Photonic band structures in one-dimensional photonic crystals containing Dirac materials

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Photonic band structures in one-dimensional photonic crystals containing Dirac materials

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Lin Wang, Li-Gang Wang

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Department of Physics, Zhejiang University, Hangzhou 310027, China

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a r t i c l e

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Article history: Received 6 March 2015 Accepted 8 May 2015 Available online xxxx Communicated by R. Wu

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We have investigated the band structures of one-dimensional photonic crystals (1DPCs) composed of Dirac materials and ordinary dielectric media. It is found that there exist an omnidirectional passing band and a kind of special band, which result from the interaction of the evanescent and propagating waves. Due to the interface effect and strong dispersion, the electromagnetic fields inside the special bands are strongly enhanced. It is also shown that the properties of these bands are invariant upon the lattice constant but sensitive to the resonant conditions. © 2015 Published by Elsevier B.V.

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1. Introduction

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Photonic crystals (PCs) have intrigued considerable attention in several decades due to unique electromagnetic properties and wide potential applications [1–3]. It is well known that the essential property of PCs is the photonic band gap (PBG) which is the result of the interference of Bragg scattering in periodical structures. Conventional Bragg gaps vary with respect to the lattice constant, the incident angle, and polarization, and defect modes are highly localized in defect layers. In the last decade, two new types of gaps, the zero-averaged refractive index gaps [4–6] and the effective zero-phase gaps [7–10], are realized in the PCs with left-handed metamaterials [11] and single-negative meta-materials [12,13], respectively. These gaps are also recognized as omnidirectional gap, which is insensitive to the change of scale length and lattice disorder. Up to now, the PCs based on the PBGs have provided various methods to manipulate photons more effectively and flexibly. On the other hand, many recent investigations show the existence of the double-conical band structures for photons according to the analogy between electronic band structures in graphene and photonic band structures in two-dimensional PCs [14–20]. The touching point in double-conical band structures is called as Dirac point (DP), and near the DP, the dispersion is linear [21]. Recently, one finds that the DP with a double-cone structure for optical fields is realizable in negative–zero–positive index metamaterials (NZPIMs) [22], and several novel optical transport properties near the DP are also demonstrated, such as Zitterbewegung effect

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E-mail address: [email protected] (L.-G. Wang). http://dx.doi.org/10.1016/j.physleta.2015.05.015 0375-9601/© 2015 Published by Elsevier B.V.

[23–26], optical nonlocality [27], tunable transmission gaps, Bragglike reflections, and negative or positive GH shifts [28–30]. Motivated by these studies, here our aim is to investigate the band structures of 1DPCs containing Dirac media (which refers to the bulk materials possessing a DP). We find that there are two kinds of passing bands (an omnidirectional narrow passing band and a special band) which result from the interaction of the evanescent and propagating waves. Due to the interface effect, the electromagnetic fields inside the special bands are strongly enhanced. The properties of the special bands are sensitive to the resonant conditions but invariant upon the lattice constant. The whole paper is organized as follows. In Section 2, it presents the theoretical model and formula on the wave propagating in the 1DPCs. In Section 3, the properties of the photonic bands and gaps, the transmission and the electromagnetic fields inside the 1DPCs containing Dirac materials are analyzed. Last, a summary is given in Section 4.

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2. Theoretical model and formula

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For simplicity, consider a 1DPC of the structure ( A B ) N with N the number of the period, as shown in Fig. 1. Assume that the layers A with thickness d A are made of NZPIMs, whose relative permittivity and permeability satisfy the Drude model as follows,

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2 ε A = a − ωep /(ω2 + i γe ω),

(1)

2 mp /(

(2)

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where ωep and ωmp are the controllable electronic and magnetic plasma frequencies, respectively, and γe and γm stand for the corresponding damping rates (relating to the absorption). In

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μA = b − ω

2

ω + i γm ω),

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Fig. 1. Schematic representation of a periodic structure ( A B ) , where A denotes NZPIMs and B denotes the vacuum or dielectric media.

