1. Introduction

Non-Hermitian optics [1,2] has been widely studied in recent years since the discovery and realization of many significant wave phenomena [3–18], known as unidirectional invisibility [2–4], the coexistence of laser and anti-laser [5–7], and exceptional points [8–10]. These interesting wave effects have been investigated in the optical framework of parity-time (PT) symmetry, which obeys a special inversion symmetry including both space and time. Originally, the concept of PT symmetry was proposed in quantum physics [19,20] and later perfectly translated to classic wave domain, such as optics [21,22] and acoustics [23]. In optics, PT symmetry is implemented by spatially modulating loss and gain that satisfy a complex refractive index distribution $n(x) = {n^\ast }( - x)$, that is, a balanced loss and gain profile is required. Different from the loss and gain of PT symmetric materials individually distributed in different media, conjugate metamaterials (CMs) [24–33] involve simultaneously both loss and gain elements in a single medium and their relative permittivity and permeability are complex conjugates of each other in their phases, given as $\varepsilon ^{\prime} = \varepsilon {e^{ - i\alpha }},\mu ^{\prime} = \mu {e^{i\alpha }}$, where $\varepsilon$ and $\mu$ are positive numbers, $\alpha \in [0,2\pi ]$ is the phase factor of CMs. A typical feature of CMs is to support unattenuated propagation of electromagnetic waves [24], as they have real refractive indices. Such CM slab with $\alpha \to \pi$ could be served as a subwavelength-resolution lens [26]. Especially, more attention has been paid to CMs with $\alpha \textrm{ = }\pi \textrm{/2}$ or $\alpha \textrm{ = 3}\pi \textrm{/2}$, which are called purely imaginary metamaterials (PIMs). In addition to PT symmetric media, PIM has been demonstrated as an alternative material platform to achieve the coexistence of laser and anti-laser (coherent perfect absorption (CPA)) in the planar slab [27–30] and cylindrical structure [31]. Recently, the concept and phenomena of PIMs were extended to acoustics [32] and similar wave phenomena were realized. In previous works, PIMs were mainly used to study laser and CPA [27–30] and perfect absorption was simply discussed in a lower-index PIM slab, but the underlying physical explanation for laser and perfect absorption has not been clearly revealed [29]. In addition, a single linearly polarized wave was considered to study the wave modes in a PIM slab, which would miss some other unknown wave phenomena. Although great efforts have been made in the study of higher-index PIMs or the PIMs supporting propagation waves [24–33], but the fundamental wave modes as well as their inherent connection for different polarizations have not been deeply investigated in a lower-index PIM slab, especially for the incidence beyond the critical angle.

In this work, we will study a lower-index PIM ($n = \sqrt {\varepsilon ^{\prime}\mu ^{\prime}} \in (0,1)$) slab by considering incident wave from air in a new way of directly analyzing the reflection and transmission coefficients of both linear polarizations. We find that the relative impedance Z at the interface of PIM and air plays a key role in the study of a PIM slab, which can be used not only to explain the physical mechanism of laser mode and perfect absorption in previous works, but also to study new findings for the incidence beyond the critical angle. For simplicity, $\varepsilon = \mu = n$ is firstly set for PIMs to study wave scattering behaviors and the PIM with $\varepsilon \ne \mu$ is discussed later. For the incidence within the critical angle, i.e., the propagation waves in the PIMs, we find that the phase compensation from the PIM-air interface is responsible for the generation of the laser modes, which can explain the laser modes in previous works [27–30]. While it is the amplitude compensation mechanism responsible for the laser modes occurring at the incidence beyond the critical angle. Correspondingly, as the time reversal counterpart of laser modes, CPA modes are also obtained by considering two coherent incidences beyond the critical angle. In addition, perfect attenuator and perfect amplifier, i.e., the reflectionless wave modes, are found for the incidence beyond the critical angle. We find the perfect attenuator is caused by the single positive wave component in the PIM slab that can exponentially reduce the incident wave without any reflection, which can explain the reason of perfect absorption in previous work [29]. While the perfect amplifier is induced by the negative wave component in the PIM slab, which can exponentially enhance the incident wave without any reflection. Note that these wave modes, including laser mode, CPA mode, perfect amplifier and perfect attenuator, don’t indicate eigenstates of a PIM slab, but are extraordinary wave scattering phenomena resulting from special types of solutions to boundary conditions of a PIM slab. Furthermore, we clearly give out the relationship between these extraordinary wave modes within/beyond the critical angle and incident polarizations, and discuss the influence of thickness and material parameters of the PIM slab on these extraordinary wave modes beyond the critical angle, with more optical behaviors revealed.

