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Abstract
The dispersion relation and the angle-resolved reflection spectra of triangular-lattice photonic crystal slabs of the C6v symmetry were examined by the finite element method. The Dirac-cone dispersion relation on the Γ point of the reciprocal space was confirmed. The reflection spectra showed unique selection rules that agreed with the analytical calculation by the k · p perturbation theory. The distortion of the liner dispersion relation of the Dirac cones due to diffraction loss was also reproduced well by the numerical calculation, while we found distortion-free Dirac cones materialized with E2-symmetric modes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In our recent study [1], we reported on the photonic crystal slabs in SOI (silicon-on-insulator) wafers that materialized the isotropic linear dispersion relation, or the photonic Dirac cone, in the mid-IR (infrared) region. The specimens were fabricated by electron beam (EB) lithography and the Dirac-cone dispersion relation was confirmed by angle-resolved reflection spectroscopy with our home-made high-resolution apparatus. The reflection spectra agreed quite well with our numerical calculation and the selection rules derived by the $\textbf {k}\cdot \textbf {p}$ perturbation theory. Our specimens consisted of a square lattice of circular air cylinders fabricated in the top silicon layer of the SOI wafers, so they had the $C_{4v}$ (regular square) symmetry. The Dirac cone was materialized by the accidental degeneracy of two eigenmodes on the $\Gamma$ point that had the $E$ and $B_{2}$ symmetries. As predicted by the $\textbf {k}\cdot \textbf {p}$ perturbation theory [2], the Dirac cone was accompanied by a flat band, which was also confirmed by the angle-resolved reflection spectra.
The linear dispersion relation on the $\Gamma$ point was first found in the microwave region [3–5], which is known as CRLH (composite right-/left-handed) transmission lines, and later extended to more general cases and higher dimensions by the tight-binding approximation [6–8] and the $\textbf {k}\cdot \textbf {p}$ perturbation theory [2,9]. When the constituent eigenmodes on the $\Gamma$ point have a finite lifetime due to diffraction loss, etc., the linear dispersion is distorted and the group velocity calculated from the slope of the dispersion curve diverges [10,11]. The linear dispersion relation on the $\Gamma$ point can be applied to steerable antennas [12], cloaking [13], and waveguides with sharp bends [14], etc. However, the Dirac cone in the square-lattice photonic crystal slab of the $C_{4v}$ symmetry suffers from the distortion from the linear dispersion relation, which is caused by the finite lifetime of the $E$ modes due to its diffraction loss [10,11], which may reduce their performance in the above-mentioned applications.
However, as we show in this paper, this problem can be resolved by using photonic crystal slabs of the $C_{6v}$ (regular hexagon) symmetry, since the $E_{2}$-symmetric modes that materialize the Dirac cone are free from diffraction loss. So, we can expect purely linear dispersion relations. In addition, we can materialize double Dirac cones as well, which is another unique feature of the $C_{6v}$ symmetry [2,8]. In this paper, we report on a theoretical study on their dispersion relation, angle-resolved reflection spectra, and the selection rules of the reflection peaks.
This paper is organized as follows. In Sec. 2, we derive the Dirac-cone dispersion relation and the selection rules for reflection peaks based on the $\textbf {k}\cdot \textbf {p}$ perturbation theory and group theory. The analytic form of the distorted Dirac cone is derived as well. In Sec. 3, the structural parameters of the photonic crystal slabs are presented for materializing the Dirac cones and double Dirac cones. The linear dispersion relation and the selection rules for the reflection peaks are confirmed by numerical calculation by the finite element method (FEM). The shape of the distorted Dirac-cone dispersion is also reproduced by the FEM calculation. In addition, the null coupling between the internal eigenmodes and the external plane waves due to the mismatching of their spatial symmetries is confirmed by examining the incident-angle dependence of the reflection-peak intensity. A brief summary is given in Sec. 4.
