Non-Hermitian doping of epsilon-near-zero media

We start considering the 2D problem geometry in Fig. 1A, consisting of an arbitrarily shaped host region filled by an ENZ medium (with relative permittivity ${\epsilon }_h$), doped with a core−shell cylindrical particle with interior and exterior radii $R_i$ and $R_e$, and corresponding relative permittivities ${\epsilon }_i$ and ${\epsilon }_e$, respectively. All materials are nonmagnetic ($\mu =1$), and the entire structure is embedded in vacuum. Throughout our study, we assume transversely magnetic polarized excitation (with z-directed magnetic field) and implicit $\exp \left(-i\omega t\right)$ time-harmonic dependence, with geometry and field quantities invariant along the z direction.

The ENZ host medium can be described via a Drude-type model,with ${\omega }_{ph}$ and ${\mathrm{\Gamma }}_h$ denoting the plasma radian frequency and collision factor, respectively. As experimentally demonstrated in ref. 15, at microwave frequencies, such behavior can be mimicked by a near-cutoff waveguide. At higher (e.g., terahertz and optical) frequencies, reliance could be made on naturally occurring ENZ materials such as silicon carbide (39) and indium tin oxide (40) which, however, tend to exhibit sensible losses (see also the general discussion in ref. 41). Nevertheless, loss-compensated or even slightly active ENZ materials can be, in principle, attained via composite materials featuring plasmonic waveguides embedded in a gain medium (37) or plasmonic nanoshells with quantum dots (42). In what follows, we assume that losses in the ENZ host are negligible or somehow compensated, so that the assumption ${\epsilon }_h\approx 0$ underlying the general theory in ref. 15 holds.

For the core (lossy) material, we consider instead a standard Lorentz−Drude model,with ${\widehat{\epsilon }}_i$, ${\omega }_{0i}$, and ${\mathrm{\Gamma }}_i$ being the high-frequency limit, center radian frequency, and collision factor, respectively. Finally, for the gain medium, we assume a standard approximate linearized model (43) for dielectric materials doped with fluorescent dye molecules (modeled as four-level atomic systems), namely,where ${\omega }_a$ denotes the emitting radian frequency, $\mathrm{\Delta }{\omega }_a$ is the bandwidth of the dye transition, ${\widehat{\epsilon }}_e$ is the host medium relative permittivity, ${\sigma }_a$ is a coupling strength parameter, ${\overline{N}}_0$ is the total dye concentration, ${\tau }_{jl}$ is the relaxation times for the state transitions, and ${\mathrm{\Gamma }}_p={\sigma }_{abs}{I}_p/\left(hf\right)$ is the pumping rate (with ${\sigma }_{abs}$ being the absorption cross-section, $I_p$ being the pump intensity, $h$ being the Planck constant, and $f$ being the pump frequency). Similar models also hold for gain materials based on quantum dots (44, 45).

The theory below, valid for the steady-state response, is derived through a time-harmonic wave-scattering formalism. Accordingly, we consider only the complex relative permittivity values at a single operational frequency, ${\epsilon }_h$, ${\epsilon }_i={{\epsilon }^{\prime }}_i+i{{\epsilon }^{″}}_i$ and ${\epsilon }_e={{\epsilon }^{\prime }}_e+i{{\epsilon }^{″}}_e$, with the prime and double-prime tagging the real and imaginary parts, respectively; in view of the assumed time convention, positive and negative imaginary parts correspond to loss and gain, respectively. However, we will take into account the full dispersive models in Eqs. 1–3 when dealing with stability issues (SI Appendix).

