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Abstract
Dual-band unidirectional reflectionlessness and coherent perfect absorption (CPA) are demonstrated in a non-Hermitian plasmonic waveguide system based on near-field coupling between a single resonator and the resonant modes of two resonators showing an electromagnetically induced-transparency-like (EIT-like) effect. The non-Hermitian plasmonic system consists of three metal-insulator-metal (MIM) resonators coupled to a MIM plasmonic waveguide.
© 2017 Optical Society of America
1. Introduction
Inspired by the notion of non-Hermitian Hamiltonians with parity-time symmetry [1, 2], various novel phenomena in a family of artificial structures with balanced gain and loss have been studied, such as nonreciprocal light propagations [3,4], lasers [5–8], optomechanically induced transparency [9], absorbers [10–12], and unidirectional reflectionlessness [13–16]. Unidirectional reflectionlessness has attracted considerable attention because of the structures obtained, which will likely substitute for optical filters, sensors, and diodes. A deeper understanding of the nature of non-Hermitian Hamiltonians has allowed non-parity-time symmetric systems with unbalanced gain and loss to be proposed [17–28], in which unidirectional reflectionless phenomena were successfully realized in passive (no gain) optical waveguides [21], large optical multilayer structures [22], periodic ternary-layered materials [23], two-layer slabs [24], metasurfaces [25], and plasmonic waveguides [26–28]. The plasmonic waveguide attracted increasing attention [26–39] because the surface plasmon polaritons (SPPs) can control light at the nanoscale beyond the diffractive limit and can be easily integrated into photonic circuits.
In 2015, unidirectional reflectionless propagation was realized by Huang et al. [26] in a non-PT symmetric plasmonic waveguide cavity system, which required two resonators with identical resonant frequencies and that one of the resonators had a smaller width compared to the other. In 2016, they developed a system with unbalanced gain and loss to increase the reflectivity difference between the forward and backward directions close to unity [27]. By adjusting the far-field phase coupling between two resonators, Zhang et al. showed the single-band unidirectional reflectionless phenomena based on Fabry-Pérot resonance in plasmonic waveguides [28]. This is much easier to fabricate through experiment, because it is not doped by gain or loss media and does not demand that one of the resonators has a smaller width compared to the other [28]. Thus far, most of researches focus on the single-band unidirectional reflectionlessness.
In this work, we present a scheme for realizing dual-band unidirectional reflectionlessness and coherent perfect absorption (CPA) based on near-field coupling between a single resonator and the resonant modes of two resonators exhibiting an electromagnetically induced-transparency-like (EIT-like) effect in a non-Hermitian plasmonic waveguide system. The system consists of three metal-insulator-metal (MIM) resonators coupled to a MIM plasmonic waveguide. The reflectivities for the forward (backward) direction are 0.84 (0) and 0 (0.92) at the two exceptional points (EPs) [40–43], respectively. The forward and backward absorptivities are about 0.99 and 0.96 with quality factors of about 67.6 and 41.7, respectively. Our system is not doped by gain or loss media and does not demand that one of the resonators has a smaller width compared to the other one, hence it is not only more compact than that based on far-field coupling but also much easier to fabricate through experiments.
2. Results and discussions
Figure 1 shows the schematic of the non-Hermitian plasmonic waveguide system. The coupled resonators A and C are placed above the waveguide and resonator B is placed below the waveguide. The lengths lA, lB, and lc of resonators A, B, and C were 250 nm, 270 nm, and 530 nm, respectively. w represents the widths of the MIM waveguide and resonators. The interval e between resonators A and C is 10 nm and the distance d between resonators A and B is variable. The dielectric of the resonators and waveguide is air and the metal is silver. Both length L and width W of the structure are 2000nm in simulation. The relative permittivity of silver is described by the Drude model with plasmon frequency ωp = 1.366×1016rad/s and collision frequency ωc = 3.07×1013 Hz [44,45]. The amplitudes of the incoming and outgoing waves of the resonator are denoted by S+ij and S−ij (i, j = 1, 2, 3), respectively. The numerical simulation is carried out by employing a finite-integration package (CST Microwave Studio). And the boundary conditions of system are magnetic (Ht = 0) in x- and z-directions and electric (Et = 0) in y-direction.
Fig. 1 Schematic of an MIM waveguide side coupled to resonators A and B, and resonator A coupled to resonator C. The geometric parameters of the structure are lA = 250 nm, lB = 270 nm, lC = 530 nm, e = 10 nm, w = 50 nm, L = W = 2000nm, and d is variable.
