Sufficient condition for the existence of interface states in some two-dimensional photonic crystals

Xueqin Huang, Meng Xiao, Zhao-Qing Zhang, and C. T. Chan

Phys. Rev. B 90, 075423 – Published 22 August 2014

Abstract

There is no assurance that interface states can be found at the boundary separating two materials. While a strong perturbation typically favors wave localization, we show on the contrary that in some two-dimensional photonic crystals (PCs) possessing a Dirac-like cone at k = 0 derived from monopole and dipoles excitation, a small perturbation is sufficient to create interface states. The conical dispersion together with the flat band at the zone center generates the existence of gaps in the projected band structure and the existence of single mode interface states inside the projected band gaps stems from the geometric phases of the bulk bands. The underlying physics for the existence of an interface state is related to the sign change of the surface impedance in the gaps above and below the flat band. The established results are applicable for long wavelength regimes where there is only one propagating diffraction order for an interlayer scattering.

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Received 6 March 2014
Revised 12 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.075423

Authors & Affiliations

Xueqin Huang, Meng Xiao, Zhao-Qing Zhang, and C. T. Chan^*

Department of Physics and Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

^*Corresponding author: phchan@ust.hk

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Issue

Vol. 90, Iss. 7 — 15 August 2014

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Images

Figure 1
(a) The bulk band structure for $ɛ = 12.5, R = 0.2 a$ . The linear bands meeting at k = 0 has a conical dispersion and a quasilongitudinal band that is flat along $Γ X$ . The inset is a schematic picture of a 2D PC consisting of an array of dielectric cylinders. (b) The projected band structure for $k_{| |}$ along the [01] direction. Bulk allowed states are shaded in red. The symbols UC, LC, and QL label the projected bands that are derived from the bulk bands in the upper cone, lower cone, and the quasilongitudinal bands, respectively. White marks gaps where surface states can form. The conical dispersion guarantees that there are gaps in the projected band structure above and below the bulk allowed states generated by the flat quasilongitudinal band (labeled by QL).
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Figure 2
(a) Schematic picture of an interface along the [01] direction constructed by two semi-infinite 2D PCs with a square lattice. The lattice constant for both PCs are $a$ . (b) The band structures of 2D PCs with parameters that are close to the conical dispersion condition at k = 0. The red circles are for the PC with $ɛ_{1} = 10, R_{1} = 0.205 a$ , the blue squares for the PC with $ɛ_{2} = 12.5, R_{2} = 0.22 a$ . (c) The projected band structures of these two PCs along the interface direction $(\bar{Γ} \bar{X})$ with red color for the PC with $ɛ_{1} = 10, R_{1} = 0.205 a$ and blue color for the PC with $ɛ_{2} = 12.5, R_{2} = 0.22 a$ . The green lines are the interface states. (d) The electric field distribution of the eigenmode of one interface state at the frequency $0.524 c / a$ and $k_{| |} = 0.6 π / a$ [labeled by a black star on the green line in (b)] computed by COMSOL. The eigenmode is seen to be localized at x = 0 which marks the interface separating two PCs each modeled by a 15-layer slab. In the COMSOL computation, periodic boundary condition is applied along the $y$ direction. The far ends of the unit cells are terminated by perfectly matched layers in the $x$ direction.
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Figure 3
The pass and forbidden band regions in the projected band structures along the interface direction $(\bar{Γ} \bar{X})$ labeled with the imaginary part of the surface impedance $[Im (Z)]$ , calculated using scattering theory (ignoring interlayer evanescent coupling, see text) for (a) $ɛ_{1} = 10, R_{1} = 0.205 a$ and (b) $ɛ_{2} = 12.5, R_{2} = 0.22 a$ . The purple color marks regions where higher order diffractions play an important role in scattering theory. (c) The projected band structures and interface states calculated by full-wave (COMSOL) calculation. Green lines represent the interface states determined using COMSOL. Black circles are calculated analytically by surface impedance shown in (a) and (b). The dashed line marks for $k_{| |} = 0.6 π / a$ .
