1. Introduction

In recent years, optical properties of subwavelength metallic nanostructures, which are frequently termed metamaterials, have attracted much attention. The key of metamaterials is to utilize the localized electromagnetic (EM) resonances, especially surface plasmon polaritons (SPPs), to achieve effective optical properties unavailable from natural media. By engineering the geometry of the constituent elements we can control the interaction of light with metamaterials. A variety of novel optical effects have been proposed and demonstrated, including extraordinary optical transmission (EOT) [1], negative index of refraction [2] and cloaking [3], to name only a few.

Resonant wavelength λ_R and width Γ_R are two intrinsic features of any EM resonance, which are of critical importance for metamaterials because they determine the performance of practical applications. With the advances of metamaterials, nowadays people have switched their attentions to complex structural designs that contain multiple constituent elements. The wavelengths λ_R, widths Γ_R, and darknesses of the hybridized EM resonances can be controlled by manipulating the coupling among the constituent elements. Although a full-wave three-dimensional numerical simulation can provide us with detailed information of the EM resonances, it is always desired that we could develop some simple but instructive analytical formulae to model the effect of the mutual coupling, especially when λ_i and Γ_i of the EM resonance in each constituent element are prior known. A nice example is the hybridization model, an EM analogue of molecular orbital theory [4], which can briefly predict the resonant frequencies of the coupled eigenstates in complex nanostructures of arbitrary shapes. By analogy with a quantum system, Hamiltonian operators are utilized in this model to characterize the interaction of various localized EM resonances [4].

From the analogy with quantum physics we can understand and explain almost all the hybridized optical effects in complex metamaterials. However, attention must be drawn here because the EM resonances in metamaterials are not ideally analogous to the eigenstates of closed quantum systems. Absorption loss could not be eliminated and ignored when metal is presented, especially in the frequency region of visible and infrared. It is well known that when a quantum system is subjected to dissipation [5–7], it becomes an open system and could not be characterized by a Hermitian Hamiltonian. A non-Hermitian Hamiltonian must be utilized. The resonant wavelengths λ_R and widths Γ_R are greatly influenced by the presence of loss in such an open quantum system, and many unique topological phenomena unavailable from closed quantum systems can be observed [8].

Among all the unique properties of open quantum systems, a singular phenomenon termed exceptional points (EPs) has attracted great attention in the past decade [5]. This effect emerges when we study the eigenstates as functions of complex interaction parameters, i.e. to find the solutions of the corresponding non-Hermitian Hamiltonian matrix. For a two-state scenario, the solutions would lie on two different Riemann sheets. The two eigenstates, including their eigenvalues and eigenfunctions, would coalesce at a branch point singularity, i.e. EP, when continued analytically into the two-dimensional (2D) plane of parameters [5,6]. An EP is associated with a level repulsion, and the existence of an anti-crossing (crossing) in λ_R must accompany a crossing (anti-crossing) in Γ_R[7]. The emergence of level crossing or anti-crossing depends on the strength of the coupling between the two basis states. A strong coupling leads to a level anti-crossing and a wave function exchange, while a weak coupling leads to a crossing of the energy levels [7, 9]. As a singularity of two Riemann sheets, EPs possess many unique properties, especially in the associated chirality [10], geometric phase [11,12] and topological structure [13].

Although EPs were initially proposed to explain the unique properties of open quantum systems, the extension of similar concept to classic systems, i.e. the classic analogue of EPs, have also been pioneered in optical structures mostly with a geometric size greater than wavelength. Wiersig studied the scarlike modes in optical microcavities, and showed that long-lived states can be formed near the avoided resonance crossings [14]. Dietz et al. discussed the Rabi oscillations at EPs in microwave billiards, and proved that a 2 × 2 matrix model describes the quantum echoes very well [15]. Based on the fact that quasibound states in an open system do not form an orthogonal and complete basis, Wiersig et al. demonstrated the formation of asymmetric scattering and nonorthogonal mode patterns in optical microspirals [16]. Ryu et al. proved that the evolution of energy levels is characterized by the avoided crossing intrinsically in EPs, which explains the highly asymmetric near-field intensity pattern of the resonant mode in coupled nonidentical microdisks that leads to an unidirectional far-field emission [17]. Song et al. proposed to improve the optical confinement in subwavelength nanostructures via external mode coupling [9]. Scenarios in a microwave billiard with time-reversal invariance violation [18], in deformed microdisk cavities [19], in lasers [20], and the feasibility in realizing unidirectional light propagation [21] have also been discussed.

