Analytical investigation of unidirectional reflectionless phenomenon near the exceptional points in graphene plasmonic system

Ze-Tong Li; Xin Li; Gui-Dong Liu; Gui-Dong Liu; Ling-Ling Wang; Qi Lin; Qi Lin

doi:10.1364/OE.499904

1. Introduction

The isomorphism between the Schrödinger equation and the light wave equation has sparked increased interest in the study of optical systems since Bender suggested that the real spectra could be obtained across a broad category of complex non-Hermitian Hamiltonians with parity-time (PT) symmetry [1,2]. To illustrate the characteristics of non-Hermitian Hamiltonians, numerous remarkable phenomena have been explored, such as unidirectional reflectionless propagation [3]. This unique phenomenon was firstly observed experimentally in a gainless optical waveguide in 2013 by Feng et al. [4]. It is formed at the exceptional point (EP), enabling light to spread in the desired direction [4,5,6–9]. The physics mechanism behind the conditions for its realization is the merge of the eigenvalues and the eigenstates of the scattering matrix in the complex regime. Recently, introducing surface plasmon polaritons (SPPs) into metamaterials has spurred extensive research, such as conducting research on electromagnetically induced transparency [10–12] or integrated into photonic circuits which achieve the creation of ultracompact optical devices [13–20]. Many promising potential applications are suggested based on the dynamic control of the EP, including coherent perfect absorber [21–23], laser [24] and optical sensing [25–28]. In most of these studies, the relevant factors affecting radiation are calculated by means of fitting [3,29].

Graphene, with remarkable properties including strong confinement, low resistance damping and so on, has garnered significant attention in the scientific research field [11,30–32]. Its electrons display a unique linear energy-momentum relationship over a broad range of energies, behaving like massless relativistic particles with an energy-independent velocity [33]. The band structure of atomically thin graphene yields a distinctive electric field effect that can be altered by electrostatic gating to modify the Fermi energy of the material [34,35]. The monolayer graphene with reflectivity under 20% has not received much attention at first [36–39]. Until Semenenko et al. experimentally discovered the radiative rate increase when the edges of the graphene nanoribbons (GNRs) are clustered together [40]. Then Zhao et al. used this property to achieve the reflectivity of up to 90% in a one-dimensional array of GNRs, where the system is operating in the state that the radiative loss rate of the resonances is greater than the intrinsic one [41]. Inspired by the previous back, can we realize unidirectional reflectionless propagation in double-layer GNRs and conduct a quantitative analysis of it?

In this paper, we achieve the unidirectional reflectionless phenomenon by using the double-layer GNRs and get the factors influencing the radiative loss and the intrinsic loss through theoretical calculation. In addition, the dynamic control of exceptional points (EPs) can be achieved based on adjusting the Fermi energy and vertical distance between nanoribbons. The observed unidirectional reflectionless phenomenon is quantitatively depicted by using the coupled mode theory (CMT) in conjunction with the transfer matrix approach, which exhibits high agreement with the finite-difference time-domain (FDTD) simulations.

2. Structure and method

Figure 1(a) exhibits the schematic representation of the designed structure of graphene nanoribbons, which consists of a double-layer of vertically separated GNRs array. The optical characteristics of the plasmonic system can be derived from CMT [42]. S⁽ⁱ⁾_{+, in} and S⁽ⁱ⁾_{−, out} (i = 1, 2) are denoted as the amplitudes of the incident lights from the forward and outgoing lights from the backward of the system, where i represents the i-th nanoribbon, “+” or “−” indicate the propagation of wave on the orientation of the ± y-axis and the subscripts “in” or “out” correspond to the incident or outgoing waves, respectively. Then the temporal-normalized mode amplitude for i-th GNRs a_i can be written as

(1)$$\frac{{\textrm{d}{a_i}}}{{\textrm{d}t}} = ( - j{\omega _i} - {\gamma _{\textrm{o},i}} + {\gamma _{\textrm{r},i}}){a_i} + {e^{j{\theta _i}}}\sqrt {{\gamma _\textrm{r}}_{,i}} S_{ + ,\textrm{in}}^{(i)} + {e^{j{\theta _i}}}\sqrt {{\gamma _\textrm{r}}_{,i}} S_{ - ,\textrm{in}}^{(i)}, $$

where the ω_i is the resonant frequency of corresponding GNRi, γ_o,i, γ_r,i is the internal loss rate and the external radiative decay rate of the resonances, respectively. θ_i stands for the coupling coefficient characterizing the phase.