2. Extraordinary wave modes in a lower-index PIM slab

Let us consider a PIM slab with thickness of d in air, and the parameters of PIMs are given as $\varepsilon ^{\prime} = \varepsilon i,\mu ^{\prime} ={-} \mu i(\varepsilon = \mu > 0)$, i.e., CMs with $\alpha \textrm{ = 3}\pi \textrm{/2}$ for discussion (similar results can also be achieved for CMs with $\alpha \textrm{ = }\pi \textrm{/2}$). When the waves with transverse electric (TE) polarization (the electric field is along the z direction) or transverse magnetic (TM) polarization (the magnetic field is along the z direction) are incident on the PIM slab, the total electric (magnetic) field in the incident area ($x < 0$) could be expressed in terms of incident and reflected waves, i.e., ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } _1} = ({e^{i{k_x}x}} + r{e^{ - i{k_x}x}}){e^{i\beta y}}\hat{z}$, where ${k_x} = {k_0}\cos \theta$ and $\beta = {k_0}\sin \theta$ are the wavevectors along the x and y directions, ${k_0} = 2\pi /\lambda$ is the wavevector in air, $\theta$ is the incident angle, r is the reflection coefficient. In the PIM slab ($0 \le x \le d$), the total field is composed of forward and backward waves, i.e., ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } _2} = ({a_p}{e^{i{{k^{\prime}}_x}x + i\beta y}} + {a_n}{e^{ - i{{k^{\prime}}_x}x + i\beta y}})\hat{z}$, where ${k^{\prime}_x} = {k_0}\sqrt {\varepsilon ^{\prime}\mu ^{\prime} - {{\sin }^2}\theta }$ is the wavevector in the PIM along x direction, ${a_p}$ and ${a_n}$ are the corresponding coefficients of forward (positive) and backward (negative) wave components. In the transmitted side ($x > d$), the transmitted wave is ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } _3} = t{e^{i{k_x}(x - d) + i\beta y}}\hat{z}$. By using Maxwell equations to get the tangential magnetic (electric) field and matching the boundary conditions at the interfaces of $x = 0$ and $x = d$, these unknown coefficients are calculated as,

In the above equations, $\varphi = {k^{\prime}_x}d$ is the propagation phase in the slab, Z is the relative impedance of the PIM-air interface and it is ${Z_E} = {k_x}\mu ^{\prime}/{k^{\prime}_x}$ (${Z_M} = {k_x}\varepsilon ^{\prime}/{k^{\prime}_x}$) for TE (TM) polarization. For the incidence within the critical angle ($\sqrt {\varepsilon ^{\prime}\mu ^{\prime}} > \textrm{sin} \theta$), $Z$ is a purely imaginary number, and it is a real number for the incidence beyond the critical angle ($\sqrt {\varepsilon ^{\prime}\mu ^{\prime}} \le \textrm{sin} \theta$).

By carefully observing the above equations, we find that when the denominator of the reflection (transmission) coefficient is zero, then $r \to \infty$ and $t \to \infty$, implying that laser mode can happen. The resonant condition of the laser mode is easily obtained and simplified as,

 

Fig. 1. Schematic diagrams of extraordinary wave modes in a PIM slab. (a) laser mode via phase compensation within the critical angle. (b) laser mode via amplitude compensation beyond the critical angle. (c) perfect amplifier in a given PIM beyond the critical angle, which works for TE polarization. (d) perfect attenuator in a given PIM beyond the critical angle, which works for TM polarization.

