## Abstract

We investigate the exceptional points in a two-layer cylindrical waveguide structure consisting of absorbing and non-absorbing dielectrics. We show that, by tuning the core to total radius ratio and the refractive index of the core layer in such a structure, the complex effective indices of two waveguide modes can coalesce so that an exceptional point is formed. We show that the sensitivity of the effective index of the waveguide mode to variations of the refractive index of the material filling the shell layer is enhanced at the exceptional point. In addition, we show that larger sensitivity enhancement is obtained for smaller perturbations. Our results could potentially contribute to the development of a new generation of chip-scale exceptional-point-enhanced optical waveguide devices for modulation, switching, and sensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Exceptional points (EPs), associated with the coalescence of both the eigenvalues and the corresponding eigenstates of open quantum systems described by non-Hermitian Hamiltonians, have attracted a lot of attention in recent years [1–5]. EPs have been investigated in ultrasonic cavities [6], mechanics [7], electronic circuits [8], and molecular systems [9]. EPs in non-Hermitian optical systems can lead to numerous interesting phenomena such as power oscillation [10,11], optical isolation [12], unidirectional light reflection [13–15], perfect absorption [16–18], enhanced sensing [19–22], directional lasing [23,24], neuromorphic photonics [25], slow light [26], topological waveguiding [27], and others [28–35]. In optical waveguide systems, EPs have been observed in coupled waveguide structures with parity-time ($PT$) symmetry. In such structures one of two parallel waveguides experiences gain, while the other one experiences an equal amount of loss [11,36]. Recently, Alaeian $et$ $al.$ reported observing EPs in a single three-layer cylindrical coaxial waveguide consisting of a dielectric layer sandwiched between two silver layers after introducing balanced gain and loss azimuthally within the dielectric layer [37]. However, these $PT$-symmetric optical systems are hard to implement in practice, because the required periodic modulation of the refractive index profile necessitates careful tuning of both the real and imaginary parts of the refractive indices of the constituent materials. EPs have also been observed in non-$PT$-symmetric structures with unbalanced gain and loss [29,38–41]. In particular, Doppler $et$ $al.$ and Ke $et$ $al.$ showed that EPs exist in coupled waveguide structures without gain media [29,38]. This is due to the fact that EPs exist in a larger family of non-Hermitian Hamiltonians, since $PT$-symmetry is not a necessary condition for the existence of EPs [40,41]. Designing single non-$PT$-symmetric waveguides to exhibit EPs could be essential for developing compact and easy-to-implement exceptional-point-enhanced optical devices.

In this paper, we investigate the EPs in a two-layer cylindrical waveguide structure consisting of absorbing and non-absorbing dielectrics. The effective indices of the propagating waveguide modes are the eigenvalues of the non-Hermitian system, and the corresponding propagating waveguide modes are the system eigenfunctions. We show that, by tuning the core to total radius ratio and the refractive index of the core layer, the complex effective indices of two waveguide modes can coalesce so that an EP is formed. We investigate the properties of the realized EP of the structure, as well as the associated physical effects of level repulsion and crossing. We show that the sensitivity of the effective index of the waveguide mode to variations of the refractive index of the material filling the shell layer is enhanced at the EP. In addition, we show that larger sensitivity enhancement is obtained for smaller perturbations. We also find that, as the loss in the shell layer increases, the sensitivity increases. Our results could potentially contribute to the development of a new generation of chip-scale exceptional-point-enhanced optical waveguide devices for modulation, switching, and sensing.

The remainder of the paper is organized as follows. In Section 2, we review the transfer matrix method which can account for the behavior of the proposed two-layer cylindrical waveguide structure. In Subsection 3.1 we use this theory to show that the two-layer cylindrical waveguide structure when properly designed exhibits an EP. The realized EP of the structure and its topological properties are then investigated in Subsection 3.2. In Subsection 3.3 we investigate the sensitivity of the effective index of the waveguide mode of the structure to variations of the refractive index of the material filling the shell layer at the EP. Finally, our conclusions are summarized in Section 4.

