1. Introduction

Reciprocity is an important consequence in the classical physical world, which states that the transmission for the forward propagation and backward propagation remain the same [1]. The scattering matrix of a linear reciprocal system is symmetric for both permittivity and magnetic tensors [2,3]. Recently there have been reports on reciprocity breaking through external magnetic field [4,5], nonlinearity [6–8] and spatiotemporal modulation [9] in acoustics and photonics. Additionally, the development in nonreciprocity generates many applications such as absorbers [10–13] and isolators [14,15]. In nonlinear optics, the resonant frequency of a cavity based on Kerr materials whose permittivity is field-intensity-dependent is derived from the input intensity [16]. The asymmetric field distribution for opposite propagation induces different resonance frequency shifting due to Kerr nonlinearity, leading to nonreciprocal transmission. [17–20].

In open multimode systems, including resonant and scattering systems, the eigenvalues and corresponding eigenstates coalesce at exceptional points (EPs) for unique combination parameters like geometric parameters, material refractive index, or excitation wavelength. A crucial characteristic of EPs is the strong response to the external perturbations. The enhancement in sensitivities on surrounding environment perturbations means that some EPs-based devices can be used as biosensors [21], thermal sensors [22], environmental refractive indices sensors [23], and nanoparticle sensors [24]. In addition, higher orders EPs (more than one) can further amplify the responses of perturbations [25,26].

Recently, strong nonreciprocal transmission and an EP at a specific wavelength are both obtained with and without nonlinear effect in an epsilon-near-zero (ENZ) materials [27]. However, the need to precisely balance gain and loss leads to structural complexities that can be challenging to fabricate. There is also a theoretical study on double EPs for forward incident in a non-Hermitian heterostructure [28]. To date, structures which integrate with higher-order EPs for one incident direction and nonreciprocal transmission have not yet been proposed. In this work, we studied the double EPs formation for forward incident and nonreciprocal transmission of a metal-nonlinear-metal-insulator-metal heterostructure. In the linear case which is defined as the situation when the input intensity is too low to induce the nonlinear effect, by engineering each layer thickness double EPs occur at particular wavelengths for forward propagation and meanwhile the transmission for the two opposite propagation directions (forward and backward) are reciprocal. Conversely, when the input power is at a considerable large value such that the nonlinear effect becomes significant, which is defined as the nonlinear case, a significant nonreciprocal effect is obtained due to the asymmetry field confinement inside the nonlinear material for the two propagation directions. The theoretical modelling of the behaviours of transmission for forward, backward incident, and the characteristics of optical bistability using nonlinear coupled mode theory (NCMT) agree with the simulation results. The findings indicated that both reciprocal and nonreciprocal transmission can be implemented in a single device, but cannot be achieved simultaneously. Our work opens up new possibilities in multifunctional devices for high-sensitive optical sensing and high-power optical switching.

2. Theory

We proposed a non-Hermitian metal-nonlinear-metal-insulator-metal heterostructure that consists a couple cavities system (Fig. 1). The first layer from the top view side is denoted by ${l_1}$ with layer thickness ${L_1}$, then the next layer is ${l_2}$, follow by ${l_3}$, ${l_4}$ and ${l_5}$ with layer thickness ${L_2}$, ${L_3}$, ${L_4}$, and ${L_5}$ respectively. In which ${l_1}$, ${l_3}$ and ${l_5}$ are silver layers with complex permittivity, ${l_2}$ is the nonlinear layer with ${\varepsilon _2} = {\varepsilon _l} + {\chi ^{(3)}}{|E |^2}$, where ${\varepsilon _l}$ is the linear term, ${\chi ^{(3)}}$ is the third-order nonlinear susceptibility, E is the local electric field intensity, and ${l_4}$ is the dielectric layer. Due to the complex permittivity of silver layers, the proposed system is non-Hermitian [25]. Inside the cavity, the permittivity of the nonlinear material is not a constant but influenced by input power. However, when the change in permittivity, $\Delta \varepsilon = {\chi ^{(3)}}{|E |^2}$ is very subtle ($|{\Delta \varepsilon /{\varepsilon_2}} |\le {10^{ - 4}}$), the nonlinear effect can be neglected; this regime is called linear regime. On the contrary, when the incident power is sufficiently strong, the nonlinear term needs to be taken into consideration; this regime is called nonlinear regime. The light with frequency $\omega $ incident from ${l_1}$ is defined as forward incident, while incident from ${l_5}$ is defined as backward incident. In the following subscripts f and b represent the forward and backward incident direction. ${l_2}$ and ${l_4}$ can also be considered as two coupled cavities (cavity a and cavity $b$) with resonance frequencies ${\omega _a}$ and ${\omega _b}$, of which their coupling strength $\kappa $ correlated with ${L_3}$ [28,29]. The decay rate for the two cavities are ${Y_a}$ and ${Y_b}$ [30]. To simplify the analysis, in our configuration we assumed ${\omega _a} = {\omega _b} = {\omega _0}$ and we used $\tilde{a}$ and $\tilde{b}$ to denote the field amplitudes in two cavities.

