1. Introduction

The effective medium theory [1,2] has provided a powerful tool in the fields of metamaterials [3] and photonic crystals (PhCs) [4] to explain the band dispersions at low frequencies. It has been discovered that when the effective permittivity and permeability accidentally cross zero at the same frequency, a Dirac point would appear at the Brillouin zone center with conical dispersions in its vicinity [5]. Moreover, there are simultaneously flat bands crossing the Dirac point, which coincide well with the mathematical condition of the longitudinal electromagnetic modes, i.e. zero permittivity or permeability component. Therefore, the effective medium theory can perfectly explain both the Dirac conical dispersion as well as the associated flat bands of longitudinal modes. Such unique dispersions and effective parameters have been observed in various frequency regime of electromagnetic and acoustic waves [5–16] and may have many important applications such as photonic chips [8–10], cloaking [11] and coherent perfect absorption [12,13], etc.

Recently, by introducing loss into the PhC with Dirac-like conical dispersion (Dirac cones plus the flat bands of longitudinal modes), the emergence of the spawning rings of exceptional points (EPs) out of the Dirac cones has been observed [17–21]. Inside the ring of EPs, the real parts of the eigen-frequencies have degenerate flat dispersions, while the imaginary parts split into two branches. Outside the ring of EPs, the situations are just the opposite. The real parts split, while the imaginary parts tend to degeneracy. Additional flat bands of longitudinal modes also emerge in the band structure of such non-Hermitian PhCs, but was not incorporated into the Hamiltonian model. Very recently, extensive efforts have been made to establish an effective medium theory for the non-Hermitian PhCs [22–24]. However, so far, a neat explanation of the ring of EPs from the effective medium point of view is still lacking.

In this work, we extend the local effective medium theory [5] to the non-Hermitian regime. In this picture, we assume that the effective parameters are independent of the wave vector, but dependent on both the real and imaginary parts of the angular frequency, i.e. $\omega = {\omega _R} + i{\omega _I}$. Here, ${\omega _R}$ describes the oscillating frequency, and ${\omega _I}$ describes the decaying or amplifying rate of waves. Recently, it has been demonstrated that ${\omega _I}$ can introduce the role of “effective gain” or “effective loss”, which can be conveniently controlled by increasing or decreasing the intensity of incident waves at a certain rate [25,26]. Here, by extending the frequency dispersion from ${\omega _R}$ to ${\omega _I}$, we have established a simple local effective medium which can not only explain the rings of EPs but also predict the occurrence and position of the additional flat bands in the band structure. Based on this picture, we further show two other unique types of band dispersions. One type has Dirac-like cones in both the ${\omega _R}$ and ${\omega _I}$ band structures, and the other type has a band gap and a point of quadratic degeneracy in the ${\omega _R}$ and ${\omega _I}$ band structures, respectively. Both dispersions have been verified by designing non-Hermitian PhCs. Our work paves a road for the manipulation of EPs and Dirac cones in the framework of non-Hermitian effective medium.

We start from the Hermitian PhC model (i.e. without loss or gain) with accidental Dirac-like cone occurring at the Brillouin zone center [5]. The PhC can be characterized by effective permittivity $\varepsilon (\omega )$ and permeability $\mu (\omega )$ as a linear function of frequency:

Equation (1) holds for the Hermitian system without loss or gain. When loss/gain is introduced into the system, the eigen-frequency turns into a complex one as $\omega = {\omega _R} + i{\omega _I}$. Here we consider the time variation term ${e^{ - i\omega t}}$ such that ${\omega _{_I}} > 0$ (or ${\omega _{_I}} < 0$) indicates the amplification (or attenuation) of electromagnetic wave energy. Generally, the effective permittivity $\varepsilon (\omega )$ and permittivity $\mu (\omega )$ are complex and dispersive, which can be expressed as $\varepsilon (\omega ) = {\varepsilon _R}(\omega ) + i{\varepsilon _I}(\omega )$ and $\mu (\omega ) = {\mu _R}(\omega ) + i{\mu _I}(\omega )$. In a small frequency regime near the Γ point, we propose the following linear relation between the effective parameters and the eigen-frequency in a similar form of Eq. (1) with complex ${A_\varepsilon }$ and ${A_\mu }$:

In order to verify the validity of Eq. (2), we demonstrate a practical case of PhC by using the finite-element software COMSOL Multiphysics. First, we construct a Hermitian PhC with a Dirac-like cone. The PhC is composed of a square array of dielectric cylindrical rods in free space. The unit cell is illustrated by the inset graph in Fig. 1(a). The lattice constant is a. The radius and relative permittivity of the dielectric cylindrical rods are ${R_a} = 0.2a$ and ${\varepsilon _A} = 12.5$, respectively. For transverse-magnetic (TM, electric field along the rods) modes, such a PhC has a Dirac-like cone with Dirac point frequency ${{{\omega _D}a} / {2\pi c}} = 0.541$, due to the accidental degeneracy of the monopole mode and the degenerate dipole modes at the Γ point [5]. Then, we assume that the ${\varepsilon _A}$ is changed to be complex as ${\varepsilon _A} = 12.5 + 0.5i$ in the frequency regime of interest. The positive imaginary part indicates loss in the rods and turns the PhC into a non-Hermitian system. The band structures in terms of ${\omega _R}$ and ${\omega _I}$ are shown in Figs. 1(a) and 1(b), respectively. They are calculated along the $\Gamma - {\rm X}$ direction. It is seen that the Dirac-like conical dispersion evolves into the interesting phenomenon of a spawning ring of EPs [17].

 

Fig. 1. [(a) and (b)] Complex band structures of a PhC composed of cylindrical rod with ${\varepsilon _A} = 12.5 + 0.5i$ in background medium of ${\varepsilon _B} = 1$. (a) and (b), respectively, show the real and imaginary parts of eigen-frequencies for given real Bloch wave vectors based on numerical calculations. (c) Real and (d) imaginary parts of the effective parameters of this PhC. Retrieved (e) real and (f) imaginary parts of eigen-frequencies based on the effective parameters. The gray solid line denotes the longitudinal mode.

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This emergence of the ring of EPs is usually understood within a k.p approximation by constructing the effective Hamiltonian ${H_{eff}} = \left( {\begin{array}{cc} {{\omega_D}}&{{v_g}|\textbf{k}|}\\ {{v_g}|\textbf{k}|}&{{\omega_D} - i{\gamma_d}} \end{array}} \right)$ with complex eigen-frequencies as [17],

From the aspect of the effective medium model with parameters and dispersion described by Eqs. (2) and (3), the solution of dispersion-less ${\omega _R}$ within the ring of EPs is ${\omega _{{\varepsilon _R}}} = {\omega _{{\mu _R}}} = {\omega _D}$. Therefore, we immediately have ${\varepsilon _R} = {\mu _R} = 0$ when ${\omega _R} = {\omega _D}$. At the Brillouin zone center (i.e. $|\textbf{k}|= 0$) within the ring of EPs, we can obtain two solutions of Eq. (3), i.e., ${\omega _I} = {\omega _{{\varepsilon _I}}}$ (upper frequency) and ${\omega _I} = {\omega _{{\mu _I}}}$ (lower frequency), which correspond to ${\varepsilon _I} = 0$ and ${\mu _I} = 0$, respectively. The ring of EPs can be obtained at the frequency of (see Supplement 1 for detailed derivation),

At the ring of EPs, the $\varepsilon ({\omega _R},{\omega _I})$ and $\mu ({\omega _R},{\omega _I})$ are purely imaginary, but with opposite signs. This result is consistent with previous effective medium studies of similar systems [23,24]. We note that the signs of the imaginary parts of $\varepsilon ({\omega _R},{\omega _I})$ and $\mu ({\omega _R},{\omega _I})$ are defined under ${\omega _I} < 0$. Therefore, they do not indicate the real gain or loss, but an “effective gain” or “effective loss” relative to the decaying behavior defined by ${\omega _I} < 0$ [26,27].

