We report a general method to design a unique type of a non-Hermitian electromagnetic double-near-zero index medium by a two-dimensional photonic crystal. The synergy of a nonsymmorphic glide symmetry of the lattice, a period-doubling of the unit cell, and the non-Hermitian perturbation of the photonic crystal induces a phase transition in the eigenvalue spectrum. Near the Brillouin zone center, such a photonic crystal is effectively an anisotropic double-near-zero index medium. Along the direction of interest, the real parts of its effective permittivity and permeability are simultaneously near zero, while the imaginary parts of the effective parameters are nonzero values with opposite signs, leading to a real and positive effective refractive index. This medium enables many fascinating applications such as an angular sensor, a coherent perfect absorber, and a laser.

With one or more constitutive parameters vanishing, a zero-index medium (ZIM) supports unique wave propagation behaviors. Owing to the zero refractive index, waves propagating in a ZIM do not accumulate any phase retardation. In the recent decade, there has been an upsurge of exciting discoveries of wave phenomena in the ZIM such as wave tunneling,1–6 cloaking of objects,7–16 and tailoring of wavefronts.17,18 A single-zero medium, with only one constitutive parameter near zero, usually has a significant impedance mismatch with the background. A double-zero medium (DZM), with both constitutive parameters near zero, may overcome this obstacle thanks to its finite effective impedance.7–14,19–22 A DZM is not directly available in nature and, therefore, can be achieved via artificial materials. Two widely accepted routes to realize the DZM consist in accidental degeneracy8–11 and photonic doping effect.23–28 

On the other hand, non-Hermitian systems have inspired perspectives that are beyond the paradigm of Hermitian physics.29–38 The spatial modulation of gain and/or loss is particularly interesting in ZIMs, where the effects of relatively low levels of gain and/or loss can be dramatically enhanced.39,40 The combination of non-Hermiticity and ZIMs leads to a variety of interesting phenomena, for instance, wave confinement and guiding,41,42 unidirectional transparency of electromagnetic waves,43 wave collimation,44–46 non-Hermitian doping,47 and CPA-Laser.48 Although some realistic designs for non-Hermitian ZIMs were proposed recently,49,50 there are some limitations in its applications. For example, the complex Dirac-like cones49–51 rely on the accidental degeneracy of eigenstates, which are sensitive to the material and geometry of the building blocks, and almost fix the normalized frequency of exceptional rings,49,50 which are unfavorable for potential applications.

In this work, we propose a design of non-Hermitian electromagnetic double-near-zero media (DNZM) in a two-dimensional photonic crystal (PC). Protected by glide symmetries in the PC, two bands below the first bandgap are doubly degenerate at Brillouin zone boundaries.52–54 Then we perturb the unit cell55,56 and add non-Hermicity. The non-Hermitian perturbations deform the doubly degeneracy and spawn a spindle-like contour of exceptional points (EPs). The PC exhibits complex and real spectra, when the wave vector is inside and outside this spindle-like domain, respectively. The effective permittivity and permeability are purely imaginary numbers inside the domain near the Brillouin zone center and are complex numbers outside the domain. We demonstrate some potential applications associated with the PC such as an angular sensor, a coherent perfect absorber, and a laser.

We start with a two-dimensional PC as illustrated in the inset of Fig. 1(a). The PC has a unit cell comprising two square silicon (ε=12.5) blocks embedded in silica with permittivity εb=2.25. The side length of silicon blocks is Ls=0.29a, where a is the lattice constant. The PC has a glide symmetry Gij=Mi|a2ĵ:i,ji,j+a2, which transforms one block to the other through a reflection over the i=x,y plane followed by a translation of a/2 along j=(y,x).52–54 Along high symmetry lines of the first Brillouin zone, the band structure for transverse electric (TE) polarization is calculated by the electromagnetic wave module in COMSOL Multiphysics and plotted in Fig. 1(a) in dimensionless frequency 2πc/a, where c is the wave speed in the host silica. Induced by the glide symmetry, the two bands below the first bandgap are doubly degenerate along the Brillouin zone boundaries.52–54 By doubling the unit cell along the x-direction, shifting two blocks in the middle along opposite vertical directions by dh=0.01a, and considering non-Hermitian perturbations by adding alternatively distributed gain and loss, with relative permittivity εg,l=2.250.26i, respectively, in the host medium, we obtain a new unit cell with length 2a in the x-direction and width a in the y-direction, as illustrated in the inset of Fig. 1(b). The real part of the band structure of the perturbed PC is shown in Fig. 1(b). We are particularly interested in the dispersion relation in the vicinity of point “A.” If we do not shift the two middle blocks and do not add non-Hermitian perturbation, i.e., just artificially double the original unit cell by grouping two units together, the branches along the original ΓX direction should be folded, and the original X point at the Brillouin zone boundary [marked by “X” in Fig. 1(a)] is folded to the center of the Brillouin zone, i.e., point A in Fig. 1(b). Around this point, the dispersion relation of the unperturbed PC is a semi-Dirac cone,57 which is linear along the ΓX direction, but quadratic along the ΓY direction. We would like to emphasize that unlike the previous works based on the Dirac-like cone,49,50 the existence of such a semi-Dirac cone is determined by the glide symmetry. In other words, the key role of the glide symmetry is to induce the “deterministic” double degeneracy in the band diagram. Compared to other lattices that do not have the glide symmetry, the double degeneracy in this case is robust against perturbations in materials and geometries. Then the shift of blocks and the non-Hermitian perturbation are applied to engineer the band structure in order to achieve the desired non-Hermitian DNZM. Figure 1(c) is an enlarged view of the band structure near the Brillouin zone center. The red and black dots represent the real and imaginary parts of the band structure, respectively. The eigenfrequencies are complex conjugate pairs near zero kx(kx<0.015π/a) but real for large kx. The transition points are the so-called exceptional points,49–51 forming a spindle-like contour as shown in Fig. 1(d).