 

Fig. 1. The metal-nonlinear-metal-insulator-metal heterostructure, also can be considered as a coupled cavities system. Each layer from top view side to bottom are defined as ${l_1}$, ${l_2}$, ${l_3}$, ${l_4}$ and ${l_5}$. ${l_1}$, ${l_3}$ and ${l_5}$ are silver. ${l_2}$ is the nonlinear material layer and ${l_4}$ is dielectric material. Light incident from ${l_1}$ is defined as forward incident, and from the opposite side is defined as backward incident.

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In the nonlinear regime a shift of the resonance frequency is observed due to change in permittivity of the material. For the heterostructure, the NCMT equation describing the forward and backward incident respectively in frequency domain, can be written as [31]:

First, consider the linear regime, that is, nonlinear effect is not significant therefore the nonlinear term ${\gamma _a}{|{\tilde{a}} |^2}/{P_a}$ in Eq. (1) can be neglected. For forward incident, we have $\tilde{b} = {\eta _{ea}}{\chi _t}$, where ${\chi _t}$ is the effective susceptibility:

Then the output field ${B_{out}} = \sqrt {{\kappa _{eb}}} \tilde{b}$, and the forward transmission is:

Similarly, we can obtain the backward transmission $|{{t_b}} |$ where $|{{t_f}} |= |{{t_b}} |= |t |\propto |{{\chi_t}} |$ . Thus, in the linear regime the transmission is reciprocal, and the transmission spectra is related to the spectra of $|{{\chi_t}} |$. In other words, the characteristics of the transmission can be described by the $|{{\chi_t}} |$. From Eq. (3), we derive the transmittance:

For nonlinear regime, we have:

Because forward ($|{{{\tilde{a}}^{nl}}_f} |$) and backward ($|{{{\tilde{a}}^{nl}}_b} |$) incident is not the same, with subscripts ${t_f}$ and ${t_{b.}}$ represent forward and backward transmission respectively; then the difference between ${\chi _t}^{nl}$ of the two incident directions ${\chi _{tf}}^{nl} \ne {\chi _{tb}}^{nl}$ induces the transmittance nonreciprocity.