2. Theory
2.1 Dispersion relation
We analyze the electromagnetic eigenmodes of photonic crystal slabs of the $C_{6v}$ symmetry in this paper. We assume a triangular array of cylindrical air holes fabricated in the top silicon layer of SOI wafers, since we successfully fabricated such structures in our recent study [1]. Figure 1(a) shows the top view of the assumed structure. Figure 1(b) shows the first Brillouin zone of the triangular lattice. Each vertex of the hexagonal first Brillouin zone is called a $K$ point, and the middle point of each side of the hexagon is called an $M$ point.
Fig. 1. (a) The triangular lattice of cylindrical air holes with the $C_{6v}$ symmetry. $a$ and $r$ denote the lattice constant and the radius of the air hole. (b) The first Brillouin zone of the triangular lattice, where three highly symmetric points are denoted by $\Gamma$, $K$, and $M$.
We will derive the selection rules for the reflection peaks of Dirac cones in Sec. 2.2 and their distortion by diffraction loss in Sec. 2.3. For this purpose, here we summarize the formulation of the Dirac-cone dispersion relation by the $\textbf {k}\cdot \textbf {p}$ perturbation theory. The eigenvalue problem for the magnetic field ($\textbf {H}$) of periodic systems is given by
When we impose a periodic boundary condition to make our problem well-defined, ${\cal L}$ is a Hermitian operator in the Hilbert space of complex vector fields. Then, $\textbf {u}_{\textbf {k}n}$ is an eigen function of operator ${\cal L}_\textbf {k}$ defined by
We assume according to the situation of our problem that $\{\textbf {u}_{0l} \vert \ l=1,2,\ldots ,M \}$ are degenerate and denote their eigenvalue by $\lambda _{D} = \omega _{D}^{2}/c^{2}$, where $\omega _{D}$ is the frequency of the vertex of the Dirac cone, or Dirac point. By the degenerate perturbation theory, the first-order solution for $\textbf {u}_{\textbf {k}l}$ ($l=1,2,\ldots ,M$) is obtained by diagonalizing the matrix $\mathrm {C}_\textbf {k}$ whose $ij$ ($1\le i, j \le M$) element is given by
The eigenmodes on the $\Gamma$ point are irreducible representations of the point group $\cal {G}$ of the photonic crystal [15]. For the present case, they are the irreducible representations of the $C_{6v}$ point group, that is, the $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $E_{1}$, and $E_{2}$ representations. The former four are one-dimensional (non-degenerate) and the latter two are two-dimensional (doubly degenerate). As was proved in [2], Dirac cones are materialized by the accidental degeneracy of two eigenmodes with particular combinations of their symmetries, which are listed in the first column of Table 1. The first four combinations yield a Dirac cone and a flat band. The last combination yields double Dirac cones.
Table 1. The combination of mode symmetries for materializing photonic Dirac cones and the selection rules for reflection peaks for Dirac cones (DC) and flat bands (FB) in triangular photonic crystal slabs with the $C_{6v}$ symmetry.
We can prove that each element of matrix $\mathrm {C_\textbf {k}}$ is invariant by any symmetry operation of the point group $\cal {G}$ [2]. By using this property, we can derive many relations among different matrix elements, which determine the structure of $\mathrm {C_\textbf {k}}$ nearly uniquely. We present two examples here to support the derivation of the selection rules in Sec. 2.2. We start with the double Dirac cone, which is a distinct feature of the $C_{6v}$ symmetry. As shown by the $\textbf {k}\cdot \textbf {p}$ perturbation theory in [2], we can obtain the dispersion relation of the double Dirac cone on the $\Gamma$ point by the accidental degeneracy of an $E_{1}$ mode and an $E_{2}$ mode. Matrix $\mathrm {C}_\textbf {k}$ for this case is given by
The second example is the accidental degeneracy of an $E_{2}$ mode and a $B_{1}$ mode, which will be examined numerically in Sec. 3. By diagonalizing
We should note that the linear dispersion derived in this section is correct in the vicinity of the Dirac point because it is a consequence of the first-order perturbation calculation. When we go apart from the Dirac point, higher-order terms like a parabolic term will appear, so the Dirac-cone dispersion and the flat-band dispersion gradually deviate from the linear relation. Other dispersion curves show a parabolic behavior around the $\Gamma$ point due to the time-reversal symmetry.