The general theory of photonic doping of ENZ media (15) is not limited to doping particles characterized by real-valued constitutive parameters, and therefore holds for our scenario of interest. Accordingly, as depicted in Fig. 1B, for an exterior observer, the structure in Fig. 1A can be effectively replaced by a homogeneous region of the same shape and ENZ character, but exhibiting an effective (generally nonunit) relative permeability ${\mu }_{eff}$. For our core−shell doping particle, the effective permeability can be calculated analytically in closed form; the exact expression is not given here, for brevity, and can be found in SI Appendix together with its detailed derivation (SI Appendix, Eqs. S1–S7). To illustrate the basic idea underlying our approach, here we focus on its approximation,valid in the limit of electrically small radii $k{R}_{i,e}\ll 1$ (see SI Appendix for details). In Eq. 4, $A$ denotes the total area (host + dopant regions) of the structure in Fig. 1A, and $k=\omega /c=2\pi /\lambda$ denotes the vacuum wavenumber (with $c$ and $\lambda$ being the corresponding speed of light and wavelength, respectively). In particular, we observe that the denominator of the fraction in Eq. 4 vanishes forWithin the limits of the implied approximation, Eq. 5 provides a condition that the core and shell relative permittivities need to fulfill in order to give rise to a pole-type singularity in the effective permeability. Interestingly, such a condition can be satisfied for real-valued frequencies also in the presence of complex-valued permittivities, that is, of suitably tailored gain and loss in the core and shell regions. This represents a key difference with respect to the cylindrical doping particle configuration considered in the original study (15), which can be recovered from Eq. 5 by assuming either $R_e=R_i$ or ${\epsilon }_e={\epsilon }_i$. In this latter case, it can be readily verified that the pole condition can be satisfied for real-valued frequencies only in the absence of gain and loss.

The above reasoning is not limited to the electrically small regime; for larger structures, the complex pole condition can no longer be expressed in a simple closed form and needs to be found numerically. From a phenomenological viewpoint, such singularity implies the possibility to achieve broad tunability of the effective permeability over the entire complex plane, by changing the core and shell radii and/or the corresponding permittivities.

This is quantitatively illustrated in Fig. 2 for a parameter configuration featuring ${\epsilon }_h=0.001$, $A=0.496{\lambda }^2$, $R_i=0.14\lambda$, $R_e=0.17\lambda$, and ${\epsilon }_e=5-i0.5$. For these parameters, the pole condition (calculated from the exact expression of ${\mu }_{eff}$ in SI Appendix, Eq. S7 for higher accuracy) yields ${\epsilon }_i=5.114+i0.287$. Fig. 2A shows, in the complex ${\epsilon }_e$ plane, a number of circles centered at the pole (magenta cross), whereas Fig. 2B shows the corresponding mappings (with approximate circular shapes) in the complex ${\mu }_{eff}$ plane, with the one-to-one correspondence illustrated via color coding. It can be observed that increasingly smaller circles (i.e., closer to the pole singularity) in the complex ${\epsilon }_e$ plane are mapped onto increasingly larger circle-like curves in the complex ${\mu }_{eff}$ plane, thereby demonstrating the anticipated broad tunability (in sign and amplitude) of the effective permeability over the entire complex plane. Fig. 2 C and D shows two representative cuts (at fixed ${{\epsilon }^{\prime }}_e$, as a function of ${{\epsilon }^{″}}_e$) from which it can be observed that arbitrary combinations of positive and negative real and imaginary parts are possible, in principle. It is worth pointing out that the gain excursions considered are compatible with values attainable from the model in Eq. 3 by assuming typical parameters available in the topical literature for organic dyes (46) and quantum dots (44, 45). To give an idea, at near-infrared wavelengths, the gain level necessary to fulfill the pole condition above is compatible with the organic dye LDS798 (by Exciton), featuring ${\omega }_a={\omega }_c=2\pi c/{\lambda }_a$ (with λ_a = 777 nm), $\mathrm{\Delta }{\omega }_a=2\pi c\mathrm{\Delta }{\lambda }_a/{\lambda }_a^2$ (with Δλ_a = 56 nm), ${\sigma }_a=6\pi c^3\eta /\left({\tau }_{21}{\omega }_a^2\sqrt{{\widehat{\epsilon }}_e}\right)$ (with $\eta =0.48$), ${\overline{N}}_0=6.1\cdot {10}^{18}{\text{cm}}^{-3}$, τ₂₁ = 50 ps, τ₁₀ = 100 fs, τ₃₂ = 100 fs, and Γ_p = 2.070 × 10⁹ s⁻¹ (46).

Moreover, we highlight that the parameters above constitute only one example that satisfies the pole condition, and that there are infinite combinations that could also work with much lower levels of gain and loss. For instance, maintaining the same geometrical and host medium parameters as above, and enforcing ${\epsilon }_e=5-i0.05$ (i.e., an order-of-magnitude lower level of gain), we obtain, from SI Appendix, Eq. S7, ${\epsilon }_i=5.109+i0.0287$ as an alternative pole condition.

