## Abstract

We analyze the steady-state transmission of high-momentum (high-*k*) electromagnetic waves through metal-semiconductor multilayer systems with loss and gain in the near-infrared (NIR). Using a semi-classical optical gain model in conjunction with the scattering matrix method (SMM), we study indium gallium arsenide phosphide (InGaAsP) quantum wells as the active semiconductor, in combination with the metals, aluminum-doped zinc oxide (AZO) and silver (Ag). Under moderate external pumping levels, we find that NIR transmission through Ag/InGaAsP systems may be enhanced by several orders of magnitude relative to the unpumped case, over a large angular and frequency bandwidth. Conversely, transmission enhancement through AZO/InGaAsP systems is orders of magnitude smaller, and has a strong frequency dependence. We discuss the relative importance of Purcell enhancement on our results and validate analytical calculations based on the SMM with numerical finite-difference time domain simulations.

© 2015 Optical Society of America

## 1. Introduction

Metal-dielectric (MD) interfaces support optical surface waves whose effective wavelength is less than the smallest wavelength achievable in dielectrics [1]. The coupling of multiple MD interfaces enables even smaller effective wavelengths, or conversely higher effective indices, or so-called high-*k* propagation, of the optical waves [2–4]. The phenomenon of large effective indices and propagating high-*k* modes lies at the heart of sub-diffraction-limited imaging devices [5–7], sensors based on plasmonic resonances [8,9], as well as applications of hyperbolic metamaterials (HMMs), including asymmetric transmission devices [10], nonlinear optics [11], and lifetime reduction of dye molecules [12,13] and quantum dots [14]. Inherent to the increase in the effective index in MD systems, is a concomitant increase in dissipation losses [15, 16]. The tradeoff between optical confinement and losses is a hallmark of plasmonic and hyperbolic media, and may be considered one of the greatest challenges to the widespread realization of integrated plasmonic technology [17,18].

Strategies for reducing losses in plasmonic and hyperbolic metamaterials have therefore been proposed, including clever design of passive structures and incorporation of active, i.e. gain media [19–22]. An exemplary case of the former strategy is the fishnet metamaterial [23, 24], whereby air voids in an otherwise continuous MD multilayer open new channels for optical transmission. Even in these structures though, gain media are still necessary to obtain low-loss transmission at optical frequencies, as was demonstrated via infiltrated optical dyes in the core of the structure [25].

While dye molecules are straightforward to model as non-interacting two-level systems and convenient for proof-of-concept experiments [25–27], an ideal gain media in plasmonic systems would be electronically addressable [28]. In this case, semiconductor heterostructures become strong candidates for incorporation into plasmonic systems. III–V semiconductors emitting in the near-infrared (NIR) have been studied in conjunction with surface plasmon amplification [29], and successfully implemented in sub-wavelength metal-coated and plasmonic sources [30–32]. However, the incorporation of active III–V materials in multilayer MD systems with hyperbolic dispersion remains to be demonstrated in the NIR.

In this work, we go beyond effective medium theory [33], and apply the scattering matrix method [34] (SMM) to analyze the possibility for using indium gallium arsenide phosphide (InGaAsP) multiple quantum wells (MQWs) as a gain media in NIR MD systems. InGaAsP is a mature quaternary III–V compound that is epitaxially grown on indium phosphide (InP) substrates. This immediately poses challenges for its incorporation in a multilayer MD system. We envision deposition of metallic thin films into finely patterned trenches between InGaAsP pillars formed by nanoimprint lithography and subsequent reactive ion etching. We assume that the smallest layer thicknesses enabled by this process are 30nm. The resulting MD system, shown schematically in Fig. 1, supports volume plasmon polaritons [4] (VPPs) with large effective indices and large loss in the absence of external pumping, the hallmark of strongly coupled MD interfaces. These high-*k* VPPs propagate normal to the epitaxial growth direction, which is a novel multilayer configuration, potentially more suitable to waveguide-integrated HMMs, compared to the more common large-area multilayers which are conformal to the wafer substrate. In the following, we show that under moderate external pumping conditions, MD systems composed of InGaAsP MQWs may support transmission of high-index modes enhanced by several orders of magnitude, relative to the unpumped case.