2.2 Selection rules
In this section, we derive the selection rules for reflection peaks of photonic crystal slabs of the $C_{6v}$ symmetry. We assume a configuration for the angle-resolved reflection spectroscopy as shown in Fig. 2. The tilt angle $\theta$ is measured from the $z$ axis normal to the slab surface and the azimuthal angle $\phi$ is measured from the $x$ axis. When the electric field of the incident wave is parallel (perpendicular) to the incident plane, it is called p- (s-) polarized.
Fig. 2. Configuration of the incident plane wave for the angle-resolved reflection measurement. $\theta$ and $\phi$ denote the tilt angle from the normal ($z$) direction and the azimuthal angle from the $x$ axis, respectively.
We start with the double Dirac cone materialized by the $E_{1}$ and $E_{2}$ modes. We denote the eigen functions by $\textbf {H}_\textbf {k}^{(\pm , j)}$ ($j=1,2$), where $+(-)$ denotes the upper (lower) Dirac cone and $j$ specifies one of the two eigen functions for the same eigen frequency. It is easy to show that they are given as follows:
Now, let us consider an incident plane wave with tilt angle $\theta$ and azimuthal angle $\phi$ as illustrated in Fig. 2. As will be shown in the following, two cases of $\phi = 0^{\circ }$ and $\phi = 90^{\circ }$ are sufficient for the symmetry assignment of each reflection peak. By examining the transformation by $\sigma _{x}$ and $\sigma _{y}$, we can easily show that $\textbf {H}_\textbf {k}^{(\pm ,j)}$ has the following symmetries.
When we consider the selection rule for reflection peaks due to the polarization of the incident wave, we should note that the spatial symmetry of the electric field, which we denote by $\textbf {E}_\textbf {k}$, is generally different from that of the magnetic field, since the former is a genuine vector whereas the latter is an axial vector. So, the magnetic field and the electric field show opposite symmetries for the mirror reflection because it changes the right-hand system to the left-hand system and vice versa [16]. For the p-polarized incident wave, its electric field is parallel to the incident plane, so it is symmetric for the mirror reflection on the incident plane. It has the same symmetry as $\textbf {E}_\textbf {k}^{(\pm ,2)}$ for $\phi =0^{\circ }$ and $\textbf {E}_\textbf {k}^{(\pm ,1)}$ for $\phi =90^{\circ }$, respectively. So, the p-polarized incident wave can excite these modes. On the other hand, the electric field is perpendicular to the incident plane for an s-polarized wave, so it is anti-symmetric for the mirror reflection. Thus the s-polarized incident wave has the same symmetry as $\textbf {E}_\textbf {k}^{(\pm ,1)}$ for $\phi =0^{\circ }$ and $\textbf {E}_\textbf {k}^{(\pm ,2)}$ for $\phi =90^{\circ }$, respectively. So, the s-polarized incident wave can excite these modes. In conclusion, both p- and s-polarized waves can excite an upper and a lower branch of the double Dirac cone.
Next, let us examine the Dirac cone and the flat band materialized by the combination of an $E_{2}$ mode and a $B_{1}$ mode. When we diagonalize $\mathrm {C}_\textbf {k}^{(E_{2},B_{1})}$ in Eq. (14), we obtain the following wave functions:
These selection rules are summarized in Table 1, in which the selection rules for the flat band and Dirac cone materialized by other combinations of mode symmetries are shown as well. Those selection rules can be derived in the same manner starting from the C$_\textbf {k}$ matrix for each mode-symmetry combination [2,17].