For validation purposes, we consider a parameter configuration tailored so as to yield a targeted ${\mu }_{eff}=-1-i10$, and show, in Fig. 2 E and F, the numerically (finite-element) computed near-field distributions pertaining to the actual and effective structures, respectively, excited by an elementary electric dipole. As can be observed, the field distributions in the exterior vacuum region are essentially identical, as are the corresponding (far-field) radiation patterns shown in Fig. 2G.

Finally, we note that, along the lines of what developed in ref. 15 for cylindrical dielectric doping particles (see supplementary note 3 therein), the lossy, dispersive character of the ENZ host in Eq. 1 could be directly incorporated in the model in terms of suitable correction factors in the effective permeability. However, in view of the more complex (core−shell) geometry considered here, such extension is rather cumbersome and less physically insightful, and we leave it to future developments.

The above premises suggest a variety of possible applications for engineering of unconventional and/or reconfigurable optical responses. For instance, the broad tunability of the effective permeability achievable by solely acting on the gain parameter (as illustrated in Fig. 2 C and D) opens up interesting perspectives in reconfigurability mechanisms based on optical pumping. As also suggested in previous related studies (16), for an easier interpretation of the optical response, it is expedient to consider slab-type configurations, as shown in Fig. 3A, made of periodic arrangements of square unit cells, under plane wave illumination. For such a geometry, Fig. 3B shows the reflectance and transmittance responses, as a function of the gain parameter ${{\epsilon }^{″}}_e$, for a configuration optimized so as to exhibit strong sensitivity to small variations of the gain (see SI Appendix for details). As can be observed in the reflectance response, around the deep minimum, variations of the gain parameter of ∼20% (which could be induced by suitable optical pumping) are sufficient to switch from an almost perfect impedance matching to a quite strong impedance mismatch, with over four-order-of-magnitude variations in the reflectance (from $\sim {10}^{-4}$ to $\sim 1$). In fact, values of reflectance moderately larger than 1 (∼2.8) can be obtained with ∼30% gain variations. As shown in Fig. 3 C and D, this mechanism essentially relies on the transition from an epsilon-and-mu-near-zero (impedance matching) condition to strongly mismatched configurations characterized by significantly larger (although still small in absolute value) effective permeabilities. Fig. 3 E and F shows the field intensity distributions pertaining to two such distinct states, from which the absence and presence, respectively, of a standing wave can be observed in the incidence region.

It is worth pointing out that gain variations of the same order are much less effective in more conventional configurations that do not exhibit effective magnetism. For illustration, in SI Appendix, Fig. S1, we consider a benchmark bilayer configuration composed of a gain layer with relative permittivity the same as the shell material in Fig. 3 (and thickness adjusted so as to equalize the overall concentration) paired with an antireflection lossy layer, showing that gain variations of the same order as in Fig. 3 yield much milder reflectance variations, with values that always remain below ∼0.1.

Also in this case, we remark that the gain levels in Fig. 3 are in line with those attainable via organic dyes (43, 46) and quantum dots (44, 45) at infrared wavelengths. For these parameters, the gain level scales almost linearly with the pumping rate (and, hence, intensity), and, therefore, mild (∼20%) variations appear to be feasible. Moreover, alternative optimized structures operating with lower gain levels can be engineered, as illustrated in SI Appendix, Fig. S2.

The possibility of relying on an arbitrary complex-valued permeability enables the engineering of unconventional optical responses. For instance, it is possible to obtain signal amplification in conjunction with near-zero reflectance or transmittance.

