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Abstract
Non-Hermitian systems have recently attracted significant attention in photonics due to the realization that the interplay between gain and loss can lead to entirely new and unexpected features. Here, we propose and demonstrate a non-Hermitian Faraday system capable of non-reciprocal omni-polarizer action at the exceptional point. Notably, both forward and backward propagating light with arbitrary polarization converge to the same polarization state. Leveraging the robustness and non-reciprocity of the non-Hermitian Faraday system, we realize an omni-polarized Faraday isolator that can effectively isolate any polarized light without the need for a polarizer at the incident port of backward propagation. Remarkably, under the given parameter configuration, the isolator achieves a maximum isolation ratio of approximately 100 dB and a minimum isolation ratio of around 45 dB for various polarized light, accompanied by near-zero insertion loss. Furthermore, our research reveals the remarkable tolerance of the non-Hermitian Faraday isolator to nonlinear effects. This unique characteristic allows us to harness nonlinear effects to achieve various optical functions, all while maintaining excellent isolation performance. The proposed non-Hermitian Faraday system paves the way for the realization of magnetically or optically switchable non-reciprocal devices.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Lately, non-Hermitian systems have raised considerable attention in photonics, given that optical gain and loss can be integrated as nonconservative ingredients to create artificial materials and structures with altogether new optical properties [1–5]. These non-Hermitian systems have demonstrated various intriguing phenomena, as evidenced by a range of recent theoretical and experimental studies, such as loss-induced transparency [6], unidirectional invisibility [7], lasing mode selection [8], non-reciprocal transmission [9,10] and directional lasing [11]. Importantly, these phenomena are typically realized at or in close proximity to an exceptional point (EP). In addition, EPs can not only be approached or swept across, but can also be dynamically encircled [12–15]. Adiabatic parametric evolution around EP provides interesting schemes for asymmetric mode-switching and omni-polarizer action in photonics [16–19]. Very recently, omni-polarizer action had been realized by emulating EP encirclements and fiber-based photonic emulator [20,21]. These omni-polarizers exhibit remarkable robustness, that arbitrary polarization state passes through the omni-polarizer will evolve to a specific polarization state. This unique property of non-Hermitian optical systems provides significant advantages for various optical devices.
Optical isolators based on non-reciprocal transmission are the key photonic elements in optical telecommunications and optical information processing. Various methods have been developed to achieve optical isolation, including the use of magneto-optical (MO) effect [22–25], topology structures [26,27], parity-time symmetry breaking [28,29]. Among these approaches, the Faraday effect is the most mature and widely utilized technology for designing and preparing high-performance optical isolators [30,31]. It is well known that the ideal Faraday systems is a non-reciprocal Hermitian system. However, by introducing gain and loss into the Faraday system, it becomes possible to create a non-reciprocal and non-Hermitian system that exhibits robustness. The properties of such a system will be fundamentally different from those of the non-reciprocal Hermitian system [32–34]. Despite the significant potential, to the best of our knowledge, there are few studies on the non-Hermitian Faraday systems.
In this work, a non-Hermitian Faraday system is constructed by introducing non-Hermitian terms into the Faraday system. The variation of the eigenstates and polarization evolution of the system with the parameters of MO materials are studied theoretically and numerically. At the EP, both forward and backward propagating light in the system converge to the same polarization state. This intriguing behavior gives rise to a new type of omni-polarizer, which exhibits strong robustness, that both forward and backward propagating light with arbitrary polarization will evolve to the same eigenstate. Benefits from the robustness and non-reciprocity of the non-Hermitian Faraday system, an omni-polarized Faraday isolator that can isolate any polarization state is realized simply by adding a polarizer at the incident port of forward propagation. The isolation properties under various polarization states, geometric and material parameters have been quantitatively analyzed. Under the given parameter configuration, the isolator achieves a maximum isolation ratio (IR) of approximately 100 dB and a minimum IR of around 45 dB for various polarized light, accompanied by near-zero insertion loss (IL). Additionally, we conducted a comprehensive analysis of the impact of nonlinear effects on the performance of the non-Hermitian Faraday isolator. Our findings provide evidence of the high tolerance of the non-Hermitian Faraday isolator to nonlinear effects. Even with strong nonlinear effects, the isolator maintains its excellent isolation properties. It achieves a maximum IR of approximately 100 dB and a minimum IR of greater than 43 dB, while keeping the absolute value of IL less than 2 dB. Our results strongly indicate that nonlinear effects can be effectively utilized to achieve a diverse range of optical functionalities, all while maintaining excellent isolation performance.
