, the PPCs have attracted tremendous interests of researchers during past 10 years.

As time goes on, the PPCs have been extensively studied in the theory and experiment, especially, for one-dimensional (1D) and 2D cases. The properties of PBGs and defect modes of 1D and 2D PPCs are investigated by many researchers even as the external magnetic field is considered (Qi 2012; King et al. 2013; Zhang et al. 2015a, b; Shiverhwari 2011). In recently, the many interesting phenomena in the PPCs are analyzed in detail such as evanescent wave (Li et al. 2011), surface plasmon modes (Zhang and Liu 2014), surface modes (Shukla et al. 2015) and nonreciprocal propagation (Ardakani 2014). The novel properties for the different ways to form the PPCs also are studied (Fu et al. 2013; Hamidi 2012; Qi and Zhang 2011). However, in those works, the properties of PBGs are focused on the PBGs since the CPBGs are hardly observed in the conventional 1D or 2D PPCs. In other words, not all of PBGs in the 1D and 2D cases are the CPBGs. Thus, the 1D and 2D PPCs have to be redesigned for obtaining CPBGs and OBGs, such as introducing the new filler into 2D case (Zhang et al. 2015c), using heterostructure (Zhang et al. 2013a), introducing matching layer (Zhang et al. 2012a) and arranging aperiodic structure (Zhang et al. 2012b) to the 1D cases. In theory, the 3D PPCs (Joannopoulos et al. 1995; Zhang et al. 2013b) are more suitable to achieve the OBGs compared to the 1D and 2D cases. However, the 3D PPCs have to suffer from fabrication defects or irregularities, and 1D PPCs also have many problems in realizing the devices for its large thickness and finite height system (Li and Xia 2001; Haas et al. 2006). As mentioned by (Li and Xia 2001) and Hass et al. (2006), the so-called 2.5D PCs structure becomes a good choice in obtaining or enlarging the OBGs since the shortcomings appearing in the 1D and 3D cases can be overcome as the EM waves are incident with an off-plane angle. The larger OBG has potential application in the omnidirectional high reflector, all-dielectric coaxial waveguide, omnidirectional mirror fiber, omnidirectional high-quality filter, antenna substrates. Obviously, in order to obtain the OBG from 2D PPCs, the properties of 2D PPCs with off-plane wave vector has to be investigated.

In this paper, the aims are to investigate the features of OBG in the 2D PPCs with triangular lattices by the plane wave expansion (PWE) method, which are composed of the non-magnetized plasma cylinders inserted into the homogeneous and isotropic dielectric background. We also further study the properties of OBG for the two novel PCs configurations. Breaking the symmetrical profile of PPCs also is a good way to enlarge the OBGs according to the practical applications. Introducing the anisotropic dielectric (uniaxial material) into the 2D PPCs also can enlarge the OBGs. We assume the time-harmonic variable is e −jωt(ω is the angular frequency and \(j = \sqrt { - 1}\)), and c is the light speed in vacuum.

2 Theoretical model and numerical method

In Fig. 1, the schematic structure of 2D PPCs and the first irreducible Brillouin zone are plotted. As shown in Fig. 1, we consider the lattice constant and the radius of inserted plasma cylinders are a and R, respectively. We also assume the relative dielectric functions for the dielectric background and plasma cylinders are ε a and ε p , respectively. One also can see from Fig. 1 that the high symmetry points for the first irreducible Brillouin zone are Г = (0, 0), J = (\(2\sqrt 3 \pi /3a\), 0), and X = (\(2\sqrt 3 \pi /3a, 2\pi/3a\)) (Li and Xia 2001; Feng and Arakawa 1996). Obviously, the EM wave are obliquely incident into the 2D PPCs with an angle θ, where the angle between the incident wave vector k and wave vector in the in-plane case is θ. It means that the wave vector k has a component in z direction, which is k z  = ωsinθ/c. As we know, ε p can be expressed as (Ginzburg 1970)

$$\varepsilon_{p} = 1 - \frac{{\omega_{p}^{2} }}{{\omega (\omega + j\nu_{c} )}}$$
(1)

where ν c and ω p are the plasma collision frequency and plasma frequency, respectively. In order to compute the OBGs of proposed 2D PPCs, the modified PWE method is used (Sakoda 2001; Kuzmiak and Maradudin 1997). The details of how to compute the band structures for the 2D PPCs with off-plane case can be found in the Ref. (Zhang et al. 2016).