We further verify the effective medium parameters based on the eigenmode analysis [28,29]. Based on the band structures and the wave impedance regarding to eigenmodes, we obtain the complex effective parameters $\varepsilon (\omega )$ and $\mu (\omega )$ with respect to complex eigen-frequency $\omega $ (see details in Supplement 1). We find that nearby the Brillouin zone center ${\varepsilon _R}$ and ${\mu _R}$ vary almost linearly with ${\omega _R}$, while ${\varepsilon _I}$ and ${\mu _I}$ vary almost linearly with ${\omega _I}$, which is in accordance with Eq. (2). Through fitting the linear relations, the relevant parameters in Eq. (2) are retrieved as ${A_{{\varepsilon _R}}} = 3.645(a/2\pi c)$, ${A_{{\mu _R}}} = 10.749(a/2\pi c)$, ${A_{{\varepsilon _I}}} = 3.686(a/2\pi c)$, ${A_{{\mu _I}}} = 10.653(a/2\pi c)$, ${\omega _{{\varepsilon _R}}} \approx {\omega _{{\mu _R}}} \approx {\omega _D} = 0.541({{{2\pi c} / a}} )$, ${\omega _{{\varepsilon _I}}} ={-} 0.00447({{{2\pi c} / a}} )$, ${\omega _{{\mu _I}}} ={-} 0.0098({{{2\pi c} / a}} )$. The corresponding effective permittivity and permeability are plotted in Figs. 1(c) and 1(d) as functions of ${\omega _R}$ and ${\omega _I}$, respectively. It is observed in Figs. 1(c) and 1(d) that ${\varepsilon _R} = {\mu _R} = 0$ at ${\omega _R} = {\omega _{{\varepsilon _R}}} = {\omega _{{\mu _R}}} = {\omega _D}$, while ${\varepsilon _I} = 0$ at ${\omega _I} = {\omega _{{\varepsilon _I}}}$ and ${\mu _I} = 0$ at ${\omega _I} = {\omega _{{\mu _I}}}$. This result is consistent with the analysis by using the effective medium picture based on Eqs. (2) and (3). Moreover, we can see that ${\mu _R} = {\mu _I} = 0$ at ${\omega _R} = {\omega _D}$ and ${\omega _I} = {\omega _{{\mu _I}}}$. This perfectly explains the occurrence and position of additional flat band of longitudinal modes that emerges at ${\omega _R} = {\omega _D}$ and ${\omega _I} = {\omega _{{\mu _I}}}$ in the ${\omega _R}$ and ${\omega _I}$ band structures. Here, we note that the condition of ${\varepsilon _R} = {\varepsilon _I} = 0$ cannot induce flat bands of longitudinal modes because here we consider the TM polarization where the electric field is in the out-of-plane direction and cannot be longitudinal with the in-plane wave vector.

It is convenient to reconstruct the complex band structure by using Eq. (3) with the fitted parameters. Figures 1(e) and 1(f) show the retrieved ${\omega _R}$ and ${\omega _I}$, respectively. The flat bands are plotted as horizontal gray lines according to the condition of ${\mu _R} = {\mu _I} = 0$. Clearly, the reconstructed band structures match well with those of the PhC obtained by finite-element numerical calculations (Figs. 1(a) and 1(b)). Note that the slight difference between the simulation results (Figs. 1(a) and 1(b)) and effective medium prediction (Figs. 1(e) and 1(f)) is caused by the linear approximation in the effective medium description. The dispersion-less ${\omega _R}$ within the ring of EPs, the shape of the ring, as well as the additional flat bands are all well predicted by the non-Hermitian effective medium theory. In the Supplement 1, we also show that the non-Hermitian effective medium gradually converges to the previous Hermitian one when the loss/gain in the system is reduced to zero.

