Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems

Surface impedance is an important concept in classical wave systems such as photonic crystals (PCs). For example, the condition of an interface state formation in the interfacial region of two different one-dimensional PCs is simply Z_SL +Z_SR=0, where Z_SL (Z_SR)is the surface impedance of the semi-infinite PC on the left- (right-) hand side of the interface. Here, we also show a rigorous relation between the surface impedance of a one-dimensional PC and its bulk properties through the geometrical (Zak) phases of the bulk bands, which can be used to determine the existence or non-existence of interface states at the interface of the two PCs in a particular band gap. Our results hold for any PCs with inversion symmetry, independent of the frequency of the gap and the symmetry point where the gap lies in the Brillouin Zone. Our results provide new insights on the relationship between surface scattering properties, the bulk band properties and the formation of interface states, which in turn can enable the design of systems with interface states in a rational manner.


I. INTRODUCTION
Impedance is a very important and useful concept in wave physics as it is the parameter that governs how a wave is scattered or reflected when it encounters an interface. As such, it characterizes how a material couples with waves coming from outside. On the other hand, the bulk band structure characterizes how waves can travel inside a periodic system. These quantities should be related in some way. We are going to establish that for a periodic multilayer film, commonly referred to as 1D photonic crystals, the surface impedance is related to the Zak phase 1 of the bulk bands. As the existence of interface states is determined by the surface impedance, this means that the existence of localized states at an interface is determined by the geometric phases of the bulk crystals. It is well known that interface states can exist in a quantum system when the topological properties of two semi-infinite systems on each side of the interface are different [2][3][4][5] . A famous example is the SSH model for polyacetylene [6][7][8] . In such systems, it was shown that an interface state exists when the Zak phase of the occupied band on one side of the chain is different from that on the other side, which can be obtained through gap inversion [9][10][11] . The purpose of this work is to find a general connection between the existence of an interface state in a photonic system and the bulk band topological properties as well as the surface impedances of the two systems on each side of the boundary. The analog between photonic systems and quantum systems has already been discussed recently [12][13][14][15] . Based on this analog, Zak phase can also be defined in photonic crystals (PCs).
For 1D binary PCs, we found a rigorous relation that relates the existence of an interface state to the sum of all Zak phases below the gap on either side of the interface. This relation holds for any 1D PCs with inversion symmetry, including those with graded refractive indices. Similar to the "bulk-edge correspondence" found in topological insulators [2][3][4] , the "bulk-interface correspondence" found here provides not only a tool to determine the existence of interface states in a photonic system but also the possibility of designing a photonic system with interface states appearing in a set of prescribed gaps.

A. Impedances and Zak phases of 1D photonic crystals and their relationship
Let us consider a dielectric AB layered structure as shown in Fig. 1(a). A plane wave from free space incidents normally on the semi-infinite 1D PC on the right, and the reflection coefficient of the electric field, x E , is given by R r .
When the frequency of the incident wave is inside the band gap of this system, the incident wave will be totally reflected, and we have  is the reflection phase. We define a surface impedance, SR Z , of the semi-infinite PC as the ratio of the total electric field to the total magnetic field on the right hand side of (RHS) the boundary, i.e., , where z= 0 defines the boundary. The impedance SR Z and the reflection coefficient R r are related by:  16,17 . But the question is then how can we -design or control the value of the surface impedance. We will show that the sign of the surface impedance for frequencies inside a band gap is in fact determined by the geometrical phase of the bulk bands. In the following, we derive a rigorous relation between the surface impedance and the Zak phase of the PC.
The band structure of a dielectric binary PC shown in Fig. 1(a) can be obtained from the following relation 18 : is shown in Fig. 1 where the integer l is the number of crossing points under the th n gap (in Fig. 1(b), the crossing point is at the 7 th band gap).
The Zak phase of the lowest 0 th band is determined by the sign of