Fig. 1. (a) The schematic of the proposed structure comprises a double-layer of vertically separated GNRs array. Where the width of the graphene sheet w = 0.9 µm, the horizontal gap between two adjacent sheets d = 0.05 µm and the period of this structure L = w + d = 0.95 µm. h represents the separation between two nanoribbons. Two pieces of gold foil are clamped to the side of the graphene nanoribbons, which is only used to adjust its Fermi level. (b) Cross sections of the proposed multiple cycles with double-layer GNRs in the metamaterials. The gray area represents the amplitude (S⁽ⁱ⁾_{+, in} and S⁽ⁱ⁾_{−, out}) of incident rays in different directions.

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According to the conservation of energy, we can derive the relation between the incoming wave and the outgoing wave for the i-th layer.

(2)$$S_{ - ,\textrm{out}}^{(i)} = S_{ - ,\textrm{in}}^{(i)} - {e^{j{\theta _i}}}\sqrt {{\gamma _{\textrm{r},i}}} {a_i}, $$

(3)$$S_{ + ,\textrm{out}}^{(i)} = S_{ + ,\textrm{in}}^{(i)} - {e^{j{\theta _i}}}\sqrt {{\gamma _{\textrm{r},i}}} {a_i}. $$

Based on the initial harmonic solution assumption, for the incident light frequency ω, the mode amplitudes a_i all contain the factor e^−jωt, which means da_i/dt = −jωa_i. Substituting this conclusion into the coupled mode equation together with Eq. (1), we can get the solution about the i-th layer incoming waves and outgoing waves:

(4)$$S_{ + ,\textrm{out}}^{(i)} = \frac{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}} - {\gamma _{\textrm{r},i}}}}{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}}}}S_{ + ,\textrm{in}}^{(i)} + \frac{{ - {\gamma _{\textrm{r},i}}}}{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}}}}S_{ - ,\textrm{out}}^{(i)}, $$

(5)$$S_{ - ,\textrm{in}}^{(i)} = \frac{{{\gamma _{\textrm{r},i}}}}{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}}}}S_{ + ,\textrm{in}}^{(i)} + \frac{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}} + {\gamma _{\textrm{r},i}}}}{{j({\omega _i} - \omega ) + {\gamma _{\textrm{o},i}}}}S_{ - ,\textrm{out}}^{(i)}. $$

Then the transition matrix for single-layer GNRs can be obtained by converting the above equation to the matrix form:

(6)$${T_s}^i = \left( {\begin{array}{cc} {1 - \frac{{j{\gamma_{\textrm{r,}i}}}}{{\omega - {\omega_i} + j{\gamma_{\textrm{o,}i}}}}}&{ - \frac{{j{\gamma_{\textrm{r,}i}}}}{{\omega - {\omega_i} + j{\gamma_\textrm{o}}_{,i}}}}\\ {\frac{{j{\gamma_{\textrm{r,}i}}}}{{\omega - {\omega_i} + j{\gamma_{\textrm{o,}i}}}}}&{1 + \frac{{j{\gamma_{\textrm{r,}i}}}}{{\omega - {\omega_i} + j{\gamma_{\textrm{o,}i}}}}} \end{array}} \right). $$

Thus the phase shift φ that accumulated between two plasmonic nanoribbons can be derived from:

(7)$${V^i} = \left( {\begin{array}{cc} {\textrm{exp} (j\phi )}&0\\ 0&{\textrm{exp} ( - j\phi )} \end{array}} \right). $$

And we describe the transmission characteristics of SPPs by the transmission matrix M, which manifests the variation characteristics of the incident wave and outgoing wave of the system. The transfer equation can be expressed as

(8)$$\left( {\begin{array}{{c}} {S_{ + ,\textrm{out}}^{(i)}}\\ {S_{ - ,\textrm{in}}^{(i)}} \end{array}} \right) = M\left( {\begin{array}{{c}} {S_{ + ,\textrm{out}}^{(1)}}\\ {S_{ - ,\textrm{in}}^{(1)}} \end{array}} \right) = T_s^i{V^{i - 1}}T_s^{i - 1}{V^{i - 2}} \cdots T_s^2{V^1}T_s^1\left( {\begin{array}{{c}} {S_{ + ,\textrm{out}}^{(1)}}\\ {S_{ - ,\textrm{in}}^{(1)}} \end{array}} \right). $$