## 2. Theory

Our proposed two-layer cylindrical waveguide structure consists of absorbing and non-absorbing dielectrics (Fig. 1). The general solution for the modal fields of a circularly symmetric cylindrical waveguide can be written in the following form [42]

where $\textbf {f}=(\textbf {e},\textbf {h})$ is a six-component electromagnetic field vector that only depends on $r$, $\beta$ is the complex wavevector of the waveguide mode, and $m$ is the azimuthal mode number. By expanding the fields in terms of cylindrical wave functions [42–45], the electromagnetic fields in each region can be expressed as follows: The fields in the core layer ($r<a_1$) with refractive index $n_1$ are given by## 3. Results

We optimize the core to total radius ratio $\chi =\frac {a_1}{a_2}$ (Fig. 1), and the refractive index of the core layer $n_1$, to obtain a core-shell cylindrical waveguide structure as in Fig. 1 which exhibits an EP. Equation (10) is numerically solved using Newton’s method [47] to calculate the effective index $n_\textrm {eff}$ of the propagating modes. Since various classes of materials such as semiconductors and oxides have high refractive indices up to 3.4 [48], the refractive index of the core layer in our structure $n_1$ is optimized within the range from 2 to 3.4. The shell layer is filled with an active material with refractive index $n_2=2.02+j\kappa$, corresponding to silicon dioxide doped with CdSe quantum dots [49–51]. The imaginary part $\kappa$ of the refractive index can be modified with an external control beam [49,51]. In addition, we choose $k_0a_2=10$, so that the proposed waveguide structure supports multiple propagating modes.

#### 3.1 Location of exceptional points

Here we choose $\kappa =0.05$ for the imaginary part of the refractive index $n_2$ of the material filling the shell layer (Fig. 1), which is within the range of experimentally achievable values [50,51]. By solving Eq. (10) using Newton’s method, we find that an EP emerges for $n_1=2.875$, $\chi =\frac {a_1}{a_2}=0.514$, and azimuthal mode number of $m=6$. At this EP, the real and imaginary parts of the effective indices of two waveguide modes coalesce at $n_\textrm {eff}=1.497+j0.046$.

Similar to $PT$-symmetric quantum systems, $PT$-symmetric optical systems possess two different phases [11,21]. The first phase is the $PT$-unbroken phase with real eigenvalues. The second phase is the $PT$-broken phase characterized by complex conjugate eigenvalues with equal real parts but opposite imaginary parts. The transition between the two phases takes place at the EP where the two eigenvalues coalesce. Figure 2(a) shows a particular curve that we found in the $\chi -n_1$ design parameter space, where either the real parts or the imaginary parts of the effective indices of the two waveguide modes coincide. We parameterize this parameter curve by $\Gamma =-sgn(\Delta \chi )\sqrt {(\Delta \chi )^2+(\Delta n_1)^2}$, where $\Delta \chi =(a_1-a_\textrm {1,EP})/a_2=\chi -\chi _\textrm {EP}$, $\Delta n_1=n_1-n_\textrm {1,EP}$, and $sgn$ is the sign function. In Figs. 2(b) and 2(c) we show the evolution of the real and imaginary parts, respectively, of the eigenvalues $n_\textrm {eff}$ of the two waveguide modes along the particular curve parametrized by $\Gamma$ in the $\chi -n_1$ design parameter space. We observe that for $\Gamma >0$, the two eigenvalues have equal real parts but different imaginary parts, which is analogous to the $PT$-symmetric broken regime. When $\Gamma$ decreases to zero, the two eigenvalues coalesce and the EP is formed. For $\Gamma <0$, the real parts of the eigenvalues bifurcate and their imaginary parts merge, which actually resembles the $PT$-symmetric unbroken regime. Note that the imaginary part of the eigenvalues for $\Gamma <0$ is non-zero, and the imaginary parts of the eigenvalues for $\Gamma >0$ are not conjugate [Fig. 2(c)]. This is due to the fact that our proposed waveguide structure (Fig. 1) is a non-Hermitian optical system without gain, which is different from conventional $PT$-symmetric non-Hermitian optical systems with balanced gain and loss [11,21].