The proposed system [Fig. 1] is analyzed by temporal coupled-mode theory (CMT) [34–36], and the time evolutions of amplitudes a and b of resonators A and B can be obtained by the following equations:
where ωA(B) and γA(B) are the resonance frequency and intrinsic loss of resonator A (B), respectively. ΓA(B) and ΓC represent the decay rates resulting from energy escape into the waveguide and resonator C, respectively. The direct coupling coefficient between resonant modes A and B is k. By utilizing the conservation of energy, the outgoing waves of the resonators can be expressed as follows: and thus, the propagation waves in the waveguide should satisfy the following formulas: where the phase difference φ = ω Re(neff) d/c, in which neff and c represent the effective refractive index of the propagating SPPs [46] and the light velocity in vacuum, respectively. The incoming and outgoing waves of the resonator C should meet a steady-state relation S−13 = δS+13eiθ(0 < δ < 1), where δ and θ represent the amplitude attenuation and phase shift between the incoming and outgoing waves of the resonator C, respectively. The phase shift θ can be expressed as θ = 2lCω Re(neff)/c + ϕ with the additional phase shift ϕ of the resonator C. The direction of the incident SPPs from the left (right) is defined as the forward (backward) direction. Hence, the complex coefficients for forward transmission tf, backward transmission tb, forward reflection rf, and backward reflection rb are expressed as: The transmissivity, forward (backward) reflectivity, and forward (backward) absorptivity are T = |t|2, Rf(b) = |rf(b)|2, and Af(b) = 1 − |t|2 − |rf(b)|2, respectively.Figure 2(a) shows the reflection and transmission spectra of resonators A and C, and the single resonator B obtained using numerical simulation. It was found that the transmission spectrum of resonators A and C had an EIT-like linetype, which is similar to that reported in [36, 47, 48]. The transparency peak, left dip, and right dip appear at frequencies of 190.64, 177.2, and 203.76 THz, respectively. The transmission dip of the single resonator B is at 187.6 THz. The reflection spectra of the forward and backward propagating SPPs, where d = 42 nm and phase difference φ = 0.075π between resonators A and C and the single resonator B, are depicted in Fig. 2(b) based on numerical simulation (solid line) and analytical calculation (solid sphere), respectively. According to Fig. 2(b), the results obtained by analytical calculation are in good accordance with those from numerical simulation. From the numerical simulation results, the forward (backward) reflectivity reaches 0.84 (0) at 174.96 THz and the backward absorptivity reaches 0.96. Additionally, the forward (backward) reflectivity is close to 0 (0.92) at 196 THz, and the forward absorptivity reaches 0.99. Hence, dual-band unidirectional reflectionlessness and CPA are simultaneously realized based on near-field coupling between resonators A and C, and the single resonator B.
Fig. 2 (a) Reflection and transmission spectra versus frequency ω for resonators A and C (solid line) and the single resonator B (dotted line) based on numerical simulation and (b) forward and backward reflectivities for the coupled system of the three resonators A, B, and C as functions of frequency ω where d = 42 nm and phase difference φ = 0.075π based on numerical simulation (Num) and analytical calculation (Analy). The parameters related to the analytical calculation are γA = 1.588 THz, γB = 1.38 THz, ΓA = 55 THz, ΓB = 42 THz, ΓC = 3.4 THz, δ = 0.97, ϕ = 1.156π, and k = 1.6 THz.
To clearly exhibit the dual-band unidirectional reflectionlessness when d = 42 nm, the forward and backward z-component distributions of the magnetic field at 174.96 THz are shown in Figs. 3(a) and 3(b), respectively, and the corresponding distributions at 196 THz are depicted in Figs. 3(c) and 3(d), respectively. It is evident that the z-component distributions of the magnetic field between resonators A and C are antisymmetric in Figs. 3(a) and 3(b) and symmetric in Figs. 3(c) and 3(d) based on plasmonic hybridization [49]. The antisymmetric and symmetric modes are similar to the modes of left and right dips of the EIT-like transmission spectrum in Fig. 2(a), respectively. From Figs. 3(a) and 3(b), the z-component distributions of the magnetic field between resonators A and B are nearly in-phase and anti-phase, which results in constructive interference and destructive interference in the forward and backward direction, respectively. That is, the strong and weak reflections occur in the forward [Fig. 3(a)] and backward [Fig. 3(b)] direction, respectively. Similarly, the weak reflection occurs in the forward direction [Fig. 3(c)] and the strong reflection occurs in the backward direction [Fig. 3(d)]. In other words, the dual-band unidirectional reflectionless phenomena appear at frequencies of 174.96 and 196 THz when d = 42 nm because of the near-field constructive and destructive interference between the resonant modes of resonators A and C, and resonator B.
Fig. 3 z-component distributions of the magnetic field of the SPPs when d = 42 nm at a frequency of 174.96 THz [(a), (b)] and 196 THz [(c), (d)] in the forward and backward directions.
Furthermore, for the phase shift φA−C of the symmetric mode between resonators A and C at a frequency of 196 THz, the total phase difference φall is equal to φB(A−C) − φA−C(B) + 2φ in the forward (backward) direction [29, 44], where φB and φ are the phase shifts of resonator B and the SPP propagation between resonators A and C, and B, respectively. In the case of phase shift φ′A−C of the antisymmetric mode at a frequency of 174.96 THz, the forward and backward phase differences φ′all are φ′A−C − φB + 2φ and φB − φ′A−C + 2φ, respectively. Both phase differences φall and φ′all are close to π when φ = 0.075π in the backward and forward directions, respectively, which results in the reflections being approximately equal to 0. The z-component distributions of the magnetic field for resonators A and B are anti-phase (∼ π), thus, there is no reflection in the backward [Fig. 3(b)] and forward [Fig. 3(c)] directions.
