Luminescent hyperbolic metasurfaces

When engineered on scales much smaller than the operating wavelength, metal-semiconductor nanostructures exhibit properties unobtainable in nature. Namely, a uniaxial optical metamaterial described by a hyperbolic dispersion relation can simultaneously behave as a reflective metal and an absorptive or emissive semiconductor for electromagnetic waves with orthogonal linear polarization states. Using an unconventional multilayer architecture, we demonstrate luminescent hyperbolic metasurfaces, wherein distributed semiconducting quantum wells display extreme absorption and emission polarization anisotropy. Through normally incident micro-photoluminescence measurements, we observe absorption anisotropies greater than a factor of 10 and degree-of-linear polarization of emission >0.9. We observe the modification of emission spectra and, by incorporating wavelength-scale gratings, show a controlled reduction of polarization anisotropy. We verify hyperbolic dispersion with numerical simulations that model the metasurface as a composite nanoscale structure and according to the effective medium approximation. Finally, we experimentally demonstrate >350% emission intensity enhancement relative to the bare semiconducting quantum wells.


Supplementary
. Measured pump PA in LuHMS with gratings and tolerance of PA to fabrication variability. Reduction of pump PA of PL in LuHMS samples by incorporation of grating couplers, showing consistency in reduction of PA with gratings of periods ranging from 170nm to 520nm. (a-c) Three samples exposed to increasingly intense electron beam doses. Figure 11. Enhancement of PL spectra. (a,b) PL spectra and (c,d) intensity enhancement spectra in LuHMS (a,c) without and (b,d) with SiO x insulation layer, under reverse excitation and fixed average pump power of 5 mW. Relative to InGaAsP MQW and a single, flat Ag/InGaAsP MQW interface, the nanostructured Ag/InGaAsP MQW system exhibits roughly 3.5x and 1.25x stronger PL intensities across the emission spectrum. Total PL signal of flat and nanostructured Ag/InGaAsP MQW is increased by ~1.25x in the presence of SiO x , suggesting that quenching is reduced via insulation layer. with SiO x insulation layer under reverse excitation. Relative to InGaAsP MQW and a single, flat Ag/InGaAsP MQW interface, the nanostructured Ag/InGaAsP MQW system exhibits roughly 3.0x and 1.25x stronger PL intensities across the range of studied pumping power. Total PL signal of flat and nanostructured Ag/InGaAsP MQW is increased by ~1.25x in the presence of SiO x , suggesting that quenching is reduced via insulation layer. Figure 13. Comparison of PL spectra before and after Ag deposition. Total PL spectra of nanostructured InGaAsP MQW (a,b) before and (c,d,) after Ag deposition for pump polarized (a,c) parallel and (b,d) normal to metacrystal Bloch vector, with average pump power as a parameter. Spectra are normalized to peak value before Ag deposition. Figure 14. Principle and realization of meta-gain media. (a) Simultaneous co-optimization of pump properties, electronic density of states (DOS) and optical DOS enables, in principle, tailor-made emission spectra with gain properties beyond those of constituent materials. Evolution of PL spectrum with power for (b) control InGaAsP MQW, (c) LuHMS with normal-polarized pump, and (d) LuHMS with parallel-polarized pump. (Electronic and optical DOS schematics were modified from 1 and 2 , respectively.)

