We propose a topological description for gaps of one-dimensional symmetric all-dielectric photonic crystals (PCs). It is shown that, in the propagating direction, the effective electromagnetic parameters of PCs can be derived from one unit cell with mirror symmetry. Besides, at the frequencies of gaps, these symmetric PCs can be described as photonic insulators with effective negative permittivity or negative permeability. Moreover, based on the mapping of Maxwell's equations to the Dirac equation and the band inversion achieved by tuning the material and structural parameters, we demonstrate that the gaps of PCs with effective negative permittivity or negative permeability possess different topological orders. Lastly, we show that a bound state is robust against the disorder under a zero-average-effective-mass condition in a heterostructure made of two PCs with different topological orders.
© 2016 Optical Society of America
Properties of electromagnetic (EM) materials can be described by their permittivity () and permeability () . Metamaterials in which and can be flexibly tuned possess unprecedented abilities to manipulate the EM waves . The novel phenomena realized by metamaterials include negative refraction [3–5], cloaking [6,7] and black hole mimicking , etc. Generally, EM materials can be classified into four quadrants according to their signs of and , i.e., double-positive materials (,), double-negative materials (,) [3–5], single-negative (SNG) materials including -negative (ENG) materials (,) and -negative (MNG) materials (,) . Because of in double-positive and double-negative materials, the EM waves in these materials are propagating waves. In contrast, the EM waves decay exponentially in the SNG materials since. In analogy to the conducting materials in which electrons can transport, the double-positive and double-negative materials can be regarded as photonic conductors . Similarly, SNG materials can be regarded as photonic insulators (PI)  which strongly reflect the EM waves. Although both ENG and MNG materials support the evanescent waves, the signs of imaginary impedance inside them are opposite. Utilizing this property, Alu and Engheta revealed that an interface mode can exist in a paired structure made of ENG and MNG metamaterials under certain conditions. At the frequency of this interface mode, light can tunnel through the ENG/MNG pair, which can be utilized to design a subwavelength cavity [9,11]. In visible frequency, metals below plasma frequency are natural ENG materials. However, the fabrication of optical MNG materials is still a great challenge. In 2008, Guo et al. found that the gaps of all-dielectric PCs  can mimic an effective ENG or MNG material . In a heterostructure made of two different PCs that act as an ENG material and a MNG material, Guo et al. also obtained the interface mode and the optical tunneling phenomenon in visible frequency. On the other hand, Tan et al.  showed that the ENG and MNG metamaterials possess different topological orders by mapping explicitly Maxwell's equations to the Dirac equation in one dimension. The topological property of ENG and MNG metamaterials are also experimentally demonstated based on the microwave transmission lines with subwavelength unit cells. As a result, the interface mode in the ENG/MNG pair previously proposed by Alu and Engheta is actually a solution of one-dimensional (1D) topological excitation, namely the bound state . Very recently, Xiao et al. have studied the band structure of 1D all-dielectric PCs from the view point of Zak phase and surface impedances of structure and extended the results to phononic crystals [15,16]. By calculating different bands, they found that in some bands the Zak phases are topologically nontrival (for instance, ) while in other bands the Zak phases are zero. The topologically nontrivial Zak phase can be tuned in different bands by changing the material and structural parameters. When the Zak phase of the upper band changes from 0 to and that of the lower band changes from to 0, or vice versa, band inversion occurs. Inspired by the work introduced above, a series of questions arise: do the effective ENG and MNG material mimicked by the all-dielectric PCs have different topological orders? If the answer is yes, then is the interface mode that appears in the photonic heterostructure composed of two different PCs also a solution of 1D topological excitation?
