1 Introduction

A considerable amount of work has been devoted to the study of the topological properties of photonic structures.^[1] The word topological means that what is at stake are the properties of a structure that are stable under a continuous variation of the parameters defining it. For instance, the existence of a band gap is a topological property since a not too large variation of the properties (e.g., the size of the basic cell, the values of the electromagnetic parameters) does not close the gap. In some cases, the topological properties can be characterized by an integer number, a quantity that obviously remains constant over small continuous variations.^[2] Of course, for larger variations it may happen that, for example, the gap closes, which can lead to a change in the integer number. This will be called a topological transition. First attempts to find topological properties in photonic structures were made by mimicking the field of topological insulators:^[3] The time-reversal invariance was broken by the use of gyromagnetic materials controlled by a magnetic field. The devices considered there were quite complicated and specific. A major breakthrough was made when it was realized that topological effects could be obtained in purely dielectric structures.^[4] As a matter of fact, a topological effect can be obtained for very simple structures: one dimensional stratified media exhibit boundary modes that are topologically protected.^[5] These can be analyzed using theoretical tools that were developed a long time ago^[6] and which have been given a second look in the context of geometric phases: in ref. [7], it was shown that the properties exhibited in ref. [6] could be interpreted using the mathematical apparatus developed by ref. [8] in view of the results obtained by Berry in ref. [9]. Mathematically speaking, this comes under the domain of vector bundles endowed with a connection.^{[10, 11]} Simon introduced in ref. [8] the now celebrated connection 1-form $$A(k)=\langle u_k,\nabla _k u_k \rangle$$, where $$u_k$$ is a Bloch mode. This connection is often called the Berry connection, while it is in fact a specific case of the Levi–Civita connection.^[12] Zak's article was probably the first work applying the concept of geometric phase to Bloch waves. Recently, there was an interest in the possibility of extending these results to non-Hermitian Hamiltonians^[13-16] and, in the context of photonic crystals, to media with losses.^[17] All the preceding results were obtained by generalizing the Levi–Civita connection for a non-Hermitian bundle. In the present work, we propose a new approach to the topological properties of 1D periodic structures. We show that the topological properties can be analyzed without reference to the Levi–Civita–Simon–Berry connection. We introduce a function of the wavenumber that presents poles and zeros. The arrangement of the poles and zeros characterizes the topological properties of the medium. It turns out that this pole and zero structure extends naturally to the situation when losses are present in the materials out of which the structure is made, that is to the non-Hermitian situation.

The article is organized as follows: In the first section, we recall the elements of the theory of wave propagation in 1D structures, comprising Bloch waves. In the second section, we develop our approach and introduce the function that will prove to be a clue to the understanding of the topological properties. In the third section, we make the link with the usual approach using the geometrical phase when the mediums under consideration are lossless. Finally, we extend the approach to the situation where loss is added. Throughout the sections, numerical illustrations are provided in order to clarify the somewhat abstract statements. Finally, in the conclusion, we show that our approach can also handle a structure with some disorder and comment on the extension to higher dimensions.

2 Wave Propagation in a 1D Structure

An example of a 1D medium is depicted in Figure 1. We have chosen to represent a 1D stratified photonic crystal, but the results that we obtain apply to continuously varying structures, as well as to acoustic structures and to quantum waves in a 1D potential. For definiteness, we proceed by using the vocabulary of electromagnetism in the following: However, we will occasionally use the word “potential” generically to designate the permittivity (permeability) or the quantum confining potential. With an abuse of notation, we will talk of the “potential $$V(x)$$” to denote generically these quantities. We will further make the hypothesis that there is an inversion symmetry in the medium, that is, an origin can be chosen in such a way that $$V(x)=V(-x)$$.

From the general theory of ordinary differential equation,^[19] there exists a so-called “resolvent matrix” $$\mathcal {R}(x,y)$$ such that $$U(x)=\mathcal {R}(x,y) U(y)$$.

Due to the fact that the Bloch waves depend on both the direct and indirect variables $$x$$ and $$\theta$$ this representation of the Bloch bundle is not easy to handle. We present another, simpler, representation of the bundle.

In order to make the discussion less arid, we shall now give a numerical illustration of the concept that we deal with (the code, written in Matlab, is available upon request). We consider the case of a binary medium, where the period is made of two homogeneous layers of relative permittivity $$\varepsilon _1$$ and $$\varepsilon _2$$ and width $$d_1$$ and $$d_2$$ (see Figure 1). Let us denote $$\nu _j=\sqrt {\varepsilon _j},\, j=1,2$$.