Interestingly, we find that the radius of the exceptional ring can be manipulated and even shrinks into a point by simply engineering the loss/gain in the different components of the non-Hermitian PhC, as indicated by Eq. (6). In this case, the parameters of effective media have ${\omega _{{\varepsilon _R}}} = {\omega _{{\mu _R}}} = {\omega _D}$ and ${\omega _{{\varepsilon _I}}} = {\omega _{{\mu _I}}} = {\omega _{DI}}$. This indicates that Dirac-like cones are formed in both ${\omega _R}$ and ${\omega _I}$ band structures. In other words, the Dirac-like cone is formed in a complex frequency space of $\omega = {\omega _R} + i{\omega _I}$, rather than only in the real frequency space. Through some derivation, we obtain dispersions ${\omega _R} = {\omega _D} \pm \frac{\beta }{{{\omega _D}}}k$ and ${\omega _I} = {\omega _{DI}} \pm \frac{{\beta {\omega _{DI}}}}{{{\omega _D}^2}}k$ near the complex Dirac-like point $({{\omega_D},{\omega_{DI}}} )$, where $\beta = {c / {\sqrt {{A_\varepsilon }{A_\mu }} }}$. This Dirac-cone like phenomenon in both real and imaginary frequency domain is absent in Hermitian systems like loss-less PhCs.

We note that this amazing phenomenon can be achieved in fully passive PhCs. For demonstration, we take the PhC model in Fig. 1 as an example, but the relative permittivities of the cylindrical rods and the background medium are changed to ${\varepsilon _A} = 12.5 + 0.33i$ and ${\varepsilon _B} = 1 + 0.0264i$, respectively. Here, we find that through varying the imaginary parts of ${\varepsilon _A}$ and ${\varepsilon _B}$, the zero-crossing positions of effective ${\varepsilon _I}$ and ${\mu _I}$ can be efficiently tuned, i.e., ${\omega _{{\varepsilon _I}}}$ and ${\omega _{{\mu _I}}}$ are changed. While the slope of ${\varepsilon _I}$ and ${\mu _I}$ with respect to ${\omega _I}$ remains unchanged, i.e., ${A_{{\varepsilon _I}}}$ and ${A_{{\mu _I}}}$ remain unchanged (see Supplement 1). Therefore, here we can only engineer the imaginary parts of ${\varepsilon _A}$ and ${\varepsilon _B}$ to realize the condition ${\omega _{{\varepsilon _I}}} = {\omega _{{\mu _I}}} = {\omega _{DI}}$.

The corresponding complex band structures are calculated using software COMSOL Multiphysics, as plotted in Figs. 2(a) and 2(b). Interestingly, the ring of EPs disappears and a Dirac cone occurs in both the ${\omega _R}$ and ${\omega _I}$ band structures (more specifically, ${{{\omega _R}a} / {2\pi c}} = {{{\omega _D}a} / {2\pi c}} = 0.541$ in the ${\omega _R}$ spectrum, and ${{{\omega _I}a} / {2\pi c}} ={-} 0.00714$ in the ${\omega _I}$ spectrum). The effective parameters of this PhC are retrieved and found to have linear dispersions satisfying Eq. (2) as the previous case. The relevant parameters are ${A_{{\varepsilon _R}}} \approx {A_{{\varepsilon _I}}} = 3.645(a/2\pi c)$,${A_{{\mu _R}}} \approx {A_{{\mu _I}}} = 10.6(a/2\pi c)$, ${\omega _{{\varepsilon _R}}} \approx {\omega _{{\mu _R}}} = {\omega _D} = 0.541({{{2\pi c} / a}} )$ and ${\omega _{{\varepsilon _I}}} \approx {\omega _{{\mu _I}}} = {\omega _{DI}} ={-} 0.00714({{{2\pi c} / a}} )$, respectively. The real and imaginary parts of $\varepsilon$ and $\mu$ are plotted in Figs. 2(c) and 2(d), respectively. It is seen that ${\varepsilon _R} = {\mu _R} = 0$ at ${\omega _R} = {\omega _D}$, and ${\varepsilon _I} = {\mu _I} = 0$ at ${\omega _I} = {\omega _{DI}}$. The retrieved complex band structures based on these effective parameters (Figs. 2(e) and 2(f)) match well with the simulation results (Figs. 2(a) and 2(b)), confirming the validity of the effective medium picture. The additional flat band of longitudinal modes now crosses the complex Dirac point $({{\omega_D},{\omega_{DI}}} )$ in the ${\omega _R}$ and ${\omega _I}$ band structures, because ${\mu _R} = {\mu _I} = 0$ at the complex Dirac point. This finding shows that the spawning ring of EPs can shrink to a complex Dirac point in a complex Dirac-like cone in a fully passive PhC.