Fig. 1
figure 1

Schematic structure of such 2D PPCs and the first irreducible Brillouin zone for computing the band structures

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3 Numerical results and analysis

In the following calculations, the frequency region is normalized by a constant ω 0 = ωa/2πc and the initial values of ω p and ν c are defined as ω p  = 0.2ω 0 and ν c  = 0.003ω p , respectively. The 1225 plane waves and 30 k-vector also are utilized to calculate the OBGs to obtain a enough accuracy (Sözüer et al. 1992).

3.1 The properties of OBG in the proposed 2D PPCs

In Fig. 2, we plot the band structures for such PPCs with different incidence angle θ as ε a  = 12.96, R = 0.46a, ω p  = 0.2ω 0 and ν c  = 0.003ω p . The red shaded regions indicate the OBGs. Those dielectric cylinders are GaAs, whose the refractive index is equal to 3.6. It can be seen from Fig. 2 that not only there is an OBG which is located between third and fourth bands above the flatband region, but also one flatband region can be observed. The OBG covers from 0.4830 to 0.5038 (2πc/a), and the flatband region runs from 0 to 0.2 (2πc/a). In the flatband region, the group velocity of EM wave is every low since the existence of surface plasmon modes (Chern et al. 2006; Zhang et al. 2013b, 2015c), which are localized at the interface between the inserted plasma cylinders and the dielectric background. The similar phenomenon also can be observed in the 3D PPCs (Zhang et al. 2013b) or the PCs with metal (Chern et al. 2006) constituents. One also can see from Fig. 2a–d) the CPBGs of PPCs are located at 0.4174–0.5038 (2πc/a), 0.4476-0.5264 (2πc/a), 0.4707–0.5416 (2πc/a) and 0.483–0.55492 (2πc/a) as θ = 0°, 45°, 66° and 89°, respectively. From those results in Fig. 2, the OBG for proposed PPCs can be achieved. Obviously, the band edges of OBG can be determined by the upper edge of CPBG for θ = 0° and the lower edge of CPBG for θ = 89°. In order to investigate the properties of OBG in the proposed 2D PPCs, the band structures are plotted in Fig. 3a as the values of ω p are different. We can see from Fig. 3a that if ω p  = 0, the plasma cylinders can be looked as the air cylinders. With increasing the incident angle θ, the band edges of CPBG are increased in the higher frequency regions, and the OBG (the area between two black dash lines) also can be obtained, which runs from 0.4753 to 0.4881 (2πc/a). If the plasma is introduced into such PCs (ω p  = 0.2ω 0), the edges of CPBGs will shift to higher frequency regions, whose OBG (the area between two green dash lines) spans 0.4830 to 0.5038 (2πc/a). Compared the results in Fig. 3a, the OBG is enlarged, whose bandwidth is increased to 0.008 (2πc/a) as the plasma is introduced into the conventional dielectric-air PCs. To further study, in Fig. 3b, we present band structures for such PPCs with similar case to Fig. 2 except that the ε a is different. As shown in Fig. 3b, if the insert rods are air and ε a  = 11.56, there are not OBG and the CPBG will be closed at θ = 81°. If the air cylinders are replaced by the plasma ones, the OBG can be observed, which is located at 0.5157–0.5215 (2πc/a). As mentioned above, compared to the conventional dielectric-air PCs, not only the OBG can be easier obtained in the PPCs but also the frequency range of OBG can be enhanced. Thus, the OBG can be obtained in the proposed PPCs.

