Metal-dielectric nanolaminates represent a class of hyperbolic metamaterials with uniaxial permittivity tensor. In this study, we critically compare permittivity extraction of nanolaminate samples using two techniques: polarized reflectometry vs. spectroscopic anisotropic ellipsometry. Both Au/MgF2 and Ag/MgF2 metal-dielectric stacks are examined. We demonstrate the applicability of the treatment of the multilayered material as a uniaxial medium and compare the derived optical parameters to those expected from the effective medium approximation. We also experimentally compare the effect of varying the material outer layer on the homogenization of the composite. Additionally, we introduce a simple empirical method of extracting the epsilon-near-zero point of the nanolaminates from normal incidence reflectance. The results of this study are useful in accurate determination of the hyperbolic material permittivity and in the ability to tune its optical properties.

Hyperbolic metamaterials possess unique properties due to extreme anisotropy in the uniaxial permittivity tensor. The hyperbolicity occurs when one of the diagonal tensor components is opposite in sign from the other two components. This unique material dispersion leads to novel optical properties, such as negative refractive index in a waveguiding geometry,1 nominally unbounded density of photonic states,2 and the existence of an ENZ (epsilon-near-zero) wavelength range for one of the permittivity components.3 

Metamaterial development relies on nanofabrication of features much smaller than the wavelength, traditionally requiring expensive techniques, such as electron beam lithography, with difficulty in scaling up samples to practical sizes. Notably, there have been recent demonstrations of large-area fabrication of two types of hyperbolic metamaterials: anodized alumina membranes, filled with metal nanocolumns4,5 and metallo-dielectric laminates.6 The uniaxial permittivity and hyperbolicity in both these types of materials arises from the effective-medium homogenization of their dielectric properties. The laminated metal/dielectric hyperbolic metamaterials are particularly attractive because their fabrication can be performed in a simple thermal evaporator over large sample areas.

Previous studies have treated nanolaminate materials as uniaxial media and have extracted their optical constants utilizing either polarized reflectometry6 or spectroscopic ellipsometry.7 Correspondingly, the effective medium approximation (EMA) theory has been used to derive the values of the permittivity components of the uniaxial tensor.8 However, the validity of the EMA theory has been questioned recently in theoretical treatments due to non-locality of the dielectric constant and the contribution of spatial dispersion or impedance singularities in reflection.9–12 Experimentally, application of the EMA approach to parameter extraction has produced unexplained resonances in the out-of-plane permittivity component.13 Additionally, EMA-based material representation was found to either underestimate or overestimate the Purcell effect.14,15

In this study, we perform a critical assessment of the experimental derivation of the uniaxial permittivity in metal-dielectric laminates, using two techniques: polarized reflectance and spectroscopic ellipsometry. We assess the extent of the homogenization of the layers, as measured by anisotropic spectroscopic ellipsometry, and compare the extracted parameters to the predictions of EMA approximation. When applying the EMA model, we verify the impact of optical constants of thin-film metal constituent layers on the prediction of the EMA. We test the effects of varying the outer layer of the multilayer stack on parameter extraction. Additionally, we introduce a simple empirical method of extracting the ENZ point of the nanolaminates from normal incidence reflectance and validate the results against those obtained with ellipsometry and derived from the EMA theory. The results of the study are important for the accurate determination of and the ability to tune optical properties of the hyperbolic material properties, enabling novel device design.

Effective permittivity of the multilayer laminates can be homogenized using Maxwell-Garnett formulas.14 Similar expression can also be obtained from the characteristic matrix formalism.8 For the permittivity components parallel and perpendicular to the laminate layers, one obtains

ε||=ρεm+(1ρ)εd,
(1)
1ε=ρεm+(1ρ)εd.
(2)

In the above, ρ = tm/(tm+td) is the filling fraction of the metal.

For a Drude metal, such as Au or Ag, the dielectric constant εm can be written as a function of frequency ω

εm(ω)=εm,r+iεm,i=εbωp2ω2+γ2+iγωp2ω3+γ2ω,
(3)

where εm,r and εm,i are real and imaginary parts of the metal's dielectric constant, respectively, εb is the static term due to bound charge and γ is the damping coefficient. As an example, for silver metal, εm,r is small and negative in the near-ultraviolet part of the spectrum, becoming progressively more negative towards higher wavelengths into the visible range.