 

Fig. 2. [(a) and (b)] Complex band structures of a PhC composed of cylindrical rod with ${\varepsilon _A} = 12.5 + 0.33i$ in a background medium of ${\varepsilon _B} = 1 + 0.0264i$. (a) and (b), respectively, show the real and imaginary parts of eigen-frequencies for given real Bloch wave vectors based on numerical calculations. (c) Real and (d) imaginary parts of the effective parameters of this PhC. Retrieved (e) real and (f) imaginary parts of eigen-frequencies based on the effective parameters. The gray solid line denotes the longitudinal mode.

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Besides the case of ${\omega _{{\varepsilon _R}}} = {\omega _{{\mu _R}}} = {\omega _D}$ and ${\omega _{{\varepsilon _I}}} \ne {\omega _{{\mu _I}}}$ (ring of EPs) and the case of ${\omega _{{\varepsilon _R}}} = {\omega _{{\mu _R}}} = {\omega _D}$ and ${\omega _{{\varepsilon _I}}} = {\omega _{{\mu _I}}} = {\omega _{DI}} \ne 0$ (complex Dirac-like cones) discussed above, the non-Hermitian PhC can also possess effective parameters satisfying ${\omega _{{\varepsilon _R}}} \ne {\omega _{{\mu _R}}}$ and ${\omega _{{\varepsilon _I}}} = {\omega _{{\mu _I}}} = {\omega _{DI}}$. In this case, the Dirac-like cone in the ${\omega _R}$ band structure would split into a band gap, while the Dirac-like point in the ${\omega _I}$ band structure would be transformed into a point of quadratic degeneracy [30]. Through some derivation, we obtain dispersions ${\omega _R} = {\omega _{{\varepsilon _R}}} + \frac{\alpha }{{{\omega _{{\varepsilon _R}}}^2({{\omega_{{\varepsilon_R}}} - {\omega_{{\mu_R}}}} )}}{k^2}$, ${\omega _R} = {\omega _{{\mu _R}}} - \frac{\alpha }{{{\omega _{{\mu _R}}}^2({{\omega_{{\varepsilon_R}}} - {\omega_{{\mu_R}}}} )}}{k^2}$ in the ${\omega _R}$ spectrum, and ${\omega _I} = {\omega _{DI}} - \frac{{2\alpha {\omega _{DI}}}}{{{\omega _{{\varepsilon _R}}}^3({{\omega_{{\varepsilon_R}}} - {\omega_{{\mu_R}}}} )}}{k^2}$, ${\omega _I} = {\omega _{DI}} + \frac{{2\alpha {\omega _{DI}}}}{{{\omega _{{\mu _R}}}^3({{\omega_{{\varepsilon_R}}} - {\omega_{{\mu_R}}}} )}}{k^2}$ in the ${\omega _I}$ spectrum, which indicates a point of quadratic degeneracy. Here $\alpha = {{{c^2}} / {{A_\varepsilon }{A_\mu }}}$.

