1. Introduction

Idealized linear energy-conserving systems with time-reversal symmetry are Hermitian, described by a Hamiltonian with real eigenvalues and the evolution of states controlled by a unitary operator [1]. However, dissipation processes are ubiquitous in nature, breaking time-reversal symmetry and giving rise to non-Hermitian systems. Non-Hermitian systems can manifest exceptional points (EPs), the singularities in parameter space where several eigenvalues and eigenvectors become degenerate. Systems with EPs have recently attracted more and more attentions due to their fundamentally difference from Hermitian.

In photonics, EPs can control light generation and propagation [2–8], such as controlling lasing modes [9–12], enhancing the sensitivity [13–16], loss-induced transparency [17], unidirectional invisibility [18,19], phase transition [20,21], coherent perfect absorbers [22–25], optical isolation [26] and unique topological features [27–33]. Interestingly, EPs in non-Hermitian metasurfaces can be evident with slight structural adjustments and allow for diverse applications [34–38]. The dark-bright mode-coupled plasmonic systems in non-Hermitian metasurfaces are important for the research and potential applications of functional and tunable photonic devices with unusual properties [39–44]. However, theoretical studies on EPs of reflection, transmission and absorption are less available.

In this paper, in the dark-bright mode-coupled plasmonic systems, the variations of EPs of reflection, transmission and absorption spectra with radiation loss and absorption loss of bright mode and absorption loss of dark mode are numerically investigated using temporal coupled-mode theory (TCMT), and an assumption is given using the representation transformation theory. The intermediate representation (IR) is firstly proposed and related to the reflection spectrum, while the normal representation (NR) is associated with the absorption spectrum. In the region far from EPs, the IR (or NR) describes the reflection (or absorption) spectrum well. Near EPs, modified formulas similar to the representation transformation theory are given. In order to verify the correctness of the assumption, two metasurfaces (controlling radiation loss of bright mode and absorption loss of dark mode, respectively) are designed. And the simulation results of the two metasurfaces are in good agreement with the assumption and it is found in the near-infrared and visible-light band that the absorption loss of the dark mode is linearly related to the EPs of reflection, transmission and absorption spectra, while the radiation loss of the bright mode is only linearly related to the EPs of the absorption spectrum. These laws can help to manipulate the splitting of spectral lines for reflection, transmission and absorption by adjusting the radiation loss and absorption loss of bright mode, the absorption loss of dark mode and the coupling coefficients between two resonant modes. This research provides a guiding scheme for the design of micro and nano photonics devices.

2. Theory

Schematic diagram of dark-bright mode-coupled plasmonic systems is shown in Fig. 1, in which a dark mode (with resonant frequency ${f_q}$ and absorption loss ${\gamma ^{\prime}_q}$) interacts with a bright mode (with resonant frequency ${f_p}$, radiation loss ${\gamma _p}$ and absorption loss ${\gamma ^{\prime}_p}$). The complex amplitudes of the incoming and outgoing waves are defined as $S_j^ + ,S_j^ - ,(j = 1,2)$ at the j-th port, then the dynamical master equation and outgoing wave equation can be written as

In addition, ${\kappa _{pp}}$ and ${\kappa _{qq}}$ are expressed as the on-site corrections by the perturbation. Because ${\kappa _{pp}},{\kappa _{qq}}$ are far less than ${f_p},{f_q}$, for simplicity, ${\kappa _{pp}},{\kappa _{qq}}$ are chosen to be 0. ${\kappa _{pq}} = {\kappa _{qp}} = \kappa$ are described as the near-field coupling coefficients between the two resonant modes, and X is expressed as the far-field interaction between two resonant modes. It is worth noting that since the dark mode is completely radiation-free, ${\gamma _q},X,{d_{1q}},{d_{2q}},{k_{q1}},{k_{q2}}$ are all chosen to be 0. The cross-absorption losses ${\gamma ^{\prime}_{pq}} = {\gamma ^{\prime}_{qp}}$ are so small that are chosen to be 0 for simplicity. ${C_{jk}}(j,k = 1,2)$ are described as the S-parameter matrix element of the background system. Due to the presence of the substrate, there are ${C_{11}} ={-} {C_{22}} = {r_{Qz}},{C_{12}} = {C_{21}} = {t_{Qz}}$ with ${r_{Qz}} = (1 - n)/(1 + n)$ and ${t_{Qz}} = 2\sqrt n /(1 + n)$ denoting the reflection and transmission coefficients of the substrate [46] with n being the refractive index of the substrate. And the asymmetry coefficient is found to be $\eta = \sqrt n$.