For the double-layers periodic GNRs, the transfer equation is

(9)$$M = T_s^2 \times {V^1} \times T_s^1. $$

The transmission matrix of the two-port system can be expressed as

(10)$$S_{ + ,\textrm{out}}^{} = {M_{11}}S_{ + ,\textrm{in}}^{} + {M_{12}}S_{ - ,\textrm{out}}^{}, $$

(11)$$S_{ - ,\textrm{in}}^{} = {M_{11}}S_{ + ,\textrm{in}}^{} + {M_{12}}S_{ - ,\textrm{out}}^{}. $$

When the light comes from a certain direction and S_-,in = 0 is met, we can obtain the transmission coefficient t_b (T = |t|²) and r_f (R_f = |r_f|²). When S_+,in = 0, the t_b and r_b (R_b = |r_b|²) can be got. And the reflection coefficients of forward direction r_f and backward direction r_b as follows:

(12)$${r_\textrm{f}} ={-} \frac{{{M_{21}}}}{{{M_{22}}}},\begin{array}{cc} {}&{} \end{array}{r_\textrm{b}} = \frac{{{M_{12}}}}{{{M_{22}}}},\begin{array}{cc} {}&{} \end{array}t = {t_\textrm{f}} = {t_\textrm{b}} = \frac{1}{{{M_{22}}}}. $$

The system should conform to the energy conservation limit at the circumstance that gain in the medium is absent and the scattering loss is not overly substantial. Consequently, the use of Eq. (12) is permitted based on the determinant of M equals unity. In the Hermitian case, the same transmission coefficients always lead to the same reflection coefficients for the unitary of the scattering matrix, where this unitary can be broken in the non-Hermitian case. Different reflection coefficients are allowed in different directions as the transmission coefficients maintain equal, so the unidirectional reflectionless phenomenon caused by destructive interference of degenerate light modes is an extreme case that the reflectance in one direction reaches zero at EPs.

Additionally, the scattering matrix S is given to characterize the optical characteristics of the proposed structure with the corresponding eigenvalues provided. After the coalescence of eigenvalues of the scattering matrix branches, the PT-symmetry is disrupted by the emergence of EPs.

(13)$$S = \left( {\begin{array}{cc} t&{{r_\textrm{b}}}\\ {{r_\textrm{f}}}&t \end{array}} \right),\begin{array}{cc} {}&{} \end{array}{s_ \pm } = t \pm \sqrt {{r_\textrm{f}}{r_\textrm{b}}} . $$

Following the theoretical derivation above, we take the two-dimensional FDTD method to carry out the numerical simulation. The surface plasmon resonances can be excited at a specific frequency in GNRs when light propagates along the y-axis direction. The x-axis directions utilize the periodic boundary conditions, while the y-axis directions are employed with perfectly matched layers (PML). In the FDTD simulation, the frequency-dependent dielectric constant of a graphene sheet is represented by an effective dielectric layer with thickness H = 1 nm [43]:

(14)$${\varepsilon _\textrm{r}}(\omega ) = 1 + j\left( {\frac{{\sigma (\omega )}}{{\omega {\varepsilon_0}H}}} \right), $$

And the source shape is set as a plane wave with a frequency range from 18.73 to 66.54 THz for monolayer GNRs and 11.81 to 145.7 THz for double-layer GNRs, respectively. The Fermi energies of double-layer GNRs are set as E_F1 = 0.6 eV and E_F2 = 0.8 eV. To guarantee the convergence of the numerical simulation, we adopt the maximum mesh size as Δx = 1.0 nm and Δy = 0.2 nm. And the graphene conductivity σ(ω) can be modeled via the Drude model [44]:

(15)$$\sigma (\omega ) = \frac{{jD}}{{\pi (\omega + \frac{j}{\tau })}}, $$

(16)$$D = \frac{{\pi {e^2}N}}{m}, $$

where ω represents the angular frequency of incident light. The scattering time τ is set as 1.25 × 10⁻¹² s to reduce the intrinsic loss while maintaining consistency with known experimental results [45–53]. The Drude weight D is a function of the carrier density N, the electron charge e and the effective electron mass m. As graphene is a two-dimensional isotropic material, the effective electron mass in the x-axis (m_x) is approximately equivalent to the rest mass of the electron (m₀).

