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Abstract
We measured angle-resolved reflection spectra of triangular-lattice photonic crystal slabs fabricated in a silicon-on-insulator wafer in the mid-infrared region. We achieved a high angle-resolution measurement by means of our homemade optical setup integrated in the sample chamber of an FT-IR spectrometer. By examining the reflection peak frequency as a function of the lateral component of the wave vector of the incident light and applying the selection rules expected from the spatial symmetry of electromagnetic eigenmodes in C6v-symmetric structures, we successfully obtained the dispersion relation and the mode symmetry of the photonic crystal slabs, which agreed well with numerical calculations by the finite element method. We also found the redistribution of diffraction loss between A1- and E1-symmetric modes, which was caused by the Dirac-cone formation due to their effective degeneracy.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the first predictions of complete photonic band gaps [1–3], extensive studies have been reported for photonic crystals (PCs). In addition to fascinating phenomena presented in the relatively early stages of the investigation [4–6] such as high-Q micro cavities for the Purcell effect and Rabi splitting [7], and quantum localization of super radiance [8], many studies have been reported in recent years from novel points of view including PC surface-emitting laser (PCSEL) [9,10], Dirac-cone dispersion relation [11–13], cloaking by vanishing effective refractive index [14], and topological PCs and topological edge modes [15,16].
We recently reported the Dirac-cone dispersion relation in square-lattice PC slabs fabricated in silicon-on-insulator (SOI) wafers by electron beam (EB) lithography [17]. Our home-made high-resolution optical set-up enabled accurate measurement of angle-resolved reflection spectra in the mid-infrared region to observe the Dirac-cone dispersion [18]. At the same time, because the electromagnetic eigenmodes of PCs with high spatial symmetries possess selection rules regarding the propagation direction and the polarization of the incident light, we can utilize them to examine the dispersion and the mode symmetry in a purely experimental manner. A complete theoretical analysis of the selection rules of Dirac-cone dispersion in triangular-lattice PCs of the C6v symmetry was presented in our recent publication [19].
In this paper, we report on an experimental study of angle-resolved reflection spectra of triangular-lattice PC slabs fabricated in an SOI wafer. We will show that we can obtain the dispersion relation and the mode symmetry by analyzing the spectra with the selection rules. We also report on the redistribution of diffraction loss between A1- and E1-symmetric modes, which originated from the Dirac-cone formation by effective degeneracy.
This paper is organized as follows: In Section 2, we describe the methods for sample fabrication, angle-resolved reflection measurement, and the numerical calculations of the dispersion relation and the reflection spectra. We also summarize the symmetry and the number of modes on the Γ point of the first Brillouin zone, which are predicted by the group theory based on the zone-folding of dispersion curves of planar wave guides without periodic modulation of their refractive index. In Section 3, we present the angle-resolved reflection spectra by our experimental study, from which we derive the dispersion relation and obtain the mode symmetry in the vicinity of the Γ point. The reflection spectra and the dispersion curves agree well with the numerical calculations by the finite element method (FEM). We also discuss the redistribution of the diffraction loss. From these results, we prove that the selection rules are a powerful tool for the analysis of the dispersion relation of PC slabs. A brief summary is given in Section 4.
2. Methods
2.1 Sample fabrication
PC slabs consisting of a triangular array of circular air holes were fabricated in a 400 nm-thick top Si layer of an SOI wafer (SOITEC) by electron beam (EB) lithography. We used Elionix ELS-7000 for EB exposure and Zenon ZEP520A as an EB resist. The top Si layer was etched by inductively coupled plasma reactive ion etching (ICP-RIE) with an etchant comprised of a mixture of Ar and Cl2. Five PC slabs with different design radii, r, of air holes, 500, 530, 560, 590, and 620 nm, were fabricated on the same SOI wafer, whose photograph is shown in Fig. 1(a). The lattice constant, a, and the air-hole depth, d, of the PC slabs were designed as 2400 and 240 nm, respectively. The top and side views of the SEM image are shown in Figs. 1(b) and 1(c). The surface area of each PC was 3.4 mm ( = 1435 a) by 3.5 mm ( = 1680 $\sqrt 3 $a/2).
