Anomalous Light Scattering by Topological PT-symmetric Particle Arrays

Robust topological edge modes may evolve into complex-frequency modes when a physical system becomes non-Hermitian. We show that, while having negligible forward optical extinction cross section, a conjugate pair of such complex topological edge modes in a non-Hermitian -symmetric system can give rise to an anomalous sideway scattering when they are simultaneously excited by a plane wave. We propose a realization of such scattering state in a linear array of subwavelength resonators coated with gain media. The prediction is based on an analytical two-band model and verified by rigorous numerical simulation using multiple-multipole scattering theory. The result suggests an extreme situation where leakage of classical information is unnoticeable to the transmitter and the receiver when such a -symmetric unit is inserted into the communication channel.

Parity-time (PT )-symmetric quantum mechanics 1 has opened up a new direction in searching for unconventional states of matter. Non-Hermitian PT -symmetric systems have been a subject of intense studies because they exhibit unusual properties, such as PT -phase transitions 2 . While realization in quantum systems could be in doubt, several unconventional PT -symmetry related phenomena, such as transition from real-frequency modes to conjugate pair of complex-frequency modes, have been applied in classical photonic systems [2][3][4][5][6][7][8][9][10][11][12] . Recently, many efforts have been put on extending topological band theory [13][14][15][16][17][18][19][20][21] to non-Hermitian PT -symmetric systems. For example, there have been different theoretical approaches to generalize topological invariants using bi-orthonormal basis [22][23][24][25] , redefining the inner product 26 , or using the global Berry phase 27 . Topological transition in the bulk of non-Hermitian system has also been realized 28 . It has been proposed that eigenstates of these non-Hermitian system associated with exceptional points could lead to new physics and applications, such as realization of Majorana zero modes 29,30 and single mode lasers 7,8,31 .
Topological edge modes may evolve into complex-frequency modes when the system becomes non-Hermitian [32][33][34] . The physical consequence of such complex topological edge modes, which decay in both space and time, is obscure so far. In this paper, we suggest a way to realize complex-frequency topological edge modes in PT -symmetric photonic systems. We show that a conjugate pair of topological complex-frequency edge modes in a PT -symmetric photonic systems can be realized through observing an anomalous sideway scattering by an array of subwavelength resonators coated with gain media. This suggests an extreme situation where leakage of classical information is unnoticeable to the transmitter and the receiver when such a PT -symmetric unit is inserted to the communication channel.
Scientific RepoRts | 6:38049 | DOI: 10.1038/srep38049 k z x y x y z E k is the eigenvalue, u is the right eigenvector, k is the Bloch wave vector, and d is the lattice constant.
It should be noted that h x (k), h y (k), and h z (k) are periodic real-valued functions with period of 2π/d in k, and they form a  h-vector space as shown in Fig. 1(a). The subscripts x, y, and z refer to the directions in the  h-vector space (not the real space), while the expressions are determined by the actual approximation. Each system is represented as a closed loop  in Fig. 1(a) as the parameter k varies from − π/d to π/d. This closed loop is associated with a winding number which determines the Zak geometric phase of a band.
The Zak phase of each band (+ or − ) is usually expressed as 18,20,21,[36][37][38][39] where w is the (integer) winding number of  h(k) about the h z axis as k varies. It can also be shown that all even (odd) integer values of w are equivalent and Zak phase is thus usually quantized as zero or π. Since there exist different approaches in defining the Zak phase, we also provide a comparison to the results in bi-orthonormal basis (see Supplementary Information).
Zak phase can be used to classify the bulk band topology due to gauge invariance. More importantly, its value can help us predict the existence of the edge modes when all bulk states are in the unbroken PT -symmetric phase. For a large but finite PT -symmetric system, the weakly coupled edge modes also guarantee the existence in form of conjugate pair. In this case, the system can be classified (according to the bulk states) as three regimes summarized in Fig. 1(b). In this paper, we focus on the complex edge modes in regime III which can be simultaneously excited by a plane wave. In general, when  symmetry is broken, a linear combination of the edge modes can be excited so that the symmetry of the scattered waves may not follow the spatial symmetry of the plane wave. In the next section, we use a concrete example to show that when parity () and time-reversal ( ) symmetries are broken but PT symmetry is not broken, the spatial symmetry of the scattered waves can be nearly opposite to that of the excitation plane wave, which finally gives rise to a negligible forward optical extinction but non-zero sideway scattering.

