1. Introduction

There has been recently a great deal of interest in understanding the physical mechanism of optical resonances in various nano-structures, e.g. metamaterials [1–4]. The enthusiasm arises due to the facts that the resonances in metamaterials can be geometrically or dynamically manipulated, that the discrete nature of them can be utilized to simulate many quantum effects, and that the sub-wavelength size of the element enables an effective medium approach for various applications. Among all the structures investigated so far, coaxial and cylinder holes are of great importance [5–14] because they are excellent platforms in supporting optical spin-orbit interaction (SOI) [13–18]. To be more explicit, in a coaxial or cylinder hole the discrete helical optical modes L_m are characterized by the topological indexes m in the transverse phases exp(−jmθ), which can be understood as the orbital angular momentum (OAM) component [11–13]. Due to a conservation rule over the polarization (spin angular momentum, SAM) and OAM of photons, transmission of a normally incident optical wave through a coaxial or circular hole relies on the excitation of bright L_±1 modes [5–7, 13]. Polarization state of transmitted field can be controlled by manipulating the relatively weights of L_±1. Helical modes with m ≠ ±1 are dark and can be utilized to store light energy for slow-light applications [12,19,20] because they have giant Q factors.

The L_±1 modes possess many celebrated merits. For example, they are energy degenerated and can be directly excited by a normal incidence. They can be converted into opposite circular polarized output via the SOI effect [9, 12, 13]. Also they can interact with dark helical modes via a proper introduced grating, from a defect inside the element, or via aperture scattering. Consequently, the L_±1 modes are appropriate candidates in simulating and testing quantum effects of coupled atomic levels in the spirit of the quantum-optic analogies [21].

In this article we propose a scheme in simulating the coherent population trapping effect of three-level configuration [19–21] by using helical modes. Coupled three-level configuration in atomic system has shown great impact over the interaction of photons with atoms, for example, in refractive index enhancement, electromagnetic induced transparency (EIT), and lasing without reversion (LWI) [19–21]. Here we consider the scenario when the two bright L_±1 modes couple with the central dark one L₀. Since the topological index of L₀ is just between those of L_±1, their mutual interaction forms a cascaded process if the coupling mechanism provides the necessary mismatch Δm of ±1. A unique feature of our investigation presented here is that we utilize parity-time $(\mathrm{P}\mathcal{T})$ symmetry, an important concept of non-Hermitian optics [22–43], as the coupling mechanism.

Non-Hermitian optics including the $\mathrm{P}\mathcal{T}$ symmetry has been shown to render many novel phenomena such as time-reversed lasing or coherent perfect absorber, unidirectional reflectionless, single-mode lasing, loss-induced lasing, and optical isolation [26, 27, 30–32, 36–43]. Here a $\mathrm{P}\mathcal{T}$-symmetric grating is utilized to produce nonreciprocal coupling strengths in the forward and backward transitions among L₀ and L_±1. This mechanism is different from others, e.g. the non-reciprocity originating from twisting a set of coupled fibers [44]. By developing a non-Hermitian matrix approach we show that a special hybridized mode L_c can be obtained. The formation of L_c is similar to the coherent population trapping in atomic system [21], a mechanism that plays a key role in EIT, LWI, stimulated Raman adiabatic passage technique (STIRAP), and adiabatic light transfer via dressed state [19–21, 45]. Formed by the interference of different interaction pathways among L_±1 and L₀, there is no population trapped in L₀. Populations in L_±1 are determined by parameters of the $\mathrm{P}\mathcal{T}$-symmetric grating. This effect can be used for a polarization-independent optical spin filtering purpose, i.e. the polarization state of transmitted wave is independent of the polarization of incidence but solely determined by the $\mathrm{P}\mathcal{T}$-symmetric grating. Finally, we provide numerical simulation in a well-designed metamaterial. From the variation of transmission spectra and polarization of field versus parameters of the $\mathrm{P}\mathcal{T}$-symmetric grating we show that the coherent-trapped mode L_c is readily to be observed in future experiments. Discussion and conclusion are provided in the last two sections.