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Compared with the traditional resonant cavities, e.g., planar mirror resonant cavities, where the lasing condition is required by the phase change of a round trip equal to integer times of $2\pi$, i.e., $\Delta \varphi \textrm{ = 2}\varphi \textrm{ = }2\pi q$, the working mechanism of the laser mode in the PIM is homologous. Interestingly, the laser mode can still be realized beyond the critical angle. In such a condition, ${k^{\prime}_x} = i\alpha (\alpha = {k_0}\sqrt {{{\textrm{sin} }^2}\theta - \varepsilon ^{\prime}\mu ^{\prime}} )$ turns into a purely imaginary number, implying that when the wave $\psi$ propagates from the left boundary of the PIM slab to the right one, it undergoes an amplitude decay, yet with the phase unchanged, i.e., $\psi ^{\prime} = \psi \tau$ with $\tau = \textrm{exp} ( - \alpha d)$, as schematically shown in Fig. 1(b). Fortunately, an enhanced reflection with $|\Re |> 1$ could occur at the PIM-air interface, which enables amplitude compensation for the decayed wave $\psi ^{\prime}$ (see the red circle with small size). Similarly, if the decayed wave can be restored to its original state ($\Re \psi ^{\prime} = \psi$) or the opposite state ($\Re \psi ^{\prime} ={-} \psi$), and then the steady wave oscillation can happen in two different cases (see the yellow lines in Fig. 1(b)), which lead to symmetric and anti-symmetric laser modes. The dispersion relationships of the laser modes in the PIM beyond the critical angle are given as,

In the considered PIM ($\varepsilon ^{\prime} = \varepsilon i,\mu ^{\prime} ={-} \mu i$), ${Z_E} = {k_x}\mu ^{\prime}/{k^{\prime}_x}$ is a negative real number, which can realize a laser mode of TE polarization in a PIM slab beyond the critical angle, while it is impossible to achieve a laser mode of TM polarization as ${Z_M} = {k_x}\varepsilon ^{\prime}/{k^{\prime}_x}$ is a positive real number. As the time inversion of laser modes, CPA modes could be deduced from the transmission and reflection coefficients, i.e., $t \pm r = 0$. Similarly, the general dispersion relationships of the CPA modes are calculated as,

For the incidences beyond the critical angle, the conditions of the CPA modes are given as,

In addition to the laser and CPA modes, the reflectionless effect can happen in the PIMs for the incidence beyond the critical angle. By observing Eq. (1), obviously, $r = 0$ happens at $Z ={\pm} 1$, which is realized at the incidence beyond the critical angle. For $Z ={-} 1$, which is only realized for TE polarization, we find that ${a_p}\textrm{ = }0$ and ${a_n}\textrm{ = }1$ (see Eq. (3) and Eq. (4)), implying that only the negative wave component (backward wave) is supported in the PIM slab (${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } _2} = {a_n}{e^{ - i{{k^{\prime}}_x}x + i\beta y}}\hat{z}$ with ${k^{\prime}_x} = i\alpha$) to perfectly magnify the incident wave. As a result, an exponential enhancement can occur in the PIM slab and the transmitted wave with an enhanced amplitude is $t = \textrm{exp} (\alpha d) = {\tau ^{ - 1}}$. We call this reflectionless amplifier mode as perfect amplifier and the perfect amplifier of TE polarization can be realized at $Z ={-} 1$, as schematically indicated in Fig. 1(c). For $Z = 1$ (it is only available for TM polarization), the coefficients are ${a_p} = 1$, ${a_n} = 0$ and the internal field is ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } _2} = {a_p}{e^{i{{k^{\prime}}_x}x + i\beta y}}\hat{z}$ with ${k^{\prime}_x} = i\alpha$. Therefore, only the positive wave component (forward wave) is supported in the PIM slab, which can perfectly weaken the incident wave, and the transmission amplitude of the incident wave through the PIM slab is $t = \textrm{exp} ( - \alpha d) = \tau$. We call this reflectionless attenuator mode as perfect attenuator, which only happens for TM polarization at $Z = 1$, as schematically shown in Fig. 1(d).

