Electromagnetic parity-time-symmetric (PT-symmetric) systems, comprising a lossy medium and its time-reversed gain counterpart, have attracted significant attention in recent research [1–7]. This interest arises from their potential as experimentally accessible platforms for investigating a variety of extraordinary wave phenomena that are typically unattainable in Hermitian systems, such as exceptional points (EPs) [8–10], unidirectional reflectionless propagation [11,12], enhanced sensitivity [13–16], coherent perfect absorption-lasing (CPAL) [17–19], and advanced control of lasing [20,21]. In the past decade, electromagnetic PT symmetry has been extensively explored across different structural frameworks, including coupled waveguides and cavities [8–10,22–27], periodic lattices [28–30], and metasurfaces [31–42]. Notably, PT-symmetric metasurfaces have emerged as highly fertile environments for observing a plethora of fascinating phenomena [31–42], such as unidirectional negative refraction and subdiffraction focusing [32–34], unidirectional cloaking [35,36], sensing [37,38,43], electromagnetically unclonable functions [44], superdirective leaky radiation [45], teleportation [39], and electromagnetic impurity-immunity [40,41]. However, practical realization of these effects is often hindered by the demanding requirements for high gain. Typically, for a gain metasurface with a thickness ∼10−3λ0 (λ0 is free-space wavelength), the gain tangent—defined as the absolute value of ratio of imaginary and real parts of complex permittivity—can reach values as high as ∼102 [32,40]. Despite efforts to alleviate this challenge, such as utilizing multilayered structures [33] and asymmetric environments [42], many effects (e.g., the aforementioned unidirectional negative refraction and electromagnetic impurity-immunity) remain far from reality due to the difficulty in realizing high-gain metasurfaces.

In this work, we present a viable strategy to overcome the challenge of high-gain requirements by establishing low-gain generalized PT-symmetric (GPT-symmetric) systems via transformation optics [46–49] and non-Hermitian doped zero-index materials (ZIMs) [50–59]. Such GPT-symmetric systems exhibit an unbalanced loss–gain profile but display similar phase transitions and scattering properties to conventional PT-symmetric systems [16,42]. Here, we first employ transformation optics to transform the high-gain metasurface into a bulky gain medium of near-zero refractive index and subsequently realize it through a non-Hermitian doped ZIM; thus a low-gain GPT-symmetric system is constructed. We find that the required gain tangent of the dopants within this GPT-symmetric system is only ∼10−1 for the emergence of coalesced EP and its associated electromagnetic impurity-immunity, with potential further reduction in an asymmetric environment. To validate this approach, we demonstrate a microwave implementation using three-dimensional photonic crystals (PhCs) in full-wave simulations. Our findings offer a feasible strategy for significantly alleviating the requirements on gain materials in PT-symmetric systems.

We start from a PT-symmetric model, comprising a lossy metasurface (left) and a gain metasurface (right), separated by a nonmagnetic slab (relative permittivity ε, thickness d) within a nonmagnetic background medium (relative permittivity εb), as depicted in Fig. 1(a). Both metasurfaces are nonmagnetic and satisfy PT-symmetric condition εms(−x)=εms(+x)*, where εms(−x) and εms(+x) denote the complex relative permittivities of the lossy and gain metasurfaces, respectively. The thickness of both metasurfaces, denoted as dms, is much smaller than the free-space wavelength λ0, i.e., dms≪λ0. Here, we consider waves with a time variation item of e−iωt. This PT-symmetric system can be characterized using a scattering matrix that describes the relationship between the incoming and outgoing waves: S=(trRrLt),(1)where rL(R) is the reflection coefficient for left (right) incidence and t is the transmission coefficient, which is identical for both left and right incidences in this reciprocal system. The scattering matrix S encompasses two eigenvalues, that is, s1,2=t±rLrR. The analytical formula of the eigenvalues is presented in Appendix A.1.