In Section 2, we introduce the optical gain model employed, in conjunction with the SMM. We present the main results of the report in Section 3, namely, the NIR transmission characteristics of multilayer MD systems based on InGaAsP MQWs. We investigate InGaAsP in combination with aluminum-doped zinc oxide (AZO), which is considered a low-loss transparent conducting oxide and alternative plasmonic material for NIR applications [35–37], and in combination with silver, which is well known as the noble metal with highest conductivity [38]. In Section 4, we discuss the relative importance of Purcell enhancement on our results. We also validate the analytical SMM results with numerical finite-difference time-domain (FDTD) simulations. We then discuss the limitations of our results, as well as potential extensions. We conclude the report in Section 5.

## 2. Methods

The quaternary III–V compound that we consider is an In* _{x}*Ga

_{1−}

*As*

_{x}*P*

_{y}_{1−}

*MQW system where (x=0.564, y=0.933) and (x=0.737, y=0.569) for the wells and barriers, respectively. The nominal room-temperature bandgap energies of the barrier and well materials are 0.953eV (*

_{y}*λ*=1.3

_{G}*µ*m) and 0.774eV (

*λ*=1.6

_{G}*µ*m), respectively. This system can be epitaxially grown on an InP substrate and has been experimentally utilized in near-infrared subwavelength semiconductor lasers [32,39].

In the dipole-approximation the peak magnitude of the optical gain is governed by the transition matrix element, *M _{T}*. Because MD multilayers only support propagation of TM-polarized modes [3], we consider TM-polarized light emission from the quantum wells. To justify the use of a steady-state analysis and the non-inclusion of dynamic nonlinear effects, and for simplicity, we concern ourselves with moderate pumping levels, about two times above the transparency inversion density. Consequently, we may assume that most of the transitions occur at the band-edge such that recombination between electrons and holes with vanishing transverse momentum dominates [40]. In this limit, only transitions from the conduction band to the light-hole band contribute to the matrix element for TM-polarization [40]. We approximate the bandstructure of InGaAsP with parabolic conduction and valence bands, characterized by the conduction and light-hole effective masses of 0.0481

*m*

_{0}and 0.0537

*m*

_{0}, respectively [41], where

*m*

_{0}=9.1×10

^{−31}kg is the free electron mass. The magnitude of the matrix element used is given by [42, 43] |

*M*|

_{T}^{2}=(24.9eV)

*m*

_{0}/3, which corresponds to a dipole length of |x|

*≈*2.8nm at

*λ*

_{0}=1500nm.

Employing this gain model we obtain complex permittivity values for a 10nm InGaAsP QW at carrier densities representative of absorbing (Abs) and moderately inverted (Inv) states, *N*=1.0×16cm^{−3} and *N*=5.0×18cm^{−3}, respectively. At *λ*_{0}=1500nm, the imaginary parts of these values are *ε _{D,Abs}*”=Γg(

*N*=1.0×16cm

^{−3})=+0.139 and

*ε*”=Γg(

_{D,Inv}*N*=5.0×18cm

^{−3})=−0.143, where Γ=1/3 is the relative area of wells in the InGaAsP heterostructure. Using Kramers-Kronig relations, the real parts at

*λ*

_{0}=1500nm are ${\epsilon}_{D,Abs}^{\prime}=11.997$ and ${\epsilon}_{D,Inv}^{\prime}=11.881$, respectively. For the sake of simplicity, we used a carrier-independent but frequency dependent real permittivity, which is ${\epsilon}_{D}^{\prime}=11.914$ at

*λ*

_{0}=1500nm, and justified because the transmission properties studied are dominated by loss and gain. Relating the imaginary permittivity to the linear gain coefficient [30], $g=-{k}_{0}{\epsilon}_{D}\u201d{\left({\epsilon}_{D}^{\prime}\right)}^{-1/2}$, the value of

*ε*” = ±0.14 corresponds to a loss/gain per unit length of

_{D}*±*1700cm

^{−1}at 1500nm. The transparency (Tra) condition [40],

*ε*”=0, for In-GaAsP at

_{D,Tra}*λ*

_{0}=1500nm corresponds to about

*N*=2.0×18cm

^{−3}. The complex permittivity values for AZO [36] and Ag [38] at

*λ*

_{0}=1500nm are −0.392+i0.139 and −122.190+i3.115, respectively. Additional details on the gain model may be found in the references [40,44].

