Dirac-cone photonic surface states in three-dimensional photonic crystal slab

Wei Zhong; Xiangdong Zhang

doi:10.1364/OE.19.013738

1. Introduction

In the past two decades, a great deal of effort has been devoted to the study of photonic crystals [1–5]. Since a photonic crystal (PC) can have a spectral gap in which electromagnetic wave propagation is forbidden in all directions, it offers the possibility of controlling the flow of photons in a way analogous to electrons in a semiconductor. It has potential applications in optoelectronic devices. Surface states can exist in truncated photonic crystal with the surface terminated by incomplete as well as complete cells [6–13]. The existence of surface states can directly affect the performance and efficiency of the PC in applications. Thus, it is important to study them. Since the pioneering discuss of surface modes of truncated three-dimensional (3D) PC by Meade et al. [6], many investigations have been done [7–13]. Recently, the manipulation of photons at the surface of 3D PC has been demonstrated experimentally [14].

On the other hand, topological insulators have been subject of recent studies because of their fundamental importance and great potential for future applications [15–18]. Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator, but have a Dirac-cone surface state. Electrons on their surfaces can be described by the relativistic Dirac equation for massless fermions and exhibit a host of unusual properties [15–26]. Some materials from bulk to thin film such as ${Bi}_{x} {Sb}_{1-x}$ , ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$ have been experimentally and theoretically demonstrated as three-dimensional topological insulators [19–26]. However, all these investigations are for electrons. The problem is whether or not the materials can be constructed to observe Dirac-cone surface states for photons on their surfaces?

Motivated by such a problem, in this work we develop a multiple-scattering method in conjunction with supercell calculations to study the electromagnetic surface states in 3D photonic PC slab. In contrast to the plane-wave expansion in conjunction with supercell method [6], our technique can handle complex structure including metal elements exactly and reach rapid convergence. Using our technique, we have investigated various 3D photonic structures and found that the Dirac-Cone photonic surface states can be observed in some 3D PC slabs.

2. Theory

We consider a 3D PC slab consisting of microspheres embedded in a homogeneous medium with the permittivity ε and the magnetic permeability μ. The slab sample is infinite in the x and y direction and of finite thickness in the z direction. Since the system are periodic in the x and y directions, the problem can be reduced to a supercell calculation. Thekey point of the supercell method is to design an appropriate auxiliary infinite periodic superstructure in order to apply the Bloch theorem. The auxiliary superstructure is formed by the infinite periodic translation of the supercell along both x and y axes. For example, Fig. 1 (a) describes the schematic picture of the 3D slabs in diamond [111] planes with 7 monolayers. The xy plane is the [111] surface of the structure and the z axes corresponds to the normal direction of the [111] plane. Here “A”, “B” and “C” denote the different monolayer number. The monolayer consists of microspheres in triangular lattice with lattice constant a, and the primitive unit cell of the monolayer only includes one sphere. Every monolayer possesses the same pattern, only malpositions are arranged among the “A”, “B” and “C” monolayers. So thesupercell of the present structure contains N=7 spheres. The pink lines mark the nearest neighbors of the spheres. Such a structure can be seen more clearly from the comparison with the schematic structure of the diamond in the textbook as shown in Fig. 1(b). The light gray plane in Fig. 1(b) is the [111] plane of the diamond, and the Cartesian-coordinates inside the figure (red mark) correspond to those in Fig. 1 (a). That is to say, the schematic picture in Fig. 1(a) is another display of the diamond structure shown in Fig. 1(b). The Brillouin zone for this lattice is shown in Fig. 1(c). There are two kinds of surface in the present structure, which correspond to the upper and lower surface of the supercell in Fig. 1 (a), respectively. The upper surface in Fig. 1 (a) (CAABB…) is called type I in the present work, while the lower surface (CCBB….) is regarded as type II. If we assume that the relative location of the jth sphere in the supercell is ${\vec{δ}}_{j}$ and consider a plane electromagnetic wave of angular frequency with the electric field component $\vec{E} (r, t) = Re [\vec{E} (r) \exp (- i ω t)]$ incident on the system, the total scattered field can be given by using the Bloch theorem [27–29]:

\begin{array}{l} {\vec{E}}_{s c} (\vec{r}) = \sum_{j = 1}^{N} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [\frac{i}{q} b_{j l m}^{+ E} \nabla \times \sum_{{\vec{R}}_{n}} \exp (i \vec{k} • {\vec{R}}_{n}) h_{l}^{+} (q r_{n j}) {\vec{X}}_{l m} ({\hat{r}}_{n j}) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} + b_{j l m}^{+ H} \sum_{{\vec{R}}_{n}} \exp (i \vec{k} • {\vec{R}}_{n}) h_{l}^{+} (q r_{n j}) {\vec{X}}_{l m} ({\hat{r}}_{n j})] \end{array}

with a corresponding expression for

{\vec{H}}_{s c} (\vec{r})

obtained according to the transformation:

E \to H

,

H \to E

and

ε \to - μ

. Where

q = \sqrt{ε} ω / c

and

{\vec{r}}_{n j} = \vec{r} - ({\vec{R}}_{n} + {\vec{δ}}_{j})

, c is the velocity of light in vacuum,

\vec{k}

is the Bloch vector and

{\vec{R}}_{n}

represents a two-dimensional (Bravais) lattice vector;

h_{l}^{+}

is the spherical Hankel functions,

{\vec{X}}_{l m}

are vector spherical harmonics;

b_{j l m}^{+ P}

(P = E, H) are the scattered coefficients of the jth sphere in the supercell, which are determined by the incident plane wave and the scattered wave from all the other spheres in the system. The wave scattered from all the other spheres can be expanded into a series of incident vector spherical waves around the

j^{'} t h

sphere as [27, 28]

{\vec{E}}_{j^{'} s c}^{'} (\vec{r}) = \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (\frac{i}{q} {b^{'}}_{j^{'} l m}^{E} \nabla \times j_{l} (q r_{n j^{'}}) {\vec{X}}_{l m} ({\hat{r}}_{n j^{'}}) + {b^{'}}_{j^{'} l m}^{H} j_{l} (q r_{n j^{'}}) {\vec{X}}_{l m} ({\hat{r}}_{n j^{'}}))

and a corresponding expression for the magnetic field can also be obtained similarly. The

{b^{'}}_{j l m}^{P}

coefficients in the above expressions are to be determined by the following equation [27–29]:

\sqrt{l (l + 1)} {b^{'}}_{j l m}^{E} = - \frac{q}{j_{l} (q r)} \int \vec{r} • {\vec{E}}^{'}_{j s c} Y_{l m}^{*} (\hat{Ω}) d \hat{Ω},

\sqrt{l (l + 1)} {b^{'}}_{j l m}^{H} = \frac{q}{j_{l} (q r)} \int \vec{r} • {\vec{H}}^{'}_{j s c} Y_{l m}^{*} (\hat{Ω}) d \hat{Ω} {(\frac{ε ε_{0}}{μ μ_{0}})}^{- 1},

where

Y_{l m} (\hat{Ω})

denotes as usual a spherical harmonic, inserting Eq. (1) into the above relations and using the addition theorems:

\begin{array}{l} h_{l}^{(1)} (q | \vec{r} - {\vec{R}}_{j} |) Y_{l m} (\hat{Ω} (\vec{r} - {\vec{R}}_{j})) \\ = \sum_{l^{'} = 0}^{\infty} \sum_{m^{'} = - l^{'}}^{l^{'}} g_{l m, l^{'} m^{'}} ({\vec{R}}_{i} - {\vec{R}}_{j}) j_{l^{'}} (q | \vec{r} - {\vec{R}}_{i} |) Y_{l^{'} m^{'}} (\hat{Ω} (\vec{r} - {\vec{R}}_{i})) \end{array}