2.3 Distortion by diffraction loss
In Sec. 3, we numerically examine the distortion of the linear Dirac-cone dispersion by the diffraction loss. To obtain its qualitative estimation, we extend our calculation in Sec. 2.1 to a lossy case by introducing a decay rate of the lossy mode to the $\textbf {k}\cdot \textbf {p}$ perturbation matrix $\mathrm {C}_\textbf {k}$. Because the eigenvalue of $\mathrm {C}_\textbf {k}$ is not the eigenfrequency $\omega _\textbf {k}$ but $\lambda _\textbf {k}$ (=$\omega _\textbf {k}^{2}/c^{2}$), it is convenient to introduce another matrix $\mathrm {D}_\textbf {k}$ that is defined as follows according to the Taylor expansion of the eigenfrequency in the vicinity of the $\Gamma$ point in Eq. (13) [10]:
For highly symmetric photonic crystals, the coupling between most of the internal eigenmodes and the external plane waves is forbidden on the $\Gamma$ point by their symmetry mismatching. For photonic crystals of the $C_{6v}$ symmetry, only $E_{1}$ modes couple to plane waves coming from or going to the direction normal to the surface, since those plane waves also have the $E_{1}$ symmetry [16]. For sufficiently low frequencies, the only diffraction channel is one that is normal to the surface (the zeroth-order diffraction). So, the diffraction loss takes place only for the $E_{1}$ modes.
As an example of a lossy Dirac cone, here we deal with the combination of an $E_{1}$ mode and an $A_{1}$ mode. Because the lifetime of the $E_{1}$ mode is finite due to the diffraction loss, we include its decay rate $\gamma$, which is the imaginary part of the eigen frequency of the $E_{1}$ mode (Im $\omega _{E_{1}}$), in the calculation by the $\textbf {k}\cdot \textbf {p}$ perturbation. Thus, the matrix to be diagonalized is given by
Equation (28) has a unique feature that the real part of the three solutions is degenerate for $k < k_{0}$, whereas their imaginary parts are different from each other. The wave number of the exceptional point, $k_{0}$, is given by
For $k > k_{0}$, the imaginary part of the eigen frequency is smaller by a factor of two for the Dirac cone than for the flat band. So, we may expect that the reflection peaks for the former are sharper than for the latter. This point will be examined numerically in Sec. 3.We note here that we started the analysis of the dispersion relation by assuming the first-order correction in $\textbf {k}$ to the $\textbf {k}\cdot \textbf {p}$ perturbation operator and carried out the first-order degenerate perturbation. So, we should obtain the eigenvalue correction linear in $\textbf {k}$. However, the distorted Dirac-cone dispersion in Eq. (28) is apparently not linear, which is, of course, a consequence of the introduction of the decay term into matrix $\mathrm {D}_\textbf {k}$. At present, the mathematical correctness of this treatment is not proved yet, although the analytical form thus obtained agrees quite well with numerical calculation, as will be shown in Sec. 3.
3. Numerical results
The dispersion relation and the reflection spectra were calculated by FEM with commercial software COMSOL. We assumed the Bloch boundary condition on the unit cell in the lateral ($x$ and $y$) directions and the PML (perfectly matched layer) absorbing boundary condition in the vertical ($z$) direction at 6.2 $\mu$m above the top silicon layer and at 3.2 $\mu$m below the boundary between the SiO$_2$ layer and the Si substrate. Because the thickness of the Si substrate was much larger than relevant wavelengths, it was assumed to be infinitely thick.
We start with the combination of the $E_{2}$ and $B_{1}$ modes. Figure 3 shows the dispersion relation of the Dirac cone with a flat band that was materialized by the degeneracy of an $E_{2}$ mode and a $B_{1}$ mode on the $\Gamma$ point. The structural parameters for their accidental degeneracy are listed in the figure caption. All bands found in this figure were TE (transverse electric)-like modes. Note that all bands shown in Fig. 3 and other figures are located above the light line, so they can be excited by incident waves unless the coupling is forbidden by symmetry reasons. We can clearly see two straight lines and a flat band as predicted by the first-order $\textbf {k}\cdot \textbf {p}$ perturbation theory on the $\Gamma$ point around 4825 cm$^{-1}$. Both $E_{2}$ and $B_{1}$ modes are free from diffraction loss because of the mismatching of their spatial symmetry with the possible diffracted wave as mentioned in Sec. 2.1, so their lifetime is infinite for lossless materials with real dielectric constant. Thus, we obtain a genuine Dirac cone as shown in Fig. 3. In addition to the Dirac cone and the flat band, there are $B_{2}$, $E_{1}$, and $A_{1}$ modes in the nearby frequency range.
