1. Introduction

Quantum systems can, in general, be divided into non-Hermitian and Hermitian. The former is an open system, in which energy is not conserved and thus it has complex eigenvalues, while the latter is a close system which has real energy eigenvalues and correspondingly a unitary time evolution. Exceptional points (EPs) are singularities in the spectrum where eigenvalues and corresponding eigenstates coalesce. Specifically, EPs are the critical points where the spectra are at real-to-complex transitions. Over the last decade or so, EPs in non-Hermitian system has gained many attentions and made tremendous progress, of which, optics is a good platform for studying the non-Hermitian system [1–5]. A particular branch of non-Hermitian system is parity-time symmetry system, which described as an open coupled system could exhibit real spectra while commuting with combined parity and time reversal operator [6]. Both the real and imaginary parts of eigenvalues coincide spontaneously, and their eigenvectors become completely parallel at EPs [7]. However, theoretically EPs of non-Hermitian system can be obtained not only through the coupled resonant matrix but can be also obtained from the scattering matrix [8,9]. EPs from the former is in the form of discrete spectrum of a non-Hermitian Hamiltonian operator, while EPs from the latter one is in continuous spectrum format of a non-Hermitian -operator [10]. The two-port optical scattering matrix, S, can be described by $S = \left( {\begin{array}{cc} t&{{r_b}}\\ {{r_t}}&t \end{array}} \right)$, which described the relationship of input and output in an open photonic system [11,12], where ${r_b}$ and ${r_f}$ are the complex reflection coefficients of backward and forward incidences, and t is the transmission coefficient; the backward and forward transmission should be the same [13] while the reflections can be asymmetric. The eigenvalues of scattering matrix are $\Omega \pm{=} t \pm \sqrt {{r_b}{r_f}} $. At EPs, the two eigenvalues and the relative eigenstates of the scattering matrix S can coalesce. In other words, when one of the ${r_b}$ or ${r_f}$ equals to 0 the EP degeneracy is formed [14–17]. In this case, the unidirectional reflectionless propagation of the system occurs. For example, unidirectional reflectionless may take place at the spontaneous PT-symmetry breaking point of some optical structures with complex period refractive index distribution [18–20]. However, there are also non-Hermitian systems that have been proposed whereby unidirectional phenomena are not attained in PT-symmetric structures [21–24]. Nevertheless, for the latter systems, usually only single wavelength dependent EP is observed in one incident direction. Two wavelength dependent EPs being observed in the same direction at the same time, have yet been reported.

In addition, the realization of asymmetric transmission is another thriving research area owning to its fundamental applications in optical information processing. Traditionally, the realization of asymmetry transmission adopts two main approaches: light-matter interactions and light manipulation using artificial structures. In the former approach, magneto-optic [25] and nonlinear mediums [26] are used to demonstrate asymmetric transmission. For the latter approach, asymmetric grating structure [27] and photonic crystals have been reported [28]. For an ideal asymmetric transmission device, high contrast ratio between forward direction to backward direction is need, that is, almost 100% transmission for light propagating from forward direction, whereas there is no light can pass from the backward direction [29].

In this work, we designed a Grating-Insulator-Metal-Insulator-Metal (GIIM) heterostructure, which are in effect coupled Fabry-Pérot cavities, showing two wavelength dependent EPs in grating periodicity equals to 150 nm, and having asymmetric transmission in larger grating periodicities ($\ge 400\textrm{ nm}$). Through the simulation, we are able to explain some of the novel phenomena. For example, the reflection dips around the resonant frequency gradually looks coalesced with increasing thickness of the partition metal layer by using coupled harmonic oscillators theory as well as double Fabry-Pérot cavities composed with three-mirrors resonator [30]. In the GIIM design with grating periodicity equaling to 150 nm, we found that it is possible to obtain two EPs simultaneously for an incident direction in strong coupling regime by only tuning the geometric parameters of the structure rather than the refractive index profile of cavities, which can lead to dual-unidirectional invisibilities. In the GIIM design with grating periodicities $\ge 400\textrm{ nm}$, the simulation results showed that the transmissions for forward and backward incident directions are no longer the same. Furthermore, within a specific wavelength range the total transmittance (viz.: transmitted power to input power) for forward incident which includes fundamental mode and higher-order diffractive modes is high, whereas the light incident from backward direction is low due to the absence of higher-order diffractive modes, which induced asymmetric transmission.