3. Results and discussion

In order to achieve high reflection in monolayer GNRs with the specific resonant frequency ω_i, it is vital to take into account any possible limits related to the process of radiation decay and intrinsic loss. Achieving high reflection requires the monolayer GNRs structure to be designed to work in the over-coupled regime where the γ_r is much larger than γ_o. Then we define the radiative quality factor Q_r,i = ω_i/(2γ_r,i) and the intrinsic loss quality factor Q_o,i = ω_i/(2γ_o,i). In order to investigate the radiative rate γ_r, it is essential to establish a connection between the radiative power and the surface current density J_x(x) of the plasmonic resonator. Through the precise application of the surface boundary condition in combination with the Maxwell's equations, the J_x(x) can be derived:

(17)$${J_x}(x) ={-} [{{H_z}(x,y = {0^ + }) - {H_z}(x,y = {0^ - })} ], $$

which allows us to ascertain the z-component of the magnetic field (H_z) and the y-component of the electric field (E_y) that characterizes the resonance. Furthermore, the monolayer structure displays periodicity and mirror symmetry about the center of the GNRs at x = 0. Thus the surface current can be mathematically represented into a Fourier series, and n is the order of Fourier expansion:

(18)$${J_x}(x) = J_x^{(0)} + \sum\limits_{n = 1}^\infty {J_x^{(n)}\cos \frac{{2n\pi }}{L}} . $$

The period-averaged radiated power P_rad is merely determined by the E_x⁽⁰⁾ and H_z⁽⁰⁾ as the lattice constant is subwavelength. Then we can derive from Eq. (17):

(19)$${P_{\textrm{rad}}} = \int_{ - P/2}^{ + P/2} {\textrm{Re(}} E_x^{(0)}H_z^{(0)\ast })\textrm{d}x = \frac{L}{4}\sqrt {\frac{{{\mu _0}}}{{{\varepsilon _0}}}} {|{J_x^{(0)}} |^2}. $$

The present discussion takes into account the possibility of radiation propagating both upward and downward into space. µ₀ is the permeability of the vacuum. All higher-order components exhibit an evanescent nature and hence nonradiative, having a significant impact on the period-averaged stored energy in one second W. It can be elucidated by the concept of a kinetic inductance that W contains the energy stored within the electromagnetic field and the electron kinetic energy [54].

Our aim is to minimize the proportion of evanescent waves in the near field W in comparison to the P_rad. Hence, we define the radiative quality factor Q_r,i as

(20)$${Q_{\textrm{r,}i}} = \frac{{{\omega _i}W}}{{{P_{\textrm{rad}}}}},\begin{array}{cc} {}&{} \end{array}\frac{{{P_{\textrm{store}}}}}{{{P_{\textrm{rad}}}}} = \frac{{\sum\nolimits_{n = 1}^N {{{|{J_x^{(n)}} |}^2}} }}{{{{|{J_x^{(0)}} |}^2}}}. $$

Where P_store represents the period-averaged stored power. From the relationship between electromagnetic energy storage and radiation, reducing the Q_r can be described as maximizing the ratio of the zeroth order Fourier component J_x⁽⁰⁾ to the higher order J_x⁽ⁿ⁾, n > 0. Considering the effect of higher order components on the order of Q_r, only the first five terms of the Fourier series are taken. By utilizing Eq. (20), the calculated radiative quality factors Q_r varying with the Fermi energy are shown by the blue line in Fig. 2(c) that Q_r,1 = 4.02 for E_F,1 = 0.6 eV and Q_r,2 = 4.65 for E_F,2 = 0.8 eV. Through the fitting to the absolute value of the current at different Fermi energies in Fig. 2(a), the absolute values of the Fourier components of the surface current increase as the Fermi energy increases when maintaining a fixed w. However, the relative contribution of all higher-order components continues to decline, which clearly suggests that the proportion of the P_rad relative to the P_store is on the rise. Furthermore, the impact of relevant factors w and d on the relative contribution of J_x⁽ⁿ⁾ can be determined by decomposing Eq. (18) into a Fourier series.

Fig. 2. (a) Absolute value of the x-component of the current J_x(x) for an array of GNRs with E_F = 0.6 eV, and the gray bars represent the GNRs. The red arrows indicate the direction of the surface current during different grating periods. (b) The Fourier components of the surface current |J_x⁽ⁿ⁾| as a function of Fermi energy E_F. (c) The blue curve represents the radiative quality factor Q_r calculated by Eq. (20). (d) Intrinsic quality factor (Q_o) as a function of Fermi energy and frequency corresponding to the dielectric environment of ɛ_r1 = ɛ_r2 = ɛ_air = 1.