2. Omni-polarizer action
Because the permittivity tensor corresponds to the Hamiltonian for plane waves in polarization space [35], for z-direction magnetized isotropic materials, the permittivity tensor satisfying Faraday effect and non-Hermitian is characterized as
Bismuth-substituted yttrium iron garnet (Bi: YIG) has been shown to have a large magneto-optical coefficient and is considered an attractive Faraday optical material [36,37]. The ${\varepsilon _{xy}}$ can be greater than 0.3 and the theoretically predicted ${\varepsilon _{xy}}$ is much greater than that obtained by experiment [38]. Thus, the MO material used in this work is Bi: YIG, and the value of ${\varepsilon _{xy}}$ is set to ${\varepsilon _{xy}} = \mu = 0.1$. Garnet materials, including YIG, possess a cubic crystal structure and exhibit isotropic behavior. Consequently, the diagonal elements of the dielectric tensor are identical, that is, ${\varepsilon _{xx}}\; = \; {\varepsilon _{yy}}\; = \; {\varepsilon _{zz}}\; = \; {\varepsilon _0}\; = \; {n^2}$. The dielectric constant ${\varepsilon _0}$ of the diagonal element is set to 4.84, which is the typical refractive index of YIG near the wavelength of 1.5 µm [39]. Then the permittivity tensor is expressed as
For a plane wave propagating along the z-axis, the form of the permittivity tensor can be expressed as
The Stokes parameters expressed on the Poincaré sphere are employed for the graphical representation of the eigenstates and polarization states. Figure 1 shows the evolution of eigenstates in the system as $|{{\gamma}/\mu } |$ varies. For $|{{\gamma}/\mu } |= \infty $, the eigenstates are ${({1,1} )^\textrm{T}}$ and ${({1, - 1} )^\textrm{T}}$, where the gain eigenstate is ${({1,1} )^\textrm{T}}$ and the loss eigenstate is ${({1, - 1} )^\textrm{T}}$ [42]. When $|{{\gamma}/\mu } |> 1$, with the decrease of $|{{\gamma}/\mu } |$, the two eigenstates ${{\textbf V}_ \pm^R} $ gradually converge from ${({1,1} )^\textrm{T}}$ and ${({1, - 1} )^\textrm{T}}$ towards ${({0,1} )^\textrm{T}}$, as depicted in Fig. 1(a). At $|{{\gamma}/\mu } |= 1$, the EP occurs and the eigenstates coalesce at ${({0,1} )^T}$, as displayed in Fig. 1(b). Subsequently, upon further reduction of $|{{\gamma}/\mu } |$ below 1, the eigenstates ${{\textbf V}_ \pm^R} $ split from ${({0,1} )^T}$ into ${({1, - i} )^T}$ and ${({1,i} )^\textrm{T}}$. When $|{{\gamma}/\mu } |$ reaches 0, the eigenstates are ${({1, - i} )^\textrm{T}}$ and ${({1,i} )^\textrm{T}}$. Importantly, the evolution of eigenstates in the proposed system remains consistent for both forward and backward propagation.
Fig. 1. Poincaré spheres describe the evolution of eigenstates of light propagating through the medium described by Eq. (2) when (a) $|{\gamma /\mu } |> 1$, (b) $|{\gamma /\mu } |= 1$, and (c) $|{\gamma /\mu } |< 1$. The red and blue axes correspond to gain and loss. The eigenvectors of the dielectric tensor are represented by the colors purple, yellow, green and cyan, respectively. The arrowheads indicate the evolution direction of the eigenstates.