 

Fig. 1. Schematic diagram of dark-bright mode-coupled plasmonic systems coupled with two ports. The arrows are expressed as the incoming and outgoing waves. The dashed lines are reference planes for the wave amplitudes in the ports.

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According to the above, Eqs. (1) and (2) are simplified as

Some of the parameters are not fully independent, but are rather linked together via certain relations. Based on time-reversal symmetry and energy conservation, it has been gotten that

Solve Eqs. (4)–(7) and the reflection and transmission complex amplitudes are written as

It is well known that the resonant frequencies ${f_p},{f_q}$ of the two modes are only related to the resonant positions, so ${f_p} = {f_q}$ are set in the numerical calculation. In this case, as shown in Eqs. (8)–(10), the split properties of dark-bright mode-coupled plasmonic systems are completely determined by ${\gamma _p},{\gamma ^{\prime}_p},{\gamma ^{\prime}_q},\kappa$. In the following, all quantities have been divided by the central frequency $\bar{f} = ({f_p} + {f_q})/2$, such as ${\bar{\gamma }_p}\textrm{ = }{\gamma _p}/\bar{f}$, ${\bar{\gamma ^{\prime}}_p}\textrm{ = }{\gamma ^{\prime}_p}/\bar{f}$, ${\bar{\gamma ^{\prime}}_q}\textrm{ = }{\gamma ^{\prime}_q}/\bar{f}$ and $\bar{\kappa }\textrm{ = }\kappa /\bar{f}$. Then the spectral lines can be plotted by Eqs. (8)–(10) with ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q},\bar{\kappa }$. Fix ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ and increase $\bar{\kappa }$ from 0.001 (the step size is 0.001) in the numerical calculation, at the same time, the spectral lines plotted by Eqs. (8)–(10) are also changing. When the spectral lines are just split, the critical splitting coupling coefficient ${\bar{\kappa }_E} = {\kappa _E}/\bar{f}$ is recorded. Vary ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ and record ${\bar{\kappa }_E}$ repeatedly, then the variations of ${\bar{\kappa }_E}$ with ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ are shown in Fig. 2 eventually. Any point in Fig. 2 indicates that when ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ are fixed, how big the value of ${\bar{\kappa }_E}$ is when the spectral lines are just split. If $\bar{\kappa } < {\bar{\kappa }_E}$, the spectral lines are in the degenerate state. Here, the critical splitting coupling coefficient ${\bar{\kappa }_E}$ is defined as the EPs of reflection, transmission and absorption spectra. It can be seen that for any ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$, there is ${\bar{\kappa }_{ER\_\textrm{S}}} < {\bar{\kappa }_{ET\_\textrm{S}}} < {\bar{\kappa }_{EA\_\textrm{S}}}$. As ${\bar{\gamma }_p}$ increases, ${\bar{\kappa }_{ER\_\textrm{S}}}$ is slightly smaller, ${\bar{\kappa }_{ET\_\textrm{S}}}$ is almost constant, and ${\bar{\kappa }_{EA\_\textrm{S}}}$ is obviously larger. As shown in Fig. 2 ((G), (H), (I)), as ${\bar{\gamma ^{\prime}}_p}$ increases, ${\bar{\kappa }_{ER\_\textrm{S}}},{\bar{\kappa }_{ET\_\textrm{S}}}$ are slightly smaller and ${\bar{\kappa }_{EA\_\textrm{S}}}$ are clearly smaller. As shown in Fig. 2 ((A), (D), (G)), as ${\bar{\gamma ^{\prime}}_q}$ increases, ${\bar{\kappa }_{ER\_\textrm{S}}},{\bar{\kappa }_{ET\_\textrm{S}}},{\bar{\kappa }_{EA\_\textrm{S}}}$ are clearly larger. It can also be seen that the effects from ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_q}$ are more pronounced, and Fig. 2(A) is the most linear case.