Combined with the presented optical gain model, we use the SMM as implemented by Krayzel et al. [34] to study the transmission properties of an idealization of the structure of Fig. 1. In comparison to simpler methods such as effective medium theory [33], Bloch’s theorem [22], and the transfer matrix method [20, 21, 45], the SMM is well known as a superior method for analyzing systems with evanescent waves [46,47] and strong loss/gain [48]. The idealized structure includes loss and gain, is infinite in the transverse (x–y) plane and surrounded by a uniform medium in the longitudinal (z) direction. To couple to the high-*k* modes supported by the MD multilayer of Fig. 1, the surrounding medium takes the form of a prism. The prism behaves as a numerical simplification to the physically equivalent grating-coupling technique [10, 12, 14, 49, 50], which is required for practical excitation of the high-*k* modes supported by each system.

To determine a suitable value of the prism permittivity, *ε _{P}*, we first determine the bounds of the high-

*k*transmission windows in each MD system. This is achieved using Bloch’s theorem [51] in the absence of losses by locating the regions in momentum space wherein the real part of the Bloch vector is purely real. For MD systems with 30nm layer thicknesses and

*λ*

_{0}=1500nm, Fig. 2(a) and 2(b) show the real (solid blue curve) and imaginary (dotted blue curve) parts of the Bloch vector, indicating that transmission windows exist when 0≤

*k*≤2.83

_{x}*k*

_{0}and 4.91

*k*

_{0}

*≤k*≤6.56

_{x}*k*

_{0}in AZO/InGaAsP and Ag/InGaAsP systems, respectively. In the limit of effective medium theory (dashed red curves), Fig. 2(a) and 2(b) are representative of type-I and type-II HMMs, respectively. Based on these high-

*k*windows, we set

*ε*=64 for the SMM calculations.

_{P}## 3. Results

Transmission and reflection at *λ*_{0}=1500nm through a 10-period AZO/InGaAsP structure with 30nm layer thicknesses, is shown as a function of incident angle in Fig. 3(a) and 3(b), respectively. The resonances extending from normal incidence to *θ _{inc}*=20° translate into a high-

*k*window that extends to roughly

*k*=8sin(20°)

_{x}*≈*2.8

*k*

_{0}, consistent with the prediction based on Bloch’s theorem of Fig. 2(a). The effect of inverting the carrier population in absolute terms appears most pronounced near normal incidence and becomes weaker as the incident angle increases. The strong absorption in this system is apparent in the local reflection minimum around

*θ*=7.5°. The lack of corresponding transmission maximum indicates strong dissipation, even when the semiconductor is inverted.

_{inc}Transmission and reflection through the AZO/InGaAsP system is best appreciated when contrasted to the behavior of the Ag/InGaAsP system. In Fig. 3(c) and 3(d), transmission and reflection for the latter system are shown, with identical conditions to the AZO-based structure. The transmission window spans approximately 38.5° to 54.5°, or about 5.0*k*_{0}*≤k _{x}*≤6.5

*k*

_{0}, again consistent with Bloch’s theorem of Fig. 2(b). The effect of inverting the carrier population becomes dramatic in both absolute and relative terms. For several resonances between 39° and 47°, or between 5.0

*k*

_{0}and 5.7

*k*

_{0}, the transmission and reflection exceed unity, indicating active behavior of the system.