Fig. 2
figure 2

The band structures for the proposed 2D PPCs with different incidence angle θ as ε a  = 12.96, R = 0.46a, ω p  = 0.2ω 0 and ν c  = 0.003ω p . a θ = 0°, b θ = 45°, c θ = 66° and d θ = 89°, respectively. The red shaded regions indicate the OBGs

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Fig. 3
figure 3

The band structures for the two cases with similar case to Fig. 2 except that the ε a are different. a ε a  = 12.96, and (b) ε a  = 11.56, respectively

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In Fig. 4, the effects of R on the CPBGs and the relative bandwidth of OBG for the proposed 2D PPCs with ε a  = 12.96, ω p  = 0.2ω 0 and ν c  = 0.003ω p as θ = 0° and θ = 89°. The shaded region indicates the OBG. As shown in Fig. 4a, both edges of CPBGs for θ = 0° and 89° will shift to higher frequency regions, and their bandwidths will be increased first and then decreased as the value of R/a is increased. The lower edge of OBG is determined by the lower edge of CPBG for θ = 89°, and its upper edge is dependent on the upper of CPBG for θ = 0°. If R/a < 0.42 or R/a > 0.47, the OBG cannot be found. Compared to the CPBGs, the similar trends for the edges of OBG also can be seen in Fig. 4. In other words, the relative bandwidth of OBG also will be increased first and then decreased with the increasing value of R/a. As shown in Fig. 4b, the relative bandwidth of OBG will increase first and then decrease with the increasing value of R/a. The maximum bandwidth and relative bandwidth of OBG are 0.0208 (2πc/a) and 0.0421, which can be observed at the case of R/a = 0.46, respectively. Compared to the case of R/a = 0.47, the frequency range and relative bandwidth of OBG are increased to 0.0078 (2πc/a) and 0.0183, respectively. As mentioned above, we can choose a optimal value of R to obtain a larger OBG. Thus, R can be acted as an important parameter to manipulate the OBG, and the larger OBG can be observed in the high-R region.

Fig. 4
figure 4

The effects of R on the CPBGs and the relative bandwidth of OBG for the proposed 2D PPCs with ε a  = 12.96, ω p  = 0.2ω 0, and ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of OBG

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In Fig. 5, we plot the effects of ω p on the CPBGs and relative bandwidth of OBG for the proposed 2D PPCs with ε a  = 12.96, R = 0.46a and ν c  = 0.003ω p as θ = 0° and θ = 89°. One can see from Fig. 5a that the trends for both edges of CPBGs for θ = 0° and 89° are similar to those in Fig. 4. With the increasing value of ω p /ω 0, the edges of two CPBGs will shift to higher frequency regions, and their bandwidths also increase first and then decrease, whose maximum values are 0.1099 and 0.0680 (2πc/a), which can be found at the cases of ω p /ω 0 = 0.39 and 0.33, respectively. Consequently, the edges of OBG also will upward to the higher frequencies, and its bandwidth also increases first and then decreases as the value of ω p /ω 0 is increased. It also can be seen from Fig. 5 that if ω p /ω 0 > 0.55, the OBG will not exist. The largest OBG can be obtained at the case of ω p /ω 0 = 0.36, whose frequency range is 0.02993 (2πc/a). In this case (as shown in Fig. 5b), the maximum relative bandwidth of OBG also can be observed, which is 0.0572. The trend of its relative bandwidth also increases first then decreases with increasing ω p /ω 0. Compared with the case of ω p /ω 0 = 0.21, the frequency range and relative bandwidth of OBG are increased to 0.0085 (2πc/a) and 0.0138, respectively. Increasing ω p means that the real part of ε p becomes smaller, and such effect is similar as increasing R. In other words, the averaged dielectric constant of present PPCs is changed (Joannopoulos et al. 1995). Thus, OBGs can be tuned and the larger OBG can be easily observed in the low-ω p region. It is worth to be noticed that the OBG will not be enlarged with continuously the increasing value of ω p .