In this paper, we describe the gaps of 1D symmetric all-dielectric PC according to the effective medium approach (EMA)  and study the topological property of the gaps in Dirac-type mass inversion situations. Based on the effective parameters derived from one unit cell, we show that a symmetric PC at different frequencies of gaps can be described as an ENG or a MNG PI. By mapping explicitly 1D Maxwell's equations to the Dirac equation, an ENG (MNG) PI can be described by negative (positive) effective mass. Besides, a topological transition point is found in the process of band inversion by tuning the material and structural parameters of PCs. During the band inversion, the PC changes from an ENG PI to a MNG PI, or vice versa, which indicates that an ENG PI and a MNG PI possess different topological orders. Moreover, an interface mode appears at the boundary separating an ENG PI and a MNG PI. This interface mode is actually a bound state in 1D case, which is robust against some kind of disorder. The paper is organized as follows. In Sec. 2, we study the effective parameters of 1D symmetric all-dielectric PCs at normal incidence. In Sec. 3, from the Dirac equation and the band inversion, we demonstrate that an ENG PI and a MNG PI have different topological orders. In Sec. 4, we discuss the influence of disorder on the bound state that occurs in a system composed of the ENG PI and the MNG PI. Finally, a conclusion is given in Sec. 5.
2. Effective parameters of 1D symmetric all-dielectric PCs
EMA is used to describe the macroscopic properties of microstructures. For metamaterials, since the size of a unit cell is in the subwavelength scale, EMA works well and the effective and (denoted byand , respectively) of structure obtained from EMA can describe the homogenization properties of metamaterials . However, the effective medium descriptions for all-dielectric PCs encounter difficulties because the size of a unit cell in an all-dielectric PC is comparable with the wavelength. Nevertheless, in a special direction, namely in the propagating direction, EMA can still be used to capture the essential properties of photonic gaps . Of course, for a 1D all-dielectric PC with asymmetric unit cells, the macroscopic properties of PC cannot be characterized by one unit cell . However, for a 1D all-dielectric PC with symmetric unit cells, at normal incidence, we show that the effective parameters in the frequencies of gaps can be derived from one unit cell and they are independent of number of periods, which is important for the topological description of photonic gaps.
We consider a symmetric unit cell , as is shown in Fig. 1. The , of single layer material are and , where the subscript represents material or , respectively. The total length of the symmetric unit cell is . Throughout the paper the losses of dielectric materials are neglected. The background is air. Suppose that a plane EM wave launches into the structure at normal incidence. With the transfer matrix method , the characteristic matrix of the jth layer is given by17]. In contrast, for an asymmetric unit cell such as , the two corresponding diagonal matrix elements are not equal (i.e., ) and cannot be considered as single-layered equivalent material. Therefore, in the propagation direction, the characteristic matrix of a symmetric unit cell can be defined asEq. (6), we have
In this paper we focus on the frequency region corresponding to the reflection gaps. When , we have . However, when , the EM wave in the symmetric unit cell contains both propagating component and attenuation component due to strong reflection. In this case the effective phase can be written as , where and describe the propagating and attenuation components in the unit cell, respectively. Substituting into Eq. (6), we haveEq. (9) we can obtain and , respectively. Then, the effective refractive index of the homogenized material is given by and thereby, . Since the reflection phase plays an important role in the topological property of gaps, we need study whether the neglect of will affect the reflection coefficient. The reflection coefficient of the structure can be written as Eqs. (8) to (10), we see that the reflection coefficient is not changed after homogenization. Of course, the neglect of will ignore the oscillations of fields that are associated with the phase of EM waves. Nevertheless, our homogenization method is still useful for capturing the topological property of gaps including band inversion from the EM parameters of and .
