1. Introduction

High-momentum (high-k) modes are particularly attractive for a range of technologically important applications, such as plasmonic sensors [1], superresolution imaging techniques [2–4], and others [5], where the low phase velocity and/or our ability to manipulate light propagation at a deep subwavelength scale is of critical importance [6–9]. Low-momentum (low-k) modes, which are characterized by the superluminal phase velocity, are also of interest for many applications, such as metatronics applications [10] or the electromagnetic field concentrators, wavefront converters, and cloaking [11]. In addition, a strong electric field, because of the boundary condition for the normal component of the displacement, gives rise to strong optical nonlinearities which, in turn, can result in a number of interesting effects [12,13]. In the case of macroscopically homogeneous media, for which a homogenization procedure can be applied, high-k modes can be described in terms of a very high permittivity, and low-k modes - in terms of a very low permittivity (epsilon-near-zero, or ENZ).

Low loss and hence long-range plasmonic modes are of relevance for a number of integrated optoelectronic devices. The imaginary part of the mode wave vector k close to zero is highly desirable. Because the energy dissipation is unavoidable in realistic plasmonic structures, the use of gain media like dyes or semiconductor quantum structures, which allow to compensate intrinsic loss of metals and semiconductor plasmonic materials, has received acceptance in recent years. Such metamaterials (MMs) exhibit effects similar to electromagnetically induced transparency [14–17].

Yet, in spite of a number of theoretical and experimental studies addressing the issue of loss compensation in various plasmonic structures (see, e.g., [18–25]), there are many challenging opportunities and problems to be solved. Even in the simplest (1d) case the problems of combining full loss compensation with the high-k and low-k regimes remain unconsidered. Volume plasmon modes with a very high momentum and corresponding giant permittivity can be achieved in metal/dye-doped dielectric lattices, but such an approach involves the use of very thin dielectric layers [9]. On the other hand, ENZ regime in the visible range has been realized for metal/dye-doped polymer nanolaminates, but the achieved imaginary part of the longitudinal component of the effective permittivity tensor could not be smaller than −0.04 [21]. The use of dyes is limited for such applications because of their bleaching, low damage threshold, and difficulties associated with current-injection pumping. Due to the limitations on the gain coefficient g for dyes, as g ∝ Imϵ/n, the negative imaginary part of their permittivity cannot be very high, that makes the problem of compensation of the plasmonic material loss difficult. Meanwhile, it can reach about −0.1 – −0.3 for semiconductor structures [23]. Among those, in the telecommunication range, transparent conducting oxides (TCO) and especially InGaAsP are worth to be noticed [26].

The aim of this paper is to examine the feasibility of the full loss compensation for high-k and low-k plasmonic modes for 1d metal/semiconductor MMs. In Sec. 2 we first describe possible solutions of the dispersion equation imposing the condition of the full loss compensation. Then, in Sec. 3 and Sec. 4 we analyze the high-k and low-k solutions, respectively. Sec. 5 contains our conclusions.

2. Loss compensation in layered structures in terms of Rytov’s equation

A schematic of the structure (1d infinite bi-layer MM) is shown in Fig. 1. While there exist a number of homogenization techniques for such MMs [27], we adopt a rigorous treatment of electromagnetic wave propagation in 1d lattices using Rytov’s dispersion equation [28]. Although this equation has been considered in many papers (see, e.g., [29–32]), finding its solutions is not a trivial task.

 

Fig. 1 The unit cell of the structure under consideration.