2. Theory

Figure 1 schematically shows a coaxial element within our interest. Here we do not consider a cylinder hole because it does not support L₀ [7, 12–14]. The coaxial element supports a serial of helical modes L_m and let us pay attention to the three ones with the lowest topological indexes m. The L₀ mode is a transverse electromagnetic mode without cutoff, with a resonant frequency ω₀. The L_±1 modes are degenerated in resonant frequency ω₁ with opposite helical phases of exp(∓jθ). By properly designing geometry of the coaxial element, i.e. its inner radius r₁, outer radius r₂, thickness h, and dielectric constant ε_d inside, we can manage their cutoff wavelengths and resonant frequencies [6, 7, 14] to make ω₀ ~ ω₁ while being far away from other modes. When the coaxial elements are periodically arranged, e.g. in a square lattice with a period of a, existence of these resonant modes can be observed from the extraordinary optical transmission (EOT) spectra [1, 2, 5–7, 12–14]. Examples about the dependence of EOT spectra versus geometric parameters r₁, r₂, a and h, can be found in [5,7,12–14] and references therein.

 

Fig. 1 (a) In a coaxial element the optical eigenmodes are helical modes L_m with different topological indexes m. (b) The modes L_±1 and L₀ can be cascadingly coupled with each other by a $\mathrm{P}\mathcal{T}$-symmetric grating in the form of Δε = ε_A exp(−jθ) + ε_B exp(+jθ), which renders nonreciprocal back and forth transitions among L_±1 and L₀. (c) The hybridized mode can be detected by measuring the spectra of optical transmission and the associated polarization state of field.

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Here a weak $\mathrm{P}\mathcal{T}$-symmetric modulation Δε is introduced into the coaxial element (r₁ < r < r2 and −0.5h < z < 0.5h), as

To develop a proper theory in describing the cascaded coupling, it is necessary to emphasize that unlike former literatures [43], here L_m is propagating along z direction, perpendicular to the circular symmetric cross-section. Resonance of L_m inside the coaxial element, especially these of the bright L_±1 ones, can be observed from EOT [1, 2, 5–7, 12–14]. A coupled mode theory in a matrix form which can determine the eigenfrequencies and the eigenfunctions is desired.

Here we develop a coupled-mode matrix by using a perturbation approach over the Maxwell’s equations. Within the coaxial element the Maxwell’s equations can be expressed as

The $\mathrm{P}\mathcal{T}$-symmetric grating mixes the initial orthogonal modes L_m together. To find the solution of Eq. (2) for nonvanishing ε_A and/or ε_B, we then approximately express the hybridized eigenmode

Since the perturbation is very weak, i.e. ε_A,B ≪ ε_d, and the overlap integral Λ is also smaller than 1, we can assume that ω associated with ε_A,B in Eqs. (4) to (6) equals to$\overline{\omega }=({\omega }_0+{\omega }_1)/2$. Now the eigensolution is given by a non-Hermitian matrix

The product of AB is always a real number. Because the structure is mirror symmetric with respect to the central plane z = 0, the waveguide resonances e_m(r, z) is either symmetric or anti-symmetric [1,6,7,46,47] in the z direction. Parameters κ, A, and B are thus generally real.

The cascaded interaction described by Eq. (8) gives three solutions of ω. An example for ω₀ = 0.90ω₁ is shown in Fig. 2. The first two solutions are

 

Fig. 2 (a) Spectra of eigenfrequencies versus B, when ω₀ = 0.90ω₁ and $A=0.08{\omega }_1^2$. At the exceptional point of $AB=-{({\omega }_1^2-{\omega }_0^2)}^2/8$ the ω₊ and ω₋ modes coalescence. Below this value the phase is broken, and the eigenfrequencies are complex. (b) Variations of the magnitude of c_±1 and c₀ versus B for the coherent-trapped helical mode (ω_c = ω₁).