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Figure 4
(a) Left panel denotes the Brillouin zone of a square lattice, the blue line represents $k_{x}$ running from $- π / a$ to $π / a$ with a fixed $k_{| |}$ . The right panel shows the coordinate for calculating the Zak phase, the origin is located on the left boundary of the unit cell. The bulk band structure with a fixed $k_{y} = k_{| |} = 0.6 π / a$ for the PC with (b) $ɛ_{1} = 10, R_{1} = 0.205 a$ and (c) $ɛ_{2} = 12.5, R_{2} = 0.22 a$ . The Zak phases of the bulk bands are labeled with green. The characters of the bulk gaps are labeled by the sign of $Im (Z)$ with blue color for $Im (Z) < 0$ and red color for $Im (Z) > 0$ .
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Figure 5
(a) The band structures of 2D PCs with parameters that are close to the conical dispersion condition at k = 0. The red circles are for the PC with $ɛ_{1} = 18, R_{1} = 0.16 a$ , and the blue squares for the PC with $ɛ_{2} = 18, R_{2} = 0.165 a$ . (b) The projected band structures of these two PCs along the interface direction $(\bar{Γ} \bar{X})$ with red color for the PC with $ɛ_{1} = 18, R_{1} = 0.16 a$ and blue color for the PC with $ɛ_{2} = 18, R_{2} = 0.165 a$ . The green line in the common band gap represents the interface states. (c) The electric field distribution of the eigenmode of one interface state at the frequency $0.544 c / a$ and $k_{| |} = 0.22 π / a$ [labeled by a black star on the green line in (b)] computed by COMSOL.
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Figure 6
(a) The band structures of 2D PCs with parameters that are close to the conical dispersion condition at k = 0. The red circles are for the PC with $ɛ_{1} = 12.5, R_{1} = 0.22 a$ , and the blue squares for the PC with $ɛ_{2} = 12.5, R_{2} = 0.225 a$ . (b) The enlarged projected band structures of these two PCs along the interface direction $(\bar{Γ} \bar{X})$ with red color for the PC with $ɛ_{1} = 12.5, R_{1} = 0.22 a$ and blue color for the PC with $ɛ_{2} = 12.5, R_{2} = 0.225 a$ . The green line in the common band gaps represents the interface states. (c) The electric field distribution of the eigenmode of one interface state at the frequency $0.496 c / a$ and $k_{| |} = 0.144 π / a$ [labeled by a black star on the green line in (b)] computed by COMSOL.
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Figure 7
(a) The band structures of 2D PCs with parameters that are close to the conical dispersion condition at k = 0. The red circles are the band structures for the PC with $ɛ_{1} = 10, R_{1} = 0.205 a$ , and the blue squares for the PC with $ɛ_{2} = 18, R_{2} = 0.17 a$ . For both of the band structures, the frequencies of dipole bands are higher than those of the monopole bands at k = 0. (b) The projected band structures of PCs along the interface direction $(\bar{Γ} \bar{X})$ with red color for the PC with $ɛ_{1} = 10, R_{1} = 0.205 a$ and blue color for the PC with $ɛ_{2} = 18, R_{2} = 0.17 a$ . There are five common band gaps in both of the two projected band structures. The green lines in the common band gaps represent the interface states at the interface created by the two PCs. Here $a$ is the lattice constant of the PCs.
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Figure 8
(a) The bulk band structures of two PCs with different parameters that are close to the conical dispersion condition at k = 0. The red circles are the band structures for the PC with $ɛ_{1} = 12.5, R_{1} = 0.22 a$ , and the blue squares for the PC with $ɛ_{2} = 20, R_{2} = 0.205 a$ . For both of the band structures, the frequencies of dipole bands are lower than those of the monopole bands at k = 0. (b) The projected band structures of PCs along the interface direction $(\bar{Γ} \bar{X})$ with red color for the PC with $ɛ_{1} = 12.5, R_{1} = 0.22 a$ and blue color for the PC with $ɛ_{2} = 20, R_{2} = 0.205 a$ . There are four common band gaps in both of the two projected band structures. The green lines in the common band gaps represent the interface states at the interface created by the two PCs. Here $a$ is the lattice constant of the PCs.