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In the case of in-plane propagation (k_⊥ = 0), as shown in Fig. 1, only one diagonal component of the effective permittivity tensor describes the nonlocal response [24]:

In the following, we impose the condition of Im ϵ_eff = 0, that corresponds to the case of the full loss compensation with either purely real or purely imaginary k_║. Because Eq. (2) contains many parameters, it is difficult to analyze the entire parameter space. To simplify our consideration, in the subsequent development we assume that the permittivity of the plasmonic component is a constant and fix it at ϵ₁ = −122.19 + i3.115, that corresponds to silver at λ = 1.5 µm [25]. To extrapolate our results to other spectral ranges, we note important scaling properties of Eq. (2). Let ${\mathit{ϵ}}_1^{\star }=r{\mathit{ϵ}}_1$ and $k_0^{\star }=s{k}_0$ with real r and s. Then, as is easy to see from Eqs. (1) and (2), the following relation holds for ϵ_eff:

For a gain medium, we take ${\mathit{ϵ}}_2=12+i{\mathit{ϵ}}_2^{″}$, that approximatelly corresponds to InGaAsP multiple quantum wells at the same wavelength, neglecting the free carrier dependence of $\mathrm{Re}{\mathit{ϵ}}_2\equiv {\mathit{ϵ}}_2^{\prime }$ and assuming that $\mathrm{Im}{\mathit{ϵ}}_2\equiv {\mathit{ϵ}}_2^{″}$ is a negative parameter which depends on the gain coefficient g as ${\mathit{ϵ}}_2^{″}=-g{n}_2/k_0$, where $n_2=\mathrm{Re}\sqrt{{\mathit{ϵ}}_2}$. For the parameters chosen, the conditions ${\mathit{ϵ}}_1^{″}\ll \left|{\mathit{ϵ}}_1\right|$ and ${\mathit{ϵ}}_2^{\prime }<|\mathrm{Re}{\mathit{ϵ}}_1|$ are met. We note that recently light emission from Ag/InGaAsP multiple quantum well structures in the near-IR has been demonstrated experimentally [33,34].

Assuming that k₀d ≪ 1 (weak nonlocality), Eq. (2) can be expanded into a power series that yields an approximate analytical solution for the effective permittivity with nonlocal corrections [35]. In the general case, the solutions of Eq. (2) can be represented as a family of curves in the 3d space of $\left({\mathit{ϵ}}_{eff};{\mathit{ϵ}}_2^{\prime };f\right)$, where f is the filling factor of the gain medium defined as f = d₂/d with d = d₁ + d₂.

As a first step, we consider the behavior of the function F(f, ϵ_eff) at fixed ${\mathit{ϵ}}_2^{″}$. In Figs. 2 and 3, we show the function |F(f, ϵ_eff)| in the logarithmic scale at ${\mathit{ϵ}}_2^{″}=-0.2$ and d = 100 nm for the TM and TE polarizations, respectively.

 

Fig. 2 The function |F(f, ϵ_eff)| (logarithmic scale) at Im(ϵ₂)=−0.2 and d = 100 nm for the TM polarization.

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Fig. 3 The function |F(f, ϵ_eff)| (logarithmic scale) at Im(ϵ₂)=−0.2 and d = 100 nm for the TE polarization.

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The solutions of Eq. (2), which correspond to the condition of ln |F| → −∞, lie within fractures which form a system of intersecting branches. Although ϵ₁, ϵ₂, and d are fixed in these calculations, such a behavior of the function F(f, ϵ_eff) is typical. For a finite range of values of ϵ_eff, if it includes its negative values, the number of solution branches depends on the lattice period d.

While an infinite number of solution branches can exist, only two of them, which correspond to positive values of ϵ_eff, are of interest for us. They are shown by the arrows in Figs. 2 and 3 as A and B. Branches A and B correspond to even and odd modes, respectively. In what follows, we define ${\mathit{ϵ}}_{eff}^{\left(A\right)}$ and ${\mathit{ϵ}}_{eff}^{\left(B\right)}$ as ${\mathit{ϵ}}_{eff}^{\left(A\right)}={\left(k^A/k_o\right)}^2$ and ${\mathit{ϵ}}_{eff}^{\left(B\right)}={\left(k^B/k_o\right)}^2$, where k^A and k^B are the solutions of Eq. (2) for branches A and B, respectively.

