## Abstract

We propose a scheme to achieve a controllable unidirectional reflectionless propagation at exceptional point (EP) in a non-ideal parity-time metasurface consisting of two silver ring resonators. The unidirectional reflectionless propagation can be manipulated by simply adjusting the angle of incident wave and the distance *s* between two silver rings based on the far field coupling. In addition, the angle of incident wave in a wide range of ∼25° is available to achieve the unidirectional reflectionless propagation. Moreover, the unidirectional reflectionless propagation at EP is insensitive to the polarization of incident wave due to the two-ring structure.

© 2017 Optical Society of America

## 1. Introduction

It is well known that Hermitian Hamiltonians have real spectrums in quantum mechanics. However, Bender *et al.* [1] proposed non-Hermitian Hamiltonians that can still exhibit entirely real spectrums if they have parity-time (PT) symmetry. Thereafter, PT symmetry has been extensively studied in various fields, such as quantum field theories [2], open quantum systems [3], non-Hermitian Anderson models [4], Lie algebras [5], electronics [6], acoustics [7–11] and so on. Particularly, researches on PT symmetry have been perfectly extended to optics domain from quantum mechanics in the last decade, such as optical lattices [12–16], photonic structure [17–20], plasmonics [21–23], waveguide couplers [24–31], Bragg gratings [32–34], microcavities [35–42], and so on.

In fact, the highlight of PT symmetric system is the presence of EP where a series of fascinating phenomena, such as optical isolation [12], power oscillation [25, 26], loss-induced transparency [27], nonreciprocal light propagation [26, 32, 33, 35], coherent perfect absorption [43], nonlinear effects [44, 45], laser-absorber [46–53], unidirectional reflectionlessness [30, 34, 38, 54] occur. Especially, the unidirectional reflectionless light propagation near the PT symmetry breaking point has been researched extensively. For example, in 2012, Feng *et al.* [34] experimentally demonstrated an unidirectional reflectionless optical metamaterial near the spontaneous PT symmetry phase transition point where reflection from one side was significantly suppressed. Their results provided the feasibility for creating on-chip PT metamaterials and optical devices based on PT properties. Huang *et al.* [30, 38] constructed a non-PT-symmetric plasmonic waveguide-cavity system to form an EP and realize the unidirectional reflectionless propagation at optical communication wavelength. Their structure would open new possibilities to potentially applications in highly compact unidirectional integrated nanoplasmonic devices. In 2016, Yang *et al.* [54] reported the unidirectional reflectionless phenomenon in periodic ternary layered material with the dimension of 30*µ*m based on balanced gain and loss. Their work has great potential applications in on-chip integrated optics. However, so far the work on the PT properties in a periodic metasurface with the sub-wavelength nanoscale has rarely been mentioned.

In this work, a non-ideal PT sub-wavelength nanoscale metasurface system without gain consists of two silver ring resonators embedded in photopolymer is firstly designed to research the unidirectional reflectionless propagation properties at EP based on far field coupling. The results based on numerical simulations and analytical calculations consistently indicate that the unidirectional reflectionless propagation can be manipulated by adjusting the angle of incident wave and distance *s* between two silver rings. Furthermore, the structure is polarization independent due to the two-ring structure. Besides, the presence of non-ideal PT metasurface structure near EP can act as an absorber with absorption of ∼96% and quality factor (Q-factor) of ∼41.

## 2. Structure

Figure 1 shows the schematic of unit cell of the unidirectional reflectionless non-ideal PT metasurface structure that place on a glass substrate. Two silver ring resonators with the thickness *h* are embedded in photopolymer. The periods *t* of unit cell in both *x* and y directions are 640nm. The inner radius *r*_{1} (*r*_{2}) and outer radius *R*_{1} (*R*_{2}) of up (down) ring are 61nm (63nm) and 122nm (170nm), respectively. The distance *s* between two rings is variables. The incident wave is in *x* − *z* plane and has an angle *θ* with +*z* axis. The permittivities of photopolymer and glass substrate are 2.4025 and 2.25, respectively. Moreover, the permittivity of silver in near-infrared can be described by Drude model with plasmon frequency *ω _{pl}* = 1.366 10

^{16}rad/s and collision frequency

*ω*= 3.07 × 10

_{c}^{13}Hz [57]. The numerical simulations are carried out by using a finite-integration package (CST Microwave Studio).