FIG. 1.

(a) Band structure of a two-dimensional square lattice PC. The unit cell shown in the inset is constructed by silicon square blocks and silica host. The dashed lines represent glide planes. (b) The real part of band structure of the PC when doubling the unit cell shown in (a) along the x-direction via shifting two silicon blocks by dh and considering alternately distributed gain and loss in the host. In the inset, the dashed lines represent the interface between gain and loss. (c) The real part (red) and imaginary part (black) of the band structure near the Brillouin zone center (point A) of the PC in (b). (d) The imaginary part of the band structure in the kxky plane forms a spindle-like contour of EPs near the middle of Brillouin zone.

FIG. 1.

(a) Band structure of a two-dimensional square lattice PC. The unit cell shown in the inset is constructed by silicon square blocks and silica host. The dashed lines represent glide planes. (b) The real part of band structure of the PC when doubling the unit cell shown in (a) along the x-direction via shifting two silicon blocks by dh and considering alternately distributed gain and loss in the host. In the inset, the dashed lines represent the interface between gain and loss. (c) The real part (red) and imaginary part (black) of the band structure near the Brillouin zone center (point A) of the PC in (b). (d) The imaginary part of the band structure in the kxky plane forms a spindle-like contour of EPs near the middle of Brillouin zone.

Close modal

From a non-Hermitian effective medium theory point of view, the PC can be described by its effective parameters.49,50 For TE polarization, the surface impedance of the PC is defined as

Z=EzHy,
(1)

where Ez and Hy are the averaged electric and magnetic fields at the boundary of the unit cell, respectively. The effective permittivity and permeability of the photonic crystal can be described as follows:49 

εz=kxZωε0,μy=kxZωμ0.
(2)

Figures 2(a) and 2(c) show the real parts of εz and μy, while Figs. 2(b) and 2(d) correspond to their imaginary parts, respectively. Figures 2(e) and 2(f) correspond to real and imaginary parts of Z, respectively. The surface impedance at the EP is ZEP=162.07iΩ. For kx<0.015π/a, the real parts of effective εz and μy approach zero simultaneously, while both possess non-zero imaginary parts attributed to the non-Hermitian engineering. Since the effective εz and μy are purely imaginary numbers, the PC behaves like a complex conjugate medium (CCM) as its effective refractive index n=εzμy is real. We should emphasize that there are two pairs of effective parameters for kx<0.015π/a, which is consistent with the band structure shown in Fig. 1(c). Take kx=0, for example, the effective parameters calculated from the two eigenstates are εz=0.148i, μy=0, and εz=0, μy=0.028i, corresponding to the eigenfrequencies with positive and negative imaginary parts, respectively. However, the eigenstates with complex eigenfrequencies do not couple with incident waves with real frequency; it is the effective properties at the EP, at which eigenfrequencies become real, which rule the response of the PC.49 

FIG. 2.

The effective parameters of the PC near the Brillouin zone center along the kx direction. (a) The real part and (b) the imaginary part of the relative permittivity εz. (c) The real part and (d) the imaginary part of the relative permeability μy. (e) The real part and (d) the imaginary part of the impedance Z.

FIG. 2.

The effective parameters of the PC near the Brillouin zone center along the kx direction. (a) The real part and (b) the imaginary part of the relative permittivity εz. (c) The real part and (d) the imaginary part of the relative permeability μy. (e) The real part and (d) the imaginary part of the impedance Z.