With

\begin{array}{l} g_{l m, l^{'} m^{'}} ({\vec{R}}_{i} - {\vec{R}}_{j}) = \sum_{l^{″} = 0}^{\infty} \sum_{m^{″} = - l^{″}}^{l^{″}} 4 π {(- 1)}^{(l - l^{'} - l^{″}) / 2} {(- 1)}^{m^{'} + m^{″}} h_{l^{″}}^{(1)} (q | {\vec{R}}_{i} - {\vec{R}}_{j} |) Y_{l^{″} - m^{″}} (\hat{Ω} ({\vec{R}}_{i} - {\vec{R}}_{j})) \\ • \int Y_{l m} (\hat{Ω}) Y_{l^{″} m^{″}} (\hat{Ω}) Y_{l^{'} - m^{'}} (\hat{Ω}) d Ω \end{array}

we can obtain

Fig. 1 (a) Schematic structure of the 3D photonic crystal slab and supercell with 7 monolayers. Here A, B and C denote the different monolayers in the supercell. The pink lines mark the nearest neighbors of the spheres. (b) Schematic structure of the diamond, and the light gray plane is the (111) plane of the diamond. The Cartesian-coordinates inside the figure (red marks) correspond to those in (a). (c) The first Brillouin zone (BZ) of the diamond structure and the corresponding surface 2D BZ. W, K and L represent the symmetric points in the 3D BZ; $\bar{Γ}$ , $\bar{K}$ and $\bar{M}$ are the symmetric points in the surface 2D BZ.

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{b^{'}}_{j l m}^{P} = \sum_{j^{'} = 1}^{N} \sum_{P^{'} = E, H} \sum_{l^{'} m^{'}} Ω_{j l m, j^{'} l^{'} m^{'}}^{P P^{'}} b_{j^{'} l^{'} m^{'}}^{+ P^{'}} .

Here the propagator $Ω_{j l m, j^{'} l^{'} m^{'}}^{P P^{'}}$ can be expressed as

\begin{array}{l} Ω_{j l m; j^{'} l^{'} m^{'}}^{E E} = Ω_{j l m; j^{'} l^{'} m^{'}}^{H H} \\ = {(ψ_{l} ψ_{l^{'}})}^{- 1} • [2 α_{l}^{- m} α_{l^{'}}^{- m^{'}} Z_{j^{'} l^{'} m^{'} - 1; j l m - 1} + m m^{'} Z_{j^{'} l^{'} m^{'}; j l m} + 2 α_{l}^{m} α_{l^{'}}^{m^{'}} Z_{j^{'} l^{'} m^{'} + 1; j l m + 1}], \end{array}

\begin{array}{l} Ω_{j l m; j^{'} l^{'} m^{'}}^{H E} = - Ω_{j l m; j^{'} l^{'} m^{'}}^{E H} \\ = (2 l + 1) {(ψ_{l} ψ_{l^{'}})}^{- 1} • [\begin{array}{l} - 2 α_{l^{'}}^{- m^{'}} γ_{l}^{m} Z_{j^{'} l^{'} m^{'} - 1; j l - 1 m - 1} + m^{'} ζ_{l}^{m} Z_{j^{'} l^{'} m^{'}; j l - 1 m} \\ + 2 α_{l^{'}}^{m^{'}} γ_{l}^{- m} Z_{j^{'} l^{'} m^{'} + 1; j l - 1 m + 1} \end{array}], \end{array}

with

ψ_{l} = \sqrt{l (l + 1)},

α_{l}^{m} = \frac{1}{2} \sqrt{(l - m) (l + m + 1)},

γ_{l}^{m} = \frac{1}{2} {[(l + m) (l + m - 1)]}^{1 / 2} / {[(2 l - 1) (2 l + 1)]}^{1 / 2},

ζ_{l}^{m} = {[(l + m) (l - m)]}^{1 / 2} / {[(2 l - 1) (2 l + 1)]}^{1 / 2},

Z_{j l m, j^{'} l^{'} m^{'}} = \sum_{{\vec{R}}_{n}}^{†} g_{l m, l^{'} m^{'}} ({\vec{δ}}_{j} - {\vec{δ}}_{j^{'}} - {\vec{R}}_{n}) e^{i \vec{k} • {\vec{R}}_{n}},

\begin{array}{l} g_{l m, l^{'} m^{'}} (\vec{r}) = \sum_{l^{″} = 0}^{\infty} \sum_{m^{″} = - l^{″}}^{l^{″}} 4 π {(- 1)}^{(l - l^{'} - l^{″}) / 2} {(- 1)}^{m^{'} + m^{″}} h_{l^{″}}^{(1)} (q r) Y_{l^{″} - m^{″}} (\hat{Ω} (\vec{r})) \\ • \int Y_{l m} (\hat{Ω}) Y_{l^{″} m^{″}} (\hat{Ω}) Y_{l^{'} - m^{'}} (\hat{Ω}) d Ω . \end{array}