3. Analytical calculation and numerical demonstration

Based on the above theoretical discussion, we show the dispersion relationships of laser modes, perfect amplifier and perfect attenuator for both TE and TM polarizations, which are respectively shown in Fig. 2(a) and Fig. 2(b), where the thickness of the PIM slab is $d = \lambda$. The laser modes for TE polarization can both exist within or beyond the critical angle (the black dashed curve in Fig. 2(a)). In particular, the perfect amplifier happens between the odd laser mode and even laser mode. For TM polarization, the laser modes can only happen within the critical angle and only the perfect attenuator can happen beyond the critical angle (see Fig. 2(b)). As the PIM is set with $\varepsilon = \mu$, the perfect amplifier at ${Z_E} ={-} 1$ and perfect attenuator at ${Z_M} = 1$ can happen at the same incident angle. Thus, the dispersion relationship of the perfect amplifier of TE polarization is identical with that of the perfect attenuator of TM polarization. In addition, the corresponding CPA modes of TE and TM polarizations are displayed in Fig. 2(c) and Fig. 2(d), where we see that there are no CPA modes beyond the critical angle for TE polarization and both odd and even CPA modes are obtained for TM polarization. Similarly, owing to the PIM with $\varepsilon = \mu$, the dispersion relationships of CPA modes of TM polarization are identical with these of laser modes of TE polarization (see Eq. (6) and Eq. (8)). In fact, the dispersion relationships of laser and CPA modes in Fig. 2 could be also clearly observed by analyzing trajectories of poles and zeros of the S-matrix of the PIM slab [9,28], which is a more general way of studying laser and CPA modes in a non-Hermitian optical system.

 

Fig. 2. Dispersion relationships (n vs $\theta$) of extraordinary wave modes in a PIM slab ($d = \lambda$). (a) laser modes and perfect amplifier of TE polarization. (b) laser modes and perfect attenuator of TM polarization. (c) CPA modes of TE polarization. (d) CPA modes of TM polarization.

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To further demonstrate these wave modes in the PIMs beyond the critical angle, numerical simulations using COMSOL MULTIPHYSICS are performed in Fig. 3. We take the case of a PIM slab with $\varepsilon = \mu \textrm{ = }0.5$ and $d = \lambda$ to confirm these extraordinary wave modes. For TE polarization, they are the odd laser mode, the even laser mode and the perfect amplifier, which approximately happen at the incident angles of $\theta \textrm{ = 35}\textrm{.6}^\circ$, $\theta \textrm{ = 41}\textrm{.3}^\circ$ and $\theta \textrm{ = 39}\textrm{.2}^\circ$, respectively, as indicated in Fig. 2(a). The corresponding simulated results are displayed in Fig. 3(a), Fig. 3(b) and Fig. 3(c), where the incident wave with electric amplitude of 1.0 V/m is incident from the left side. Clearly, by observing the field patterns in Fig. 3(a) and Fig. 3(b), giant outgoing waves (see the color bar) are seen in the incident and transmitted sides, as also indicated by the power fluxes (see the black arrows). In particular, the symmetric and anti-symmetric phase distributions in the PIM slab indicate that they are even and odd modes. Figure 3(c) shows the simulated result of the perfect amplifier and an enhanced amplitude of 11.4 appears in the transmitted side, which is consistent with the analytical result of ${\tau ^{ - 1}} = 11.4$. While for TM polarization, the extraordinary wave modes in the PIM slab beyond the critical angle are the odd CPA, the even CPA and the perfect attenuator, which also occur at $\theta \textrm{ = 35}\textrm{.6}^\circ$, $\theta \textrm{ = 41}\textrm{.3}^\circ$ and $\theta \textrm{ = 39}\textrm{.2}^\circ$. To achieve odd CPA (see Fig. 3(d)), the two incoming waves must be coherent with anti-symmetric amplitude, e.g., the magnetic fields from the left and right sides are 1.0 A/m and −1.0 A/m, respectively. By observing the field pattern and power fluxes (black arrows), the incoming waves are completely absorbed in the PIM slab, as there are no outgoing waves. Likewise, two incoming waves with symmetric amplitudes (e.g., 1.0 A/m and 1.0 A/m) are used to realize an even CPA, as shown in Fig. 3(e). For the perfect attenuator, the incident wave with magnetic amplitude of 1.0 A/m is absorbed by the PIM slab and the amplitude of the transmitted wave is reduced as $\textrm{0}\textrm{.092}$, which agrees with the analytical result.