B. Changing the sign of impedance by tuning pass a topological transition point
To have a guaranteed existence of an interface state, one need to make sure that surface impedance on the left and right half space is of opposite sign at one common gap frequency. One possible way (but not the only way) is to "tune the system parameters across a topological transition point" as elaborated below. To demonstrate this idea, we simply tune the parameter a  used in Fig. 1 6 When the value of a  is further increased, the gap opens again and accompanied by a change of sign in the surface impedance as well as a switch of the Zak phase in bands 6 and 7. This represents a topological phase transition, which occurs when two bands cross each other. Thus, by constructing an interface with PC1 on the one side and PC2 on the other side, we should see an interface state inside the gap 7. This is verified in our numerical study of the transmission spectrum of a system consisted of a slab of PC1 (with 10 unit cells) on the one side and a slab of PC2 (with 10 unit cells) on the other side embedded in vacuum. Fig. 2(a) shows clearly a resonance transmission due to an interface state around 5/ c   in gap 7. Such a topological phase transition represents a classical analog of the SSH model in electronic systems 6-8 although impedance is not usually considered in electrons.
The above example is a manifestation of a topological phase transition arising from band crossing in photonic systems.
It should be pointed out that the occurrence of the band crossing shown in Fig. 1 does not change sign. The variation of  with respect to a  can be seen as follows. In Fig. 1 , the frequency at which two bands meet in gap 7 is 7 5/ c   (see Fig. 1(b)), which is also the frequency where

C. Relationship between the Zak phase and the symmetry properties of the edge states
We will give a physical interpretation of the Zak phase in an isolated band by using the symmetries of the two edge states at the two symmetry points of the Brilliouin Zone.
As we have seen, the topological property of the band structure changes every time when a band crossing occurs as   The blue dash lines marked in Figs. 3(c)-3(h) indicate the position of the origin (z=0), which is the center of slab A. According to this rule, it is easy to see from Figs. 3(e) and 3(g) that the Zak phase of the 6 th band of PC1 is zero as the wave functions of the points M and N are both non-zero at the origin, whereas the value changes to π in PC2 because the wave function at point Q becomes zero after band crossing.
For the same reason, the Zak phase of the 7 th band in PC2 is also changed after band crossing.
The band inversion can also be seen from the switching of two edge states across the gap. For example, the wave functions at points L and Q have nealy the same distribution, i.e., the wave functions are both zero at the origin and with larger amplitudes in the B slab, whereas for points M and P the absolute values of the wave functions are both at maximum at the origin and their amplitudes are nearly the same in slab A and slab B. However, the wave functions at points N and R are nearly the same, not affected by the band crossing. This is also true for points K and O. Thus, it is precisely the switching of two edges states at gap 7 that gives rise to different Zak phases in PC1 and PC2 for both bands 6 and 7. Similar behavior has been reported in the electronic system 9-11 .

D. Relationship between the sign of impedance and the symmetry properties of the edge states
The sign of the imaginary part of the surface impedance, i.e.,  , can also be related to the symmetries of the two edge states. It is well known that, the amplitude of the wave function of the band edge states at the origin (z=0) is either zero or maximum 21 as also shown in Fig. 3

E. Existence of interface states
As we have mentioned before, the occurrence of band crossing at a particular gap (say, the th n gap) appears simultaneously for all gaps which are integer multiples of the th n gap. However, we should emphasize that "gap inversion" is just one way but not the only way to achieve an interface state. As an example, we consider a system consisting of 10 unit cells of "PC3"  Fig. 4(a).

