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Abstract
In this study, we propose the application of non-Hermitian photonic crystals (PCs) with anisotropic emissions. Unlike the ring of exceptional points (EPs) found in isotropic non-Hermitian PCs, the EPs of anisotropic non-Hermitian PCs appear as symmetrical lines about the Γ point. The formation of EPs is related to the non-Hermitian strength and the real spectrum appears in the ΓY direction. The PCs have been validated as the complex conjugate medium (CCM) by effective medium theory (EMT). Conversely, EMT indicates that the effective refractive index has a large imaginary part along the ΓX direction, which forms an evanescent wave inside the PCs. Consequently, coherent perfect absorption (CPA) and laser can be achieved in the directional emission of the ΓY. The outgoing wave in the ΓX direction is weak, which can significantly reduce the losses and electromagnetic interference. The non-Hermitian PCs enable many fascinating applications such as signal amplification, collimation, and angle sensors.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The Dirac cone underlies many unique electronic properties of graphene and topological insulators [1–4]. However, Dirac cone dispersions are not limited to graphene but can also be found in classical wave periodic systems such as photonic crystals [5–9]. In 1987, Yablonovitch and John independently proposed the concept of photonic crystals composed of materials with different refractive indices arranged periodically [10,11]. In recent years, photonic crystals with Dirac dispersion have shown novel physical properties. Huang et al. have achieved a double zero-index medium (the effective permittivity and permeability are near zero) by designing a Dirac cone in C4-symmetric PCs [12]. Since effective medium parameters approach zero for every propagation direction, PCs with Dirac dispersion behave like an isotropic near-zero index medium. The near zero-index PCs mean that the impedance can be tuned to match that of the external medium [13] and have interesting optical applications, including electromagnetic wave manipulation [14], beam focusing [15], and electromagnetic cloak [16]. Besides, it was noted that Dirac cone dispersion can change with lattice symmetry. He et al. have realized semi-Dirac dispersion in rhombic lattice and demonstrated that PCs behave as an anisotropic zero medium [17], solving the directional radiation of waves in zero-index PCs. Yasa et al. have demonstrated the formation of semi-Dirac dispersion in microwave experiments, and the transmission efficiency of incident waves in different directions vary greatly in anisotropic PCs [18]. Note that these research works are based on the Hermite system without material loss or radiation leakage.
Over the past two decades, open systems described by a non-Hermitian Hamiltonian have become a subject of intense research [19–23].These non-Hermitian systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems. Besides, the non-Hermitian systems exhibit a rich array of novel phenomena, such as non-Hermitian skin effects, nonreciprocity, and topological transitions. In recent years, it was suggested that some non-Hermitian perturbations (gain or loss) could deform a Dirac cone and spawn a ring of exceptional points [24–27]. For the two-dimensional (2D) PCs, a ring of EPs can be achieved by introducing complex permittivity, whose imaginary part means loss or gain [28]. In the PT-symmetric systems, the complex permittivity exhibits a symmetric spatial profile of $\varepsilon (x)=\varepsilon ^*(-x)$ [29,30]. The scattering of electromagnetic waves by boundaries can produce CPA and laser effects [31,32]. When the amplitude and phase of the two incident waves are well-tuned (in-phase excitation), the outgoing waves will disappear, and the resulting effect may lead to perfect absorption. Conversely (out-of-phase excitation), the outgoing waves by pumping the incident waves are greatly enhanced, which can be regarded as a laser mode. Such non-Hermite PCs behave like CCM [33]. A CCM was first proposed by Bai et al. [34], who used a lattice of core-shell cylinders made of loss and gain materials, respectively. In addition, Cui et al. showed that CCM in a wide frequency range could be realized using non-PT symmetric PCs with a ring of EPs [35]. The PCs with a ring of EPs represent an isotropic CCM, supporting the waves to continue propagating in all directions. To confine wave propagation in specific directions, it is advisable to introduce confining materials at the boundaries of the PCs. However, the above-discussed limitations may complicate the fabrication and utilization of non-Hermitian PCs. Furthermore, PCs with anisotropic transmission can effectively control the emission direction of light waves, enabling signal collimation. Therefore, it is necessary to explore methods for designing anisotropic non-Hermitian PCs.