Generically, the monodromy matrix can be put in diagonal form. However, for $$(k_0,z) \in \Delta \setminus \Delta _0$$, a particular situation happens: the eigenvalues $$\pm 1$$ have a multiplicity of 2 but the eigenspace is of dimension 1. The points where this happens are called ramification points. The set $$\Delta _0$$ where $$\mathcal {M}=\pm I_2$$ corresponds to a non-generic situation that is a topological phase transition, as will be shown in the following. Indeed, this corresponds to two bands touching at one point $$z=\pm 1$$. This singularity can be removed by an infinitesimal variation of the parameters. It separates two topological phases stable under small variations of the parameters far enough from this singularity.

3 Topological Characterization Using Poles and Zeros

Let us consider for now the usual case of a real potential, or lossless media. The eigenvectors $$U^{\pm }(x_0)$$ of $$\mathcal {M}$$ are the boundary values of the generalized Bloch waves at an origin $$x_0$$ that can be chosen at will. Changing the value of $$x_0$$ amounts to changing the basic cell to $$[x_0,x_0+1]$$. It is equivalent to a change of gauge in real space whenever an infinite medium is considered. When finite structures are considered, different choices of the origin will correspond to different physical properties, as shall be seen later on.

The eigenvectors $$U^{\pm }$$ are complex conjugated quantities, when the potential is real and for $$k_0 \in B$$. Using the resolvent matrix, the value of a Bloch mode at any point $$x$$ in the period is obtained by using the relation: $$U^{\pm }(x)=\mathcal {R}(x,x_0)U^{\pm }(x_0)$$. From these considerations, we conclude that the Bloch eigenspace at a point $$(k_0,z)$$ is entirely determined by the eigenvectors $$U^{\pm }(x_0)$$. Therefore, the Bloch bundle, that is, the collection of all the eigenspaces as $$z$$ describes $$S^1$$, is isomorphic to the vector bundle of eigenvectors of the monodromy matrix $$\mathcal {M}$$. As we have already said, it is a complex vector bundle over $$S^1$$, and therefore it is trivial, which means that there exists a non-vanishing section, that is, a continuous parametrization of a Bloch mode all around the Brillouin zone. We recover here one of the conclusions of ref. [7]. Still, there can be topological properties provided that we take into account the hypothesis that the potential has an inversion symmetry $$\sigma$$: $$V(x)=V(-x)$$.

Two different origins, $$x_0$$ and $$x_1$$ can be chosen such that $$V(x)=V(-x)$$. These two points are such that $$x_1-x_0=1/2 \mod {(}1)$$ (see Figure 1, where the two origins are indicated as dashed lines). Let us assume that the boundary values are chosen at one of the two points such that $$V(x)=V(-x)$$, that is, we fix the gauge in real space. This point is now the new origin, $$x=0$$.

For the eigenvectors $$U^{\pm }$$ of the monodromy matrix $$\mathcal {M}$$, the inversion symmetry $$\sigma$$ acts as $$\sigma (U^{\pm })=\sigma _z U^{\pm }$$, where $$\sigma _z$$ is the Pauli matrix $$\def\eqcellsep{&}\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$$. This is so because, under the change $$x \rightarrow -x$$, the derivative changes sign and the wave propagates backward; the corresponding monodromy matrix is $$\mathcal {M}^{-1}$$. The inversion symmetry implies that if $$U$$ is an eigenvector of the monodromy matrix with the eigenvalue $$z$$ then $$\sigma _z U$$ is an eigenvector of the monodromy matrix but with the eigenvalue $$1/z$$. As a consequence, we have the following result:

For $$k_0 \in B$$, it holds, projectively, $$U^*=\sigma _z U$$ ($$*$$ denotes complex conjugaison). When we say that two vectors are equal “projectively,” we mean that they are collinear. Given a vector $$U=(u(0;k_0,z),u^{\prime }(0;k_0,z))^t$$, the vector space generated by $$U$$ is denoted $$\tilde{\chi }=[u(0;k_0,z):u^{\prime }(0;k_0,z)]$$. This is the standard notation for an element of the projective space $$\mathbb {C}P_1$$ which is the set of all complex lines going through the origin in the complex plane $$\mathbb {C}^2$$. This space is equivalent to the Riemann sphere, that is, the complex numbers $$\mathbb {C}$$ together with a point $$\infty$$ at infinity.