Substituting Eq. (3) into Eqs. (1) and (2), it is easy to show that the real in-plane permittivity of the multilayer laminate will have a zero-crossing (ENZ point) when εm,r = −(1 − ρ)εd/ρ and will display metallic behavior for wavelengths above the ENZ point. At the same time, the out-of-plane permittivity will display a resonance occurring at a wavelength where εm,r = −ρεd/(1 − ρ). The strength of this resonance is limited by the damping constant γ of Eq. (3). For wavelengths above the ENZ point and sufficiently far away from this resonance, the composite material will exhibit hyperbolicity. The above-described behavior near the ENZ point is illustrated in Figure 1 for Ag metal,16 a dielectric with εd = 2 and ρ = 0.45. For this set of parameters, the ENZ point occurs near 370 nm. Additional hyperbolic region occurs when εm,r is negative and close to zero. In Figure 1, this second hyperbolic region occurs between 330 and 340 nm. By contrast to the first hyperbolic region, here, the real component of the in-plane permittivity is positive and the out-of-plane component is negative.

FIG. 1.

In-plane (||) and out-of-plane (⊥) components of the uniaxial polarizability tensor of laminate hyperbolic metamaterial as predicted by EMA theory in the vicinity of the ENZ point. Here, Ag is the metal layer, εd = 2 and the filling fraction is 0.45.

FIG. 1.

In-plane (||) and out-of-plane (⊥) components of the uniaxial polarizability tensor of laminate hyperbolic metamaterial as predicted by EMA theory in the vicinity of the ENZ point. Here, Ag is the metal layer, εd = 2 and the filling fraction is 0.45.

Close modal

To verify convergence of optical response of a multilayer stack to that of a uniaxial material described by EMA, we calculate reflectivity for a stack comprised of Au/MgF2 layer pairs of equal thickness. We keep the total thickness of the stack fixed at 200 nm and compute s- and p-reflectance at 45° incident angle and a 600 nm wavelength for an increasing number of bilayers. The calculations are performed using a commercial thin film package (WVASE from Woollam Corporation). For the refractive indices of Au and MgF2 thin films, literature values are used.16,17 As plotted in Figure 2(a), the optical response saturates past 5 bilayers and approaches an asymptote that is very close to the prediction of an EMA theory (shown as dashed lines in the figure).

FIG. 2.

(a) Homogenization of a Au/MgF2 multilayer stack reflectance as compared to EMA predictions. Solid lines and symbols are based on thin film interference theory for an increasing number of bilayers. Dashed lines indicate the limit as predicted by EMA. (b) and (c) Comparison of reflectance of a bare homogenized stack vs. reflectance of the homogenized stack with a 20-nm thick outer layer for Au/MgF2 stack (b) and Ag/MgF2 stack (c). Solid lines—homogenized uniaxial material. Dotted-dashed lines—addition of a 20-nm metal layer on top of the homogenized material. Dashed lines: addition of a 20 nm MgF2 layer on top of the homogenized material.

FIG. 2.

(a) Homogenization of a Au/MgF2 multilayer stack reflectance as compared to EMA predictions. Solid lines and symbols are based on thin film interference theory for an increasing number of bilayers. Dashed lines indicate the limit as predicted by EMA. (b) and (c) Comparison of reflectance of a bare homogenized stack vs. reflectance of the homogenized stack with a 20-nm thick outer layer for Au/MgF2 stack (b) and Ag/MgF2 stack (c). Solid lines—homogenized uniaxial material. Dotted-dashed lines—addition of a 20-nm metal layer on top of the homogenized material. Dashed lines: addition of a 20 nm MgF2 layer on top of the homogenized material.

Close modal

For a homogenized multilayer stack, it should make no difference whether the outer layer is a dielectric or a metal, according to the EMA formulation. This assumption is plausible for layers that are much thinner than the wavelength, but should eventually break down as the layers become thicker. To test this assumption, we modeled s-polarized 45° reflectance from a homogenized EMA layer with an additional thin film, either dielectric or metal, placed on top of the homogenized slab. For the EMA layer, we assumed a 200-nm thick slab with uniaxial dielectric constants described by Eqs. (1) and (2) and ρ = 1/2. We performed the modeling for both the Au/MgF2 (Figure 2(b)) and Ag/MgF2 (Figure 2(c)) metallo-dielectric stacks. We assumed a 20-nm film thickness for either the metal or the MgF2 additional layer. From the results of the calculations, at wavelengths ≥650 nm the response is the same for bare homogenized layer and such layer with either a metal or a dielectric thin film on top. Thus, homogenization can be achieved when the layer thicknesses are sufficiently smaller than the wavelength. On the other hand, for the wavelength below 650 nm, the response for the extra metal layer shows opposite reflectivity trend vs. the response for the extra dielectric layer, and the magnitude of the discrepancy increases towards lower wavelengths. Since the calculations of Figure 2(b) are based purely on thin film interference, this discrepancy is attributed to extra accumulated phase upon reflection as the film thickness becomes a significant fraction of the wavelength.