 

Fig. 2. Schematic diagram of variations of ${\bar{\kappa }_E}$ with ${\bar{\gamma }_p}$ when ${\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ takes the values 0.01, 0.1 and 0.19 in the numerical calculations, respectively. ${\bar{\kappa }_{ER\_\textrm{S}}} = {\kappa _{ER\_\textrm{S}}}/\bar{f}$, ${\bar{\kappa }_{ET\_\textrm{S}}} = {\kappa _{ET\_\textrm{S}}}/\bar{f}$ and ${\bar{\kappa }_{EA\_\textrm{S}}} = {\kappa _{EA\_\textrm{S}}}/\bar{f}$ are expressed as the variations of ${\bar{\kappa }_E}$ for reflection (red solid triangle), transmission (green solid circle) and absorption (blue solid rectangle) spectral find-peak results (S), respectively. ${\bar{\kappa }_{ER\_\textrm{F}}} = {\kappa _{ER\_\textrm{F}}}/\bar{f}$, ${\bar{\kappa }_{ET\_\textrm{F}}} = {\kappa _{ET\_\textrm{F}}}/\bar{f}$ and ${\bar{\kappa }_{EA\_\textrm{F}}} = {\kappa _{EA\_\textrm{F}}}/\bar{f}$ are expressed as the variations of ${\bar{\kappa }_E}$ for reflection (red underlined), transmission (green solid line) and absorption (blue dotted line) after fitted (F), respectively. All quantities have been divided by the central frequency $\bar{f} = ({f_p} + {f_q})/2$.

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In order to clarify why the splits in reflection, transmission and absorption spectra show such differences, it is started from the theory to find the physical mechanism behind. The splits refer to the variations of the corresponding eigenvalues from one to two values and it is required to diagonalize the corresponding matrixes. The master Eq. (4) of TCMT is composed of the near-field coupling matrix, the absorption loss matrix and the radiation loss matrix, then two representational transformations have been performed. The first is to diagonalize the sum of near-field coupling term and absorption loss term in Eq. (4) to obtain a set of eigenvalues:

To make Eq. (11) have a non-zero solution, i.e., to find the following determinant to be zero:

The eigenfrequencies are solved as

This representation is called the intermediate representation (IR).

It can be seen that the eigenfrequencies vary from two values to one when the part under the root sign is equal to 0. In this case, $4{\kappa ^2}\textrm{ = }{({{{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} )^2}$, i.e.,

As shown in Eq. (14), when $\kappa \le |{{{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} |/2$, the spectra are in the degenerate state. Only when $\kappa > |{{{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} |/2$, the spectral lines are just split.

The second is to diagonalize the sum of near-field coupling term, absorption loss term and radiation loss term in Eq. (4) to obtain another set of eigenvalues:

To make Eq. (15) have a non-zero solution, i.e., to find the following determinant to be zero:

The eigenfrequencies are solved as

This representation is called the normal representation (NR).

It can be seen that the eigenfrequencies vary from two values to one when the part under the root sign is equal to 0. In this case, $4{\kappa ^2}\textrm{ = }{({{\gamma_p} + {{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} )^2}$, i.e.,

As shown in Eq. (18), when $\kappa \le |{{\gamma_p} + {{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} |/2$, the spectra are in the degenerate state. Only when $\kappa > |{{\gamma_p} + {{\gamma^{\prime}}_p} - {{\gamma^{\prime}}_q}} |/2$, the spectral lines are just split.

Because the normal representation considers more than the intermediate representation a radiation loss matrix, it is supposed that the radiation loss brings about the difference in the critical splitting coupling coefficient ${\bar{\kappa }_E}$ of the reflection spectrum and the absorption spectrum. Based on the derivations above, we can compare the connections of the eigenfrequencies of two representations with the resonant frequencies of reflection and absorption spectra by Eqs. (8), (10), (13) and (17). Here, for the convenience of description, the real parts of the eigenfrequencies of IR (or NR) are defined as ${\bar{f}^{\textrm{IR}}} = Re ({\Omega ^{\textrm{IR}}})/\bar{f}$ (or ${\bar{f}^{\textrm{NR}}} = Re ({\Omega ^{\textrm{NR}}})/\bar{f}$). Based on Eqs. (13) and (17), variations of ${\bar{f}^{\textrm{IR}}},{\bar{f}^{\textrm{NR}}}$ (blue and green solid curves) with $\bar{\kappa }$ can be plotted in Fig. 3, respectively. Based on Eqs. (8) and (10), the reflection (or absorption) spectra can be plotted. We can find the variations of the number of resonant frequencies as $\bar{\kappa }$ increase. The variations of the resonant frequencies of the reflection (or absorption) spectra ${\bar{f}_{FR}} = {f_{FR}}/\bar{f}$ (or ${\bar{f}_{FA}} = {f_{FA}}/\bar{f}$) with $\bar{\kappa }$ are indicated by red hollow circle (or triangle), respectively, as shown in Fig. 3.

