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Abstract
We study bulk optical modes in media with a parity-time symmetric permittivity tensor via the k-surface approach. We show that k-surfaces support multiple exceptional points (EPs), and the off-axial EPs can be manipulated by tuning the tensor elements. They merge into diabolic points if they have opposite handednesses, and annihilate each other otherwise. The underlying physical mechanisms and the potential applications are discussed.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The similarities between Schrödinger equation and Maxwell’s equations enable people to transfer the concepts between quantum physics and optics via the quantum-optical analogies [1]. This approach also applies to the recent enthusiasm for non-Hermitian open quantum systems [2–6] where the energy might not be conserved. By noting that loss and gain can be readily manipulated in optics, in the past decade non-Hermitian optics, especially parity-time ($\mathcal {PT}$) symmetry and exceptional points (EPs), has attracted extensively attention in the optical society [7–29]. The abnormal phase has been shown to lead to many intriguing phenomena such as loss-induced transmission [8], unidirectional invisibility [12], coherent absorption [10,11], enhanced sensors [18,19], topological energy transfer [16], slow light [20,21], and even topology [17].
The most concise way in understanding $\mathcal {PT}$ symmetry and EPs is the matrix approach [4,5,25], where the diagonal elements represent the initially isolated resonances, while the off-diagonal elements are their mutual interaction coefficients. By diagonalizing a non-Hermitian matrix, it is relatively easier to understand the phase diagram of the $\mathcal {PT}$ symmetric system and the coalescence of eigenvectors at EPs [4,5,25]. Since many features of the optical response and interaction in optics can also be represented by a matrix or a tensor, it is then natural to ask what happens if the similar concept of non-Hermitian is introduced to them.
In this article we pay attention to the permittivity tensor $\bar {\bar {\varepsilon }}$ and investigate the novel features when a $\mathcal {PT}$ symmetry is introduced. Unlike in former work [13], here we emphasize the three-dimensional (3D) nature of the optical response and analyze the bulk modes inside. From the dispersion and polarization of waves propagating along various directions via the $k$-surface approach, the $\mathcal {PT}$ phase in the full 3D spatial space is analyzed. Being different from literatures about Voigt waves in lossy anisotropic crystals [29–32], from the $k$-surface approach we show that even when the $\mathcal {PT}$ symmetric permittivity tensor $\bar {\bar {\varepsilon }}$ is a simple monoclinic one, the bulk medium supports multiple EPs especially some off-axial ones that have not been noticed before. The off-axial EPs can be manipulated by tuning the dielectric components of the permittivity tensor. They merge into diabolic points if they have opposite handednesses, and annihilate each other if they have the same handedness. The underlying physical mechanisms and the potential applications are discussed.
The structure of this article is organized as following. In Section 2 we develop the theoretic frame about bulk modes via the $k$-surface approach based on the electric displacement $D$. We predict some features on the $k$-surface and argue that EPs could exist at two symmetric planes. In Section 3 we present a series of calculations that prove the existence of off-axial EPs and their dependence on the parameters of the permittivity tensor $\bar {\bar {\varepsilon }}$. The coalescence and annihilation of these off-axial EPs are analyzed. Discussion is made in Section 4.