As an example, we change the relative permittivities of PhC model studied in Fig. 1 to ${\varepsilon _A} = 12 + 0.33i$ and ${\varepsilon _B} = 1.04 + 0.02978i$. The corresponding complex band structures are plotted in Figs. 3(a) and 3(b). Evidently, a band gap appears in the ${\omega _R}$ band structure (Fig. 3(a)), while a point of quadratic degeneracy is seen in the ${\omega _I}$ band structure (Fig. 3(b)). The effective parameters are retrieved and found to have linear dispersions. The relevant parameters are obtained as ${A_{{\varepsilon _R}}} \approx {A_{{\varepsilon _I}}} = 3.804(a/2\pi c)$, ${A_{{\mu _R}}} \approx {A_{{\mu _I}}} = 10.380(a/2\pi c)$, ${\omega _{{\varepsilon _R}}} = 0.539({{{2\pi c} / a}} )$, ${\omega _{{\mu _R}}} = 0.550({{{2\pi c} / a}} )$ and ${\omega _{{\varepsilon _I}}} \approx {\omega _{{\mu _I}}} = {\omega _{DI}} ={-} 0.00759({{{2\pi c} / a}} )$. The real and imaginary parts of $\varepsilon$ and $\mu$ are presented in Figs. 3(c) and 3(d), respectively. The retrieved complex band structure by using these effective parameters are presented in Figs. 3(e) and 3(f), which match well with the simulation results (Figs. 3(a) and 3(b)). The band edges in the ${\omega _R}$ band structure are determined by ${\omega _R} = {\omega _{{\varepsilon _R}}}$ and ${\omega _R} = {\omega _{{\mu _R}}}$. The additional flat band of longitudinal modes is induced by ${\mu _R} = {\mu _I} = 0$ at the frequency of ${\omega _R} = {\omega _{{\mu _R}}}$ and ${\omega _I} = {\omega _{{\mu _I}}} = {\omega _{{\varepsilon _I}}}$. This finding shows that quadratic degeneracy can be realized in the ${\omega _I}$ band structure by simply designing ${\omega _{{\varepsilon _I}}} = {\omega _{{\mu _I}}} = {\omega _{DI}}$.

 

Fig. 3. [(a) and (b)] Complex band structures of a PhC composed of cylindrical rod with ${\varepsilon _A} = 12 + 0.33i$ in background medium of ${\varepsilon _B} = 1.04 + 0.02978i$. (a) and (b), respectively, show the real and imaginary parts of eigen-frequencies for given real Bloch wave vectors based on numerical calculations. (c) Real and (d) imaginary parts of the effective parameters of this PhC. Retrieved (e) real and (f) imaginary parts of eigen-frequencies based on the effective parameters. The gray solid line denotes the longitudinal mode.

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Table 1 summarizes the features of band structures for different types of parameters discussed in Figs. 1–3 based on the non-Hermitian medium description. Interesting characteristics of complex band structures including the ring of EPs, complex Dirac-like cones, quadratic degeneracy and flat bands of longitudinal modes are observed by simply adjusting the loss in the PhC. We note that the many of the predicted phenomena are clearly beyond the description of the Hamiltonian model for the ring of EPs. The non-Hermitian effective medium theory, however, can interpret all the above phenomena and is thus universal.

Table 1. Features of band structures of non-Hermitian PhCs with loss

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The above discussions are within the picture of complex eigen-frequencies and real Bloch wave vectors. In fact, we have also investigated the validity of the non-Hermitian effective medium theory under the other picture of real eigen-frequencies and complex Bloch wave vectors. The results also prove the validity of the non-Hermitian effective medium theory, and the details are presented in the Supplement 1.

In summary, we have proposed a local non-Hermitian effective medium theory that can accurately predict the spawning ring of exceptional points in photonic crystals with loss/gain. Based on this theory, we also predict the existence of complex Dirac-like cones with Dirac conical dispersions, the quadratic degeneracy and the additional flat bands in the band structures of real and imaginary frequencies. By engineering non-Hermitian photonic crystals, we have numerically realized such effective media with complex Dirac-like cones or quadratic degeneracy. Our work thus opens a gate for exploring the rich physics in non-Hermitian effective media.