The evolution of topological singularities between real- and complex-frequency domains and the engineering of photonic bands for Hermitian and non-Hermitian photonic crystals

Singularity annihilation, generation, and evolving (SAGE) lead to the topological phase transition (TPT) in electronic, photonic and acoustic systems. Traditionally the singularity study of Hermitian systems is only focused on the real frequency domain. In this work, we systematically investigate the complicated SAGE in complex frequency domain (CFD) for one-dimensional (1D) Hermitian and non-Hermitian systems and a more general picture is revealed. First, we study the abnormal phenomenon that one singularity evolves from the first band to the zero frequency and then into the pure imaginary frequency for Hermitian 1D photonic crystals (PhCs). New results, e.g. the general condition for the singularity at zero frequency, the stricter definition of the Zak phase of first band and the phenomenon that more singularities are pushed from first band into the imaginary frequency, are found. Second, a general evolving picture of SAGE in CFD for Hermitian systems is constructed. Complicated processes of singularities in CFD are observed, such as the SAGE not only on the real frequency axis but also on the imaginary frequency axis, the closed evolving loops for singularities which connected imaginary-frequency axis and real-frequency axis. Even more, when the PhCs is degenerated since the permittivity on one kind layer becomes same as the neighbor layer, the singularities on the integral reduced frequency will move to infinite far away and come back with half-integral shift. Third, when gain or absorption is introduced in, the SAGE on a tilted axis is also observed. The phenomenon of one singularity moving back to real frequency axis for non-Hermitian systems means that the stable states with resonance could be realized. Such complicated and general singularity evolving picture in CFD opens a new window for the studies of TPT and the rich new topological phenomena could be expected. Besides the theoretical importance, the evolution of singularity can also be used to engineer the band properties of PhCs. Some novel applications, such as the super-broadband sub-wavelength high-transmission layered structure and the broadband deep-sub-wavelength absorber, are proposed.

Because of the non-interaction and robust features, photonic crystals (PhCs) have been thought to be a perfect platform to investigate the topological physics, where different topological invariants of electronic systems have found their counterparts [2,26,27]. Different topological invariants are widely used to describe the non-triviality of different dimensional systems, for example, the Zak phase for 1D systems with spatial inversion symmetry (SIS) [28], the Chern number for 2D-3D systems with breaking time-reversal symmetry [7] and the spin Chern number for topological insulators [22] with time-reversal symmetry. Beside the topological invariants, the singularity, which is thought as the position of the topological charge, is another very essential sign of non-triviality. Generally, it is very hard to study the topological systems by the singularity directly since its position is related with the gauge choosing, although the observable phenomena from topological non-triviality are gauge independent. Recently, for 1D binary PhCs, Chan's group [29] finds that the topological singularity can be directly positioned by the zero-scattering condition of a unit cell based on the transfer-matrix gauge and the non-zero Zak phase is from a π-phase jump of the Bloch states at the singularity. Hence, the study from the singularity evolution of low dimensional systems becomes a new and intuitive path to study the topological phase transition (TPT) of low-dimensional systems. Several interesting works are done based on this path for 1D PhCs with synthetic dimensions for different topics, such as the TPT, the Weyl points and the nodal lines [30][31][32][33][34], etc.
Very recently, it is reported that, at near-zero frequency range where the wavelength could be several orders larger than the scale of the unit, there could be non-trivial topology for 1D Hermitian PhCs [44]. Based on the 1D dielectric PhCs model with SIS, by tuning the structural parameters, it is found that one singularity can be pushed to zero frequency and then to the pure imaginary frequency domain. Even more, the topology of first band and first gap is still non-trivial after the singularity being pushed to imaginary frequency. These conclusions are very interesting and beyond the traditional effective medium theory [45] which is widely believed to be correct for very-low-frequency range and topologically trivial.
However, despite the TPT which is signified by the singularity evolving between different bands on RFD is well studied, the more general picture of singularity evolving in CFD for 1D Hermitian or non-Hermitian PhCs is uncovered to the best of our knowledge. Some basic questions are still waiting to be answered. The first is that 'Is the phenomenon observed in [44]. in 1D Hermitian systems just a special case of a more general picture of singularity evolving in the CFD for Hermitian and non-Hermitian systems?' . If answer is 'yes' , then we need to answer more questions, e.g. 'Comparing the discontinuous evolving on the RFD, such as the sudden annihilation of pair of singularities, is the singularity evolving picture in CFD continuous?' and 'Are there more complicated evolving paths for the singularities in the CFD?' , etc. After all, if we find that the TPT in the RFD is only a small section of a complicated picture in a larger domain (i.e. CFD), the understanding of the origin of TPT would be quite different. Even more, with the insight of singularity evolving in CFD, we can control the evolving paths of singularities by tuning certain structural or material parameters of our systems and find a new method to employ TPT on the design novel photonic devices from special topological properties.