Fig. 1. (a) Photograph of the five PCs fabricated in an SOI wafer. (b) Top and (c) cross-sectional side views of the SEM image of one of the five PCs with a design r = 560 nm. The top Si layer and the SiO2 layer are 400 nm and 3 µm thick, respectively.
2.2 Angle-resolved reflection measurement
Angle-resolved reflection spectra were measured by using our home-made high-resolution set-up, which was placed in the sample chamber of a Fourier Transform Infrared (FT-IR) spectrometer (JASCO 6800) with a high-intensity ceramic light source. Its schematic diagram is shown in Fig. 2(a), whose details were reported in our recent publication [9]. The measurement configuration is illustrated in Fig. 2(b), where the first Brillouin zone of the triangular lattice and its highly symmetric points (Γ, K, and M points) are shown as an inset. The tilt angle from the normal (z) direction and the azimuthal angle from the x axis are denoted by $\theta $ and $\phi $, respectively.
Fig. 2. (a) Schematic diagram of our home-made optical set-up for the angle-resolved reflection measurement. M: mirror, PM: parabolic mirror, BS: beam splitter, S: specimen. (b) Configuration of the incident plane wave. The tilt angle from the normal (z) direction and the azimuthal angle from the x axis are denoted by $\theta $ and $\phi $, respectively. The polarization of the incident wave is denoted by p (s) when its electric field is parallel (perpendicular) to the incident plane. The inset shows the first Brillouin zone of the 2D triangular lattice.
2.3 Theory
The dispersion relation and the angle-resolved reflection spectra were calculated by the finite element method (FEM) with the commercial software, COMSOL. The Bloch boundary condition was imposed on the unit cell in the lateral (x and y) directions. The perfectly matched layer (PML) absorbing boundary condition was imposed in the vertical (z) direction. The results of the calculations will be presented in Section 3.
The selection rules for reflection peaks are summarized in Table 1, which was obtained for six irreducible representations of the C6v point group. They represent the symmetries of eigenmodes on the Γ point [6]. Note that the electric field is a genuine vector and the magnetic field is an axial vector, so their symmetry properties are generally different. We refer to the symmetry of the magnetic field in Table 1. E1- and E2-symmetric modes are doubly degenerate and the other four modes are nondegenerate. Among these irreducible representations, only E1-symmetric modes are active to the incident plane wave from the normal (z) direction because this incident wave conserves the lateral component of the wave vector when it excites the eigenmodes on the Γ point and it is symmetry-matched only to the E1-symmetric modes [6].
Table 1. Selection rules for reflection peaks
Modes on the Γ point connect to those on adjacent points in the first Brillouin zone according to the compatibility relations [6], so we can tell the symmetry of the latter by a purely analytical manner. In the Γ-K and Γ-M directions, all eigenmodes are symmetric or antisymmetric about the incident plane. So, when we use linearly polarized incident light whose electric field is parallel (p) or perpendicular (s) to the incident plane, there are additional selection rules as listed in Table 1.
In addition, we can tell the symmetry and the number of eigenmodes on the Γ point by using a group theoretical analysis based on the zone-folding of dispersion curves of planar wave guides without periodic modulation of the refractive indices, which relies on the reduction of reducible representations given by the superposition of the planar waveguide modes [6]. In the present case, as we will show in Section 3, the relevant eigenmodes mainly originate from the transverse electric (TE) modes of the SOI wafer and the reciprocal lattice vector for the zone-folding is the second smallest one. In this case, we can prove by the reduction procedure that we have E1-, E2-, A1-, and B1-symmetric modes [6]. An exception is the emergence of an A2-symmetric mode, which originates from the zone-folding of a transverse magnetic (TM) waveguide mode by the smallest reciprocal lattice vector.
3. Results and discussion
3.1 Angle-resolved reflection spectra
For all PC specimens, we found three peaks for normal incidence (θ = 0 deg) in the spectral range from 2,000 to 3,500 cm−1, which are attributed to E1-symmetric eigenmodes according to the selection rule for the Γ point (see Table 1). From this observation, we can conclude that there are three groups of eigenmodes that have different origins. By comparing their frequencies with those of the waveguide modes of the SOI wafer, we found that the first peak around 2,300 cm−1 and the second peak around 3,150 cm−1 originated from the lowest TE and TM bands, respectively, which were folded into the first Brillouin zone with the smallest reciprocal lattice vector, whereas the third peak around 3,280 cm−1 originated from the lowest TE band, which was folded into the first Brillouin zone with the second smallest reciprocal lattice vector.