To quantify the effect of the inverted carrier population on the behavior of the MD systems, we define the relative transmission enhancement factor (*TEF*) as

*TEF*for the AZO-based multilayer at several different incident wavelengths. This system exhibits significant dispersion due to the proximity of the plasma frequency of AZO to the NIR. For shorter wavelengths, enhancement is negligible, indicating that realistic gain levels cannot compete with the strong damping at the plasmon resonance. As the wavelength increases to 1550nm, a modest

*TEF*appears, increasing with the incident angle. However, this relative enhancement must be celebrated cautiously. The improvement in the

*TEF*with increasing angle of incidence is offset by a drop in the absolute transmission of Fig. 3(a). Consequently, the overall effect of carrier inversion is quite small in the AZO/InGaAsP system.

Again, these results are best appreciated by contrasting to the Ag/InGaAsP system, the *TEF* of which is shown in Fig. 4(b). Relative enhancements close to a factor of 10 are observed over the range of incident angles supported by this system, with several prominent peaks exceeding factors of 100 near the edges of the transmission window. In comparison to the AZO/InGaAsP system, the *TEF* has weak dispersion and is consistently larger in magnitude, indicating that the effect of the inversion in the Ag/InGaAsP system is much stronger.

Finally, the dependence of the average *TEF* on the number of periods for the Ag/InGaAsP system is shown in Fig. 4(c), where the *TEF* is averaged over the angular spectrum of the transmission window. The *TEF* is seen to increase exponentially with the length of the system. If the structure were used as an absorption modulator at a wavelength of 1500nm, for example, extinction ratios of 30dB were obtainable in a device length of just 2.1*µ*m. Thus, if efficient coupling to this system is achieved, tunable and/or active, pump-dependent behavior may be feasible, enabling extremely compact and potentially electronically addressable optical amplifiers, sources, mixers, and modulators.

## 4. Discussion

Prior work on gain-compensated plasmonic systems has emphasized that the Purcell effect has a deleterious role [28, 45]. Specifically, while the carrier densities required for significant improvement in transmission appear practically achievable, the required current densities may be extremely high [28]. This is a consequence of the large optical density of states in plasmonic and HMMs and the related high rates of spontaneous emission [52]. We stress, however, that the Purcell effect in MD systems is most significant near the plasma frequency of the constituent metal and becomes less important as the frequency decreases. Using the expression [28] *F _{P}*=1+

*π*Γ

*k*(

_{z,D}k_{x}ω*dk*/

_{x}*dω*)/(

*ε*)

_{D}k_{0}^{3}for the Purcell factor,

*F*, at a single MD interface, where

_{P}*k*is the longitudinal wave-vector component in the dielectric, and

_{z,D}*ω*and

*k*

_{0}are the angular frequency and vacuum wave-number, respectively, we immediately notice that the Purcell enhancement in the AZO/InGaAsP system is significantly greater than unity due to the proximity of the AZO plasma frequency to the NIR (Appendix 1). On the other hand, the Purcell factor for the Ag/InGaAsP system is on the order of unity because

*dk*/

_{x}*dω*≈0 (Appendix 1). Therefore, while Purcell enhancement may cause the already stringent gain requirements for the AZO/InGaAsP system to increase, they may be considered a minor concern for the Ag/InGaAsP system.

To validate the SMM results on MD systems, we performed numerical FDTD simulations (Lumerical) of the transmission through a 10-period Ag/InGaAsP system with 30nm layers under the prism coupling configuration and with material parameters identical to the SMM. Periodic boundary conditions and perfectly matched layers (PML) were enforced in the transverse and longitudinal directions, respectively. For each simulation, the center wavelength, *λ _{C}*, of the finite-bandwidth pulse was tuned to match the wavelength of the monochromatic SMM calculations,

*λ*

_{0}. Figures 5(a) and 5(b) compare transmission through this system at 1500nm and 1550nm, respectively, over a narrow angular range. The SMM is seen to consistently agree with FDTD results within a factor of two. Over the complete transmission window of the Ag/InGaAsP system, agreement between the SMM and FDTD worsens slightly, particularly at the second resonance of the transmission window (Appendix 2). The discrepancy between the SMM and FDTD results may be caused for several reasons, including imperfect PML, insufficient meshing, and the linewidth of the source. We provide FDTD simulation details in Appendix 2 and conclude that the finite-bandwidth of the FDTD source is the most likely reason for the discrepancy.