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Their characteristics are determined by the magnitude of AB. If AB is positive, we get two split branches. When AB is zero, ω₊ = ω₁ and ω₋ = ω₀. When AB becomes negative, the two branches approach each other, and merge into ${[({\omega }_1^2+{\omega }_0^2)/2]}^{0.5}$ at the exceptional point (EP) of $AB=-{({\omega }_1^2-{\omega }_0^2)}^2/8$. Below this value the phase is broken and the eigenfrequencies are complex. All these features are general properties of a$\mathrm{P}\mathcal{T}$-symmetric two-level system [22–26,28–32,40–42], and would not be discussed furthermore in this article.

Our attention is paid to the third eigenmode L_c [see the blue line in Fig. 2(a)], whose eigenfrequency is given by

This solution is independent of A and B, and always coincides with that of the L_±1 modes. As for the eigenfunction, from Eq. (8) we can find

The variations of magnitudes of c_m versus B are shown in Fig. 2(b). We can see there is no population trapped in L₀ due to a destructive interference between the two transition pathways of L₊₁ ↔ L₀ and L₀ ↔ L₋₁. Such a phenomenon is a close analogue of the coherent trapping effect that takes place in a three-level atom [19,20], or the population transfer effect via STIRAP [21, 45]. Modes L_±1 are equivalent to the two metastable levels, while L₀ is the third dark one [19–21, 45]. Parameters A and B are equivalent to the Rabi frequencies that stimulate the transitions of L_±1 ↔ L₀ [20, 21, 45]. Note that unlike atomic systems, here the transitions are nonreciprocal, that the strengths A and B in the forward and backward transitions of L_±1 ↔L₀ are different from each other [see Fig. 1(b)]. Magnitudes of A and B can be artificially tuned by changing the values of ε_A and ε_B in the $\mathrm{P}\mathcal{T}$-symmetric grating [see Eqs. (9) and (10)].

The existence of the coherent-trapped helical mode L_c can be observed from the EOT spectra of metamaterials [1, 2, 5–7, 12–14]. Characteristics on the populations c_±1 can be detected from the polarization state of transmitted wave. Since SOI would convert the bright L_±1 modes into far-field optical waves with opposite circular polarizations [12–14], that L₊₁ (L₋₁) is associated with left (right)-handed circular polarization with S₃ = +1 (S₃ = −1), where S₃ is the normalized Stokes parameter, it is readily to show from Eqs. (13) to (15) that the polarization of transmitted optical wave is

In other words, the polarization state of EOT peak is solely determined by ε_A,B other than the polarization of incidence. Such an effect can be utilized to manipulate the polarization state of optical field, as demonstrated below.

3. Simulation

We provide a metamaterial design and simulate the EOT spectra to demonstrate the existence of coherent-trapped helical mode L_c. The simulation is based on COMSOL Multiphysics software. The metamaterial is made of a square lattice of coaxial elements in a perfect electrical conductor (PEC) plate, with r₁ = 0.5mm, r₂ = 0.7mm, h = 0.6mm, ε_d = 12, and period a = 2.5mm. When the $\mathrm{P}\mathcal{T}$-symmetric grating is absent, i.e. ε_A = ε_B = 0, a single EOT peak at ω₁ = 77.04 GHz is obtained. By checking the variation of field pattern with the polarization of incidence we confirm that it is associated with the excitation of L_±1 modes, in agreement with [5,7,12–14].

Then, we introduce a $\mathrm{P}\mathcal{T}$-symmetric grating in the form of Eq. (1) into the coaxial element. Figure 3 shows the results when ε_B varies from −2 to 2 for ε_A = 1. The incident wave is right-handed circularly polarized $(\widehat{x}+j\widehat{y})/\sqrt{2}$. We can see three EOT branches are obtained. One branch is relatively flat and briefly around ω_c = ω₁. This branch is the coherent-trapped mode L_c within our interest. As for the other two branches ω₊ and ω₋, they merge into a single EP-like point at ε_B ~ −1.3. When ε_B equals zero, the ω₊ branch degenerates with ω_c. All these features are in consistence with these shown in Fig. 2(a).

