Unidirectional reflectionless propagation of near-infrared light in resonator-assisted non-parity-time symmetric waveguides

The unidirectional reflectionless (UR) light propagation is investigated in the waveguide coupled to gain and loss resonators by using a developed coupled mode-scattering matrix theory. The results show that there is almost no reflection in the case of the backward incidence, but total reflection in the case of the forward incidence under the condition of balancing gain and loss in the gain resonator for the proposed waveguide when the indirect coupling phase θ ranges from 0.8 rad to 2.3 rad and from 4 rad to 5.5 rad. Moreover, the coherent perfect absorption (CPA) can be observed at the same time. Especially, the UR light propagation appears when the absolute value of detuning δ is smaller than 1 × 1013 rad s−1. Based on the findings above, we propose a metal–insulator–metal non-parity-time symmetric plasmonic waveguide and obtain the UR plasmonic propagation and CPA. The theoretical results are in excellent agreement with the finite-difference time-domain simulations. These results will provide a new pathway for the realization of unidirectional propagation and absorption of light at the nanoscale.


Introduction
Hermitian Hamiltonians have great significances for observable physical variables due to its real eigenvalues. However, non-Hermitian Hamiltonians exist more universally in physical systems, and thus it is crucial to explore the non-Hermitian Hamiltonian with real eigenvalues. Until 1998, Bender et al had proposed non-Hermitian Hamiltonians with parity-time symmetry (PT-symmetry), which exhibit real eigenvalues [1]. Subsequently, the PT-symmetry was studied in various systems, such as quantum fields [2], acoustics [3,4], non-Hermitian Anderson models [5], electronics [6], Lie algebras [7], and so on. Based on the similarity between the dispersion equation and the schrödinger equation, Ganainy et al constructed optical gain and loss regions and achieved the PT-symmetric optical system [8]. The optical system presented real eigenvalues when the gain-loss ratio was less than the PT-symmetric threshold. The threshold could be regarded as the PT-symmetric phase transition point or exceptional point (EP). At the EP, a number of novel optical phenomena were reported, such as optical isolation [9][10][11], nonlinear effect [12,13], unidirectional reflectionless (UR) [14,15], coherent perfect absorption (CPA) [16,17], loss induced transparency [18,19]. Especially, the UR and CPA were widely studied in both the PT-symmetric and non-PT-symmetric systems [17,[20][21][22][23][24][25][26][27][28][29]. Feng et al experimentally demonstrated the UR near the PT-symmetric EP, providing the feasibility for optical PT-based unidirectional devices [20], the demonstrated UR at EP on chip confirms the feasibility of creating complicated on-chip parity-time metamaterials and optical devices. Zhang et al achieved a dual-band UR in the ultracompact non-Hermitian plasmonic waveguide [21], there was no need to dope with any gain or loss, which allowed for simpler fabrication of the waveguide structure. Jin et al achieved the UR in the non-PT-symmetric metasurfaces based on the far field coupling [17], they achieved the polarization independent UR and CPA due to the two-ring structure. Rivolta et al analyzed the UR in the dual-and quadruple-resonator coupled waveguides through the coupled mode theory and transfer matrix theory [22], they could achieve a broadband unidirectional invisibility with only two resonators and observed tunable rich dispersions for these anisotropic transmission resonances with four resonators. Sakhdari et al achieved the low-threshold lasing and CPA in the generalized PT-symmetric optical structures [26], they shown that the concept of generalized PT-symmetry may help to reduce the threshold gain in achieving newly discovered PT-enabled applications. Sarısaman et al reported the broadband CPA in the PT-symmetric two-dimensional materials [27], they shown that a two-dimensional Weyl semimetal is more effective than graphene in obtaining the optimal conditions. Huang et al demonstrated broadband total light absorption in the non-PT-symmetric waveguide systems by cascading multiple unit cell structures, they realized the absorption of ∼100% in a wide range of frequencies [28]. Jin et al found that the time-reversal symmetry, pseudo-Hermiticity, and generalized inversion symmetry can protect the symmetric transmission and/or reflection, but the particle-hole symmetry, chiral symmetry, and sublattice symmetry do not [29]. These previous results show that UR could be only observed in the case of resonant detuning δ = 0 when the indirect coupling phase θ is equal to single value. The results are not conducive to designing reliable unidirectional devices. Therefore, it is of great significance to realize UR in the case of a continuously varying phase when the resonant detuning δ = 0.
Here, we study the UR in a metal-insulator-metal PT-symmetric plasmonic waveguide by using the coupled mode-scattering matrix theory and finite-difference time-domain simulation. We can see that UR can be realized under the condition of a continuously varying phase θ when the absolute value of resonant detuning δ < 1 × 10 13 rad s −1 . Especially, the CPA can be observed in the non-PT symmetric resonator-coupled waveguide.