The close, but albeit non-exact, agreement between the SMM and FDTD in the static case suggests that FDTD may be useful for studying the dynamics of periodic MD systems with loss and gain. Throughout this analysis we have assumed a steady-state gain model, which implies continuous wave external pumping conditions. A logical extension of the present work is therefore an FDTD study incorporating spatial dependence of the gain within each semiconductor layer, as well as the use of a self-consistent gain model that takes into account transient and nonlinear effects [53, 54]. The inhomogeneously broadened lineshape of InGaAsP MQWs however poses challenges in this regard, as a simple 4-level system [55] does not capture the electronic density of states responsible for the broadening. A potential solution to this problem is the use of a superposition of 4-level systems, each with a slightly different transition frequency. The added accuracy of this approach, however, would come at the expense of large memory and processing requirements. Nonetheless, the utility of the FDTD method in studying transient and nonlinear effects is one of our motivations for comparing SMM and FDTD results under steady-state conditions.

## 5. Conclusion

We have analyzed transmission of high-*k* waves in the presence of loss and gain through metal-InGaAsP systems. We have shown that, under moderate pumping, Ag/InGaAsP may support transmission of high-*k* modes enhanced by several orders of magnitude, relative to the un-pumped case, and absolute transmission on the order of unity. While the magnitude of the enhancement factor for the AZO/InGaAsP system is significant, the absolute transmission is rather negligible. We have further shown that the Purcell effect does not have a strong deleterious role for the Ag/InGaAsP system. Finally, we have compared results based on the SMM with numerical FDTD simulations. The results provide a rigorous foundation for further theoretical work on the nonlinear dynamics of multilayer MD systems based on InGaAsP MQWs and justification for experimental efforts to circumvent the confinement-loss tradeoff fundamental to plasmonic and hyperbolic metamaterials.

## Appendix 1: Use of Bloch’s Theorem and the TMM in analysis of MD systems

To date, most work on the analysis of MD systems and HMMs has relied upon the transfer matrix method (TMM). For example, greater-than unity transmission through Ag/TiO_{2} multilayers was predicted via TMM by Cortes et al [52]. This result is unphysical because loss and gain were omitted. In the presence of loss and gain, Savelev et al. used the TMM method to analyze transmission through Ag/PMMA systems [45]. While the reported transmission is plausible, no attempt was made to validate the results with numerical methods.

In addition to transmission, the TMM is routinely used to calculate the complex amplitude reflectivity of HMMs, upon which the photonic density of states and emission lifetimes are often calculated. This was done, for example, by Galfsky et al. on quantum dots in Ag/Al_{2}O_{3} multilayers [14], by Lu et al. and Ferrari et al. on dyes in Ag/Si systems [12,56], and by Cortes et al. for Ag/TiO_{2} [52]. The TMM was also used to elucidate the presence of volume plasmon polaritons in Au/Al_{2}O_{3} multilayers by Zhukovsky et al. [52], and to calculate the bandstructure of HMMs in the presence of gain [57].

While useful qualitatively, the TMM is known to become increasingly numerically unstable as the structure increases in length [46], as strong loss or gain is incorporated [48], and as evanescent modes are considered [58]. The essential difference between the TMM and SMM lies in the fact that the matrices of the former contain exponential functions with arguments of different sign [46–48, 58, 59]. For large propagation lengths or large values of the imaginary component of the wave-vector, one of the exponentials will diverge. The matrices of the SMM on the other hand contain exponentials with arguments of the same sign, which prevent the instabilities that limit the utility of the TMM. Given these facts, it is imperative to use a more stable method for calculating absolute transmission of high-*k* modes in the presence of loss and gain, as well as relative transmission enhancement.