In this work, we systematically investigate the topological singularity evolution in CFD for 1D Hermitian and non-Hermitian PhCs and a general picture is constructed. First, two general models of 1D Hermitian PhCs with SIS are introduced, the layered dielectric model (LDM) and the periodic modulation model (PMM), which have different advantages for topological singularity study. Second, by tuning structural or material parameters of 1D PhCs, we find that more singularities can be pushed to the zero frequency and then into the pure imaginary frequency domain. The general condition for a singularity exactly at zero frequency is obtained. Even more, to determine the topological non-triviality of first band and first gap of 1D PhCs, we present a general definition of the Zak phase of first band, with singularity passing through the zero frequency into the imaginary frequency. Third, the general singularity-annihilation-generation-evolution (SAGE) processes in CFD is explored. It is found that a pair of singularities can collide and annihilate with each other on the real frequency axis. If we expand RFD to CFD, such SAGE is revealed as a section of a continuous process that two singularities collide and then leave the real frequency axis in opposite directions and move into the CFD. The reversal process can be thought as the generation of a pair of singularities on real frequency axis. Actually, the phenomenon of singularity passing through zero frequency into the imaginary frequency [44] is a only special case of the SAGE in CFD. Surprisingly, we observe such SAGE not only on the real frequency axis, but also on the imaginary frequency axis. Complicated evolving paths of singularities, e.g. the closed loops between real frequency axis and imaginary frequency axis, are also observed. Another interesting phenomenon is that, when the complicated LDM (4-layers LDM) is degenerated into a simpler one (2-layers LDM) if we tune the permittivity of the central layer passing the permittivity of its two neighbor layers, the singularities at every integral reduced frequencies will move to infinity in the imaginary frequency direction, then, the new singularities will come back from infinity with half-integral shift. Forth, when the absorption or gain is introduced in, our models become non-Hermitian and the singularities will evolve on a tilted axis in CFD. After the two singularities leaving the tilted axis, one singularity can be back to the real axis, which means a static and resonant mode can be realized in non-Hermitian systems which is very helpful to design novel devices. At last, we demonstrate that the evolving of singularities provides a new method to engineer the photonic bands and some novel devices, such as the super-broadband sub-wavelength high-transmission layered structure and the broadband deep-sub-wavelength absorber, could be designed. The rich performance of singularity evolving in CFD broaden our comprehension of the origin of TPT and open a new window for researchers to study the topology for other waves (e.g. matter waves and acoustic waves) and for higher dimensional systems.

Two types of 1D PhCs Models
In this work, we introduce two types of 1D PhCs models with SIS. One is the LDM with N dielectric layers in the unit cell, while the other is the PMM which is the homogeneous dielectric background with different orders of periodic modulation. In this work, the relative permeability of all materials is supposed to be same as vacuum µ r = 1. For LDM, two examples with N = 4 and N = 6 are shown in figures 1(a) and (b). The first example is called 'ABCBA-model' , for which the starting point of unit cell is set at the central point of layer-A and its unit cell has two obvious SIS centers at the central points of layer-C and layer-A. The dielectric constants and the widths of different four layers are denoted by respectively. Similarly, we can construct the second example with N = 6, which is called as 'ABCDCBAmodel' with the dielectric constants and the widths of different layers as Different from the LDM, the dielectric distribution of PMM can be expressed as: ε(x) =ε + δ 1 cos(Gx + θ 1 ) + δ 2 cos(2Gx + θ 2 ) + · · · + δÑ cos(ÑGx + θÑ) where, G = 2π/Λ and Λ is the length of unit cell,ε is the average dielectric constant, δ i and θ i are the modulation strength and the original phase of ith-order modulation,Ñ denotes the largest order of periodic modulation for a particular model. Theoretically, all 1D periodic dielectric systems can be expressed by the PMM. But in this work, we only get to third-order (Ñ = 3) modulation for the topological singularity research. To keep SIS of this model, we have to set the original phase as θ i = 0 or θ i = π. Since θ = π is equal to the effect by setting θ i = 0 and changing the sign of δ i , we set all θ i = 0 in this work. One example of PMM is shown in figure 1(c) withÑ = 2 which means δ i = 0 for all i ⩾ 3. Two types of models have different advantages and disadvantages. For LDM, it could be strictly solved by simple 1D transfer-matrix method (TMM) [46,47], hence the analytical equations can be obtained for the topological singularities and can help to find the evolution tracks of singularities in parameter-space. However, LDM also has disadvantages, e.g. we can not obtain the physically clear meaning for some critical conditions of singularity evolving. To make up these disadvantages, we introduce PMM to study of singularity evolving of 1D PhCs in this work. We will see that PMM can help us to obtain deeper insight for the evolving processes of singularities and especially for some critical conditions. So, based on the comparison of the results from both LDM and PMM, we not only can follow the evolving tracks of singularities easily in parameter-space, but also can give the clear explanation of the critical conditions.

The evolving of singularity in pure imaginary frequency domain
In this section, similar as the previous work in [44], we will carefully study the whole evolving process of singularity from the second band to the first band and then to the pure imaginary frequency domain for both LDM and PMM, and find the general condition for singularity at the zero-frequency which is the critical condition for the singularity evolving from the RFD into the pure imaginary frequency domain. Then, the evolving of singularity in pure imaginary frequency and the topology of first band and first gap in the process are investigated. We find that the topological effect from the singularity in the pure imaginary frequency can be concluded by a new Zak phase on zero-frequency point. Therefore, a general definition of Zak phase of  first band can be obtained. At the last of this section, we will show that more singularities can be tuned to first band and then to the imaginary frequency domain by changing some structural or material parameters for both LDM and PMM and the general topological properties of the first band and first gap are discussed.