PT -symmetric plasmonic particle array
We consider a spatially periodic plasmonic structure in which the electromagnetic modes are represented by the eigenvalue problem equivalent to Eq. (1) and takes the form similar to Bergman's representation [40][41][42][43][44][45][46] : where k, p k , and ε 1 are the Bloch wave vector, eigenvector, and eigenvalue respectively. The problem corresponds to a dimer array formed by two types of coated spherical plasmonic nanoparticles with alternative separations, s and d − s, as shown in Fig. 2(a). The coating shells are non-dispersive dielectrics, and are gain-loss-balanced. It with complex eigenvalues Figure 1. Classification of topological bands and edge modes of a PT -symmetric system described by Eq. (1). (a) Two types of paths in parameter space. The blue and the red path have winding number 1 and 0, respectively. (b) Table summarizing the three regimes according to the bulk and the edge state frequencies, including the Zak phase. In regimes I and III, the bulk states have real eigenvalues and the entire bulk dispersion relation is in the unbroken PT -symmetric phase. In regime II, some bulk states are associated with complex eigenvalues and thus part of the bulk states are in the broken PT -symmetric phase.
should be noted that there have been previous studies on plasmonic PT -symmetric systems 47,48 . Here, we focus on the topological edge modes 20 .
Eq. (5) comes from the coupled dipole equation 20,45,49,50 , which is obtained by approximating each nanoparticle as a point dipole scatter, using the Bloch's theorem, and taking the quasistatic approximation (see Methods). The dipole moments on nanoparticle A and B are denoted by the vector p k = (p k;A , p k;B ) T , while the matrix A k is related to dipole couplings and the polarizabilties. Eigenvalue ε 1 is also the dielectric function of the plasmonic core, and is mapped to the plasmon frequency ω via the Drude model 20,45,49,50 , given by in which ω p and 1/τ are the plasma frequency and electron collision frequency for the plasmonic core. The finite plasmonic particle array has no  and  symmetries but PT -symmetry when the plasmonic cores are lossless (1/τ = 0). A detailed discussion on symmetry operators is given in the Methods.

Complex edge modes
The longitudinal mode (x component) frequencies of a finite PT -symmetric particle array when 1/τ = 0 are shown in Fig. 2(b). We focus on the longitudinal oscillation since it only weakly couples with the light line. As described by the table in Fig. 1(b), there are three regimes (I to III) in Fig. 2(b). In regime I, two bands can be clearly identified, and all ω are real. This implies all the eigenvalues ε 1 are real, so the entire bulk dispersion relation is in unbroken PT -symmetric phase. Regime III is similar to regime I except that there are two complex topological edge modes supported in the band gap in regime III. The frequencies of these two topological edge modes form a complex conjugate pair, which is associated with the horizontal blue line and the red lines at Im(ω/ω p ) = ± 0.002. The existence of these complex edge modes are due to γ = π, which is classified as the (non-trivial) regime III. The evaluation of Zak phase for different gain/loss parameters are provided in Supplementary Information. The two edge modes are eigenmodes with simultaneous complex frequency and complex wavevector, meaning that they decay or grow in both space and time. If the array is long enough, we have the following closed form solution (see Methods): n t n d ; , ω 0 is the complex frequency of an edge mode, and p n;σ (t) represents the time-domain dipole moments in the nth unit cell. Both of the edge modes decay spatially, and the mode patterns at point D are shown in Fig. 2(c) for reference. Although the relation between spatially decaying modes and broken time-reversal symmetry has been studied 51 , the physical consequence of modes that decay in both space and time is obscure. In the next section, we will see that these modes can give rise to an anomalous scattering phenomenon.

