Letter Cite This: ACS Photonics XXXX, XXX, XXX−XXX
pubs.acs.org/journal/apchd5
Toward Silicon-Based Metamaterials Sergey V. Li,† Yuri S. Kivshar,†,‡ and Mikhail V. Rybin*,†,§ †
Department of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia Nonlinear Physics Center, Australian National University, Canberra ACT 2601, Australia § Ioffe Institute, St. Petersburg 194021, Russia
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‡
ABSTRACT: We study periodic lattices of silicon nanorods and introduce the concept of a phase diagram that characterizes a transition between the regimes of photonic crystals and the dielectric metamaterials when the lattice spacing and operational wavelength vary. We find the conditions when a hexagonal periodic lattice of silicon nanorods can operate as a metamaterial described by averaged parameters. In general, we reveal that the metamaterial regime can be achieved for dielectric permittivity exceeding the value ε = 14, being commonly available for semiconductors in both visible and near-infrared frequency ranges. Thus, advanced semiconductor technologies can offer a versatile platform for novel designs of all-dielectric Mie-resonant metadevices. KEYWORDS: photonic phase diagram, silicon photonics, photonic crystals, metamaterials
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on silicon nanorods that can be described by averaged parameters in a wide frequency range. Silicon became a popular material for the fabrication of photonic structures.17 Although the dielectric permittivity of silicon is about ε ≈ 13 in the infrared frequency range, this is insufficient for creating bulk metamaterials that require larger values ε, as discussed previously in ref 16. However, siliconbased metasurfaces are frequently employed for many applications18−20 because the conditions for metasurfaces are much weaker due to the nearly normal propagation direction of light.21 At the same time, a design of bulk silicon-based metamaterials is highly desired for many applications requiring a near-zero refractive index.22 In this paper, we show that, by engineering the lattice geometry and employing the specific frequency dependence of the silicon permittivity, we are able to realize the silicon-based bulk metamaterials. We employ a hexagonal lattice, being the densest arrangement of cylindrical nanorods, that supports the Mie-type magnetic dipole modes (see Figure 1a) and decrease the minimum value of the nanorod permittivity. Also, silicon is a semiconductor material with the indirect transition for λ < 1130 nm and direct transition for λ < 365 nm. In the range of indirect transitions, 600 nm < λ < 1000 nm, the real part of permittivity grows, while the losses remain relatively weak.23 The frequency-dependent complex permittivity of silicon does not allow to apply any approach based on the analysis of bandgap diagrams with real frequencies and wave vectors. Those approaches were used earlier for constructing the photonic phase diagrams,16 with the purpose to distinguish whether the structure is photonic crystal or metamaterial by
rtificial structured media with resonant constituents are commonly associated with the concept of electromagnetic metamaterials.1 They keep a promise for advanced applications in photonics such as invisibility cloaking2,3 and flat lenses with the subwavelength focusing.4,5 In contrast to magnetic resonances, natural materials demonstrate a strong response to an electric field up to the optical frequencies.6 Metallic resonators are usually employed as the basic structural elements of such metamaterials allowing to achieve optically induced magnetic permeability different from unity,7 but this approach imposes severe restrictions on a design and functionality of optical metamaterials due to the Ohmic losses and heating.8 A common way to solve these problems is to employ two-dimensional metasurfaces9−11 instead of bulk metamaterials, for minimizing the effect of scattering losses and absorption. The recent trend in the field of metamaterials is to employ dielectric structural elements (the so-called “meta-atoms”), which support geometric Mie-type magnetic resonances.12−14 Dielectric meta-atoms are larger in size, that makes challenging to fit a sufficient number of structural elements into a finite-size volume compared to a wavelength of light. Thus, such structures become very similar to photonic crystals where the Bragg resonances, originating from the extended Bloch modes of a periodic structure, cannot be described by local effective parameters.15,16 Thus, the main question in the field of metamaterials (and, more generally, in the feld of meta-optics) remains if we can create periodic photonic structures from dielectric resonant meta-atoms that can operate as bulk metamaterials being described by averaged parameters made of commonly used materials. In this paper, we answer positively to this important and long-standing question and demonstrate how to design all-dielectric metamaterials based © XXXX American Chemical Society