To elucidate the behavior of the proposed structure transmittance under the nonlinear effect, we calculated ${|{{\chi_t}} |^2}$, ${|{{\chi_{tf}}^{nl}} |^2}$ and ${|{{\chi_{tb}}^{nl}} |^2}$ with different incident powers ${P_{in}}$ (Fig. 2). To illustrate Eq. (5), we let ${\gamma _a} = {\gamma _b} = 1 \times {10^{13}} - 9 \times {10^{13}}i\textrm{ rad/s}$, $\kappa = 1.5 \times {10^{14}}\textrm{ rad/s}$, resonance wavelength ${\lambda _0} = 500\textrm{ nm}$, ${P_a} = 500$, ${P_{in}} = 3 \times {10^{27}}$ (Fig. 2(a)) and ${P_{in}} = 5.5 \times {10^{27}}$ (Fig. 2(b)). In Fig. 2, the blue dashed line is ${|{{\chi_t}} |^2}$, which is the linear regime, the red and black solid lines are ${|{{\chi_{tf}}^{nl}} |^2}$ and ${|{{\chi_{tb}}^{nl}} |^2}$, respectively. Two splitting peaks aroused from strong coupling are observed and we defined the peaks in the longer wavelength range as Peak I while the peaks in the shorter wavelength range are defined as Peak II. From Fig. 2, the peaks for the forward and backward incident have a red-shift bending with respect to the changing of permittivity. The bending in Peak I for forward incident is more obvious than that of the backward incident. The difference between the degree of bending for the two directions increases with the increasing of incident power. Furthermore, Peak II bend less obvious than those of Peak I.

 

Fig. 2. The spectrum of ${|{{\chi_t}} |^2}$, ${|{{\chi_{tf}}} |^2}$ and ${|{{\chi_{tb}}} |^2}$ as the function of incident wavelength with the setting of ${\gamma _a} = {\gamma _b} = 1 \times {10^{13}} - 9 \times {10^{13}}i\textrm{ rad/s}$, $\kappa = 1.5 \times {10^{14}}\textrm{ rad/s}$, ${\lambda _0} = 500\textrm{ nm}$, ${P_a} = 500$, ${P_{in}} = 3 \times {10^{27}}$ (a) and ${P_{in}} = 5.5 \times {10^{27}}$ (b). The blue dashed line is the ${|{{\chi_t}} |^2}$, the red line is the ${|{{\chi_{tf}}} |^2}$ and the black line is the ${|{{\chi_{tb}}} |^2}$.

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We then investigated ${|{{\chi_{tf}}^{nl}} |^2}$ as a function of ${P_{in}}$ at $\lambda = 540\textrm{ nm}$(Fig. 3). An optical bistability for forward incident is obtained; with increasing incident power ${|{{\chi_{tf}}^{nl}} |^2}$ increases gradually until ${P_{in}}$ reached to a switch-on threshold (red dot in Fig. 3), where the effective susceptibility has a sharp increase (red arrow). Similarly, the upper branch also exhibits a sharp drop (blue arrow) at switch-off threshold (blue dot in Fig. 3) with decreasing ${P_{in}}$. The difference between the two thresholds is the hysteresis width. Moreover, the middle branch with negative gradient is inherently unstable thus a small perturbation can cause the solution to switch to the upper or lower branch [32].

 

Fig. 3. The optical bistability for forward incident of the proposed heterostructure. The red dot is the switch-on threshold and the blue dot is the switch-off threshold.

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3. Numerical simulation results and discussions

The simulation has shown that there are several notable optical characteristics have emerged from the metal-dielectric/nonlinear-metal-insulator structure within an incident power range. In the simulation, for ${l_2}$ the linear term ${\varepsilon _l}$ is 2.2, the third-order nonlinear susceptibility ${\chi ^{(3)}}$ is $4.4 \times {10^{ - 18}}\textrm{ }{\textrm{m}^\textrm{2}}\textrm{/}{\textrm{V}^\textrm{2}}$ [33], whereas for ${l_4}$ the refractive index is $n = 1.47$. Silver can be described by Drude model, where the high-frequency dielectric constant is assumed as 3.7, the plasma frequency is $2.2 \times {10^{15}}$ Hz and the damping coefficient equals to $4.3524 \times {10^{12}}$ Hz [34]. The refractive index of the substrate is 1.52 and the incident angle equals to 0 degree.

