Supplementary Information for Negative refraction in semiconductor metamaterials

Transcription

1 Supplementary Information for Negative refraction in semiconductor metamaterials A.J. Hoffman *, L. Alekseyev, S.S. Howard, K.J. Franz, D. Wasserman, V.A. Poldolskiy, E.E. Narimanov, D.L. Sivco, and C. Gmachl * To whom correspondence should be addressed; ajhoffma@princeton.edu Included in this document: Supplementary Discussion 1 Supplementary Discussion 2 Supplementary Discussion 3 Supplementary Figures

2 Supplementary Discussion 1 Theoretical model of layers semiconductors with strong anisotropy Here we provide additional details for the theoretical model of our strongly anisotropic metamaterials: The anisotropy presented in the main text, ε < 0 < ε, is realized in n + -i-n + semiconductor heterostructures with appropriately thick layers that have alternating positive and negative dielectric constants. The thickness of the layers is chosen to be sufficiently large so that quantization of the energy levels in the system is irrelevant, but also thin enough, much smaller than the wavelength, so that the effective medium approximation is applicable. An effective permittivity tensor for heterostructures composed of n + -InGaAs and i-inalas is obtained by using a constant permittivity for the AlInAs layers and the Drude model for the doped InGaAs layers. Thus, ε AlInAs = ε -AlInAs and ε InGaAs (ω) = ε -InGaAs [1 - ω p 2 /(ω 2 iω/γ)], where ε -AlInAs = and ε -InGaAs = are the high frequency permittivities of their respective materials, ω p is the plasma frequency, and γ = 0.1x10-12 s -1, is the damping parameter; ε InGaAs = 0 at λ o, the wavelength corresponding to ω p, and the value of λ o decreases with increasing doping density, n d. The effective permittivity tensor for the layered system is uniaxial, and the two components, ε and ε are related to the isotropic permittivities ε InGaAs and ε AlInAs as follows: ε ε + ε 2 InGaAs AlInAs =, ε 2ε = ε InGaAs InGaAs Far from the plasma frequency of the InGaAs layers, ε approximately equals ε, and the sample behaves as an isotropic material with an effective permittivity approximately equal to (ε InGaAs +ε AlInAs )/2. As the wavelength increases and approaches the plasma frequency of the isotropic InGaAs layers, ε decreases, eventually becoming negative for λ > λ o. Figure 1a (main text) shows a plot of ε and ε as a function of wavelength. ε + ε AlInAs AlInAs

3 Supplementary Discussion 2 Discontinuity of the Brewster angle Here we present additional discussion and theoretical calculations on the discontinuity of the Brewster angle at the spectral location where ε = 0. Using the values for ε and ε shown in Fig. 1a (main text) and a transfer matrix method, we calculate the reflectance as a function of wavelength and incident angle for both polarizations for a half-infinite slab of material. Supplementary Fig. 1a displays several reflectance-versus-wavelength curves of the calculated data for the TM polarization on a logarithmic scale. The strong minima in the 58 and 69 degree curves result from the Brewster angle conditions being met at these wavelengths and incident angles. The different spectral locations of the minima are due to the frequency dependent permittivity of the metamaterial. The lack of a similar, well-defined minimum in the 64 degree curve shows a discontinuity of the Brewster angle and is characteristic of a material with ε < 0 < ε and the wavelength onset of the negative refraction regime. Isotropic, highly-doped materials, with an electric resonance at the same wavelength do not exhibit such feature. Rather, for these materials, the Brewster angle changes continuously with wavelength. Supplementary Fig. 1b depicts the TM reflectance, R TM, as a function of wavelength and incident angle in a log-scale colorplot. The Brewster angle, which is marked by the solid white line, exhibits a discontinuity at λ o = 8.8 μm. This wavelength corresponds to ε 0 and marks the beginning of the interval where negative refraction will occur. The reflectance of the TE polarization (R TE ) does not exhibit a Brewster angle or any discontinuities, and hence serves as a suitable spectral reference; a color plot for the TE polarization is shown in supplementary Fig. 1c.