Fig. 3. Dirac cone with an auxiliary flat dispersion surface (flat band) materialized by the degeneracy of an $E_{2}$ mode and a $B_{1}$ mode. The vertical axis is the wavelength/frequency of the electromagnetic eigenmodes and the horizontal axis is the wave vector in the first Brillouin zone. The dispersion relation is plotted in the $\Gamma$-to-K and $\Gamma$-to-M directions. M/10, for example, implies that the horizontal axis is magnified by 10 times. Structural parameters: Lattice constant $a = 1.50\ \mu \mathrm {m}$, air-hole radius $r = 325.4\ \mathrm {nm}$, air-hole depth $d=400\ \mathrm {nm}$. The thickness of the top Si layer and that of the SiO$_{2}$ layer of the SOI wafer were assumed to be 400 nm and 3 $\mu$m, respectively, according to available specimens. The refractive indices of Si and SiO$_{2}$ were 3.427 [18] and 1.440 [19].
Figure 4 shows their angle-resolved reflection spectra calculated by FEM, where four spectra with different tilt angles by $0.5^{\circ }$ steps are drawn with an equal spacing in the vertical direction. Figures 4(a) and 4(b) show the reflection spectra for the incident wave tilted in the $\Gamma$-to-K ($\phi =0^{\circ }$) and $\Gamma$-to-M ($\phi = 90^{\circ }$) directions, respectively. A distinct feature of these figures is two peaks that diverge linearly from each other with increasing $\theta$ and a nearly $\theta$-independent peak sandwiched by the former two peaks around 4825 cm$^{-1}$. This feature agrees quite well with the Dirac cone and the flat band found for the dispersion relation in Fig. 3. In addition, the Dirac-cone peaks are observed only for s-polarization (p-polarization) for $\phi =0^{\circ }$ ($\phi =90^{\circ }$), whereas the flat-band peak is observed for the opposite polarization. These properties agree perfectly with the selection rules for the ($E_{2}$, $B_{1}$) mode combination given in Table 1.
Fig. 4. Angle-resolved reflection spectra of the Dirac cone with a flat band materialized by the accidental degeneracy of an $E_{2}$ mode and a $B_{1}$ mode. The structural parameters assumed for the calculation are the same as Fig. 3. Reflection spectra with an s- (p-) polarized incident wave are drawn with a blue (red) color. In each figure, the upper and lower limits of the reflection spectrum for $\theta =0^{\circ }$ are 1 and 0, respectively. Other spectra are drawn on the same scale and shifted by unity from each other in the vertical direction. (a) $\phi = 0^{\circ }$ ($\Gamma$-to-K direction), (b) $\phi = 90^{\circ }$ ($\Gamma$-to-M direction).
Another feature is that these three peaks vanish at $\theta =0^{\circ }$, which is caused by the symmetry mismatching between the $E_{2}$/$B_{1}$ modes and the external plane wave coming from the normal direction [16]. The latter has the $E_{1}$ symmetry, so it can only couple to the $E_{1}$-symmetric modes of the photonic crystals slab. When we further examine Figs. 4(a) and 4(b), we observe that the peak around 4775 cm$^{-1}$ due to an $A_{1}$ mode vanishes at $\theta =0^{\circ }$, whereas the peak around 4895 cm$^{-1}$ due to an $E_{1}$ mode survives, since the latter has the same symmetry as the incident wave. We should note that the reflection peaks of the $E_{1}$ mode are much broader than other peaks because of the lifetime broadening due to the diffraction loss. We should also note that the p- and s-polarizations give the same spectra for normal incidence ($\theta =0^{\circ }$), so it is represented by a single blue line in Figs. 4(a) and 4(b).