 

Fig. 3. Variations of eigenfrequencies and resonant frequencies of reflection spectra (A),(B) and absorption spectra (C),(D) with near-field coupling coefficients $\bar{\kappa }$ when ${\bar{\gamma }_p} = {\bar{\gamma ^{\prime}}_p} = {\bar{\gamma ^{\prime}}_q} = 0.01$ (A), (C), ${\bar{\gamma ^{\prime}}_p} = {\bar{\gamma ^{\prime}}_q} = 0.01,{\bar{\gamma }_p} = 0.2$ (B), (D). All quantities have been divided by the central frequency $\bar{f} = ({f_p} + {f_q})/2$.

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As shown in Fig. 3(A) and (C), when ${\bar{\gamma ^{\prime}}_p} = {\bar{\gamma ^{\prime}}_q} = 0.01 = {\bar{\gamma }_p}$, the almost identical results are given by ${\bar{f}^{\textrm{IR}}},{\bar{f}^{\textrm{NR}}},{\bar{f}_{FR}},{\bar{f}_{FA}}$. As shown in Fig. 3(B) and (D), when ${\bar{\gamma ^{\prime}}_p} = {\bar{\gamma ^{\prime}}_q} = 0.01,{\bar{\gamma }_p} = 0.2$, the almost identical results are given by ${\bar{f}^{\textrm{IR}}},{\bar{f}_{FR}}$ (or ${\bar{f}^{\textrm{NR}}},{\bar{f}_{FA}}$). However, ${\bar{f}_{FR}}$ is distinctly different from ${\bar{f}_{FA}}$, and the increase of ${\bar{\gamma }_p}$ brings delayed splitting of the absorption spectrum. Therefore, it is further supposed that ${\bar{\gamma }_p}$ may be related to ${\bar{\kappa }_{EA\_\textrm{S}}}$. The assumption is in good agreement with Fig. 2(A), in which ${\bar{\gamma }_p}$ and ${\bar{\kappa }_{EA\_\textrm{S}}}$ are linearly correlated. Moreover, the results show that when ${\bar{\gamma ^{\prime}}_p}$ and ${\bar{\gamma ^{\prime}}_q}$ are small, the resonant frequencies of the reflection (or absorption) spectra are well explained by ${\bar{f}^{\textrm{IR}}}$ (or ${\bar{f}^{\textrm{NR}}}$) in the region far from EPs. But near EPs, because of the singularity of EPs, the original representation transformation theories are no longer applicable, so Eqs. (14) and (18) should be modified. It is found in the spectral find-peak results that ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_p},{\bar{\gamma ^{\prime}}_q}$ have different roles and ${\bar{\gamma ^{\prime}}_p}$ likes a perturbation. Thus, for reflection, ${\bar{\gamma ^{\prime}}_q}$ is dominant while ${\bar{\gamma ^{\prime}}_p}$ brings little effect and the related expression is

Similarly, for transmission, ${\bar{\kappa }_{ET\_\textrm{F}}}$ is very similar to ${\bar{\kappa }_{ER\_\textrm{F}}}$, but always slightly larger than ${\bar{\kappa }_{ER\_\textrm{F}}}$. The associated expression is ${\bar{\kappa }_{ET - \textrm{F}}}\textrm{ = }{\bar{\gamma ^{\prime}}_q}/2 + {C_q}$, where ${C_q}$ is mainly contributed by ${\bar{\gamma ^{\prime}}_q} - {\bar{\gamma ^{\prime}}_p}$. Finally, ${\bar{\kappa }_{EA\_\textrm{F}}}$ is mainly related to ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_q},{\bar{\gamma ^{\prime}}_p}$. When ${\bar{\gamma }_p} > {\bar{\gamma ^{\prime}}_p}$, the ${\bar{\gamma }_p}$ is dominant and the associated expression is