Received: August 13, 2018 Published: November 21, 2018 A
DOI: 10.1021/acsphotonics.8b01126 ACS Photonics XXXX, XXX, XXX−XXX
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for calculating the complex bandgap diagrams, and construct effective phase diagrams for the silicon-based periodic photonic structures. In photonics, we usually consider periodic structures either as photonic crystals or metamaterials. The typical band diagrams for these structures are shown in Figure 1b and c, respectively. We start from the case of photonic crystals. Figure 1b shows a crossing of two light cones (gray dashed lines) with origins at the Γ point (k = 0) and Γ′ point (k = 2π/a for a square lattice with the lattice constant a) points. The cone with origin at the Γ′ point appears due to periodicity in the reciprocal space. Modulation of the dielectric index leads to the scattering of light between these two modes, and it removes the branch degeneracy at the X point on the surface of the Brillouin zone. Another type of the band diagram is observed for metamaterials (Figure 1c). The metamaterial properties are related to local resonances (in particular, Mie resonances) that manifest themselves as flat branches in band diagrams due to an uncertainly of the wavevector for the localized modes (horizontal orange dashed line in Figure 1c). The coupling of the free photon (described by the light cone) with the Mie resonance removes the degeneracy of these modes. As a result, the dispersion becomes similar to the polariton-type band diagrams in the solid states.25 The avoid-crossing is related to the local Mie resonance rather than the structure periodicity. The qualitative deference between the band diagrams allows us to identify the type of photonic structure by calculating and analyzing the second branch. An interplay between the Mie and Bragg resonances leads to a broadening of the bandgap, similar to the so-called resonant photonic crystals (e.g., periodically arranged quantum wells).26 Thus, the formation of photonic bandgap diagrams can be understood as a synergy between local Mie and extended Bragg resonances.27 However, at certain conditions, the Mie gap splits off making the structure to behave like a metamaterial.15,16,28 The common believe is that a transition between photonic crystals and metamaterials is gradual, that is, a “quantity” of metamaterial in a periodic structure is decreasing, and the
Figure 1. Difference between the regimes of photonic crystals and metamaterials. (a) An artistic view of dielectric nanorods arranged in periodic lattices, which operate in two different photonic regimes, in accord with the values of their parameters (such as a type of lattice, lattice spacing, nanorod radius, and dielectric permittivity of the material). Red cylinders correspond to the metamaterial regime, green cylinders correspond to the photonic crystal regime. Here, we consider the TE-polarized electromagnetic waves when the magnetic field is directed along the nanorod axes. (b) Typical bandgap diagram of a photonic crystal: the interaction between two light cones (gray dashed lines) results in a frequency gap (shaded in green) at the boundary of the Brillouin zone (for example, in the X point). (c) Typical bandgap diagram of a metamaterial: a coupling of the almost flat Mie-resonance branch (orange dashed line) and the light cone (gray dashed line) leads to a polariton-like feature and the Mie gap (shaded in orange).
analyzing the second dispersion branch; however, this dispersion branch is not well-defined for the case of complex permittivity. Here we employ the inverse dispersion method24
Figure 2. Dramatic change in the magnetic field pattern depending on the operational photonic-crystal or metamaterial regimes (or the corresponding phase). Simulated field patterns for the lowest frequency of the second band vs nanorod radius. An incident Gaussian beam propagates along the Γ-X direction (a−d) or along the Γ-M direction (e−h). The structure boundary is marked by a gray dashed line; arrows show the incident direction, nanorod permittivity ε = 25. Simulations are performed within the multiple scattering theory approach. B