First, we will investigate the evolving of singularity from the second band to the first band and then to the pure imaginary frequency domain for PMM. The topological singularity can be strictly verified in several methods [29,44]. The simplest method is according to the reflection coefficient |r| of the system. The singularity, zero-scattering of the unit cell of the periodic system, is the zero reflectional point (i.e. |r| = 0), which can also be predicted by the phase vortex point in the reflection phase map [44]. We setÑ = 2, which means that δ i = 0 when i ⩾ 3. In our calculation, average dielectric constant and the second order modulation strength are set asε = 2 and δ 2 = −0.2, respectively, and the δ 1 is the tuning parameter during the process. The absolute value of reflection coefficient |r| for five PhCs with different δ 1 are shown in figure 2. We show that with the increase of δ 1 from −0.3, the singularity from the second band moves to the lower frequency as shown in figure 2(a), and the first gap will close when δ 1 = 0 as shown in figure 2(b). Continuing increasing δ 1 , the singularity will appear on the first band and the first gap opens again as shown in figure 2(c). When we increase δ 1 further, the singularity will move to the bottom of the first band and reach the zero frequency at δ 1 = 0.2 as shown in figure 2(d). For δ 1 > 0.2, the singularity will disappear in the RFD and evolve into the pure imaginary frequency domain as shown in the inset of figure 2(e). Then, we will study the general condition for the singularity at zero frequency. Based on ABCBA-model of LDM, Xiong et al [44] have find that the condition for the zero-frequency singularity as For more complex case of ABCDCBA-model, we find that the zero-frequency singularity satisfies the condition whose detailed derivation is shown in appendix A. As we have mentioned, all models of LDM can be expressed by PMM. Based on PMM, we find that the general condition for zero-frequency singularity is: In fact, for both LDM and PMM models with SIS, this condition can also be described as: where,ε is the average dielectric constant and ε(x = 0 − ) is the dielectric constant of background material since the origin point x = 0 is the interface between the background material and the PhCs. Physically, this condition can be understood intuitively. Since the wavelength λ → ∞ for near-zero frequency, the wave only can sense the average dielectric constantε according to effective-medium theory [45] and the zero-scattering can be realized when equation (3) is satisfied. We note that, if we can tune all δ i in equation (2) freely, the zero-frequency singularity condition can have multi-solutions, which is also one advantage from the discussion based on PMM and very important for our further study. Next, we will carefully investigate the evolving of singularity in pure imaginary frequency which is missed in previous work. In figure 3(a), we show that the singularity on pure imaginary frequency will move away from the zero frequency point with the increase of δ 1 from 0.2, based on the PMM with all other parameters same as that in figure 2. We find that, when δ 1 is tuned to 0.8, the singularity evolves to the infinity on the imaginary frequency axis. If we tune the δ 1 > 0.8, the singularity will not come back again. The critical condition for the singularity moving to the infinity on the imaginary frequency axis for PMM satisfies: which can also be written as: The similar phenomenon can also be seen in ABCBA-model. As shown in figure 3(b) for ABCBA-model, the singularity on imaginary frequency axis will move away from zero frequency with the increase of ε B and it will evolves to infinity when ε B → ε A . The critical condition ε B = ε A corresponds to the case that the ABCBA-model degenerates into a 1D binary PhCs (i.e. AB-model). Also, the singularity will not come back again on the imaginary frequency axis for ε B > ε A .

The topology of the first band and the first gap
It is reported in [44]. that the first band and first gap keep topological nontrivial when the singularity evolves from the first band to the imaginary frequency domain. So, it seems that the singularity in the imaginary frequency domain needs to be considered when we discuss the topological properties of Hermitian systems. However, as we have shown, the singularity can evolve to infinity on the imaginary frequency axis when the parameter is tuned to a critical value and will not come back when the parameter of our model is tuned over the critical value. Such evolution process of singularity determines whether we can find the singularity on the imaginary frequency axis. Then, some interesting questions appear, such as 'Does the topological properties of the first band and the first gap will change in the process?' and 'How can we signify the topological properties of Hermitian systems in such process of singularity evolution?' .
To answer these questions, we need to investigate the topological properties of the first band and the first gap in the process of singularity evolution on the imaginary frequency axis. A simple and direct method to judge the topological nontriviality of first gap of a PhCs is to check the existing of topological edge-state when it is spliced with a topologically trivial PhCs with same gap region. The PhC-A with δ 1 = −0.3, which is used in figure 2, is topologically trivial for the first band and the first gap. First, we tune the parameter δ 1 from −0.3 of PhC-A to δ 1 = 0.3 and obtain the model of PhC-E, which is also used in figure 2 and it is topologically nontrivial for the first band and the first gap since the singularity on the second band of PhC-A moves to the first band and then to the pure imaginary frequency in the tuning process. The nontriviality of PhC-E is also demonstrated by figure 4(a) in which a topological edge-state, the near zero reflection in the first gap, is clearly shown when we splice PhC-E with PhC-A. Second, we can tune the parameter δ 1 from 0.3 of PhC-E to δ 1 = 0.81, which is larger than the critical value δ 1c = 0.8 for the singularity to infinity on the imaginary frequency axis, and obtain the model of PhC-F. We know that there is no topological singularity on the imaginary frequency axis now for PhC-F since the singularity has moved to infinity without coming back. However, when we splice PhC-F with PhC-A together, the topological edge-state is still in the first gap as shown in figure 4(b). So the PhC-F is still topologically nontrivial for the first band and the first gap even if the singularity has moved to infinity and can not be found on the imaginary frequency axis. In other words, the topological nontriviality of PhCs is not affected when the singularity moves to infinity and disappears on the imaginary frequency axis. Then, how to distinguish the topologically-nontrivial case of PhC-F from the topological trivial cases is a question waiting to answer, since, for both of them, there is no singularity on the first band and the imaginary frequency axis.
In the previous work [29], the binary-layered model (AB-model) is studied and the Zak phase of the first band (called '0th band' in the work) is defined as: where, z α is the surface impedance of α − layer. However, the definition is obtained from the transfer-matrix in the work [29] and there is no detailed discussion for the physical origin of the Zak phase of first band. Also, since all singularities are on the higher bands for AB-model and there is no complex singularity-evolving process, e.g. the singularity from the second band to the first band and to the pure imaginary frequency axis, the discussion of the topology change of first band is absent in the work [29] obviously. Next, first we will investigate the physical origin of the topology of the first band of AB-model. Then, a general definition of the Zak phase of the first band of complicated models, e.g. LDM and PMM, is presented which can be used to determine the topology of the first band and the first gap of PhCs.