Nevertheless, the TMM does correctly predict the bounds to the transmission resonances in *k*-space. Because it is simpler to implement than the SMM, it is useful for studying the dispersion of MD systems. Figure 6 shows transmission contours with loss and gain omitted calculated via TMM with explicitly evanescent fields for 10-period 30/30nm MD systems. Qualitatively, we observe that the AZO/InGaAsP system of Fig. 6(a) exhibits little dispersion (*dλ*_{0}/*dk _{x}*≈0), while the Ag/InGaAsP system of Fig. 6(b) is highly dispersive (

*dλ*

_{0}/

*dk*≫0). This is a direct consequence of the fact that the plasma frequencies of AZO and Ag lie within and outside the NIR, respectively. The greater-than-unity transmission in the absence of gain is a quantitatively unphysical result and motivates the use of the SMM for absolute transmission and relative transmission enhancement calculations.

_{x}## Appendix 2: Discrepancy between SMM and FDTD

The discrepancy in absolute transmission between the SMM and FDTD calculations is almost negligible at the high-*k* end of the transmission window, but becomes significant at small values of *k _{x}*, in particular at the second angular resonance of Fig. 7(a), which is calculated at an angular resolution of 7 samples per degree. Potential reasons for this discrepancy include FDTD errors caused by imperfect PMLs, insufficiently small mesh size, and the finite bandwidth of the source. The bandwidth of the plane wave source used in the FDTD simulations was 200nm at a center wavelength of 1500nm or 1550nm. The timestep Δ

*t*was 0.00310473 fs, and was determined by the spatial step Δ

*z*and meeting the stability criterion that

*c*Δ

*t*/Δ

*z*< 1, where

*c*is the speed of light in vacuum. In all of the simulations we used 100 mesh cells per wavelength with a minimum mesh step of 0.1 nm and a grading factor of 1.41421. Bloch boundary conditions were used in the direction parallel to the layer interfaces and uniaxial anisotropic PML were used in the direction normal to the layer interfaces. The PML was polynomial order 3 and consisted of 12 layers with

*κ*=2,

*σ*=0.25, and

*α*undefined, where

*κ*,

*σ*, and

*α*are parameters for expressing Maxwells Equations in stretched-coordinate form [60].

Generally, PMLs are optimized to function at normal incidence and are known to produce unwanted reflections for broadband sources at oblique incidence, particularly for non-center frequencies [60]. To test this potential problem, we increased the PML thickness from 12 layers to 40 layers. However, we found negligible change in the transmission spectra. To test potential meshing errors, we reduced the mesh size from the default minimum of 4nm to 0.1nm in both the direction normal and parallel to the layer interfaces, but also found negligible change in transmission.

We did however find that the transmission spectra between SMM and FDTD most closely match when the center wavelength, *λ _{C}*, of the FDTD source coincides with the wavelength of the monochromatic SMM source. Figure 7(b) shows transmission spectra through the same Ag/InGaAsP multilayer at

*λ*

_{0}=1550nm, calculated by the SMM and with broadband FDTD sources having center wavelengths of 1500nm and 1550nm. The discrepancy between the SMM and FDTD result increases as the center wavelength deviates from 1550nm, suggesting that the finite bandwidth of the FDTD source, relative to the monochromatic SMM source, is the most plausible cause for the general discrepency between the SMM and FDTD results. Potential problems with FDTD simulations involving gain media also include convective and absolute instabilities [61], though these do not hamper our simulation results because we have assumed pumping levels to achieve carrier densities only a factor of two above the transparency condition [40].

## Acknowledgments

The authors wish to thank Prof. Antione Moreau for helpful discussions. This work was supported by the Office of Naval Research Multidisciplinary Research Initiative ( N00014-13-1-0678), the National Science Foundation (NSF) ( ECE3972 and ECCS-1229677), the NSF Center for Integrated Access Networks ( EEC-0812072, Sub 502629), the Defense Advanced Research Projects Agency ( N66001-12-1-4205), and the Cymer Corporation.

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