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nanorods is assumed to be constant, and the real parts of permittivity are equal to 4 and 30 for photonic crystals and metamaterials, respectively. By using the direct and inverse dispersion methods, we calculate the band diagrams for the wave vector in the Γ-X direction, which corresponds to the lowest Bragg bandgap. Here, we use the expansion over 128 by 128 plane waves for the direct method and 25 by 25 waves for the inverse dispersion method. The band diagram of the lowindex structure (see Figure 3a) demonstrates a typical behavior
system transforms eventually into a photonic crystal. However, this intuitive picture is not correct. A qualitative change in the band diagrams (see Figure 1b,c) results in a sharp change in the patterns of the electromagnetic fields. Figure 2 compares the field patterns in the scattering by prism-shaped photonic structures in both Γ-X and Γ-M directions. The frequencies correspond to the lowest frequency of the second band (μ-near zero in the metamaterial regime). We vary the rod radius r to change the type of the band diagram with the transition taking place at r ≈ 0.19a. When r ≥ 0.190a, the field is homogeneous, and it does not depend on the lattice orientation (Figure 2). Thus, r ≥ 0.19a corresponds to the metamaterial regime. Just a small change of parameters leads to a dramatic modification in the patterns. Indeed, when r ≤ 0.189a, for the Γ-X direction we observe a stripped pattern typical to the case of photonic crystals29 (the neighboring unit cells are out of phase). For the Γ-M direction the bandgap stops the propagation. Therefore, from one side of the critical value r ≈ 0.190a, the structure behaves as a photonic crystal, but from the other one, as a metamaterial. And we indeed observe a sharp transition between the manifestations of these two regimes. A common approach for calculating the bandgap diagrams is to consider the vector Helmholtz equation as an eigenproblem for ω2, which is usually normalized by c2 (see ref 29). This approach is very attractive because of a close analogy with the Schrödinger’s equation in quantum mechanics and also because there exists a variety of analytical and numerical approaches for its solutions well developed to date. However, a disadvantage of these methods is following. It is a quite complicated task to calculate the band diagram of a structure, when its constituents have a frequency dependent dielectric permittivity ε(ω), because the operator becomes dependent on the eigenvalue. Additionally, the common methods are losing their appeal in the case where the permittivity has an imaginary part, because of the operator ceased to be Hermitian. There are several alternative methods (e.g., see ref 24 and references therein), which allow one to overcome these problems. We notice here that the non-Hermitian inverse dispersion method for plane-wave basis solves an equivalent mathematical problem by using the same equations that are solved by the common methods,30 which we will refer to as direct ω(k) methods below. In our work we analyze and compare the results obtained by both these approaches. Realizations of the direct method31,32 allow a fast calculation of bandgap diagrams with a large number of plane waves. It allows to find real eigenfrequencies as functions of real wave vector ω(k), assuming that the constituent permittivities have constant real values. Also, we can calculate the bandgap diagrams with complex eigenwavevectors as functions of real frequencies k(ω) by means of the inverse dispersion method.24 Besides the propagating modes, the complex bandgap diagrams reveal evanescent modes with the branches occupying the frequency range inside bandgaps. The analysis of complex bandgap diagrams requires a different criterion (relative to that discussed in Figure 1b,c) to identify the type of photonic structure. A criterion for complex diagrams has been suggested earlier;24 however, its relation to the criterion discussed in Figure 1b,c is unclear. Here we apply both the direct ω(k) and inverse k(ω) methods for the analysis of metamaterial and photonic crystal regimes. We consider photonic structures composed of square lattices of dielectric nanorods with the ratio of the radius to the lattice spacing r/a = 0.3. The dielectric permittivity of
Figure 3. Phase criteria for the lossless and lossy structures. (a, b) Band diagrams of a square lattice of nanorods with r/a = 0.3 and lossless permittivity ε = 4 (the photonic crystal phase) and 30 (the metamaterial phase). Data obtained by the direct method are shown by black solid curves. Results of the inverse-dispersion method are shown by cyan circles. (c, d) Band diagrams calculated by the inverse dispersion method for the similar structure as in (a, b), but with the nanorod permittivity having an additional imaginary part ε'' = 0.01ε' (cyan solid curves). Black dashed curves show the results for lossless structures.