Fig. 5
figure 5

The effects of ω p on the CPBGs and the relative bandwidth of OBG for the proposed 2D PPCs with ε a  = 12.96, R = 0.46a and ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of OBG

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3.2 Enhanced the OBGs by new PCs configurations

As we know, not only changing the parameters of PPCs can enlarge the OBG but also the OBG can be enhanced by breaking the symmetrical profiles of PPCs. In Fig. 6, schematic views of the two new PCs configurations are present. As shown in Fig. 6a, the 2D PPCs with triangular lattices are composed of two different sizes of plasma cylinders, whose radius are R 1 and R 2. Obviously, if R 1 = R 2, the topology of PPCs will be same as those in Fig. 1 (the conventional structure). For convenience, such topological structure is named type-A structure. As shown in Fig. 6b, the other new configuration is that the rotated-square plasma rods are immersed into dielectric background with triangular lattices whose side length is d and filling factor is same as it in Fig. 1. In the other words, \(d^{2} = \pi R^{2}\). φ represents the angle between axes of the square cross section and the 2D lattices. In those cases, if the band structures are computed by the conventional PWE method, the equations for computing are complicated (Zhang et al. 2015c). Thus, the band structures can be calculated by a modified PWE method with mesh grid technique (Zhang et al. 2015c). The details of such technique can be seen in Ref. (Zhang et al. 2015c). In order to achieve a sufficiently high accuracy, the unit cell is meshed by 140 × 140 grids. In Fig. 7, the band structures of CPBGs for different PCs configurations as the incidence angle θ = 0° are plotted. The red regions represents CPBGs. As shown in Fig. 7, there are two CPBGs for 2D PPCs with the conventional structure, which are located at 0.3657–0.3964 (2πc/a) and 0.4325–0.4419 (2πc/a). The parameters are the same as those in Fig. 2 except for R = 0.42a. If the 2D PPCs are with type-A structure as R 1 = 0.42a and R 2 = 0.47a, those two CPBGs are covered 0.3825–0.4343 (2πc/a) and 0.4718–0.4948 (2πc/a). Similarly, the positions of such two CPBGs also will be changed to 0.3695–0.3847 (2πc/a) and 0.4208–0.4581 (2πc/a) as the 2D PPCs with type-B structure as φ = 30°. Compared the results in Fig. 7, we can know that if the PPCs with type-A structure, the first (1st) CPBG can be enlarged, whose bandwidth is increased to 0.0211 (2πc/a). If the PPCs with type-B structure, the second (2nd) CPBG also can be enhanced, whose bandwidth is increased to 0.0279 (2πc/a). Thus, the CPBGs of PPCs for θ = 0° can be enlarged by the two new PCs configurations.

Fig. 6
figure 6

Schematic views of two configurations of 2D PPCs to enhance the OBGs. a type-A structure, and (b) type-B structure

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Fig. 7
figure 7

The band structures of CPBGs for different PCs configurations as the incidence angle θ = 0°. a the 2D PPCs with similar case to Fig. 2 except for R = 0.42a, (b) the 2D PPCs with type-A structure as R 1 = 0.42a and R 2 = 0.47a, and (c) the 2D PPCs with type-B structure with similar filling factor to (a) as φ = 30°

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According to the results in Fig. 7, the 1st OBG also may be enlarged as the PPCs with type-A structure, and the 2nd OBG may be enhanced as the PPCs with type-B structure. In order to prove this point, in Figs. 8 and 9, we plot the band edges of CPBGs a function of incident angle θ for the 2D PPCs with such two different configurations as ε a  = 12.96, ω p  = 0.2ω 0 and ν c  = 0.003ω p . As shown in Fig. 8, if R 1 = 0.47a and R 2 = 0.46a, the band edges of 1st CPBGs move to higher frequency regions as θ is increased, and there is an OBG (the area between two green dash lines) which runs from 0.4967 to 0.5188 (2πc/a). If R 1 = R 2 = 0.46a, the band edges of CPBGs move to higher frequency regions, and there is an OBG (the area between two black solid lines) which spans from 0.4830 to 0.5038 (2πc/a). Compared with the results in Fig. 8, the 1st OBG can be enlarged as the PPCs with type-A structure which is increased to 0.0013 (2πc/a). Similar phenomenon also can be observed in Fig. 9. The 2nd OBG can be obtained as the PPCs with type-B structure and φ = 30°, which is located from 0.4465 to 0.4581 (2πc/a). However, for the conventional 2D PPCs, the 2nd OBG does not exist and the CPBG will be closed at θ = 31° (A point in Fig. 9). In other words, upper and lower edges of the CPBG will coincide at point A. Compared with the 2nd OBG in the conventional case, the 2nd OBG for the PPCs with type-B structure is enhanced obviously, and its bandwidth is increased to 0.0116 (2πc/a). Thus, the OBGs can be widened as the PPCs with such two configurations. It means that we can enhance the different OBGs by the present PCs structures according to the practical needs.