In this paper, we propose a non-Hermitian PC with anisotropic emissions. We reveal the unique characteristics of wave propagation in this PC, including the significant amplification of the signal and collimation [36], and demonstrate typical applications, such as angle sensors [37], coherent perfect absorbers [38,39], and lasers [40]. The non-Hermitian Hamiltonian model shows that the real spectrum appears in the $\Gamma$Y direction, while a complex spectrum appears in the $\Gamma$X direction. From the EMT perspective [41–43], the effective refractive index is real for the $\Gamma$Y direction, and the PC behaves as a homogeneous CCM, which supports CPA and laser effects. However, for the $\Gamma$X direction, EMT indicates that the effective refractive index has a large imaginary part, resulting in the formation of an evanescent wave inside the PCs [44]. Numerical simulation results confirm that coherent perfect absorption-lasing can be achieved in the directional emission of the $\Gamma$Y. The outgoing wave in the $\Gamma$X direction remains weak, which can significantly reduce losses and electromagnetic interference caused by the leakage waves. Significantly, we present a complete theoretical framework for the design of anisotropic non-Hermitian PCs. Our design is not constrained by closed boundary conditions, and the direction of light wave emission is effectively restricted, which has the potential to broaden the applications and development of non-Hermitian PCs.
2. Non-Hermitian PC design
The structure of our design is shown in Fig. 1, a two-dimensional PC unit cell composed of a C2-symmetric rectangular dielectric rod. The relative permittivity of the rectangular dielectric rods in domains A1 and A2 (the red region) and domain B (the blue region) are $\varepsilon _A$ and $\varepsilon _B$, respectively. The air (domain C, the gray region) with relative permittivity $\varepsilon _C$, and the relative permeability $\mu$ of all media is 1.0. The structural parameters are $b_1 = 0.5a$, $b_2 = 0.3a$, $b_3 = 0.523a$, $b_4 = 0.19a$, and $b_5 = 0.085a$, where "$a$" is the lattice constant of the unit cell. To construct a non-Hermitian photonic crystal, we introduce the imaginary part of the relative permittivity to represent a certain amount of loss or gain. Due to the rotational symmetry (rotation of $90^{\circ }$) of the PC around the unit cell’s center, the rectangular rod retains its appearance, the relative permittivity of the rectangular rod takes the form: $\varepsilon _A=10.04+i \gamma$, $\varepsilon _B=10.04+i \ell _r \gamma$. The real part of the relative permittivity is 10.04 (indium phosphide), and the positive (negative) sign of $\gamma$ indicates that the rectangular area consists of a loss or (gain) medium. The remaining air portion does not add non-Hermitian perturbation and its relative permittivity $\varepsilon _C=1$. Here, $\ell _r$ is defined as the loss-gain ratio, and the positive sign of $\ell _r$ indicates that the red and blue regions are either both loss or gain medium. In contrast, a negative sign means that one region is loss and the other is gain.
Fig. 1. The representation of a C2-symmetric unit cell consists of three rectangular dielectric rods with structural parameters $b_1 = 0.5a$, $b_2 = 0.3a$, $b_3 = 0.523a$, $b_4 = 0.19a$, and $b_5 = 0.085a$. The relative permittivity parameters used are $\varepsilon _A=10.04+i \gamma$, $\varepsilon _B=10.04+i \ell _r \gamma$ and $\varepsilon _C=1$. $\gamma$ represents that the rectangular area consists of a loss or (gain) medium and $\ell _r$ is defined as the loss-gain ratio.
When the value of $\gamma$ is zero, the entire PCs become the common Hermitian system. We show the band structures of Hermitian PC in Fig. 2(a). The two dispersion bands (second and thirds bands) form a semi-Dirac point at the $\Gamma$ point, which is linear along the $\Gamma$Y direction but involves a quadratic band along with a flat band for the $\Gamma$X direction. Fig. 2(b) is a zoomed-in view of the semi-Dirac dispersion near the $\Gamma$ point (marked by a red rectangle in Fig. 2(a)). Because the second dispersion term of the second band near the $\Gamma$ point is small, it is partly degenerate with the flat band. The formation of the flat band results in the disappearance of the dispersion in the $\Gamma$X direction, implying that the group velocity of the system is zero. For a given Bloch wave [45], the averaged velocity $\mathbf {V}_e$ of the energy flow (the average is taken upon a lattice cell) is equal to the group velocity $\mathbf {V}_g$
Fig. 2. (a) Band structures of Hermitian PC $(\gamma =0)$ computed with COMSOL. (b) The zoomed-in view of a part of the second and third bands in (a) near the $\Gamma$ point (marked by the red rectangle). (c,d) Electric field $E_z$ of bands 2 and 3, respectively. The black arrows indicate the in-plane magnetic field vector, whose magnitude is proportional to the length of the arrows.
