Photonic crystal (PC) system has aroused huge interest in the field of optics for its interesting optical properties. Even defects, cavities or specially designed PCs can support localized/trapped modes which can lead to some niche applications like optical fibers, lasers of different kind, couplers, filters, etc.^[1] Like electronic bands in crystal, PCs have photonic bands that may possess band gaps also. A two-dimensional (2D) or three-dimensional (3D) PC possessing photonic band gap forbids the propagation of electromagnetic wave of certain frequencies that lies within that gap. The creation of an appropriate defect will allow light of certain frequencies to be trapped^[2] or to propagate which might not be allowed within the crystal due to band gap restrictions. Band gap guidance is well used in PC fibers where out-of-plane guidance is relevant.^[3]

Interestingly, two photonic bands of the PC possess an accidental degeneracy and have a linear dispersion relation at that point, a photonic Dirac cone (DC) arises where the density of states vanishes, and this phenomenon is similar to that of the electronic DC formed in graphene.^[4] This special photonic band structure is unique due to its exciting properties like exceptional transmission, zitterbewegung of photon, Klein tunnelling, anti-localization, pseudo-diffusion, quantum Hall effect,^[5–7] etc. This kind of structure has been used to create scatter-free waveguide, unidirectional filter,^[8] lenses of arbitrary shape,^[9] surface emitting lasers,^[10] etc. After the pioneering findings of the existence of photonic DC at the zone boundary of two-dimensional PC by Haldane and Raghu,^[11] and the honey-comb lattice by Ochiai and Onoda,^[12] Huang et al.^[13] showed such a condition is possible for a medium of which effective permittivity (ε_eff) along with effective permeability (μ_eff) is zero at a particular frequency. Sakoda showed that such a conical dispersion exists at zone centre for a three-dimensional (3D) triangular lattice,^[14] and at zone boundary for a two-dimensional (2D) square lattice.^[15] Dirac points in a 2D core-shell PC^[16] have also been reported recently. By creating the appropriate defect, trapping of light^[17,18] with frequencies at the Dirac point has been shown. Recently, Dirac point based guidance in optical fiber has also been demonstrated.^[19] They have considered out-of-the plane photonic bands where a true Dirac point with vanishing photonic density of states (DOS) is created and some open wide Dirac spectra have been noticed. There are other linear dispersion regions where DOS is not at all zero and named as Dwarf Dirac cone (DDC)^[19] which does not possess such an open Dirac spectrum. They did not consider these points since those points reside within the radiation continuum and should not support localized mode except that there exist some embedded eigenvalues.^[20,21] These points do not exist within the band gap and hence the radiation losses cannot not be suppressed. In the present paper, the structures that are used do not support total internal reflection either. What we have done in the present work is to trap those frequencies at the Dwarf Dirac point (DDP) by creating suitable defects and observe that such a linear dispersion of DDC has some inherent properties of effective and better suppression of radiation mode in order to achieve the confinement of light compared with the other points in the continuum. This property is apparently unusual but evidenced from the quality factors (Q value) of the structure obtained from k-point sampling in the first Brillouin zone along with frequency sampling in and around a close vicinity of DDP. Moreover as mentioned, this confinement does not rely on the band gap guidance^[3,23] mechanism, not on total internal reflection. Study of such cases indicates a new mechanism hence induces new tuneable characteristics suitable for application in sensors or in lasers.

A 2D PC is defined by a periodic dielectric function ε(x_∥), which is periodic in a particular plane and invariant in the direction perpendicular to the plane of periodicity. The schematic diagram of the unit cell of a 2D triangular lattice of air holes embedded in a solid dielectric matrix is shown in Fig. 1(a). Here we denote the diameter of air holes and the pitch (center-to-center distance between two consecutive air holes) by d and Λ respectively.

Fig. 1.

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	Fig. 1. (left) Schematic diagram of the unit cell of a triangular lattice PC and (right) various components of wave vectors. Here k∥ is the in-plane wave vector, which lies in the kx−ky plane. The angle between effective k vector (keff) and kx–ky plane is defined as off-plane angle α.

The propagation of electromagnetic wave in such a PC is governed by Maxwell’s electromagnetic wave equation. One can start with either a wave equation with an electric field or a wave equation with a magnetic field. In this study we solve the wave equation for a magnetic field which is given by

In search of a localized mode formed by a defect within a supercell of a photonic crystal, it is known that for an N_xN supercell, the band structure is highly folded where each of the photonic bands of a unit cell is replaced by N² number of bands and the bound/localized state appears as a flat band^[17] superposed on the folded PBS calculated by PWM. The analogy of this flat band in the electronic case may be the appearance of an extra n-type or p-type band for a doped semiconductor. Besides, the finite element method (FEM) eigenvalue search shows a highly concentrated field around the defect. The concentrated field is traceable by sampling the entire k-path in the first Brillouin zone at the same frequency (flat band). Here, the defect usually acts like a cavity which supports a particular frequency. This frequency depends on the size or radius of the defect. When an incident radiation (a plane wave) is coupled with the frequency supported by the defect, a resonance occurs and an intense energy is trapped in the defect.