For AB-model, it is widely known that the Zak phase of nth-band can be obtained by the integral of Wilson-loop´G /2 −G/2 ⟨u n,k+δk |u n,k ⟩dk, where |u n,k ⟩ is the normalized periodic part of the Bloch function. The integral is trivial except a phase jump at the topological singularity, which contribute a π to Zak phase of the nth-band. However, there is no detailed study of the Zak phase for the first band of PhCs. For the Zak-phase of first band (n = 1) of AB-model, can we find similar singularity with a phase jump at zero frequency? We check the periodic Bloch functions |u H,k ⟩ with k = ±δk near zero frequency with the gauge from the transfer-matrix [29], where δk is a small value. A π phase jump at the zero-frequency point is found for some PhCs. In figures 5(a) and (d), the distribution of refractive index of two AB-models with ε A > ε B and ε A < ε B in a cell is shown, respectively. The real and imaginary parts of Bloch functions of magnetic fields  c) and (e), (f) for two models with δk = 0.02π/Λ. First, for all cases, we find that the real part of magnetic field is going to the limit 0 while the imaginary part is close to ±1 when δk → 0. So, we can neglect the real part and just take care of the imaginary part of the fields if the frequency is extremely close to zero. Second, we find that |u H,−δk ⟩ = −|u H,δk ⟩ for ε A > ε B case by comparing figures 5(b) and (c), while |u H,−δk ⟩ = |u H,δk ⟩ for ε A < ε B case by comparing figures 5(e) and (f). So, if we follow the traditional definition of Zak phase from the integral of Wilson-loop, the non-zero Zak phase of first band will be obtained for the case with ε A > ε B since the discontinuity of |u H ⟩ at the zero frequency. In other words, the zero-frequency point is a special singularity for the first band and it must be taken into account to understand the topological nontriviality of first band.
Then, for more complicated models with SIS, such as LDM and PMM, we will find a general definition of Zak phase of the first band which is also essential for the topology of first gap. Now, we know that three basic phenomena need to take care for the study of Zak phase of first band for complicated models with SIS, the singularity from higher bands to the first band, the passing of the singularity from the first band to the imaginary frequency through the zero frequency, the singularity moving to the infinity and disappearing from the imaginary frequency axis.
First, we need to generalize the contribution of zero-frequency singularity from AB-model to the complicated models. As we have discussed, when the frequency is approaching zero the wavelength diverges and the average dielectric constant is the effective dielectric constant for the wave inside the model. So, for complicated models with the contribution of zero-frequency singularity, based on our numerical results we generalize the equation (6) by: where, θ Zak ω=0 is defined as the contribution of the zero-frequency singularity, ε x=0 − and µ x=0 − are the relative permittivity and permeability of background material since the PhCs are at the right side of origin point,ε andμ are average permittivity and permeability of PhCs which are defined asε =´Λ 0 ε(x) dx/Λ,μ = Λ 0 µ(x) dx/Λ. Equation (7) exactly allows the average impedance matching to the background material. This generalization is checked by numerical methods and we find that it is correct for complicated models. Since the relative permeability of all materials is suppose same as vacuum µ r = 1 in this work, it is clear that if ε x=0 − >ε, there is a topology singularity at zero frequency, otherwise, no singularity at zero frequency.
Then, we will discuss the sum of total contributions for the Zak phase of the first band, which determines the topology of the first band and the first gap. In figure 4(a), we have shown the process that the topological property of the first band has be changed when a common singularity, which is defined by the resonance for a unit cell of PhCs, moves from second band to the first band. It indicates that we need to at least consider two

PhCs Zak phases
PhC contributions for the topology of the first band, (a) the contribution of zero-frequency singularity which is determined by equation (7), (b) the contribution from common singularity or singularities on the first band.
If there are odd number of singularities, including both zero-frequency singularity and the common singularities on the first band, the total Zak phase of first band is θ Zak 1st = π, otherwise, θ Zak 1st = 0. Combining these two kinds of singularities, the total Zak phase of the first band can described as: where, θ Zak represents the contribution of common singularities on the first band. At last, we need to answer the question 'What has happened for the topology of first band when a common singularity passes the zero frequency into the imaginary frequency domain and disappears on the infinity on the imaginary frequency axis?' . The result is that the topology will not change in these processes, as we have demonstrated in figures 4(a) and (b) numerically by checking the existing of topological edge state. Next, we will explain the result according our generalized definition of the total Zak phase θ Zak 1st of first band. According to equation (3), we find that, when a common singularity is passing zero frequency, ε x=0 −μ/εµ x=0 − is exactly equal to 1. In other words, when a singularity passing zero frequency into the imaginary frequency domain, the sign of sign[1 − ε x=0 −μ/εµ x=0 − ] also changes, so that θ Zak ω=0 in equation (7) changes. So, two changes happen simultaneously in this process, i.e. one is the disappearing of a common singularity on the first band and the other is the changes of θ Zak ω=0 . Since two simultaneous changes will contribute zero to the total Zak phase according to equation (8), the topology of the first band and the first gap will not change in the process. Further more, when the singularity evolving in the imaginary frequency (including the disappearing at infinity), two contributions of Zak phase keep same, therefore, the topology of the first band and the first gap will not change during this process.
Specifically, both of the PhC-A and PhC-F discussed in figure 4(b) without singularity on the first band and the imaginary frequency domain, but the sign of equation (7) are different and consequently the topology of the first band and first gap for PhC-A and PhC-F are different. Similar analyses can also be applied to other PhCs shown in figure 2. We calculate the Zak phase of the first band of these PhCs according to equation (8) and results are shown in the table 1. The results of Zak phase of these PhCs are consistent with the existing of edge state in figure 4. We also emphasize that our general definition of Zak phase based on the integral of Wilson-loop for the first band in this work is self-consistent with the cases of simple AB-model [29].