3. Two band non-Hermitian Hamiltonian model
In principle, the eigenfrequencies and eigenfields of non-Hermitian PC can be obtained by the Helmholtz equation. However, using the Bloch states of the Hermitian system as a basis, we can construct a non-Hermitian Hamiltonian [47], and effective eigenequation can be obtained (Section S1 in Supplement 1)
It can be seen that the imaginary part of the non-Hermitian Hamiltonian $\boldsymbol {H}$ comes from the component $\boldsymbol {H}_1$, in Eq. (6),
Here, $\tau _m\left (m=m^{\prime }\right )$ is defined as the non-Hermitian strength of the $m$ state, and $\tau _{m^{\prime } m}\left (m \neq m^{\prime }\right )$ represents the coupling between two eigenmodes. Note that the integration of the eigenmode of region C without non-Hermitian perturbations is not involved in Eq. (7). However, region C should be considered in the computation of the orthonormalized eigenmodes. Substituting Eq. (5) into Eq. (4), we obtain
According to the Eqs. (2)-(8), the non-Hermitian Hamiltonian model can be deduced by the eigenfrequencies and the eigenfields of the Hermitian system $(\gamma =0)$.
Considering only the contribution from the second and third bands near the semi-Dirac point, the subscript $m$ and $m^{\prime }$ in Eqs. (3)-(8) can be represented by the numbers 2 or 3. For convenience, we can omit the subscript $k$ for simplicity and rewrite Eq. (2) as $\boldsymbol {H} \cdot \psi =W_n \psi$, where $W_n=\left (\omega _n / c\right )^2$. Thus, the non-Hermitian Hamiltonian model can be written as a $2 \times 2$ matrix
Then the eigenvalues of the non-Hermitian Hamiltonian model can be given by
In Eq. (11), the eigenvalues $W_{2,3}$ are real at $\tau _{2,3}=0$. To achieve the non-Hermitian strength is zero, Eq. (7) can be written as
So, we need a specific value of the loss-gain ratio $\ell _r$ to satisfy Eq. (12), and the value of the loss-gain ratio $\ell _r$ can be solved by
To determine the value of $\ell _r$ in Eq. (13), the PC system must also satisfy the following conditions
Amplitude distribution of eigenmodes in different regions (A1, A2 and B) as shown in Fig. 3(a). Due to the eigenmode symmetry near the $\Gamma$ point, the eigenmode profiles of the A1 and A2 regions are similar, which behaves as $F_{\mathrm {A} 1, m m}=F_{\mathrm {A} 2, m m}$. In addition, $F_{\Omega, m m}$ does not change with the Bloch wave vectors in the $\Gamma$X direction because of the formation of the zero-dispersion bands. Note that the numerical difference between $F_{\Omega, 2 2}$ and $F_{\Omega, 3 3}$ gradually decreases along the $\Gamma$Y direction, and achieves $F_{\Omega, 22}=F_{\Omega, 33}$ when $k_y \leq -0.02 \pi / a$. Then Eq. (13) can be used to determine the loss-gain ratio $\ell _r$, whose value is very close to a constant (about $-1.6269$) shown in Fig. 3(b). Substituting $\ell _r=-1.6269$ into Eq. (12), we can obtain the non-Hermitian strength of the second and third bands shown in Fig. 3(c). It can be seen that the values of $\tau _m$ tend to zero when $k_y \leq -0.02 \pi / a$, which means that the non-Hermitian Hamiltonian $\boldsymbol {H}$ can have real eigenvalues. However, the values of $\tau _m$ are non-zero when $k_y>-0.02 \pi / a$, which leads the eigenvalues to have imaginary parts. Here, the yellow dot corresponds to a phase transition point in the eigenvalue spectrum, called an exceptional point (EP). According to Eq. (11), the EP emerges at $\Delta =0$ when the values of $\tau _m$ are zero. Near the $\Gamma$ point, the $W_{2,3}^{(0)}$ can be represented by a small $k_y$