In order to analyze the transmission characteristics of this system using transfer matrix approach, the convenient basis of horizontally ($\theta = 0^\circ $) linearly polarization (LP) and vertically ($\theta = $ 90$^\circ $) LP are employed: ${({1,\; 0} )^\textrm{T}}\; \textrm{and}\; {({0,1} )^\textrm{T}}$ [35]. From the relation between the incident wave ${E_{\textrm{inc}}} = [{{E_{H\textrm{I}}}\; ,\; {E_{\textrm{VI}}}} ]_{\textrm{LP}}^\textrm{T}$ and the transmitted wave ${E_{\textrm{tra}}} = [{{\textrm{E}_{\textrm{HT}}}{\; },{\; }{\textrm{E}_{\textrm{VT}}}} ]_{\textrm{LP}}^\textrm{T}$ in LP basis, transfer matrix ${\textbf M}$ can be expressed using structural and material parameters as ${E_{\textrm{tra}}} = {\textbf M}{E_{\textrm{inc}}}$,
Next, for several systems characterized by specific $|{\gamma /\mu } |$ values, the evolution of polarization states is analyzed. In the broken phase, specifically when $|{\gamma /\mu } |= 2/\sqrt 3 $, the eigenvalues in Eq. (4) are two different complex numbers. Correspondingly, the eigenstates are two non-orthogonal LP states that experience equal gain and loss respectively. The Poincaré sphere in Fig. 2(a) illustrates the evolution of polarization in this scenario. Since ${{\textbf V}_ +^R} $ is the gain eigenstate and ${{\textbf V}_ -^R} $ is the loss eigenstate, any polarization eigenstate, apart from ${{\textbf V}_ -^R} $, will be amplified and rotate towards the gain eigenstate ${{\textbf V}_ +^R} $, as described in the yellow and purple trajectory on the Poincaré sphere. Additionally, the direction of polarization state evolution within the region between the two eigenstates ${{\textbf V}_ \pm^R} $ (indicated by purple) is opposite to that observed in other regions. When $|{{\gamma}/\mu } |= 1$, an EP occurs and the eigenstates coalesce at ${({0,1} )^\textrm{T}}$, as stated in Eq. (4). In this scenario, incident light with arbitrary polarization state will be amplified and rotate towards ${({0,1} )^\textrm{T}}$ in the same direction as it propagates through the medium. This behavior is shown in Fig. 2(b). Nevertheless, in the exact phase of $|{{\gamma}/\mu } |= 1/3$, the eigenvalues are two different real numbers and the corresponding eigenstates are two elliptic polarization states. The polarization precesses in circles around the nonorthogonal eigenstates ${{\textbf V}_{1,2}^R} $, as depicted in Fig. 2(c).
Fig. 2. Poincaré spheres describes the evolution of polarization of light propagating through the medium described by Eq. (2) when (a) $|{\gamma /\mu } |= 2/\sqrt 3 $, (b) $|{\gamma /\mu } |= 1$ and (c) $|{\gamma /\mu } |= 1/3$. The eigenvectors of the dielectric tensor are shown in cyan, yellow, green and purple, respectively. The arrowheads indicate the evolution direction of the eigenstates.