## 3. Results and discussions

To verify the unidirectional reflectionless phenomena, we make use of the scattering matrix to analyze the reflection spectra. The scattering properties corresponding to the presence of non-ideal PT metasurface system in Fig. 1 can be given by the transfer matrix *T*_{all} [57, 58] in a certain frequency *ω*

*T*

_{p}are the transfer matrices for up (down) silver ring and the phase shift of wave propagation from up silver ring to down silver ring. Γ

_{1(2)}and

*ω*

_{1(2)}are the dissipative losses and resonant frequency of up (down) silver ring, respectively.

*γ*

_{1(2)}is the width of resonance for up (down) ring coupled with the incident wave.

*ϕ*is accumulated phase shift of wave propagation between the two rings and it depends on distance

*s*.

*ω*is the frequency of incident wave. Moreover, the phase shift

*ϕ*

_{1(2)}for up (down) ring can be expressed as [57, 58]

In the present structure, the phase difference *ϕ _{all}* between two silver rings is composed of three parts: the phase shift of up silver ring, the phase shift of down silver ring and the phase shift between two silver rings. In other words,

*ϕ*is equal to

_{all}*ϕ*

_{1(2)}−

*ϕ*

_{2(1)}+ 2

*ϕ*in excitation along +

*z*(−

*z*)

Additionally, the transmission and reflection coefficients can be defined by Eq. (1), as

*S*

Figure 2 shows the simulated (analytical) reflections in excitation along +*z* and −*z* of the incident wave by optimizing the distance *s* (phase *ϕ*). The black solid and red solid lines show the simulated reflection spectra in excitation along +*z* and −*z* corresponding to distance *s* = 485nm, 455nm, 444nm, 430nm and 395nm, respectively, in Figs. 2(a)–2(e). From Figs. 2(a)–2(e), the reflection peak values in excitation along +*z* increase gradually with the decreasing distance *s* from 485nm to 395nm, and the reflection peaks have the blue-shift. On the contrary, the reflection peak values in excitation along −*z* decrease gradually with the decreasing distance *s* from 485nm to 395nm. The reflectivities in excitation along +*z* and −*z* for *s* = 485nm (395nm) are ~0 (0.8) and ∼0.67 (0) at wavelength 1441nm (1387nm), respectively. These phenomena indicate that the bilateral unidirectional reflectionlessness at wavelength 1441nm and 1387nm appear with the appropriate distance *s*, as shown in Figs. 2(a) and 2(e). Based on Eq. (5), the reflection spectra for excitation along +*z* and −*z* are shown in Fig. 2. The blue dot and dark cyan dot lines depict the analytical reflection spectra corresponding to different phase *ϕ*. The results from Fig. 2 indicate the good agreement between numerical simulation (based on distance *s*) and analytical calculation (based on phase *ϕ*). The bilateral unidirectional reflectionless phenomena can be obtained in our structure. The relevant fitting parameters Γ_{1(2)} and *γ*_{1(2)} are shown in Fig. 2(f) by using the analytical calculation. We notice that Γ_{1(2)} and *γ*_{1(2)} nearly remain invariant with the variation of *s*.