In this study, we characterized three hyperbolic multilayer stacks as well as their individual constituent films. All the samples were prepared at Norfolk State University by thermal evaporation of thin films onto glass substrates. Metal/MgF2 multilayers were produced by alternating deposition of metal and MgF2 with the sequential change of the source boat. Baseline chamber pressure was <10−6 Torr. Sample thickness was controlled by deposition time. After deposition, the overall thickness of the multilayer structure was verified to make sure that it met predictions. Scanning electron microscopy cross-sectional analysis of the multilayer samples was performed to confirm thicknesses of individual layers. The samples were measured at Norfolk State University using polarized reflectometry setup and, thereafter, shipped to MIT Lincoln Laboratory for ellipsometric measurements. Upon arrival to MIT Lincoln Laboratory, the samples were kept in a nitrogen-purged box to avoid contamination effects due to long-term storage. Ellipsometric measurements are extremely sensitive to monolayer effects of contamination and, thus, we carefully monitored any potential appearance of such effects. (One of the samples with Ag outer layer did exhibit contamination, as manifested by significantly reduced reflectance, but the results of this sample are not presented in the study.)

Sample 1 consisted of 7 layers of Au/MgF2 films, with Au being the outer layer. The nominal thickness of each layer for this sample was 15–20 nm. Sample 2 consisted of 7 bilayers of Ag/MgF2 films, with Ag being the outer layer. For this sample, the nominal thickness of each Ag layer was 25 nm and the nominal thickness of each MgF2 layer was 35 nm. Finally, sample 3 consisted of 6 bilayers of Ag/MgF2 films, with MgF2 being the outer layer. For this sample, the nominal thickness of each Ag layer was 25 nm and of each MgF2 layer was 35 nm.

Reflectometry measurements to extract anisotropic permittivity followed the procedure outlined in an earlier publication.5 In that study, for every wavelength of interest, s-polarized and p-polarized reflectance from the surface of the sample were measured. Reflectivity for s-polarization is given in terms of incident angle θ by

Rs=| sin(θθts)sin(θ+θts) |2,
(4)

where

θts=arcsin(sinθε||).
(5)

Reflectivity for p-polarization is given as

Rp=| ε||tanθtptanθε||tanθtp+tanθ |2,
(6)

where

θtp=arctanεsin2θε||εε||sin2θ.
(7)

The reflectance formulae are based on Fresnel equations and are exact, subject only to the following assumptions: (a) the sample transmission is negligibly small; (b) the material is a uniaxial semi-infinite medium with the optic axis perpendicular to the surface. The angular dependent reflectance measurements in this work were performed with a spectrophotometer reflectance attachment at either 543 nm or 550 nm. As an example, experimental reflectometry data for sample 1 are shown in Figure 3 along with the data fit utilizing Eqs. (4)–(7). We stress that the modeling approach does not assume any specific EMA theory; it only relies on the assumption that the modeled material is uniaxial and homogeneous.

FIG. 3.

Typical polarized reflectance from a Au/MgF2 laminate stack vs. angle of incidence at 550 nm. Symbols are experimental data, whereas solid lines are fit to the theory (see Eqs. (4)–(7) in the main text).

FIG. 3.

Typical polarized reflectance from a Au/MgF2 laminate stack vs. angle of incidence at 550 nm. Symbols are experimental data, whereas solid lines are fit to the theory (see Eqs. (4)–(7) in the main text).