To understand the underlying formation mechanisms of the EPs in our proposed two-layer cylindrical waveguide system, the transverse field intensity distributions $|e_\phi |^2$ along the curve $\Gamma$ [Fig. 2(a)] are shown in Fig. 3. More specifically, Figs. 3(a) and 3(b) show the field intensity distributions $|e_\phi |^2$ for the waveguide modes with effective indices $n_\textrm {eff}=1.5658+j0.0373$ and $n_\textrm {eff}=1.5658+j0.0552$, respectively, for $\Gamma = 0.35$ (Fig. 2). We observe that one eigenmode is mostly localized in the core region [Fig. 3(a)] and will therefore be referred to as the core mode, while the other eigenmode is mostly localized in the shell layer [Fig. 3(b)] and will therefore be referred to as the shell mode. Figure 3(c) shows the field intensity distribution $|e_\phi |^2$ for the degenerate mode with effective index $n_\textrm {eff}=1.497+j0.046$ at the EP ($\Gamma = 0$ in Fig. 2). We observe that for the waveguide mode at the EP the field intensity is strong in both the core and shell regions. Figures 3(d) and 3(e) show the field intensity distributions $|e_\phi |^2$ for the waveguide modes with effective indices $n_\textrm {eff}=1.3423+j0.0413$ and $n_\textrm {eff}=1.4133+j0.0413$, respectively, for $\Gamma = -0.35$ (Fig. 2). We observe that hybrid modes are formed due to the strong intermode interaction between the two eigenmodes as the core to total radius ratio $\chi$ further increases along the curve parametrized by $\Gamma$ [Fig. 2(a)]. The EP is therefore the point of transition between the strong intermode interaction phase and the weak intermode interaction phase. Similar kinds of EPs have been observed in two-dimensional concentric circular cavity structures [6,52].

In addition, the occurrence of EPs in our proposed two-layer waveguide structure (Fig. 1), when the real and imaginary parts of the eigenvalues $n_\textrm {eff}$ of the two waveguide modes evolve along the curve parametrized by $\Gamma$ [Fig. 2(a)], can be illustrated with a two-level model. The corresponding non-Hermitian Hamiltonian is given by [5]

#### 3.2 Topological properties of the exceptional points

In this subsection, to gain deeper insight into the exceptional point of the structure, we investigate the associated phenomena of level repulsion and crossing, as well as the cross conversion of the eigenvalues by encircling the EP.

It is well known that non-Hermitian quantum systems can experience a transition from level repulsion to level crossing or vice versa by the slipping of EPs over the parameter space [1–3,9]. To provide further evidence of the occurrence of an EP in the waveguide structure of Fig. 1, we calculate the real and imaginary parts of the effective indices of the propagating waveguide modes below ($n_1= 2.865$) and above ($n_1= 2.885$) the critical refractive index of the core layer ($n_1= 2.875$) as a function of the core to total radius ratio $\chi$ (Fig. 4). We observe that the effective indices show level repulsion in the real parts and level crossing in the imaginary parts when the refractive index of core layer is below ($n_1 = 2.865$) the critical value of the EP. In contrast, the effective indices show level crossing in the real parts and level repulsion in the imaginary parts when the refractive index of core layer is above ($n_1 = 2.885$) the critical value of the EP.

Another unique topological property of EPs is that, when encircling an EP in parameter space, the eigenvalues will switch their positions after a closed loop of $2\pi$ because of the square root behavior of the singularity [1,4,29,30,46,53]. In our cylindrical optical waveguide structure (Fig. 1), we consider the parameter space of the refractive index of the core layer $n_1$ and the core to total radius ratio $\chi$. The EP is encircled by a circular loop with a radius of $\rho$ in the $n_1-\chi$ plane and is located at the center of this circle ($n_1=2.875, \chi =0.514$) [Fig. 5(a)]. We choose the radius $\rho$ to be 0.01 to ensure that only the EP that we obtained by optimizing the structure is embedded inside this circle. We vary the refractive index of the core layer $n_1$ and the core to total radius ratio $\chi$ in the counterclockwise direction along the circular loop from the initial position A with $n_1=2.885, \chi =0.514$ [Fig. 5(a)]. That is, we vary the refractive index of the core layer $n_1$ and the core to total radius ratio $\chi$ so that

where $\theta$ is adiabatically varied from $\theta =0$ to $\theta =2\pi$.Figures 5(b) and 5(c) show the trajectories of the real and imaginary parts of the effective indices of the waveguide modes for the structure of Fig. 1, as the path of the refractive index of the core layer $n_1$ and the core to total radius ratio $\chi$ traces the circular loop of Fig. 5(a). We indeed observe that the two complex effective indices switch their values after one loop of 2$\pi$ in the $n_1-\chi$ plane. The cross conversion of the two eigenvalues shown in Figs. 5(b) and 5(c) further confirms the existence of the EP in our proposed two-layer cylindrical optical waveguide of Fig. 1. In contrast, if there is no EP in the closed loop, the eigenvalues will return to their initial values at the end of the loop [1]. Note that the cross conversion of the two eigenvalues implies the existence of an EP inside the loop. Taking advantage of this unique topological property of EPs, the cross conversion of the two eigenvalues indicates the existence of the EP inside the loop without the need to locate the exact parameters at which the EP occurs [1]. Practical realization of the two-parameter control in our structure in order to encircle the EP, can be achieved by adiabatically varying the two parameters (refractive index of the core layer $n_1$ and core to total radius ratio $\chi$) along the direction of propagation $z$ [38]. More specifically, by setting $\theta =2\pi z/L$ in Eq. (13), we obtain $n_1 = n_{1,EP}+\rho cos(2\pi z/L), \chi =\chi _{EP}+\rho sin(2\pi z/L)$. The length parameter $L$ should be large, so that the variation along the direction of propagation $z$ is slow [38].