For the TM polarization, branch A gives the traveling-wave solutions of Eq. (3), because for it k is always real and hence ϵ_eff is positive. Indeed, this branch starts at ϵ_eff = ϵ₂, when f = 1 and $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}=0$, and then monotonically ϵ_eff → ∞ when f approaches zero [9] (here and below the superscript (0) denotes the value of the permittivity of the gain medium which satisfies the dispersion equation (2) under the condition of Imϵ_eff = 0).

The numerical solutions of Eq. (3) (branch A) in the form of the dependences ϵ_eff (f) and corresponding $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}\left(f\right)$ at different values of d, including the local approximation (the quasistatic limit d → 0), are presented in Fig. 4. In the local approximation, the effective permittivity takes for this mode the well-known form

 

Fig. 4 The dependence of ${\mathit{ϵ}}_{eff}^{\left(A\right)}$ (green curves) and corresponding ${\mathit{ϵ}}_2^{\left(0\right)}$ (blue curves) vs f at different values of d for the even TM mode.

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As can be seen, nonlocality reduces the effective permittivity. At the same time, it can result in increasing as well as decreasing the gain coefficient needed to achieve the transparency condition (Imϵ_eff = 0) as compared to the case of local approximation. It is interesting to note that in the local approximation, there is a cut-off at $f=f_c={\mathit{ϵ}}_2^{″}/\left({\mathit{ϵ}}_2^{″}-{\mathit{ϵ}}_1^{″}\right)$ such that ϵ_eff → +∞ if $f\to f_c^{+}$ and ϵ_eff → −∞ if $f\to f_c^{-}$. However, no cut-off takes place when nonlocality is taken into account (see Appendix).

Branch B arises as the solutions of Eq. (4). The numerical solutions of this equation are presented in Fig. 5, where we do not show the region of large negative values of ${\mathit{ϵ}}_{eff}^{\left(B\right)}$. This branch includes both the traveling $({\mathit{ϵ}}_{eff}^{\left(B\right)}>0)$ and evanescent $({\mathit{ϵ}}_{eff}^{\left(B\right)}<0)$ mode solutions. For this branch, ${\mathit{ϵ}}_{eff}^{\left(B\right)}=\infty$ in the limit of f → 1, then it reduces, intersects the line of ${\mathit{ϵ}}_{eff}^{\left(B\right)}=0$, and becomes negative as f becomes smaller. One can show that the solutions for this branch can exist only when f lies within an interval of [f_min, 1]. As d → 0, this interval shrinks, and in the quasistatic limit the solutions disappear. This means that branch B cannot be described within the framework of the local approximation. Indeed, in the local approximation, Eq. (8) cannot provide ϵ_eff = 0, except for the trivial case when ϵ₁ = 0 or ϵ₂ = 0. At the same time, taking ϵ_eff = 0 in Eq. (4), one has the equation

 

Fig. 5 The dependence of ${\mathit{ϵ}}_{eff}^{\left(B\right)}$ (green curves) and corresponding ${\mathit{ϵ}}_2^{\left(0\right)}$ (blue curves) vs f at different values of d for the odd TM mode.

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For the TE-polarization, there is only one branch (A), which can give the traveling wave solution. As is seen from Fig. 3, this branch includes also evanescent mode solution, which can occur at smaller values of the parameter f.

3. High-momentum modes

As can be seen from Fig. 5, when dealing with branch B, to obtain ϵ_eff higher than 300, $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$ must be about −0.3 or lower, and this value only slightly depends on the lattice period d. This requires very high gain coefficient, that can involve technical difficulties. At the same time, for branch A, ϵ_eff ≈ 300 can be achieved at ${\mathit{ϵ}}_2^{″}\approx -0.25$ (see Fig. 4). Thus, the use of branch A is more preferable than the use of branch B, if we want to reach the very high permittivity at reasonable values of the gain coefficient.