Consequently, the physical properties of the present system are explored. The non-Hermitian scattering matrix S of the system can be described by [21]:
The eigenvalues of S are . And the corresponding eigenstates for rb ≠ 0 are not orthogonal. When is 0, two eigenvalues coalesce and an EP appears. This is to say, when rf = 0, rb ≠ 0 (or rb = 0, rf ≠ 0), unidirectional reflectionlessness occurs.The real and imaginary parts of the eigenvalues s± as a function of frequency ω when φ = 0.075π are depicted in Figs. 4(a) and 4(b), respectively. As expected, two EPs occur at frequencies of 174.96 and 196 THz, respectively. It can be clearly seen that two real parts of s± coalesce with each other and two imaginary parts cross at the two frequency points of 174.96 and 196 THz, which corresponds to eigenvalue s+ = s−. At the two EPs, t is complex and is 0. Therefore, dual-band unidirectional reflectionlessness is obtained at two EPs.
Fig. 4 Real (a) and imaginary (b) parts of eigenvalues s± of the scattering matrix S as a function of frequency ω when φ = 0.075π. s+ and s− are represented by a blue dashed line and red short dotted line, respectively. The positions of CPAs are marked by two black arrows and green arrows. All parameters γA, γB, ΓA, ΓB, ΓC, δ, ϕ and k are same as that in Fig. 2(b).
On the premise of complex t and , one real and the other complex eigenvalues can be obtained by choosing some appropriate parameters, which correspond to the phenomenon of CPA [12]. From Fig. 4(b), both imaginary parts of the eigenvalues s± are equal to 0, marked by two black arrows and green arrows, where CPAs appear. For example, CPAs of the backward propagating SPPs emerge at frequencies of 174.7 and 175.3 THz (near 174.96 THz), respectively, where both reflections [Fig. 2(b)] and transmissions approach 0. In this case, absorptivities of about 0.96 with quality factors of 41.7 are obtained. The quality factor is defined as f/Δf, where f and Δf are resonant frequency and full width half maximum, respectively. Similarly, for the forward propagating SPPs, the high absorptivities of about 0.99 with quality factors of 67.6 are obtained at frequencies of 195.8 and 196.3 THz (in the vicinity of 196 THz). In conclusion, four unidirectional CPAs are realized in the vicinities of two EPs in an ultracompact non-Hermitian plasmonic waveguide system.
Subsequently, we show the dependencies of the real [Figs. 5(a) and 5(c)] and imaginary [Figs. 5(b) and 5(d)] parts of the eigenvalues s± on frequency ω and the phase φ. The horizontal magenta dashed line expresses the case of Fig. 4 corresponding to the phase φ = 0.075π. We notice that the real and imaginary parts of the eigenvalues s± change abruptly around 174.96THz and 196THz. At the two frequencies, when the real and imaginary of s+ are equal to that of s−, respectively. Hence, dual-band unidirectional reflectionlessness appears at two EPs (φ = 0.075π). Apparently, the results from Fig. 5 take our insight into the relations of real and imaginary parts of s± with the phase φ and the resonance frequency ω.
Fig. 5 The real [(a), (c)] and imaginary [(b), (d)] parts of the eigenvalues s± as the function of the frequency ω and the phase φ. The horizontal magenta dashed line corresponds to the case of phase φ = 0.075π. All parameters γA, γB, ΓA, ΓB, ΓC, δ, ϕ and k are same as that in Fig. 2(b).
3. Conclusion
Dual-band unidirectional reflectionlessness was realized at two EPs based on near-field coupling in a non-Hermitian plasmonic waveguide structure. The forward (backward) reflectivity was approximately 0.84 (0) and the backward (forward) was about 0 (0.92) at a frequency of 174.96 THz (196 THz). Large reflectivity differences between the two directions were obtained. Dual-band undirectional CPA near the two EPs was also exhibited. The forward and backward absorptivities were about 0.99 and 0.96 with quality factors of about 67.6 and 41.7, respectively. There was no need to dope with any gain or loss and one resonator did not need to have a smaller width compared to the other resonators, which allowed for simpler fabrication of the plasmonic waveguide structure. In addition, our structure can be realized based on the current nanofabrication technology [50]. This work will be useful in the design of dual-band filters, sensors, and plasmonic diode-like devices.
Funding
National Natural Science Foundation of China (Grant No. 11364044); National Natural Science Foundation of China-the Special Fund of Theoretical Physics (Grant No. 11347122); The Education Department of Jilin Province Science and Technology Research Projects (Grant No. 2015-09 and JJKH20170455KJ).
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