1. Introduction

Stemming from quantum mechanics [1–7], PT symmetry is an important feature of a group of non-Hermitian systems which render exotic physical phenomena beyond the conventional Hermitian ones in both quantum and classical physics [1–16]. With the balanced gain and loss sufficiently strongly coupled, the system renders real eigenvalue spectrum as a Hermitian system does. However, the fascinating physics emerges just as the equilibrium is broken. The critical points across which the system enters the symmetry-broken phase are the so-called EPs [12]. Therefore, it is a fundamental task to find the EPs before the exotic physics can be revealed. EPs can be retrieved by tuning physical parameters of the system in the phase space. Either increasing the gain and loss or decreasing the coupling strength can lead to the emergence of EPs and the symmetry-broken phase. It is also seen in some papers that EPs exist at some points of the dispersion curves. Such EPs are always accompanied by complex conjugate wavenumbers in the symmetry-broken region while the frequency is treated as the tuning parameter and is kept real through the whole spectrum [17–19]. However, it is noteworthy that the gain and loss in this kind of systems are generally constants and added to dielectric layers. This is very different from the frequency-dependent loss or gain of metallic layers. Up to our knowledge, few papers have utilized loss or gain in metallic layers to form the PT symmetric system. It is still interesting to investigate the PT symmetry formed by the balanced gain and loss in disperive metallic layers. Meanwhile, transformation optics (TO) has provided an easy way of conceiving functional structures from the geometric point of view [20–22]. It is also a useful method in describing physical phenomena in various length scales [23], from celestial mechanics [24,25] down to nano-optics and plasmonics [26–28]. With the so-called complex coordinate transformations [29], it is now rather convenient to conceive non-Hermitian or even PT symmetric structures by geometrical mappings [30]. Similar concepts have also occurred in designing the so-called “Pseudo-Hermitian System” with TO theory [31].

Quasinormal-mode (QNM) theory emerges as a profound method in describing open systems [32]. QNMs per se are modes with complex frequencies which is able to depict the time dependent attenuation or amplification of energy of the system. As an analogy of the normal mode expansion in conservative systems, QNM expansion has been well established and applied to various linear classical wave systems and gravitational systems [33]. Recently, QNM theory has been with the rapid development of the research on nanophotonics and non-Hermitian loss-gain systems [34], so that the conventional theoretical framework would be kept consistent with the cutting-edge of the latest research [35]. It is generally considered that QNMs are more of a theoretical tool in analysing open systems. Nevertheless, a recent work which introduced the concept of “virtual PT symmetry” ingeniously utilized a decaying QNM as the input signal and generated the so-called “virtual gain” within the transmission line model [36]. This pioneering work indicates that QNMs are also revealing its potential application in advanced classical wave systems besides its conventional use as a theoretical tool for open system analysing. This has become part of the motivation of conceiving this paper.

In this paper, we focus on the study of the QNMs that exists in a PT symmetric structure obtained via the complex-coordinate transformation. Not only will the QNM be depicted in terms of its dependence of the geometry and the constitutive parameters of the structure, but the physical mechanism of its existence will also be briefly discussed, aiming at a rather full understanding of QNMs in the model.

2. Transformation optics (TO) framework

Comparing with Ref. [30], we consider a more general coordinate transformation:

Here, $(x^{\prime},y^{\prime},z^{\prime})$ and (x, y, z) are the Cartesian coordinates of the auxiliary vacuum space and the physical space. The corresponding Jacobian matrix is formulated as:

Here, the subscript denotes the partial derivatives. The deduced constitutive parameters in the physical space are written as:

Without loss of generality, we aim to focus on two semi-infinite homogeneous slabs with PT symmetry. In this case, the assumed three functions p, q, and s in the coordinate transformation (CT), should firstly be set to

Equation (3a) reduces to

As we focus on TM modes which is the intrinsic feature of the surface plasmon polaritons (SPPs) at the metal-insulator interface, only the following three parameters are involved

In the optical regime, it is reasonable to set ${\mu _{yy}} = 1$. Meanwhile, as mentioned above, we aimed at homogeneous media, i.e. ${\varepsilon _{xx}} = {\varepsilon _{zz}}$. The two conditions give rise to the following constitutive parameter in the physical space:

Here the constants p, q and s are substituted by $\eta = {{qs} / p}$, with p, q, s satisfying $ps = q$ and ${p^2} = {s^2}$.

Up to this stage, based on the complex coordinate transformations, the PT symmetric semi-infinite slabs have been geometrically established (Fig. 1).

 

Fig. 1. Schematic illustration of the TO framework. Vacuum is the auxiliary space before coordinate transformation. After performing ${r^{\prime}} = {F}({r})$, the system is transformed to the M-I-aM structure, the transformed physical space exhibits PT symmetry.