High-k modes (branch A), which correspond to the extraordinary high effective permittivity ϵ_eff, were addressed in [9]. Here, we consider this issue in more detail. First, we elucidate the relationship between ${\mathit{ϵ}}_{eff}^{\left(A\right)}$, Imϵ₂, and the background permittivity, ${\mathit{ϵ}}_2^{\prime }$, at fixed d and different values of d₂. In Fig. 6, we show the dependences of $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}({\mathit{ϵ}}_2^{\prime })$ and ${\mathit{ϵ}}_{eff}^{\left(A\right)}({\mathit{ϵ}}_2^{\prime })$ at d = 90 nm for several values of d₂. The presented results evidence that a decrease in the background permittivity ${\mathit{ϵ}}_2^{\prime }$ leads to lowering both the transparency threshold $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$, that can be considered as a useful effect, and the effective permittivity ϵ_eff, that is detrimental from the practical point of view. In a similar fashion, a decrease in the thickness d₂, which corresponds to reducing f, leads to lowering both ${\mathit{ϵ}}_2^{\left(0\right)}$ and ${\mathit{ϵ}}_{eff}^{\left(A\right)}$.

 

Fig. 6 The dependence of ${\mathit{ϵ}}_{eff}^{\left(A\right)}$ (green curves) and corresponding $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$ (blue curves) vs Reϵ₂ for the even TM mode at d = 90 nm.

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Within the framework of our approach, as large as desired in-plane momentum can be obtained. In fact, however, it is bounded above by gain saturation, as well as due to the intrinsic nonlocality of the dielectric responses and crystallinity of the MM constituents [9].

Finally, it should be noted that Eq. (2) at high ϵ_eff can be approximately solved with respect to f. For this case, its closed-form analytical solutions for both branches, A and B, are given in Appendix.

4. Low-momentum modes

Low-momentum plasmonic modes can occur for the TM polarization (branch B) as well as for the TE polarization (branch A). However, in the limit of ϵ_eff → 0, Eqs. (4) and (5) (which describe the corresponding modes) coincide, yielding Eq. (9).

To determine the conditions for the existence of zero-momentum modes, we numerically solve Eq. (9) for different values of the background permittivity ${\mathit{ϵ}}_2^{\prime }$. The results are shown in Fig. 7. They evidence that a decrease in Reϵ₂ lowers the transparency threshold $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$, while a decrease in the lattice period d elevates it. Such a low transparency threshold as $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}\approx -0.117$ can be achieved at ${\mathit{ϵ}}_2^{\prime }=12$ and f ≈ 0.8026, d = 200 nm (k₀d = 0.8378), that is feasible with today’s technology. It is noteworthy that as the lattice period d becomes large, the value of $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$ only slightly depends on the background permittivity ${\mathit{ϵ}}_2^{\prime }$.

 

Fig. 7 The dependence $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}\left(k_{0}d\right)$ (blue curves) and corresponding f(k₀d) (green curves) at which ϵ_eff = 0 for different values of Reϵ₂.

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Although Eq. (9) is valid for both the TM and TE polarizations, the behavior of the solutions of Eq. (2) in the vicinity of the point of ϵ_eff = 0 is different. In particular, contrasting the TE polarization case, the TM case soulions of Eq. (2) are highly sensitive with respect to small changes in the material and geometrical parameters. For example, at d = 100 nm, one obtains ${\mathit{ϵ}}_{eff}^{\left(A,B\right)}=0$ as ${\mathit{ϵ}}_2^{″}=-0.2526$ and f = 0.8952. Meanwhile, ${\mathit{ϵ}}_{eff}^{\left(B\right)}=1$ as ${\mathit{ϵ}}_2^{″}=-0.2562$, f = 0.8965 (the TM polarization) and ${\mathit{ϵ}}_{eff}^{\left(A\right)}=1$ as ${\mathit{ϵ}}_2^{″}=-0.2358$ f = 0.905 (the TE polarization).

5. Discussion

The existence of two solution branches of Rytov’s dispersion equation for the TM polarization does not come as a surprise. In photonic crystals, it can manifest itself in the splitting of the incidence beam due to the appearence of an additional wave, as was shown for the metallic nanorod MMs [36,37]. However, the additional wave is characterized by a non-vanishing component of the quasi-microscopic electric field in the longitudinal direction E_║, which has a symmetrical distribution, in contrast to the antisymmetrical distribution of its transverse component [28]. This means that the odd TM mode cannot be excited by a normally incident transverse plane wave. In other words, only one (even) mode can be excited at normal light incidence. The coexistence of the two modes and mode mixing become possible for obliquely incident light only. This problem calls for further consideration and is out of the scope of the present work.