Model and theory
As shown in figure 1, the proposed waveguide consists of an optical waveguide side-coupled with gain and loss resonators. The dissipation in the left resonator is caused by the intrinsic and coupling losses, and the gain in the right cavity is attributed to the gain medium. The coupled mode-scattering matrix theory can be used to analyze the spectral responses in the proposed waveguide [17,30,31]. Here, L is the indirect coupling distance between the gain and loss resonators. κ 0 (κ 1 ) is the coupling coefficient between the loss (gain) resonator and the bus waveguide. The incoming and outgoing waves are depicted by S n± (n = 1, 2, 3, 4). The subscripts '±' represent the forward and backward propagating directions of light, respectively.
Based on the coupled waveguide model mentioned above, the corresponding coupled mode equations can be described as [32][33][34][35][36][37][38][39][40] − jωa 1 Where, a 1 and a 2 are complex amplitudes of the resonant modes in the loss and gain cavities, ω is the angular frequency of the incident light wave, ω 1 and ω 2 are angular frequencies of the loss and gain resonant modes, Γ is the gain coefficient of the gain resonator. 1/τ i1 and 1/τ i2 are decay rates of the loss and gain resonant modes due to the intrinsic loss, 1/τ w1 and 1/τ w2 are decay rates of the loss and gain resonant modes due to the energy escaping into the waveguide from the resonators, respectively. Here, we assume that the bus waveguide is lossless, according to the energy conservation law, the relationships among S n± could be written as Where, θ is the indirect coupling phase between the loss and gain resonators. According the scattering matrix theory, the scattering matrix equation can be expressed as where, t is the transmission coefficient of the proposed waveguide system, r f (r b ) is the reflection coefficient in the case of the forward (backward) incidence, respectively. Here, the scattering matrix s can be described as with Thus, the transmission (T), reflection (R f ) and absorption (A f ) in the case of the forward incidence, reflection (R f ) and absorption (A b ) in the case of the backward incidence can be expressed as T = |t| 2