In Hermitian systems, the scattering matrix S is unitary, i.e., SS+=I, resulting in unimodular eigenvalues (|s1,2|=1) at any real frequency. Nevertheless, in non-Hermitian systems, the scattering matrix is generally nonunitary at real frequencies, leading to nonunimodular eigenvalues (|s1,2|≠1). Notably, under the PT-symmetric condition, i.e., (P^T^)S(P^T^)=S−1, the eigenvalues can exhibit either unimodular or nonunimodular behavior, depending on the system parameters. The transition of eigenvalues from |s1,2|=1 to |s1,2|≠1 correlates with the transition from the exact PT-symmetric phase to PT-broken phase, with the phase transition point referred to as the EP [7].

To investigate the phase diagram of the PT-symmetric metasurface model, we compute the scattering matrix eigenvalues and plot their absolute values as a function of the magnitude of the imaginary part of the metasurfaces’ relative permittivity (|Im(εms)|) for different values of ε, as presented in Fig. 1(b). Here, we fix Re(εms)=1 and dms=10−3λ0 for the metasurfaces, d=λ0 for the sandwiched slab, and εb=1 for the background (i.e., air). It is observed that with increasing |Im(εms)|, the eigenvalues are initially unimodular, then undergo a bifurcation, and eventually return to unimodular values. Within this phase diagram, two EPs emerge when |Im(εms)| satisfies the following condition [40]: |Im(εms)|=εb±εk0dms,(2)where k0 is the wave number in free space. The “±” sign indicates two types of EPs. It is crucial to emphasize that the physical nature of the two EPs is distinct. The PT-symmetric metasurfaces function as a pair of unidirectional antireflection coatings at the left EP associated with smaller |Im(εms)|, while they behave as a pair of coherent perfect absorber and laser at the right EP associated with larger |Im(εms)| [40]. At EPs, two scattering matrix eigenvalues and their associated eigenvectors coalesce simultaneously. The eigenvalues become s1,2=1, and, under this circumstance, we have rLrR=0 and t=1, indicating unidirectional reflectionless propagation [11].

Notably, as ε approaches zero, the region of PT-broken phase shrinks, and the two types of EPs tend to coalesce into one EP at |Im(εms)|=εbk0dms.(3)

Such a coalesced EP holds particular significance as it gives rise to the unprecedented phenomenon of electromagnetic impurity-immunity effect—perfect wave transmission regardless of impurities embedded inside the sandwiched slab [40]. The underlying physics can be elucidated through the simulation results in Fig. 1(c). In this simulation, a planar wave with electric field polarized along the z direction is normally incident on the PT-symmetric metasurface from the left side under the coalesced EP condition [|Im(εms)|=159]. The simulation is performed using the software COMSOL Multiphysics. The color map shows the distribution of Ez/E0, while the line profiles display |E|/E0 (red) and |H|/H0 (blue) along the simulation model’s edge, where E0 and H0 are the amplitudes of the incident electric and magnetic fields, respectively. Figure 1(c) showcases near-unity |E|/E0 and |H|/H0 in air regions, indicating negligible reflection and near-perfect transmission. Notably, the magnetic field experiences a rapid decline within the lossy metasurface, reaching near-zero values within the sandwiched slab. This zero-magnetic field environment inhibits the excitation of nearly all cavity modes within impurities positioned inside the sandwiched slab [60]. Consequently, the perfect wave transport behavior is unaffected by impurities of various materials or shapes. A numerical proof is presented in Fig. 1(d), where a spherical dielectric impurity (relative permittivity 10, radius 0.2λ0) is positioned inside the sandwiched slab. No discernible impact on the wave transmission is observed, confirming the electromagnetic impurity-immunity effect.

It is essential to note that while the electromagnetic impurity-immunity is intriguing, its practical realization presents significant challenges due to the excessively high gain requirements. As seen from Fig. 1(b), the coalesced EP occurs at |Im(εms)|∼159, indicating that the gain tangent of the gain metasurface must exceed 102. Mathematically, the high-gain demand arises from the fact that the required |Im(εms)| is proportional to 1/dms [Eq. (3)]. Physically, achieving electromagnetic impurity-immunity requires the magnetic field within the sandwiched slab to be reduced to a near-zero value by the lossy metasurface, and such near-zero magnetic field must be fully restored by the gain metasurface [Figs. 1(c) and 1(d)]. Given the ultrathin thickness of metasurfaces, a large value of |Im(εms)| is necessary to rapidly diminish or restore the magnetic field within ultrathin metasurface. Although gain metasurfaces could be potentially realized using photopumped graphene at terahertz frequencies [37,38,42,43] or mimicked by negative-resistance converters in the radio-frequency region [44], practical realization of such a high-gain metasurface is still extremely challenging, if not impossible.