Since the odd TM polarized mode does not couple to the normally incident transverse wave, this makes it impossible to get ϵ_eff = 0 in this regime. For example, ${\mathit{ϵ}}_{eff}^{\left(B\right)}=0$ at d = 200 nm, when ${\mathit{ϵ}}_2^{\left(0\right)}\approx 12-i0.1167$.and f ≈ 0.8016. At the same time, for such parameters our calculations give ${\mathit{ϵ}}_{eff}^{\left(A\right)}\approx 14.84-i0.12$. This yields the effective refractive index $n_{eff}=\mathrm{Re}\sqrt{{\mathit{ϵ}}_{eff}^{\left(A\right)}}\approx 3.85$, as the zero-momentum TM polarized mode gives no contribution to both the effective permittivity and effective refractive index. To check this, we performed RCWA simulation [38] to calculate the normal transmission and reflection by a slab of the thickness h consisting of altenating metal and dielectric layers with the above parameters (not shown here). The results showed that both the transmission and reflection are oscillating functions of h with the oscillation period of about 390 nm. This is approximately equal to a half wavelength in a homogeneous slab with n = 3.85, as 1500 nm/3.85 ≈ 389.6 nm.

As was noted in Sec. 3, the proper choice of the background permittivity Reϵ₂ can be considered as a trade-off between the transparency threshold $\mathrm{Im}{\mathit{ϵ}}_2^{\left(0\right)}$ and the mode momentum (the effective permittivity ϵ_eff). For a fixed gain, however, higher ϵ_eff can be obtained at smaller Reϵ₂ and f values, as is seen in Fig. 6. A decrease in Reϵ₂, in turn, can be accomplished by tuning the composition of ternary or quaternary semiconductor compounds (see, e.g., [39]).

6. Closing remarks

In this paper, we provided a guideline for the practical implementation of extraordinary high- and low-momentum lossless plasmonic modes dealing with the simplest 1d bi-layer MM. In particular, for the TM polarization we showed that Rytov’s dispersion equation gives rise to two traveling mode solution branches. One of the branches is an ordinary solution which yields the well-known result for the effective permittivity in the quasistatic limit (the local approximation). In contrast, the other branch cannot be described within the local approximation framework. It represents an extraordinary (additional) wave, which results from strong optical nonlocality and cannot be excited at normal incidence. Furthermore, we demonstrated that both solution branches can describe plasmonic modes with infinitely high in-plane momentum, but the ordinary mode can be generated at a smaller gain coefficient providing lower transparency threshold. A decrease in the semiconductor background permittivity allows one to lower the transparency threshold, but it also lowers the nonlocal effective permittivity.

For the TE polarization, there is only one traveling mode, which can be excited at normal incidence and is capable to provide the low in-plane momentum. In the limit of zero momentum, this solution coincides with that for the TM polarization.

It is well known that the background permittivity sufficiently depends on the compound composition. Thus, one can conclude that tuning the composition of ternary and quaternary compounds, such as, e.g., InGaAs, InGaN, InAlN, AlGaN, InGaAsP, InGaAsAl and others, used for fabricating semiconductor quantum structures, could greatly improve the efficiency of loss compensation and facilitate the design of 1d MMs, supporting plasmonic modes with extraordinary high- and low momentum.

Finally, we would like to make a few remarks on the limitations of our study. Dealing with nonlocal response of the 1d MM, we did not take into account the nonlocal optical response of constituent phases. In some cases, especially for metals, this intrinsic nonlocality cannot be neglected [40]. Furthermore, when dealing with the ENZ regime, giant enhancement of the local electric field can occur that, in turn, can result in various nonlinear effects. All these issues remain open for further consideration.