In order to analyze the unidirectional reflectionlessness in the non-ideal PT metasurface system better, the z-component distributions of electric field of two silver rings for the incident wave at two EPs are shown in Fig. 3. When *s* is 485nm, both silver rings at EP (1441nm) are excited simultaneously by the incident wave of excitation along +*z*(−*z*) and the induced currents are the same (opposite) directions as shown in Figs. 3(a) (3(c)) and 3(b) (3(d)), which means that the phase differences of induced currents of two silver rings are approximately 2*π* (Figs. 3(a) and 3(b)) and *π* (Figs. 3(c) and 3(d)), respectively. Therefore, the reflections at wavelength 1441nm are ∼ 0 (black solid line) and a high value (red solid line) corresponding to the phase difference of ∼2*π* in excitation along +*z* and ∼*π* in excitation along −*z*, respectively, based on the Fabry-Pérot (FP) resonant coupling as shown in Fig. 2(a). When the distance *s* reduces to 395nm, the unidirectional reflectionless propagation appears again. It can also be seen from Figs. 3(e)–3(h), when *s* = 395nm at the second EP (1387nm), the phase differences between induced currents of two silver rings in excitation along +*z* and −*z* are ∼*π* and ~2*π*, respectively, and the corresponding reflections are a high value (black solid line) and ~0 (red solid line), respectively, as shown Fig. 2(e). Therefore, the unidirectional reflectionless phenomena appear at two EPs (1441nm and 1387nm) for *s* = 485nm and 395nm, respectively. Based on Eq. (4) and some appropriate parameters, *ϕ _{all}* of about 2

*π*can be obtained, which indicates that the directions of induced currents of both silver rings at two EPs are the same as shown in Figs. 3(a)–(b) and Figs. 3(g)–3(h). Therefore, we have confirmed the unidirectional reflectionless phenomena in our non-ideal PT metasurface system further by analytical calculation.

Then we discuss the influences of distance *s* and accumulated phase shift *ϕ* between two silver ring resonators on the bilateral reflectionlessness in our structure by using the numerical simulations (Figs. 4(a) and 4(b)) and analytical calculations (Figs. 4(c) and 4(d)). From Figs. 4(a) and 4(b), the low reflection peaks have blue-shifts with the decreasing *s* in excitation along +*z* and −*z*. Moreover, the low reflection region (Fig. 4(a)) in the ranges of wavelength from 1421nm to 1463nm and distance *s* from 470nm to 500nm in excitation along +*z* corresponds to the high reflection region (Fig. 4(b)) in excitation along −*z*. Similarly, the low reflection region (Fig. 4(b)) in the ranges of wavelength from 1367nm to 1401nm and distance *s* from 380nm to 410nm in excitation along −*z* corresponds to the high reflection region (Fig. 4(a)) in excitation along +*z*. Therefore, the bilateral unidirectional reflectionless propagations can be manipulated by adjusting distance *s* in our non-ideal PT metasurface structure in a wide wavelength range. According to Eq. (5), the dependency of reflection on phase *ϕ* and wavelength in excitation along +*z* and −*z* are shown in Figs. 4(c) and 4(d), respectively. A comparision of the Figs. 4(a)–4(b) with Figs. 4(c)–4(d) shows the consistency of results based on numerical simulations and analytical calculations, and the dispersions of reflection versus phase *ϕ* match well with that versus distance *s* in the same direction.

Next, we discuss the dependency of reflection on the incident angle of excitation along +*z* and −*z*. Figures 5(a) and 5(c), and 5(b) and (5(d) show the reflections as the functions of incident angle and wavelength for *s* = 485nm (395nm) in excitation along +*z* and −*z*, respectively. Increasing the angle of incident wave in excitation along +*z* (Fig. 5(a)) and −*z* (Fig. 5(d)), red-shifts occur gradually in the low reflection region. When incident angle is smaller than 25°, high reflections appear in the same region for excitation along −*z* (Fig. 5(b)) and +*z* (Fig. 5(c)), respectively. From Figs. 5(a)–5(b) and Figs. 5(c)–5(d), the unidirectional reflectionless phenomena appear corresponding to the incident angle in a wide range of nearly ±25°. Not only that, the unidirectional reflectionless phenomena can be controlled by adjusting incident angle from 0 to ∼40° for *s* = 485nm and 395nm, respectively. Actually, when the incident angle is 0 for *s* = 485nm (395nm), the unidirectional reflectionless phenomena appear, while when the incident angle is ∼40°, the reflections in both excitation along ±*z* are nearly the same, thus the unidirectional reflectionless phenomena disappear.