Based on Eqs. (19) and (20), the variations of ${\bar{\kappa }_{ER\_\textrm{F}}},{\bar{\kappa }_{ET\_\textrm{F}}},{\bar{\kappa }_{EA\_\textrm{F}}}$ are indicated by red dashed line, green solid line and blue dotted line respectively in Fig. 2. It is showed that modified formulas are in good agreement with the spectral find-peak results. In the fitting results, ${\bar{\gamma ^{\prime}}_q}$ and ${\bar{\kappa }_{ER\_\textrm{F}}},{\bar{\kappa }_{ET\_\textrm{F}}},{\bar{\kappa }_{EA\_\textrm{F}}}$ are linearly related, while ${\bar{\gamma }_p}$ is only linearly related to ${\bar{\kappa }_{EA\_\textrm{F}}}$. As ${\bar{\gamma }_p}$ increases, ${\bar{\kappa }_{ER\_\textrm{F}}}$ is slightly smaller and ${\bar{\kappa }_{ET\_\textrm{F}}}$ is almost constant as before.

In the near-infrared and visible-light band, which is shown in Fig. 2(A) and Fig. 3, the absorption losses of bright mode and dark mode are both very small, so Eq. (19) is simplified as

Equation (21) means that only a small value of $\bar{\kappa }$ is required for reflection and transmission spectra to split. And Eq. (20) is simplified as

Equation (22) means that absorption spectrum will split if $\bar{\kappa }$ reaches half of ${\bar{\gamma }_p}$. Here, based on Eqs. (21) and (22), we can give a definition for the “far” and “near”. That is, the range of “near” is about $\bar{\kappa } - {\bar{\kappa }_E}\textrm{ = } \pm ({\bar{\gamma ^{\prime}}_q} + {\bar{\gamma ^{\prime}}_p})/2$. The regions other than “near” are all regions of “far”. Moreover, in some areas of Fig. 2, ${\bar{\kappa }_{ER\_\textrm{F}}},{\bar{\kappa }_{ET\_\textrm{F}}},{\bar{\kappa }_{EA\_\textrm{F}}}$ are very close. It is because that ${\bar{\gamma ^{\prime}}_p}$ is relatively large and ${\bar{\gamma }_p},{\bar{\gamma ^{\prime}}_q}$ are relatively small in these areas. And when ${\bar{\gamma }_p} < < {\bar{\gamma ^{\prime}}_p}$, the bright mode becomes a continuous state, which isn’t a mode. Therefore, it cannot be explained by two-modes and two-ports TCMT. And ${\bar{\kappa }_{ER\_\textrm{F}}},{\bar{\kappa }_{ET\_\textrm{F}}},{\bar{\kappa }_{EA\_\textrm{F}}}$ are very close, which cannot be explained by Eq. (19)–(22).

3. Structures and simulations

As above, ${\gamma ^{\prime}_q}$ and reflection, transmission and absorption EPs are linearly related, while ${\gamma _p}$ is only linearly related to absorption EPs. And it is well known that ${\gamma ^{\prime}_q}$ can be varied by the absorption loss of the material, while ${\gamma _p}$ can be varied by the structure size [49]. In the following, two metasurfaces are designed, where the material absorption loss of the dark mode and the structural size of the bright mode can be adjusted to observe the variations of reflection, transmission and absorption EPs, respectively.

3.1 Structural design and optical properties

First, a unit cell of metasurface I is composed of two gold nano-bars, as shown in Fig. 4(A), (B). The bars with thicknesses (t) of 35 nm are grown on the Si₃N₄ film with a thickness (h) of 330 nm. Without loss of generality, the period P of the lattice is chosen to be 730 nm. The length and width of the bars are 210 nm and 150 nm, respectively. The asymmetry degree $d = P/2 - w - s$ is defined as the lateral displacement of each bottom bar along the x-axis.

 

Fig. 4. Schematic diagrams of the structural parameters, (A) perspective and (B) top views of a unit cell of metasurface I, (E) perspective and (F) top views of a unit cell of metasurface II. Reflection, transmission and absorption coefficients of metasurface I when (C) $d = 10$ nm and (D) $d = 150$ nm, metasurface II when (G) $d = 0$ nm and (H) $d = 80$ nm. The metasurfaces I and II are illuminated by normally incident lights under y-polarization.