For $$k_0 \in B$$, the ratio $$\chi (k_0,z)=u(0;k_0,z)/u^{\prime }(0;k_0,z)$$ satisfies the relation $$\chi ^*(k_0,z)=-\chi (k_0,z)$$ and therefore, thanks to Equation (18), it is purely imaginary in the conduction bands for a real potential.

Let us show that these can be combined to provide a single function.

At the ramification point, the curves cross at a right angle, which makes it seemingly impossible to follow one eigenvalue by continuity from $$B$$ to $$G$$. However, this degeneracy is in fact directly linked to the stringent condition that the potential be real, since it is a consequence of the fact that the monodromy matrix has eigenvalues $$\pm 1$$ at the boundaries of the conduction bands. This degeneracy can be lifted by using the limiting absorption principle, that is, by adding a small imaginary part $$\delta$$ either to the potential or to the frequency. Indeed, if we replace $$k_0$$ by $$k_0+i \delta$$, the equation $$tr(\mathcal {M}(k_0))=\pm 2$$ is replaced by the equation $$tr(\mathcal {M}(k_0+i\delta ))=\pm 2$$. Therefore the crossing points are no longer real (generically). This is what is done in Figure 5 where a small imaginary value of $$10^{-2}$$ was added to $$k_0$$. The real part of $$k_0$$ is used as a blow-up parameter to plot the eigenvalues as curves in $$\mathbb {R}^3$$. It is seen that the curves no longer cross and therefore the eigenvalues can be followed individually.

Since the eigenvalues can be distinguished, it is also possible to follow the eigenvectors by continuity, that is to resolve the ramification points. Therefore, the same holds for the functions $$\chi ^{\pm }$$. For these functions, another extension problem appears when there is a pole. When a small imaginary part is added to $$k_0$$, the poles of $$\chi ^{\pm }$$ gain a small imaginary part and the restrictions of these functions to the real axis of $$k_0$$ are now continuous. It is illustrated in Figure 6 where the values of $$\chi ^{\pm }$$ are plotted on the Riemann sphere by stereographic projection. We conclude the following:

There is a unique function $$\chi (k_0,z)$$ corresponding to eigenspaces associated with eigenvalues lower than 1 in modulus for $$k_0 \in G$$.

At the heart of the topological properties captured by the function $$\chi$$ are the poles and zeros that it possesses. A zero is associated with the eigenvector $$(0,1)^t$$ and a pole to the eigenvector $$(1,0)^t$$. We have the following result:

For $$k_0\in B$$, since $$U^*(k_0,z)=U(k_0,1/z)$$, we have that either $$\chi (k_0,z)$$ is null at $$z=\pm 1$$ or it has a pole.

The values of $$k_0$$ for which there can be a zero or a pole are characterized as follows. Assume that there is a zero $$k_{\text{Z}}$$ in the interior of a band. Then $$\chi ^{\pm }$$ are both null and therefore the eigenvectors of $$\mathcal {M}(k_{\text{Z}})$$ are linearly dependent and the eigenvalues $$z$$ and $$1/z$$ are equal. Consequently, $$tr(\mathcal {M})=\pm 2$$. If $$tr(\mathcal {M})$$ is not transversal there, that is, if the derivative of $$tr(\mathcal {M})$$ is null, then it is a non-generic point that can be removed by an infinitesimal variation of the parameters. Similarly, if there is a pole of $$\chi$$ then $$U=(1,0)^t$$ with eigenvalue $$z$$. The other eigenvector is $$\sigma _z U=(1,0)^t$$. Therefore there is a degeneracy and $$z^2=1$$ and therefore $$tr(\mathcal {M})=\pm 2$$. We can then conclude the following:

The only poles and zeros of the function $$\chi$$ are generically situated at band edges only, that is, at points where $$tr(\mathcal {M})=\pm 2$$ and $$\mathcal {M}\ne \pm I_2$$. They are real when the potential is real.

The position of poles and zeros is the clue to understanding the topological properties of the structures. Indeed, a topological transition is characterized by the closing and re-opening of a gap through a continuous variation of the parameters defining the structure. The value of $$k_0$$ for which the lower and upper bands touch is a critical point of $$f(k_0)=\operatorname{Tr}(\mathcal {M}(k_0))^2-4$$, indeed, there it holds $$f(z)=0$$ and $$f^{\prime }(z)=0$$. Therefore: $$\mathcal {M}=\pm I_2$$. Since at this point the eigenvectors are $$(1,0)^t$$ and $$(0,1)^t$$, this means that a pole and a zero of $$\chi$$ merge. As a consequence, there is always a pole and a zero at the boundaries of a band gap. In other words, when the topological transition takes place, a pole and a zero change places and therefore they have to merge. We conclude that:

When the potential is real, the topological transitions take place at $$\mathcal {M}=\pm I_2$$.