2. Theory
Our investigation focuses on the spatial dispersion of a plane wave propagating in a $\mathcal {PT}$-symmetric medium at a given angular frequency $\omega$. It is a reasonable choice because $\mathcal {PT}$ symmetry should follow the causality principle and could take place only at isolated frequency points [22]. We neglect the magnetic response and assume that the effective permittivity tensor in the $xyz$ rectangular coordinates is a simple $\mathcal {PT}$-symmetric one for monoclinic crystals [32], as
To find the dispersion and polarization of bulk modes in such a bulk medium, we should resort to Maxwell’s equations. Referring to the methodology proposed by Berry and Dennis [30] and assuming that all the field quantities are proportional to $\exp (-i\omega t-i\boldsymbol{k}\bullet \boldsymbol{r})$, the variation of $n$ versus $k$ can be found rigorously from
Since the classical refractive index ellipsoid only suitably applies to a Hermitian medium with orthogonal normal modes, here we would utilize the iso-frequency wavevector surfaces ($k$-surfaces) to characterize the spatial dispersion of the bulk mode.Defining a unit vector $\hat {\boldsymbol{k}}_0$ to represent the direction of propagating, as
from Eqs. (2) and (3) we can get [29]Now expressing the components of the unit vector $\hat {\boldsymbol{k}}_0$ as
where $k_x^2+k_y^2+k_z^2=1$, and substituting Eq. (1) into Eq. (5), the secular equation about the eigenvalue $n$ and the normalized eigenvector $\boldsymbol{D}_{[k_xk_yk_z]}=[D_x, D_y, D_z]^T$ readsEquation (8) generally gives complex solutions because $\textbf {M}$ is non-Hermitian. Since we are interested in EPs, let us first analyze their existence conditions. Because the trivial solution of Eq. (8) is $1/n=0$, and at an EP the other two nontrivial solutions are degenerated and are real, the determinant of the square matrix $\textbf {M}-\varepsilon _w^2n^{-2}\textbf {I}$ should be in the form of $n^{-2}(n^{-2}-a^2)^2=0$, where $a$ is real and $\textbf {I}$ is the unity matrix. It is then readily to find that it requires
which is satisfied only when Consequently, EPs only exist in the $k_x=k_y$ or $k_x=-k_y$ planes.Based on the symmetry of $\bar {\bar {\varepsilon }}$ and Eq. (11), we can briefly check the scenarios when $\hat {\boldsymbol{k}}_0=[0,0,1]^T$ and when $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,\pm 1,0]^T$, respectively. When $\hat {\boldsymbol{k}}_0=[0,0,1]^T$, the wave propagates along the longitudinal direction $\hat {\boldsymbol{z}}$, i.e. $[001]$. The matrix $\textbf {M}$ is given by
When $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,\pm 1,0]^T$, the matrix $\textbf {M}$ reads
With the existence of EP$_z$ at $\hat {\boldsymbol{k}}_0=[0,0,1]^T$ and two real solutions at $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,\pm 1,0]^T$, off-axial EPs might exist in the $k_x=\pm k_y$ planes with a coalescence of complex eigenvalues $n$. Here we use the term of off-axis to emphasize that the orientations of these EPs are deviated from the longitudinal direction $z$ [13]. Numerical calculations in solving Eq. (8) are required to provide distinct evidence on the existence of these EPs, as shown in the next section.
3. Calculation and analysis
Equation (8) enables us to investigate the variation of $n$ among different propagating directions in the bulk medium. Note that our simulation and discussion are about the electric displacement $D$, so some features demonstrated in this article are different from the common knowledge about the electric field $E$. Also, in our $k$-surface approach we use spherical coordinates to define the direction of $\hat {\boldsymbol{k}}_0$, and the length along the $\hat {\boldsymbol{k}}_0$ direction is proportional to the real (Re) and imaginary (Im) parts of $n$ as defined in Eq. (4). This approach provides an intuitive illustration about the bulk mode dispersions. We also analyze the polarization degree of the normal modes. Color of each point shown in the figures of this article is given by the third Stokes parameter $S_3$ on $D$ defined in the spherical coordinates,
When $S_3=\pm 1$ the mode is right-(left-) handed circularly polarized. When $S_3=0$, the normal mode is linearly polarized.3.1 Tunable Off-axial EPs
Figure 1 provides a 3D schematic of $k$-surfaces with parameters $\varepsilon _w=4$, $\gamma =2$ and $\varepsilon _d=2$. Projected contours at selected $k_z$ values are also shown to reveal the topology of the $k$-surfaces. Because the imaginary components of the two nontrivial solutions of $n$ are opposite with each other, here we only keep the positive one.
Fig. 1. A schematic of the (a) real and (c) imaginary parts of the $k$-surfaces with parameters $\varepsilon _w=4$, $\gamma =2$ and $\varepsilon _d=2$. (b) and (c) show the projected contours at chosen $k_z$ values. EPs are labeled out by arrows. In our $k$-surface approach the lengths along the $\hat {\boldsymbol{k}}_0$ direction is proportional to Re{n} and Im{n}, respectively.
From Fig. 1 we can see usually there are two sets of $k$-surfaces for $\textbf {Re}\{n\}$. The imaginary part has four lobes and might intersect with each other. Features characterizing the non-Hermitian nature of the optical response can be also seen from the $k$-surfaces. For example, as shown in Fig. 2 about the spatial dispersion in the $k_x=\pm k_y$ planes, once $\textbf {Re}\{n\}$ has two non-degenerated values, $\textbf {Im}\{n\}$ equals zero. Only when the two solutions of $n$ have equal real part, does the system fall into the broken-$\mathcal {PT}$ phase with $\textbf {Im}\{n\}\neq 0$.