Unlike these classic systems [14–20] investigated so far, metamaterials are subwavelength in nature [9]. By comparing a complex metamaterial with an open quantum system we can find that they possess many similarities. They both contain or support two or more otherwise independent resonances. Quantum confinement effect determines the frequencies of these resonances, and loss destroys the coherence of their mutual coupling. For metamaterials, the loss is mainly related to the non-radiative absorption by the collision of oscillating free electrons in metal. Radiative loss is prohibited because the high-order diffraction modes are evanescent. It is well known that proper elements in the Hamiltonian matrix that represent loss are necessary in rending the absence of self-adjointness of the Hamiltonian and in determining the emergence of level anti-crossing or crossing in EPs [7]. Consequently, if the loss in a complex metamaterial is taken into account, it is feasible to observe EP-like effects. To propose and prove such a feasibility is the main purpose of this paper. The observation of EPs in complex metamaterials would contribute to the progress of metamaterials toward many potential applications by transferring the concepts from open quantum systems to optical metamaterials.

In this paper, we pay attention to one of the simplest complex metamaterials that can be modeled by a two-state non-Hermitian quantum system. This metamaterial is a dual metallic grating (DMG) that contains two one-dimensional single subwavelength metallic gratings (SMGs), as shown in Fig. 1. Each constituent SMG supports an EOT effect via the excitation of waveguide resonance [22, 23], and can be equivalent to a meta-atom. The two meta-atoms are longitudinal displaced by G and lateral displaced by L. The angle of incidence is φ₀. This structure has been shown to support novel optical properties, including the emergence of a forbidden transmission window [22, 23]. Here we are interested in the topological structure of the transmission spectra and the feasibility in observing an EP-like phenomenon. It seems that we should directly study the wavelengths λ_R and broadenings Γ_R of the eigenstates. However, the distributions of field profile and phase in a complex subwavelength structure can not be directly detected. On the other hand, transmission spectra have been widely utilized to discover the existence of eigenstates and their general optical properties in various nanostructures, for example, the dispersion relation in photonic crystals. Consequently, here we pay attention directly to the transmission spectra. Such an investigation is also of great importance and interest to the progress of EOT and the related applications of metamaterials.

 

Fig. 1 Schematic view of the unit cell of a dual metallic grating. The first (second) grating, meta-atom 1 (2), has a thickness of h₁(h₂). The two meta-atoms are longitudinal displaced by G in the z direction, and lateral displaced by L in the x direction. The angle of incidence is φ₀.

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In Section 2, we briefly discuss why a DMG can be considered as an analogue of a two-state quantum system. In Section 3, we study the transmission spectra of DMGs in the parameter space spanned by (L, φ₀). We show that similar to EPs in an open quantum system, anti-crossing (crossing) and crossing (anti-crossing) in the peaks and widths of the EOT branches can be obtained. Other fingerprints of EPs, including the exchanges of field profiles, the variation of phase difference between the fields in the two constituent meta-atoms, and the topological structure when circling EP, are analyzed. Discussion and summary are made in Section 4.