To address the challenge of high-gain requirements, we first employ the strategy of transformation optics to transform the ultrathin gain metasurface into a bulky slab. As depicted in Fig. 2(a), we stretch the metasurface along the x direction via the coordinate transformation x′=κx, y′=y, and z′=z [61,62], where the metasurface resides in space {x,y,z} and the bulky slab occupies the space {x′,y′,z′}. Thus, the stretched slab is characterized by a relative permittivity of εms′=diag(εmsκ,εms/κ,εms/κ), a relative permeability of μms′=diag(κ,1/κ,1/κ), and a thickness of dms′=dmsκ, where κ is the stretching ratio. For simplicity, we consider normal incidence, which requires only the z (or y) component of the permittivity (or permeability) tensor to be considered. In this scenario, the stretched metasurface can be regarded as an isotropic medium with εms′=εms/κ and μms′=1/κ. Here, we set κ=103 to balance the trade-off between achieving near-zero εms′ and μms′ and maintaining a manageable thickness dms′. A smaller κ would result in εms′ and μms′ not being sufficiently close to zero, while a larger κ would yield an excessively large thickness dms′. Thus, the stretched metasurface is characterized by a near-zero refractive index (εms′=10−3εms, μms′=10−3) and a thickness of dms′=103dms.

Figure 2(b) illustrates the model after the stretching spatial transformation, where parity symmetry is broken. Interestingly, despite the fact that this model no longer satisfies the PT symmetry as εms(−x)≠εms′(+x)*, the original non-Hermitian properties remain unaltered by the spatial transformation [63]. Based on the transfer matrix method [64,65], it can be proven that for normal incidence, the transfer matrix of the entire model remains unchanged after the stretching spatial transformation. Consequently, the scattering properties also remain unchanged. Therefore, here we term this model as the GPT-symmetric model. Within this GPT-symmetric model, the coalesced EP and its associated electromagnetic impurity-immunity persist. To demonstrate this, we resimulate the models studied in Figs. 1(c) and 1(d) by substituting the original ultrathin gain metasurface with a stretched slab characterized by εms′=10−3−0.159i, μms′=10−3, and dms′=λ0, as presented in Figs. 2(c) and 2(d). Clearly, the perfect wave transmission that is immune to embedded impurities persists at the coalesced EP. By contrast, in the absence of (stretched) metasurfaces, almost all the incident waves are reflected, as indicated by standing waves in the left air region [Fig. 2(e)]. Furthermore, the transmittance through the models in Figs. 2(d) and 2(e) as a function of the relative permittivity εimp of the embedded impurity is calculated and plotted in Fig. 2(f). It is evident that the transmission in the presence of GPT symmetry remains near-perfect, regardless of εimp, while it is very low in the absence of GPT symmetry. The low transmission is attributed to the sandwiched slab being an epsilon-near-zero (ENZ) medium with divergent impedance, greatly mismatched with free space [66]. Remarkably, the GPT symmetry employed here can eliminate the divergent impedance at the coalesced EP, transforming the ENZ medium into a unidirectional impedance-matched medium, immune to any embedded impurities.

Although the absolute value of Im(εms) is significantly reduced through the utilization of transformation optics, the gain tangent remains unchanged, remaining on the order of 102. The challenge of high gain demand persists. It is noteworthy that the stretched gain metasurface represents a unique form of ZIM with nearly pure imaginary permittivity. Fortunately, the theory of photonic doping enables us to transform a loss/gain-less ZIM into such a unique ZIM via gain dopants [55–58]. Remarkably, we find that the gain tangent of gain dopants can be significantly reduced to the order of 10−1, as we will elucidate in the following.