Fig. 2. Projected contours of the (a) real and (b) imaginary components of the $k$-surfaces in the $k_x=\pm k_y$ plane. EPs are labeled out. In the $k_x=-k_y$ plane Im{n}=0 because Re{n} has two values and the system is within the conserved $\mathcal {PT}$ phase.
The most interesting feature on the $k$-surfaces is the existence of six EPs that form three pairs. As shown in Figs. 1 and 2, they are within the plane of $k_x=k_y$, in consistence with our analysis made in section 2.
Special attention should be paid to the four off-axial EPs (including the polarization $S_3$) not along the $k_z$ direction. From Fig. 2 we can see they have a $C_2$ rotational symmetry about the $k_z$ axis, and a mirror reflection antisymmetry with respect to the $k_z=0$ plane. Their propagating directions and eigenvectors also possess the corresponding symmetry. For example, if the eigenvector $[D_x, D_y, D_z]^T$ and propagating direction $[k_x,k_y,k_z]$ of EP above the $k_z=0$ plane (EP$_\alpha$ in Fig. 2) are known, then the other EP$_\alpha$ is characterized by $[-k_x,-k_y,k_z]$ and $[D_x, D_y, -D_z]^T$. This pair of EPs has a polarization state of $S_3=1$. As for the other pair of off-axial EPs (EP$_\beta$ in Fig. 2), their $\hat {\boldsymbol{k}}_0$ vectors are given by $[k_x,k_y,-k_z]$ and $[-k_x,-k_y,-k_z]$, and the eigenvectors are $[D_x, D_y, -D_z]^T$ and $[D_x, D_y, D_z]^T$, respectively. The polarization of them is $S_3=-1$. These off-axial EPs have not been noticed before, and will be the focus of our investigation in the present article.
To test the dependence of EPs on the parameters of $\bar {\bar {\varepsilon }}$ we make a series of calculations. We find that the off-axial EPs (EP$_\alpha$ and EP$_\beta$) preserve their polarizations while change the orientations continuously with $\varepsilon _d$. Figure 3 shows some examples about the projected contours of the $k$-surfaces in the $k_x=\pm k_y$ planes for different $\varepsilon _d$ values. With increased $\varepsilon _d$ value, EP$_\alpha$ and EP$_\beta$ moves toward EP$_z$ at $k_z=1$ and $k_z=-1$, respectively. When $\varepsilon _d=4$, they overlap with EP$_z$ [see Fig. 3(b)]. With further increased $\varepsilon _d$ value, they switch into the $k_x=-k_y$ plane and move toward each other. The imaginary components also change their patterns accordingly because they are nonzero only in the plane supporting off-axial EPs.
Fig. 3. Projected contours of the real and imaginary components of the $k$-surfaces in the $k_x=\pm k_y$ plane for (a) $\varepsilon _d=3.5$, (b) $\varepsilon _d=4$, and (c) $\varepsilon _d=5$, respectively. When $\varepsilon _d=4$ the imaginary component is not shown because it is zero for all $k_x=\pm k_y$.
3.2 Mergence of EPs at $k_z=0$ in forming a diabolic point
Above we show that the off-axial EPs are sensitive to the value of $\varepsilon _d$, and they shift to opposite directions when $\varepsilon _d$ varies. Since EP$_\alpha$ and EP$_\beta$ have opposite circular polarizations, it is then a very interesting question as what happens when they overlap.
Because the off-axial EPs are mirror-reflectional antisymmetric with respect to the $k_z=0$ plane, when $\varepsilon _d$ increases from 2 (Fig. 1) they should first meet in the $k_z=0$ plane, where the eigenvalues $n$ are given by Eqs. (16) and (18). Since an EP requires a coalescence of eigenvalues, the two solutions should be identical. It leads to the conditions of
With $\varepsilon _w=4$ and $\gamma =2$ Eq. (21) gives $\varepsilon _d=8/3$ at $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,1,0]$, and $\varepsilon _d=8$ at $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,-1,0]$, respectively.We confirm above prediction. For example, as shown in Fig. 4, when $\varepsilon _d=8/3$ EP$_\alpha$ and EP$_\beta$ merge together at $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,1,0]$. Now all the solutions within the $k_x=k_y$ plane are complex except for the points of $\hat {\boldsymbol{k}}_0=[0,0,1]$ and $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,1,0]$. Solutions in the $k_x=-k_y$ plane are all real.