2. Feasible of the analogy

The analogy between a DMG and a two-state quantum system can be understood straightforwardly by considering the near-field coupling via evanescent modes. Proper discussion can be found in Refs. [22, 23]. Here we would like to provide a further discussion, and show that the transmission spectra of a DMG is similar to the energy levels of a two-state quantum system with a 2×2 Hamiltonian matrix. The discussion is based on the fact that the interaction of light with a metamaterial can be satisfactorily modeled by Lorentzian oscillators [24, 25]

Consider the structure shown in Fig. 1, and label the two meta-atoms by ’1’ for the first SMG that interacts with the incident radiation, and ’2’ for the second one that radiates EM wave toward the backside of the structure, respectively. Because each SMG supports an EOT effect, we can assume that each meta-atom possesses an intrinsic resonance with frequency ω_α and broadening γ_α, where α =1 and 2. Assume that the coupling coefficient of meta-atom α to plane wave in free space is g_α. From the equation of

The DMG structure shown in Fig. 1 can be modeled in a similar way by considering two linearly coupled Lorentzian oscillators [24, 25]. The excitation of the two oscillator-like meta-atoms are governed by the equation set of

The denominator of t₀ briefly tells the positions ω_R of the transmission peaks and the broadenings γ_R. From Eq. (5) we can see that the denominator is similar to the secular equation in quantum physics for the eigenvalues ω₀ and eigenstates of a 2 × 2 Hamiltonian matrix, as

From above analysis we can see that the transmission spectrum through a DMG is in analogy with the energy levels in a two-state open quantum system. By properly choosing the geometric parameters so as to engineer the absorption loss, it is then feasible to observe many classic analogue of quantum effects, especially EPs, in the transmission spectra of DMGs.

3. Simulation

To observe EPs in the transmission spectra of DMGs, we must study how the wavelengths λ_R and widths Γ_R of the EOT branches vary in a 2D plane of parameters. The available free parameters of a DMG include the longitudinal displacement G, the lateral displacement L, and the angle of incidence φ₀. Here, we consider the situation when the 2D plane of parameters is formed by L and φ₀. The longitudinal displacement G is fixed to be a constant throughout this paper.

We investigate the transmission spectra of DMGs shown in Fig. 1. The period d is 1000 nm, and the width a of the slits is 100 nm. The metal is chosen to be silver (Ag) with a Drude dispersion [22, 23], and the wavelength within our interest is from 1200 nm to 1700 nm. The lateral displacement G equals 100 nm. The corresponding thicknesses of the two meta-atoms are h₁ = 500 nm and h₂ = 510 nm, respectively. Transmission properties for p-polarized incidence are investigated by full-field three-dimensional finite element optical simulations (COMSOL Multiphysics 4.3a).

3.1. Topological structure

We calculate the transmission properties of DMGS at different values of L and φ₀. To see how the transmission peaks develop in the 2D plane of parameters (L, φ₀), in Fig. 2 we present the transmission spectra versus L at four different angles of incidence φ₀. Figure 2(e) is a simple diagram that helps us to understand the topological structure of the sheets formed by the wavelengthes of the EOT peaks.

 

Fig. 2 Transmission spectra of DMGs at (a) φ₀ = 4°, (b) φ₀ = 5°, (c) φ₀ = 9°, and (d) φ₀ = 12°, respectively. Plot (e) is a simple diagram of the topological structure of the EOT peaks in the 2D plane of parameters (L, φ₀).

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From Fig. 2 we can see an interesting topological structure exists in the transmission spectra, as diagrammatically shown in Fig. 2(e). Two transmission branches versus L can be observed for any value of φ₀. In general, an anti-crossing between the two transmission branches can be observed, with an avoided transmission gap near L_c = 150 nm and λ_c = 1375 nm. However, the anti-crossing changes sharply at a critical angle of incidence φ_c = 4.7°, where the two transmission branches coalesce and only one peak exists. Note that when φ₀ is smaller than φ_c, e.g. φ₀ = 4° as shown in Fig. 2(a), it seems that there was an anti-crossing of parameter L versus wavelength λ, and a broad-band prohibition of transmission takes place [22, 23]. However, by examining the transmission spectra very carefully we find that a peak is still observable in the broad-band region of low transmission [22]. In other words, a single transmission peak persists there, albeit with a low amplitude T.