Figure 3(a) shows the schematic graph of a loss/gain-less ZIM (εZIM=μZIM→0) doped with a cylindrical nonmagnetic dopant (relative permittivity εd, radius of circular cross section rd) oriented along the z direction. According to the photonic doping theory [55–58], this doped ZIM can be homogenized as an effective uniform ZIM, characterized by an effective relative permittivity εeff that strongly depends on the dopant, and a dopant-independent near-zero effective relative permeability, i.e., μeff→0. For general dopants, the expression for εeff can be written as [55–58] εeff=−iωε0EcSxy∮Hd·dl,(4)where ε0 and ω are the permittivity of vacuum and the angular frequency, respectively. Ec is the uniform electric field within the ZIM. Sxy is the area of the ZIM together with dopants on the xy surface. The term ∮Hd·dl denotes the line integral of magnetic fields along the boundaries of all dopants on the xy plane. Concerning the cylindrical dopant with a circular cross-section, Eq. (4) can be reformulated as [56] εeff=2πrdεdk0SxyJ1(k0εdrd)J0(k0εdrd),(5)where J0(x) and J1(x) are the zeroth- and first-order Bessel functions, respectively.

Figures 3(b) and 3(c) respectively display the computed log(|Re(εeff)|) and log(|Im(εeff)|) as functions of Re(εd) and |Im(εd)| for the dopant with rd=0.35λ0 based on Eq. (5). Two specific points, P1 at εd=3.06−0.91i and P2 at εd=10.18−0.85i, correspond to effective parameters εeff=0.00023−0.159i and εeff=0.00021−0.159i, respectively, which approximately match the ideal value in Fig. 2 (i.e., 0.001−0.159i). We note that when |Re(εeff)|≪|Im(εeff)|, the impact of variations in the real part of εeff is negligible. In both cases, the non-Hermitian doped ZIM effectively functions as the stretched gain metasurface studied in Fig. 2. As demonstrations, we replace the stretched gain metasurface with the doped ZIM, and the simulation results are presented in Figs. 3(d) and 3(e), corresponding to points P1 and P2, respectively. Evident impurity-immune behavior is observed, demonstrating the effectiveness of the non-Hermitian doped ZIM as the gain metasurface. It is noteworthy that the gain tangent at P2 is less than 10−1, signifying a significant reduction in the required gain tangent for the emergence of coalesced EP and its associated impurity-immunity effect. In fact, there are infinite solutions of εd if its value can be infinitely large due to the oscillating Bessel functions. In principle, the required gain tangent can be further reduced.

The reduction in demand for gain tangent significantly facilitates the practical realization of electromagnetic impurity-immunity. Figure 4 illustrates a potential practical implementation operating at microwave frequencies. In this configuration, the lossy metasurface on the left is implemented using conductive films, such as graphene, indium tin oxide films, and metallic filaments, which naturally exhibit large imaginary permittivity at low frequencies [37,38,42,43,67–69]. For a conductive material with a sheet resistance Rs measuring the in-plane resistance for a film of arbitrarily sized square shape, its relative permittivity can be expressed as 1+iZ0/(k0dmsRs). Considering the condition of coalesced EP [Eq. (3)], the required sheet resistance of the conductive film is given by Rs=Z0.(6)

The conductive films with a sheet resistance of Z0 can be readily obtained in microwave experiments [67,68].