Vanishing optical extinction cross section at resonance
PT -symmetric particles and arrays have been shown to support very interesting properties [52][53][54][55][56][57] . Here, we study an unusual zero optical extinction in a situation where the topological complex modes are excited. To excite the topological edge modes in regime III, we introduce a x-polarized electrical plane wave with magnitude E 0 and frequency ω, as shown in Fig. 2(a). The excited frequency-domain dipole moments on the 2N nanoparticles where S(ε 1 ) is a square matrix depending on ε 1 , and explicit expression is shown in Eq. (21). In Fig. 3, we show the corresponding extinction cross section calculated in dipole approximation 58,59 where k 0 = ω/c, c, and ε 0 are, respectively, the photon wavenumber, the speed of light, and the permittivity in free space. In Fig. 3(a), the peak at ω = 0.529ω p is due to the conjugate pair of edge modes, which drops drastically when Im(ε 3 ) slightly increases from 0 [ Fig. 3(a)] to 0.025 [ Fig. 3(b)]. Surprisingly, the two edge modes are strongly excited in Fig. 3(b) (as shown by the normalized dipole moment in the inset) while the corresponding optical extinction cross section is vanishing. To explain the vanishing C ext in Fig. 3(b), we define and evaluate the symmetry operators (see Methods for details). The time reverse operator  turns p i into its complex conjugate ( On the other hand, the inversion operator  flips the direction of p i and reverse the position order of the nanoparticles ( ). Consequently, their combination, transforms a column vector p i and a general 2N × 2N matrix A i, j in the way that . Since the matrices A and S(ε 1 ) commute with PT when ε 1 is real (see Methods for a proof), we have when the incident field is uniform (i.e., E 0i = E 0 ). The result is consistent with the actual dipole moments (according to Eq. (8)) as shown in the insets of Fig. 3(b) by setting E 0 = 1. At the frequencies when only complex eigenmodes are excited (i.e., no divergence of extinction due to real eigenmode), we have which means that the simultaneously excited edge modes give vanishing C ext in the perfect PT situation. It should be noted that when real eigenmodes are excited, the C ext should normally diverge at the resonance frequencies. In the case of complex eigenmode being excited, the C ext would be finite. In discussion above, we conclude that extinction cross section is vanishing even if the complex topological edge modes are excited. Due to finite absorption and radiation effects, we found that the net extinction cross section could be below zero, which means that there is net finite amount of energy emitted from the array of particles. As shown in Fig. 3(b), it is clear that an anti-symmetric response of the array is shown, which will contribute to net radiation but the field is orthogonal to the plane wave such that it does not contribute to forward or backward scattering along the propagating direction of incident wave.

Verifications by multiple scattering theory
We present the verifications by Multiple Scattering Theory (MST) 60,61 which includes up to the 5th order multipoles as well as dynamic interactions in Figs 4 and 5. The scattering and extinction cross sections of PT -symmetric particle arrays calculated by MST are shown in Fig. 4. Remarkably, it is shown in Figs 4 and 5 that the simultaneously excited edge modes (ω ~ 0.52ω p ) provide strong scattering cross section but negligible forward extinction. This is due to the net emission of energy associated with the slightly imbalance between gain and loss (b) Integrated field intensity P abs of the mostly excited particle when ε 3 = 1.5 + 0.025i, where P abs is proportional to ω∫ |E(r)| 2 d 3 r over the particle volume. The dashed peak in (b) at ω = 0.5188ω p indicated that the edge modes are strongly excited simultaneously. Other parameters are a = 0.125d, b = 0.175d, d = 100nm, N = 16, and 1/τ = 0.0001ω p .
since there exists additional radiation loss in realistic situation beyond the quasi-static dipole approximation in previous sections.
Although the forward extinction is nearly zero due to anti-symmetric response, there is finite sideway scattering. To show this, we plot the electric field patterns of the PT -symmetric particle array at edge mode frequency in Fig. 5. The x-and z-components of the electric field are plotted in Fig. 5(a) and (b), respectively. It is observed that the distortion on the E x component of the electric field is small, which is due to the fact that E x mostly represents the light transmitted or reflected in the forward or backward directions.
In Fig. 4(b), we show the power spectrum absorbed/emitted by the mostly excited particle, which also represents the integrated electric field intensity in each particle. The dashed peak at ω = 0.5188ω p for the case of s = 0.6d is produced by the simultaneously excited edge modes. A comparison with the case of s = 0.4d shows the significant contribution from the edge modes. This provides another piece of evidence that the edge modes are strongly excited even though forward extinction is vanishing.
Moreover, Fig. 5(b) shows that E z is close to symmetric with reference to an inversion about the center of the array, which represents a nearly anti-symmetric response. Field patterns of the same particle array without gain/ loss are also provided in Supplementary Information.