In the following, we focus on the third group and present detailed analyses of reflection spectra for r = 530 and 560 nm. First, Fig. 3 shows the reflection spectra for r = 530 nm, where the azimuthal angle of the incident light was chosen to excite eigenmodes along the Γ-K (ϕ = 0 deg) and Γ-M (ϕ = 90 deg) directions. For each azimuthal angle, the reflection spectra were measured with s- and p-polarized incident waves. In each panel of Fig. 3, nine spectra measured from θ = 0 to 4 deg are presented, where adjacent spectra are shifted vertically by 0.5.
Fig. 3. Angle-resolved reflection spectra of the specimen with r = 530 nm. Red (blue) lines denote spectra measured with an s- (p-) polarized incident wave. Arrows on the bottom of each panel show approximate locations of eigenmode frequencies on the Γ point, which are obtained by interpolating the peak frequency as a function of the lateral component of the incident wave vector in Fig. 4(a).
As we mentioned, there is a peak at 3,280 cm−1 originating from an E1-symmetric mode for θ = 0 deg. When the tile angle is increased, this peak shifts and several new peaks appear. Their peak frequencies are plotted as functions of the lateral component of the wave vector of the incident light in Fig. 4(a), where the horizontal axis denotes the wave vector in the Γ-K and Γ-M directions, and M/10 and K/10 imply that the horizontal axis is magnified by ten times. s- and p-active peaks are marked by red and blue colors, respectively. Note that the third and fourth highest bands are nearly degenerate at 3,270 cm−1, although their peaks disappear on the Γ point. Also note that their two bands are active to s and p polarizations for each direction. These features agree with the properties of the E2-symmetric mode (see Table 1).
Fig. 4. Dispersion relation for r = 530 nm obtained (a) from Fig. 3 and (b) by numerical calculations. The horizontal axis is the wave vector in the first Brillouin zone. Red (blue) color denotes s- (p-) active bands. Refractive indices were assumed to be 3.427 for Si [20] and 1.440 for SiO2 [21].
There are three more bands in Fig. 4(a), two of which should be attributed to A1- and B1-symmetric modes as we mentioned in Section 2.3, which are accompanied by two s-active bands in the Γ-K direction and by an s-active band and a p-active band in the Γ-M direction according to Table 1. Then, the remaining band is a p-active band in each direction. By consulting Table 1 again, we conclude that the remaining bands should connect to an A2-symmetric mode on the Γ point. The peak position of this A2-symmetric mode, or its eigenfrequency, strongly depends on the air-hole radius, r. When r is decreased, its eigenfrequency decreases and moves away from the frequencies of the third group. So, we can conclude that the A2-symmetric mode belongs to the second group, which originated from the lowest TM band of the SOI wafer.
We also calculated the dispersion relation by FEM. By comparing the observed and calculated eigen frequencies of E1-symmetric modes, we found that there is a discrepancy of about 40 cm−1 between them, which should be attributed to calculation and fabrication errors, ambiguity in the refractive indices, and ambiguity in the structural parameters of the SOI wafer. We obtained a good agreement between calculations and observations when we assumed a larger value for the air-hole depth, d, by 20%, which is 288 nm. Figure 4(b) shows the dispersion relation thus calculated for r = 530 nm, where the symmetry of eigenmodes on the Γ point was obtained by examining their field distributions by the FEM calculations. Both dispersion curves and symmetry assignment agree well between the observations and calculations.
Next, Fig. 5 shows the reflection spectra for r = 560 nm. We analyzed their features as before to obtain the dispersion relation and assign the mode symmetry as shown in Fig. 6(a). Because the averaged refractive index was decreased by increasing the air-hole radius, the eigen frequencies generally increased, which is apparent when we compare them on the Γ point with those in Fig. 4(a). In addition, it is observed that the order of the modes on the Γ point have changed, which was caused by the difference in the radius-size dependence of their eigen frequency. These features are well reproduced by numerical calculations as shown in Fig. 6(b).