#### 3.3 Applications of EPs in two-layer cylindrical optical waveguides

It has been demonstrated both theoretically and experimentally that EPs can enhance the sensing of nanoparticles [19,21]. More specifically, the frequency splitting at an EP is proportional to the square root of the external perturbation, whereas the response to the external perturbation at a conventional diabolic point is linear [3,19,21]. In addition, the performance of optical absorption switches is strongly dependent on the sensitivity of the imaginary part of the effective index of the waveguide mode to variations of the imaginary part of the refractive index of the active absorbing material [51]. In this subsection, we show that for our proposed two-layer cylindrical waveguide structure of Fig. 1 the sensitivity of the imaginary part of the effective index of the waveguide mode to variations of the imaginary part of the refractive index of the active absorbing material is enhanced at the EP.

When a variation of $\Delta \kappa$ is introduced into the imaginary part $\kappa$ of the refractive index of the active absorbing material filling the shell layer of the waveguide (Fig. 1), the waveguide mode at the EP is perturbed and split into two modes. We define the effective index splitting $\Delta n_\textrm {eff}$ as the difference between the effective indices of these two waveguide modes divided by two. Figure 6(a) shows the imaginary part of $\Delta n_\textrm {eff}$ as a function of the variation $\Delta \kappa$ at the EP for $k_0a_2=10$, $n_1=2.875$, $n_2=2.02+j0.05$, $\chi =0.514$, and $m=6$. Similar to nanoparticle sensors in $PT$-symmetric optical systems based on second order EPs [21,22], we observe a square-root dependence of the effective index splitting on the variation $\Delta \kappa$. As shown in the log-log plot of Fig. 6(b), a fitting straight line (red line) with a linear slope of $\sim \frac {1}{2}$ confirms the square-root dependence of the imaginary part of the effective index splitting Im($\Delta n_\textrm {eff})$ on the refractive index variation $\Delta \kappa$.

To characterize the sensing capability of the proposed two-layer cylindrical waveguides at EPs, we define the sensitivity $\frac {Im({\Delta n_\textrm {eff}})}{\Delta \kappa }$. In addition, we consider as a reference waveguide a single-layer cylindrical waveguide with its core filled with the same active absorbing material as the one filling the shell of the waveguide in Fig. 1. We then define the sensitivity enhancement as

When a small variation of $\Delta \kappa >0$ is introduced into the imaginary part $\kappa$ of the refractive index of the material filling the shell layer (Fig. 1), $n_{r,\textrm {eff}}$, ${n^+_{i,\textrm {eff}}}$, and ${n^-_{i,\textrm {eff}}}$ in Eq. (12) change to ${n_{r,\textrm {eff}}}+\Delta n_{r,\textrm {eff}}$, ${n^+_{i,\textrm {eff}}}+\Delta n^+_{i,\textrm {eff}}$, and ${n^-_{i,\textrm {eff}}}+\Delta n^-_{i,\textrm {eff}}$, where $\Delta n_{r,\textrm {eff}}$, $\Delta n^+_{i,\textrm {eff}}$, and $\Delta n^-_{i,\textrm {eff}}$ are also small. The eigenvalues of the Hamiltonian corresponding to the two-level model [Eq. (11)] then change to