The sandwiched ENZ slab and non-Hermitian doped ZIM on the right can be implemented using three-dimensional PhCs. It has been demonstrated that a PhC operating at the band-edge frequency could serve as an effective ENZ medium [70–72], and a PhC exhibiting Dirac-like cones at the center of the Brillouin zone can function as an effective ZIM with both effective permittivity and permeability approaching zero [52–54,57,73]. Therefore, two types of PhCs, denoted as type I and type II, are designed. Both types of PhCs consist of a simple cubic lattice of core-shell spheres with a lattice constant of a. The core is made of perfect electric conductor (radius rc), and the shell is made of a dielectric material (relative permittivity εs, radius rs). For type I PhC, rc=0.138a, εs=29.2, and rs=0.4a; for type II PhC, rc=0.126a, εs=30, and rs=0.4a, as depicted by the right insets in Fig. 4(a). Here, we set a=8mm. In practice, these core-shell spheres can be fixed in certain positions using foam (relative permittivity ∼1) [67]. The high-permittivity dielectric shells can be realized using polymer-matrix ceramic composites comprising high-permittivity ceramics, such as SrTiO3 [74], and low-permittivity organic polymer materials [75]. The band structures of both types of PhCs are presented in left panel graphs in Figs. 4(b) and 4(c) based on the software COMSOL Multiphysics. A photonic band gap is observed for type I PhC, while a Dirac-like cone arises as a consequence of sixfold degenerate modes for type II PhC. The band-edge frequency of type I PhC and the Dirac-point frequency of type II PhC are designed to coincide at fa/c=0.2678, where c is the speed of light in free space and f is the eigenfrequency. Based on averaged eigenfields [76,77], the effective parameters of the two types of PhCs can be obtained as follows: εI(II),eff=−kx2πfε0⟨Hy⟩⟨Ez⟩,(7a)μI(II),eff=−kx2πfμ0⟨Ez⟩⟨Hy⟩,(7b)where Ez (or Hy) is the eigen-electric (or magnetic) field on the yz plane for modes along the ΓX direction. ⟨…⟩ denotes the average of eigenfields on the yz surface of the PhC unit cell. kx is the x component of the Bloch wave vector. μ0 is the permeability of vacuum. The right panel graphs in Figs. 4(b) and 4(c) show the computed relative permittivities (εI,eff and εII,eff) and relative permeabilities (μI,eff and μII,eff) of both types of PhCs. It is observed that εI,eff≈0 and εII,eff≈μII,eff≈0 at fa/c=0.2678 (or f=10GHz), indicating the single-zero (double-zero) property of type I (II) PhC.

By utilizing conductive films and two types of PhCs, we construct a GPT-symmetric model for achieving electromagnetic impurity-immunity, as depicted in Fig. 4(a). The sandwiched type I PhC slab and the type II PhC slab on the right are each composed of 3×3×3 PhC units. Within the type I PhC slab, a cubic impurity (relative permittivity εimp=10, side length a) is embedded at the center. Meanwhile, the type II PhC slab is doped with a nonmagnetic cylinder of square cross section (side length a), oriented along the z direction. Through meticulous parameter optimization, the relative permittivity of the cylindrical dopant is adjusted to εd=15.9−1.8i to induce the coalesced EP (Appendix A.2). We note this gain dopant could be realized at microwave frequencies by incorporating active elements [78,79]. Figure 4(d) presents the simulated Ez/E0 under the illumination of a planar wave incident from the left air region at fa/c=0.2678 (or f=10GHz), showing near-perfect transmission. In contrast, when the conductive film and type II PhC slab are removed, evident reflection occurs [Fig. 4(e)]. Furthermore, we compare the transmittance through the GPT-symmetric model [Figs. 4(d)] and the model without GPT symmetry [Fig. 4(e)] as a function of εimp of the embedded impurity, as shown in Fig. 4(f). Robust near-unity transmission is observed in the GPT-symmetric system. Meanwhile, the transmission is generally low in the absence of GPT symmetry, except for a peak induced by impurity resonance. These results validate the coalesced EP-induced impurity-immunity effect and demonstrate the feasibility and effectiveness of the proposed practical implementation. It is worth noting that the performance can be further improved using high-permittivity materials to construct PhCs, which can function as highly homogeneous ZIMs [74].

Further mitigation of the demand for gain materials can be achieved in an asymmetric environment. Figure 5(a) illustrates a GPT-symmetric model in an asymmetric environment. This model resembles that shown in Fig. 2(b), with the exception that the background media on the left and right sides are no longer identical. We assume the relative permittivity of the background medium on the left (right) as εb1 (εb2). Then, the scattering matrix for this model can be expressed as follows: S=(t′rRrLt′),(8)where t′=(εb2εb1)1/4tL=(εb1εb2)1/4tR in reciprocal systems; rL/R (tL/R) represents the reflection (transmission) defined on the tangential electric field. In Hermitian systems, |rL/R|2+|t′|2=1 due to energy conservation. Consequently, this scattering matrix is also unitary, i.e., SS+=I. The two eigenvalues are s1,2=t′±rLrR. Notably, EP occurs when rLrR=0. In particular, when the lossy metasurface satisfies |Im(εms)|=εb1k0dms, according to Eq. (3) by setting εb=εb1, the required |Im(εms′)| of the stretched gain metasurface on the right for the emergence of coalesced EP is |Im(εms′)|=εb2k0dms′.(9)

Equation (9) indicates a decrease in |Im(εms′)| with smaller εb2, as observed in Fig. 5(b). Clearly, the asymmetric environment leads to a reduction in the demand for gain materials.