It is clear that the parameters of and derived from one unit cell are independent of the periodic number when a number of unit cells construct a PC. In particular, in the periodic boundary condition, Eq. (7) is also the dispersion equation of the infinite-periodic PC (). Supposing that the parameters of the symmetric unit cell are, , , and , we calculate the band structure of the infinite-periodic PC in Fig. 2(a). For convenience, we number the gaps at the right label. In Fig. 2(a) there are seven open gaps indicated by the gray regions in the normalized frequency region ranging from 0 to 3. One gap, i.e., the 7th gap is closed at the normalized frequency 2.5. Then, based on our homogenization method, the corresponding and in open gaps are given in Fig. 2(b). It is seen that for frequencies within the 2nd, 3rd and 6th gaps, only is negative, while for frequencies within the 1st, 4th, 5th and 8th gaps, only is negative. In the above paragraph we have demonstrated that the homogenization will not change the reflection coefficient by analyzing the matrix elements. Here we further numerically verify that, within the gaps, the reflection phases of the PC are the same as those of the homogenized material (HMM) with and . From Fig. 2(c) we see that the reflection phases (denoted by ) of the finite-periodic PC are overlapped with those of homogenized materials with and given in Fig. 2(b). Moreover, it is known that the interval of reflection phase is in an ENG PI, and in a MNG PI . Therefore, within the 2nd, 3rd and 6th gaps, the PC can be considered as an ENG PI, while within the 1st, 4th, 5th and 8th gaps, the PC can be considered as a MNG PI. To obtain more understanding of the band gaps, in Fig. 2(d) we also give the variation of with frequency. In Fig. 2(d), territory corresponds to the gap region. The quick oscillation of whose quick periodicity (the “spectral range” between gap centers) is related to the Bloch phase, while the envelope is a slowly varying function (dashed line), that can occasionally intersect the (gap closure) limit. The number of the gaps and that of the gap closures before the function goes into the territory is closely related to the property of gaps, see Eq. (4) in Ref .
Here we confirm that in our homogenization we only neglect the oscillating propagation component of fields that is associated with . We compare the electric field distributions in the HMM and in the finite-periodic PC , respectively, for frequencies within gaps. We consider two normalized frequencies in Fig. 2: within the 1st gap and within the 6th gap. The interfaces of symmetric unit cells are indicated by the white lines. For or , it is seen from Fig. 3(a) or Fig. 3(b) that the electric fields in the PC decay exponentially with periodic oscillation. The periodic oscillation comes from the propagating component in the PC. The number of oscillations or peaks in one unit cell depends on the serial number of the gaps (i.e., ). We see that there is only one peak in one unit cell in Fig. 3(a) and six peaks in one unit cell in Fig. 3(b). On the other hand, the electric fields in the HMM also decay exponentially but without oscillation. Moreover, the values of electric fields at the interfaces of symmetric unit cells in the PC are equal to those at the same positions in the HMM. Therefore, if only considering the attenuation component, the decaying electric field distributions in the HMM is consistent with those in the PC. Similar phenomena can be found in other gaps. Moreover, from Fig. 3(a) at and Fig. 3(b) at (the high frequency), it is seen that our homogenization method is not restricted by the value of frequency so long as the frequency locates in the gaps. In a word, although the propagating component is neglected, our homogenization method is still very useful for capturing the topological property of photonic gaps in symmetric PCs.
3. Topological description for photonic gaps
According to the discussion above, at the frequencies of gaps, symmetric PCs can be described as an ENG PI or a MNG PI. In the following parts, we will discuss the topological property of photonic gaps from Dirac equation and the band inversion.
Dirac equation is a particle wave equation which can be used to describe the topological insulator in electronic system. Investigations have shown that there are two different topological configurations in Su-Schrieffer-Heeger (SSH) model for polyacetylene  which can be obtained through band inversion. The electronic gaps in these two configurations are related with the positive or the negative effective mass in Dirac equation which correspond to different topological orders.
Recently, people have found that Dirac equation for describing the topological insulator in electronic system can be extended to photonic insulator . An explicit mapping between the Maxwell's equations and the Dirac equation in one dimension has been revealed . Assuming that the electric and magnetic fields are and , respectively, a 1D plane EM wave with the frequency propagating throughout an optical media will satisfy the following Maxwell's equations
It is worthy to point out that the 7th gap in Fig. 2(a) is closed at the normalized frequency 2.5. This closed point appears when the two band-edge states touch each other. In electronic systems, band inversion indicates a closing or reopening of the gap when the two band-edge states interchange each other [23,24]. The closing point of the gap is also called a topological transition point. In a similar way, the gap closing and reopening in the symmetric PCs are analogous to the band inversion process in electronic systems. By tuning the system parameters, the gap in a SNG PI will possess different topological characteristics while crossing the topological transition point. Therefore, we believe that the normalized frequency 2.5 corresponding to the 7th gap is a topological transition point and the topological transition behaviours can be observed by closing or reopening the 7th gap.