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3. QNMs: finding exceptional points (EPs) with causality

As seen from Eq. (8), the two semi-infinite slabs form the PT symmetric model. In this section we adopt realistic dispersive permittivities and search for the EPs. $\eta $ is set to the Drude model permittivity of metals:

 

Fig. 2. The dispersion curves of the MIaM systems calculated by sweeping real frequency. The main figure is for the silver MIaM. The inset is for gold MIaM. The horizontal axis stands for the propagation constant $\beta $, which is normalized by ${k_\textrm{p}} = {{{\omega _\textrm{p}}} / c}$. Here the plasma frequency for silver (gold) is set to ${\omega _\textrm{p}}$ = 9 eV (8.9 eV) and c is the speed of light in vacuum. The vertical axis is the frequency denoted by its corresponding photonic energy $\hbar \omega $. The small discrepancy of the two cases could be observed from the starting point of the upper branch of the dispersions, where the starting point for gold MIaM is a bit lower than the silver case. The coalescence point of the two branches is shifted left a bit for the gold case. Neither case exhibits EPs in the real frequency spectrum.

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Therefore, based on Eq. (11), we alternatively swept real wavenumber (propagation constant) $\beta$ to search for EPs. For simplicity, the horizontal and vertical coordinates of EPs are written as (${\beta _{\textrm{EP}}}$, ${\omega _{\textrm{EP}}}$). In Fig. 3, $\beta $ is swept from 0 to 15k_p for silver MIaM systems. Here, ${k_\textrm{p}} = {{{\omega _\textrm{p}}} / c}$, ${\omega _\textrm{p}}$ is the plasma frequency and c is the speed of light in vacuum. Take the vacuum gap 2d = 10 nm in Fig. 3(a) for example. When the two blue branches coalesce at $\beta = 9.12{k_\textrm{p}}$, $\omega $ switches from real values into complex conjugates, which indicate that the point ${\beta _{\textrm{EP}}} = 9.12$ ${k_\textrm{p}}$ is the EP of the MIaM system. When $\beta > \,{\beta _{\textrm{EP}}}$, the complex $\tilde{\omega }$ (from now on, the tilde represents complex values) indicates that the supported SPP modes are with time-dependent attenuation or amplification. They are the QNMs of the PT symmetry broken phase. Varying the gap size 2d (10 nm, 20 nm, 100 nm) reveals that ${\beta _{\textrm{EP}}}$ is inversely proportional to the gap size. This can be qualitatively explained as: the PT symmetry is easier to be broken down as the coupling strength is weakened when increasing the gap size. At the EP, ${\omega _{\textrm{EP}}} \approx {{{\omega _\textrm{p}}} / {\sqrt 2 }}$ hold for all the three different cases, which is the characteristic surface plasmon frequency ${\omega _{\textrm{SP}}}$ of the planar interface and is thus independent of the gap size. For all the three cases with different gap sizes in Fig. 3(a), the imaginary parts of $\tilde{\omega }$ ‘s of the QNMs approach the same asymptotic value $\gamma /2$, ($\gamma $ is the damping factor of the Drude model of silver). Therefore, for all the QNMs in the symmetry broken phase, $\tilde{\omega } \to {\omega _{\textrm{SP}}} \pm \textrm{i}{\gamma / 2}$ when $\beta \to \infty $. In Fig. 3(b), we investigate the dependence of the QNMs on the damping (amplifying) factor when the gap size is fixed. It can be observed that larger damping (amplifying) induces smaller ${\beta _{\textrm{EP}}}$. This is also an intuitive conclusion because the fixed coupling strength would not be able to balance too large attenuation and amplification. When the electric field of the surface modes becomes sufficiently large, the nonlinearity of metals would emerge and lead to second harmonic generations (SHG) [41,42]. However, it should also be importantly noted that the above description is based on the ideal Drude model. Further considerations should be taken because real metallic materials would not offer a constant amplification. Saturation effect [43] of the amplification factor would emerge as the amplitude of the modes increases. This would confine the whole system within finite amount of energy so that no singularity exists.

 

Fig. 3. (a) Dispersion curves of the silver MIaM of different gap sizes. Blue (black, green) solid line and red (pink, orange) dash dot line represent the real and imaginary part of complex frequency of the QNMs with the gap size set to 2d = 10 (20, 100) nm. The horizontal axis denotes the propagation constant $\beta $ normalized to ${k_\textrm{P}}$ = ${{{\omega _\textrm{P}}} / c}$. The vertical axis denotes the frequency with its corresponding photon energy in unit eV. (b) Dispersion curves of the silver MIaM of different Drude damping factors.

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4. Conclusion

In conclusion, starting from the complex coordinate transformation we have investigated the existence of EPs and QNMs in a dispersive MIaM system with PT symmetry. It has been demonstrated that the EPs can’t be found by sweeping the real frequency spectrum. By extending the frequency to the complex regime, the EP and the QNMs in the symmetry broken phase emerge. The MIaM is a very typical example which indicates the importance of considering the causality principle before searching for EPs in a realistic dispersive PT symmetric system. On the other side, owing to the rapid development of the research on PT symmetry in photonic systems, QNMs have been regarded as the input signal to realize the so-called virtual PT symmetry [36]. This is a rather new application of the modes in the PT symmetry broken phase. When the transmission line circuit with uniform decaying factor ${\gamma _0}$ is input by a time-decaying signal with a complex frequency $\tilde{\omega } = \omega - \textrm{i}{\gamma / 2}$, viz the QNM in this paper, the circuit (transmission line TL) renders the spatially amplifying time-average power flow when ${\gamma / 2} > {\gamma _0}$. It can be expected that the time-decaying (amplifying) characteristics of QNMs would be utilized to achieve further exotic phenomena in future.