Close modal

In addition to extracting anisotropic permittivity at several select wavelengths utilizing angular-dependent reflectivity, we developed an easy method to determine the ENZ point of the lamellar structures from spectroscopic normal incidence reflectance. We first determine the ENZ points from the wavelength-dependent in-plane permittivity components calculated using EMA (see Figure 4(a)) for the following structures: 1—Ag thick film, 2, 3—Ag/MgF2 stack with metal fill fractions ρ = 0.2 and 0.41, respectively, and 4, 5—Au/MgF2 stack with metal fill fractions ρ = 0.2 and 0.41, respectively. Reflectance spectra at normal incidence for the same set of samples, calculated using EMA, are shown in Figure 4(b). Inspecting the reflectance spectra, we observe that the sharp onset of increased reflectance lies in the vicinity of ENZ points determined using EMA (this onset also corresponds to the maximum value of the first derivative of the reflectance spectrum). We also observe that reflectance saturates at about twice its value at the ENZ wavelength. Thus, the ENZ point for a multilayered structure with any fill fraction can be easily obtained by first determining the onset of sharply increased reflectance (which is the maximum of the first derivative of the reflectance spectrum) and then measuring the wavelength at which the value of reflectance is ∼22% of saturation (or of twice the value of onset of increased reflectance). This procedure is illustrated in Figure 4(b) with vertical lines for traces 2 and 5 indicating the ENZ points. By plotting the ENZ wavelengths determined from calculated in-plane permittivity against those determined from reflectance spectra of metallo-dielectric stacks with varying metal fill fractions, we observe a good agreement with a discrepancy less than 9 nm (Figure 4(c)). For silver-based structures, the ENZ wavelength occurs where the first derivative of the corresponding reflectance spectrum reaches its maximum. Albeit a purely empirical method, our recipe for determining ENZ points is simple and effective. In the section below, we will compare ENZ points determined from this reflectance-based method with those determined from spectroscopic ellipsometry.

FIG. 4.

(a) Effective medium in-plane real permittivity components, calculated for the following structures: 1—thick silver film; 2 and 3—lamellar Ag/MgF2 structure with metal fill fractions ρ = 0.41 and 0.25, respectively; 4 and 5—lamellar Au/MgF2 structure with metal fill fractions ρ = 0.41 and 0.25, respectively. (b) Reflectance spectra calculated using EMA for the same samples as in (a). The vertical dashed lines intersect the reflectance curves at the ENZ point for samples 2 and 5. (c) Epsilon-near-zero wavelengths (λENZ) determined from the spectra of epsilons for silver (black markers) and gold (orange markers) based lamellar structures with varying metal fill fractions, plotted against the same determined from effective medium reflectance spectra.

FIG. 4.

(a) Effective medium in-plane real permittivity components, calculated for the following structures: 1—thick silver film; 2 and 3—lamellar Ag/MgF2 structure with metal fill fractions ρ = 0.41 and 0.25, respectively; 4 and 5—lamellar Au/MgF2 structure with metal fill fractions ρ = 0.41 and 0.25, respectively. (b) Reflectance spectra calculated using EMA for the same samples as in (a). The vertical dashed lines intersect the reflectance curves at the ENZ point for samples 2 and 5. (c) Epsilon-near-zero wavelengths (λENZ) determined from the spectra of epsilons for silver (black markers) and gold (orange markers) based lamellar structures with varying metal fill fractions, plotted against the same determined from effective medium reflectance spectra.

Close modal

Spectroscopic ellipsometry was used to characterize both multilayered hyperbolic materials as well as single layer constituent films. A commercial spectroscopic ellipsometer was used (Woollam, model M-2000) and the data were processed with WVASE software. The ellipsometric measurements were performed over a wavelength range from 250 nm to 1000 nm for seven incidence angles from 45° to 75°. The spot size of the ellipsometer beam was 1 mm in diameter for these measurements. In addition to ellipsometric measurements, near-normal-incidence reflectance data were obtained with a fiber-optic-based reflectance probe system over a wavelength range from 250 nm to 1000 nm. These data were fit together with ellipsometric data in order to assist in parameter extraction. With multiple data types and seven incidence angles, our data set overdetermines the four dielectric permittivity components for the uniaxial material model for each wavelength. Additional constraint on the extracted data was imposed by enforcing Kramers-Kronig consistency through the use of oscillator models for the permittivity dispersion.

1. Thin film optical constants

Before ellipsometric characterization of the full metamaterial stack, we measured optical constants of typical films, comprising the stack: an Au film and a thin MgF2 film on glass substrates. Figure 5(a) shows the real and imaginary parts of the dielectric constant of the Au witness film, as compared with literature data of evaporated Au films, obtained by Johnson and Christy.16 From the data, the permittivity of our Au film agrees quite well with the literature data. We also obtain optical constants of our thin MgF2 films (Figure 5(b)). The dielectric constants of these films are also consistent with previous measurements in the literature.17 

FIG. 5.