Furthermore, the poles and zeros are continuous functions of the parameters and disappear only when they merge. As a consequence:

The ordered sequence of poles and zeros is a topological invariant. It determines the positions of the forbidden and conduction bands

This sequence is called the poles and zeros pattern. For a given band gap, the various possibilities are zero–zero, zero–pole, pole–zero, pole–pole.

When losses are added, the situation is more complicated, because in that case the poles and zeros pattern is contained in the lower part of the complex plane. When the potential is complex, the function $$\chi$$ is no longer purely imaginary in the conduction bands nor real in the band gaps. It turns out that the function $$\chi$$ is defined over $$F$$ and takes values in the projective space, assimilated to the Riemann sphere. This result is immediate whenever the eigenspaces are not degenerated, since the entries of the monodromy matrix are holomorphic.

4 The Berry–Zak Phase and the Triviality of the Bundle

We note that following our approach, the Berry–Zak phase is very easy to compute, as it suffices to consider the form of the monodromy matrix at the boundaries of the conduction bands.

5 The Bulk-Boundary Correspondence

Let us consider a structure made of two semi-infinite photonic crystals, Phc1 and Phc2, put side by side, such as depicted in Figure 7, and characterized by the monodromy matrices $$\mathcal {M}_1$$ and $$\mathcal {M}_2$$. Our point is to investigate under what conditions a boundary mode can exist and how it can be characterized topologically by means of the properties of the $$\chi$$ function.

We can thus conclude the following:

The link with the poles and zeros pattern can now be deduced. Recall that at the boundaries of the gaps the functions $$\chi$$ necessarily have a zero or a pole and that they have a constant sign within a band gap. This means that, provided $$\chi _1$$ and $$\chi _2$$ have opposite signs in a band gap, when the pattern inside a gap is pole–zero for one structure and zero–pole for the other, the functions $$\chi _1$$ and $$-\chi _2$$ necessarily cross and there is necessarily an edge mode.

A few words are in order as to the fact that the existence of a mode is linked in a very strict way to the symmetry property of the potential, as far as the theoretical analysis is concerned. An edge mode inside a gap corresponds to a pole of the scattering matrix (i.e., of the reflection and the transmission coefficients). Since we expect the pole to be a continuous function of the parameters it seems paradoxical that it could disappear suddenly when the symmetry is broken since it can be broken in a continuous fashion by moving continuously the origin of the basic cell. To resolve this apparent paradox, one should recall that the reflection and transmission coefficients are defined for two finite structures put side by side (see ref. [21] for the definition of the reflection coefficient for a semi-infinite photonic crystal). When two semi-infinite photonic crystals are considered, the mode should be evanescent in both photonic crystals away from the edge and this is a strict condition. When finite photonic crystals are considered (containing each N periods), and with an plane wave incident field, there necessarily are anti-evanescent waves in addition to evanescent waves in order to fulfill the boundary conditions. Therefore, breaking the symmetry condition does not kill suddenly the edge mode, rather its wavenumber is shifted toward the edges of the band gap and, as the number of periods N tends to infinity, the edge mode disappears continuously by being absorbed at the edges of the band gap.

Let us illustrate these results numerically. The two band structures corresponding to each photonic crystal are given in Figure 8. A finite structure made of 10 periods of each photonic crystal is considered. The transmission spectrum for an incident plane wave is plotted in Figure 8. The existence of an edge state is detected as a peak inside the band gap for the forbidden band situated inside the interval of $$k_0/2\pi \in [2.4, 2.6]$$.

In Figures 9 and 10, we have plotted $$\chi _1$$ (in red) and $$-\chi _2$$ (in blue), as well as the commutator of $$\mathcal {M}_1$$ and $$\mathcal {M}_2$$ in green. The positions of the band gaps are indicated by the dashed blue and dot-dashed red curves (with a value of 1 in the band gaps and 0 in the conduction bands). As can be seen in Figures 9 and 10, the edge state (appearing as a peak inside the band gap in the transmission spectrum, cf. Figure 8) corresponds to a value of $$k_0$$ for which the functions $$\chi _{1}$$ and $$-\chi _2$$ cross and the commutator of the monodromy matrices is null. All the other zeros of the commutator do not correspond to a crossing of $$\chi _1$$ and $$-\chi _2$$.