Supplementary Note 1. Description of the Ag/InGaAsP MQW system under the effective medium approximation (EMA) and Bloch's theorem.
The EMA is a powerful tool for gaining intuition about complex composite media 3 . Essentially, the EMA transforms the periodic, inhomogeneous Ag/InGaAsP system into an anisotropic, homogeneous material, with properties governed by those of the constituent materials and their respective ratio. We here show that, under both the EMA and Bloch's theorem, the Ag/InGaAsP system exhibits hyperbolic dispersion for a wide range of ratios throughout the telecommunication and near-infrared frequency range.
We first approximate our fabricated structure as a one-dimensional infinitely periodic system. Bloch's theorem may then be invoked to determine the range of transverse momentum states supported by the system, Δk ┴ . This is done by solving for the Bloch vector, K B , of the system, given by 4 In Eq. (1) a is the length of one period, equal to the sum of the Ag and InGaAsP layer thicknesses, t M and t D , respectively. In Eqs. (2)-(3) the complex, frequency dependent dielectric function of Ag, ε M , is based on experimental data 5 , whereas that of InGaAsP, ε D , is based on a combination of experimental 6 and theoretical data 7,8 . Furthermore, ε D also depends upon the free carrier density, N, which is controlled by external pumping. The longitudinal wave components within the Ag and InGaAsP layers, k ||,M and k ||,D , respectively, are related to the conserved transverse component, k ┴ , and vacuum wavenumber, k 0 =2π/λ 0 by Experimentally, we verified that t M ≈t D ≈40nm. However, to account for sample nonuniformities we allow the material ratio, ρ=t M /t D , to vary, while keeping the period fixed to a=80nm. The zeroth-order EMA of Eq. (1) is where   In Eq. (6) the negative root is selected because the condition  is required to satisfy the causality constraint 9,10 . To most clearly illustrate the broadband hyperbolic dispersion of the Ag/InGaAsP system, Supplementary Figs. 1(a) and 1(b) show the real part of the effective permittivity elements parallel and normal to K B , respectively. Hyperbolic dispersion exists for all values of (λ 0 ,ρ) such that || 0    , which occurs throughout the plotted parameter space except for ρ<0.1. Two primary sources of losses in the system are evident from Supplementary Figs. 1(c) and 1(d), which show the imaginary part of the effective permittivity elements parallel and normal to K B , respectively, with N=1x10 16 cm -3 . Firstly, as the Ag fraction increases, Ohmic losses increase. Secondly, absorption at the band-edge of InGaAsP MQW leads to an abrupt increase in losses at λ 0 ≈1.55 μm.
To illustrate the limitations of the EMA, Supplementary Fig. 2 shows the solutions to Eq.
(1) and (6) with losses omitted, for ρ=0.3, ρ=0.5, and ρ=0.7 at the pump wavelength of λ 0 =1064 nm and the emission wavelength of λ 0 =1550 nm. The solid (dashed) blue curves correspond to real (imaginary) parts of K B , whereas the red curves correspond to the purely real k B,EMA . Optical states with transverse momentum exceeding that of the constituent MQW, k ┴ /k 0 >3.5, are clearly present. As the wavelength increases from 1064 nm to 1550 nm, the EMA more closely matches the complete solution, as expected. For all wavelengths, the EMA performs best at K B =0, also as expected for a local (zeroth order) theory. As the Ag ratio increases, the EMA becomes quite poor, especially at the pump wavelength. Nonetheless, the existence of non-zero real solutions to Eq. (1) shows that the Ag/InGaAsP MQW system supports hyperbolic dispersion over a large region of the (λ 0 ,ρ) parameter space, covering the wavelengths of the pump and all MQW emission.

Supplementary Note 2. Experimental setup and properties of control InGaAsP MQW.
A simplified schematic of the experimental setup used for characterization of the fabricated samples is shown in Supplementary Fig. 3. The half-wave plate closest to the pump (HWP 1) was first rotated to maximize the TM || polarized pump exiting the polarizing beam splitter (PBS). HWP 2 functioned as a variable polarization rotator. The dichroic mirror (DM) used also rotated the polarization state of the pump by a fixed 90°, such that the polarization state incident on the sample was normal to that exiting the HWP 2. While the DM reflected orthogonal polarization states exiting HWP 2 almost equally, the pump transmission through the DM was highly polarization dependent, differing by several orders of magnitude. Therefore our setup could not be used directly to measure pump reflection and we relied upon the measured photoluminescence (PL) signal to indicate the effective pump absorption. To measure polarization of the PL, the PL was passed through a linear polarizer (LP) before reaching the detector. For measurements of total PL, the LP was removed. To remove inconsistencies associated with changing the focal plane of the sample, all sample were focused such that the detected signal at the wavelength of 1550 nm was maximized.
The InGaAsP MQW wafer that we used showed no pump polarization anisotropy (PA), however, partially polarized emission was observed. Supplementary Fig. 5(a) shows the total PL induced by parallel and normal-polarized pumps at several pump powers. Supplementary Fig.  5(b) quantifies the pump PA, which is close to unity over the entire spectrum, indicating that the PL is independent of pump polarization. Supplementary Fig. 5(c) shows the total PL, along with the PL resolved into parallel and normal polarization components. A clear difference is observed. The degree-of-linear-polarization (DOLP) quantifies this difference in Supplementary  Fig. 5(d), which shows that the emission is predominantly normal polarized.
To understand the origin of the peaks in the emission spectra, we theoretically calculated the spontaneous emission and gain spectra of a 10 nm InGaAsP QW, according to the method outlined in 7 . A valence band offset of 0.55(E G,B -E G,W ) was used where, E G,B and E G,W are the bandgap energies of the barrier and well materials, respectively, both depending on temperature and carrier density 7 . Supplementary Fig. 8 shows emission peaks at ~1550 nm, ~1450 nm, and ~1350 nm, which arise from transitions between the first conduction and heavy-hole, first conduction and light-hole, and second conduction and heavy-hole subbands, respectively. As the pumping strength is increased the spectra blue-shift due to filling of higher energy states. For pump powers used in our experiment, the peak at 1550 nm dominates the control MQW, suggesting that experimental carrier densities do not exceed 3x10 18 cm -3 .
We also characterized emission from the InGaAsP MQW after etching into nanostructures, but prior to Ag deposition. Supplementary Figs. 13(a) and 13(b) show that the spectra of MQW prior to Ag deposition for parallel and normal pump polarizations are nearly identical, indicating that etching has negligible effect on the response of the material to different pump polarizations. After Ag deposition, however, the spectra show a strong dependence on pump polarization, as shown in Supplementary Figs. 13(c) and 13(d). Therefore the presence of Ag, and the consequent hyperbolic dispersion, is necessary to achieve extreme polarization anisotropy.