of branches for the photonic crystal regime with the gap between 0.43 < a/λ < 0.48 (compare to Figure 1b). For the diagram corresponding to the high-index structure, we observe the polariton-like feature (see Figure 3b and compare it with Figure 1c). The light cone crosses the flat Mie branch (a/λ ≈ 0.225). Because of the similar anticrossing for the negative wavevectors, the branch does not start from a/λ ≈ 0.225 at the Γ point. As a result, the Mie gap is formed between a/λ ≈ 0.225 and 0.270. We notice here an excellent agreement between the data obtained by the direct and inverse dispersion methods. Most of high-index materials have strong frequencydependent complex dielectric permittivities leading to complex bandgap diagrams. It was recognized earlier24 that the analysis of complex bandgap diagrams could make possible to distinguish metamaterial and photonic crystal regimes. We add a small imaginary part ε″ to the dielectric permittivity, which is related to the real part ε′ as ε′′ = αε′, where α is a small real number. The bandgap diagrams for α = 0.01 are shown by cyan lines in Figure 3c,d. The analysis of the wavevector real part is sufficient for our goals although it takes complex values. For convenience, the branches of the lossless structure (α = 0) are shown by the black dashed curves in Figure 3c,d as well. C
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calculated by the inverse-dispersion method for the same structure is shown by circles, demonstrating that both criteria agree well. This result is very important because we can construct a phase diagram for lossy photonic structures. Next, we consider in detail the phase diagram of the photonic structures with the hexagonal lattice. First, we find that this phase diagram is similar to that for a photonic structure with a square lattice (dashed curve in Figure 4). The left boundary is obtained from the requirement that the Mie gap should be lower than the lowest Bragg gap. For the constant nanorod radius, the permittivity defines the frequency of the Mie mode, while the distance between the neighboring nanorods is described by the parameter r/a. We can satisfy this requirement either by increasing dielectric permittivity for fixed r/a or by increasing r/a for the constant permittivity. However, the phase diagram also reveals the existence of the right boundary. When the coupling between the Mie modes located in the neighboring nanorods becomes strong, they form a collective mode with a strong spatial dispersion. Therefore, for the metamaterial regime we have to decrease the value of r/a or increase permittivity of nanorods thus selecting the hexagonal lattice as an optimal choice. Optimization of the nanorod density appears to be very important for engineering metamaterials for the optical wavelengths, since permittivity is limited. Indeed, the minimum value of permittivity for a square lattice of nanorods is equal to 19, while for a hexagonal lattice, it is about 14, being much closer to the permittivity of semiconductors. Also, we consider in more details the role of the parameter α = 0.01 that defines the imaginary part of permittivity, which we introduce for constructing the phase diagram by means of the inverse dispersion method. We find that, when the imaginary part of permittivity grows, an error in the phase boundary is increasing. For example, for the imaginary part of dielectric permittivity described by α = 0.35, this deviation is about 5%. Now we move to the main result of our study and this paper. Silicon has a frequency dependent permittivity, which is close to 12 almost in the entire infrared frequency spectrum, and it starts to grow for the wavelength less 1000 nm. In particular, the real part of the silicon dielectric index is 19 at 490 nm, what makes it possible to observe the metamaterial regime experimentally even for a square lattice. However, the imaginary part of the dielectric index describing absorption grows as well (the coefficient α ≈ 0.1 at 400 nm). We employ here the inverse dispersion method to calculate the phase diagrams for hexagonal lattices of silicon nanorods with the values of the dielectric index reported in the
We start from the case of a photonic crystal. The two lowest branches do not demonstrate a photonic bandgap anymore as they appear in the band diagrams in Figure 3a. Instead, the branches reveal an intersection behavior in the vicinity of the Brillouin zone boundary (that is at the X point) around the frequency a/λ = 0.46. For metamaterials, the situation differs qualitatively. With a growth of frequency, the branches approach at the X point until a/λ ≈ 0.225. Then they start to deviate after avoiding crossing at a/λ ≈ 0.230 and go in the opposite directions to the Γ and Γ′ points. In contrast to the case of photonic crystals, the branches have large imaginary parts after the repulsion and, therefore, they cannot be calculated by the direct method. Below, we use this criterion for the bandgap diagrams of photonic structures with complex permittivity to distinguish the metamaterial and photoniccrystal regimes. Now, we have a complete tool for constructing the phase diagrams of periodic photonic structures. Such a diagram allows us to show that both criteria described in Figure 3 agree with each other. Also, we employ the hexagonal lattice that has the densest packing in two-dimensional geometry. The phase boundary calculated by the direct method for the hexagonal lattice is shown by a red line in Figure 4. The phase boundary
Figure 4. Phase diagram that allows differentiating metamaterials and photonic crystals. The boundary is calculated for both hexagonal (red line) and square (black dashed line) lattices, the circles mark the data obtained by the direct and inverse dispersion methods, respectively. Green and red shadings correspond to the photonic crystal and metamaterial regimes of a hexagonal lattice, respectively.