Figure 5(c) provides a verification example, where εb1=1 and εb2=0.6. In this model, the lossy metasurface remains unchanged compared to the model in Fig. 2(b). Meanwhile, the |Im(εms′)| for the stretched gain metasurface on the right is set to 0.123 [marked by the dot in Fig. 5(b)]. The constant |E|/E0 and |H|/H0 in the left air region signify the zero-reflection property despite the presence of an impurity within the sandwiched slab. Additionally, we implement the stretched gain metasurface using a non-Hermitian doped ZIM. As shown in Fig. 5(d), the ZIM is doped with a cylindrical gain dopant with rd=0.35λ0 and εd=10.18−0.65i. The simulation confirms the impurity-immune wave propagation in an asymmetric environment.

Compared to the characteristics of the model in a symmetric environment [Fig. 3(d)], the required gain tangent for the gain dopant is reduced. Hence, the asymmetric environment provides an additional avenue for further reducing the requirement for the gain tangent. This can be understood intuitively. Due to the distinct impedances between the background media on both sides, the transmittance is εb2/εb1 at the coalesced EP, which differs from the unity transmittance in a symmetric environment. In a symmetric environment, the near-zero magnetic field within the sandwiched ENZ slab must be fully restored to its incident value by the stretched gain metasurface to maintain unity transmittance. Interestingly, in an asymmetric environment with εb2<εb1, the transmittance is less than unity, indicating that the magnetic field does not need to be fully restored by the stretched gain metasurface, as observed in Figs. 5(c) and 5(d). Consequently, the required gain tangent value of the stretched gain metasurface is reduced. In a sense, this GPT-symmetric model functions as a non-Hermitian unidirectional impurity-immune antireflection coating, eliminating the impedance mismatch between the two background media.

The pursuit of impurity-immune wave transport is a significant research area in physics due to its potential to ensure robust high transmission of energy against disturbances, holding paramount significance across numerous applications. Recent investigations into topological insulators have unveiled strategies for achieving impurity-immunity in surface waves [80,81]. Our findings demonstrate a viable pathway for attaining impurity-immunity in bulk waves within GPT-symmetric systems with low-gain requirement. This impurity-immunity emerges as a consequence of coalesced EP, showcasing additional extraordinary properties at the phase transition point in non-Hermitian systems.

It is noteworthy that, in addition to the aforementioned EPs, CPAL is another significant property within PT- and GPT-symmetric metasurfaces. PT- and GPT-symmetric metasurface systems operating at the CPAL point exhibit many interesting effects and applications, such as electromagnetic sensors with unprecedented sensitivity [38,43], electromagnetically unclonable functions [44], and superdirective leaky radiation [45]. Our proposed approach for lowering the gain tangent requirement could be extended to these scenarios to facilitate these effects and applications. Additionally, we note that when operating at the CPAL point, the scattering responses of PT- and GPT-symmetric metasurface systems are usually highly sensitive to variations in the loss–gain parameter [38,43,44]. In contrast, for our systems operating at EPs, the scattering responses and the associated electromagnetic impurity-immunity effect are stable in the presence of variations in loss–gain parameter, demonstrating good robustness against manufacturing imperfections (see Appendix A.3).

In summary, we have demonstrated a low-gain GPT-symmetric system manifesting the rare property of electromagnetic impurity-immunity via the approaches of transformation optics and non-Hermitian doping of ZIMs. Compared to the required gain tangent of gain metasurface in PT-symmetric systems, the required gain tangent of the dopants in ZIMs is substantially reduced by three orders of magnitude, while the effective gain tangent of the entire non-Hermitian doped ZIM remains almost unchanged. The required gain tangent can be further reduced in an asymmetric environment. A practical implementation utilizing conductive films and PhCs is validated through full-wave simulations. This work not only provides a viable approach for significantly reducing requirements on gain materials but also unveils promising pathways for advanced wave manipulation with GPT symmetry.