Let us now see what happens when complex values of $$k_0$$ are used. In Figure 11, we plot the absolute value of $$\chi$$ in the complex strip around the real axis. The poles are indicated by bright spots and the zeros by blue spots. We see clearly that both photonic crystals have the same poles and zeros pattern, except for the gap around $$k_0/2\pi =2.5$$ where one of the structure has the pattern zero–pole and the other one the pattern pole-zero. In this band gap, there is a boundary mode. In Figure 12, we have plotted the band structure and transmission spectrum when the right-hand side photonic crystal does not fulfill the symmetry condition that $$V(-x)=V(x)$$. As explained in the discussion above, the edge mode is still present but is shifted toward the boundary of the band gap. By varying the values of the permittivities and the width of the layers, it is possible to obtain a phase diagram for the topological properties, that is, the poles and zeros pattern (cf. Figure 13). It can be seen that the regions of interest are separated by a line corresponding to the closing of the band gap.

Our point is now to study the situation where the potential is complex, that is, an imaginary part is added to the permittivities.

The new poles and zeros pattern is given in Figure 14 for both the photonic crystals Phc1 and Phc2, as defined in Figure 8 but with a small imaginary part of 0.01 added to the permittivities of the layers of Phc1 (Phc2 remains lossless). There, it can be seen that the poles and zeros have moved toward the lower part of the complex plane of wavevectors. Still, the poles and zeros pattern is preserved. There is a continuous transformation, mathematically speaking, a homotopy, between the poles and zeros structures of both photonic crystals. When a pole and zero exchange places by continuously varying a parameter, there is a configuration for which the pole and the zero have the same real part. This amounts to saying that the gap closes there. Consequently, as far as the poles and zeros pattern remains close to the real axis, these patterns continue to characterize topologically the structures. Indeed, when plotting the transmission spectrum for the finite structure with losses (cf. Figure 15), a mode can be seen to remain (as an enlarged peak) in the same band gap. Of course, the life-time of the mode is now much shorter due to material losses added to radiative losses.

6 Conclusion and Possible Extensions

It is customary to analyze the topological properties of one dimensional structures by using the familiar concept of the Zak phase, directly linked to Berry's connection. We have shown here that another approach can be put forward, by using the poles and zeros of a function defined for all energies and not only for that corresponding to propagating modes. By using this tool, the extension of the topological classification of Bloch waves to lossy or non-Hermitian situations is straightforward and avoids the difficulties encountered when trying to extend the Berry connection approach to non-Hermitian systems. Moreover, our approach can be applied to structures in which a small amount of disorder is introduced. While a complete exposition goes beyond the scope of this work, let us give one numerical example. The structure is the same as that used in the previous simulations and defined in Figure 8, but a small amount of disorder is now added to the widths of all the layers. More precisely, we start with ten periods of the three-layers structure and then we add some disorder to the width of the layers. The disorder is obtained by modifying the width of the layers by an amount of $$\pm 0.01$$ at most with a uniform law. We thus deal with 30 layers for each structure. The entire structure contains 60 layers. A typical realization of the media is given in Figure 16.

The monodromy matrix is now computed by taking into account the entire structure since the periodicity is lost. The Bloch diagrams and the transmission spectrum are given in Figure 17.

The functions $$\chi _1$$ and $$\chi _2$$ are given in Figure 18.

A mode is obtained as can be seen in Figure 17: A peak is seen inside the transmission spectrum around $$k_0/2\pi =2.5$$ inside a common band gap of both photonic crystals. In this bang gap, the curves $$\chi _1$$ and $$-\chi _2$$ cross with inverted pole and zero patterns (see Figure 18). This shows that the method also works in that situation.

A natural question is to investigate whether it is possible to extend this approach to higher dimensional structures. It is a general rule that the methods employed in dimension 1 are difficult to extend to dimensions 2 or 3. While the extension of our approach seems difficult to extend to higher dimensions for continuous models, the situation seems more accessible when considering tight-binding models, where the proposed approach could give possible theoretical insights for handling the case of complex energies and for analyzing the Bloch variety.^[23] By tight-binding model, we have in mind a structure such as an inverted contrast 2D photonic crystal where holes are filled in such a way as to design a periodic superstructure made of weakly coupled cavities. We are currently investigating the properties of such a structure which is amenable to a discrete description.

Conflict of Interest

The authors declare no conflict of interest.