 

Fig. 3. Simulated electric-field patterns (E_Z) for the odd laser mode (a), even laser mode (b) and perfect amplifier (c). Simulated magnetic-field patterns (H_z) for the odd CPA mode (d), even CPA mode (e) and perfect attenuator (f). The arrows in the insets denote the directions of energy flow. In all the cases, $n = 0.5$, $d = \lambda = 1$.

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4. General feature of extraordinary wave modes

Based on theoretical calculation and numerical simulation, we have analyzed and demonstrated the extraordinary wave modes in the PIMs beyond the critical angle. These modes and their interrelationships have intriguing features, which could be used to design some optical devices. Firstly, we find the most suitable parameters of the PIMs to obtain maximum efficiency (maximum value of $\alpha = {k_0}\sqrt {{{\textrm{sin} }^2}\theta - {n^2}}$) of perfect amplifier/attenuator, e.g., corresponding to maximum transmission of perfect amplifier in a PIM slab with a given thickness. Analytical and numerical calculations show the relationship between $\ln (|t |)$, $|r |$ and the incident angle for the perfect amplifier as shown in Fig. 4(a), where the thickness of the PIM is $d\textrm{ = 2}\lambda$. Both results consistently reveal the reflectionless effect and amplified transmission of the perfect amplifier, and the maximum transmission of $t = 182.16$ occurs at $49.94^\circ$, corresponding to the PIM with $n = 0.6436$. This value of PIM is also the best parameter to obtain perfect attenuator with maximum absorption efficiency. In the following, we will use this PIM to reveal some features of these wave modes (i.e., laser mode, perfect amplifier and perfect attenuator) by only considering single side incidence. Figure 4(b) shows the dispersion relationships of perfect amplifier, laser modes and the PIM thickness. We can see that with the increase of d, the position of the perfect amplifier is unchanged, as it only depends on the PIM parameter. While the positions of odd and even laser modes gradually trend to the position of the perfect amplifier, i.e., $\theta \textrm{ = }49.94^\circ$. This is explained from Eq. (7), when d is larger, the left side of the equations tends to −1. As a result, the resonant angles of both laser modes will be close to that of the perfect amplifier. As no reflection and extreme high transmission can respectively happen for perfect amplifier and laser modes, this feature could be potentially used for designing a highly sensitive angular sensor. The corresponding reflection on a logarithmic scale versus the incident angle for the PIM slab with different thickness is shown in Fig. 4(c), where the reflection dip is unchanged with the thickness. Extreme high reflection can happen in two different angles, corresponding to the odd and even laser modes, and they are gradually close to the reflection dip with the increase of the thickness. For $d = 1.5\lambda$, the incident angles of the reflection peaks are $49.31^\circ$ (odd) and $50.46^\circ$ (even). For $d = 1.75\lambda$ and $d = 2\lambda$, the angles are $49.62^\circ$ (odd), $50.22^\circ$ (even) and $49.78^\circ$ (odd), $50.09^\circ$ (even), respectively. We define the quality factor of $\gamma = {\theta _2}/({\theta _3} - {\theta _1})$ to describe the convergence, where ${\theta _1}$, ${\theta _2}$, ${\theta _3}$ are the incident angles of the odd laser mode, perfect amplifier and even laser mode. For the thicknesses of $d = 1.5\lambda$, $d = 1.75\lambda$ and $d = 2\lambda$, $\gamma$ are $43.43$, $83.23$ and $161.09$ respectively. The $\gamma$ increases with d and can go to infinity. A highly sensitive angular sensor using a PIM slab could be designed, with the sensitivity depending on its thickness.