In order to validate the feasibility of the proposed non-Hermitian Faraday system, COMSOL Multiphysics is employed to construct the model shown in Fig. 3(a). As a proof of concept, the representative condition in which the EP occurs is selected for further study. The material parameters for simulation are set according to Eq. (2). In the presence of a magnetic field ${\textbf B}$ along the z-axis, both forward and backward propagating light with arbitrary polarization state evolves into the degenerate eigenstates ${({0,{\; }1} )^\textrm{T}}$. Figure 3(b) shows that the electric field polarizations at the exit ports for both forward and backward propagation are close to ${({0,{\; }1} )^\textrm{T}}$, which is consistent with theoretical calculations. Additionally, the electric field appears to be uniformly distributed, corresponding to the electromagnetic properties of bulk materials. Figure 3(c) displays the simulation results of evolution trajectory of incident light with different polarizations states on the Poincaré sphere. The arrowheads indicate the respective directions of evolution. Polarization states located on the upper and lower spherical surfaces evolve towards eigenstate ${({0,{\; }1} )^\textrm{T}}$ along symmetrical trajectories, while LP states evolve towards eigenstate ${({0,{\; }1} )^\textrm{T}}$ along the equator. This observed behavior aligns with the evolution trend of polarization depicted in Fig. 2(b), further confirming the feasibility of the proposed non-Hermitian Faraday system. These results demonstrate that the non-Hermitian Faraday system can achieve a new type of omni-polarizer. Different from the previous omni-polarizer obtained by surrounding the EP, our proposed omni-polarizer exhibits a distinctive behavior. When the same coordinate system is established for observation in different directions, the evolutions of polarization states in both the forward and backward directions of the omni-polarizer follow the same trajectory. This unique characteristic allows for the preservation of the non-reciprocity of Faraday systems. Furthermore, our new type of omni-polarizer exhibits strong robustness about input polarization and isotropic perturbations, which will provide unique advantages for optical components based on non-Hermitian Faraday systems.
Fig. 3. (a) Schematic diagram of the Faraday non-Hermitian system. The whole structure is under a static magnetic field ${\textbf B}$ and polarization conversions are also illustrated with red arrowheads. ${\varepsilon _0} = 4.84,\; \mu = \gamma = 0.1,\; {\mathrm{\Lambda }_0} = 1550\; \textrm{nm}$. (b) The electric field polarizations at forward and backward exit ports when incident light is $0^\circ $ LP ($d = 100\; \mathrm{\mu m}$). (c) Simulation results of the polarization evolution of light propagating through the medium described by Eq. (2). The purple five-pointed star, red triangle, blue triangle, orange square and cyan diamond denote the incident light with 100$^\circ $ LP, right-handed circular polarization, left-handed circular polarization, right-handed elliptical polarization and left-handed elliptical polarization, respectively. The change from large to small points on each trajectory depictes the evolution trend as the propagation distance gradually increases.
3. Omni-polarized Faraday isolator
Then it is demonstrated that the novel omni-polarized Faraday isolator can be realized based on non-reciprocal property of the non-Hermitian Faraday system. In order to achieve optical isolation, an isotropic global loss parameter q is introduced into the dielectric tensor shown in Eq. (2). The modified dielectric tensor is expressed as:
Isolation can be achieved by simply adding a polarizer that transmits horizontally polarized electric field at the incident port of forward propagating (defined as the transmission direction). As depicted in Fig. 4(a), for forward propagating, $0^\circ $ linearly polarized light will evolve into 90$^\circ $ LP. On the other hand, for backward propagating (defined as the isolation direction), any polarized light will also eventually evolve into 90$^\circ $ LP and is blocked. Besides, unlike traditional Faraday isolators that can only isolate specific linearly polarized light, requiring polarizers at both incident ports, the omni-polarized Faraday isolator can isolate any polarized light without the need for a polarizer at the incident port of backward propagating direction. This advantage stems from the strong robustness of non-Hermitian Faraday system. The specific comparison is provided in the inset of Fig. 4(a).
COMSOL Multiphysics is employed to perform simulation, and the results are shown in Fig. 4(b)-4(f). The corresponding IL and IR calculation formula is as follows:
Fig. 4. (a) Schematic diagram of omni-polarized Faraday isolator based on the Faraday non-Hermitian system. ${\varepsilon _0} = 4.84,\; \mu = \gamma = 0.1,\; {\mathrm{\Lambda }_0} = 1550\; \textrm{nm}$. The inset provides a schematic diagram of a traditional Faraday isolator for comparison. (b) The electric field polarizations at forward and backward exit ports when the incident light is 0$^\circ $ and 90$^\circ $ LP, respectively ($d = \; $100 µm). The curves of IL obtained from calculation and simulation with parameters settings $\mu = \gamma = 0.1,\; q = 0$ (c) and $\mu = \gamma = 0.1,q = 5.66 \times {10^{ - 3}}$ (d). (e) The curves of IR obtained from calculation and simulation varying with the polarization state of incident light within the range of ${\pm} $89.9$^\circ $. The dashed line represents the IR of circularly polarized light, and it intersects with linearly polarized light at ${\pm} $45$^\circ $. The parameters settings are the same with (d). (f) The curves of IR obtained from calculation and simulation varying with propagation distances for $0^\circ $ linearly polarized incident light.