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The dispersion relationship of the quasi-guided mode on monolayer GNRs can be derived by combining the Maxwell equations with the boundary conditions that take into account the conductance of the graphene layer [55], which is depicted in Fig. 2(d).

(21)$$\frac{{{\varepsilon _{\textrm{r}1}}}}{{\sqrt {{\beta ^2} - \frac{{{\varepsilon _{\textrm{r}1}}{\omega ^2}}}{{{c^2}}}} }} + \frac{{{\varepsilon _{\textrm{r}2}}}}{{\sqrt {{\beta ^2} - \frac{{{\varepsilon _{\textrm{r}2}}{\omega ^2}}}{{{c^2}}}} }} ={-} \frac{{j\sigma }}{{\omega {\varepsilon _0}}}, $$

where β is the in-plane wave vector in graphene, ɛ_r1 and ɛ_r2 are the dielectric constants of the dielectric materials above and below the GNRs, respectively. Consider the surrounding dielectric environments and set the permittivity as ɛ_r1 = ɛ_r2 = ɛ_air = 1. The real part of the effective refractive index demonstrates a direct correlation with the Fermi energy, indicating that the resonant wavelength is significantly impacted by the Fermi energy of the graphene sheet. Then utilizing Q_o = Re(n_eff)/Im(n_eff) to calculate the intrinsic quality factors of GNRs. Figure 2(d) depicts the relationship between the Q_o and Fermi energy within the band range of 10.0 to 13.2 THz, which indicates that the Q_o is significantly influenced by the E_F and exhibits a slightly positive correlation with the E_F at a specific frequency. When the scattering time τ = 1.25 × 10⁻¹² s, we calculate that Q_o,1 = 84.86 for E_F = 0.6 eV and Q_o,2 = 97.44 for E_F = 0.8 eV.

The analytically calculated reflection spectra of monolayer GNRs with various Fermi energies are represented in Fig. 3 with the parameters setting w = 0.9 µm and d = 0.05 µm. Due to the dramatic tunability of Fermi energy, as the Fermi energy increases, the resonance positions with high reflection exhibit a blue shift. As the graphene is a two-dimensional carbon material whose electronic behavior is determined by Fermi energy levels. When the Fermi level is at the energy gap position, the absorption of light by the graphene strip is weak and the reflectivity is low. However, when the Fermi level is in the conduction or valence band, the reflectivity of graphene is higher. This will directly affect the numerical distribution of surface current density J_x(x) and the specific values of different orders J_x⁽ⁿ⁾ after Fourier expansion. Further, the radiative quality factor Q_r and external radiative decay rate γ_r,i are also affected. Higher Fermi levels produce higher radiation mass factors. Therefore, appropriately adjusting the Fermi energy of monolayer GNRs allows for precise control over the resonance characteristics of plasmonic modes.

Fig. 3. Numerically computed reflection for single-layer GNRs with width w = 0.9 µm, gap d = 0.05 µm and various Fermi energies. The inset represents the numerical reflection spectra.

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Based on the calculation of the radiative and intrinsic quality factor in the CMT with the scattering matrix method, Fig. 4 describes the transfer properties of the double-layer GNRs structure for forward and backward incidence, which shows good consistency with the analytical results. Our computations have revealed that the utilization of GNRs possessing energy values of 0.6 eV and 0.8 eV can yield a remarkably low radiation quality factor, potentially as low as a single digit. As the vertical separation of the two layers of GNRs decreases, the reflectivity depression shows a blue shift in the frequency domain, accompanied by periodic fluctuations in intensity, which is illustrated in Fig. 4(a). The reduction in reflectance reaches its peak when the incident light is from the forward direction, while it reaches its minimum when the light is traveling in the opposite direction for the non-reciprocity of the system. The numerical and theoretical reciprocal transmission spectra for incidence in two directions are depicted in Figs. 4(a) and (b), respectively.

Fig. 4. (a) The numerical reflection spectra for the backward and forward direction and transmission spectra in the double-layer GNRs system with different distances h, respectively. (b) The theoretical reflection spectra for the backward and forward direction and transmission spectra. The parameters of the system w = 0.9 µm, d = 0.05 µm and the Fermi energies of the two nanoribbons E_F,1 = 0.6 eV and E_F,2 = 0.8 eV.