where $C_g=2 \omega _0 v_g / c^2$, $W_0=\left (\omega _0 / c\right )^2$, and $\omega _0$ is the eigenfrequency at the semi-Dirac point, and $v_g$ is the group velocity of the linear dispersion of the Hermitian PC. Substituting Eq. (15) into Eq. (11), the positions of the EP can be given by where the value of $\kappa$ is shown in Fig. 3(d). It can be found that $\kappa$ is almost a constant (about 0.05272) in the $\Gamma$Y direction. According to Eq. (16), the non-Hermitian perturbation $\gamma$ can change the value of $k_p$. EP separates the unbroken $\Delta >0$ and broken $\Delta <0$ phases. When the condition of $\tau _m=0$ is satisfied, the upper bound of $k_p$ is $-0.02 \pi / a$, and the corresponding $\gamma =0.3$. In addition, $\kappa$ is always zero in the $\Gamma$X direction, which means the decoupling of the two eigenmodes at the semi-Dirac point. Combined with Fig. 2(b), the linear term disappears near the $\Gamma$ point as the coupling integral $\kappa$ disappears [48]. It can be deduced that the disappearance of the linear term also has an important impact on the formation of EP. For the $\Gamma$X direction, considering that the eigenfrequency of the flat bands is $\omega _0$ in Fig. 2(b), and $\tau _2=-\tau _3$ in Fig. 3(c). The eigenvalues of Eq. (11) can be written asEquation (17) shows that the eigenvalues will form a complex conjugate pair, which leads to the absence of the real spectrum of non-Hermitian PC in the $\Gamma$X direction. Now, we can verify the accuracy of the non-Hermitian model. As shown in Fig. 4(a) and 4(b), we show the band structures along the Y-$\Gamma$-X direction. Black lines show the theoretical results, and circles show the simulation results. The bands calculated using the non-Hermitian Hamiltonian model Eq. (11) (lines) agree well with the result of simulations (circles). Additionally, we show the shape of these EPs in Bloch k-space in Fig. 4(c) and 4(d). The orange surfaces (theoretical results) and dots (simulation results) indicate that these EPs form two symmetrical straight lines near the center of the Brillouin zone. The eigenfrequencies inside the EPs line form complex-conjugate pairs, and outside the EPs line, the system can obtain the real spectra. Therefore, our non-Hermitian PC will exhibit different scattering behaviors for wave vectors in $\Gamma$X and $\Gamma$Y directions.
Fig. 3. (a) Eigenmode profiles of the second and third bands. (b) The loss-gain ratio $\ell _r$, the negative values indicate that opposite non-Hermitian perturbations are added to the red and blue regions (one region is a gain medium and the other region is a loss medium). (c) Non-Hermitian strength $\tau _m$ of the second and third bands when the loss-gain ratio $\ell _r=-1.6269$. (d) The coupling between two eigenmodes in the second and third bands. The yellow dots and vertical dashed lines correspond to $k_y=-0.02 \pi / a$ in (a),(c), and (d).
Fig. 4. (a,b) The real and imaginary parts of the band structures along the Y-$\Gamma$-X direction. (c,d) The real parts and imaginary parts of the band structures near the middle of the Brillouin zone. Circles and dots are calculated by COMSOL, and black lines and orange surfaces are calculated using the non-Hermitian Hamiltonian model Eq. (11). The parameters used are $\gamma =0.3$, $\ell _r=-1.6269$, and $\kappa =0.05272$.