In open quantum systems, it is well known that when the system varies in a 2D parameter space, the coalescence of eigenvalues and the transition from anti-crossing to crossing or vice verse are fingerprints of EPs [5–8]. From the topological structure shown in Fig. 2 we observe a similar phenomenon. When φ₀ is smaller than φ_c, a coalescence of transmission branches takes place, in sharp contrast with the anti-crossing of transmission branches when φ₀ > φ_c. To prove that this coalescence is associated with an EP-like effect, we verify the transition between anti-crossing and crossing in the wavelengths λ_R and widths Γ_R, and study the exchange of field profiles I, the variation of phase difference Δθ, and the evolution of field distribution when circling EP.

3.2. Wavelengths λ_R and widths Γ_R

Within the 2D parameter space spanned by (L, φ₀), the trajectories of the transmission wavelengthes form two Riemann sheets, as diagrammatically shown in Fig. 2(e). To prove that the coalescence of the two Riemann sheets is in analogy with EPs, we should prove that there exists a transition from crossing to anti-crossing in the transmission wavelengths λ_R, together with a transition from anti-crossing to crossing in their widths Γ_R.

We analyze the transmission spectra at two angles of incidence φ₀, above and below φ_c, respectively. Figure 3 displays the curves of the transmission wavelengths λ_R and widths Γ_R versus L at φ₀ = 4.5° and φ₀ = 10°, respectively. The widths Γ_R are extracted by using the best fit with two Lorenz curves. From the results shown in Fig. 3 we can see with increased φ₀, a transition from crossing to anti-crossing in λ_R and the corresponding transition from anti-crossing to crossing in Γ_R can be clearly observed.

 

Fig. 3 Wavelengths λ_R and widths Γ_R of the transmission branches at (a) φ₀ = 4.5° < φ_c and (b) φ₀ = 10° > φ_c, respectively.

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It is proposed that when the strength of coupling in a two-state open quantum system is weak [9], a coalescence of the real part of their energy levels (frequencies) and repulse of their imaginary parts (broadenings) take place. By examining the results shown in Fig. 3 we can see this statement can be applied to the case of φ₀ = 4.5° < φ_c. The two branches of λ_R coalesce and intersect with each other at the coalescent point of L = L_c and λ = λ_c. As for the widths Γ_R, although their values fluctuate strongly near the coalescent point, they avoid each other and do not intersect. The error in the evaluated values of widths Γ_R is due to the imperfect fitting of the EOT spectrum by using two Lorenz profiles.

When the angle of incidence φ₀ increases, the strength of coupling becomes greater. As shown in Fig. 3(b) for φ₀ = 10°, now an anti-crossing is presented between the two branches of λ_R. Note that at this large angle of incidence no Wood’s anomaly takes place, because the requirement of k_d − k_c sinφ₀ > k_c is still satisfied, where k_d = 2π/d and k₀ = 2π/λ_c. All the high diffractive orders are evanescent and decay away from the surface. Now the widths Γ_R of the two transmission branches, as expected, become intersect with each other. It is a strong coupling regime, for which identity of the eigenstates will exchange after EP [9], as shown in the next subsection.

3.3. Profile of intensity I and phase difference Δθ

In quantum theory, an EP is a branch point singularity of both eigenvalue and eigenvector of a Hamiltonian [10]. As an consequence of the topological structure of Riemann sheets at this branch point, the two wave functions coalesce into one, which implies a defect of the underlying Hilbert space [13]. Phases and wave functions of the eigenstates around an EP have been shown to possess many interesting properties. For example, it is shown that a full loop in the eigenvalue plane requires two loops in the parameter plane [6, 13].

For an open quantum system, the behaviors of wave functions when varying in the 2D parameter space depend on whether the two energy levels anti-cross or cross. Consequently, under different strengths of coupling, different tendencies on the variations of wave functions can be observed. In general, we should try to vary the parameters in the space spanned by (L, φ₀), and detect the field profile and phase of the eigenstates. However, an investigation of the field distributions of the eigenstates in DMGs by encircling EP is impossible from the transmission spectra, because the incident radiation excites all the eigenstates simultaneously. The phases of the eigenstates are also fixed by the driving of the incidence. Consequently, here we study how the profile of field intensity I varies with L, for both φ₀ < φ_c where the two branches intersect with each other, and φ₀ > φ_c where the two branches avoid each other. The amplitude of I in each meta-atomic SMG can be considered as an analogue of the weight of each basis state in a quantum eigenstate.