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Figure 9
(a) The system configuration used for calculating the reflection $(r)$ and transmission $(t)$ coefficients for a one-layer PC. The cylinders are at the center of unit cell at $x = a / 2$ . (b) The moduli of reflection $(| r |)$ and transmission $(| t |)$ coefficients and the imaginary part of surface impedance $[Im (Z_{N})]$ inside the band gap of a $N$ -layer PC for considering only zero-order interlayer diffraction as a function of the number of layers of the 1D PC with a fixed $k_{| |} = 0.6 π / a$ . In the $N$ -layer slab, each layer has the configuration shown in (a) with $ɛ_{1} = 12.5, R_{1} = 0.22 a$ . The working frequency is $0.524 c / a$ . Here $Z_{0}$ is the impedance of vacuum.
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Figure 10
The projected band structures of PC along the interface direction $(\bar{Γ} \bar{X})$ with $ɛ_{1} = 10, R_{1} = 0.205 a$ calculated by two different methods: (a) for full-wave calculation and (b) for multiple scattering theory with only zero-order interlayer diffraction. The purple region in (b) represents the higher-order diffractions play an important role in the multiple scattering theory.
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Figure 11
The electric field distributions of the eigenmodes at two high symmetry points for two bands in a unit cell. (a) The schematic of $k_{x}$ from $- π / a$ to $π / a$ with a fixed $k_{| |} in$ the Brillouin zone. The relative permittivity, permeability, and radius of the cylinders of the PC are $ɛ_{1} = 10, R_{1} = 0.205 a$ , respectively. Here $a$ is the lattice constant. $P$ and $Q$ points denote two high symmetry points in the Brillouin zone. (b), (c) and (d), (e) are the electric field distributions of the eigenmodes in $P$ and $Q$ points for the quasilongitudinal band (the third band in the band structure shown in Fig. 4), respectively. (f), (g) and (h), (i) are the electric field distributions of the eigenmodes in $P$ and $Q$ points for the lower band (the second band in the band structure shown in Fig. 4), respectively. The origin of the coordinate is located at the center of the cylinder.
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Figure 12
(a) The left panel shows the 2D Brillouin zone. The blue line marks the reduced 1D zone along the $k_{x} direction$ for a fixed $k_{y} = k_{| |}$ with the $k$ points $P$ and $Q$ at the center and the boundary, respectively. The right panel shows one unit cell of the 2D PC with the origin of the coordinate at the center of the unit cell. (b) The bulk band structure of the PC with $ɛ = 10, R = 0.205 a$ . The two dashed black lines denote wave vector $k_{y} = k_{| |} = 0.2 π / a$ along the $Γ X$ and $X^{'} M$ directions, respectively. $P_{1}$ , $P_{2}$ , and $P_{3}$ denote the eigenmodes in the first, second, and third bands with wave vector at $P$ point shown in (a) with $k_{y} = k_{| |} = 0.2 π / a$ , respectively. $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ denote the eigenmodes in the first, second, and third bands with wave vector at $Q$ point shown in (a) with $k_{y} = k_{| |} = 0.2 π / a$ , respectively. (c) The Zak phases of the three lowest bands for the PC with $ɛ = 10, R = 0.205 a$ and $k_{| |} = 0.2 π / a$ are labeled with green. Here the origin of the coordinate for calculating the Zak phase is located in the middle of two neighbor cylinders (the same as Fig. 4). The characters of the bulk gaps are labeled by the sign of $Im (Z)$ with blue color for $Im (Z) < 0$ and red color for $Im (Z) > 0$ . (d) The schematic diagram showing the projected bands of two PCs along the $\bar{Γ} \bar{X}$ direction with two Dirac-like cones with a small frequency deviation. There is a common gap (between two dashed black lines) between the projected bands of the two flat bands.
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