Fig. 8
figure 8

The band edges of 1st CPBGs a function of incident angle θ for the 2D PPCs with type-A structure as the R 1 are different

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Fig. 9
figure 9

The band edges of 2nd CPBGs a function of incident angle θ for the 2D PPCs with type-B structure as the filler are different and rotated angle φ = 30°

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In order to further study the effects of parameters for two new configurations on the properties of OBGs, we display the effects of R 1 on the CPBGs and the relative bandwidth of 1st OBG for the 2D PPCs with type-A structure and ε a  = 12.96, R 2 = 0.46a, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89° in Fig. 10. The shaded region indicates the OBG. As shown in Fig. 10a, the band edges of 1st CPBGs for θ = 0° and θ = 89° shift to higher frequency regions, and their bandwidths will increase with increasing the value of R 1 /a. The similar trends for the edges of 1st OBG also can be observed but its bandwidth increases first and then decreases as the value of R 1 /a is increased. The maximum bandwidth of 1st OBG is 0.01887 (2πc/a), which can be found at the case of R 1 /a = 0.44. Compared with the results in Fig. 7a, the PPCs with type-A structure have the advantages of obtaining the larger 1st OBG, especially, in the low-R 1 /a region. As shown in Fig. 10b, the trend of relative bandwidth will increase first and then decrease with increasing R 1. The maximum value is 0.0404 at the case of R 1 /a = 0.44. Compared with the case of R 1 /a = 0.46, the relative bandwidth is increased to 0.0036.

Fig. 10
figure 10

The effects of R 1 on the CPBGs and the relative bandwidth of 1st OBG for the 2D PPCs with type-A structure and ε a  = 12.96, R 2 = 0.46a, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of 1st OBG

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In Fig. 11, we plot the effects of d on the CPBGs and the relative bandwidth of 2nd OBG for the 2D PPCs with type-B structure and ε a  = 12.96, φ = 30°, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89°. As shown in Fig. 11a, the both band edges of CPBGs for θ = 0° and θ = 89° will move to higher frequency regions, and their bandwidths will increase with the increasing value of d/a. The similar trends for the band edges of 2nd OBG also can be observed in Fig. 11a but its bandwidth will increase first and then decrease as d/a is increased. If d/a < 0.676 or d/a > 0.78, the OBG will disappear. Obviously, it can be seen from Fig. 11b that the relative bandwidth of 2nd OBG also will increase first and then decrease with increasing d. The maximum bandwidth and relative bandwidth of OBG are 0.01478 (2πc/a) and 0.03282, which can be observed at the case of d/a = 0.744, respectively. Compared with the case of d/a = 0.7, the frequency range and relative bandwidth of 2nd OBG are increased to 0.01 (2πc/a) and 0.0198, respectively. Thus, the PPCs with type-B structure can improve the properties of 2nd OBG, and the larger 2nd OBG also can be obtained in the high-d region. It means that the 2nd OBG can be manipulated by choosing a suitable value of d.

Fig. 11
figure 11

The effects of d on the 2nd CPBGs and the relative bandwidth of 2nd OBG for the 2D PPCs with type-B structure and ε a  = 12.96, φ = 30°, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of 2nd OBG