From Fig. 3B, it can be verified that, at the reflectance dip, the transmittance is nearly 1, whereas the regions featuring signal amplification in transmission are characterized by strong reflectance. This is not surprising, in view of the underlying assumption of a lossless ENZ host, which implies that the impedance matching condition is fulfilled for a real-valued effective permeability (Fig. 3 C and D). In other words, loss and gain in the core−shell doping particle must exactly balance at the impedance matching condition. This constraint can be lifted by assuming that the ENZ host exhibits gain (i.e., ${{\epsilon }^{″}}_h<0$). In this case, at the impedance matching condition, the signal amplification in transmission would be essentially dictated by the amplification constant $kL\sqrt{-{{\epsilon }^{″}}_h}$. This implies that, for small to moderately sized layers, sensible amplification generally require gain levels that, while not necessarily large in absolute value, may invalidate the underlying ENZ assumption and, in turn, the analytic derivations of the closed-form effective parameters ${\epsilon }_{eff}={\epsilon }_h$ and ${\mu }_{eff}$ given in Eq. 4 (and SI Appendix, Eq. S7). To give an idea, in SI Appendix, Fig. S3, we compare the theoretical effective parameters for a configuration featuring ${\epsilon }_h=0.01-i0.01$ with the values obtained from a conventional parameter retrieval algorithm (47). As can be observed, the differences may be substantial, up to a factor of ∼4. Nevertheless, it is important to emphasize that the doping principle remains valid also in this regime, as the structure still exhibits effective magnetism, even though the magnetic field in the ENZ region may not be uniform, and the effective properties may depend on the dopant location and not be accurately expressed analytically in closed form. Fig. 4A shows the response of a structure that was numerically optimized so as to exhibit the desired response. The optimization was carried out directly in the space of the physical (${\epsilon }_h$, ${\epsilon }_i$, ${\epsilon }_e$) and geometrical ($R_i$, $R_e$) parameters, without resorting to the analytic effective parameter representation (see SI Appendix for details). As can be observed, for a specific value of the gain level in the shell region, it exhibits a transmittance of ∼5 accompanied by near-zero reflectance; the corresponding field distribution is shown in Fig. 4B. Similar to the previous example, the gain parameter ${{\epsilon }^{″}}_e$ can be exploited to tune (via optical pumping) the reflectance response over several orders of magnitude, while maintaining an essentially constant transmittance. The extracted effective parameters are shown in SI Appendix, Fig. S4; as expected, at the impedance matching condition, the imaginary parts are negative. Also in this case, it is interesting to compare the response with that of a conventional nonmagnetic bilayer structure consisting of a gain material layer (with the same permittivity and concentration as the shell material in Fig. 4A) paired with an antireflection lossy layer. As shown in SI Appendix, Fig. S5, in this case, the transmittance is much lower (over an order of magnitude), thereby indicating the pivotal role played by the effective magnetism (and, hence, doping) in our proposed configuration.

Concerning the dual response, that is, signal amplification in reflection and near-zero transmission, a straightforward implementation may feature a gain material layer backed by a regular photonic-doped ENZ layer tuned so as to exhibit a perfect magnetic conductor response (15). However, it is possible and insightful to engineer this response with a single layer of active ENZ material doped with non-Hermitian core−shell particles. As shown in SI Appendix, Fig. S6 and the accompanying discussion, effective magnetism accompanied by gain allows unconventional tradeoffs between reflectance and transmittance, including the response of interest here. However, also in this case, it is necessary to operate in a parameter regime where the closed-form expressions for the effective parameters are not applicable, and therefore the structure needs to be optimized directly in the physical/geometrical parameter space. Fig. 4C shows an example of an optimized structure which, for varying level of gain in the core $\left({{\epsilon }^{″}}_i\right)$, exhibits an almost flat reflectance of ∼2.8 and a moderately tunable transmittance that can be as low as $\sim {10}^{-8}$, thereby yielding an almost perfect amplifying mirror. The corresponding field distribution is shown in Fig. 4D, whereas the extracted parameters are shown in SI Appendix, Fig. S7.

We now move on to a different non-Hermitian scenario, schematized in Fig. 5A, featuring an ENZ bilayer with loss ($\epsilon ={\epsilon }_h={{\epsilon }^{\prime }}_h+i{{\epsilon }^{″}}_h$, ${{\epsilon }^{″}}_h>0$) on one side and gain $\left(\epsilon ={\epsilon }_h^{\ast }={{\epsilon }^{\prime }}_h-i{{\epsilon }^{″}}_h\right)$ on the other, periodically doped with identical cylindrical (lossless) dielectric particles. Note that the host permittivity distribution obeys the PT symmetry condition (22, 23)and this condition is preserved also in the presence of the lossless doping particles, provided that they are placed symmetrically across the gain−loss interface $y=0$.