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our calculation, without loss of generality, we assume a = b = 1, γe = γm = γ = 10−5 GHz  ωep,mp , and ωep = ωmp = ω D . Under these parameters, material A has almost the linear dispersion with a DP at frequency ω D . The layers B can be the vacuum or dielectric media. For simplicity, here we take the layers B as vacuum with thickness d B . Let a plane wave be injected from vacuum into the 1DPC at an incident angle θ . From the transfer matrix method [31,32], the electric and magnetic fields between any two positions z and z +  z within the same layer are related via a transfer matrix



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M j ( z, ω) =

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j

where k z =

j

cos[k z  z] j



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k j = k0

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(3)

,

√ √

k2j

>

k2y ,

otherwise

j kz

=i



k2y

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− k2j ;

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ε j μ j is the corresponding wave vector in materials, j j q j = k z /(μ j k0 ) for TE wave and q j = k z /(ε j k0 ) for TM wave, k0 = ω/c the wave vector in the vacuum, and c is the light speed

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in the vacuum. By using the boundary conditions, the reflection and transmission coefficients for a finite 1DPC can also be obtained from the transfer matrix method [31,33,34]

[q0 x22 − q s x11 ] − [q0 q s x12 − x21 ] r (θ, ω) = , [q0 x22 + q s x11 ] − [q0 q s x12 + x21 ] t (θ, ω) =

2q0

[q0 x22 + q s x11 ] − [q0 q s x12 + x21 ]

,

(4) (5)

where q0 = q s = cos θ for the vacuum of the space z < 0 (before the incident end) and the space z > L (after the exit end), and L is the total length of the 1DPC. In the above, xi j are the matrix elements of the total matrix X N (ω) =

2N  j =1

M j (d j , ω), which repre-

sents the total transfer matrix of the fields propagating from the incident to exit ends. For an infinite periodic structure ( N → ∞), based on the Bloch’s theorem, the dispersion at any angle of incidence obeys the following relation

cos[βz ] =

1 2

Tr[ M A M B ]

= cos[k zA d A ] cos[k zB d B ] 1 qA

− (

2 qB

+

qB qA

) sin[k zA d A ] sin[k zB d B ],

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vector in the jth layers for

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cos[k z  z]

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k2j − k2y ( j = A , B) is the z component of the wave

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iq j sin[k z  z]

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j

i q1 sin[k z  z]

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(6)

where  = d A + d B is the lattice constant of the 1DPC, and βz  is the reduced phase shift of the propagating fields along the z direction. The condition of the real solution for βz requires | cos βz |  1, which corresponds to the pass bands of the 1DPCs. In the following calculation and discussion, we take ω D /2π = 10 GHz, and this corresponds to the wavelength λ D = 30 mm in vacuum. In what follows we will demonstrate the properties of

Fig. 2. Dependence of photonic band structures on the ratio of widths η = d A /d B for the structures ( A B ) N with N → ∞ under  = 0.5λ D , the ratio η from (a) to (h) is 0.1, 0.5, 1.0, 2.0, 5.0, 15.0, 50.0, and 100.0. (i) Dependence of photonic band structures on the thickness d B under d A = 0.5λ D and θ = 10◦ . The dark areas are the allowed bands, the white areas correspond to the forbidden gaps. The Dirac frequency of layers A is ω D /2π = 10 GHz.

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the PBGs, transmission spectrum, and the electromagnetic fields in the PCs. Here we only present the result of TE-polarized plane waves, and similar results can be obtained for the TM-polarized plane waves. 3. Numerical results and discussions In this section, we will demonstrate the photonic structures in the 1DPC by using the transfer matrix approach. The approximate band structures, transmission spectrum and electromagnetic fields are also shown to display the physical mechanism and the unique property of special bands. Finally, we will explore the effects of incident angle and lattice constant of photonic crystal on the photonic structures. Let us first consider the photonic band gaps in the system. Fig. 2 demonstrates the dependence of photonic band structures on the ratio of widths η = d A /d B , for the structures ( A B ) N with N under  = 0.5λ D . From Fig. 2(a) to Fig. 2(c), as the thickness of layer A increases, a PBG appears near the Dirac frequency ω D of the bulk material A. This gap opens at the inclined angle (i.e. θ = 0). The band edge of the above passing band (above the Dirac frequency) shifts into the higher frequencies and gradually becomes linear as η increases. However, the edge of the lower passing band gradu-

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Fig. 3. Dependence of photonic band structures on the ratio of widths η = d A /d B for the structures ( A B ) N with N → ∞ under  = 1.5λ D , the ratio 0.5, 1.0, 2.0, 5.0, 15.0, 50.0, and 100.0. The dark areas are the allowed bands, the white areas correspond to the forbidden gaps.