In Eqs. (14) and (15), there is a problem of lattice sum, namely “structure constants”, which is given in the Appendix [30,31]. If the external incident field is expanded in vector spherical waves and the expansion coefficients are characterized by $a_{j l m}^{0 P}$ [27–29], we have the Rayleigh identities

b_{j l m}^{+ P} = T_{j l m}^{P} (\sum_{j^{'} = 1}^{N} \sum_{P^{'} = E, H} \sum_{l^{'} m^{'}} Ω_{j l m, j^{'} l^{'} m^{'}}^{P P^{'}} b_{j^{'} l^{'} m^{'}}^{+ P^{'}} + a_{j l m}^{0 P}),

where

T_{j l m}^{P}

are the elements of the scattering matrix by the single microsphere, which can be obtained analytically. This is the basic equation for the present multiple-scattering system. The normal modes of the system may be obtained by solving the following secular equation in the absence of an external incident wave:

\det | δ_{P P^{'}} δ_{j j^{'}} δ_{l l^{'}} δ_{m m^{'}} - \sum_{l^{″} m^{″} P^{″}} Ω_{j l m, j^{″} l^{″} m^{″}}^{P P^{'}} T_{j l m, j^{'} l^{'} m^{'}}^{P} | = 0.

Here $T_{j l m, j l^{'} m^{'}}^{P} = T_{j l m}^{P} δ_{l l^{'}} δ_{m m^{'}}$ for isotropic sphere. Based on such an equation, the dispersion relation of the 3D PC slab including bulk modes and surface states can be obtained exactly through the numerical calculations. In our method, we do not take a supercell in which slabs of 3D PC alternate with slabs of vacuum as that in Ref [6]. Thus, some spurious unphysical solutions due to unphysical assumption of periodical supercell can be avoided.

3. Numerical results and discussion

Firstly, we consider the bulk band structure of 3D PC composed of metallo-dielectric spheres. An important property of the PC composed of metallo-dielectric spheres is that a complete photonic band gap is easily found in such a structure [32–34]. Figure 2 (a) shows the photonic band structures for the metallo-dielectric spheres arranged in the diamond structures. The lattice constant of the diamond is $a_{0} = \sqrt{2} a$ . The filling ratio is 0.34 and the background is assumed to be air. The metallo-dielectric spheres have metal cores modeled by dielectric constant $ε = - 200$ [33, 34] and an encapsulating $ε = 12.96$ dielectric coating with thickness equal to 2% of the sphere radius. As can be seen, such a PC possesses a complete band gap in the reduced frequency ranging from about 0.29 $ω a / 2 π c$ to 0.66 $ω a / 2 π c$ . This gap is wide enough and its lower edge appears in the very low frequency of 0.29 $ω a / 2 π c$ , therefore, the sample is a good candidate to support the surface modes.

Fig. 2 (a) Calculated photonic band structures of a 3D diamond structure consisting of metallo-dielectric spheres in air, the filling ratio is $f = 0.34$ , the metal core is modeled by $ε = - 200$ , and coating layer is 2% in radius with a dielectric constant $ε = 12.96$ . (b) The corresponding in-plane band structures for the 3D PC slab with type I surfaces and ten-monolayer thickness. Red dot lines represent the surface modes, green solid lines are the lightlines. (c) The intensity distribution of the magnetic-field for a surface state in the supercell of the yz plane for the $\bar{Γ} - \bar{M}$ direction and a frequency of f = 0.466 $ω a / 2 π c$ (blue=low, red=high).