UR on resonant
Firstly, we investigate the dependence of the indirect coupling phase θ on spectral responses of R f , R b , A f , and A b when Γ = 2 × 10 13 s −1 . Here, the angular frequencies are assumed as ω 1 = ω 2 = 1.86 × 10 15 rad s −1 meaning on resonant in the proposed waveguide. For simplification, the intrinsic losses are set as 1/τ i1 = 1/τ i2 = 1 × 10 13 s −1 , the coupling losses are set as 1/τ w1 = 1/τ w2 = 1 × 10 13 s −1 , respectively. Γ = 1/τ i2 + 1/τ w2 , representing the gain and loss in the gain resonator are completely balanced. From figure 2, we can see that spectral responses show the periodic evolution with increasing the indirect coupling phase θ. As shown in figures 2(a) and (b), the spectra of R f is obviously different from that of R b . Meanwhile, the spectrum of A f is also different from that of A b , as depicted in figures 2(c) and (d). Notably, from figure 2(b), we can see that an obvious UR appears as θ ranges from 0.8 rad to 2.3 rad and from 4 rad to 5.5 rad. This continuously varying phase θ for UR is of great significance for designing reliable unidirectional devices. We also can see that there are two asymmetric absorption peaks as shown in figure 2(c), which is caused by the asymmetric dissipation in the two resonators. Especially, from figure 2(d), we can see that the perfect absorption effect appears in the case of the backward incidence, and the largest absorption can be observed at ω = 1.86 × 10 15 rad s −1 when θ = 1.57 rad and 4.76 rad. To describe the index of UR, we introduce the contrast ratio of UR as η = |r f − r b |/|r f + r b | [20,41]. The contrast ratio η can be approximately equal to 1 at ω = 1.86 × 10 15 rad s −1 when θ = 1.57 rad and 4.76 rad, respectively.
Then, we also investigate the spectral responses of R f , R b , A f , and A b as a function of the gain coefficient Γ in the gain resonator. Here, the relative parameters are set as ω 1 = ω 2 = 1.86 × 10 15 rad s −1 , θ = 1.57 rad, 1/τ i1 = 1/τ i2 = 1 × 10 13 s −1 , and 1/τ w1 = 1/τ w2 = 1 × 10 13 s −1 . In figure 3(a), it is shown that the R f increases continuously as Γ increases. However, the R b always keeps low values, as shown in figure 3(b), and the largest contrast ratio η of UR can be realized when Γ = 2 × 10 13 s −1 . From figures 3(c) and (d), we can see that the A b can approach to 1, which means the perfect absorption phenomenon in the proposed non-PT symmetric waveguide when Γ increases to 2.0 × 10 13 s −1 . What's more, the negative  absorption can be observed in figure 3(c) when Γ > 1/τ i2 + 1/τ w2 , which is caused by the over gain in the gain resonator [42].
To clarify the physical mechanism of UR and CPA, we analyze eigenvalues of the scattering matrix s. Generally, the matrix s is non-Hermitian, whose corresponding complex eigenvalues can be expressed as s ± = t±(r f × r b ) 1/2 . The scattering matrix is in analogy to the Hamiltonian matrix in quantum systems [41,42]. The EPs can be formed by selecting proper elements for the scattering matrix leading to the generation of UR and CPA. In figure 4, we plot real and imaginary parts of the eigenvalues s ± with θ = 1.57 rad and ω 1 = ω 2 = 1.86 × 10 15 rad s −1 when Γ = 0, 1 × 10 13 s −1 and 2 × 10 13 s −1 , respectively. From figures 4(a) and (b), we can see that the eigenvalues s ± are complex numbers meaning the inexistence of EPs when Γ = 0 and 1 × 10 13 s −1 . As shown in figure 4(c), both the real and imaginary parts of eigenvalues s ± are equal to  zero at ω = 1.86 × 10 15 rad s −1 when Γ = 2 × 10 13 s −1 and θ = 1.57 rad. This point is regarded as EP, and UR and CPA appear near the EP.

UR in the presence of detuning
In this section, we study spectral responses of R f , R b , A f , and A b as a function of the resonant detuning (δ = ω 2 − ω 1 ) for the indirectly coupled resonators. The relative parameters are set as ω 1 = 1.86 × 10 15 rad s −1 , 1/τ i1 = 1/τ i2 = 1 × 10 13 s −1 , 1/τ w1 = 1/τ w2 = 1 × 10 13 s −1 , Γ = 2 × 10 13 s −1 , and θ = 1.57 rad. From figures 5(a) and (b), we can see that UR appears distinctly when the resonant detuning δ ranges from −1 × 10 13 rad s −1 to 1 × 10 13 rad s −1 , which is of great significance for realizing the reliable unidirectional devices. In addition, the CPA also can be observed when δ = 0, as shown in figure 5(d). The light absorption decreases with increasing the |δ|, as depicted in figure 5(d). Therefore, the resonant detuning δ plays an important role for UR and CPA in the proposed non-PT symmetric waveguide. In figures 5(e) and (f), we give the analysis on the corresponding eigenvalues of the scattering matrix s. Both the real and imaginary parts of eigenvalues s ± are equal to zero when δ = 0, as shown in figure 5(e). However, when δ = 1 × 10 13 rad s −1 , the eigenvalues s ± turn into complex, which means that the typical EP disappears, and thus there are no CPA phenomena in the coupled waveguide system.