Fig. 4. $k$-surfaces when $\varepsilon _d=8/3$. EP$_\alpha$ and EP$_\beta$ merge into a diabolic point in the $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,\pm 1,0]$ directions.
To check the degeneracy of EP$_\alpha$ and EP$_\beta$ at $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,1,0]$, we calculate the eigenvectors and confirm that they are still the two orthogonal ones given by Eqs. (17) and (19) with linear polarizations. Consequently, the degeneracy point is now a diabolic point other than an EP. Such an effect is in a close analog with the combination of two waves with opposite circular polarizations and the decomposition of it into two orthogonal linearly polarized components, in which the spin angular momentum of photon is conserved.
3.3 Annihilation of EPs at $k_z=1$
With further increased $\varepsilon _d$ value, each pair of EPs with the same circular polarization, e.g. the two EP$_\alpha$, can also meet with each other in the $[0,0,\pm 1]$ direction. Now, all the points in the $k_x=\pm k_y$ planes, except for $[0,0,\pm 1]$, are within a conserved $\mathcal {PT}$ phase and have real solutions, i.e. double real $k$-surfaces and zero imaginary components.
This scenario requires that Eq.(8) possesses real solutions for all possible $k_z=\pm k_y$ values. We can prove that now
The projected contours are shown in Fig. 3(b).We check the eigenvalues and eigenvectors in the $[0,0,\pm 1]$ directions. Now we still get an isolated EP$_z$ that have a single eigenvalue of $n=\sqrt {\varepsilon _w}$ and eigenvector $2^{-1/2}[1,i,0]^T$. So when the two EP$_\alpha$ (or the two EP$_\beta$) overlap with EP$_z$, they in fact annihilate each other and leave the EP$_z$ intact. By noting that the three EPs have the same circular polarizations (see Fig. 3), the annihilation of EPs with the same circular polarization is the most reasonable result because Eq. (8) has at most two nontrivial solutions and the system cannot support high-order EPs [18,23,33,34].
4. Discussion
Above we discuss the emergence of off-axial EPs, their traces on the $k_z=\pm k_y$ planes, and the mergence and annihilation of them at high-symmetric directions. We note that Voigt waves and singular optical axes in anisotropic lossy media have been discussed by various groups [29–32]. Unlike in these works, here our attention is paid to the spatial dispersion of bulk modes by using $k$-surfaces approach on electrical displacement $D$ and the emergence of off-axial EPs. We also investigate their overlapping characters, and show that the off-axial EPs merge into diabolic points if they have opposite handednesses, and annihilate each other if they have the same handedness. To the best of our knowledge, these interesting effects have not been discussed before.
We would like to emphasize that the $k$-surfaces also possess many interesting topologies other than these in Figs.1 and 4. Figure 5 shows an example when $\varepsilon _d=15$. Now the topology of the $k$-surfaces are totally different from the ones shown in Fig.1. The off-axial EPs are moved into the $k_x=-k_y$ plane. The real $k$-surfaces are strongly distorted from ellipse, which cannot be obtained from classic Hermitian crystals such as biaxial and uniaxial crystals. The imaginary $k$-surfaces also overlap and distort strongly. Detailed analysis on the unique topology is an interesting subject of future investigation.
Fig. 5. A schematic of the (a) real and (c) imaginary parts of $k$-surfaces when $\varepsilon _d=15$. (b) and (c) show the projected contours at chosen $k_z$ values. EPs are labeled out by arrows.
Before ending this article, we would like to provide a brief discussion on the possible observation of the off-axial EPs. Bulk modes can be detected by CCD if the wave can be propagated in free space. To observe an EP, its mode must refract out of a bulk medium. Considering there is an interface oriented normal to the $z$ direction that separates the bulk medium and free space, when an EP mode with refractive index $n$, propagating direction $k_0$, and eigenvector $D$ is incident into the interface, it should satisfy both the Snell’s theory and the boundary conditions. The key of the analysis is that when an EP mode of an eigenvector $[D_x, D_y, D_z]^T$ is reflected by the $xy$ plane, it is still an EP mode with an eigenvector $[D_x, D_y, -D_z]^T$. In other words, under a mirror reflection by a $xy$ plane an EP$_\alpha$ mode is converted to an EP$_\beta$ mode, and vice versa.