2. Theory

We first consider a non-Hermitian Metal-Insulator-Metal-Insulator-Metal (MIMIM) structure consisting of silver and dielectric layer. Each layer is denoted by ${l_1}$, ${l_2}$, ${l_3}$, ${l_4}$ and ${l_5}$ with layer thickness ${L_1}$, ${L_2}$, ${L_3}$, ${L_4}$ and ${L_5}$, respectively. ${l_1}$, ${l_3}$ and ${l_5}$ are silver layer, ${l_2}$ and ${l_4}$ are dielectric layer with refractive index $n = 1.5$. The complex refractive index of silver can be described by the Drude model [31], where the high-frequency dielectric constant is assumed as 3.7, the plasma frequency is $2.2 \times {10^{15}}\textrm{ Hz}$ and the damping coefficient equals to $4.3524 \times {10^{12}}\textrm{ Hz}$. The light travels through the five layers from ${l_1}$ to ${l_5}$ is defined as forward transmission while travelling from opposite direction is defined as backward transmission. The reflection coefficients can be calculated using COMSOL Multiphysics. A resonant mode splitting is observed when the partition layer thickness is below a threshold value. This is because the variation in transmissivity is particularly significant with thickness in nanometer scale; the thicker the metal layer, the lower transmissivity will be. The transmissivity of the partition layer decides the coupled strength between the two dielectric layers. To further understand the principle of mode splitting, two theoretical models were used to explain the physics: the harmonic oscillator model was used to derive the mode coupling effect and the three-mirror resonators theory to study the phase shift in reflection.

We use the coupled harmonic oscillators [32–35] to describe a coupled cavity system, which is equivalent to the MIMIM structure, as shown in Fig. 1(a). The three grey color layers are the metal layer, from left to right are ${l_1}$, ${l_3}$ and ${l_5}$ respectively. The areas between each metal layer are the dielectric layer ${l_2}$ and ${l_4}$, which can be treated as two cavities ($a$ and $b$) with resonant frequencies ${\omega _a}$ and ${\omega _b}$. Here we set ${L_2} = {L_4}$ to keep their resonant frequencies the same i.e., ${\omega _a} = {\omega _b} = {\omega _0}$. Figure 1(b) is an equivalent oscillation model of Fig. 1(a). The system is modeled as two harmonic oscillators connecting to two sides by springs with resonant frequencies ${\omega _a}$ and ${\omega _b}$. Energy pass through the partition layer, ${l_3}$, to cavity b can be seen as the energy is transferred to the second sphere through the middle spring, causing the second sphere to oscillate. In our configuration, the frequency dependent scattering loss for cavity a and b are ${Y_{a(b)}}(\omega )$ and coupling strength $\kappa (\omega )$ between the two cavities are defined in Ref. [36,37]. The $2 \times 2$ Hamiltonian has the form $H = \left( {\begin{array}{cc} {{\omega_0} - i{\gamma_a}}&\kappa \\ \kappa &{{\omega_0} - i{\gamma_b}} \end{array}} \right)$, where ${\gamma _{a(b)}}(\omega ) = {Y_{a(b)}}(\omega )/2$.

 

Fig. 1. Coupled harmonic oscillator model. (a) A coupled cavity system, which can be seen as the MIMIM structure. The three dark color parts are the metal layer, from left to right are ${l_1}$, ${l_3}$ and ${l_5}$ respectively. Cavity a and b are the dielectric layer ${l_2}$ and ${l_4}$. Two cavities, with resonant frequencies ${\omega _a} = {\omega _b} = {\omega _0}$ and decay rates ${Y_{a(b)}}(\omega )$, communicated with coupling strength $\kappa (\omega )$. Meanwhile, cavity a is excited by a light whose frequency is $\omega$ and strength is $\varepsilon$. (b) A classical oscillation model that is exactly equivalent to (a).

Download Full Size | PDF

From coupled mode theory [38,39], light with frequency $\omega $ and strength $\varepsilon $, driven from left side and couple through the partition layer to the cavity b, the field amplitudes $\tilde{a}$ and $\tilde{b}$ can also be written as linearly coupled Lorentzian oscillators [35,40]:

According to Cramer’s rule, the amplitude of cavity a in steady state is given by [41]:

Equation (3) is simplified to:

Then Eq. (5) becomes:

Finally, the imaginary part of $\chi $ can be written as:

From Eq. (7) the characteristics of $\Im (\chi )$ can quantify the resonant frequencies and linewidths [42]. The positions of the $\Im (\chi )$ peaks are where the resonant frequencies with linewidths equal to $\Im ({\omega _ + })$. Due to the complex loss values, unlike the PT-symmetry approach, these two modes will not merge at central frequency unless $\kappa (\omega ) = 0$. One of the modes, however, appear to merge with the other one if their distance is close enough due to the diminishing linewidth. Also, eigenvalues can be quantified as reflection spectra. For example, the position of the reflection dip is the resonant frequency. Thus, the shape of reflection spectra can be determined by analyzing the $\Im (\chi )$ [42]$.$ To determine the resonant frequencies, we calculate the spectrum of $\chi $ with ${L_1} = 35\textrm{ nm}$, ${L_2} = {L_4} = 1200\textrm{ nm}$, ${L_5} = 20\textrm{ nm}$ as a function of ${L_3}$ as shown in Fig. 2. In Fig. 2(a-c), ${L_3} = 20\textrm{ nm}$, $45\textrm{ nm}$ and $130\textrm{ nm}$ respectively, the black solid line (left y-axis) is the $\Re (\chi )$ and the red dashed line (right y-axis) is the $\Im (\chi )$. From Eq. (7) we know that there are two parts of $\Im (\chi )$, and there is a crossover point between them. We define the system is in the weak-coupling regime when the crossover point exceeds $- 3\textrm{ dB}$ point of one of the parts which has lower $\Im (\chi )$ peak, and crossover point is $\le - 3\textrm{ dB}$, the regime is defined as the strong-coupling regime. In strong-coupling regime (Fig. 2(a)), with the two cavities exchange energy significantly, the red dashed line splits into two peaks, and the peak position(s) of $\Im (\chi )$ is at where the two splitting resonant frequencies occur. From Fig. 2(b), if the thickness of partition layer, ${L_3}$, increases, two peaks of red curve shift closer, which means that their eigenvalues branches are merging. When ${L_3}$ increases further, from Fig. 2(c) we can see that there is only one peak left. This is because the system is in the weak-coupling regime, resulting in two resonant frequencies coalescing into one. The reflection coefficients $|r |$ are calculated using COMSOL as shown in Fig. 2(d-f) with ${L_1} = 35\textrm{ nm}$, ${L_2} = {L_4} = 1200\textrm{ nm}$, ${L_5} = 20\textrm{ nm}$ and ${L_3} = 20\textrm{ nm}$, $45\textrm{ nm}$ and $130\textrm{ nm}$. By comparing the shapes of $\Im (\chi )$ and the reflection coefficients, it is noted that the resonant frequencies derived from coupled harmonic oscillator model and COMSOL give similar splitting and merging characteristics. We used the three-mirror resonators method to further elucidate the mode splitting behavior (details of the method are in Supplement 1 S1). However, the MIMIM structure demands very fine tuning of the ${L_3}$ layer thickness in sub-nanometer in order to obtain double EP points. To overcome this constraint, we replace one of the metal layers with a grating structure; thus, added a second variable that assist in relaxing the severe thickness constraint.

 

Fig. 2. The spectrum of susceptibility with incident wavelength range from 700 nm to 800 nm in the setting of ${L_1} = 35\textrm{ nm}$, ${L_2} = {L_4} = 1200\textrm{ nm}$, ${L_5} = 20\textrm{ nm}$ and ${L_3} = 20\textrm{ nm}$ (a), 45 nm (b), 130 nm (c). The black solid line of left y axis is the real part of $\chi $ while the red dashed line of right y-axis is the imaginary part. (d-f) are the reflection coefficients $|r |$ where ${L_3} = 20\textrm{ nm}$, 45 nm, 130 nm calculated by COMSOL.

Download Full Size | PDF

3. Results and discussion

The top silver layer of the MIMIM heterostructure (Fig. 3) is replaced by a grating structure which consist of a grating layer ${l_g}$ with thickness ${L_g}$ and a plane layer ${l_1}$ with thickness ${L_1}$, so it becomes a GIIM structure; this provides an additional variable, effective refractive, to tune the incident light transmission and reflection within the cavity. To take into the effective refractive index into consideration; two parameters, fill factor, f, and periodicity, P, are introduced as means to tune the effective refractive index. The grating layer then can be considered as an effective medium layer [43,44]. In the simulation, the device is incident by plane wave with fundamental mode at angle of ${0^ \circ }$ and the refractive index of the substrate material is 1.52.

 

Fig. 3. Schematic of Grating-Insulator-Metal-Insulator-Metal (GIIM) heterostructure. The grating layer thickness is ${L_g}$, the fill factor and periodicity of top layer grating are f and P. Each layer material refractive index is the same as the MIMIM structure.