Close modal

The non-Hermitian DNZM supports interesting wave propagation properties. As mentioned earlier, both effective permittivity and permeability are purely imaginary numbers, leading to a real effective refractive index, meaning that the wave shall propagate in this medium without any amplification nor attenuation. However, we found that when a slab of such a DNZM is embedded in an ordinary material, amplification or attenuation will occur as the size of the slab changes, because of the interference caused by the transmitted and reflected waves between the two boundaries. Figure 3(a) shows the transmission and reflection spectra of a plane wave at ω=0.357(2πc/a) along the x-direction normally incident onto a slab of the DNZM as a function of the thickness of the slab. The DNZM is embedded in the silica host. Solid curves are calculated from the standard transfer matrix method with εz=0.074i and μy=0.014i, which are the values of the effective medium parameters at the exceptional point, marked by “EP” in Fig. 2. Dots are obtained from the simulation of the PC. Excellent agreement indicates the validity of the effective medium descriptions. Figure 3(a) clearly shows high transmission for slabs with thicknesses up to 70a. We note there are two dips in the reflection spectrum, one located at l0a and the other located at l=65.2a, which is around half of the wavelength in the DNZM (λ=130.4a), implying they are induced by the Fabry–Pérot resonance. It is quite remarkable that at l=41.2a, marked by a blue star in Fig. 3(a), both the transmission and reflection coefficients spike, indicating significant amplification as the wave propagates in this material. This point is located between the two reflection dips and can be attributed to the anti-resonance. From the standard transfer matrix method, one can write the expressions for the transmission and reflection coefficients as follows:

t=1ζ2exp(ikel)1ζ2exp(2ikel)andr=ζ(1exp2ikel)1ζ2exp(2ikel),
(3)

where ζ=ZeZ0Ze+Z0, Ze and Z0 are the impedance of the DNZM and the host medium, respectively, and ke is the effective wave vector in the DNZM. Since the effective relative permittivity and relative permeability are purely imaginary and having opposite signs, the impedance Ze is a purely imaginary number as well. Therefore, ζ is a complex number with magnitude equals to 1 and can be written as exp(iδ). According to Eq. (3), if 2δ+2kel=2mπ,(m=0,1,2), the denominators in t and r are vanishing, leading to the divergence of both t and r. This property is unique to the CCM with purely imaginary permittivity and permeability with opposite signs. For other cases, the magnitude of ζ is generally not 1, and therefore, the denominators will not become exactly zero.

FIG. 3.

Transmission and reflection properties of the PC when the incident plane wave comes from one side at ω=0.357(2πc/a). (a) Transmission and reflection spectra for normal incidence. Black and red dots are transmission and reflection coefficients for the PC with different numbers of layers. Solid lines are the transmission and reflection coefficients for the effective medium slab as a function of the thickness. (b) Transmittance of the PC with oblique incidence. The PC consists of 21 layers of unit cell.

FIG. 3.

Transmission and reflection properties of the PC when the incident plane wave comes from one side at ω=0.357(2πc/a). (a) Transmission and reflection spectra for normal incidence. Black and red dots are transmission and reflection coefficients for the PC with different numbers of layers. Solid lines are the transmission and reflection coefficients for the effective medium slab as a function of the thickness. (b) Transmittance of the PC with oblique incidence. The PC consists of 21 layers of unit cell.

Close modal

It is worth mentioning that the DNZM designed in this work is anisotropic. At the EP, the effective medium parameters along the orthogonal direction can be obtained in the same way described earlier and are εz=0.074i and μx=1.014. According to the dispersion relation

kx2μy+ky2μx=εzω2,
(4)

it is not difficult to conclude that for a system with purely imaginary εz and μy, while real μx, waves that can propagate must have either vanishing ky or complex valued ky. Combining the property of vanishing ky and field enhancement, we can turn a slab of the DNZM into a sensor with angle selection functionality. To illustrate this application, we plot in Fig. 3(b) the transmission of the same plane wave at ω=0.357(2πc/a) incidence onto a slab of our PC with 21 layers (please note one layer contains 2a along the x-direction) vs the angle of incidence. It clearly shows that at normal incidence, the transmission is as high as 400 and then plunges to 10% of the peak value when the angle of incidence is only 5°.