As summarized in SI Appendix, PT-symmetric gain−loss interfaces like the one in Fig. 5A can sustain surface waves resembling surface plasmon polaritons, which are exponentially bound along the y direction and propagate unattenuated along the x direction (33, 34, 48). This waveguiding mechanism can be sustained in the presence of a sufficiently high gain/loss level ${{\epsilon }^{″}}_h$, and becomes progressively less effective for reduced gain/loss levels, eventually transitioning to a leaky-type radiation mechanism below a certain threshold (34). However, we showed, in a previous study (34), that, in the ENZ regime, such a gain/loss threshold can be significantly lowered (see also SI Appendix for details). The above physical mechanism, in principle, enables very intriguing “on-demand” waveguiding effects, for instance, by deploying in a lossy background channels made of gain media, and activating them selectively via suitable optical pumping. To add further complexity and new degrees of freedom, we explore the effects of effective magnetism induced by photonic doping. As shown in SI Appendix, unlike the previous scenario in Fig. 1A, here, there is no apparent advantage in assuming a non-Hermitian core−shell doping particle, so we consider a standard photonic doping as in the original study (15). However, the presence of a PT-symmetric ENZ host implies two characteristics that render the extension of the general theory not straightforward: 1) the presence of an interface between two doped regions and 2) the generally nonconstant (in fact, exponentially decaying for the surface wave scenario of interest) magnetic field distribution inside the ENZ bilayer.

We first consider a configuration in the absence of doping (and hence of effective magnetism), with parameters ${\epsilon }_h={10}^{-5}+i{10}^{-3}$ and $d=0.25\lambda$, chosen so that the waveguiding mechanism cannot be sustained (gain/loss threshold level of $4.4\cdot {10}^{-3}$; see SI Appendix for details). Although such low losses (and small gain, for ${\epsilon }_h^{\ast }={10}^{-5}-i{10}^{-3}$) are not available in natural ENZ materials, they can be, in principle, attained via suitable loss compensation (37, 42). From the intensity map in Fig. 5B, and the corresponding transverse cut in Fig. 5G, it is apparent that the field is not confined inside the bilayer, but it is rather radiated in the exterior vacuum region. As detailed in SI Appendix, for the same gain/loss level, a magnetic response could render the waveguiding mechanism much more effective. For instance, Fig. 5C shows the ideal response of a bilayer with the same permittivity distribution as above, but now with a relative permeability $\mu =25$. The presence of a bound mode at the gain−loss interface is now evident, with the standing wave pattern attributed to the structure truncation along the x direction. The exponential localization at the gain−loss interface is better visualized in the transverse cut shown in Fig. 5F. The next step is to realize the magnetic response via photonic doping. For the assumed parameters, the general theory (15) is applicable, and, by inverting the closed-form expression of the effective permeability, we obtain the doping particle parameters ${\epsilon }_d=11.9$, $R_d=0.11\lambda$. Fig. 5D shows the corresponding intensity distribution, from which the field confinement at the gain−loss interface is apparent. For a more quantitative assessment, Fig. 5F compares a transverse cut with that pertaining to the ideal $\left(\mu =25\right)$ scenario, showing a very good agreement. This indicates that the photonic doping concept is also applicable in the presence of interfaces and fast variations of the magnetic field distribution.

It is worth stressing that, although their level is quite low $\left({{\epsilon }^{″}}_h={10}^{-3}\right)$, gain and loss are essential for the above waveguiding mechanism. To gain some insight, Fig. 5 E and G shows the intensity map and the transverse cut, respectively, pertaining to the same doping scenario as in Fig. 5F, but now with the gain switched to zero (and loss still present). Despite the still visible doping effect, the exponential localization at the interface is now totally lost, and the field distribution is completely extended in the transverse direction. Once again, the possibility of switching from a completely extended to an exponentially bound field distribution (possibly with subwavelength localization) via (de)activation of a very small level of gain is particularly intriguing for potential applications in reconfigurable nanophotonics.

We note that the above configuration differs fundamentally from other coupled chains of gain/loss particles, which can also sustain waveguiding mechanisms (49). One important difference is the sensitivity to the doping particle displacement, as a consequence of the generally nonconstant magnetic field distribution in the ENZ host regions. In SI Appendix, Fig. S8, we show that the waveguiding mechanism is quite robust with respect to particle displacements that preserve the PT symmetry condition in Eq. 6, whereas it is very sensitive to displacements that break it. This strong sensitivity could be exploited for engineering interesting mechanical actuation systems and for optical sensing purposes.

All data needed to evaluate the conclusions in the paper are available in the manuscript and SI Appendix.