Permittivity of the constituent materials for the Au/MgF2 dielectric stack, as measured by spectroscopic ellipsometry. (a) Upper curves refer to the imaginary part of the permittivity, lower curves refer to the real part of the permittivity. Solid lines are for the control Au film, dashed lines are literature reference.16 (b) Real permittivity of MgF2 thin film. (Imaginary permittivity is close to zero over the measured wavelength range). Solid line is for the control MgF2 film, dashed line is the literature reference.17 

FIG. 5.

Permittivity of the constituent materials for the Au/MgF2 dielectric stack, as measured by spectroscopic ellipsometry. (a) Upper curves refer to the imaginary part of the permittivity, lower curves refer to the real part of the permittivity. Solid lines are for the control Au film, dashed lines are literature reference.16 (b) Real permittivity of MgF2 thin film. (Imaginary permittivity is close to zero over the measured wavelength range). Solid line is for the control MgF2 film, dashed line is the literature reference.17 

Close modal

2. Uniaxial parameter extraction

For evaluation of the full metamaterial stacks, anisotropic ellipsometry data were acquired following the Jones matrix formalism. According to this treatment, reflectance from an anisotropic sample can be described with a Jones matrix, J, given in terms of complex reflectance ρ as18 

J=ρSS(AnEAspAps×AnE1).
(8)

In the above, AnE, Asp, and Aps refer to ellipsometric parameters, defined as ρpp/ρss, ρsp/ρss, and ρps/ρpp, respectively. The subscripts of complex reflectances indicate how light of one polarization is converted into another: thus “ps” indicates how p-polarized light is converted into s-polarized light, etc.

Analysis of the experimentally obtained Jones matrices for all the samples in the study revealed that off-diagonal components, Asp and Aps were zero for all wavelengths. The lack of cross-polarization terms does not necessarily imply that our material is isotropic, but is also consistent with the uniaxial material whose optical axis is perpendicular to the surface normal. In fact, permittivity extraction assuming isotropic material led to a poor fit quality of the experimental, whereas uniaxial material model produced an excellent quality fit to the data and, thus, was used in this study.

For the multilayer hyperbolic stacks, a bulk uniaxial layer was used, which involved two components of out-of-plane permittivity and two components of in-plane permittivity. Initial data fitting was performed using a point-by-point wavelength data extraction over the full spectrum. At each wavelength point, four dielectric constants were extracted from the ellipsometric data at 7 angles for each wavelength and the near-normal reflectance. This procedure produces good quality fits, but noisy data, which is not necessarily Kramers-Kronig consistent. Thus, the fit data were further refined with a generalized oscillator model. For this study, hybrid Gaussian-Lorentz oscillators were used. Typically, four to five oscillators were required to describe the data over the spectrum, most of which with central energies in the UV-to-blue spectral range. Figure 6 shows experimental and fitted data for AnE ellipsometric parameters (see Eq. (8)), utilizing the fitting procedure outlined above for the Au/MgF2 multilayer stack, sample 1. In this figure, Ψ and Δ are conventional ellipsometric angles, related to AnE through

AnE=tan(ΨAnE)eiΔAnE.
(9)
FIG. 6.

Experimental (dashed lines) and fitted (solid lines) components of the anisotropic ellipsometric parameter AnE for the Au/MgF2 laminate, sample 1. Data fitting is performed using a uniaxial Kramers-Kronig consistent set of optical constants. (a) Ψ ellipsometric parameter and (b) Δ ellipsometric parameter vs. wavelength for seven different angles of incidence. Numbers next to the curve indicate the angle of incidence.

FIG. 6.

Experimental (dashed lines) and fitted (solid lines) components of the anisotropic ellipsometric parameter AnE for the Au/MgF2 laminate, sample 1. Data fitting is performed using a uniaxial Kramers-Kronig consistent set of optical constants. (a) Ψ ellipsometric parameter and (b) Δ ellipsometric parameter vs. wavelength for seven different angles of incidence. Numbers next to the curve indicate the angle of incidence.

Close modal

Excellent fit quality is obtained at all angles and all wavelengths. These results suggest that (a) homogenization of the multilayer stack does occur and (b) uniaxial permittivity model is a good description of the experimental data.