4. CPA and laser in the $\Gamma$Y direction
In this section, we will describe the behavior of waves in the PCs by EMT. The boundary field averaging method can calculate the effective parameters of PC. When the PC is incident by TM polarized waves along the $\Gamma$Y direction, the surface impedance of the PC is defined as
Because the frequency of the incident wave in the actual experiment is a real number, the effective parameters should be calculated as a function of the real frequency. We should use the eigenfields with complex Bloch $k$ but real-valued frequencies to retrieve the effective parameters. Here, $E_z$ and $H_x$ of eigenfields are obtained by solving the complex Bloch $k$ bands. We can compute the complex Bloch $k$ bands and eigenfields via the weak-form PDE module of COMSOL Multiphysics software package (Section S2 in Supplement 1). The obtained band dispersion is shown in Fig. 5(a), we can see that there is a $k_y$ gap within the range $\left |k_y\right | a<0.02 \pi$ along the $\Gamma$Y direction, which corresponds to the broken phase in Fig. 4(a) and 4(b). Furthermore, the imaginary parts of the wave vector $k_y$ are almost zero, and the effective refractive index can be given by
where $k_0$ is the wave vector in air. Therefore, the $n_{e y}$ of the non-Hermitian PC is a real number. Fig. 5(b) shows the surface impedance corresponding to positive and negative wave vectors. According to Eq. (19), the effective parameter of the non-Hermitian PC can be calculated. As shown in Fig. 5(c) and 5(d), real parts of the effective parameters $\varepsilon _{e z}$ and $\mu _{e x}$ approach zero simultaneously at the EP frequency ($\omega =0.424(2 \pi c / a)$), while both possess non-zero imaginary parts. Because the $\varepsilon _{e z}$ and $\mu _{e x}$ are purely imaginary numbers, this non-Hermitian PC behaves like the homogeneous CCM, which the CPA or laser can be achieved at EP frequency. As shown in Fig. 6(a), We consider two counterpropagating plane waves with the same frequency $\omega$ incident on a CCM slab (orange region) with thickness $d$ embedded in air. The incident plane waves are $E_z$ polarized, and the wave vector is along the $y$ direction. Then, the electric fields in the background (air) can be expressed as [34]At $\omega =0.424(2 \pi c / a)$, the variation of the absolute values of the eigenvalues $\left |\lambda _{\pm }\right |$ with the length $d(a)$ shown in Fig. 6(b). It can be seen that $\left |\lambda _+\right |$ is close to zero when $d=12a$ and $\left |\lambda _{-}\right |$ reaches the maximum value at $d=38a$. Therefore, CPA can be achieved by setting $a_1=a_2$ (in-phase excitation), and the CCM slab has 12 layers, while the laser can be realized by setting $a_1=-a_2$ (out-of-phase excitation) and the CCM slab has 38 layers. Fig. 6(c) shows the transmission and reflection of a plane wave at EP frequency along the $y$-direction. The PC slab only contains 10 unit cells in the $x$ direction, but the number of unit cells is changed in the $y$ direction. Note that we use open boundary condition (OBC) to simulate the leaky wave along the $x$ direction. As shown in Fig. 6(d), it can be seen that the transmission $T_1$ is at least 100 times higher than the transmission $T_2$ and $T_3$, which indicates that the leaky wave in the $x$ direction is feeble. Besides, the values of reflection and transmission are almost the same at $d=12a$. Substituting $\left |\lambda _+\right |=0$ and $a_2=0$ into Eq. (25), we can obtain $R=T_1=\left |\lambda _{-}\right | / 2$. The values of reflection and transmission will peak because of the divergence of $\left |\lambda _{-}\right |$ at $d=38a$. Because the transmission efficiency of the incident wave varies greatly in different directions, the PC slab can be used as a sensor with angle selection functionality. As shown in Fig. 6(e), the transmission is as high as 130 and then drops to $10 {\% }$ of the peak value when the angle of incidence is only $5^{\circ }$. $T_2$ and $T_3$ are near zero even though the angle of incident wave changes continuously, meaning that the PC prevents the spread of wave in the $x$ direction.
The effect of the CPA is shown in Fig. 7(a), which shows the field distribution of the electromagnetic wave $E_z$ when two coherent counter-propagating waves are normally incident on the PC slab with a phase difference $\Delta \varphi =0$. The Poynting vectors are shown by the red arrows pointing inward, indicating that the incident waves are absorbed by the PC slab. Besides, the electric field outside the OBC is close to zero, meaning that the leaky wave in the $x$ direction can be ignored. We also plot the amplitudes (normalized to the incident wave) of the electric field shown in Fig. 7(b). Obviously, for the CPA case, $\left |E_z\right |$ outside the PC slab approaches $\left |E_0\right |$, meaning that the reflection at the boundary is tiny and almost all the incident energy is absorbed. On the other hand, the power flows are entirely reversed at $\Delta \varphi =\pi$. As shown in Fig. 8(a), the Poynting vectors are now all pointing outwards, indicating the energy transfer from the PC slab to the air background. The normalized amplitude of the electric field in the background is shown in Fig. 8(b). The fields outside the PC slab are significantly amplified, which are about ten times the incident waves, corresponding to a laser phenomenon. However, the amplitude of the field outside the OBC is still extremely low, meaning that the waves are not enhanced for the $x$ direction. Therefore, the waves will exhibit independent propagation behavior in different directions.