First, let us pay attention to the distributions of field intensity I shown in Fig. 4 for φ₀ = 4.5°, in which a crossing of wavelengths λ_R and an anti-crossing of widths Γ_R take place. We can see that when L varies, the distribution of intensity I also changes. Strong EM field is excited inside the subwavelength slits, which is a well known phenomenon in guided-mode mediated EOT [22,23]. The profiles of field intensity I in the two transmission branches do not exchange after EP. For the transmission peak with a smaller width Γ_R, i.e. a longer lifetime, gap SPPs are excited, as indicated by the strong field between the two meta-atomic SMGs. On the other hand, the other transmission branch has a relatively weaker field intensity between the two meta-atomic SMGs. Near EP at L_c = 150 nm, the field patterns are mixed [9], because a coalescence on wave functions takes place.

 

Fig. 4 Distributions of field intensity I in the two transmission branches at φ₀ = 4.5° for different L values. Upper (lower) row is for the branch with smaller (larger) widths Γ_R as shown in Fig. 3(a).

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When φ₀ increases and becomes greater than φ_c, an anti-crossing in λ_R is formed. Figure 5 is for the case of φ₀ = 10°. From the distributions of intensity I we can see that in sharp contrast with these shown in Fig. 4, the two transmission branches exchange their field patterns after the anti-crossing. Around the anti-crossing, there is a strong mixing of field patterns [9]. Similar to that shown in Fig. 4, the transmission peak with a smaller width Γ_R possesses a strong field between the two meta-atomic SMGs.

 

Fig. 5 Distributions of field intensity I in the two branches at φ₀ = 10° for different L values. Upper (lower) row is for the branch with larger (shorter) wavelengths λ_R as shown in Fig. 3(b).

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Consider a quantum system formed by the two meta-atoms, and assume that the initially uncoupled wave functions of the two meta-atoms are ψ₁ and ψ₂, respectively. The eigenfunction of the hybridized system should, in general, read $\left[{\psi }_1+A\text{exp}\left(i\mathrm{\Delta }\theta \right){\psi }_2\right]/\sqrt{1+A^2}$, where the positive-valued parameter A represents the factor of weight, and Δθ is the phase difference between the two basis states. The distributions of intensity I as shown in Figs. 4 and 5 only reveal the information about A. Consequently, we also study how the phase difference Δθ varies within the 2D parameter space (L, φ₀). Here Δθ is defined as the difference between the phases θ_α of field in the middle of the two SMGs [see inset of Fig. 6], as Δθ = θ₁ − θ₂. Figure 6 shows the curves of Δθ versus L for both φ₀ = 4.5° and φ₀ = 10°. For each transmission branch Δθ changes smoothly with L, which proves the reliability of our numerical simulation.

 

Fig. 6 Phase difference Δθ versus L for (a) φ₀ = 4.5° and (b) φ₀ = 10°. Inset shows the definition of phases θ₁ and θ₂, from which we can calculate Δθ = θ₁ − θ₂.

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From Fig. 6(a) we can see when φ₀ = 4.5° < φ_c, the phase differences Δθ vary strongly near EP, and a coalescence of Δθ can be observed at L = L_c. When L increases from 0 nm to 350 nm, the advances of Δθ for different branches are different. For the transmission branch with broader widths Γ_R (blue solid line), Δθ advances by ∼ −2π after EP. On the other hand, Δθ of the other branch (red dashed line) does not vary too much.

When φ₀ = 10° > φ_c, the phase differences Δθ also vary strongly near EP, see Fig. 6(b). The advance for the transmission branch with larger wavelengths λ_R (red dashed line) decreases by nearly π when L changes from 0 nm to 350 nm, while the other branch (blue solid line) initially decreases by ∼ −π till L = 200 nm, and then increases by ∼ π/2 till L = 350 nm.

3.4. Variation on a closed path circling EP

Above we show how the wavelengths λ_R, widths Γ_R, distributions of field intensity I and phase difference Δθ vary in the 2D parameter space spanned by (L, φ₀). Below, let us discuss how the system would vary when circling the EP.