Fig. 5. Angle-resolved reflection spectra of the specimen with r = 560 nm. Red (blue) lines denote spectra measured with an s- (p-) polarized incident wave. Arrows on the bottom of each panel show approximate locations of eigenmode frequencies on the Γ point, which are obtained by interpolating the peak frequency as a function of the lateral component of the incident wave vector in Fig. 6(a).
Fig. 6. Dispersion relation for r = 560 nm obtained (a) from Fig. 5 and (b) by numerical calculation. The horizontal axis is the wave vector in the first Brillouin zone. Red (blue) color denotes s- (p-) active bands.
Figure 7 is the reflection spectra calculated by FEM. We assumed r = 560 nm and d = 240 nm for this calculation. Not only the peak positions but also their spectral intensities are well reproduced, which further supports the validity of our analyses.
In our recent publication [19], we showed by the method of k-p perturbation that a distorted Dirac cone was materialized by the accidental degeneracy of an E1-symmetric mode and an A1-/A2-symmetric mode. Originally, the Dirac cone is characterized by a linear dispersion relation in the vicinity of the Γ point [11–13]. However, if the quality factor of the E1-symmetric mode is small due to a large diffraction loss, the deviation from the linear relation becomes apparent and their dispersion is characterized by an exceptional point [22,23]. In addition, the mixture of the E1-symmetric mode with the A1-/A2-symmetric mode results in the redistribution of the diffraction loss. As a consequence, the two peaks of the Dirac cone have the same width, which is one half of the original width of the E1-symmetric mode [19].
For the present specimens, exact accidental degeneracy with the E1-symmetric mode was not observed. However, the spectral width of the E1-symmetric mode of the third group was large (FWHM = 54.6 cm−1), so A1- and A2-symmetric modes whose eigen frequencies are located in this range can be regarded as effectively degenerate. This situation is materialized for the A1- and E1-symmetric modes in Figs. 6(a) and 6(b). Although there is also an A2-symmetric mode in the vicinity of the E1-symmetric mode, the former, which is a TM-like mode, has a character different from the latter, which is a TE-like mode, so their mixing is generally small and we can safely neglect it when we discuss the mode mixing. For example, the lowest peak in Figs. 5(c) and 7(c), which has the A1 origin, has a considerably large width due to the mixing with the E1-symmetric mode. Its width measured at θ = 4 deg ( = 31.4 cm−1) is approximately one half of the width of the E1-symmetric mode on the Γ point ( = 54.6 cm−1). This feature was also confirmed by numerical calculations. The Q factor of the A1-origin mode at θ = 4 deg was 123 whereas that of the E1-symmetric mode was 61. So, the spectral width of the former should be one half of the latter, which agrees well with our theoretical results of Ref. [19].
On the other hand, the experimental observation of the Dirac-cone dispersion relation was difficult for the present specimens of the triangular-lattice PCs because the photonic bands were densely present and their spectra overlapped each other. The observation of accidental degeneracy of E1- and E2- symmetric modes, which materializes double Dirac cones [13], was particularly difficult. We need such a PC slab specimen that materializes a somewhat isolated E1-E2 mode pair for this purpose. Its fabrication remains as a future challenge. When it is materialized, the effective refractive index at the Dirac point is equal to zero, so we can expect peculiar phenomena like cloaking and propagation through sharply bent waveguides [11,14].
4. Conclusion
We presented accurate measurements of angle-resolved reflection spectra of triangular-lattice PC slabs fabricated in an SOI wafer by our home-made high-resolution set-up integrated inside of an FT-IR spectrometer. We demonstrated their dispersion relation and mode symmetries by combining the observed peak positions and the selection rules expected for the C6v-symmetric structure. Thus, we proved that the selection rules are a powerful tool to distinguish PC bands of different symmetry origins. Although accidental degeneracy of modes was not observed in the present specimens, an effective degeneracy of an E1-symmetric mode and an A1-symmetric mode was materialized, which resulted in the redistribution of the diffraction loss from the former to the latter. This feature was confirmed by observed spectral widths and calculated Q factors.
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.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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