In order to discuss the relevant physics phenomena at two EPs in detail, we can calculate the eigenvalues of scattering matrix *S* as

*t*is real or complex [43]. In our metasurface system,

*t*is complex, which corresponds to a non-ideal PT system. From Eq. (7), one can determine that the non-ideal metasurface system is symmetry when

*r*

_{+z}

*r*

_{−z}> 0 and the symmetry is broken when

*r*

_{+z}

*r*

_{−z}< 0. The EP occurs in

*r*

_{+z}

*r*

_{−z}= 0. This means that when

*r*

_{+z}or

*r*

_{−z}is 0, two eigenvalues coalesce, which indicates that the unidirectional reflectionlessness appear at the EP. According to Eq. (7), the real and imaginary part curves of eigenvalue

*λ*

_{1(2)}with respect to the wavelength are shown in Fig. 6 for

*ϕ*= 3.31,

*π*and 2.88, respectively. Obviously, the real and imaginary parts of two eigenvalues coalesce at a non-zero point (EP) at 1441nm and 1387nm for

*ϕ*= 3.31 and 2.88, respectively, according to the Figs. 6(a)–6(b) and 6(e)−6(f). In this case,

*t*is the complex and

*r*

_{+z}

*r*

_{−z}is 0. This means that the phase transition from non-ideal PT symmetry to symmetry breaking appears at EP. Hence, by appropriately adjusting the phase difference between two silver rings, the unidirectional reflectionlessness can be obtained in our non-ideal PT metasurface system. And from Figs. 6(c)–6(d), the imaginary parts of

*λ*

_{1}and

*λ*

_{2}coalesce at the zero point at wavelength 1416nm for

*ϕ*=

*π*. In this case,

*t*is real and

*r*

_{+z}

*r*

_{−z}is bigger than 0. That is to say, the phase transitions from Hermitian to non-Hermitian occur at wavelength 1416nm.

In fact, our non-ideal PT metasurface system is not only applicable to the unidirectional reflectionless propagation at EPs, but also can act as an absorber near EPs where coherent perfect absorptions (CPAs) emerge, as shown in Figs. 6(a)–6(b) and Figs. 6(e)–6(f). From Figs. 6(b) and 6(f), the imaginary part curves of *λ*_{1} and *λ*_{2} have two intersections near EP with the zero axis line (blue), respectively. When *r*_{+z}*r*_{−z} < 0, one of eigenvalues for non-ideal PT symmetric broken system is real and the other is complex, in this case, two CPAs near each EP appear. Moreover, due to that the phase differences *ϕ _{all}* of induced currents for two silver ring resonators are ~2

*π*near two EPs based on FP resonant coupling, the corresponding reflection close to 0. Together with the low transmissions near EP, the absorptions near 1441nm are ~83% with Q-factor of ~32 for

*ϕ*= 3.29 according to

*A*= 1 −

*R*−

*T*. While for

*ϕ*= 2.878 and 2.9, the absorptions at 1388nm and 1385nm (near EP 1387nm) are ∼96% with Q-factor of 41. Therefore, the present non-ideal PT metasurface system is applicable to light absorption within a narrow spectral range. In addition, our structure can be fabricated by using standard nanofabrication procedures [59].

## 4. Conclusion

We have designed a non-ideal PT metasurface system consisted of two silver ring resonators to realize the unidirectional reflectionless propagation based on far field coupling. The unidirectional reflectionless propagation can be manipulated by adjusting the angle of incident wave and the distance *s* between two silver ring resonators. Moreover, the bilateral unidirectional reflectionless propagations is insensitive to a wide range of incident angle (~25°). And the structure is polarization independent due to the two-ring structure. Besides, the ultra-narrow-band absorptions of ∼96% with Q-factor of ∼41 appear near EPs in non-ideal PT symmetric broken metasurface system. We believe that the unidirectional reflectionless non-ideal PT metasurface will have potential applications to the filter, sensor and highly compact intergrated nanophotonic devices.

## Funding

National Natural Science Foundation of China (Grant No. 11364044); the Education Department of Jilin Province Science and Technology Research Project (Grant No. 2015-09 and JJKH20170455KJ).

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