Discussion
The anomalous scattering phenomenon demonstrated in dipole theory and rigorous multiple-multipole scattering theory in this paper suggests an extreme situation where the forward wave and backward scattered wave are nearly unaffected by the inclusion of the PT -symmetric chain. If we consider a classical wave signal propagating in a waveguide or in free space, it might suggests that the scattered wave (i.e., the leakage of signal) can be unnoticeable to the transmitter and the receiver. If the incident wave carries some signals with small bandwidth within the bandgap, the receiver (who measures the transmitted wave) and the transmitter (who send the signals) will not see a difference before and after the PT -symmetric chain is inserted into the communication channel. This effect could suggest an approach to probe information in a two-way communication channel without affecting the original physical signals. It should be noted that this conclusion is made using purely classical theory and it may not be valid in the case where quantum effect is important. If that is the case, it may provide a method to demonstrate collective quantum effect.
In conclusion, we have predicted an anomalous light scattering property associated with the complex topological edge modes supported in a near-PT -symmetric plasmonic system. Due to its special PT -symmetric properties, such an anomalous scattering field is almost anti-symmetric when the system is excited by a symmetric plane wave. This resonantly scattered field has no contribution to the transmission and reflection parallel to the incident light, which means that the total forward extinction and feedback to the source is near zero. Since the resonant scattering is associated with the topological zero modes (which maps to a fixed frequency), the frequency at which the anomalous scattering occurs could be robust against weak perturbations. The result suggests an extreme situation where leakage of classical information is unnoticeable to the transmitter and the receiver. It should be noted that plasmonic particles are not the only candidates that support the anomalous scattering phenomenon. The main requirement to achieve such phenomenon is that the system is PT -symmetric and edge modes are guaranteed to be in the PT -broken phase. The effect should be general for any PT -symmetric array of resonators with gain and loss, such as dielectric ring resonators.

Methods
Analytical evaluation of Zak phase. We evaluate the Zak phase analytically. The Zak phases discussed in all figures are evaluated using Eq. 3. The derivation of Eq. 3 (in usual basis) is provided here. For reference, the Zak phase in bi-orthonormal basis are also provided in Supplementary Information. To derive Eq. (4), we consider the eigenvalues and the right eigenvectors of Eq. (1), which are where the normalizing factor is defined as , we still can evaluate the Zak phase of each band as those defined in Hermitian systems according to Eq. (3). We also provide another version of Zak phase defined in bi-orthonormal basis in Supplementary Information as a Using Eq. (13), we can write the integrant in the integral of Eq.
is real for all k, and by defining ξ , the second term can be simplified . As a result, the Zak phase in Eq. (3) becomes for all k, its argument is bounded (− π/2 < ξ(k) < π/2), which means . As a result, Eq. (14) gives Eq. (4).

Formulation of eigenvalue problem for finite array. The eigenvalue problem in the form similar to
Bergman's representation is an important step to connect the actual problem to the topological band theory. Our formulation of eigenvalue problem is based on coupled-dipole method. It should be noted that obtaining resonant Scientific RepoRts | 6:38049 | DOI: 10.1038/srep38049 frequencies through this eigenvalue problem is much faster than searching complex non-trivial solutions directly using Eq. (15).
To do this, we group terms with ε 1 in Eq. (15). The first and the second line are multiplied by c 3 ε 1 + c 4 and ε + ⁎ ⁎ c c 3 1 4 with respectively. By factoring out ε 1 , we have The above equations can be vectorized into the following matrix equation: Multiplying Eq. (17) by N −1 , we have Formulation of eigenvalue problem for infinite array. The formulation of the 2-band eigenvalue problem for an infinite array is again based on coupled-dipole method with the use of Bloch's theorem. We first put E 0 = 0 in Eq. (15). We write p n;A = p k;A e iknd and p n;B = p k;B e iknd , and Eq. (15)  where G k;AB = 2(b/d) 3 [e −ikd /(1 − s/d) 3 + 1/(s/d) 3 ]. To obtain an eigenvalue problem with ε 1 as the eigenvalue, we group terms with ε 1 . Eq. (23) is then vectorized to the matrix equation Multiplying the matrix equation by − N k 1 , we recover the eigenvalue problem shown in Eq. (5): . For the case that the plasmonic cores are lossless, time reverse operation turns the energy gaining dielectric into energy losing dielectric and vice versa, which means the time reversed particle array is effectively the same as the particle array obtained by swapping the positions of nanoparticles A and B.
Analysis of parity-time PT symmetry of matrix. The analysis of PT symmetry is again in the basis of coupled-dipole method. Here we would like to show that S(ε 1 ) in Eq. (8) Secondly, if matrixes A and B commute with PT , then their combination, AB, also commutes with PT : We see if ε 1 is real, we have ε =ε − ( ) ( ) 1 1 1
Analytical solutions of topological edge modes. The analytical solutions of the topological edge modes are based on coupled-dipole method in quasi-static approximation. For simplicity, we consider a semi-infinite plasmonic particle array with only one edge on the left. The protected edge modes are required to satisfy both the boundary condition and the dispersion relation with complex wave vectors kd = ± π + iκd 20,64 , where  κ ∈ . By assuming the dipole moments are in the Bloch's form with complex wave vectors, we have ; . When k is real, it gives the dispersion relation; when kd = ± π + κd, it becomes As only decaying dipole moments are physical (κ ≥ 0), there is no solution for κ unless s > 0.5d, which is the case with non-trivial Zak phase. By putting Eq.