Next, we examine the dispersion relation and the reflection spectra of the Dirac cone and the flat band distorted by the diffraction loss for the ($E_{1}$, $A_{1}$) mode combination. Because the decay rate $\gamma$, and hence, the quality factor $Q$ of the $E_{1}$ mode ($= \ \mathrm {Re} [\omega _{E_{1}}] / 2 \mathrm {Im} [\omega _{E_{1}}]$) substantially depend on the structural parameters, we can design the amount of the distortion as shown in the following. The dispersion relations calculated by FEM for two different structural parameters are presented in Figs. 5(a) and 5(b). The quality factor of the $E_{1}$ mode was 15,800 for Fig. 5(a) and 245 for Fig. 5(b). This big difference in the quality factor was brought about mainly by the depth of the air hole, which significantly affected the diffraction efficiency. Due to the small diffraction loss in Fig. 5(a), the deviation from the linear dispersion of the Dirac cone is negligibly small. On the other hand, there is an appreciable distortion in Fig. 5(b) due to the large diffraction loss. Note that the Dirac point frequency is larger in Fig. 5(b) because the volume of the air hole is larger so that the averaged refractive index is smaller.
Fig. 5. Dirac cone with a flat band materialized by the accidental degeneracy of an $E_{1}$ mode and an $A_{1}$ mode. (a) Structural parameters: $a = 1.5\ \mu \mathrm {m}$, $r = 412$ nm, $d=200$ nm. Quality factor of the $E_{1}$ mode: 15,800. A nearby $E_{2}$ band is also shown. (b) Structural parameters: $a = 1.5\ \mu \mathrm {m}$, $r = 450$ nm, $d=333$ nm. Quality factor of the $E_{1}$ mode: 245. The Dirac cone is distorted by the diffraction loss. A nearby $A_{2}$ band around 4895 cm$^{-1}$ is shown by a dotted line. (c) Comparison between the numerical results of panel (b) and analytic form (Eq. (28)) of the dispersion relation of the Dirac cone and the flat band.
The line shape of the distorted Dirac cone is close to the prediction by the $\textbf {k}\cdot \textbf {p}$ perturbation theory in Sec. 2.3. Figure 5(c) shows a comparison between the numerical result and the analytic form calculated for the structural parameters of Fig. 5(b). Among the three parameters in Eq. (28), the decay rate $\gamma$ was obtained from the imaginary part of $\omega _{E_{1}}$. $\omega _{D}$ was also specified by the numerical result. $\vert b'\vert$, which determines the slope of the Dirac cone at a sufficiently large distance from the $\Gamma$ point, was also deduced from the numerical data. A considerably good coincidence between the numerical results and the analytic form was obtained, so we may conclude that Eq. (28) estimates the distorted Dirac-cone dispersion well, although its mathematical basis is not very clear yet.
The quality factor, $Q$, of the distorted Dirac cone and the flat band is given in Fig. 6 as a function of the wavenumber, where both numerical results by the FEM calculation and analytical results obtained from Eq. (28) are shown. Their agreement is good again. As predicted by the $\textbf {k}\cdot \textbf {p}$ perturbation theory, the quality factor of the Dirac-cone modes is two-times larger than the flat-band mode for $k > k_{0}$.
Fig. 6. Wavenumber dependence of the quality factor of the distorted Dirac cone and the flat band shown in Fig. 5(c). Circles denote numerical results and solid lines denote analytical results obtained from Eq. (28). The exceptional point, $k_{0}$, is denoted by a vertical dotted line. The same values were assumed for $\omega _{D}$, $\gamma$, and $\vert b' \vert$ as Fig. 5(c).
The reflection spectra for this case are presented in Fig. 7. Figures 7(a) and 7(b) show the spectra by $0.5^{\circ }$ steps whereas 7(c) shows the spectra by $0.1^{\circ }$ steps. Figures 7(a) and 7(b) look very close to each other because the Dirac-cone dispersion relation is highly isotropic. As in Fig. 4, one nearly $\theta$-independent peak (flat band) is sandwiched by two peaks (Dirac cone) that diverge with increasing $\theta$. As predicted in Table 1, the Dirac cone is active to the s-polarization, whereas the flat band is active to the p-polarization for both azimuthal angles of $\phi = 0^{\circ }$ and $90^{\circ }$. Compared with the ($E_{2}$, $B_{1}$) mode combination, the reflection peaks for the ($E_{1}$, $A_{1}$) combination of Fig. 5 are much broader, which is an important consequence of the small quality factor of the $E_{1}$ mode.