Supplementary Note 3. Numerical simulations
To better understand experimental results, we performed numerical finite-difference time-domain (FDTD, Lumerical®) simulations at pump and emission wavelengths. The LuHMS was modeled both as the exact (as-fabricated) nanostructure and by the EMA, shown schematically in Supplementary Figs. 4(a) and 4(c). The NS consists of a 100 nm tall and 40 nm wide InGaAsP pillar clad with a 20nm tall and 10 nm wide Ag layer, atop a 200 nm tall and 80 nm wide InGaAsP base. Both materials have a frequency dependent, complex-valued permittivity 5,6,8 . The EMA model consists of a 300 nm tall and 80 nm wide effective medium assuming Ag fraction of ρ=0.5. In both models periodic boundary conditions and perfectly matched layers are employed along the z-coordinate and x-coordinate, respectively. Pumping is simulated by a monochromatic plane wave of wavelength λ 0 =1064 nm and polarization parallel or normal to the z-coordinate, which is parallel to the metacrystal Bloch vector K B . Results are shown in Supplementary Fig. 4(b). The calculated absorption anisotropies of 26 and 17 for the LuHMS modeled by exact nanostructure and EMA, respectively, are in excellent qualitative and good quantitative agreement with measured values of pump PA. The close agreement between NS and EMA models further validates the use of the EMA in describing the LuHMS. Emission is simulated similarly, but with a planewave source incident from the opposite direction. Results at the emission wavelength of 1350 nm are shown in Supplementary Fig. 4(d). The simulated DOLP is calculated as the transmission anisotropy and is seen to be in good qualitative agreement with experimental values.
To confirm that extreme anisotropy is an effect of hyperbolic dispersion and not a simple artefact independent of period size, we simulated pump behavior over a large range of period lengths. Supplementary Fig. 6(a) shows absorption of the parallel and normal-polarized pump for period lengths from 20 nm to 800 nm and constant Ag fraction of ρ=0.5. Anisotropy is strongest when the EMA is most valid and the Ag/InGaAsP system exhibits hyperbolic dispersion. As the period increases, the anisotropy becomes significantly smaller, confirming that the extreme anisotropy measured in our samples results from hyperbolic dispersion enabled by deeply subwavelength structuring. This is further supported through calculation of the modes of the system, according to Eq. (1). Supplementary Fig. 6(b) shows, for the pump vacuum wavelength of 1064 nm, that only a single parallel (TM || ) polarized Bloch mode exists below a critical period length. As the period length increases, normal-polarized (TE ┴ ) modes are supported, reducing the absorption anisotropy illustrated in Supplementary Fig. 6(a). Hence, deeply subwavelength periodicity, and hyperbolic dispersion in the effective medium limit, is required to observe extreme polarization anisotropy.
To increase absorption of normal-polarized pump absorption and PL emission, we designed a wavelength-scale grating based on, both, infinitely-extended multilayer and EMA models. Supplementary Fig. 6(c) shows simulated absorption of parallel and normal polarized pumps as a function of grating depth for the Ag/InGaAsP system modeled in the EMA with a grating of period Λ G =390 nm and ρ=0.5. Pump PA decreases dramatically in the presence of a grating, consistent with our experimental observations. Similar results were found for different grating periods.