 

Fig. 3 COMSOL simulation results on the variation of EOT spectra versus ε_B in the $\mathrm{P}\mathcal{T}$-symmetric metamaterial shown in Fig. 1(c). Parameters are r₁ = 0.5mm, r₂ = 0.7mm, h = 0.6mm, ε_d = 12, a = 2.5mm, and ε_A = 1. The branch ω_c is the coherent trapped mode L_c within our interest.

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In order to prove that the excitation of L_±1 and L₀ modes dominates these EOT peaks, we analyze the distribution of field inside the coaxial elements. Figure 4 shows the results when ε_A = ε_B = 1. Three EOT peaks are present, at ω₋ = 66.8 GHz, ω_c = 76.6 GHz, and ω₊ = 78.4 GHz, respectively. From the field distributions shown in Figs. 4(b) to 4(c) we can see at most two regions of high field are present. It implies that only helical modes with |m| ≤ 1 are excited, i.e. the modes could only be L_±1 and L₀. Furthermore, the field intensity at ω_c [Fig. 4(c)] possesses a better symmetry than those of ω_±, and the two regions of high field are almost equal with each other. Such a feature hints that at ω_c the L₀ mode is very weakly excited and approaches zero, consistent with Eq. (14). Reasons for the nonzero population of c₀ and the weakly dispersion of L_c branch are discussed in the next section.

 

Fig. 4 (a) EOT spectra when ε_A = ε_B = 1, and the distribution of field intensity at (b) ω₋ = 66.8 GHz, (c) ω_c = 76.6 GHz, and (d) ω₊ = 78.4 GHz, respectively. Plots (c) and (d) share the same colormap.

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Paying attention to the L_c branch, we analyze the associated polarization state S₃ of transmitted field. The result is shown in Fig. 5. When ε_B = 0, the transmitted field is left-handed circularly polarized despite the fact that the incidence has a right-handed circular polarization. When the magnitude of ε_B increases, the polarization becomes elliptical and finally turns to linear (S₃ = 0) when ε_B = ±ε_A. If the magnitude of ε_B is larger than ε_A, the stoke’s parameter S₃ is negative. We fit the variation of S₃ versus ε_B by using Eq. (16). From the solid line shown in Fig. 5 we can see our theory agrees well with the results from COMSOL simulation.

 

Fig. 5 Variation of the polarization state S₃ of transmitted field versus ε_B on the ω_c branch. Red dots are these from COMSOL simulation and the blue solid line is given by Eq. (16).

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4. Discussion

Above we have shown that a classic analog of coherent population trapping effect can be realized in $\mathrm{P}\mathcal{T}$-symmetric coaxial metamaterials. The energy degeneracy of L_±1 modes and the utilization of L₀ are two keys of this scheme. If L_±1 have different resonant frequencies, we could not get the simple curves shown in Fig. 2(a). An avoid anti-crossing would appear around the otherwise degenerated point of ω₊ and ω_c branches at ε_B = 0, with a strong dispersion on the population c_m. As for L₀, it plays a determinant role because it is the interference between the two transition pathways between L₀ and L_±1 that renders the zero dispersion of ω_c = ω₁ and the null population in L₀. In case that the grating reads Δε = ε_A exp(−j2θ) +ε_B exp(+j2θ) other than that of Eq. (1), a directly transition pathway between L_±1 is open. The interaction now can be modeled by a 2 × 2 matrix, which is just a standard $\mathrm{P}\mathcal{T}$-symmetric two-level system [22–26, 28–32, 40–42] discussed by various authors.