More singularities on the first band and into the imaginary frequency
In the above sections, the evolving process of one singularity from higher bands to the first band then to the imaginary frequency is well investigated. However, can we realize more complicated evolving processes, e.g. more singularities move to the first band and then into the imaginary frequency? In this section, we will show the realizations of such complicated processes for both LDM and PMM.
First, we consider a ABCDCBA-model whose reflection coefficient |r| versus reduced frequency are shown in figure 6. In figures 6(a) and (b), we have shown that it is possible to push second singularity from the second band for PhC-G to the first band for PhC-H when we tune three parameters d A , ε D and d D . Further more, in figure 6(c), one singularity is pushed into the imaginary frequency while the other one is still on the first band for PhC-I. Then in figure 6(d), we can see that there is a topological edge state when we splice PhC-G and PhC-I since the topological difference for the first gap between them. So, we can push more singularities to the first band and then to the imaginary frequency. The basic rule to judge the topological triviality or nontriviality of the first band and first gap for the PhCs with more singularities on the first band is still determined by equation (8). The Zak phases of first band for PhC-G, PhC-H and PhC-I are shown in table 2.

PhCs Zak phases
PhC frequency while the other singularity is pushed from the second band to the first band, as shown in figure 7. The topological properties of the first band and first gap changes because it is a TPT process. In this section, we have carefully investigated the evolving process of singularities from higher band to the first band and then to the imaginary frequency. The general condition for the zero frequency singularity and the specific singularity-evolving processes in the pure imaginary frequency domain are discussed. Then, the general Zak phase of the first band is defined, which is determined by two aspects, i.e. the contribution of zero frequency singularity and the common singularity (or singularities) on the first band. We show that the topology of the first band and the first gap is unchanged when a singularity (or singularities) is pushed to imaginary frequency through the zero frequency, since both effects that the topological property of zero-frequency singularity is changed and one singularity is absent on the first band would cancel each other. Lastly, the phenomenon that more singularities are pushed to the first band and then into the imaginary frequency is realized for both LDM and PMM and the topology properties of these PhCs during the singularity-evolving processes are also discussed.

The evolution of topological singularities in complex-frequency domain
In the previous section 3.1, the phenomenon of singularity from the RFD into the pure imaginary frequency domain are studied. The phenomenon can also be understood as the process that a pair of singularities from positive and negative real frequency axis meet at zero frequency and annihilate each other if we only take care of the RFD when the parameter is tuned over the critical value [44]. However, at non-zero frequency on bands, we have also observed the similar phenomena of annihilating or generating of the singularities. So it is natural to wonder whether there is a more general picture to unite both phenomena and reveal more complicated mechanisms of singularity evolving on the whole CFD. In this section, we will extend the singularity study from the RFD into the whole CFD and a general picture of singularity-evolving in the CFD is revealed.
First, we consider the ABCBA-model and focus on the RFD. The relative permittivity of layer-A and layer-C are fixed as ε A = 4, ε C = 9, respectively, and the length of all layers are fixed as d A = 0.4Λ, d B = 0.5Λ, d C = 0.1Λ, where Λ is the length of the unit cell. By changing the relative permittivity of layer-B ε B , we can study the singularity-evolving process of this model, which is shown in figure 8. The reflectivity in dB (10 lg |r| 2 ) versus reduced frequency of semi-infinite PhC-L with ε B = 1.53 in figure 8(a) shows that there are two singularities in the 11th band. However, with the increase of ε B , two singularities will move to each other and will collide when ε B is about 1.5367. Two singularities annihilate each other at reduced frequency ω r Λ/2πc = 3.104 and then there is no singularity on 11th band for larger ε B , as shown in figure 8(b) with ε B = 1.545. The Zak phase of 11th band will not change because singularities in this band disappear in pairs. The reverse process by tuning the same parameter can be thought as the singularity pair generation. The phenomenon of the singularity pair annihilation-generation (i.e. SAGE) at bands is very similar as that at zero frequency, so it is natural to wonder that such phenomenon is very general if we extend our frequency domain from real frequency and pure imaginary frequency into the CFD.
Next, we will demonstrate that, in CFD, such SAGE process is very general and very interesting since it can tell us where are these singularities from. The complex frequency is defined as ω = ω r + iω i , where ω r and ω i are real numbers. We note that our system is still a Hermitian one since the dielectric constants of layers are supposed to be real, in other words, without absorptions or gains. The diagrams of reflectivity (dB) for different semi-finite PhCs in CFD (ω r , ω i ) are shown in figure 9. In figures 9(a)-(c), we repeat the pair-annihilating process shown in figure 8 in CFD. The low reflectivity points signed as the deep blue points represent topological singularities in figures 9(a) and (b), which are circled by black dots. In figure 9(c), we find that, after pair-annihilating process, two singularities have just disappeared on RFD, but in CFD they are still there and move in opposite directions leaving the real axis. In CFD, two singularities are conjugate with each other. The reflectivity at the singularity on the negative imaginary-frequency region (under the real frequency axis) is approaching infinite while the singularity on the positive imaginary-frequency region (upper the real frequency axis) is approaching zero. For clearance, the trajectories of singularities evolution are also depicted in 3D space (ω r , ω i , ε B ) in figure 10(a), in which blue circles and red circles represent singularities in real-and complex-frequency domain, respectively. Further more, if we tune the parameter reversely for the process shown in figure 10(a), it is a process for the generation of a pair of singularities at one point on the real frequency axis, and then two singularities moves in the opposite directions on the real frequency axis.