η from (a) to (h) is 0.1,

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ally shifts up and becomes flat as η increases. When η  1 (i.e. the thickness of layer B is very thin for a fixed value of ), the first lower passing band becomes flat and is almost independent of θ for a very small d B , see Figs. 2(g) and 2(h). In this sense, although the passing band becomes extremely narrow, and it can be regarded as an omnidirectional band. Moreover, the location of the narrow band is slightly located below the Dirac frequency ω D . Fig. 3 demonstrates another situation for  = 1.5λ D . When η  1, there is still a passing band below the Dirac frequency ω D . By comparing Fig. 3 with Fig. 2, it is expected that this lower passing band is sensitive to the lattice constant . Through our numerical analysis, this lower passing band will disappear in the cases of large values of  and η . However, in the case of small values of  [like Fig. 2], when η  1, in principle, there is always the lower passing band. The width of this narrow band is roughly proportional to the thickness of layers B, see Fig. 2(i). Using this property, one may fabricate the controllable all-angle filters. In Fig. 3, there is another interesting effect. For a fixed value of , under the appropriate value of η , see Figs. 3(b) and 3(d), there is a special passing band appearing near the Dirac frequency. Through the careful analysis, the condition of occurring this special passing band is that the thickness of layer B satisfies k B d B = mπ (m = 1, 2, 3 · · · is the integer). This special passing band is totally located outside of the Dirac cones. In the below, we will see that the origin of this special band is due to the interaction of the evanescent waves in the Dirac medium and the propagating waves in the dielectric medium. Fig. 4 further demonstrates the generation of the special passing band near the Dirac frequency. From Fig. 4(a) to Fig. 4(c), it is clear seen that, when d B satisfies k B d B = mπ with the suitable value of η , a very narrow passing band appears from the Dirac frequency, see Fig. 4(b); otherwise, this passing band may shift up or down depending on whether the resonant condition is satisfied or not. From Figs. 4(d) to 4(f), as the value of the resonant condition (k B d B = mπ ) decreases, the width of the special band also becomes narrow. In order to explain the above effect, we take a simple approximation. Owing to k2 = k2z + k2y → 0 when ω → ω D , we can use ik y to substitute for k zA . Therefore, the following relation can be obtained from Eqs. (6),

cos[βz ] = cosh[k y d A ] cos[k zB d B ] i qA

− (

2 qB

+

qB qA

) sinh[k y d A ] sin[k zB d B ].

(7)

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Fig. 4. Dependence of photonic band structures on the different resonant conditions. From (a) to (f), k B d B are 4.14π , 4.0π , 3.87π , 3.0π , 2.0π , and 1.0π . The dark areas are the allowed bands, the white areas correspond to the forbidden gaps. The points with the symbols “” in Fig. 4(b) will be considered later (see Fig. 7). Here  = 3λ D .

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The corresponding band structures are shown in Fig. 5(a–c). Comparing Fig. 5(a–c) with Fig. 4(a–c), we can see that the special passing band is due to the interaction of the evanescent waves in Dirac media and the propagating waves in the dielectric media. In order to further reveal the nature of the special bands, the transmission spectrum and electromagnetic fields inside a finite periodic structure are discussed below. Firstly, the amplitude and phase of transmission are plotted in Figs. 6(a) and 6(b) respectively. As θ increases, a gap opens near ω D in the transmission spectrum, and meanwhile a very narrow transmission band occurs in that gap close to ω D , see Fig. 6(a). The corresponding phase curves are shown in Fig. 6(b). It is expected that the dramatic

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Fig. 5. Dependence of approximate photonic band structures on the different resonant conditions. The resonant conditions of k B d B from (a) to (c) are 4.14π , 4.0π , and 3.87π . Other parameters are the same as in Fig. 4.

