## Abstract

We demonstrate experimentally signatures and dispersion control of surface plasmon polaritons from 1 to 1.8 µm using periodic multilayer metallo-dielectric hyperbolic metamaterials. The fabricated structures are comprised of smooth films with very low metal filling factor. The measured dispersion properties of these hyperbolic metamaterials agree well with calculations using transfer matrix, finite-difference time-domain, and effective medium approximation methods despite using only 2.5 periods. The enhancement factor in the local photonic density of states from the studied samples in the near-infrared wavelength region is determined to be 2.5-3.5. Development of this type of metamaterial is relevant to sub-wavelength imaging, spontaneous emission and thermophotovoltaic applications.

©2013 Optical Society of America

In recent years, there has been a growing interest in the use of local photonic density of states (LPDOS) enhancement of surface plasmon polariton (SPP) states in the near-infrared (NIR) [1] region (0.8-3 µm) for a vast range of applications including sub-wavelength imaging [2–8], spontaneous emission enhancement [9–11], nanobiosensing [12, 13], near-field energy transfer enhancement [14–19], and near-field thermophotovoltaics [20–26]. Therefore, the ability to control LPDOS through the SPP dispersion remains at the forefront of research. While encouraging progress has been made in the development of low-loss NIR plasmonic materials [27, 28], metamaterials can potentially offer a viable solution to the lack of low-loss plasmonic materials at NIR wavelengths.

The use of an array of cut-off waveguides to effectively reduce the plasma frequency of a metal from its bulk value, and thereby control the SPP dispersion was first proposed by Pendry et al. [29, 30], and subsequently verified experimentally at microwave frequencies [12, 31–33]. Since then, many metamaterial designs have been proposed to control SPP dispersion in the NIR range [34–36]. However, most of the proposed schemes (e.g., metal gratings and nanowire arrays) require high aspect ratios and/or submicron structural features, which are rather difficult to fabricate. To alleviate this difficulty, here we employ a design which consists of simple periodic multilayer metallo-dielectric structures to produce the metal-dilution effect. In the wavelength region and metal filling factors investigated here, the structure is referred to as a type II hyperbolic metamaterial [37], which possesses a uniaxial anisotropic dielectric tensor, with metal-like properties in the direction parallel to the surface (i.e. ${\epsilon}_{\left|\right|}<0$), and dielectric-like properties in the direction perpendicular to the surface (i.e. ${\epsilon}_{\perp}>0$). This type of structure has been used to create ultrahigh optical index cavities for broadband radiative emission control [10]. In this work, we focus on using the surface modes of hyperbolic metamaterials to create a concentrated narrow band electromagnetic energy through LPDOS control. Although the existence of SPP [38, 39] and bulk plasmon modes [40] of hyperbolic material have been previously shown, we here fully characterize the dispersion properties and demonstrate tunability as well as the validity of using effective medium approximations for Au-SiO_{2} multilayer metallo-dielectric hyperbolic metamaterials. The measured characteristics are in good agreement with calculations using the transfer matrix method (TMM), finite-difference time-domain (FDTD) approaches, and the effective medium approximation (EMA). This design offers flexibility and controllability of the SPP dispersion of the mode located at the air-metal interface.

The samples investigated in this work are composed of alternating layers of Ti-Au (1 nm of Ti as an adhesion layer and 5 nm of Au) and SiO_{2} (50, 100 or 250 nm) [see Fig. 1(a)], called sample A, B and C, respectively. The thin films were sequentially deposited on a glass substrate using thermal evaporation with a base vacuum pressure of 10^{−7} torr. A total of 5 layers (i.e., 2.5 periods) were deposited, with the top layer being Au. The dielectric functions of the constituent materials (Ti-Au, SiO_{2}, and glass substrate) were independently determined by ellipsometry measurements over the spectral range between 0.3 and 5 µm (VVASE and IRVASE from J.A. Woollam). The results reveal that the Ti layer does not introduce any noticeable change to the fitted dielectric function of the Ti-Au bilayer and therefore the bilayer is treated as a single layer in our calculations. The measured Au dielectric function shows Drude-like behavior with a negative real part of the permittivity which monotonically decreases with wavelength. The dielectric function also exhibits a monotonic increase in AC conductivity with wavelength, a clear signature that the film is above the percolation threshold [41, 42]. The reported percolation threshold thickness for Au is 6.5nm with no adhesion layer [42, 43]. The presence of Ti adhesion layer is expected to reduce the percolation thickness to a smaller value. A cross-sectional SEM image of sample B is shown in Fig. 1(b), and topographic images of the Ti, Ti-Au and SiO_{2} films obtained using atomic force microscopy are shown in Fig. 1(c)-1(e). The high quality of the films demonstrated by these images enables us to control the short wavelength limit of the SPP dispersion of Au from 1.0 µm to 1.8 µm, as we will show later.