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Second, a unit cell of metasurface II is composed of a gold nano-ring and a gold nano-bar, as shown in Fig. 4(E), (F). The bar with a thickness (t) of 30 nm is inserted into the silica film with a thickness (h) of 90 nm, and the thickness of the ring is 30 nm. Without loss of generality, the period P of the lattice is chosen to be 960 nm. The length and width of the bar are 350 nm and 70 nm, respectively. The outer and inner radii of the ring are 370 nm and 300 nm, respectively. The asymmetry degree d is defined as the lateral displacement of the bottom bar along the y-axis.

In order to investigate the optical properties of the two metasurfaces, finite-difference time-domain simulations are performed. Here, we scan the asymmetry degree of the above designed structures and observe the changes of the spectral lines. In the simulation, the refractive index of the glass substrate and the intermediate silica layer are chosen to be 1.5 [50], and that for the background is 1. The dielectric constant of gold follows the Drude model ${\varepsilon _{Au}} = 1 - \omega _p^2/({\omega ^2} + i\gamma \omega )$, where the frequencies of plasmonic resonances and the damping are ${\omega _p} = 1.374 \times {10^{16}}$ rad/s and $\gamma = 1.224 \times {10^{14}}$ rad/s [51]. The refractive index of Si₃N₄ is set to $n = {n_r} + i{n_i}$, where ${n_r}$ is chosen to be 2 and ${n_i}$ takes values of 0.02, 0.03 and 0.04.

Without loss of generality, in metasurface I, the material loss ${n_i}$ is taken as 0.03, while other parameters are as above, and the reflection spectrum (red solid line), transmission spectrum (green dashed line) and absorption spectrum (blue dotted line) of metasurface I are simulated in Fig. 4 for $d = 10$ nm (C) and $d = 150$ nm (D), respectively. It can be seen that when $d = 10$ nm, there is only one resonant frequency in every spectral line, which means that all three spectral lines are unsplit. When $d = 150$ nm, there are two resonant frequencies in every spectral line, which means that all three spectral lines are completely split. This is due to the fact that the localized surface plasmon (LSP) [42] of the bars act as the bright mode, while the waveguide mode [52] of the intermediate Si₃N₄ layer acts as the dark mode. When the structure is symmetry breaking, a plasmon-induced transparency (PIT) is motivated by the coherent interference of LSP and waveguide mode. Therefore, when d is relatively large, the coupling is stronger leading to the split of the peak.

Similarly, without loss of generality, in metasurface II, the length of the bar is taken as $l = 410$ nm, and the outer and inner radii of the ring are adjusted to $R = 400$ nm and $r = 330$ nm. The thickness of the intermediate silica film is also adjusted to $h = 150$ nm, so that the ring is submerged in the silica film, and other parameters are the same as above. The reflection spectrum (red solid line), transmission spectrum (green dashed line) and absorption spectrum (blue dotted line) of metasurface II are simulated in Fig. 4 for $d = 0$ nm (G) and $d = 80$ nm (H), respectively. It can be seen that when $d = 0$ nm, all three spectral lines are unsplit, as same as previously described. When $d = 80$ nm, the reflection and transmission spectral lines are completely split, while the absorption spectral line is unsplit as before. This is due to the fact that the electric dipole mode of the bar acts as the bright mode and the electric quadrupole mode of the ring acts as the dark mode [53]. When the structure is symmetry breaking, a PIT is motivated by the coherent interference of electric dipole mode and electric quadrupole mode. Therefore, when d is relatively large, the coupling is stronger leading to the split of the peak.

3.2 Results and discussion

As shown in Fig. 5, to verify the previous theory related to ${\gamma ^{\prime}_q}$, the reflection, transmission, and absorption spectra are simulated when the material loss ${n_i}$ in metasurface I are 0.02, 0.03, and 0.04 respectively. It can be seen that when ${n_i}$ is 0.02, 0.03 and 0.04, the position of ${d_{ER}}$ is at 25 nm, 35 nm and 45 nm (${d_{ET}}$ at 35 nm, 47.5 nm and 65 nm and ${d_{EA}}$ at 37.5 nm, 62.5 nm and 77.5 nm), respectively. Here, ${d_{ER}},{d_{ET}},{d_{EA}}$ are expressed as the critical splitting asymmetry degree ${d_E}$ for reflection, transmission and absorption spectra, respectively. For the same ${n_i}$, there is ${d_{ER}} < {d_{ET}} < {d_{EA}}$. As ${n_i}$ increases linearly, ${d_{ER}},{d_{ET}},{d_{EA}}$ increase correspondingly linearly. The absorption loss ${\gamma ^{\prime}_q}$ of the dark mode is theoretically linearly related to ${n_i}$, and the critical splitting coupling coefficient ${\kappa _E}$ is also linearly related to ${d_E}$. It means that when ${\gamma ^{\prime}_q}$ increases linearly, the critical splitting coupling coefficients ${\kappa _{ER}},{\kappa _{ET}},{\kappa _{EA}}$ also increase linearly. According to the theory, ${\gamma ^{\prime}_q}$ and ${\kappa _{ER}},{\kappa _{ET}},{\kappa _{EA}}$ are linearly related. The simulation results in Fig. 5 are in good agreement with the assumption related to ${\gamma ^{\prime}_q}$.