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To test such a conjecture, we calculate the band structures of the corresponding 3D PC slab with different thickness based on the above method. The calculated results for the in-plane band structure with type I surfaces and ten-monolayer thickness are plotted in Fig. 2(b). The green solid lines are the lightlines. Two eigenmodes (red dotted lines) inside the gap are observed clearly below the lightline. For further analysis of these modes, we investigate the eigenfield distribution of the modes. The spatial intensity distribution of the magnetic-field of the surface state in a supercell corresponding to the frequency 0.466 $ω a / 2 π c$ (S point in Fig. 2 (b)) is shown in Fig. 2(c), which is in the yz plane for the $\bar{Γ} - \bar{M}$ direction. It is seen clearly that the field is localized strongly to the surface plane of the slab, which is exponentially decaying in the air region and inside the crystal slab. This confirmed that the eigenmodes inside the gap as shown in Fig. 2(b) represent actually the surface states. The above calculations focus on only a special choice of the dielectric constant for the metallic component. In fact, the phenomena are not sensitive to such a choice. We have also adopted other values, i.e., $ε = - 10^{3} \sim 10^{4}$ or Drude-like dielectric constant for the metallic component, similar phenomena can be obtained by changing the thickness of the dielectric coating for the metallo-dielectric spheres. Without loss of generality, in the following discussions we only take a concrete value of the dielectric constant for the metallic component.

Such surface states are related to the thickness of the slab. It is a suitable sample that the crystal slab is thick enough to ensure that the surface modes at any two interfaces do not interact with each other, the dispersion curve obtained is expected to converge to the dispersion curve of the surface modes of the semi-infinite crystal. For example, when the layer number of the slab is bigger than 8 monolayers for the present system, the dispersion curves of the surface modes can converge to the results of the semi-infinite crystal within the present accuracy of our calculations (the percentage error introduced in eigenmodes is less than 0.1%). In addition, these surface states also depend on the properties of scatters and structure of the PC. Although different kinds of surface states can be obtained by tuning the structure and properties of the scatters, Dirac-cone like surface states cannot be found in such a case. However, the situation can become different if we consider the case with the type II surfaces.

Figure 3 shows in-plane band structures of the 3D PC slab with the type II surfaces and ten-monolayer thickness. Here the dielectric coating layer of the sphere is 10% in radius. The other parameters are identical with those in Fig. 2. The surface state with two linear dispersion branches below the 0.54 $ω a / 2 π c$ at the $\bar{K}$ point is observed. This is a kind of surface state with a single Dirac cone. Recently, the surface states consisting of a single Dirac cone for the electron system have been observed in some ternary compounds [35]. In fact, the present case can be regarded as an optical analogue to the case of such an electron system. These results are for the case of perfect surface. If we modify the scatters on the surface layers of the slab, many more phenomena can be observed.

Fig. 3 in-plane band structures for the 3D PC slab with type II surfaces and ten-layer thickness in diamond structure consisting of metallo-dielectric spheres in air. The coating layer is 10% in radius. The other parameters are identical with those in Fig. 2. Red dot lines represent the surface modes and green solid lines are the lightlines.

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Figure 4 (a), (b), (c), (d), (e), (f), (g) and (h) displays the evolution of the surface states when the surface layers are modified. The in-plane band structure of the 3D PC slab without modifying the surface layers is plotted in Fig. 4 (a). Here two surfaces of the slab are taken as type II and the other parameters are identical with those in Fig. 2. For such a case, the eigenmodes are similar to those in Fig. 3, the surface state with double Dirac cones cannot be found. However, the situation can be changed by tuning the thickness of the dielectric coating of the metallo-dielectric spheres on the surface layers when the filling ratio keeps unchanged. Figure 4 (b), (c), (d), (e), (f), (g) and (h) correspond to the cases with the thickness of dielectric coating 10%, 20%, 30%, 40%, 50%, 60% and 65% of the sphere radius, respectively. It is shown clearly that many more surface states appear with increase of the thickness of dielectric coating. This is because the property of the defect becomes stronger with the increase of the dielectric coating. When the thickness of dielectric coating is 30%, two surface modes touch as a pair of cones around 0.48 $ω a / 2 π c$ at $\bar{K}$ point. Such a conical singularity can be referred to as the Dirac point similar to the case of photonic graphene [36–41]. These surface states are robust against further modifying. For example, the surface state with double Dirac cones still exists when the thickness of dielectric coating changes from 30% to 60% (see Fig. 4(e), (f) and (g)), only the frequency of the Dirac point shifts. Owing to the linear Dirac-like energy dispersion for the photon on the surface, photon transport near the Dirac point on the surface of the sample can be described by the Dirac equation. This is similar to the case of electron on the surface of topological insulators.