Actually, the phenomena of SAGE are widely existing in CFD for 1D PhCs. In figures 9(d)-(f), we show that SAGE process can be at other bands, e.g. third band, when we tune certain structural or material parameters. Even more, the phenomenon that the singularity is pushed from real frequency into the pure imaginary frequency domain, studied in previous sections, could also be thought as a special case of SAGE at  As we note that SAGE processes in RFD are very general, which can be observed in complicated 1D PhCs, such as ABCDCBA-model and PMM. It's rational to wonder if such SAGE processes can only happen on the real frequency axis. Actually, it is not limited on the real frequency axis, since we surprisingly find that the SAGE process can also happen on the imaginary frequency axis. For example, for PMM withÑ = 3, we fix ε = 2, δ 1 = 0.08, δ 2 = −0.4 and we show the singularity-evolving process by changing δ 3 in figure 11. As shown in figure 11(a), two singularities on the imaginary frequency axis will move towards each other with the increase of δ 3 . Then, they will collide at imaginary reduced frequency ω i Λ/2πc = 0.134 when δ 3 = 0.3208, and then they will separate into the CFD with opposite real part, as shown in figure 11(b). So, we show that the SAGE process not only happens in real frequency axis, but also happens in imaginary frequency axis. Interestingly, continuing increasing δ 3 , these two singularities in CFD (under and above the real frequency axis) will approach the real frequency axis together and collide at real reduced frequency ω r Λ/2πc = 0.453 when δ 3 = 0.6577, and then two generated singularities on real frequency axis will move in the opposite directions. The whole trajectories of two singularities form a closed loop on the CFD, which are shown in figure 11(c).  The complicated SAGE process shown in figure 11 implies that, although the TPT processes have been widely studied which is dominated by the singularity evolving on real frequency axis, the rich evolving phenomena of singularities on whole CFD are still waiting for further study. Even more, the relation between the trajectories of these singularities and the track for tuning parameters(structural or material) in multi-parameter space need to be investigated intensively in future.
Another inspiring phenomenon for singularity-evolving in CFD is that the singularities with integral real reduced frequencies move to infinite at imaginary frequency direction and then come back from infinite with half-integral real reduced frequencies when a 1D complicated PhCs model is degenerated to a simpler PhCs model when we tune certain material parameter. For example, for a ABCBA-model with ε B < ε A , when we tune ε B → ε A , the ABCBA-model will degenerated into a binary-layered AB-model. Then we find that there are singularities with real reduced frequency coordinate at every integral number and they are moving towards the positive or negative infinite imaginary frequency with almost same imaginary frequency coordinate, as shown in figures 12(a) and (b). The special example of this phenomenon has been observed when we discuss the singularity on the pure imaginary frequency moving to infinite as shown in figure 3(b), now we can see that there are many of them. They all disappear at infinite far away when ε B = ε A . However, when we tune ε B further so that ε B > ε A , naively we guess that they will come back from far away at same position since the model becomes ABCBA-model again. But the numerical results show that the singularities come back from infinite with half-integral real reduced frequency on real frequency axis, which are shown in figures 12(c) and (d). If we increase ε B even more, we will find that these singularities will arrive at real frequency axis and generate the common singularities on RFD. Theoretically, we can prove that the coordinates of singularities on real frequency axis are integral when ε B approaching ε A from down-side, and the half-integral when ε B approaching ε A from upper-side, since there is a sign change from two sides. The theoretical proving is presented in appendix B. Originally, we know that the moving of singularities on real frequency axis between different bands can cause TPT. Now, the half-integral shifting of real frequency for singularities at PhCs degenerating process presents a new path for singularity jumping between different bands. As shown in figure 12(a), the singularities disappearing at infinite with the coordinates at 1 is from the fifth band, while the singularities moving back from infinite with the coordinates at 1/2 on real frequency axis turn out to land at the third band, as shown in figure 12(d). Such half-integral shift for singularities and its relation with the degenerated PhCs structure also reveal the deep physics behind the evolution of singularity in CFD.
Briefly, in this section, a more general picture of singularity evolving in the CFD is revealed for 1D Hermitian PhCs. First, by tuning certain parameter, we show the SAGE that the annihilation of a pair of singularities on real frequency axis can be explained as that two singularities leave real frequency axis into the CFD and the reversal process is the generation of a pair of singularities on real frequency axis. Actually, the phenomenon of singularity passing through zero frequency into the imaginary frequency in previous section can be thought as a special case of SAGE process. Second, the complicated evolving paths of singularities in CFD are investigated. We observe the SAGE process not only on real frequency axis, but also on the imaginary frequency axis, and the closed loop for singularity-evolving paths between imaginary frequency axis and real frequency axis is observed. Other novel singularities-evolving processes, such as the half-integral reduced frequency shifting of singularities when the ABCBA-model degenerates into AB-model, are also revealed. Beyond the traditional singularities-evolving picture on real frequency axis, which is usually accompanied by the TPT process, the general picture of singularity in the CFD indicates that the evolution of singularities is continuous and complicated, which is worthy for further study.

The evolution of topological singularities in complex-frequency domain for non-Hermitian systems
So far, all of our studies are based on the Hermitian systems without any gain or absorption. Next, we will introduce the imaginary part of permittivity ε ′ ′ into our 1D systems and study the evolving of the singularities for 1D non-Hermitian systems. Physically, the sign of ε ′ ′ could be positive which means the absorptive materials, or negative which means the gain materials. We will show the evolving phenomena of singularities for non-Hermitian systems in CFD. First, the singularities, which move on the real frequency axis for Hermitian systems, will evolve on a tilted axis in the CFD for non-Hermitian systems. Second, the SAGE can also appear on the tilted axis. Third, a singularity still can be back to the real frequency axis for a non-Hermitian system by tuning certain structural or material parameters, which can help us to design novel devices.