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Fig. 7. The EM fields inside the 1DPC of ( A B )20 at different frequencies. ω/2π = 9.5 GHz in the band I, 9.8 GHz at the gap II, 10.0046 GHz in the III, 10.2 GHz at the gap IV, and 10.5 GHz in the band V at θ = 5◦ . Here, the line denotes the electric field | E x | and the gray line is for the magnetic field Other parameters are the same as in Fig. 6.

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(a–e) band black | H y |.

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Fig. 6. (Color online.) Typical changes of the amplitude (a) and phase (b) of transmission t (ω, θ) as the function of the frequency ω in the structure of ( A B )20 at  = 3λ D and k B d B = 4π under different incident angles. The solid, dash and dash– dot lines correspond to θ = 5◦ , 15◦ and 20◦ , respectively.

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change of the phase may lead to the strong dispersion within this narrow passing band. Since the width of this passing band could be controlled by adjusting the value of k B d B , as discussed in the above, this property may be applied in the dispersion engineering, such as PC-based delay lines. Next we turn to consider the electromagnetic fields inside the structure. At first, the electric and magnetic fields inside the 1DPC can be expressed by [31]

E x ( z, ω) = E (i ) (0, ω){[1 + r (ω)] Q 11 ( z, ω)

+ q0 [1 − r (ω)] Q 12 ( z, ω)]},

(8)

c H y ( z, ω) = E (i ) (0, ω){[1 + r (ω)] Q 21 ( z, ω)

+ q0 [1 − r (ω)] Q 22 ( z, ω)]},

(9)

where Q α β ( z, ω) (α , β = 1, 2) are the elements of the matrix Q =

i = j −1

i = j −1

M j ( z, ω) i =1 M j (d j , ω), z =  z + i =1 di represents any position inside the 1DPC, and  z is the distance from the point within the jth layer to the interface between jth and ( j − 1)th

Fig. 8. (Color online.) The transmission as the function of the frequency ω . Solid, dash and dash–dot curves correspond to  = 3λ D , 4.5λ D , 6λ D at η = 0.5 and θ = 5◦ .

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layers. Fig. 7 plots the behaviors of the EM fields inside the structure of ( A B )20 at certain frequencies with an incident angle θ = 5◦ . The frequencies of light in Figs. 7(a) to 7(e) are ω/2π = 9.5 GHz, 9.8 GHz, 10.0046 GHz, 10.2 GHz, and 10.5 GHz. The locations of these frequencies are denoted in Fig. 4(b). From Figs. 7(a) and 7(e), when the light frequencies are located inside the band region I and V [see Fig. 4(b)], and the whole electromagnetic fields are cyclical and localized. From Figs. 7(b) and 7(d), when the light frequencies are located inside the gap region II and IV, the electromagnetic fields are decaying along the propagating distance, and the fields at the interfaces are localized but decaying. However, in Fig. 7(c), when the light frequencies are located in the special band, the fields become periodicity and strongly localized at the interfaces. Comparing Figs. 7(c) and 7(a, e), the fields in the special band are enhanced almost a hundred times larger than that of the other passing bands. Thus, the dispersion inside the special band is very strong. In the above, we show that the widths of special bands are dependent on the resonant conditions [see Fig. 4(d–f)]. Meanwhile, one unique property should be noted that the special bands keep

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invariant with the lattice constant at fixed ratio in Fig. 8, which is distincted from the Bragg gaps.

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4. Conclusion

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In summary, we have presented the photonic band structures of the 1DPCs containing NZPIMs. It is found that there exists an omnidirectional passing band and the special band near the Dirac frequency. The EM fields and transmission are demonstrated to be dissected the mechanism of the special bands which is caused by the interaction of the evanescent and propagating waves in the systems. It also shows the different properties between these special bands and other passing bands. The properties of these bands are sensitive to the resonant conditions under fixed lattice constant, and they are invariant upon the lattice constant at fixed ratio. All these characteristics may be helpful for designing the tunable photonic devices.

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Acknowledgements

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This work is supported by NSFC grants (No. 61078021 and No. 11274275), and a grant from the National Basic Research Program of China (No. 2012CB921602).

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