Download Full Size | PDF

3.1 Unidirectional reflectionless

In the mode splitting stage, the two-splitting reflection amplitude dips are not symmetrical [45,46]. There is a crossover in amplitude with respect to the incident wavelength. In Fig. 4(a), we show the reflection spectra with the setting of ${L_g} = 20\textrm{ nm}$, ${L_1} = 22\textrm{ nm}$, ${L_2} = {L_4} = 2500\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 24.71\textrm{ nm}$ and $f \times P = 100\textrm{ nm}$, and the red solid circles denote the reflection dip at the left-hand side of central frequency (mode II) while the blue open circles denote the reflection dip at the right-hand side of central frequency (mode I). It can be observed that there is an alternation between red and blue circles. At wavelength less than 620 nm, the red solid circles are higher than the blue open circles whereas above 650 nm, the blue open circles are higher than the red solid circles. They crossover at about 635 nm (see Supplement 1 S2 for details).

 

Fig. 4. (a) Reflection coefficient with ${L_g} = 20\textrm{ nm}$, ${L_1} = 22\textrm{ nm}$, ${L_2} = {L_4} = 2500\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 24.71\textrm{ nm}$ and $f \times P = 100\textrm{ nm}$. The red solid circles are the reflection coefficient for left-hand side of central frequencies. The blue open circles are the reflection coefficient for right-hand side of central frequencies. (b) Transmission coefficients for forward $|{{t_f}} |$ and backward incident $|{{t_b}} |$, reflection coefficient for forward incident $|{{r_f}} |$ with ${L_g} = 22.9\textrm{ nm}$, ${L_g} = 20.5\textrm{ nm}$, ${L_2} = {L_4} = 150\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 33\textrm{ nm}$ and $f \times P = 105.4\textrm{ nm}$. The black star-line represents the $|{{t_f}} |$, while the red circle-solid line represents the $|{{t_b}} |$. The black solid line is for the $|{{r_f}} |$. At ${\lambda _{EP1}} = 570.56\textrm{ nm}$ and ${\lambda _{EP2}} = 642.74\textrm{ nm}$ the reflection coefficients equal to 0 and form EPs.

Download Full Size | PDF

We then calculated the transmission and reflection coefficients as function of incident wavelength for forward incident with ${L_g} = 22.9\textrm{ nm}$, ${L_1} = 20.5\textrm{ nm}$, ${L_2} = {L_4} = 150\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 33\textrm{ nm}$ and $f \times P = 105.4\textrm{ nm}$ in Fig. 4(b). The black star-line represents the transmission coefficient for forward incident $|{{t_f}} |$, while the red circle-solid line represents the transmission coefficient for backward incident $|{{t_b}} |$. In this low grating periodicity case there is only the fundamental mode (0^th order) and no other diffractive modes generation for both propagation directions. The scatter matrix of this system satisfied $S = \left( {\begin{array}{cc} t&{{r_b}}\\ {{r_t}}&t \end{array}} \right)$ and the eigenvalues have the forms of $\Omega \pm{=} t \pm \sqrt {{r_b}{r_f}} $, where r is the reflection coefficient and the subscripts f and b indicate forward and backward direction respectively. Once ${r_f}{r_b} = 0$, EP degeneracies appear. The reflection coefficients are shown in Fig. 4(b), the black solid line is the forward incident $|{{r_f}} |$. The thickness of partition layer is sufficiently small such that two cavities can have strong coupling, causing mode splitting. For forward incident the reflection coefficients equal to 0 at ${\lambda _{EP1}} = 570.56\textrm{ nm}$ and ${\lambda _{EP2}} = 642.74\textrm{ nm}$ simultaneously. However, the reflections at the two ${\lambda _{EP}}$ for the backward incident are not equal to 0. The disappearances of reflections result in unidirectional reflectionless at the EP points.