Figure 5. Phase diagram of a periodic silicon nanorod structure with a hexagonal lattice. Red circles show the phase boundary, solid line is a guide for eyes only. Green and red shadings correspond to the photonic crystal and metamaterial regimes, respectively. The phase boundary for the square lattice is shown by a black dashed line. Phase diagram is plotted in the difference axes: (a) ratio r/a vs wavelength; (b) lattice spacing a vs wavelength; and (c) rod radius r vs wavelength. D
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Figure 6. Refraction of a Gaussian beam on a silicon-based metamaterial prism, for a = 167 nm and r = 50 nm. The top panels are the magnetic field profiles (red are positive values, and blue are negative values of Hz); the bottom panels are the electromagnetic field intensities. (a, b) Offresonance case, λ = 1500 nm. (c, d) Low-frequency edge, λ = 700 nm; (e, f) High-frequency edge, λ = 495 nm. Green dashed triangles mark the prism. Black arrows in the top panels show the propagation direction of the incident beam. The results are for the TE polarization.
literature.23 In the case of the frequency-dependent permittivity, we cannot employ scalability of Maxwell’s equation anymore. However, the dependence of dielectric permittivity allows a mapping of the dielectric index to the wavelength. The phase diagram of silicon-base photonic structures as a function of wavelength and r/a is shown in Figure 5a. Since longer waves correspond to higher values of the silicon permittivity, the diagram is flipped (relative one in Figure 4) and it has a bell shape. Since the real part of permittivity for silicon growths up to 42, one can also achieve the metamaterial regime even for the case of a square lattice, where the metamaterial phase spreads to 530 nm (black dashed curve in Figure 5a). However, the considerable losses for these wavelengths make it difficult to design applicable silicon-based metamaterials with a square lattice. For a hexagonal lattice, the metamaterial regime is realized up to 850 nm. Low losses in silicon for the wavelengths larger than 600 nm enable silicon-based photonics.17 In particular, silicon has ratio between real and imaginary part of permittivity the estimate α = Im(ε)/Re(ε) = 0.012 at 600 nm. For convenience, we also present the phase diagram in two other axes involving the wavelength and nanorod radius (see Figure 5b) and also the lattice spacing (see Figure 5c). These representations are useful for designing realistic photonic structures and experiments. For example, the metamaterial regime is achieved for a lattice of silicon nanorods with the lattice spacing varying from 150 to 270 nm at the wavelength about 600 nm. Figure 5b depicts the boundary value of about 300 nm for the lattice spacing when the silicon metamaterial are available. Now we consider the phase diagram in Figure 5c with a narrow stripe, which corresponds to the metamaterial regime. It means that the metamaterial operation depends on the Mie resonances, which are defined by on the nanorod radius. Figure 5c suggests that one can tailor the nanorod radius in the range of about 10 nm. Also, it is instructive to analyze a weak bend at the larger wavelengths. It is a consequence of the frequency dependence of silicon permittivity that tunes the Mie resonance as observed earlier for single silicon spheres.33 Metamaterials are known to exhibit strong modulation of the material parameters in the vicinity of resonant frequencies. We make a number of assumptions to evaluate the effective magnetic permeability μeff of a photonic structure in the
metamaterial regime. Namely, we assume that (i) Snell’s law describes the refraction at the boundaries, (ii) the effective refractive index is a product of the square roots of effective permittivity and permeability neff = εeff μeff , (iii) the effective permeability differs from unity around the magnetic Mie resonance, and (iv) the frequency dependence of the effective permittivity is weak because the electric response for the TE-polarized waves is nonresonant.34 We consider a prism being a right triangle with an acute angle of 30°. This periodic photonic structure consists of 1220 silicon nanorods of the radius 50 nm arranged in a hexagonal lattice with the lattice spacing a = 167 nm. A Gaussian beam impinges normally on the long side of the prism and it gets deflected. Figure 6 demonstrates our numerical simulation results obtained by means of the multiple scattering theory.35,36 To extract the effective permittivity, we start from an offresonant case λ = 1500 nm when the magnetic response is negligible, that is, μeff ≈ 1. Figure 6a,b shows the magnetic field patterns Hz and the intensities of the electromagnetic field, respectively. We notice that the incident beam interferes with the reflected beam leading to the interference pattern appearing in the left corner of the prism. The analysis of the beam deflection yields the value of the effective index neff = 1.39 and εeff = 1.93. In general, causality