The classic signature for the identification of the SPP mode is the presence of a reflectivity minimum at the SPP resonance frequency for *p-*polarized light using prism coupled attenuated total internal reflection (ATR) measurements. Conventionally, the ratio *R _{p} /R_{s}* is measured to reduce systematic errors, where

*R*and

_{p}*R*are reflectivities of

_{s}*p-*and

*s-*polarized light, respectively. These measurements were performed using an ATR apparatus mounted on the previously mentioned ellipsometers for accurate angle and polarization control. The results of ATR and non-ATR (i.e., with the prism removed)

*R*measurements for sample B are shown in Fig. 2(a) and 2(b), respectively. The ATR measured

_{p}/R_{s}*R*exhibits a prominent minimum [Fig. 2(a)] which is absent in the non-ATR ratio [Fig. 2(b), a signature of a SPP resonance excitation. As an additional confirmation, we obtained the absorption spectrum from the transmission and reflection measurements without prism coupling (data not shown) and did not observe any absorption feature that corresponds to the reflectivity minimum at the SPP resonance wavelength from the ATR measurements, as expected from a SPP state.

_{p}/R_{s}Figure 3(a) shows the experimentally measured SPP dispersion curve which was obtained by measuring the wavelength of the ATR minimum as a function of the angle of incidence. We simulated the dispersion curve using the TMM method as follows: Using the previously measured dielectric function for the Ti/Au bilayer as a starting point, we used the TMM method for the 5-layer stack to fit the *R _{p}/R_{s}* reflectivity curve at a single angle (usually 45 deg.) to obtain a slightly modified dielectric function for the metal layer in the full stack. In this fitting procedure all layer thickness were held at their as-designed values. The fitted curve is shown in Fig. 2(a). The fit obtained shows good agreement in the region of minimum reflectivity with slight deviation away from the minimum location. This difference is caused by inaccuracy of the Au dielectric function, and non-uniformities in thicknesses and dielectric properties of the film stack. Next, keeping all thicknesses and dielectric functions constant, we used the TMM method to generate

*R*for different angles of incidence, thereby mapping out the SPP dispersion curve shown in Fig. 3(a).

_{p}/ R_{s}We also utilized the effective medium description of the metamaterial to simulate the behavior of the multilayer samples. As noted previously, the effective properties of the multilayer metallo-dielectric metamaterial can be described by a uniaxial hyperbolic material whose dispersion relation [44] is given by, $\frac{{k}_{\perp}{}^{2}}{{\epsilon}_{\left|\right|}}+\frac{{k}_{\left|\right|}{}^{2}}{{\epsilon}_{\perp}}={k}_{0}{}^{2}$where the subscripts “||” and “$\perp $” indicate the directions parallel (in-plane) and perpendicular (out-of-plane) to the surface, respectively (notice the notation here for parallel and perpendicular are reverse of Schilling’s notation [44]), and *k* and *k _{0}* are the wavevectors in the medium and vacuum, respectively. The uniaxial anisotropic dielectric functions can be evaluated using mean field approach [45]: ${\epsilon}_{\left|\right|}=f{\epsilon}_{m}+(1-f){\epsilon}_{d}$and $\text{\hspace{0.17em}}1/{\epsilon}_{\perp}=f/{\epsilon}_{m}+(1-f)/{\epsilon}_{d},$where

*ε*and

_{m}*ε*are the dielectric functions of the metal and dielectric layers, respectively, and $f={t}_{m}/{t}_{t}$ is the metal filling factor, with ${t}_{m}$ and ${t}_{t}$being the thicknesses of metal and metamaterial period, respectively, as depicted in Fig. 1(a). For a small filling factor, the real part of ${\epsilon}_{\perp}$is almost the same as the real part of ${\epsilon}_{d}$, whereas its imaginary part becomes significantly larger than the imaginary part of ${\epsilon}_{d}$ due to the contribution from the lossy metal. On the other hand, both the real and imaginary part of ${\epsilon}_{\left|\right|}$ are reduced greatly by about a factor of

_{d}*f*when compared to${\epsilon}_{m}$. Therefore, varying the metal filling factor of the metamaterial structure offers a promising and effective way to control the SPP frequency. The trade-off of this control is a reduction of the most commonly used figure of merit (FOM) defined as the ratio of real and imaginary parts of the dielectric function. The FOM for the new effective medium plasmonic material can be related to the FOM of its metal constituent by, $\frac{\left|{\epsilon}_{//}^{\text{\'}}\right|}{{\epsilon}_{//}^{"}}=\frac{\left|{\epsilon}_{m}^{\text{\'}}\right|/{\epsilon}_{m}^{"}-\left(1-f\right){\epsilon}_{d}^{\text{\'}}/\left(f{\epsilon}_{m}^{"}\right)}{1+\left(1-f\right){\epsilon}_{d}^{"}/\left(f{\epsilon}_{m}^{"}\right)}$ where the single and double primes represent the real and imaginary parts of the dielectric functions. From this expression, we can easily see that the FOM of the effective medium is always smaller than its metal constituent. For the samples A, B and C, their peak FOMs values are reduced by a factor of 1.6, 1.8 and 2.3, respectively.