3.2 Asymmetric transmission

The transmission behaviors under different grating periodicities were investigated. When the grating periodicity was increased, the total transmittance includes the fundamental mode and higher-order diffractive modes for forward and backward incident are no longer the same due to the grating diffraction in the asymmetric dielectric environment. We derive the total transmittance of different grating periodicities for $P = 400\textrm{ nm}$ (Fig. 5(a)), $P = 420\textrm{ nm}$ (Fig. 5(b)) and $P = 440\textrm{ nm}$ (Fig. 5(c)), while the other parameters are kept unchanged with the double-EPs case, ${L_g} = 22.9\textrm{ nm}$, ${L_1} = 20.5\textrm{ nm}$, ${L_2} = {L_4} = 150\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 33\textrm{ nm}$ and $f = 0.7027$. The black solid lines represent the forward incident and the red dashed lines represent the backward incident. From Fig. 5(a), we can see that the transmittance for forward incident in the range of 430 nm to 550 nm is strong because higher-order diffractive modes are generated by the grating, and there are three distinct peaks. However, the transmittance for backward incident only have the fundamental mode. And the transmitted wave intensity is weak therefore the backward transmittance almost equal to zero in the same wavelength range as forward incident. With the increasing of the grating periodicities (Fig. 5(b) and Fig. 5(c)), the three peaks of the transmittance have a significant redshift and the peaks will merge with the peaks in the range above 550 nm. To further clarify, it should be noted that the asymmetric transmission discussed is different from the nonreciprocal transmission. The reciprocity of the proposed grating structure still holds so for example it cannot act as an optical isolator [29,47]. We then compared the contrast ratio $|{({T_f} - {T_b})/({T_f} + {T_b})} |$ [27] of $P = 400\textrm{ nm}$, $P = 420\textrm{ nm}$ and $P = 440\textrm{ nm}$ in Fig. 5(d), where the forward incident denoted by ${T_f}$ and the backward incident is denoted by ${T_b}$. It can be noted that the contrast ratio of from wavelength 430 nm to 550 nm are larger than 0.5, but they drop to zero abruptly at about wavelength at $550\textrm{ nm}$. Noteworthy, the $P = 400\textrm{ nm}$ have the largest wavelength range with extreme high contrast ratio which are close to 1.

 

Fig. 5. Transmittance under different periodicities (a) $P = 400\textrm{ nm}$, (b) $P = 420\textrm{ nm}$, (c) $P = 440\textrm{ nm}$ under ${L_g} = 22.9\textrm{ nm}$, ${L_1} = 20.5\textrm{ nm}$, ${L_2} = {L_4} = 150\textrm{ nm}$, ${L_3} = 30\textrm{ nm}$, ${L_5} = 33\textrm{ nm}$ and $f = 0.7027$. The black solid line stands for the forward incident and the red dashed line stands for the backward incident. (d) Contrast ratio $|{({T_f} - {T_b})/({T_f} + {T_b})} |$ with different periodicities $P = 400\textrm{ nm}$ (black star-line), $P = 420\textrm{ nm}$ (red close-square line) and $P = 440\textrm{ nm}$ (blue open-circle line).

Download Full Size | PDF

4. Conclusions

From coupled harmonic oscillator model, we have numerically shown that the partition layer thickness affects the energy coupling characteristics in the double cavities system. The energy coupling characteristics of the system can be deduced from the reflection spectra, in which the two reflection dips around the central frequency can merge or split. This phenomenon is dictated by the thickness of the partition layer which can be considered as an energy exchange gate. Furthermore, using a three-mirror resonators model, the characteristics of reflection dips merging and splitting can be interpreted as transition from one mode to another mode. Thus, we can design and determine the behaviors of our heterostructure system through varying the partition layer thickness. However, coupling of the reflection waves are very sensitive to variation in partition layer thickness; therefore, it is a challenge to design a model with the desired energy coupling effect using only the partition layer thickness as the only variable. Thus, we introduced a grating structure in the top metal layer (Fig. 3), and the structure can be described by a non-Hermitian scattering matrix. From the calculation, significant modes splitting is observed including double EPs when $r = 0$ is satisfied for forward incident at ${\lambda _{EP1}} = 570.56\textrm{ nm}$ and ${\lambda _{EP2}} = 642.74\textrm{ nm}$ simultaneously under a set of parameters with grating periodicity equaling to 150 nm. This leads to a dual-unidirectional reflectionless propagation at EPs. Our studies on the appearance of dual-EPs in one direction can be extended to other photonic non-Hermitian systems, such as whispering-gallery mode cavity system, plasmonic cavity system and photonic crystal cavity system. Compared with the proposed single EP sensors, the presence of double EPs can enhance the devices’ sensitivity. Two singularities provide higher monitoring of ambient refractive index perturbations. Moreover, the GIIM can be used as a bio- or drugs sensors to detect two different samples which are corresponding to different resonance wavelengths simultaneously. The theoretical work of double EPs also provides a new insight to the new generation of optical communications, optical sensing, photo-detection, and nano- photonic devices. Additionally, larger grating periodicities, for example, $\ge 400\textrm{ nm}$, can introduce asymmetric transmission. The contrast ratio of transmittance between forward incident and backward incident can reach an extreme high value (${\approx} 1$) in a large wavelength range. The wavelength range of asymmetric transmission can be modified by designing the grating periodicity.