 

Fig. 4. (a) Relationships between the transmission (on a natural logarithmic scale) and reflection coefficients of the PIM slab and the incident angle of the perfect amplifier. (b) Dispersion relationships ($d$ vs. $\theta$) of the laser modes and perfect amplifier in a PIM slab. (c) Reflection on a logarithmic scale versus incident angle for different thicknesses of the PIM. (d) Relationship between the polarization angle of the transmitted light and the thickness d of the PIM slab. For (b-d), the PIM is $n = 0.6436$.

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Besides, the transmission coefficient of the perfect amplifier (perfect attenuator) increases (reduces) exponentially with the increase of the PIM thickness. Especially, when the slab is thick enough, the transmission of the perfect amplifier becomes extremely high so that we can define the non-reflective amplifier as perfect laser. It is reasonable as the odd and even modes can converge to the perfect amplifier for a thick PIM slab by observing Fig. 4(b). As we discussed in Fig. 2, the CPA mode and perfect attenuator of TM polarization share identical dispersion relationships with these of laser modes and perfect amplifier of TE polarization. Similarly, with the increase of the thickness, the CPA modes in a PIM slab can gradually merge with the perfect attenuator, which indicates that perfect absorption realized by two coherent waves will be eventually achieved by a single-sided incident wave. In other words, the perfect attenuator will turn into perfect absorption by considering a thick PIM slab, as its transmission is extremely low. Therefore, a thick PIM slab can perfectly absorb incident TM wave and perfectly enhance incident TE wave, and similar polarization-dependent effect of perfect absorption and transmission was theoretically and experimentally found in a van der Waals crystal of hexagonal boron nitride [34]. Thanks to the different transmission response on the PIM slab, we can use the perfect amplifier and perfect attenuator to realize the function of a polarizer, i.e., it can convert any incident light into polarized light with polarization state depending on the PIM thickness. As a polarizer, we can list the Jones Matrix of the PIM slab with thickness of d as,

For the left (right) handed circularly polarized light, we set the x-direction as the long axis and the polarization angle is $\phi = {\cos ^{ - 1}}(\frac{{\textrm{exp} (2\alpha d)}}{{\textrm{exp} (4\alpha d) + 1}})$. As shown in Fig. 4(d), the polarization angle is gradually changed with the thickness. For $d = \lambda$, $\phi$ is close to zero, which means a linear polarized light is realized from circularly polarized light through the PIM slab. When the thickness is changed from 0 to $d = \lambda$, the circularly polarized light gradually becomes elliptically polarized light and finally linearly polarized light.