Fig. 7. Angle-resolved reflection spectra of the Dirac cone with the auxiliary flat dispersion surface materialized by the degeneracy of an $E_{1}$ mode and an $A_{1}$ mode. The structural parameters assumed for the calculation are the same as in Fig. 5(b). Reflection spectra with an s- (p-) polarized incident wave are drawn with a blue (red) color. In each figure, the upper and lower limits of the reflection spectrum for $\theta =0^{\circ }$ are 1 and 0, respectively. Other spectra are drawn on the same scale and shifted by unity from each other in the vertical direction. (a) $\phi = 0^{\circ }$ ($\Gamma$-to-K direction), (b), (c) $\phi = 90^{\circ }$ ($\Gamma$-to-M direction).
On the other hand, the nearby $A_{2}$ band, which has the same symmetry as the Dirac cone materialized by the ($E_{1}$, $A_{2}$) combination, is active to p-polarization. So, the reflection peaks due to the flat band and the $A_{2}$ band interfere with each other to form a Fano-type line shape around 4900 cm$^{-1}$. In addition, the interference vanishes at $\theta =0^{\circ }$, since the $A_{2}$ mode is inactive to the incident wave from the normal direction because of the symmetry mismatching.
Another interference in the reflection spectra is observed for two peaks originating from the upper and lower Dirac cones as shown in Fig. 7(c), which is plotted for small incident angles. The two peaks of the Dirac cone observed for p-polarization merge into a single peak with decreasing $\theta$. Eventually for $\theta =0^{\circ }$, the flat band (s-polarization) and the Dirac cones (p-polarization) give the same spectrum, which is represented by a single blue line. The line width of the former should be larger than the latter by a factor of two for $k > k_{0}$, since the imaginary part of the eigen frequency is larger for the former according to Eq. (28). The critical tilt angle to excite the Dirac-cone mode just at the exceptional point ($k_{0}$) is $\theta _{0}=0.27^{\circ }$ for the present case. In actuality, it is observed in the reflection spectra that the former reflection peak is broader than the latter by about a factor of two.
Finally, we examined the double Dirac cone materialized by the ($E_{1}$, $E_{2}$) mode combination. We looked for their accidental degeneracy by changing the structural parameters of the triangular lattice of air holes fabricated in the SOI wafer with a 400 nm-thick top silicon layer, which we have assumed in this paper. However, we could not find double Dirac cones whose Dirac point frequency, $\omega _{D}$, was sufficiently apart from other eigenmodes. So, we further examined two different structures. The first one was the different air-hole configuration, which is illustrated in Fig. 8(a), and the second one was the SOI wafer with a thicker top silicon layer. All bands found in Figs. 8(b) and 8(c) were TE-like modes.
Fig. 8. (a) Illustration of the unit cell consisting of six additional air holes. (b) Double Dirac cone materialized by the degeneracy of the $E_{1}$ and $E_{2}$ modes. Structural Parameters: $t=400$ nm, $a=1.5\ \mu \mathrm {m}$, $r_{1}=150$ nm, $r_{2}=45$ nm $R = 495$ nm, $d=400$ nm. (c) Double Dirac cone materialized with a thicker top silicon layer. Structural Parameters: $t=3.6\ \mu \mathrm {m}$, $a = 1.435\ \mu \mathrm {m}$, $r = 402$ nm, $d=2.33\ \mu \mathrm {m}$.
In the first case, we assumed six air holes in addition to the central one in each unit cell. By assuming the structural parameters listed in the caption of Fig. 8, we found the formation of the double Dirac cone whose dispersion relation is shown in Fig. 8(b), where the degeneracy of the $E_{1}$ and $E_{2}$ modes is achieved around 3320 cm$^{-1}$. Because the large quality factor of the $E_{1}$ mode, which is 6470, the distortion caused by the diffraction loss is negligibly small.