The results of the uniaxial permittivity extraction from the Au/MgF2 Sample 1 are shown in Figure 7(a) for the in-plane and Figure 7(b) for the out-of-plane permittivity. For the in-plane permittivity, the upper curves refer to the imaginary part, while the lower curves refer to the real part. For the out-of-plane permittivity, the upper curves refer to the real part, while the lower curves refer to the imaginary part. Within each set of curves, the solid line is the data extracted from the ellipsometric measurement, the dashed line is the values modeled with Eqs. (1) and (2) using the Au dielectric constant as measured above. In both Figures 7(a) and 7(b), the star symbols refer to the permittivity values obtained from the polarized reflectance measurements at 550 nm.

FIG. 7.

In-plane (a) and out-of-plane (b) components of the permittivity extracted from ellipsometry (lines) and polarized reflectance (star symbols). (a) Upper curves refer to imaginary permittivity component and lower curves refer to real permittivity component. (b) Upper curves refer to real permittivity component and lower curves refer to imaginary permittivity component. For all the data sets, solid lines refer to data derived from spectroscopic ellipsometry, and dashed lines are EMA predictions, using optical constants for the control Au layer.

FIG. 7.

In-plane (a) and out-of-plane (b) components of the permittivity extracted from ellipsometry (lines) and polarized reflectance (star symbols). (a) Upper curves refer to imaginary permittivity component and lower curves refer to real permittivity component. (b) Upper curves refer to real permittivity component and lower curves refer to imaginary permittivity component. For all the data sets, solid lines refer to data derived from spectroscopic ellipsometry, and dashed lines are EMA predictions, using optical constants for the control Au layer.

Close modal

The hyperbolic behavior of the multilayer stack is evident with the real part of the in-plane permittivity exhibiting “metallic” negative values and the real part of the out-of-plane permittivity exhibiting “dielectric” positive values. The in-plane permittivity becomes less negative towards shorter wavelengths, crossing zero near 490 nm at the ENZ point.13,19

EMA theory predicts the general trends of the experimental extraction. Below 500 nm for the out-of-plane permittivity, a strong resonance is observed, as discussed earlier in relation with Figure 1. The location of this resonance is close to the ENZ point for the in-plane permittivity, as predicted by the EMA theory.8 From the data of Figure 7, EMA predicts less-negative permittivity for the metallic in-plane constants and lower dielectric constant for the out-of-plane permittivity.

Comparing permittivity parameters derived from polarized reflectance vs. ellipsometry measurement (Table I and “star” symbols in Figures 7(a) and 7(b)), we find excellent agreement for the in-plane component, both for real and imaginary parts of the dielectric constant. For the out-of-plane component, the agreement is not as good for the real part of the dielectric constant. Ellipsometry-derived data are somewhat closer to the EMA prediction. Possible reasons for this poorer agreement between the methods will be discussed below.

TABLE I.

Comparison of uniaxial permittivity and λENZ of the Au/MgF2 multilayer stack sample 1, derived with reflectance and ellipsometry vs. the EMA prediction.

Methodε|| (at 550 nm)ε (at 550 nm)λENZ
Reflectance −2.8 + 1.36i 3 + 0.001i 487 
Ellipsometry −2.76 + 1.52i 6.38 + 0.38i 490 
EMA prediction −1.5 + 0.9i 5.7 + 1.1i 497 
Methodε|| (at 550 nm)ε (at 550 nm)λENZ
Reflectance −2.8 + 1.36i 3 + 0.001i 487 
Ellipsometry −2.76 + 1.52i 6.38 + 0.38i 490 
EMA prediction −1.5 + 0.9i 5.7 + 1.1i 497 

The results of the permittivity extraction for sample 2, Ag/MgF2 stack, are shown in Figure 8(a) for the in-plane permittivity and Figure 8(b) for the out-of-plane permittivity. For the in-plane permittivity, the upper traces in Figure 8(a) refer to the imaginary part and the lower traces refer to the real part of the dielectric constant. For the out-of-plane permittivity (Figure 8(b)), the upper traces refer to the real part and the lower traces to the imaginary part of the dielectric constant. In all cases, solid lines refer to parameters extracted from the ellipsometric measurements and the dashed traces refer to the prediction from EMA theory, utilizing thin-film optical constants from the literature.16 

FIG. 8.

In-plane (a) and out-of-plane (b) components of the permittivity extracted from ellipsometry (lines) and polarized reflectance (star symbols). (a) Upper curves refer to imaginary permittivity component and lower curves refer to real permittivity component. (b) Upper curves refer to real permittivity component and lower curves refer to imaginary permittivity component. For all the data sets, solid lines refer to data derived from spectroscopic ellipsometry, dashed lines are EMA predictions.