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In Fig. 12, we plot the effects of φ on the CPBGs and the relative bandwidth of 2nd OBG for the 2D PPCs with type-B structure and ε a  = 12.96, d = 0.744a, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89°. It can be seen from Fig. 12a that if φ ≥ 23°and φ ≤ 64°, the 2nd OBG for proposed PPCs can be observed. The upper edge of OBG move to lower frequencies first and then shift higher frequency regions with increasing φ, but the trend for lower edge of OBG is irregular as the value of φ is changed. If the rotated angle of inserted square plasma rods are 23°, 42°and 64°, such OBG hardly can be observed. As shown in Fig. 12b, if φ = 54°, the maximum bandwidth and relative bandwidth of such OBG can be obtained, which are 0.04061 (2πc/a) and 0.09074, respectively. According to the results in Fig. 12a, we can know that there is an optimal value of φ to achieve the largest frequency range of 2nd OBG for the PPCs with type-B structure as the filling factor is never changed. In other words, the OBG can be enlarged and manipulated by changing φ. This can be explained that changing φ means the symmetrical profile of PPCs is broken. Thus, φ is an important parameter which is needed to be chosen. Of course, the OBG also can be enhanced and tuned by introducing the anisotropic dielectric into PPCs as mentioned in Ref. (Li et al. 1999). We will study this in next part.

Fig. 12
figure 12

The effects of φ on the CPBGs and the relative bandwidth of 2nd OBG for the 2D PPCs with type-B structure and ε a  = 12.96, d = 0.744a, ω p  = 0.2ω 0, ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of 2nd OBG

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In order to investigate the effects anisotropic dielectric background on the OBG, in Fig. 13, the band structures of CPBGs for different dielectric background as the incidence angle θ = 0° are plotted. Obviously, the proposed cases are with similar case to Fig. 2 except for R = 0.47a or the dielectric background is different. As we know, the expression for the conventional dielectric can be written as

$${\varvec{\upvarepsilon}}_{{\mathbf{a}}} { = }\left( {\begin{array}{*{20}c} {\varepsilon_{xx} } & 0 & 0 \\ 0 & {\varepsilon_{yy} } & 0 \\ 0 & 0 & {\varepsilon_{zz} } \\ \end{array} } \right)$$
(2)

where ɛ xx  = n 2 x , ɛ yy  = n 2 y , ɛ zz  = n 2 z . If \(\varepsilon_{xx} = \varepsilon_{yy} { = }\) ɛ zz , the conventional isotropic dielectric can be obtained. If the value ofɛ xx is not equal to that for ɛ yy or ɛ zz , the anisotropic dielectric can be achieved. In the nature, the anisotropic dielectric can be found easily. The typical anisotropic dielectric is the uniaxial material, such as Te and Tl3AsSe3 (Li et al. 1999). For the uniaxial material, two different principal-refractive indices can be obtained, which are named ordinary-refractive and extraordinary-refractive indices (n o and n e ), respectively. In Fig. 13b, we only consider a simper case, which is that the extraordinary axis of anisotropic dielectric background (uniaxial material) is parallel to the inserted plasma cylinders (ɛ xx  = n 2 o  = 4.82, ɛ yy  =  ɛ zz  = n 2 e  = 6.22). As shown in Fig. 13a, for the conventional isotropic dielectric background (\(\varepsilon_{xx} = \varepsilon_{yy} { = }\varepsilon_{zz}\) = 12.96), there is one CPBG can be found as the incidence angle θ = 0°, which runs from 0.4346 to 0.5436 (2πc/a). The frequency range of CPBG is 0.109 (2πc/a). As a comparison, we can see from Fig. 13b that there exist two CPBGs, which are covered 0.3539–0.4648 (2πc/a) and 0.4907–0.5207 (2πc/a), respectively. The bandwidths are 0.1109 and 0.03 (2πc/a), respectively. Compared with the results in Fig. 13a, b, we can know that not only the bandwidth of CPBG can be enhanced but also the number of CPBG is increased by introducing the anisotropic dielectric (uniaxial material) into the proposed 2D PPCs. In order to further study the effects anisotropic dielectric background on the OBG, in Fig. 14, we display the band edges of CPBGs a function of incident angle θ for the 2D PPCs with the different dielectric background as R = 0.47a, ω p  = 0.2ω 0 and ν c  = 0.003ω p . The regions between the green dash lines are OBGs. As shown in Fig. 14a, the edges of CPBGs move to higher frequency region with increasing the incident angle θ. There is a OBG, which spans from 0.5288 to 0.5436 (2πc/a). The bandwidth of such OBG is 0.0148 (2πc/a). We also can see from Fig. 14b that the edges of two CPBGs for the proposed 2D PPCs with the anisotropic dielectric background shift to higher frequency region, and two OBGs can be observed. Those two OBGs are located 0.3972–0.4648 (2πc/a) and 0.5016–0.5207 (2πc/a), whose bandwidths are 0.0676 and 0.0191 (2πc/a), respectively. Compared with the results in Fig. 14, we can know that not only the OBG also can be enlarged but also the number of OBG also can be increased, when the conventional isotropic dielectric background is changed to the anisotropic dielectric. Thus, the OBG of proposed 2D PPCs can be enhanced by introducing the anisotropic dielectric.