We select 4 positions from the 2D parameter space (L, φ₀), as (0 nm, 4.5°), (350 nm, 4.5°), (0 nm, 10°), and (350 nm, 4.5°). The corresponding 8 points in the two Riemann sheets are represented by A to H, as shown in Figs. 3 to 6. To discuss which route the system would choose and how the field develops when the system varies in the parameter space circling the EP, we adopt the criterion that the field profile I and phase difference Δθ should change smoothly and continually. The status of the system is represented by the field profile and phase difference (Ψ_L,S, Δθ), where Ψ_L and Ψ_S stand for the field profiles with a long and short lifetimes, with excited and non-excited gap SPPs between the two meta-atoms, respectively. Without loss of generality, let us begin from point A at (0 nm, 4.5°):

Above analysis implies that in order to go back to the original state, we should take two loops in the parameter space (L, φ₀) around the EP. Evidently, it is closely related to the coalescence at the branch point singularity of EP, where the two Riemann sheets join with each other. A single loop only moves the system from one Riemann sheet to the other Riemann sheet, for examples, from A to E, or from D to H. However, from the transmission spectra we could not distinguish the phase ξ imposed on the whole field Ψ_L,S, as exp(− jξ)Ψ_L,S. Driven by the incident radiation, the geometric Berry phase of EP cannot be discussed here.

4. Discussion and conclusion

Above we show that just similar to open quantum systems, an EP-like topological structure can be observed in the transmission spectra of DMGs in the 2D parameter space (L, ψ₀). Some fingerprints of EPs, including the anti-crossing and crossing in EOT wavelengths λ_R and widths Γ_R, the exchange of field profiles I, and the variation of system when circling the EP, are demonstrated. The importance of the coalescence in the two Riemann sheets of EOT can be clearly understood.

It seems that we had better fit the simulation results by using some analytical formulae. However, such a task is impossible, because of the existence of frequency-dependent dispersion in Ag and the geometric dispersion of the subwavelength gratings. Dispersion exists not only in γ_α, but also in the coupling constants κ_αβ and g_α. It forbids us to use a simple model with only a few parameters to describe the EM resonances in DMGs.

In our investigation, we analyze the topological structure of the transmission spectra from the resonant wavelengths λ_R, the widths Γ_R, the profiles of field intensity I and phase difference Δθ. The results about λ_R, I and Δθ are more accurate than that of Γ_R, because the former parameters can be directly achieved at the single wavelength of λ_R, while the later one relies on the fitting of the transmission curve with two Lorenz profiles. Nevertheless, all the results, as shown in Figs. 3 to 6, are consistent with each other, especially when we study the variation of field in the parameter space (L, φ₀) circling the EP.

The importance of this investigation is that it demonstrates a classic analogue of quantum phenomenon of open quantum systems in subwavelength-scaled metamaterials. By comparing with closed quantum systems, an open quantum system, in fact, possesses many unique and unexpected phenomena [8], which can be manipulated by engineering the resonance of the basis states, their coupling strength, and the loss. The additional freedom of loss enables us to access much flexibility in pursuing novel optical effects and potential applications. A nice example is the realization of long-lived states in nanostructures [9] by Song et al.. If other classic analogue of effects in open quantum systems can be realized, they will definitely contribute to the advances of optical metamaterials toward various potential applications.

In summary, we show that complex metamaterials with multiple optical resonances in sub-wavelength constituent metallic elements possess many similarities with open quantum systems. From the analogue with a two-dimensional open quantum system, we demonstrate the classic analogue of EPs in the transmission spectra of DMGs within the parameter space spanned by the lateral displacements L and incident angles φ₀. Anti-crossing (crossing) in the resonant transmission branches λ_R with crossing (anti-crossing) of the corresponding widths Γ_R are observed. Exchanges of field patterns and phases, and the variation of field when circling the EP are demonstrated. This work highlights the potential to transfer the concepts and applications from open quantum systems to optical metamaterials.