Fig. 4 The in-plane band structures for the 3D PC slab with type II surfaces and ten-layer thickness in diamond structure consisting of metallo-dielectric spheres in air under the modification of the surface layers. (a) corresponds to the case without being modified; (b), (c), (d), (e), (f), (g) and (h) correspond to the cases with the thickness of the dielectric coating 10%, 20%, 30%, 40%, 50%, 60% and 65% of the sphere radius, respectively. The other parameters are identical with those in Fig. 2. Red dot lines represent the surface modes (all in the gap region) and green solid lines are the lightlines.

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In order to demonstrate such a phenomenon, we perform a numerical simulation of the wave transport in the finite sample by using the multiple-scattering method with the incidence of a fundamental Gaussian beam. We select a finite PC slab consisting of 16 monolayers, which has 10a length (y direction), 11a width (x direction). The Gaussian beam is focused at the center of the sample from the left side and propagates in y direction with the electric field polarized along z axis. The numerical aperture (NA) of the Gaussian beam is taken as NA = 0.3. By choosing various frequencies of the Gaussian beam, we can examine whether the guide modes exist, which corresponds to the surface states obtained above. Our calculated results identify that all modes obtained above are physical. For example, Fig. 5 (a) and (b) display the electric field intensity patterns in yz (x=0) plane and xy plane (the surface of the sample), respectively. The frequency of the Gaussian beam is taken as 0.43 $ω a / 2 π c$ corresponding to one surface mode as shown in Fig. 2.(b) and the other parameters of the sample are also identical with those used in Fig. 2. The guide mode of the wave on the surface of the sample along y direction is clearly seen from the distributions of the field intensity. If we take 0.54 $ω a / 2 π c$ in the gap region, the field intensity along propagating direction is suppressed exponentially, such a case can be found in Fig. 5 (c) and (d).

Fig. 5 Distributions of the electric field intensity on the surface of 3D finite PC sample with 16 layers under the incidence of a fundamental Gaussian beam with numerical aperture NA=0.3. The length (y direction), width (x direction) of the sample are taken as 10a, 11a respectively. (a), (c) and (e) correspond to the field patterns in yz plane, and (b), (d) and (f) to those in xy plane. (a) and (b) at 0.43 $ω a / 2 π c$ for the structure as shown in Fig. 2; (c) and (d) at 0.54 $ω a / 2 π c$ for the structure as shown in Fig. 2; (e) and (f) at 0.481 $ω a / 2 π c$ (Dirac point) for the structure as shown in Fig. 4 (d).

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In contrast, the propagating behaviors of waves near the Dirac point are different from the above two kinds of case. Figure 5 (e) and (f) show the electric field intensity patterns at 0.481 $ω a / 2 π c$ corresponding to the Dirac point as shown in Fig. 4.(d). We find that the field intensity for such a case decreases linearly along the propagating direction which is similar to the diffusion behavior of waves through a random medium. In the previous investigations [36–41], the people have found that such a phenomenon appears in 2D photonic graphene when the frequency is near the Dirac point, and demonstrated that the transport of the wave at such a case can be described by the Dirac equation. The present phenomenon is similar to that although the waves only propagate on the surface of the sample. The above discussions are only for the theoretical results. At end, we would like to give some experimental remarks on such a problem. In fact, 3D PC slab consisting of metallo-dielectric spheres can be fabricated experimentally according to the method described in Ref [33], and the physical properties of the surface states can be explored experimentally by using the method in Ref [14].

4. Conclusion

The multiple-scattering method in conjunction with supercell calculations to study the electromagnetic surface states in 3D photonic crystal slab has been developed. The photonic surface states with both single and double Dirac Cones have been found in some 3D photonic crystal slabs, which are similar to the case of the electron on the surface of topological insulators. Our findings contain two aspects of implication. On the one hand, photonic crystal slab can offer a particularly clean and controlled way to test some predictions from electron topological insulators experimentally. The experimental test in the electronic case is severely hindered by the difficulty to maintain a homogeneous electron density throughout the system. No such difficulty exists in the photonic crystal. On the other hand, our findings open the possibility to investigate the phenomena of relativistic quantum mechanics on the PC surface at will, by using optical wave and extensive application of the phenomena to optic devices.