FIG. 8.

In-plane (a) and out-of-plane (b) components of the permittivity extracted from ellipsometry (lines) and polarized reflectance (star symbols). (a) Upper curves refer to imaginary permittivity component and lower curves refer to real permittivity component. (b) Upper curves refer to real permittivity component and lower curves refer to imaginary permittivity component. For all the data sets, solid lines refer to data derived from spectroscopic ellipsometry, dashed lines are EMA predictions.

Close modal

As in the Au/MgF2 multilayers, the hyperbolic nature of the multilayer composite is evident from the negative values of the in-plane real permittivity and the positive values of the out-of-plane real permittivity. The ENZ point occurs at 374 nm, as predicted by EMA and verified by ellipsometry and normal-incidence reflectance data (see Table II). Very good agreement between ellipsometrically extracted parameters and EMA prediction is found for both in-plane and out-of-plane permittivities for wavelength above the ENZ point. As in the data for Au/MgF2 stack above, we observe strong resonance near the ENZ point in the out-of-plane permittivity values.

TABLE II.

Comparison of uniaxial permittivity and λENZ of sample 2, derived with reflectance and ellipsometry with the EMA prediction.

Methodε|| (at 543 nm)ε (at 543 nm)λENZ
Reflectance −3.99 + 0.71i 1.29 + 0.04i 374 
Ellipsometry −3.68 + 0.67i 3.0 + 0.27i 374 
EMA prediction −3.85 + 0.35i 3.75 + 0.04i 374 
Methodε|| (at 543 nm)ε (at 543 nm)λENZ
Reflectance −3.99 + 0.71i 1.29 + 0.04i 374 
Ellipsometry −3.68 + 0.67i 3.0 + 0.27i 374 
EMA prediction −3.85 + 0.35i 3.75 + 0.04i 374 

Comparing predictions between ellipsometric data and polarized reflectance at 543 nm (“star” symbols in Figures 8(a) and 8(b) and Table II), we observe excellent agreement between both components of the in-plane permittivity. Additionally, experimentally derived values agree very well with the EMA prediction. A poorer agreement for the real part of the out-of-plane permittivity is observed.

For this sample, constituent layer thicknesses were significantly higher than for sample 1, with the MgF2 layer thickness of 35 nm as opposed to 20 nm for sample 1. Nevertheless, the homogenization procedure still worked well for this sample with a 20-nm Ag outer layer. As we shall see below, the homogenization does not work well for sample 3, where the outer layer is the 35-nm MgF2 film.

For a homogenized sample in a metal-dielectric stack, the identity of the outer layer should make no impact on the optical signature from the stack. Yet, the ellipsometric data of sample 3 with the MgF2 outer laser are significantly different from that of sample 2 with the Ag outer layer. Figure 9(a) shows the Ψ angle of the AnE ellipsometric data of sample 2, as compared with the same ellipsometric parameter measured for sample 3 (Figure 9(b)). Both sets of data are shown for seven angles of incidence, from 45° to 75° (indicated as number next to traces). Solid lines are raw data, while dashed lines are model fits to the data, as will be explained below. We show an expanded wavelength range between 350 nm and 500 nm to emphasize the differences between the samples, even though the full data set was acquired up to 1000 nm. Ψ values for sample 2 with Ag outer layer drop below 45° at lower wavelength, in qualitative agreement with the behavior of sample 1. By contrast, Ψ values for sample 3 with MgF2 outer layer display distinctly different behavior below 400 nm: at higher angles of incidence, the ellipsometric data increase steeply.

FIG. 9.

Comparison of the Ψ ellipsometric parameter for the Ag/MgF2 laminate with Ag outer layer (a) vs. MgF2 outer layer (b). Solid lines are experimental data and dashed lines are fitted data. The inset pictures of each graph show the model used for data fitting. Lines next to the curves indicate angles of incidence for the measurements.

FIG. 9.

Comparison of the Ψ ellipsometric parameter for the Ag/MgF2 laminate with Ag outer layer (a) vs. MgF2 outer layer (b). Solid lines are experimental data and dashed lines are fitted data. The inset pictures of each graph show the model used for data fitting. Lines next to the curves indicate angles of incidence for the measurements.