Fig. 5. (a) The complex Bloch $k$ bands as functions of real frequencies. (b) The real and imaginary parts of the effective impedance $Z_{e y}$. Bands with positive and negative wave vectors are represented by squares and triangles, respectively. (c,d) The real and the imaginary parts of the effective parameters $\varepsilon _{e z}$ and $\mu _{e x}$, note that effective parameters obtained from the two Bloch $k$ bands are the same.
Fig. 6. (a) Schematic graph of the CCM, which is homogeneous and symmetric about $x = 0$, the effective parameters of CCM are shown in Fig. 5. (b) Values of $\left |\lambda _+\right |$ (solid red dots) and $\log \left (\left |\lambda _{-}\right |\right )$ (solid blue dots) as functions of the length of the CCM slab. (c) Schematic graph of the transmission and reflection of plane waves normally incident on the width-finite PC slab. The black dashed lines represent the open boundary condition, and the solid green lines represent the ports in (a,c). (d) The reflection and transmission for the PC slab with different lengths. (e) Transmission of the PC slab with oblique incidence and the length of the PC slab consists of 38 unit cells. The plane wave with frequency $\omega =0.424(2 \pi c / a)$.
Fig. 7. CPA. (a) The field distributions of the electric field $E_z$ in the PC slab. Poynting vectors (red arrows) for two counterpropagating plane waves (black arrows) normally incident on the PC slab in the case of $\Delta \varphi =0$, and the PC slab consists of 12 layers of the unit cell. (b) The normalized amplitude of the electric field $\left |E_z / E_0\right |$ for the case of (a), and the red dashed lines denote the boundaries of the PC slab. The two plane waves with frequency $\omega =0.424(2 \pi c / a)$.
Fig. 8. Laser. (a) The field distributions of electric field $E_z$ in the PC slab. Poynting vectors (red arrows) for two counterpropagating plane waves (black arrows) normally incident on the PC slab in the case of $\Delta \varphi =\pi$, and the PC slab consists of 38 layers of the unit cell. (b) The normalized amplitude of the electric field $\left |E_z / E_0\right |$ for the case of (a), and the red dashed lines denote the boundaries of the PC slab. The two plane waves with frequency $\omega =0.424(2 \pi c / a)$.
5. Formation of evanescent waves for the $\Gamma$X direction
We have shown in the previous subsection that the system can have are real spectrum in the $\Gamma$Y direction, and the PC behaves as a CCM with a real refractive index, which can realize CPA and laser. However, the non-Hermitian Hamiltonian model shows that the eigenvalues have imaginary parts for the $\Gamma$X direction, implying that the effective refractive index will be complex. In the actual experiments, the frequencies of the incident wave are real, so we need to establish the EMT to study the scattering properties of the wave in the $\Gamma$X direction. The effective parameters for the $\Gamma$X direction can be expressed as
Fig. 9. (a) The real parts of the complex Bloch $k$ bands as functions of real frequencies. (b) The imaginary parts of the complex Bloch $k$ bands as functions of real frequencies. Bands with positive and negative wave vectors are represented by squares and triangles, respectively.