We start from ABCBA-model by introduce the permittivity of different layers as ε m = ε ′ m + iε ′ ′ m = ε m0 * (1 + iW), where m = A, B, C and W is the absorption or gain relative strength which is chosen as ±0.1 in this example. At first, the material parameters and the structural parameters are same as those in figure 9(f) and then we tune ε B from 5.8 to 5.6 to show the singularities evolving path in CFD. The diagrams of reflectivity in dB and the evolving of singularities in CFD with W = 0.1(with absorption) and −0.1 (with gain) for the 1D semi-infinite PhCs are shown in figure 13. We find that the singularities move on a tilted axis, whose slope can be obtained by: Even more, we find that, by further tuning ε B , after the colliding of two singularities, one singularity can be back to the real frequency axis when ε B = 5.711, as shown in figures 13(c) and (g). The singularity in real axis means that the static resonant mode could be realized for both gain and absorptive materials, especially for absorptive materials that can be easily found in nature. We will show in next section that this property is very important for us to design the novel devices. The trajectory of singularities in CFD with ε B turning from 5.8 to 5.6 are shown in figures 13(d) and (h), for W = 0.1 and −0.1, respectively.

The band engineering and applications
In this section, we will show that the transmission property of bands can be engineered and some novel applications can be designed by controlling the evolution of topological singularities in real-and complexfrequency domains for 1D Hermitian and non-Hermitian PhCs.
First, we propose the super-broadband perfect transparency with the working frequency ranging from zero frequency to the maximum frequency which is about seventy percentage of first band width for a layered PhCs with ten cells. The 'perfect transparency' is defined as |r| < 10 −3 , which corresponding to the reflectivity R = |r| 2 is less than −60 dB. In previous study [44], a ABCBA-model with broadband high transmission at low frequency is presented when we tune one singularity to the zero frequency. However, in this work at section 3.1, we find that two singularities can be tuned to the first band simultaneously for ABCDCBA-model in figure 7. Therefore, it is possible to control two or more singularities at or near the zero frequency to obtain a structure with super-broadband perfect transparency, whose band-width could be five times wider than the structure in previous design [44].
In figure 14(a), we show the reflection coefficient |r| versus frequency of ten-cells structure of ABCBA-model based on the design in [44]. At the left side of the insert of figure 14(a) one singularity at zero-frequency is pointed out by circled red dots and at the right side of insert a red dashed line shows the edge perfect transparency frequency range which is stopped at 4.35 GHz. In figure 14(b), we show the reflection coefficient |r| of ten-cells structure of ABCDCBA-model in which two singularities can be tuned to the first band. To guarantee the fairness of comparing, we keep the optical path of one cell of two models same. In the insert of figure 14(b), two singularities are pointed out, one fixed at zero frequency and the other at 18 GHz. We can obtain perfect transparency at the range 0-18.74 GHz as shown in figure 14(b), which is four times wider than that in figure 14(a). Further more, to obtain even more wider frequency range of perfect transparency, we can improve the strategy of our design, e.g. moving the first singularity a little away from the zero-frequency. In figure 14(c), we show that, when the two singularity are set at 8.9 GHz and 18.8 GHz, the perfect transparency is at the range 0-20.5 GHz, which is about five times wider comparing with the figure 14(a). We emphasize that, now the perfect transparency frequency range has surprisingly covered 68.3% of first band, whose range is from zero to 30.03 GHz. Such super-broadband perfect transparency has never been reported to the best of our knowledge.
Even more, the band-engineering method by singularity tuning is also applicable for other higher-order bands. Here, we define the 'almost perfect transparency' (APT) as |r| < 10 −2 for higher bands, which corresponding the reflectivity R = |r| 2 is less than −40 dB. In figure 15, we show the high-transmission band-engineering of fifteen-cells ABCDCBA-model for different bands. In figures 15(a)-(c), we show the  reflection coefficient |r| versus frequency for the second, third and fifth band, respectively. In the inserts of figure 15, the APT frequency range can cover 50.16%, 49.38% and 36.62% for the second band, the third band and the fifth band, respectively. Such broadband APT can widely used on the design of the optical communication, the display screen, etc. At last, we emphasize that such band-engineering method based singularities is also applicable for 2D and three-dimensional (3D) PhCs.
Another application we proposed is the broadband deep-sub-wavelength absorbers. As shown in figure 13, when we introduce the absorption into the material of our PhCs, one singularity can be settled on real frequency axis, which means that the PhCs can be the perfect absorber at that frequency since of the resonance. This means that we can use such PhCs to design very good absorber. Generally, it is supposed that the absorber has the perfect electric conductor (PEC) as the background. Since the PEC background, there is no transmission, hence the absorption α can be obtained by α = 1 − R. The deep-sub-wavelength absorber is defined as that the absorber thickness L can be one order smaller than the wavelength with the 'good absorption' which means the absorption α is larger than 99%, corresponding to the reflectivity R less than −20 dB. Physically, it is generally very hard for a 'very thin' absorber to absorb 99% of EM energy.
We consider a ABCDCBA-model with the permittivity of different layers as ε m = ε ′ m + iε ′ ′ m = ε m0 * (1 + iW 1 ), where m = A, B, C, D and W 1 is the relative strength of the absorption. The reflectivity (dB) of the ten-cells PhCs with PEC as the background is shown in figure 16. In figure 16(a), the curves of reflectivity (dB) versus frequency are shown for different absorbing strength W 1 with normal incidence. We find that, with the increase of W 1 , the −20 dB range expands in both the lower and higher frequency directions. When W 1 = 2, a singularity is exactly settled at real frequency axis. In this case, the lowest frequency for 'good absorption' is at 1.43 GHz, whose wavelength is about ten times larger than the total thickness of the layered absorber L = 10 × 2 mm = 20 mm. The frequency range of good absorbtion shown in figure 16(a) for W 1 = 2 is very broad and there is a 'perfect absorption' (less than −60 dB) range from 4.52 GHz to 11.57 GHz. We have also investigated the cases with different incident angles for both s-and p-polarizations, which are shown in figures 16(b) and (c), respectively. We find that the absorber can maintain high absorption efficiency till the incidence angle approaching 60 • for both s-and p-polarizations, which demonstrates that the absorber is very robust for different incidence angles.