With above considerations we analyze and calculate the refraction features of the off-axial EPs. From Fig. 6 we can see the refractive index $n$ of EP monotonously increases with $\varepsilon _d$, and the refracted wave is generally elliptically polarized. Most noticeable features are associated with $\varepsilon _d=4$, across which point the off-axial EPs move from $k_x=k_y$ plane into the $k_x=-k_y$ plane. At this scenario EP$_\alpha$ and EP$_\beta$ overlap with EP$_z$, and the transmission coefficient reaches its maximum value. Now the refracted wave is circularly polarized, in consistent with reported literature [13].
Fig. 6. Refraction properties of EP mode into free space from a $xy$ interface. (a) the refractive index $n$ of EP, (b) its angle with respect to the $z$ axis (in the unit of radian), (c) the polarization of refracted field, and (d) the overall transmission coefficient in field intensity.
Since at an EP only a single bulk mode exists, the refraction of wave from free space into a bulk $\mathcal {PT}$ medium under the excitation of off-axial EP mode should be polarization-sensitive. Potential applications such as mode filtering and polarization manipulation are expected.
Future experimental effort about the off-axial EPs can be resorted to the design principle of coupled meta-atoms as proposed in [13], where the background surrounding and the layer distance determine the effective value of $\varepsilon _d$. The method by utilizing anisotropic optical microcavity based on monoclinic nonpolar crystals [28,30] is also very promising and deserves out further attention. As for future extended study, we can also pay attention to other anisotropic media [31]. They have sophisticated forms of permittivity tensor than the monoclinic one discussed in this article, and should have many other novel features in the $k$-surfaces with potential applications.
5. Conclusion
In summary, we study the bulk optical modes in media with a $\mathcal {PT}$ symmetric permittivity tensor via the $k$-surface approach. We show that albeit with a simple $\mathcal {PT}$ symmetric form, the $k$-surfaces support multiple EPs especially some off-axial ones not noticed before. They can be manipulated by tuning the dielectric components. Furthermore, the off-axial EPs merge into diabolic points if they have opposite handednesses, and annihilate each other if they have the same handedness. The underlying physical mechanisms and the potential applications are discussed.
Funding
National Natural Science Foundation of China (11574162, 11874228).
Disclosures
The authors declare no conflicts of interest.
References
1. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009). [CrossRef]
2. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]
3. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015). [CrossRef]
4. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
5. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and $\mathcal {PT}$ symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]
6. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in $\mathcal {PT}$-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016). [CrossRef]
7. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in $\mathcal {PT}$-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008). [CrossRef]
8. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\mathcal {PT}$-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef]
9. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
10. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011). [CrossRef]
11. S. Longhi, “$\mathcal {PT}$-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]
12. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]
13. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of $\mathcal {PT}$ symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014). [CrossRef]
14. J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91(3), 033825 (2015). [CrossRef]
15. W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in $\mathcal {PT}$-symmetric systems,” Phys. Rev. Lett. 119(7), 077401 (2017). [CrossRef]
16. S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546(7658), 387–390 (2017). [CrossRef]
17. H. Shen, B. Zhen, and L. Fu, “Topological band theory for non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120(14), 146402 (2018). [CrossRef]
18. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017). [CrossRef]
19. W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017). [CrossRef]
20. T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120(1), 013901 (2018). [CrossRef]
21. S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121(9), 093901 (2018). [CrossRef]
22. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in $\mathcal {PT}$-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014). [CrossRef]
23. Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016). [CrossRef]
24. A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25(11), 12325 (2017). [CrossRef]
25. M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]
26. Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99(7), 075413 (2019). [CrossRef]
27. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001). [CrossRef]
28. X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “$\mathcal {PT}$-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014). [CrossRef]
29. S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019). [CrossRef]
30. M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003). [CrossRef]
31. C. Sturm and M. Grundmann, “Singular optical axes in biaxial crystals and analysis of their spectral dispersion effects in β-Ga2O3,” Phys. Rev. A 93(5), 053839 (2016). [CrossRef]
32. M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017). [CrossRef]
33. K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015). [CrossRef]
34. Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Phys. 14(5), 53603 (2019). [CrossRef]