We employed PWM for calculating the PBS to identify the degeneracy point. We calculated PBS for air holes (ε_a = 1) in materials with dielectric constants ε_b = 15, 16, 17, and 18 respectively each for k_z = 0.25/Λ, 0.5/Λ, 0.75/Λ. The defect modes appearing due to the creation of defect in the PC are calculated by utilizing FEM in the frequency domain as it is in the commercially available software COMSOL multiphysics. While calculating the defect modes by FEM, we took an 11 × 11 supercell of each crystal and made eigen-frequency analysis by using Fouquet periodicity to find out the possible trapped mode in air (ε_d = 1) defect. We also calculated PBS using COMSOL to compare the result obtained by PWM. Differences in results were less than 2% and hence we assume that results in both cases are the same and use PWM for calculating PBS. Q value and mode solution calculations are also made by FEM study as implemented in COMSOL.

PCs with air holes (ε_a = 1) in four different dielectrics (ε_b = 15, 16, 17, and 18) in triangular manner are studied where only the results of ε_b = 16 in the case of k_z = 0.5/Λ are shown. Here the filling fraction (d/Λ) for each case is kept at 0.8. In the PBS (Fig. 2(a)), we obtain a linear dispersion region (marked as A) between bands 1 and 2 at the k point of the first Brillouin zone around the normalized frequency fΛ/c = 0.2131. The corresponding conical dispersion plot is shown in Fig. 2(b). The electric field distributions of the degenerate fundamental and first higher order mode at the K point are shown in Figs. 2(c) and 2(d) respectively. This case is not similar to that discussed in Refs. [17]–[19] which are for the case of a true Dirac Cone (DC), as the Dirac point discussed in the present work is not a true one but a “Dwarf Dirac cone” (DDC) (“A” in Fig. 2(a), since the photonic density of states is non-vanishing here because the upper branch (band 1) bends back to a lower frequency and thus it has a zero Dirac spectrum width.^[19] It is worth mentioning here that structure discussed by Huang et al.^[13] has triply degenerate states and appears as a zero index material (ZIM). Such a structure is different from both DC and DDC and known as a Dirac like point (DLP).^[20] Similar PBS is observed for the cases of k_z = 0.25/Λ, 0.75/Λ. Truly, this DDC maintains its linearity to some extent away from the K point (Fig. 2(b)). Iso-frequency plot in Fig. 3 shows a distorted conical nature which is due to the non-zero values of k_z. This kind of conical topology is also maintained for other non-zero k_z values.

Fig. 2.

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	Fig. 2. (a) PBS showing degeneracy at the K point (marked by A), (b) conical dispersion in the vicinity of the K point, and the electric field distributions of degenerated fundamental and first higher order mode (c) and (d) respectively.

Fig. 3.

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	Fig. 3. Iso-frequency surfaces of bands 1 and 2 at the K point (marked as ‘A’ in Fig. 2(a)) and its close vicinity for three kz values for εa = 1, εb = 16, and r = d/2 = 0.4Λ.

In order to trap this Dwarf Dirac frequency, we need to create a suitable defect in a supercell structure. The PBS of a supercell structure is a highly folded one and the defect state due to a defect appears as a flat band overlapped with the folded PBS (see Appendix A: Supplementary materials). This flat band can also be obtained from the eigen-frequency search by FEM by k-point sampling in the first Brillouin zone. The defect mode is observed as a highly concentrated electric field around the defect. Crystal structures along with the defect support optical modes of a particular frequency. The frequency of this trapped mode depends on the radius/size (r_d) of the defect. A resonance occurs when the incidence wave couples with the supported optical modes and thus light is trapped. The change in r_d, hence the trapped frequency reflects as a vertical shift of flat band in the PBS. The point of interest is to use that particular frequency where two photonic bands make such a conical topology with no band gap; the total internal reflection is not applicable here either, for the value of effective index of the surrounding crystal (clad) is higher than air defect (core). We systematically introduce an air column defect into the crystal (for ε_a = 1, ε_b = 16, and k_z = 0.25/Λ) and study (by FEM) the trapping of defect modes at different frequencies with respect to defect radius (black line in Fig. 4(b)). Each point of the black line corresponds to a defect mode supported by the crystal structure with a defect of a particular radius. The defect radius corresponding to the mode resonating at DDP is obtained by Dirac frequency calculated by PWM (red line in Fig. 4(b)).