4 a b c Supplementary Figure 1 Reflection calculations. a, Select traces of the TM reflectance (R TM ) versus wavelength for a simulated, half-infinite metamaterial with λ o = 8.8 μm. The strong minima in the 58 and 69 deg data occur at the Brewster angle. The lack of such a minimum for 64 deg, i.e. a discontinuity in the Brewster angle, is the salient feature of the transition into negative refraction. b, Colorplot of R TM versus wavelength and incident angle. The solid white curve marks the angle corresponding to the Brewster angle for each wavelength. The discontinuity at 8.8 μm marks the transition from normal refraction to negative refraction. c, Calculated transverse electric (TE) reflectance from a half-infinite slab of metamaterial with n d = 7.5 x cm -3. The TE polarization is not affected by the anisotropy of the material and therefore does not exhibit any strong features at λ o ~ 8.8 μm. The dip in reflection around 12 μm occurs because the real part of the effective permittivity for the TE polarization, ε, is approximately 0 and the imaginary part is becoming increasingly large.

5 Supplementary Discussion 3 Transmission theory and experimental results Here we present additional results and discussion concerning transmission measurements: Calculations of the transmission through a slab of material 8.08 μm thick on top of an InP substrate, n InP = 3.1, for the same dielectric tensor were also performed. Supplementary Fig. 2a displays calculated TM transmittance spectra, T TM, versus wavelength for three incidence angles, 10, 37, and 74 degrees, and Supplementary Fig. 2b is a colorplot of transmittance versus wavelength and incident angle assembled from many such curves. In both figures, the fringes are due to interference effects across the epitaxial layer, and the overall decrease in transmission with increasing wavelength is due to increasing free carrier absorption. The strong dip in the transmission around the critical wavelength λ o = 8.8 μm is due to the increasing imaginary component of the wavevector, which is related to loss, as the real part of ε approaches 0. Beyond the resonance, the increase in transmission is due to a decrease in the loss as ε becomes increasingly negative. Such behavior is indicative of the anisotropy of our metamaterial. Transmission measurements were performed on samples B and D as well as the high-doped InGaAs control sample. Spectra as a function of polarization, incident angle, and wavelength were collected for incident angles from 0 to 74 degrees in 2 degree increments. For all transmission measurements, the light was incident upon the substrate side, which was mirror polished to minimize surface scattering. The backgroundcorrected data were analyzed as the ratio of TM over TE. Experimental results for sample D are shown in supplementary Fig. 3a and the corresponding theoretical calculations are shown in supplementary Fig. 3d. The main feature of both plots is the minimum in the ratio around the transition wavelength λ o = 8.8 μm, resulting from the dip in the TM transmission. The presence of this feature and its spectral location agrees very well with the reflection measurements and our theoretical model of the metamaterial (see Fig. 2a and 2b in the main text).

6 a b Supplementary Figure 2 Transmission calculations. c, Select traces of the transverse magnetic transmittance (T TM ) for a simulated metamaterial with a thickness of 8.08 μm and λ o = 8.8 μm. The dip in transmission centered around 8.8 μm for the curves is due to the increasing imaginary component of the wavevector. d, Colorplot mapping the T TM versus wavelength and incident angle. The waviness of the graphs in (c) and (d) results from the finite thickness of the metamaterial slabs a b Supplementary Figure 3 Experimental results and theoretical calculations of transmittance. a, Measured ratio of TM/TE transmittance versus wavelength and incident angle for sample D with n d = 7.5x10 18 cm -3 and an epitaxial layer thickness of 8.08 μm. The dark blue region around 8.8 μm is a result of a dip in the TM transmittance. b, Theoretical calculations of (c) using the known material thickness and doping values.

7 Supplementary Figures a b Supplementary Figure 4 Reflection measurements for lower free carrier densities. Measured ratios of TM/TE reflectance versus wavelength and incident angle for (a) sample A with n d = 3.4 x cm -3 and (b) sample B with n d = 5.7 x cm -3. Both samples have an epitaxial layer thickness of 8.08 μm. The ratios are plotted on a logarithmic scale. The transition from positive to negative refraction for the samples occurs at 13.1 and 10.1 μm for (a) and (b) respectively. Supplementary Figure 5 Reflection measurements for a highly doped isotropic control. Measured ratios of TM/TE reflectance versus wavelength and incident angle for an isotropic control sample of highly-doped InGaAs on an InP substrate. The doping density is 5.7 x cm -3 and the epitaxial layer thickness is approximately 5.7 μm. The ratio is plotted on a logarithmic scale. No discontinuity, such as the one seen for the anisotropic metamaterials, is observed or where expected.

8 Supplementary Figure 6 Transmission measurements for the transverse electric polarization. Select curves of TE transmittance (T TE ) versus wavelength for sample D. The TE electric polarization is not affected by the anisotropy of the metamaterial and hence it displays no features at the transition wavelength, λ o = 8.8 μm. The decrease in transmission with increasing wavelength is due to an increase in the optical loss from free carrier absorption.