 

Fig. 5. (A), (E), (I) Schematic diagrams of the three kinds of the side views of metasurface I. (B)-(D), (F)-(H), (J)-(L) Schematic diagrams of the variations of the resonant wavelengths (speed of light / frequency) of reflection, transmission and absorption spectra with the asymmetry degree d when the loss ${n_i}$ of Si₃N₄ are 0.02, 0.03, and 0.04, respectively. The red solid pentagrams, expressed as the resonant wavelengths corresponding to the EPs, are connected with the red dashed lines, expressed as the variations of EPs with ${n_i}$. ${d_{ER}},{d_{ET}},{d_{EA}}$ are expressed as the asymmetry degree d corresponding to the resonant wavelengths when the reflection, transmission and absorption spectra are split.

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As shown in Fig. 6, to verify the previous theory related to ${\gamma _p}$, the reflection, transmission and absorption spectra are simulated when the length l of the bar in metasurface II are 350 nm, 410 nm and 560 nm respectively. It is noteworthy that the outer radius R of the ring is fine-tuned while l is varied in order to find the phenomenon consistent with the PIT to ensure ${f_p} = {f_q}$. Of course, the difference between the outer and inner radii of the ring are the same as before, which is still 70 nm. Furthermore, the positions of the bar and the ring on the z-axis are varied appropriately, and the corresponding side views have been shown in Fig. 6 ((A), (E), (G)). In Fig. 6(A), $l = 350$ nm and $R = 370$ nm. The bar is at the bottom of the intermediate silica layer and the ring is on the top of the silica layer, and other parameters are unvaried as before. In Fig. 6(E), $l = 410$ nm and $R = 400$ nm. The thickness of the silica layer is adjusted to 150 nm and the ring is dumped. Therefore, the bar is at the bottom of the silica layer as before, while the ring is in the center of the silica layer, and other parameters are unvaried as before. In Fig. 6(I), $l = 560$ nm and $R = 400$ nm. In this case, the thickness of the silica layer is 90 nm as before, while the bar and the ring exchange positions. The ring is at the bottom of the silica layer and the bar is on the top of the silica layer, and other parameters are unvaried as before.

 

Fig. 6. (A), (E), (I) Schematic diagrams of the three kinds of the side views of metasurface II. (B)-(D), (F)-(H), (J)-(L) Schematic diagrams of the variations of resonant wavelengths of reflection, transmission and absorption spectra with asymmetry d when the length l of gold nano-bar are 350 nm, 410 nm, and 560 nm. The red solid pentagrams, expressed as the resonant wavelengths corresponding to the EPs, are connected with the red dashed lines, expressed as the variations of EPs with l. ${d_{ER}},{d_{ET}},{d_{EA}}$ are expressed as the asymmetry degree d corresponding to the resonant wavelengths when the reflection, transmission and absorption spectra are split.

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It can be seen in Fig. 6 that when l are 350 nm, 410 nm and 560 nm, the position of ${d_{ER}}$ is around 10 nm, 10 nm and 0 nm (${d_{ET}}$ around 20 nm, 20 nm and 20 nm and ${d_{EA}}$ around 110 nm, 140 nm and 160 nm), respectively. For the same l, there is ${d_{ER}} < {d_{ET}} < {d_{EA}}$. When l increases approximately linearly, ${d_{ER}}$ is slightly smaller, ${d_{ET}}$ is constant as before and ${d_{EA}}$ increases correspondingly linearly. Theoretically, the radiation loss ${\gamma _p}$ of the bright mode is approximately linearly related to l, and the critical splitting coupling coefficient ${\kappa _E}$ is also linearly related to ${d_E}$. It means that as ${\gamma _p}$ increases linearly, ${\kappa _{ER}}$ is slightly smaller, ${\kappa _{ET}}$ is constant as before, and ${\kappa _{EA}}$ increases linearly. According to the theory, ${\gamma _p}$ is only linearly related to ${\kappa _{EA}}$. As ${\gamma _p}$ increases, ${\kappa _{ER}}$ is slightly smaller and ${\kappa _{ET}}$ is almost constant as before. The simulation results in Fig. 6 are in good agreement with the assumption related to ${\gamma _p}$.