As for the discrepancy of COMSOL simulation (Section 3) with the non-Hermitian matrix approach (Section 2), for example, the nonzero dispersion of L_c branch versus ε_B and the nonzero population c₀, they might arise from below aspects. Firstly, Eq. (8) has neglected the coupling of L_±1 with other dark modes especially L_±2. Such an assumption is a good one if ω₁ is away from other ω_m(m ≠ ±1). However, resonances in metamaterials, not only inside the coaxial elements but also at the two PEC-air interfaces, are very complex. The lattice considered in this article is a cubic one, which has a lower symmetry than that of the coaxial cross-section. There might be some other interaction pathways that could enhance the coupling of L_±1 modes with other helical modes or surface optical waves, and consequently violate the assumptions made in our theory.

Secondly, the $\mathrm{P}\mathcal{T}$-symmetric perturbation is utilized to stimulate the transitions among L_±1 and L₀. However, energy flux of helical modes are mainly propagating along the z direction. The introduced $\mathrm{P}\mathcal{T}$-symmetric perturbation also changes the effective k_z component and modifies the waveguiding resonant conditions of various L_m modes [5–7]. The resonant frequencies ω_m might slightly shift with ε_A and ε_B. This effect will be considered in our further investigation.

Thirdly, the matrix proposed in Eq. (8) is a simplified model. It only considers a static situation that the population in each mode does not change with time. Also it does not include the broadening effect. Consequently, some features on the transmission spectra, for example, the disappearance of transmission peaks when ε_B decreases further from the EP-like point, and the appearance of transmission coefficient greater than unity when |ε_B| > 1.5, are not predicted. Nevertheless, such a model is a successful one because it predicts most of the features on the EOT spectra.

The proposed coherent-trapping effect can be utilized for some obviously applications, for example, to manipulate the polarization of optical wave. In fact, besides the case shown in Section 3 we have also checked the scenarios of other polarized incidences. Similar results with Figs. (3) to (5) are obtained. It proves that the scheme proposed here is of a polarization insensitive nature, and promises a potential utilization of polarization-insensitive optical spin filters. To be more specific, with properly introduced ε_A and ε_B values, we can always get the desired S₃ state on the transmitted wave at the desired frequency of ω_c, no matter what the polarization of incidence is.

An experimental realization of the proposed coherent-trapped mode needs our further effort. By referring to literatures about optical $\mathrm{P}\mathcal{T}$ symmetry [25, 28–30, 32, 43], we believe the most feasible scheme is to utilize external fields in modulating the refractive index, e.g. the in-diffusion and nonlinear mixing technique in Fe-doped LiNbO₃ [28]. For further improvement, we can try to tune the geometric parameters so as to enhance the Q factors of L_±1 and L₀ modes. It might overcome the limitation on the available ε_A and ε_B values, enhance the transitions among L_±1 and L₀ even for a moderate $\mathrm{P}\mathcal{T}$-symmetric modulation, and improve the spin-filtering performance by reducing the line-width of the EOT spectra. Advanced theory in considering the influence of the $\mathrm{P}\mathcal{T}$-symmetric modulation over the resonance in z direction, the broadening effect in each helical mode, and the interaction with other high-m helical modes, is also applauded. It might describe some features of our COMSOL simulations that cannot be explained by the matrix proposed in Section 2, and predict other potential applications.

5. Conclusion

In summary, here we show that by coupling the two bright L_±1 modes inside a coaxial element with the dark L₀ mode via a $\mathrm{P}\mathcal{T}$ symmetric perturbation, we can get a special hybridized mode L_c via a classic analog of coherent-population-trapping effect. Resonant frequency of L_c is independent of the $\mathrm{P}\mathcal{T}$-symmetric perturbation, and the populations in L_±1 can be artificially manipulated. No population is trapped in the L₀ component. Numerical COMSOL simulation in a well-designed coaxial metamaterial verifies our analysis. This effect can be utilized to realize a polarization-insensitive optical spin filter. Discussion on potential investigations and improvements is provided.