Fig. 10(a) and Fig. 10(b) shows the effective parameters $\varepsilon _{e z}$ and $\mu _{e y}$. It clearly shows that the real parts of $\varepsilon _{e z}$ and $\mu _{e y}$ do not pass through zero. The imaginary parts of $\mu _{e y}$ are near zero, and only the imaginary parts of $\varepsilon _{e z}$ are non-zero. At the EP frequency, the imaginary part of $n_{e x}$ has a maximum value when the imaginary parts of $\varepsilon _{e z}$ reach a peak. To analyze the scattering properties of PC, we assume that a wave is normally incident along the $\Gamma$X direction to the PC with different lengths $d$. Substituting the effective parameters $\varepsilon _{e z}$ and $\mu _{e y}$ into Eq. (24), the transmission spectrum can be calculated by EMT. As shown in Fig. 10(c), we give the results of the COMSOL simulation and EMT. The results are well consistent, confirming the accuracy of the effective parameters shown in Fig. 10(a) and Fig. 10(b). The transmission decreases with the increase of length $d$, indicating that the wave attenuates continuously with the propagation distance. When the PC length $d$ is constant, the transmittance is related to the imaginary part of $n_{e x}$, whose values determine the attenuation strength of the electric field. At the EP frequency, the transmission reaches a minimum, corresponding to the largest value of the imaginary part of $n_{e x}$. But away from the EP frequency, the transmission gradually increases due to the decrease of the imaginary part of $n_{e x}$.
Fig. 10. (a,b) The real and the imaginary parts of the effective parameters $\varepsilon _{e x}$ and $\mu _{e x}$, note that effective parameters obtained from the two Bloch $k$ bands are the same. (c) The transmission spectrums of PC with different lengths are calculated using COMSOL simulation and EMT, respectively. The incident plane waves are $E_z$ polarized and come from the $\Gamma$X direction.
In Fig. 11(a), we show that a plane wave with frequency $0.424(2 \pi c / a)$ is normally incident on the PC slab, which consists of 20 layers of the unit cell. It can be found that the wave decays rapidly in the PCs, and the amplitude of the electric field of the outgoing wave is close to zero. Fig. 11(b) shows the electric field (normalized to the incident wave) in the PCs. Due to the lack of transverse modes that support wave propagation in the $x$ direction, there are some reflected waves at the boundary, which leads to $E_z$ being smaller than $E_0$ at $x<0$. In addition, the weakening of the electric field is noticeable. The amplitude of the electric field is almost zero at $x>10a$. The above results confirm the formation of evanescent waves along the $\Gamma$X direction.
Fig. 11. (a) The field distributions of the electric field $E_z$ in the PC slab. (b) The normalized amplitude of electric field $\left |E_z / E_0\right |$ for the case of (a). The incident plane waves are $E_z$ polarized with frequency $\omega =0.424(2 \pi c / a)$, and the wave vector is along the $x$ direction. As for the $y$ direction, we use the periodic boundaries to simulate an infinite PC slab.
6. Discussion
In summary, we implement a non-Hermitian PC with anisotropic emission by introducing gains and losses. The effective Hamiltonian model shows that the EP point and real spectrum appear in the $\Gamma$Y direction because of the near-zero non-Hermitian intensity $\tau _m$. The non-zero $\tau _m$ leads to the absence of the EP point and real spectrum for the $\Gamma$X direction. CMT indicates that the PC behaves as a CCM with the real refractive index for the $\Gamma$Y direction, supporting the CPA and laser effects. However, the effective refractive index is a complex number with a large imaginary part in the $\Gamma$X direction, corresponding to the formation of the evanescent waves. Combined with boundary reflection caused by the absence of transverse modes, we achieve the CPA and laser in the directional emission of the $\Gamma$Y. The outgoing waves in the undesirable direction are extremely weak even if the boundaries of the PC are open. Therefore, this non-Hermitian PC with anisotropic emissions will reduce loss and electromagnetic interference caused by leaky waves. Because our PC has significant transmission differences in different directions, it can have some typical applications [49,50], such as significant amplification of the signal and collimation and angle sensors.
Funding
National Natural Science Foundation of China (NSFC42274206, NSFC41974195); Hubei University (X202210512039, 030-017643); Open Fund of China Ship Development and Design Centre (XM0120190196); Hubei Province Innovation Special Fund (2023BAB061); Science and Technology Department of Hubei Province (2022CFB553, 2022CFA012); Hongque Innovation Center (HQ202104001).
Disclosures
The authors declare no conflicts of interest.
Author contribution. Z. L. and H. L. proposed the idea. Z. L. performed the calculation, produced all the figures, and wrote the manuscript draft. R. T. and H. C. contributed to the calculation tools. J. T. analyzed the data. R. Z. and J. J. lead the project and revised the whole manuscript thoroughly. Y. L. revised the paper.
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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