Fig. 4.

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	Fig. 4. (a) Schematic diagram of the defect created for trapping Dwarf Dirac mode, (b) variation of normalized frequency fΛ/c with normalized defect radius (rd/r) for obtaining the trapped mode resonating at the Dirac point. From panel (b), we find rd = 2.511r for εa = 1, εb = 16 , and kz = 0.50/Λ.

A similar technique is used to find the defect radii for other values of k_z. The electric field distributions of defect modes resonating at DDP of the same PC for k_z = 0.5/Λ at the k point are shown in Fig. 5(a) with the field heights in Fig. 5(c). The corresponding value of defect radius is found to be 2.511r. At the same frequency, the same concentrated electric field is observed for each of the k-points in the entire k-path. This observation by FEM justifies the existence of the flat band. Mode analysis at the resonating frequency using a perfectly matched layer confirms the mode confinement which is shown in Fig. 5(b). Realistic example for this study may be a germanium (ε_b = 16) based PC with Λ = 10 μm and r_d = 0.04 μm which can trap a frequency of 6.39 THz under the k_z = 0.5/Λ condition.

Fig. 5.

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Fig. 5. (a) Electric field distributions of defect modes resonating at Dwarf Dirac points for kz = 0.50/Λ, calculated by eigen-frequency solver. (b) Result of mode solution at the same frequency as that in the case of panel (a). (c) Electric field |E| pattern in the x–z plane for the same case. Scales in figures are in micron and the colour bar indicates the highest magnitude of electric field norm in red and the lowest in blue.

As mentioned earlier, the defect state appears as a flat band in the highly folded band structure of the supercell, and at each of the points in the entire k-point sampling of the PBS, a highly concentrated field is observed in the defect site at the same frequency. Suitable value of r_d ensures that this flat band passes through the DDP which is situated at the k point (special point of the Brillouin zone).

Now we study the Q values of those resonance modes along with that for Dirac frequency in the case of 16-1 dielectric contrast PC with k_z = 0.5/Λ, which are shown in Fig. 6(a), and a much higher Q at the Dirac point is observed. We also change r_d, to trap different frequencies at the neighborhood of Dirac frequency at the same point k and calculate their Q values (Fig. 6(b) for the same case) and again observe a much higher Q for the Dirac frequency. The same observations are also made for the other structures. These two evidences indicate that the Dirac dispersion point is efficient enough in confining light compared with other points while working in a case of zero bandgap. Earlier, light confining at a true DC has been reported.^[17–19] It is already mentioned that the present case is not similar to those in Refs. [17]–[19] and these DDCs cannot support localized modes,^[19] for these are points in the continuum. The possibility of existence of “embedded eigenvalue”^[21,22] cannot be extracted since the obtained Q values at DDPs are not infinite, but a sudden increase of confinement is definitely observed. In the case of zero band gap, some radiative channels exist which give rise to some leakage modes. Introduction of defect sites can considerably suppress these radiation/leakage modes, but the order at DDP is much higher than those at the other points in the first Brillouin zone, which indicates that the presence of such a linear dispersion can cancel such leakage modes, hence it can be concluded that such linear dispersion has an inherent ability to confine light compared with the other points in a case of zero bandgap or a non-vanishing photonic density of states. Observation of such a property of Dwarf Dirac dispersion emerges as a new phenomenon apart from the known confining mechanisms and hence attracts niche applications.

Fig. 6.

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Fig. 6. Calculated Q factors for 16-1 photonic crystal for kz = 0.5/Λ (a) at different k-points on the path Γ–K–M at the Dirac frequency and (b) for different trapped frequencies including the Dwarf Dirac point at the k point of the Brillouin zone. Insets of both panels show the confined modes that are traced for making such plots. In both cases, the Dwarf Dirac point shows better confinement than the other sample points. A similar situation occurs for other structures (Figs. S3 and S4).

In this work, we investigate the existence of Dirac point in out-of-plane condition by calculating PBS for non-zero values of out-of-plane wave vector. Numerical calculations reveal that conical dispersion is possible for non-zero out-of-plane wave vector at the K point, and our case of study comes to be a situation where a conical dispersion exists but the DOS is not vanishing. We show that such a linear dispersion has an inherent light confining property compared with those at the other points in the continuum and it is beyond the known light confining mechanism, hence it indicates a new phenomenon. This study may be useful in designing tuneable photonic devices with exciting properties and new functionalities.