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Figure 5(a) and (c) respectively portray the y-component field diagrams of the electric field of the system realizing unidirectional reflectionlessness for incidence from the forward and backward directions, where the arrow represents the incident direction of the light, and the gray line indicates the graphene sheet. We take an individual cell for analysis. By comparing Figs. 5(a) and (d) (or Figs. 5(b) and (c)), the system individual cell exhibits different electric field excitation modes for incident light in different directions while keeping h constant. For light incident in the same direction, such as Figs. 5(a) and (b) (or Figs. 5(c) and (d)), the electric field excitation pattern is the same. The contribution of the electric field in the y-axis is localized on the graphene strip sheets and has completely different patterns for incident light from different directions when holding h as a constant. The energy is almost completely confined to the two graphene sheets, while there is less energy distribution in the space between them.

Fig. 5. The y-component of electric field distributions (E_y) for both forward and backward incidence at frequency 11.42 THz [(a) and (d)] and 11.79 THz [(b) and (c)]. The arrows indicate the direction of the incident light, and the gray bars represent the one cycle of GNRs with different Fermi energies.

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Figures 6(a)-(c) exhibit a repulsion of the real parts and an intersection of the imaginary parts of s_± at EPs at 11.42 THz with φ = 1.03π for the forward incidence and at 11.80 THz with φ = 0.97π for backward incidence. We denote φ_total= φ₁ + φ₂ + 2φ as the total phase shift of the proposed system, consisting of the accumulated phase shift φ in the air and the phase shifts in the nonreciprocal ribbons φ₁₍₂₎, which can be expressed by:

(22)$${\phi _i} = \textrm{arctan}\left[ {\frac{{\textrm{Im}({{{T_{\textrm{s}21}^i} / {T_{\textrm{s}22}^i}}} )}}{{\textrm{Re}({{{T_{\textrm{s}21}^i} / {T_{\textrm{s}22}^i}}} )}}} \right] = \frac{{\omega - {\omega _i}}}{{{\gamma _{\textrm{r},i}} - {\gamma _{\textrm{o},i}}}}. $$

Here, φ_total = 1.98π for the forwards and φ_total = 2.04π for the backwards. Both φ_total approaching 2π when r_f = 0, r_b ≠ 0 (r_b = 0, r_f ≠ 0), the unidirectional reflectionlessness phenomenon appears at EP in the forward (backward) direction. Then (r_f·r_b)^1/2 reaches to zero, two eigenvalues coalesce and eventually degenerate into the real and imaginary parts of the transmission coefficient [56]. While phase φ = π with identical reflection spectra for the forward and backward direction, the real parts of s_± at 11.62 THz do not merge and the imaginary part of s₋ = 0 (see Fig. 6(b)). In this case, non-ideal PT symmetry breaks and the phase transition from Hermitian to non-Hermitian.

Fig. 6. The real and imaginary parts of eigenvalues s_± of the scattering matrix S with the h = 13.85 µm, φ = 1.03π (a), h = 12.77 µm, φ = 1.00π (b), and h = 12.00 µm, φ = 0.97π (c), respectively. The Riemannian surface plotted in the parameter space of phase and frequency successively corresponds to the real part (d) and imaginary part (e) of eigenvalues s_± of the scattering matrix S. Use white dots to indicate where EPs appears in (d) and (e).

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The Riemannian surface spanned by frequency and phase reasonably predicts the appearance of the EPs and the phase transition of the PT-symmetric phase. In Figs. 6(d) and (e), as the frequency increases and the phase decreases, the two eigenvalues of the real part of the system successively go through three different stages: coalescence of s_± into a single eigenvalue, the failure of the degeneracy with the formation of a void, and the realization of coalescence again.

4. Conclusions

In summary, we investigate both theoretically and numerically the unidirectional reflectionless phenomenon in the double-layer of GNR structure. The EPs can be regulated by altering the Fermi energy and the vertical distance between two nanoribbons. The analytic solution of the radiative quality factor can be computed based on the analysis of surface current distribution. In addition, the intrinsic loss quality factor is mainly modulated by the Fermi energy and the scattering time. The implementation of unidirectional reflectionless propagation with dynamic regulation can be utilized in adjustable sensors.

Funding

National Natural Science Foundation of China (11947062, 62205278); Natural Science Foundation of Hunan Province (2020JJ5551, 2021JJ40523).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Analytical investigation of unidirectional reflectionless phenomenon near the exceptional points in graphene plasmonic system

Abstract

1. Introduction

2. Structure and method

3. Results and discussion

4. Conclusions

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (6)

Equations (22)

Optics Express