Fig. 13
figure 13

The band structures of CPBGs for different dielectric background as the incidence angle θ = 0°. a the 2D PPCs with similar case to Fig. 2 except for R = 0.47a, (b) the 2D PPCs with similar case to Fig. 2 except for R = 0.47a and dielectric background is anisotropic. The red regions present the CPBGs

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Fig. 14
figure 14

The band edges of 1st CPBGs a function of incident angle θ for the 2D PPCs with different dielectric background. a the dielectric background is the conventional isotropic dielectric (ε a  = 12.96), (b) the dielectric background is anisotropic dielectric

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In Fig. 15, we plot the effects of n o on the CPBGs and the relative bandwidth of 1st OBG for the 2D PPCs with the anisotropic dielectric background (ɛ xx  = n 2 o ,ɛ yy  = ɛ zz  = n 2 e ) and n e  = 6.2, R = 0.47a, ω p  = 0.2ω 0, and ν c  = 0.003ω p as θ = 0° and θ = 89°. The shaded region indicates the OBG. As shown in Fig. 15a, the both band edges of CPBGs for θ = 0° and θ = 89° will shift to lower frequency regions, and their bandwidths will increase first and then decrease with the increasing value of n o . The similar trends for the band edges of 1st OBG also can be obtained in Fig. 15a. If n o  < 3.3, the 1st OBG cannot be observed. Obviously, the maximum bandwidth of 1st OBG can be found at the case of n o  = 4.8, which is 0.0676 (2πc/a). It also can be seen from Fig. 15b that the relative bandwidth of 1st OBG also will increase first and then decrease with increasing n o . The maximum relative bandwidth of such OBG is 0.1535, which can be observed at the case of n o  = 4.8. Compared with the case of n o  = 5.6, the bandwidth and relative bandwidth of 1st OBG are increased to 0.0187 (2πc/a) and 0.0334, respectively. Thus, the properties of 1st OBG for the 2D PPCs can be manipulated obviously by changing the value of n o , and the larger 1st OBG also can be obtained in the mid-n o region.

Fig. 15
figure 15

The effects of n o on the CPBGs and the relative bandwidth of 1st OBG for the proposed 2D PPCs with n e  = 6.2, R = 0.47a, ω p  = 0.2ω 0, and ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of 1st OBG

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In Fig. 16, we present the effects of n e on the CPBGs and the relative bandwidth of 1st OBG for the 2D PPCs with the anisotropic dielectric background and n o  = 4.8, R = 0.47a, ω p  = 0.2ω 0, and ν c  = 0.003ω p as θ = 0° and θ = 89°. The shaded region indicates the OBG. It can be seen from Fig. 12a that the band edges of 1st CPBG for θ = 0° hardly varies with changing the value of n e but the band edges of 1st CPBG for θ = 89° shift to lower frequency region slightly. Obviously, the frequency range of 1st CPBGs for θ = 0° is almost unchanged but that for θ = 89° is decreased with the increasing value of n e . The trends for the edges of 1st OBG also can be observed from Fig. 16a, the upper edge of 1st OBG is almost unchanged but it’s the lower edge shift to lower frequency region slightly. Thus, the bandwidth of 1st OBG will be enhanced with increasing the increasing value of n e . The maximum bandwidth of 1st OBG can be observed at the case of n e  = 9, which is 0.0639 (2πc/a). As shown in Fig. 16b, the trend of relative bandwidth will increase with the increasing value of n e . The maximum value is 0.1465 at the case of n e  = 9. Thus, maximum bandwidth and relative bandwidth of 1st OBG can be founded at the case of n e  = 9. Compared to the case of n e  = 5, those values are increased to 0.0023 (2πc/a) and 0.0056, respectively. It can be seen from the results in Fig. 16 that the 1st OBG cannot be manipulated by changing the increasing value of n e in a large frequency range. The bandwidth of 1st OBG only can be enlarged slightly by increasing n e . As mentioned above, the 1st OBG can be tuned slightly by changing n e , and the larger 1st OBG can be achieved in the high-n e region. According to the results in Figs. 15 and 16, in our case, the 1st OBG for the proposed PPCs can be tuned in a larger frequency region by changing n o compared with tuning n e .