3.1 Double EPs at linear regime

In the absent of the nonlinear effect at low incident power, the ${\chi ^{(3)}}{|E |^2}$ term can be neglected. Then the system is a metal-dielectric-metal-insulator-metal structure, and the permittivity of ${l_2}$ will only have linear term. For such a non-Hermitian system the two-port scattering matrix can be expressed as $S = \left( {\begin{array}{*{20}{c}} t &{{r_b}}\\ {{r_f}} & t \end{array}} \right)$, where t is the complex transmission coefficient, ${r_f}$ and ${r_b}$ is the complex reflection coefficient for forward and backward incident, respectively. In low input power, the transmission for forward and backward is reciprocal. The eigenvalues of the scattering matrix S are $\varOmega \pm{=} t \pm \sqrt {{r_b}{r_f}} $. When ${r_b}{r_f} = 0$, that is, ${r_f}$ or ${r_b}$ equals to 0, the two eigenvalues and its related eigenstates coalesce and form EPs, which would induce an unidirectional reflectionless propagation in one incident direction.

According to the coupled mode theory, when the coupling strength between two cavities is strong enough the eigenfrequencies of each cavity can split significantly, giving rise to two dips around central frequency of the reflection spectra. By tuning the layer thicknesses, it is possible to obtain double EPs, i.e. the reflection coefficients are both equal to 0 for the two splitting modes. We simulated the reflection coefficient using COMSOL Multiphysics for forward incident $|{{r_f}} |$ versus wavelength with ${L_1} = 34.2\textrm{ nm}$, ${L_2} = {L_4} = 400\textrm{ nm}$, ${L_3} = 12.3\textrm{ nm}$, ${L_5} = 40.4\textrm{ nm}$ (Fig. 4). Due to the relatively small value of ${L_3}$ the coupling between two cavities is strong, then the split between two dips of the reflection coefficient is about $\textrm{50 nm}$. Furthermore, we can observe that at 425.69 nm and 476.79 nm there is no reflection for forward incident ($|{{r_f}} |= 0$). Thus, in this configuration the $|{{r_f}} |$ vanish and lead to double EPs at ${\lambda _{EP1}} = 425.69\textrm{ nm}$ and ${\lambda _{EP1}} = 476.79\textrm{ nm}$ simultaneously.

 

Fig. 4. Reflection coefficient for forward incident $|{{r_f}} |$ at low incident power with ${L_1} = 34.2\textrm{ nm}$, ${L_2} = {L_4} = 400\textrm{ nm}$, ${L_3} = 12.3\textrm{ nm}$, ${L_5} = 40.4\textrm{ nm}$. At ${\lambda _{EP1}} = 425.69\textrm{ nm}$ and ${\lambda _{EP2}} = 476.79\textrm{ nm}$ the reflection coefficients equal to 0 and form EPs.

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3.2 Nonreciprocal transmission and optical bistability

In the structure, only ${l_2}$ is the nonlinear material. When the incident power is increased to a sufficiently high value, the nonlinear effect will change the permittivity significantly. The field amplitude in cavity a for different incident directions are different. Thus, this will lead to different effective permittivity of the nonlinear material for the two incident directions. We simulated the transmittance in COMSOL Multiphysics with the setting as described in Fig. 4. The results under different incident powers is shown in Fig. 5. In the linear regime, the transmittance for forward and backward incident is reciprocal. However, when the power is increased to $0.225\textrm{ GW/c}{\textrm{m}^2}$ (Fig. 5(a)), the two peaks of the transmittance exhibit bending. For Peak I, the forward incident has stronger bending than the backward incident. This is because for the forward incident the light enters cavity a first, while for the backward incident cavity a is in effect the second cavity. The change in effective permittivity of the nonlinear material for forward incident is greater than that of backward incident. The calculated results show that the nonlinearity breaks the reciprocity and leads to a nonreciprocal transmission. When the input power is increased to $0.6\textrm{ GW/c}{\textrm{m}^2}$ (Fig. 5(b)), the bending of the transmittance peaks is more prominence, especially in Peak I. In contrast, the difference between transmittances for two incident directions are not that prominence in Peak II, although, we still can observe that the bending of the backward incident slightly exceeds that of the forward curve.