By changing the dielectric permittivity, we can close or reopen the 7th gap. Here we change the permittivity of media from 3.8 to 4.2, and keep the other parameters the same as those in Fig. 2(a). For the case of , as is shown in Fig. 4(a), the () in the 7th gap is always negative (positive), which indicates that the 7th gap is an ENG gap. When approaches 4, the and of the structure are close to zero. During this process, the gap of this system remains an ENG gap. The 7th gap will be closed at the normalized frequency 2.5 when reaches 4. It should be pointed out that there are no data for the and at this closed point in Fig. 4(a). At first glance, it looks that both and are zero. However, this is a fake because at this closed point the imaginary part of is zero and we have neglected the real part of (i.e., ) in our homogenization method. Therefore, the corresponding effective parameters at this closed point cannot be obtained by our method. When further increases from 4 to 4.2, the 7th gap will reopen. At this reopening gap, the signs of and are inverted, which indicates that the reopening gap becomes a MNG gap. Obviously, the 7th gap has different topological characteristics across the transition point, which indicates that band inversion in 1D PCs is actually the gap inversion of a SNG PI.
The 7th gap also can be tuned by changing the filling fractions. As is shown in Fig. 4(b), when the filling fraction of media is smaller than 0.4, the 7th gap is an ENG gap. When the filling fraction of media approaches 0.4, the and are close to zero and the width of 7th gap becomes narrower and narrower. When the filling fraction of media reaches 0.4, the gap is eventually closed. Notice that we also cannot obtain the and at this closed point due to the same reason in Fig. 4(a). When the filling fraction of media further increases from 0.4 to 0.46, the 7th gap reopens, and the signs of and are inverted, which indicates the inversion of the 7th gap from an ENG gap to a MNG gap. During the band inversion of PCs, the Zak phase of the upper band can change from 0 to and that of the lower band can change from to 0, or vice versa [15,16]. The gap inversion from an ENG gap to a MNG gap (or vice versa) is consistent with the band inversion described by the Zak phases of pass bands.
4. Topological interface state and influence of disorder
Dirac equation reveals that a bound state exists in the heterostructure composed of positive and negative masses when , where and are effective masses of insulators with length and , respectively, and they have different topological orders . We assume that the heterostructure is composed of two types of PCs with symmetric unit cell and . The parameters of are , , , and , and the parameters of are , , and . Based on the method mentioned above, our calculations show that PC and PC exhibits gaps with overlaping normalized frequency regions, which is illustrated in Figs. 5(a) and 5(b), respectively. Moreover, within this gap region, PC and PC act as MNG and ENG PIs, respectively. We also calculate of symmetric unit cell and in Fig. 5(c). It is seen that the value of of within the gap is positive and that of is negative. Moreover, at the normalized frequency 2.028, the value of of is 0.043 and that of is −0.043, respectively. Their effective masses are equal in absolute value and opposite in signs. Therefore, when the sizes of the two PCs are identical (), the sum of total effective mass of the heterostructure becomes zero, which predict the existence of an interface state at the normalized frequency 2.028.
Now we will discuss the influence of disorders on the interface state in the heterostructure composed of two kinds of PCs with different topological orders. It should be pointed out that our system is not protected by the time reversal breaking and thereby the interface state is different from the one-way edge mode that is robust against disorders . Nevertheless, it was shown that the topological interface state in electronic system  or 1D dielectric resonator chain  is robust against effective mass distributions so long as the sum of the total effective mass is zero [for convenience, this condition is called the zero-average-effective-mass condition]. Below we will demonstrate that the interface state in the heterostructure will be robust against disorders under the condition. The total mass of PC and PC can be written as and . For simplicity, we introduce the disorder for PC only. The disorder is introduced by randomly tuning the permittivity and filling fraction of media in symmetric unit cell . Maintaining invariant and being sure that the mass of each unit is positive (MNG PI), we construct a kind of randomness configuration that the effective impedance () of each unit at frequency 2.028 is maintained after the disorder is introduced, which is illustrated in Fig. 6(a). Under this condition, we choose the effective refractive index to be , in which the value of is random. To make sure does not change, must be zero. We divide the 10 periods into 5 pairs. Each pair has a random and its counterpart . Thus the sum of will be zero and will not change. Based on this kind of randomness configuration, we tune the dielectric permittivity of from 2.2 to 2.31, and the filling fraction from 0. 40 to 0.44. We calculate 1000 randomness configurations and the average value of transmittance spectra is shown by the red solid line in Fig. 6(b).