5. Extraordinary wave modes in a PIM slab with $\varepsilon \ne \mu$

Finally, we perform the analytical calculation to address how the parameters ($\varepsilon \ne \mu$) of the PIM affect these extraordinary wave modes as displayed in Fig. 5, where the PIM is considered with $n = \sqrt {\varepsilon \mu } = 0.5$ and its thickness is $d = \lambda$. We show the dispersion relationships of the laser modes of TE wave and the CPA modes of TM wave in Fig. 5(a), in which the incident angle is changed from $\textrm{30}^\circ$ (the critical angle) to $\textrm{80}^\circ$ and $\mu$ varies from 0 to 3. We can find that the laser and CPA modes with the same parity (i.e., odd or even) only happen at the same incident angle for $\mu \textrm{ = }0.5$, i.e., $\varepsilon \textrm{ = }\mu$. While for other values of $\mu$, the laser and CPA modes occur at different angles. Similar results are also found for the perfect amplifier of TE wave and the perfect attenuator of the TM wave (see Fig. 5(b)). In addition, all odd and even laser and CPA modes can happen for $\mu \in (0.37,0.68)$, in particular, the CPA and laser modes with opposite mode parity can simultaneously happen at $\mu \textrm{ = }0.\textrm{42}$ and $\mu \textrm{ = }0.6\textrm{0}$, which happen at the same incident angle of $\theta \textrm{ = }3\textrm{9}\textrm{.2}^\circ$. The condition to achieve CPA and laser modes with opposite mode parity at the same incident angle can be deduced from Eq. (6) and Eq. (8) and given as $\theta = \arcsin \sqrt {2{n^2}/({n^2} + 1)}$. For $\mu \in (0.68,3]$, the odd CPA mode is cut off, while the odd laser mode is cut off for $\mu \in (0,0.37),$, as seen in Fig. 5(a). We perform numerical simulations to demonstrate that the CPA and laser modes with opposite mode parity can appear at the same incident angle (see Fig. 5(c) and Fig. 5(d)), where $\mu \textrm{ = }0.6\textrm{0}$. As predicted from Fig. 5(a), a laser mode is found with odd symmetric field pattern (see Fig. 5(c)) and a CPA with even symmetry is revealed in Fig. 5(d). In addition, the even and odd laser/CPA modes seem to merge at a large angle of incidence. This is due to that the “coth” term in the dispersion relation of the odd mode and the “tanh” term of the even mode tend to be equal when the refractive index is low.

 

Fig. 5. Dispersion relationships ($\mu$ vs. $\theta$) of laser/CPA modes (a) and perfect amplifier/attenuator (b), respectively. Simulated patterns for odd laser mode (c) and even CPA mode (d), where $\varepsilon ^{\prime} = 0.4167i,\mu ^{\prime} ={-} 0.6i$ and $\theta \textrm{ = }3\textrm{9}\textrm{.2}^\circ$. In all the cases, $n = 0.5$ and $d = \lambda = 1$.

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6. Conclusion

In conclusion, we have analytically and numerically demonstrated that extraordinary wave modes including laser modes, CPA modes, perfect amplifier and perfect attenuator can happen in a lower-index PIM slab beyond its critical angle, which is different from total reflection in a traditional medium. Especially, it was revealed that the laser modes and perfect amplifier (CPA modes and perfect attenuator) only work for TE (TM) polarized wave in our studied PIM with $\alpha \textrm{ = 3}\pi \textrm{/2}$, while for the PIM with $\alpha \textrm{ = }\pi \textrm{/2}$, the laser modes and perfect amplifier (CPA modes and perfect attenuator) only work for TM (TE) polarized wave, which can easily be deduced from our theory. Based on the scattering characteristics of these wave modes, we find that the PIM slab could be used to realize highly-sensitive angular sensor and polarizer. In addition, similar results were revealed in a lower-index PIM slab with $\varepsilon \ne \mu$, where we found that the laser modes and CPA modes, as well as perfect amplifier and perfect attenuator, generally occur at different incident angles. Interestingly, the laser and CPA modes with opposite mode parity can happen at the same incident angles. In fact, several theoretical schemes have been proposed to realize a PIM, including PT symmetric metasurfaces [27], core-shell structure with loss and gain media [28] and non-Hermitian photonic crystals [33], and its practical realization is promoting by considering recent advance in non-Hermitian optics [1,7]. All in all, we systematically revealed the fundamental wave modes in a lower-index PIM slab beyond the critical angle, which can enrich and complete the study of PIM.