 

Fig. 5. Transmittance for linear case (blue dashed line), forward incident (red line) and backward incident (black line) with ${L_1} = 34.2\textrm{ nm}$, ${L_2} = {L_4} = 400\textrm{ nm}$, ${L_3} = 12.3\textrm{ nm}$, ${L_5} = 40.4\textrm{ nm}$, while the incident power equals to $0.225\textrm{ GW/c}{\textrm{m}^2}$ (a) and $0.6\textrm{ GW/c}{\textrm{m}^2}$ (b).

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The optical bistability in transmittance for forward incident as a function of the incident power was looked into using COMSOL Multiphysics. Here, 483 nm is used as the incident wavelength in the simulation and computed the transmittance under different incident powers (Fig. 6). To compute the two stable branches, each simulation step of the incident power is calculated from previous step value obtained. The C1/C2 curve in Fig. 6 represents the transmittance obtained by the simulations on progressively increasing/decreasing incident power from a low/high value. From Fig. 6 a hysteresis loop is observed; when the power gradually increased to about $0.34\textrm{ GW/c}{\textrm{m}^2}$ (switch-on threshold) the transmittance jumps to the upper branch from the lower branch, as indicated by the red arrow. However, when the light decreases from higher power to lower power, the transmittance will not switch down until it reaches $0.187\textrm{ GW/c}{\textrm{m}^2}$ (switch-off threshold). At the switch-off threshold, the transmittance drops abruptly from the upper branch to the lower branch, as indicated by the black arrow. Moreover, the simulation results of Fig. 5 and Fig. 6 agree well with the theoretical modelling results discussed in Section 2.

 

Fig. 6. The optical bistability about the transmittance for forward incident as a function of ${P_{in}}$ at $\lambda = 483\textrm{ nm}$. The C1/C2 denotes the transmittance got from the increasing/decreasing incident power. The red arrow stands for the switching from lower branch to upper branch, while the black arrow stands for the switching from upper branch to lower branch.

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4. Conclusions

In conclusion, we have studied the optical properties of transmittance spectra under different incident light powers for the metal-nonlinear-metal-insulator-metal heterostructure. By using the nonlinear coupled mode theory, we showed the presence of notable optical characteristic of the transmittance spectra; there are two coupled modes characteristics: linear and nonlinear regimes; these modes are a function of the incident power. In the nonlinear regime the transmittance peaks have different degrees of bending under different powers. In addition, the gap between the forward and backward transmittance peaks can broaden or narrow with the increasing or decreasing of incident power. This indicated that the nonlinear effect can break the reciprocity of the system and induce nonreciprocal propagation. The nonreciprocal characteristic gives rise to hysteresis loop in transmittance for forward incident with respect to the incident power, and this means there is an optical bistability presence. The feature of the optical bistability have been discussed in detail through analyzing the hysteresis loop. The later numerical simulation results about the behaviours of the transmittance curves versus wavelength under different incident powers and the hysteresis loop in transmittance fit the theoretical modelling results very well. This further validates the accuracy of the theoretical model. For the linear regime, we can observe double EPs which $|{{r_f}} |= 0$ at ${\lambda _{EP1}} = 425.69\textrm{ nm}$ and ${\lambda _{EP1}} = 476.79\textrm{ nm}$ simultaneously. The proposed multi layers thin film structure is relatively simple to fabricate by comparing with the Parity-Time-Symmetric system which needs photonically doped to balance gain/loss in ENZ materials [27]; thus, these theoretical findings would not be too difficult experimentally verified. The proposed structure can provide new ideas for design of versatile optical devices as it exhibits different optical properties under different incident light intensities. For example, it can be used as high sensitivity optical sensors in low power or auto-tunable power fuse at high-power for high-power laser devices. Moreover, the concurrent realization of the double EPs, and nonreciprocal propagation in an individual device can also be further implemented in other nano-photonic platforms, such as metal-dielectric-metal (MDM) plasmonic waveguides, photonic crystal (PC) waveguides and whispering-gallery microcavities (WGM). Currently, experiments are in progress to verify the theoretical result.