requires that material parameters growth with the frequency in exception of a narrow range of the anomalous dispersion.6 The metamaterial bandgaps (see Figure 1c) correspond to the range of negative material parameters, since the refractive index takes complex values. Thus, we expect large values of the effective permeability at the low-frequency edge, and small values (less than one) at the high-frequency edge. We notice that the prism limits the range of the values of μeff to be retrieved. The total internal reflection defines the largest value of μeff observable with the prism, and a strong impedance mismatch suppresses the transmission for the near-zero values of μ (see, e.g., the absence of a transmitted beam in Figure 2c,d and g,h). At the low-frequency edge, we consider the wavelength of λ = 700 nm (see Figures 6c,d). This gives us the values of neff = 1.78 and μeff = 1.64 (here we assume εeff = 1.93). At the high-frequency edge, the wavelength is set to be λ = 495 nm. An analysis of the patterns in Figures E
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6e,f yields the values neff = 0.92 and μeff = 0.44, both less than unity. Now we discuss feasibility of an experimental realization of bulk silicon metamaterials studied above. We analyze two major points: (i) available technologies to fabricate the required silicon samples and (ii) the existence of Mie resonances in silicon nanorods for the visible spectral range. We notice that experimental samples should be created by silicon nanopillars with a high aspect ratio (see Figure 1a). We notice that over the last 10 years nanoimprint lithography, combined with reactive ion etching, have been used to fabricate arrays of silicon nanopillars.37 This method allows the fabrication of well-defined arrays with precise control in position, density, and size (the reported diameter is as small as 40 nm with the lattice spacing of 200 nm). Optimal etching parameters can lead to uniform vertical nanowires with high aspect ratios 60:1, being sufficient for experimental measurements. Nanopillar arrays are important for efficient solar cells, and a number of reviews have discussed the recent advances in the fabrication of such structures.38,39 Thus, available technologies allow the fabrication of silicon photonic structures with the parameters related to the metamaterial regime [as shown in Figure 5]. Finally, we evaluate the quality factor (Q factor) of the silicon nanorods that is defined by both radiative and −1 −1 absorption contributions: Q = (Q−1 rad + Qabs) . The radiative contribution can be evaluated from the Mie scattering spectra by fitting with the Fano formula.40 For the permittivity range ε ∈ [10−20], the Q factor of the lowest Mie mode in the TE polarization is approximated by the linear function Q = 2.6 + 0.73ε. A good evaluation for the absorption losses is Qabs = Re(ε)/(2Im(ε)) = 1/(2α).41 Function Qabs demonstrates a rapid growth with the wavelength, and at λ = 465 nm, both contributions coincide, Qrad = Qabs. Thus, in silicon resonators the radiative losses dominates over the absorption losses for the wavelengths longer than 465 nm, and the Mie resonances in silicon should be observed in experiment. Indeed, the Mie resonances have been observed in the scattering spectra of isolated silicon nanoparticles in the visible frequency range.12 Besides, the Mie-resonant silicon metasurfaces operating in the visible optical spectrum have also been reported to date (see, e.g., the table in ref 42). In summary, we have suggested, for the first time to our knowledge, a practical strategy for creating all-dielectric bulk silicon metamaterials based on the magnetic response of the resonant Mie modes. We have revealed that the geometry optimization is crucial for minimizing the threshold value of the refractive index that would allow to describe a periodic dielectric photonic structure with averaged parameters. We notice that the phase diagrams presented in Figure 5 are consistent with the realistic parameters provided by the modern fabrication techniques, and thus they can be realized experimentally in the visible and near-IR frequency ranges.
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ACKNOWLEDGMENTS Y.K. acknowledges useful discussions with W. L. Barnes, K. Busch, Th. Krauss, M. F. Limonov, and C. R. Simovski. This work was supported by the Ministry of Education and Science of the Russian Federation (Grant 3.1500.2017/4.6), the Russian Foundation for Basic Research (Grant 16-02-00461), and the Strategic Fund of the Australian National University S.L. acknowledges support from the Russian President's Scholarship.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Yuri S. Kivshar: 0000-0002-3410-812X Mikhail V. Rybin: 0000-0001-5097-4290 Notes
The authors declare no competing financial interest. F
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ACS Photonics
Letter
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DOI: 10.1021/acsphotonics.8b01126 ACS Photonics XXXX, XXX, XXX−XXX