To explicitly show the hyperbolic nature of the material, we treat the film stack as a single layer of uniaxial effective medium material, and calculate *R _{p}/R_{s}* using the TMM. This will be referred to as the single layer uniaxial TMM model from here on. The ATR reflectivity of sample B obtained from the single layer uniaxial TMM model is shown in Fig. 2(a). The dielectric functions ${\epsilon}_{\left|\right|}$ and ${\epsilon}_{\perp}$of the layer are obtained from the equations provided above. For small metal filling factors, ${\epsilon}_{\perp}$remains largely unchanged from its dielectric constituent, whereas ${\epsilon}_{\left|\right|}$ will depend sensitively on the metal filling factor. Since our samples (A, B & C) all have only 2.5 periods with three layers of metal and two layers of dielectric, the metal filling factor is expected to be 50% higher than the value of an infinite medium which is given by the ratio of the metal thickness to the period. We determine the effective metal filling factor and the effective thickness of the film stack by fitting the

*R*reflectivity curve for a single angle of incidence using uniaxial TMM model. The obtained metal filling factors for sample A, B and C are 0.118, 0.076 and 0.032 which are 31%, 58% and 63% higher than their corresponding infinite medium values of 0.09, 0.048, and 0.0196, respectively. The ${\epsilon}_{\left|\right|}$ dielectric functions obtained from this procedure are shown in Fig. 4.

_{p}/R_{s}Once the filling fraction and the dielectric functions have been determined, the SPP dispersion curves for the uniaxial effective medium are obtained by varying the angle of incidence in the TMM simulations. In addition, the SPP dispersion curves for a single layer of uniaxial effective medium material of finite thickness *t _{t}* can be obtained analytically from the transcendental equation [46]

*ε*is the dielectric function of glass substrate, and ${\epsilon}_{1}=1$ for SPP located at the air-sample interface. The analytical SPP dispersion obtained by solving the transcendental equation using an iterative numerical method shows good agreement with that obtained from the uniaxial TMM calculations as shown in Fig. 3(a). It is worth noting that the dispersion equation based on semi-infinite uniaxial material [47] can only provide qualitative agreement (data not shown).

_{sub}Since LPDOS control depends on the ability to tune the dispersion of SPP, we fabricated two other samples with 50 and 250 nm thick SiO_{2} layers (sample A and C) to demonstrate this design flexibility. The experimentally determined dispersion curves of these samples are also shown in Fig. 3(a). We obtained similarly good agreement with the measured SPP dispersion using the 5-layer TMM, the single layer uniaxial TMM model, and the analytical calculations as sample B. The values of the short wavelength limit of the SPP, near the maximum wavevector, are shown in the inset table of Fig. 3(a). From these samples, we demonstrated experimentally more than 3x increase in the SPP wavelength compared to bulk Au by changing the metal filling factor. The LPDOS can be estimated from the measured dispersion curves of the structure by observing that the LPDOS is proportional to *dk/dω* = *1/v _{g}*, where

*v*is the group velocity. The LPDOS enhancement factors with respect to free space determined from the experimental data for the three fabricated samples are plotted in Fig. 3(b). The maximum LPDOS values observed from the three samples are in the range of 2.5-3.5, where these values are limited by the angular steps and spectral resolution of the experimental data taken. We expect significantly higher enhancement factor can be realized using low-loss Ag as metal.

_{g}The identification of these resonances as SPPs is also supported by FDTD simulations using a commercially available package (Lumerical Inc.) which show the expected electric field pattern [shown in Fig. 5(a)] and enhanced electric field intensity near the surface (shown in Fig. 5(b)) for a SPP state. The reflectivity ratio on the top metal surface retrieved from the FDTD simulation results is in very good agreement with 5-layer TMM calculations (shown in Fig. 2(a)). The square of the field is enhanced by a factor 14 when the incidence angle is 45°, and the 1/e intensity decay length normalized to the SPP wavelength is 0.2 [47]. For additional confirmation that the 5-layer stack can be treated as effective medium, we performed FDTD simulation on a single uniaxial layer of effective medium using the in-plane and out-of-plane dielectric functions obtained from ellipsometry, and using a thickness equal to that of the entire multilayer. The surface normal component of the electric field pattern of the SPP mode [shown in Fig. 5(c)] obtained closely resembles that of the multilayer.

In summary, we have experimentally demonstrated SPP dispersion control using high quality hyperbolic metamaterials consisting of Au-SiO_{2} metallo-dielectric multilayer thin films with metal filling factors ranging from 0.032 to 0.118. We have found that the experimentally observed SPP dispersion can be accurately described using 5-layer TMM, the uniaxial TMM model, FDTD, and EMA calculations. The hyperbolic nature of the material is also explicitly demonstrated. In addition, we observed a 2.5-3.5 fold enhancement in the local photonic density of state for the frequency range and filling factors explored in this paper. We expect the photonic density of state enhancement can be significantly improved with the use of Ag or by annealing the samples [48, 49].

## Acknowledgments

The authors acknowledge stimulating discussions with Yong Zeng, and Young Chul Jun. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science and Office of Basic Energy Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

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