With the results in Fig. 5 and Fig. 6, it can be roughly seen that ${\gamma ^{\prime}_q}$ is linearly related to ${d_{ER}},{d_{ET}},{d_{EA}}$, while ${\gamma _p}$ is only linearly related to ${d_{EA}}$. As ${\gamma _p}$ increases, ${d_{ER}}$ is slightly smaller and ${d_{ET}}$ is almost constant as before. To more intuitively represent the relationships of ${d_{ER}},{d_{ET}},{d_{EA}}$ with ${n_i}$ and l, the simulation results are summarized and shown in Fig. 7. It can be seen in Fig. 7 that as ${n_i}$ increases linearly, ${d_{ER}},{d_{ET}},{d_{EA}}$ increase correspondingly linearly. However, when l increases approximately linearly, ${d_{ER}}$ is slightly smaller, ${d_{ET}}$ is almost constant as before and ${d_{EA}}$ increases correspondingly linearly. The simulation results in Fig. 7 are in good agreement with the assumption related to ${\gamma _p},{\gamma ^{\prime}_q}$.

 

Fig. 7. Schematic diagrams of the variations of ${d_{ER}},{d_{ET}},{d_{EA}}$ with (A) the material loss ${n_i}$ in metasurface I and (B) the length l of the bar in metasurface II, respectively. The dotted lines are expressed as the EPs of reflection (red solid triangle), transmission (green solid pentagram) and absorption (blue solid circle) spectra, respectively.

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Next, electron beam lithography (EBL) will be tentatively used to make the designed metasurfaces and the spectral lines of the metasurfaces will be observed on the optical platform. Through the spectral lines, the radiation loss and absorption loss of the bright mode and the dark mode of the metasurfaces can be fitted by TCMT. By comparing these losses with the asymmetries of the metasurfaces, the assumption can be verified. In addition, when the imaginary part of the complex frequency of the hybrid mode is 0, that is, the total loss is 0, the metasurface can be understood as a bound state in the continuum (BIC) [54]. This is reflected in the fact that the radiation loss of the metasurface is 0 and the absorption loss is very small. It can be regarded as a dark mode with high quality factor.

In a word, between dark mode and bright mode, bright mode and bright mode, or in multiple modes, the characteristics of EP of each spectral line can be studied. However, for practical realization of optical devices or metasurfaces, it is still difficult to implement the characteristics of EP, especially the high-order EP. There are two reasons: first, metasufaces with EP require high experimental processing. Second, it is difficult to achieve the coupling of only two modes in the band of interest in metasurfaces, which is always affected by the third mode. In addition, the high-order EP is easily affected by the error caused by processing, and is vulnerable to the interference of the environment, thus collapsing into a second-order EP. Therefore, theoretical research is needed to provide guidance for the design and implementation of optical devices.

4. Conclusions

In summary, in the dark-bright mode-coupled plasmonic systems, the variations of EPs of reflection, transmission and absorption spectra are numerically investigated using TCMT, and an assumption is given using the representation transformation theory. The IR is firstly proposed and related to the reflection spectrum, while the NR is associated with the absorption spectrum. In the region far from EPs, the IR (or NR) describes the reflection (or absorption) spectrum well. Near EPs, modified formulas similar to the representation transformation theory are given. In order to verify the correctness of the assumption, two metasurfaces are designed. And the simulation results of the two metasurfaces are in good agreement with the assumption and it is found in the near-infrared and visible-light band that the absorption loss of the dark mode is linearly related to the EPs of reflection, transmission and absorption spectra, while the radiation loss of the bright mode is only linearly related to the EPs of the absorption spectrum. These laws can help to manipulate the splitting of spectral lines for reflection, transmission and absorption by adjusting the radiation loss and absorption loss of bright mode, the absorption loss of dark mode and the coupling coefficients between two resonant modes. This research provides a guiding scheme for the design of micro and nano photonics devices.