To gain insight into the IR of the system, we calculated and simulated the IR of incident light with different polarization states at a propagation distance of 1 mm, as shown in Fig. 4(e). When the linearly polarized incident light varies from $- $90$^\circ $ to 90$^\circ $, the IR initially decreases, then increases, and exhibits symmetry around 0$^\circ $. The maximum IR is approximately 100 dB and the minimum IR is approximately 45 dB. In addition, the IR of circularly polarized light is identical to that of ${\pm} $45$^\circ $ linearly polarized light. Figure 4(f) is the curves of IR obtained from calculation and simulation for 0$^\circ $ linearly polarized incident light, varying with propagation distances. It is evident that as the propagation distance increases, the IR gradually increases as well. However, the rate of increase becomes progressively slower. These calculation and simulation results serve as strong evidence of the accuracy and feasibility of the proposed omni-polarized Faraday isolator.
4. Impact of nonlinearity
In traditional isolators, nonlinear effects are typically avoided as they can significantly impair the isolator's effectiveness. However, in the case of our proposed non-Hermitian Faraday isolator, which incorporates a gain medium, it becomes necessary to consider nonlinear effects such as saturable absorption. Here, we have included an analysis of the potential impact of isotropic nonlinear effects on the isolation effect. In simpler terms, nonlinear effects in the material can alter both the real and imaginary parts of the dielectric tensor. Studying the influence of isotropic nonlinearity on the isolation effect is essentially equivalent to analyzing the impact of the change in real and imaginary parts of the diagonal of the dielectric tensor. It is worth noting that these changes caused by nonlinear effects are typically very small in magnitude. However, to demonstrate the robustness of our non-Hermitian Faraday system, we deliberately investigate the impacts of larger values of $\mathrm{\Delta }{\varepsilon _0}$ (change in the real part) and $\mathrm{\Delta }q$ (change in the imaginary part) on IL and IR. Specifically, we consider values of $\mathrm{\Delta }{\varepsilon _0} = $0.0441 ($\mathrm{\Delta }n \approx 0.01$), 0.45 ($\mathrm{\Delta }n \approx 0.1$) and $\mathrm{\Delta }{\varepsilon _0} ={-} $0.0439 ($\mathrm{\Delta }n \approx{-} 0.01$), $- $0.43 ($\mathrm{\Delta }n \approx{-} 0.1$), $\mathrm{\Delta }q = $ ${\pm} 1 \times {10^{ - 5}}$ and $\mathrm{\Delta }q ={\pm} 1 \times {10^{ - 4}}$. The results, depicted in Fig. 5, indicate that the presence of nonlinear effects leads to a slight decrease in IL and IR, which remains within an acceptable range. When $\mathrm{\Delta }{\varepsilon _0} = $0.0441, $- $0.0439 or $\mathrm{\Delta }q ={\pm} 1 \times {10^{ - 5}}$, the performance of the isolator remains essentially unchanged compared to the linear state. The IL remains close to 0, and the minimum IR stays around 45 dB. Similarly, for $\mathrm{\Delta }{\varepsilon _0} = $ 0.45, $- $0.43 or $\mathrm{\Delta }q ={\pm} 1 \times {10^{ - 4}}$, the absolute value of IL is still less than 2 dB, while keeping the minimum IR remains greater than 43 dB. In summary, reducing the real part or increasing the imaginary part will result in an increase in the IR, accompanied by an increase in IL. Conversely, increasing the real part or decreasing the imaginary part will lead to a decrease in IR, and the corresponding IL becomes negative, indicating the achievement of gain in forward propagation and isolation in the backward propagation. Additionally, with increasing propagation distance, the differences and impacts caused by nonlinearity become more significant. These findings suggest that isotropic nonlinear effects have a relatively minor impact on the isolation performance of our non-Hermitian Faraday isolator. Furthermore, they demonstrate the robustness of our non-Hermitian Faraday system. It is worth noting that the exceptional tolerance of the non-Hermitian Faraday isolator to nonlinear effects enables us to achieve a wide range of functions based on these effects, all while maintaining excellent isolation performance. This unique characteristic expands the potential applications of the isolator and significantly enhances its versatility in various optical systems.