For comparison, the transmittance spectrum of the heterostructure without disorder is also given in Fig. 6(b), as is shown by the scattered open circles. As predicted above, at the normalized frequency 2.028, an interface state with nearly unit transmittance appears in the gap. Moreover, at 2.028, the scattered open circles are overlapped with the red solid line. Therefore, after the disorder is introduced, the frequency as well as the transmittance of the interface state is not changed. By contrast, the transmittance of the pass band is deformed noticeably by the disorder. This is because the disorder in unit cells changes the interference conditions for the formation of the pass band. To demonstrate that the interface state is protected by the condition, we also study the case when the condition is not satisfied. For example, we use the same randomness of the filling fraction in Fig. 6(a) but keep the dielectric permittivity of unchanged. In this randomness configuration, . We do the same in other randomness configurations. Then, we calculate 1000 randomness configurations and the average value of transmittance spectra is shown by the blue short dashed line in Fig. 6(c). For comparison, the transmittance spectrum for the case of is also plotted in Fig. 6(c), as is shown by the red solid line. A magnification around 2.028 is given for a better view. It is seen that the transmittance of the interface state is significantly reduced when the condition is broken. Therefore, the interface state will be robust against disorders when it is protected by the condition.
We study the topological description for gaps of 1D symmetric all-dielectric PCs. Based on the effective parameters derived from one unit cell of PCs, the PCs at the frequencies of gaps can be described as an ENG PI or a MNG PI. Based on the Dirac equation and the band inversion, we reveal that an ENG PI and a MNG PI have different topological orders. When we construct a heterostructure composed of two PCs with different topological orders and same thicknesses, a bound state exists when the effective masses of the two PCs cancel each other. Moreover, the bound state is robust against the disorder of mass distribution in case that the sum of effective masses is fixed to be zero.
National Basic Research Program of China (No. 2011CB922001); National Natural Science Foundation of China (NSFC) (No. 11264003, No. 11234010, No. 11474220); Innovation Program of Shanghai Municipal Education Commission (Grant No. 14ZZ040); Project of outstanding young teachers' training in higher education institutions of Guangxi.
The authors would like to acknowledge Yueyun Hu and Yong Sun of the Tongji University for their valuable discussions
References and links
1. J. D. Jackson and J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).
2. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (John Wiley & Sons, 2006).
4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]
8. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
9. A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE. T. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]
10. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004). [CrossRef] [PubMed]
12. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton university, 2011).
13. J. Guo, Y. Sun, Y. Zhang, H. Li, H. Jiang, and H. Chen, “Experimental investigation of interface states in photonic crystal heterostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(2), 026607 (2008). [CrossRef] [PubMed]
15. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 021017 (2014). [CrossRef]
16. M. Xiao, G. C. Ma, Z. Y. Yang, P. Shen, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 4(2), 021017 (2014).
17. H. A. Macleod, Thin-film Optical Filters (CRC, 2001).
18. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]
19. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).
20. J. Y. Guo, Y. Sun, H. Q. Li, Y. W. Zhang, and H. Chen, “Optical tamm states in dielectric photonic crystal heterostructure,” Chin. Phys. Lett. 25(6), 2093–2096 (2008). [CrossRef]
21. M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013). [CrossRef]
22. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
23. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011). [CrossRef]
24. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]
25. S. Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters (Springer Science & Business Media, 2012).
26. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6, 6710 (2015). [CrossRef] [PubMed]