Close modal

While the data of Figure 9(a) can be fit with the homogenized uniaxial model, the same model clearly could not be applied to the data of sample 3 of Figure 9(b). However, a good fit to that data could be obtained adding a single 35-nm MgF2 film layer to the top of the uniaxial stack (shown as the inset of Fig. 9(b)). The breakdown of homogenization at lower wavelengths is consistent with qualitative predictions of Figure 2(b).

For all three samples described above, the ENZ point was extracted from the normal incidence reflectance and compared to that extracted with spectroscopic ellipsometry. The agreement between these two extraction methods (Figure 10, Tables I and II) is within a few nanometers.

FIG. 10.

λENZ determined from experimental reflectance spectra vs. the same determined from ellipsometry for samples 1, 2, and 3.

FIG. 10.

λENZ determined from experimental reflectance spectra vs. the same determined from ellipsometry for samples 1, 2, and 3.

Close modal

The results presented here demonstrate permittivity homogenization for a multilayered metal/dielectric medium. The homogenized material is uniaxial, as confirmed by anisotropic spectroscopic ellipsometry measurements, and has a hyperbolic dispersion above its ENZ point. We also confirm experimentally that layer thicknesses much smaller than the wavelength are required to ensure homogenization. Both sputtered and evaporated metal films become discontinuous below metal thickness of ∼20 nm—this may ultimately limit the lowest wavelength at which multilayer homogenization may be possible. It has been recently demonstrated, however, that by providing ultrathin wetting layers, continuous Ag metal films down to 5 nm can be deposited.20 Such fabrication techniques may ultimately extend material homogenization to lower wavelengths.

Dielectric constants of constituent thin films need to be measured in order to predict homogenized properties of the stack. Using the Au permittivity of the control film in Eqs. (1) and (2) for the effective EMA permittivity yielded good fit of the experimental results. If ultrathin metal films are used for constituent layers, we might expect their optical constants to change due to the effects of size of grain boundaries and the degree of crystallinity.21 

The role of ambient contamination cannot be ignored when studying optical properties of multilayer films, comprised of nanometer thick Ag layers. Silver is known to undergo tarnishing in ambient environment, through surface oxide or sulfide formation.22,23 Thus, a densified dielectric outer layer or another stable surface passivation layer may be required to protect Ag-based multilayer structures. Alternatively, recent results demonstrated that graphene layers can serve as an excellent protection against tarnishing of Ag-based plasmonic structures.24 

In comparing reflectance-based vs. ellipsometer-based parameter extraction, we find that in-plane permittivity results show excellent agreement between the two methods, while out-of-plane permittivity results do not agree as well, especially for the real part of the permittivity. The discrepancy may be based on experimental challenges of measuring out-of-plane permittivity. We also speculate that the out-of plane permittivity is more sensitive to breaking the validity of the EMA approximation.

We have presented a critical comparison of permittivity extraction for multilayered nanolaminated hyperbolic metamaterials utilizing two measurement techniques: polarized reflectometry and spectroscopic ellipsometry. Neither technique assumes a specific EMA model, only that the measured material is homogeneous and uniaxial. We have tested parameter extraction for Au/MgF2 and Ag/MgF2 multilayers. Additionally, we examined the effect of varying the outer layer (Ag vs. MgF2) on the optical properties of the nanolaminates.

Anisotropic ellipsometry data are well-described by a uniaxial material model. Experimentally derived permittivities confirm the hyperbolic nature of the material and the existence of the ENZ point. Above the ENZ point, good qualitative agreement of the derived optical properties with the EMA model is obtained. Agreement between reflectance and ellipsometry-derived data is excellent for the in-plane permittivity; it is somewhat poorer for the out-of-plane real component of the permittivity. Additionally, we have demonstrated a simple method for extracting ENZ wavelength with normal incidence spectral reflectance and showed a good correlation between this method and that utilizing spectroscopic ellipsometry.

For a sufficiently thick MgF2 outer layer (35 nm), the homogenization fails at lower wavelengths, in qualitative agreement with simulations. The ellipsometric data can still be well-described by a model, as long as the extra outer layer is explicitly added to the homogenized stack model.

The Norfolk State University portion of this work was supported by the NSF PREM grant DMR-1205457, NSF IGERT grant DGE-0966188, NSF HRD-1345215, AFOSR grants FA9550-09-1-0456 and FA9550-14-1-0221, and ARO grant W911NF-14-1-0639. The Lincoln Laboratory portion of this work was sponsored by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract No. FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.

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