Fig. 5. Simulation results of IL and IR with nonlinearity. The curves of IL varying with propagation distance when there are (a) change in the real part, (b) change in the imaginary part. The curves of IR varying with the polarization state of incident light when there are (c) change in the real part, (d) change in the imaginary part. The parameters settings are $d = 1000\; \mathrm{\mu m},\; \mu = \gamma = 0.1,\; q = 5.66 \times {10^{ - 3}}$.
For the experimental realization, one approach is to construct non-Hermitian systems hierarchically by combining different materials [21], which provides us with significant flexibility in experimental design. As long as the transmission matrix remains unchanged, the effective dielectric tensor can be represented in the same manner. For the non-Hermitian Faraday system, we can select MO materials from a range of materials that exhibit a relatively strong Faraday effect. Bi: YIG is considered an attractive Faraday optical material due to its large magneto-optical coefficient. Therefore, in this work, we utilize Bi: YIG as the MO material. We can combine isotropic gain media with anisotropic loss media [43,44]. To maintain the system at an EP, we need to choose isotropic materials for the laser gain medium to avoid birefringence caused by the Pockels effect. Suitable laser gain media can include gases, liquids, or solid materials with a cubic crystal structure. For instance, commonly used materials such as carbon dioxide, dyes, or Er:YAG can serve as the laser gain medium. Conversely, loss can be introduced by employing anisotropic semiconductor media or metals. By controlling the intensity of the external magnetic field and pump light, we can construct a non-Hermitian Faraday system that closely approaches the EP infinitely.
5. Conclusion
In summary, a non-Hermitian Faraday system capable of non-reciprocal omni-polarizer action at EP have been systematically studied theoretically and numerically. A new type of omni-polarizer is obtained, wherein any polarized light propagating in both forward and backward directions converges to the same polarization state. Benefits from the inherent robustness of the non-Hermitian Faraday system, any polarized light can be effectively isolated by introducing a polarizer at the incident port of forward propagation. This innovative device is referred to as the omni-polarized Faraday isolator. The isolation properties of the non-Hermitian Faraday isolator have been quantitatively analyzed under various polarization states, as well as geometric and material parameters, utilizing both transfer matrix theory and COMSOL Multiphysics software. Furthermore, the non-Hermitian Faraday isolator demonstrates remarkable tolerance to nonlinear effects. Even in the presence of nonlinear effects, the isolator continues to exhibit exceptional isolation performance. It can achieve a maximum IR of approximately 100 dB and a minimum IR of about 45 dB, while maintaining a near-zero IL. Simultaneously, the effective utilization of nonlinear effects expands the capabilities of the isolator, enabling it to achieve a wide range of optical functions and further enhance its versatility and utility. These findings open up promising opportunities for device applications, including magnetically or optically switchable omni-polarizers and omni-polarized isolators.
Funding
National Natural Science Foundation of China (11874229, 61875093, 62205158); China Postdoctoral Science Foundation (2022M711709); Natural Science Foundation of Tianjin Municipality (06TXTJJC13500, 20JCQNJC01410); Science and Technology Commission of Tianjin Binhai New Area (BHXQKJXM-PT-ZJSHJ-2017003); 111 Project (B07013); Program for Changjiang Scholars and Innovative Research Team in Nankai University (IRT_13R29); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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