In Fig. 7(a) we show the evolution of the imaginary part of the effective indices $n_\textrm {eff}$ of the two waveguide modes along the particular curve parametrized by $\Gamma$ in the $\chi -n_1$ design parameter space, where either the real parts or the imaginary parts of the effective indices of the two waveguide modes coincide. We show results for different values of the imaginary part $\kappa$ of the refractive index of the active material filling the shell layer (Fig. 1). We found that for $\kappa =0.03$, $\kappa =0.05$, and $\kappa =0.07$ the EPs, corresponding to $\Gamma =0$, are located at ($n_1=2.928, \chi =0.505$), ($n_1=2.875, \chi =0.514$), and ($n_1=2.829, \chi =0.523$), respectively, in the $\chi -n_1$ plane. We observe that, as the loss in the shell layer increases, the difference between the imaginary parts of the effective indices of the two waveguide modes for $\Gamma >0$ increases. This result can indicate two points. Firstly, the variation $\Delta n^-_{i,\textrm {eff}}$ is positive when the variation of $\Delta \kappa >0$ is introduced into the shell layer. Since the factor $n^-_{i,\textrm {eff}}$ is also positive, thus the effective index splitting is mostly contributed by its imaginary component when the variation of $\Delta \kappa$ is sufficiently small [Eq. (17)]. Secondly, the imaginary part of the effective index splitting can be enhanced by increasing the loss in the shell layer based on Eq. (17). Thus, based on Eq. (17) we expect that, as the loss in the shell layer increases, the sensitivity of the structure will increase. Figure 7(b) shows the sensitivity $\frac {Im({\Delta n_\textrm {eff}})}{\Delta \kappa }$ at the EP as a function of the imaginary part $\kappa$ of the refractive index of the active material filling the shell layer of the waveguide (Fig. 1). In all cases, the variation in the imaginary part of the refractive index of the material filling the shell layeris $\Delta \kappa =5 \times 10^{-4}$. As expected, we observe that the sensitivity increases as the loss in the material filling the shell layer increases.

In practice, our proposed two-layer cylindrical waveguide structure can be realized using oxides such as Cu$_2$O for the material filling the core of the waveguide (Fig. 1). Cu$_2$O is lossless and has a refractive index very close to 2.875 at the wavelength of $\lambda _0 =610$ nm [54,55]. In addition, as mentioned above, our choice for the imaginary part of the refractive index $n_2$ of the material filling the shell layer ($\kappa =0.05$) is within the range of experimentally achievable values [50,51].

## 4. Conclusions

In this paper, we investigated the EP of a two-layer cylindrical waveguide structure consisting of absorbing and non-absorbing dielectrics. We used the transfer matrix method to account for the behavior of the structure. The effective indices of the propagating waveguide modes are the eigenvalues of the non-Hermitian system, and the corresponding propagating waveguide modes are the system eigenfunctions. We showed that, by tuning the core to total radius ratio and the refractive index of the core layer, the complex effective indices of two waveguide modes can coalesce so that an EP is formed. The underlying formation mechanism of the EP in our proposed two-layer cylindrical waveguide structure is associated with the interaction between two propagating modes supported by the waveguide. We found that the interaction between the two modes undergoes a transition from weak intermode interaction to strong intermode interaction, which resembles the phase transition from the $PT$-symmetric phase to the $PT$-broken phase in optical $PT$-symmetric systems.

We then investigated the realized EP of the structure, as well as the associated physical effects of level repulsion and crossing. We found that the level repulsion to level crossing transition in the real parts of the effective indices, and the level crossing to level repulsion transition in the imaginary parts of the effective indices occur when the refractive index of the core layer is crossing over the EP. In addition, we found that, when encircling the EP in the parametric space, the eigenvalues switch their positions after a closed loop of $2\pi$.

Finally, we found that the sensitivity of the effective index of the waveguide mode to variations of the refractive index of the material filling the shell layer is enhanced at the EP. In addition, we showed that larger sensitivity enhancement is obtained for smaller perturbations. We also found that, as the loss in the shell layer increases, the sensitivity increases.

As final remarks, we note that the proposed two-layer cylindrical structures can be fabricated by combining chemical vapor deposition and atomic layer deposition [56–58]. Even though here we investigated EPs for azimuthal mode number $m=6$, EPs exist for other azimuthal mode numbers as well. As an example, an EP exists for $n_1=2.450$, $\chi =0.462$ when $m=7$. Selective excitation of the individual modes of a multimode waveguide can be realized using fiber Binary phase spatial light modulators [59]. In addition, here we investigated the sensitivity of the effective index of the waveguide mode to variations of the refractive index of the material filling the shell layer, and showed that the sensitivity is enhanced at the EP. Similarly, we expect that the sensitivity to variations of the refractive index of the material surrounding the two-layer cylindrical structure will also be enhanced at the EP. Thus, our results could potentially contribute to the development of a new generation of chip-scale exceptional-point-enhanced optical waveguide devices for modulation, switching, and sensing.

## Funding

National Natural Science Foundation of China (61605252); Natural Science Foundation of Hunan Province (2017JJ3375); National Science Foundation (1254934).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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