Appendix

In this appendix, we provide the explicit expressions for “structure constants” by Ewald’s treatment of lattice sums for the present slab system. From Eqs.(14) and (15), it is rewritten as

D_{l m} = \sum_{{\vec{R}}_{n}} \begin{matrix} † \end{matrix} e^{i \vec{k} • {\vec{R}}_{n}} h_{l} (q | \vec{δ} - {\vec{R}}_{n} |) Y_{l m} (\hat{Ω} (\vec{δ} - {\vec{R}}_{n})) .

The † means that ${\vec{R}}_{n} = \vec{0}$ should be omitted in the series when $\vec{δ} = \vec{0}$ . Here we use the Ewald’s methods to calculate the lattice sums [30]. Ewald’s methods are fairly standard in the solid state physics where they are used to evaluate structure constants in electron scattering theory [30]. The case of the 2D periodicity in 3-dimensional system has been treated by K. Kambe [31]. For the present system, the “structure constants” can be given by using the Ewald’s methods [31]:

D_{l m} = D_{l m}^{(1)} + D_{l m}^{(2)} + D_{l m}^{(3)} .

Where

\begin{array}{l} D_{l m}^{(1)} = \frac{1}{i q {(- 1)}^{l - m}} \frac{1}{A q^{l}} \frac{i^{m + 1}}{2^{l}} {[(2 l + 1) (l - | m |)! (l + | m |)!]}^{\frac{1}{2}} \\ • \sum_{{\vec{K}}_{n}} e^{i (\vec{k} + {\vec{K}}_{n}) • {\vec{δ}}_{x y} - i m φ_{\vec{k} + {\vec{K}}_{n}}} • \sum_{n = 0}^{l - | m |} \frac{1}{n!} {(Γ_{K})}^{2 n - 1} Δ_{K}^{n} \sum_{s = n}^{\min (2 n, l - | m |)} (\begin{matrix} n \\ 2 n - s \end{matrix}) \frac{δ_{z}^{2 n - s} {(| \vec{k} + {\vec{K}}_{n} |)}^{l - s}}{(\frac{l - | m | - s}{2})! (\frac{l + | m | - s}{2})!}, \end{array}

D_{l m}^{(2)} = - \frac{i}{q} \frac{1}{\sqrt{2 π}} \sum_{{\vec{R}}_{n}} \begin{matrix} † \end{matrix} e^{i \vec{k} • {\vec{R}}_{j}} Y_{l m} (\hat{Ω} (\vec{δ} - {\vec{R}}_{n})) \frac{{| \vec{δ} - {\vec{R}}_{n} |}^{l}}{q^{l}} \int_{\frac{1}{η}}^{\infty} ξ^{l - \frac{1}{2}} e^{- \frac{1}{2} ({| \vec{δ} - {\vec{R}}_{n} |}^{2} ξ - \frac{q^{2}}{ξ})} d ξ,

D_{l m}^{(3)} = [- \frac{1}{\sqrt{4 π}} - \frac{i}{q π} \sum_{j = 0}^{\infty} {(\frac{1}{2 η})}^{\frac{1}{2}} \frac{{(\frac{q^{2} η}{2})}^{j}}{j! (2 j - 1)}] δ_{l, 0} δ_{m, 0} δ_{\vec{δ}, 0},

where A represents the area of the super-cell in x-y plane, $Γ_{K} = \sqrt{q^{2} - {| \vec{k} + {\vec{K}}_{n} |}^{2}}$ and $\vec{δ} = {\vec{δ}}_{x y} + δ_{z} \vec{z}$ is the relative position of the sphere in the supercell.

Δ_{K}^{n} = \int_{e^{- π i} \frac{Γ_{K}^{2}}{2} η}^{\infty} ξ^{- \frac{1}{2} - n} e^{- ξ + \frac{Γ_{K}^{2} δ_{z}^{2}}{4 ξ}} d ξ .

From the expressions of Eqs.(A16-18), we find that the value of $D_{l m}$ seems to be dependent on the parameter η. In fact, the calculated results show that the $D_{l m}$ is independent of the value of η. When we choose a appropriate value for η, $D_{l m}^{(1)}$ and $D_{l m}^{(2)}$ are roughly equal which are independent of it.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.10825416) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.

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Dirac-cone photonic surface states in three-dimensional photonic crystal slab

Abstract

1. Introduction

2. Theory

3. Numerical results and discussion

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (23)

Optics Express