Although Fig. 3(a) implies that a slab of our non-Hermitian DNZM is not convenient for absorbing a wave incident from one side, it can be used to engineer the interference of two counter propagating coherent waves and function as a coherent perfect absorber (CPA) and a laser. Consider a slab of the DNZM embedded in the silicon host. Two counterpropagating plane waves are normally incident from the left and right [as illustrated in Fig. 4(a)]. The amplitudes of the incoming waves (in1,in2)T and outgoing waves (out1,out2)T are related by the scattering matrix of the slab49,58–60

S=t1r2r1t2,
(5)

where t1,t2 and r1,r2 represent the transmission and reflection coefficients of the slab from left and right, respectively. At frequency ω=0.357(2πc/a), the DNZM takes the previously mentioned effective medium parameter, i.e., ε=0.074i and μ=0.014i. The eigenvalues of the scattering matrix can be calculated, and their dependences on the thickness of the slab at a fixed frequency are depicted in Fig. 4(b), which shows that the zero of one eigenvalue (at l=24.1a) does not coincide necessarily with the pole of the other eigenvalue (at l=41.2a). The zero and the pole of the eigenvalues of the scattering matrix correspond to the occurrence of CPA and lasing, respectively. Figure 4(b) implies that at a fixed thickness, CPA and lasing will not happen simultaneously by simply changing the phase of the incident coherent beams. This is different from the balanced zero-index material, where the absolute values of the purely imaginary permittivity and permeability are identical.48 Nevertheless, the vanishing real parts of the effective parameters enforce the existence of the CPA and lasing. In Figs. 5(a) and 5(b), we plot the electric field distribution of two counterpropagating coherent plane waves incident normally onto the left and the right boundaries of the PC slab with 12 layers. The thickness of the PC l=24a, close to the green star in Fig. 3(a), where the transmission and reflection coefficients are equal and the zero eigenvalue in Fig. 4(b). When the phase difference between the incident waves is Δφ=0(π), the electric field in the PC concentrates in the loss (gain) region, and the PC behaves as a CPA (laser). Here, the amplification is not the maximal supported by the PC as from the plot of the eigenvalue of the scattering matrix, and the optimal lasing mode should occur at l=41.2a. Figures 5(c) and 5(d) are the same as Figs. 5(a) and 5(b), when the PC is replaced by its effective medium. Again, the agreement between the results of the PC and its effective medium demonstrates the validity of our effective medium description.

FIG. 4.

Scattering properties of the non-Hermitian DNZM. (a) Schematic graph of the normally incoming and outgoing plane waves to and from a slab of the DNZM. (b) Eigenvalues of the scattering matrix as a function of the thickness of the slab.

FIG. 4.

Scattering properties of the non-Hermitian DNZM. (a) Schematic graph of the normally incoming and outgoing plane waves to and from a slab of the DNZM. (b) Eigenvalues of the scattering matrix as a function of the thickness of the slab.

Close modal
FIG. 5.

The PC can behave as a coherent perfect absorber or a laser. (a) and (b) The electric field distributions when two-plane waves with a phase change (a) Δφ=0 and (b) Δφ=π normally incident from opposite sides of the PC with 12 layers of unit cells. The frequency of plane waves is ω=0.357(2πc/a). (c) and (d) The electric field distributions when the PC in (a) and (b) is replaced by an effective medium with εz=0.074i and μy=0.014i, respectively.

FIG. 5.

The PC can behave as a coherent perfect absorber or a laser. (a) and (b) The electric field distributions when two-plane waves with a phase change (a) Δφ=0 and (b) Δφ=π normally incident from opposite sides of the PC with 12 layers of unit cells. The frequency of plane waves is ω=0.357(2πc/a). (c) and (d) The electric field distributions when the PC in (a) and (b) is replaced by an effective medium with εz=0.074i and μy=0.014i, respectively.

Close modal

To conclude, we propose a general method to realize an anisotropic non-Hermitian DNZM in a two-dimensional PC. By considering a period-doubling of the unit cell and the non-Hermitian perturbation in a PC with glide symmetry in the lattice, we obtain complex conjugate pairs of eigenfrequencies near the Brillouin zone center, where the PC can be effectively regarded as a non-Hermitian DNZM. We unveil the unique features of the wave propagation in a slab of such a medium, including significant amplification of the signal and collimation, and demonstrate some typical applications of the non-Hermitian DNZM such as angular sensors, coherent perfect absorbers, and lasers. We would like to point out that the choice of gain and loss is not unique for a non-Hermitian double-zero medium, but it indeed affects the effective refractive index and subsequently affects the thickness of the slab l, when CPA or lasing occurs. To avoid l involving fraction of a lattice and to constrain the imaginary part within 10% of the real part, we set the current gain/loss profile. Although both gain and loss are dispersive in reality, our primary interest, at this stage, lies in the special properties of a non-Hermitian DNZM, which only requires a particular set of values of gain and loss at certain frequencies. In principle, our method may be extended to 3D structures with glide symmetry. Different from some previous works, our method does not depend on the accidental degeneracy of eigenstates and, therefore, is robust with respect to the selection the material and geometry, which, in turn, offers certain tunability on its working frequency.

This work was partially supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award Nos. OSR-CRG2020-4374 and OSR-2016-CRG5-2950 and the KAUST Baseline Research Fund under No. BAS/1/1626-01-01.

The authors declare that they have no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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