F. Generalization to other waves
Finally, we want to stress that the results obtained above for PCs also hold for other one-dimensional systems with inversion symmetry such as acoustic waves. Because of inversion symmetry, the wave functions at two edges of an We should mention that the electric field is taken as the scalar field in this work. If the magnetic field is chosen as the scalar field, Eq. (4) still holds. The sign of imaginary part of the surface impedance is an intrinsic property of the PC and should not depend on the choice of field. The Zak phase of an isolated band also remains unchanged because it depends on the symmetry properties of two edge states of the band. The change of field from electric to magnetic changes the symmetry properties of both edge states, and therefore, keeps the Zak phase unchanged. However, the Zak phase of the 0 th band will change sign, but the outcome will be the same as the effect will be canceled by the change of the factor  

III. CONCLUSION
In summary, we showed that the surface impedance in the band gaps of a 1D photonic crystal is determined by the geometric Zak phases of the bulk bands. In particular, each photonic band gap has a character that is specified by the sign of the impedance which is related to the Zak phases of the bulk band below the band gap through Eq. (4). As the surface impedance determines the existence of interface states at the boundary of PCs, the existence of the interface states can be determined by the bulk band geometric phases. This correspondence between surface impedance and bulk band properties gives us a deterministic recipe to design systems with interface states.

Acknowledgement
This work is supported by Hong Kong RGC through AOE/P-02/12. Xiao Meng is supported by the Hong Kong PhD Fellowship Scheme. We thank Prof. S.Q. Shen and Prof. Vic Law for stimulating discussions.

APPENDIX
In this appendix, we will give some mathematical details mentioned in the main text. We then give several additional examples in support of the statements made in the main text.
Let us consider a dielectric AB layered structure with the relative permittivity, relative permeability, refractive index, relative impedance and width given by a . The unit cell length is ab dd    and the relative permittivity and permeability of the slabs are positive and non-dispersive. We will employ several ancillary parameters    is the ratio of optical path in the slab A and B,  is the phase delay in a unit cell, and  reflects the impedance mismatch between the slab A and B and is always larger than 1 when the impedances of slab A and slab B are not the same.

APPENDIX A: BANDS CROSSING CONDITION
Here we will prove, when ab zz  , the necessary and sufficient condition for two bands to cross (either at zone center or zone boundary) is given by being a rational number, i.e., 12 / mm Sufficient condition: The band dispersion relation of dielectric AB layered structure is given by 18   Following the same idea, we can also prove that

APPENDIX B: ZAK PHASE OF EACH BAND
In this appendix, we will show that, if one isolated band (excluding the 0 th band) contains the frequency point  at which   sin / 0 bb n d c   , then the Zak phase of this band must be π (if we set the origin of the system at the center of A slab). Proof: The Zak phases of isolated bands depend on the choice of origin. We choose the origin to be at the center of A slab.
To prove this assertion, we adapt the standard transfer-matrix method described in Ref 18 . The mathematical details can be found in Ref. [18]. Here, we adopted some changes in notations. Knowing the eigen field distribution, the Zak phase of each bands can be further calculated with Eq.  The above proof can be easily extended to the case when the system is dispersive [12][13][14] .
the relation   2arctan nn     , it is straight forward to show that, n  is a monotonic decreasing function of frequency from  to 0 or from 0 to  depending on the sign of n  .
In the main text, we argue that, once sgn of the left and right PCs are different inside the common band gap, there must be an interface state, here we will give another example. In Fig. 7(b), we choose the 2 nd common band gap in Fig. 4 of the main text as an example. The solid black, red and blue lines show the imaginary parts of relative impedances of PC3, PC4 and the sum of those two, respectively, inside the 2 nd common band gap. The solid black and red lines are both monotonic decreasing function of frequency, and their sum must also be a monotonic decreasing function of frequency from positive to negative. Thus there must exist some frequency point at which the blue line crosses the 0, corresponding to an interface state as showed in Fig. 7(a), where resonant transmission is observed inside the common band gap.

APPENDIX E: EXTENTION OF EQ. (4)
In this appendix, we will show that, Eq. Eq. (4) in the main text also applies when the dielectric function is a continuous function of z. In Fig. 9, we considered a system consist of 20 unit cells of "PC6" (   The transmission spectrum of system is given in Fig. 9 (a), where the boundary between two PCs is set at 0 z  . The band structures (Solid black line) of PC6 and PC7 are given in Fig. 9