Fig. 16
figure 16

The effects of n e on the CPBGs and the relative bandwidth of 1st OBG for the proposed 2D PPCs with n o  = 4.8, R = 0.47a, ω p  = 0.2ω 0, and ν c  = 0.003ω p as θ = 0° and θ = 89°. a The CPBGs for the proposed 2D PPCs with different θ, and (b) the relative bandwidth of 1st OBG

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It is worth to be noticed that the plasma cylinders/square rods can be realized, which are mentioned in this paper. In order to obtain the plasma cylinders/square rods, we can use the Helium as a working gas to generate the plasmas material, (obviously, the tubes with double-shell structures), and the plasma cylinders or square rods with double-shell structures are ionized by the direct current. The plasma cylinders/square can be realized by the the commercial quartz discharge tubes, which are argon-filled to a pressure of 550 Pa with added mercury (to obtain the larger plasma frequency). The details of this part will be proposed in another paper.

4 Conclusions

In summary, the features of CPBGs and OBGs in the 2D PPCs with triangular lattices are theoretically investigated in detail by the PWE method as the incidence wave is off-plane, which are containing the non-magnetized plasma cylinders inserted into the homogeneous and isotropic dielectric background. According to the theoretical simulation results, some conclusions can be obtained. If the plasma is introduced into 2D dielectric PCs, not only tunable OBGs can be obtained but also a flatbands region can be observed since existing the surface plasmon modes, whose frequency range is less than ω p . In such region, the group velocity of EM wave is every low. Compared to the conventional 2D dielectric-air PCs, the OBG is easier achieved and the larger OBG also can be observed in the 2D PPCs with same structure. In other words, the 2D PPCs are more suitable than the conventional 2D dielectric-air PCs to realize the omnidirectional reflector or mirror. The effects of R and ω p on the properties of OBG also are studied in calculation, respectively. The calculated results demonstrated that the OBG can be manipulated obviously by changing R and ω p . Increasing R and ω p , the band edges of CPBGs and OBG shift to higher frequency regions, and their bandwidths and relative bandwidths will increase first and then decrease, whose maximum values can be found in the high-R and low-ω p regions. As a comparison, two new configurations of 2D PPCs also are proposed to enlarge the OBGs since the symmetrical profile of PPCs is broken. The simulated results reveal that the 1st OBG can be enhanced as the PPCs with type-A structure, and the 2nd OBG can be broadened as the PPCs with type-B structure. For type-A structure, if R 2 remains a constant and unchanged, the larger 1st OBG can be observed in the high-R 1 region compared to the conventional 2D PPCs. For type-B structure, the 2nd OBG can be enhanced obviously in the high-d and high-φ regions. Compared to changing d, the OBG is more sensitive to the value of φ. However, it is worth to be noticed that the two newly proposed configurations cannot help to enhance all of OBGs in the conventional PPCs, and we have to choose the optimizations according to the practical applications. Introducing the anisotropic dielectric (uniaxial material) into the proposed 2D PPCs to enlarge the OBG also is studied. The effects of n o and n e of uniaxial material on the properties of OBG also are investigated in theory, respectively. The computed results shown that the OBG can be manipulated obviously by changing n o but it will be tuned slightly by changing n e . Compared with changing n e , the OBG can be tuned in a larger frequency region by changing n o . The larger 1st OBG can be obtained in the mid- n o or high- n e regions. As mentioned above, those theoretical results may open a new scope to design the omnidirectional reflector or mirror based on the 2D PPCs.