Conclusion
In summary, based on two kinds of models of 1D Hermitian and non-Hermitian PhCs, we have investigated the general evolving pictures of the topological singularities in CFD. First, the phenomenon that singularity evolves from the first band into the pure imaginary frequency is investigated. Some new results, such as the general condition for singularity at zero frequency and more singularities can be pushed to the zero frequency and then into the pure imaginary frequency domain, are obtained. Even more, the general definition of the Zak phase of the first band is presented to determine the topology of the first band and the first gap. Second, the general singularity-evolving picture in CFD are constructed. In the new picture, the SAGE process on real axis can explained by a continuous evolving process of singularities in CFD. Actually, The phenomenon that the singularity passing through the zero frequency into the pure imaginary frequency is a special case of the SAGE process. Surprisingly, such SAGE process is observed not only on the real frequency axis, but also on the imaginary frequency axis, and the closed loops for the singularities to move from imaginary frequency axis to the real frequency axis are observed. Other complicated singularities-evolving phenomena in CFD are also observed, such as the half-integral shift of real reduced frequencies for singularities as they moves to infinite at imaginary frequency direction and then come back, when the ABCBA-model is degenerated into the AB-model. Third, the singularity-evolving in 1D non-Hermitian systems with gain or absorption is also studied. The SAGE phenomenon is observed in a tilted axis for non-Hermitian 1D PhCs and one singularity of a pair can move back to the real frequency axis, which means that the static resonant mode can be realized. Besides the theoretical importance for topological study, the evolving of singularities in CFD is helpful to engineer photonic bands and some novel applications, such as the super-broadband sub-wavelength high-transmission layered structure and the broadband deep-sub-wavelength absorber, are demonstrated. Beyond the traditional singularities-evolving in real frequency axis, which is usually accompanied by the TPT process, this work opens a new window to study the topological evolving of the 1D systems from the view of the singularity evolving in CFD.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). .
According to equation (13), the critical condition for singularity at zero frequency for ABCDCBA-model can be computed by setting T 12 = 0. So we obtain: We find that equation (14), the critical zero frequency singularity for ABCDCBA-model, is just equivalent to the equation (3), which is the general condition for the zero frequency singularity.
Appendix B. The half-integral reduced frequency shifting of singularities when ABCBA-model degenerates to AB-model In figure 12, we show that the singularities will move to the positive (or negative) infinite imaginary frequency with real integral reduced frequency when ε B → ε − A and will come back from the infinite imaginary frequency with half-integral shift of real reduced frequency when ε B > ε A . In this appendix, we will give a mathematical proof of the phenomena.
We set the center of layer-A as the origin and the matrix element T 12 of the transfer matrix T of ABCBA-model can be written as [34]: where, At the critical condition ε B ≈ ε A , we set ε A = ε B + δ, where δ is a small number. When ε A > ε B , δ > 0, otherwise, δ < 0. The following approximation can be obtained: where, n = −1/(n A n B ), m = (n 2 B + n 2 C )/2(n B n C ). According to equation (15), the trajectory of singularities can be obtained by setting T 12 = 0: Considering the CFD, the wave vector k = (k r + ik i ). When δ → 0, we suppose the imaginary wave vector of singularities satisfy k i >> 1 and the relationship between k i and δ can be obtained later. So, tan(kn C d C ) = tan(k r n C d C ) + i * tanh(k i n C d C ) Substituting equation (19) into equation (18), the trajectory of singularities when δ → 0 can be obtained: Considering the wave vector k = (k r + ik i ), equation (20) can be divided into two equations: Divide the equation (22) by equation (21), and the take the approximation tanh(k i n B d B ) ≈ i, we can obtain:  (25) and (28) compared with the results of strict TMM. (a), (b) is the real and imaginary reduced frequency of singularity versus δ when δ > 0, whose real reduced frequency equals to 1. Blue circles are from strict TMM and black lines are from equations (25) and (28), respectively. (c), (d) is the real and imaginary reduced frequency of singularity versus δ when δ < 0, whose real reduced frequency equals to 0.5. Red asterisks are from strict TMM and black lines are from equations (25) and (28), respectively.
So, sin(k r n B d B ) = 0 and cos(k r n B d B ) = 1 or −1. According to equation (21), because F 1 is a constant, the sign of cos(k r n B d B ) depends on the sign of δ. That is, when δ > 0 (ε B < ε A ), cos(k r n B d B ) = 1, otherwise, cos(k r n B d B ) = −1. So, real wave vector k r can be described by a piecewise function when ε B ≈ ε A : Next, we will derive the relationship between k i and δ. We have supposed k i is a large number, so that cosh(k i n B d B ) ≈ sinh(k i n B d B ) = exp(k i n B d B )/2. The equation (21) can be rewritten as: So, imaginary wave vector k i and imaginary reduced frequency ω i Λ/2πc satisfy: Now, we have obtained the real and imaginary reduced frequency of singularities when δ → 0 from equations (25) and (28), respectively. To verify the correct, we compare the results of equations (25) and (28) with the results of strict TMM in figure 17. In figures 17(a) and (b), in which δ > 0, the real and imaginary reduced frequency of singularity obtained from equations (25) and (28) are labeled by black line, which is perfectly agree with the results from strict TMM (blue circle). Also, the excellent agreements between equations (25) and (28) (black line) with TMM (red asterisk) are also found in figures 17(c) and (d), in with the real reduced frequency shifts half when δ < 0.