Supplementary Note 4. Directional propagation properties of LuHMS
It is well known that energy propagation in media with hyperbolic dispersion is highly directional, forming resonance cones 11,12 . The resonance cone half-angle determines the principal direction of energy propagation and, in the EMA, is defined as where the angle is measured relative to the metacrystal axis. The half-angle may also be directly described with the Poynting vector, where the parallel wave vector component is calculated by EMA or Bloch's theorem. On the other hand, the principle direction of the wave vector is given by which is known to be counterposed to the Poynting vector in media with hyperbolic dispersion 13 . Supplementary Fig. 9(a) shows a schematic of the multilayer, defining the resonance cone angle and effective permittivity at this angle. In Supplementary Fig. 9(b) the dispersion of the imaginary effective permittivites is shown, while in Supplementary Fig. 9(c) the wavelength dependence of Eqs. (9)-(11) for our LuHMS made of Ag and InGaAsP MQW is shown. Regardless of the technique used to calculate the resonance cone angle, the angle of principal energy flow increases monotonically with wavelength, suggesting a mechanism for the observed difference in PL spectra of the LuHMS relative to the control MQW. As the angle increases, the wave is directed more normal to the metacrystal axis and therefore experiences more attenuation.
Relative to the control MQW, shorter wavelengths are more likely to be detected because they propagate closer to the metacrystal axis and therefore experience less attenuation than longer wavelengths. For convenience, the dispersion of the imaginary effective permittivity elements are shown in Supplementary Fig. 9(b) from which we may quantitatively estimate the effect of directionality.
The attenuation, α, of a plane wave is directly proportional to the imaginary part of the permittivity. From an elementary model of the QW, assuming parabolic conduction and valence bands 8 , we calculate an effective attenuation using the following relation We find that emission at 1350 nm experiences ~35% less attenuation than emission at 1550 nm, shown in Supplementary Fig. 9(d). Thus we believe the wavelength dependence of the principal direction of energy propagation is observed as a blue-shifting of peak emission in the LuHMS relative to the control MQW for the same pumping conditions. This blue-shifting occurs independently of the inhomogeneous broadening associated with filling of electronic states according to the Pauli Exclusion Principle.

Supplementary Note 5. Coupling between high-k states and vacuum states without a grating
In conventional multilayer structures, a grating is necessary to efficiently couple the highk states supported by the HMM to vacuum for detection [14][15][16] . By rotating the optical axis of the multilayer 90°, excitation and emission of high-k states becomes possible without the need of a grating. Supplementary Fig. 7(a) shows a schematic of the LuHMS, with wave-vector components parallel, k || , and normal, k ┴ , to the optical axis specified. Coupling of the pump beam at normal incidence (along the k ┴ axis) into the LuHMS from vacuum, and coupling of normal emission from the LuHMS into vacuum, requires conservation of k || . Supplementary Figs. 7(b) and 7(c) present the wave-vector diagram of the LuHMS at λ 0 =1064 nm λ 0 =1550 nm, respectively, with losses omitted for clarity. Coupling occurs for HMM states with k ┴ /k 0 >k bulk , such that k || /k 0 <1. The black and green curves describe the states (±k || ,k ┴ ) supported in vacuum and bulk MQW, respectively, while HMM states are described by blue and red curves, calculated by Bloch's Theorem and EMT, respectively. Supplementary Fig. 7(b) shows that a pump beam, consisting of a finite angular bandwidth centered at normal incidence, will excite a range of highk ┴ states in the HMM. Considering an experimental excitation half-angle of 24°, the angular bandwidth of the pump beam, in terms of wave-vector components (±k || ,k ┴ ), is (-0.41, 0.91)k 0 < ɸ < (0.41, 0.91)k 0 . Looking at Supplementary Fig. 7(b), conservation of k || then tells us that bulk and Bloch-HMM states in the ranges of 3.68<k ┴ /k 0 <3.70 and 5.46< k ┴ /k 0 <5.48, respectively, may be excited without use of a grating. Similarly, Supplementary Fig. 7(c) shows that emission over the same angular bandwidth, allows states in the ranges 3.42< k ┴ /k 0 <3.45 and 4.76< k ┴ /k 0 <4.77 to out-couple from the bulk and Bloch-HMM, respectively, into vacuum without a grating.