UR in a non-PT symmetric plasmonic waveguide
As an example, we design a non-PT symmetric plasmonic waveguide for verifying the generation of UR. Figure 6(a) shows the schematic diagram of the plasmonic waveguide, which is composed of a metal-insulator-metal plasmonic bus waveguide side-coupled with a gain ring-shaped resonator and a loss ring-shaped resonator. Here, the metal is set as silver, whose permittivity ε m can be defined by the Drude model ε m (ω) = ε ∞ − ω p /(ω 2 + jωγ p ). In the model, ε ∞ = 3.7 is the relative permittivity at the infinite frequency, ω p = 1.38 × 10 16 rad s −1 is the bulk plasmon frequency, and γ p = 2.73 × 10 13 rad s −1 stands for the damping rate [32]. The bus waveguide is assumed as air. The left and right ring-shaped resonators are filled with CdSe quantum dots (ε 1 = 4.0804 -j0.6) and InGaAsP (ε 2 = 11.38 + j0.41), respectively [28,43]. As shown in figure 6(a), r Li and r Lo stand for the inside and outside radii of the loss ring-shaped resonator, respectively. r Gi and r Go are the inside and outside radii of the gain ring-shaped resonator, respectively. l is the distance between the loss and gain resonators. The structural parameters are set as w = 50 nm, r Gi = 40 nm, r Go = 65 nm, r Li = 75 nm, r Lo = 125 nm l = 280 nm. The spectral responses in the waveguide structure can be simulated by using the finite-difference time-domain simulation. In the simulations, the effective area is divided into uniform Yee cells with Δx = Δy = 2 nm and Δt = Δx/2c (c is the velocity of light in vacuum) [44,45]. The perfectly matched layer can be set for the boundary conditions in the simulation [46].
In figures 6(b)-(d), we plot the T, R f , R b , and A f in the proposed non-PT symmetric plasmonic waveguide when δ = 0, 0.8 × 10 13 rad s −1 , and 4.4 × 10 13 rad s −1 , respectively. We can see that the theoretical calculations are consistent with the simulation results. From figure 6(b), we can see that the transmission spectrum exhibits a distinct dip at the wavelength of 1410 nm when δ = 0. Meanwhile, the reflection R f reaches the maximum at the wavelength. However, the reflection R f is equal to zero, and CPA appears at 1410 nm in the case of the backward incidence. In figure 6(c), the UR also can be observed, but the CPA phenomenon disappears and the quality factors of spectra decrease when δ = 0.8 × 10 13 rad s −1 (r Li = 74 nm and r Lo = 124 nm). The spectra of T, R f , R b , and A f show asymmetry when δ = 4.4 × 10 13 rad s −1 (r Li = 70 nm and r Lo = 120 nm) as depicted in figure 6(d). Compared with the case when δ = 0, the R f shows an obvious reflection peak when δ = 4.4 × 10 13 rad s −1 , denoting the disappearance of UR.

Conclusions
In summary, we have studied the dependences of the indirect coupling phase θ, gain coefficient Γ, and resonant detuning δ on T, R f , R b , A f , and A b in the waveguide side-coupled with gain and loss resonators by using the developed coupled mode-scattering matrix theory and finite-difference time-domain simulation. It is shown that the UR light propagation can be realized in the case of the backward incidence under the condition of balancing gain and loss in the gain resonator when the indirect coupling phase θ ranges from 0.8 rad to 2.3 rad and from 4 rad to 5.5 rad. However, the total reflection appears in the case of the forward incidence at the same time. Moreover, the CPA can be observed in the non-PT symmetric case for the proposed waveguide. And we find that the negative absorption appears when Γ > 1/τ i2 + 1/τ w2 . Especially, the reflection in the case of the backward incidence is approximately equal to zero when the absolute value of the detuning δ between the two resonant modes is smaller than 1 × 10 13 rad s −1 . At last, we have also proposed a metal-insulator-metal PT-symmetric plasmonic waveguide and obtained UR and CPA phenomena. These results may be of great significance for the realization of reliable unidirectional devices