In the second case, we assumed a 3.6 $\mu$m-thick top silicon layer. Because the $E_{1}$ mode has a large diffraction loss for this case, the double Dirac cone discussed in Sec. 2.1 is distorted as the Dirac cone formed by the ($E_{1}$, $A_{1})$ mode combination. By introducing the decay term for the $E_{1}$ mode to the $\mathrm {D}_\textbf {k}$ matrix of the ($E_{1}$, $E_{2}$) mode combination, we can easily obtain the analytical expression of its complex dispersion relation:
Figure 8(c) shows the dispersion relation calculated by FEM for this case. Other structural parameters to materialize the degeneracy of the $E_{1}$ and $E_{2}$ modes are also listed in the caption of Fig. 8. Because the $E_1$ mode is lossy, it is clearly observed that the double Dirac cone is distorted just around the $\Gamma$ point whose line shape is similar to that of the Dirac cone with the ($E_{1}$, $A_{1}$) mode combination in Fig. 5. In addition, one of the two straight lines that compose the upper and lower double Dirac cones deviates from linear behavior, particularly in the $\Gamma$-to-K direction. This is caused by the repulsion between these modes and nearby modes with the same symmetry, the latter of which are not shown in Fig. 8.
The angle-resolved reflection spectra for this case are shown in Fig. 9. For both azimuthal angles of $0^{\circ }$ and $90^{\circ }$, the double Dirac cone is active to both p- and s-polarizations, so there are two peaks for each case, which agrees with the selection rule listed in Table 1. Note that two reflection spectra of p- and s-polarizations are the same for $\theta =0^{\circ }$, so it is represented just by the blue color. The quality factor of the $E_{1}$ mode is 1470, and the reflection peaks have a big natural line width corresponding to this quality factor.
Fig. 9. Angle-resolved reflection spectra of the double Dirac cones materialized by the degeneracy of an $E_{1}$ mode and a $E_{2}$ mode. The structural parameters assumed for the calculation are the same as Fig. 8(c). Reflection spectra with an s- (p-) polarized incident wave are drawn with a blue (red) color. (a) $\phi = 0^{\circ }$ ($\Gamma$-to-K direction), (b) $\phi = 90^{\circ }$ ($\Gamma$-to-M direction).
Here, we note three things. First, we could successfully confirm the Dirac-cone and the double Dirac-cone dispersion relation, which were predicted by the $\textbf {k}\cdot \textbf {p}$ perturbation theory, by numerical calculation of the dispersion relation. Second, the angle-resolved reflection spectra calculated by FEM agreed quite well with the dispersion relation and the selection rules. So, the latter is a powerful tool for the mode assignment when we analyze experimental data. Finally, the reflection peak width also agreed well with the numerical results and the prediction of the $\textbf {k}\cdot \textbf {p}$ perturbation theory. The spectral features of the distorted Dirac cone expected from the $\textbf {k}\cdot \textbf {p}$ perturbation theory, in particular, were well reproduced by the reflection spectra. So, these features can be used for the practical confirmation of the distorted Dirac cone in future experimental studies.
4. Conclusion
The dispersion relation and the angle-resolved reflection spectra were calculated by FEM for Dirac cones with a flat band and double Dirac cones materialized by the accidental degeneracy in triangular-lattice photonic crystal slabs of the $C_{6v}$ symmetry. Structural parameters for materializing the Dirac cones were obtained for the triangular lattice of circular air cylinders in the top Si layer of SOI wafers. The calculated reflection spectra agreed quite well with the dispersion relation by FEM and the selection rules derived by the $\textbf {k}\cdot \textbf {p}$ perturbation theory. The distortion from the linear dispersion due to diffraction loss and the wavenumber dependence of the quality factor were quantitatively evaluated, which also agreed with the prediction by the $\textbf {k}\cdot \textbf {p}$ perturbation. In comparison with our previous study on square-lattice photonic crystal slabs, we newly found distortion-free Dirac cones materialized with $E_2$ symmetric modes, whereas both Dirac cones with $E_1$ symmetric modes and double Dirac cones are influenced by the diffraction loss.
Funding
Innovative Science and Technology Initiative for Security by Acquisition, Technology & Logistics Agency (ATLA), Japan. (JPJ004596).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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