Letter Cite This: Nano Lett. 2018, 18, 3978−3984
pubs.acs.org/NanoLett
Nonlinear Wavefront Control with All-Dielectric Metasurfaces Lei Wang,† Sergey Kruk,*,† Kirill Koshelev,†,§ Ivan Kravchenko,∥ Barry Luther-Davies,‡ and Yuri Kivshar†,§ †
Nonlinear Physics Centre and ‡Laser Physics Centre, Australian National University, Canberra, ACT 2601, Australia § ITMO University, Saint Petersburg 197101, Russia ∥ Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States
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S Supporting Information *
ABSTRACT: Metasurfaces, two-dimensional lattices of nanoscale resonators, offer unique opportunities for functional flat optics and allow the control of the transmission, reflection, and polarization of a wavefront of light. Recently, all-dielectric metasurfaces reached remarkable efficiencies, often matching or out-performing conventional optical elements. The exploitation of the nonlinear optical response of metasurfaces offers a paradigm shift in nonlinear optics, and dielectric nonlinear metasurfaces are expected to enrich subwavelength photonics by enhancing substantially nonlinear response of natural materials combined with the efficient control of the phase of nonlinear waves. Here, we suggest a novel and rather general approach for engineering the wavefront of parametric waves of arbitrary complexity generated by a nonlinear metasurface. We design all-dielectric nonlinear metasurfaces, achieve a highly efficient wavefront control of a third-harmonic field, and demonstrate the generation of nonlinear beams at a designed angle and the generation of nonlinear focusing vortex beams. Our nonlinear metasurfaces produce phase gradients over a full 0−2π phase range with a 92% diffraction efficiency. KEYWORDS: Metasurfaces, nonlinear optics, third-harmonic generation, Mie resonances, wavefront control
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mechanism with which to control the forward-to-backward ratio of generated nonlinear signals. All-dielectric nanostructures supporting Mie resonances have recently been suggested as an important pathway for the enhancement of the nonlinear efficiency beyond the limits associated with plasmonics.21 Indeed, more-efficient thirdharmonic generation in individual Si and Ge nanoresonators has been recently demonstrated by several groups, showing a huge enhancement of the conversion efficiency by optical pumping in the vicinity of the magnetic dipole Mie mode or composite resonances.22−24 Si and Ge are of a paramount interest for nonlinear photonics because they offer superior nonlinear characteristics while allowing for a potentially lowcost complementary metal−oxide−semiconductor-compatible (CMOS) fabrication.25 Conversion efficiencies of the order of 10−6 have been achieved experimentally in both individual Si and Ge nanoparticles and metasurfaces.26 These results clearly illustrate the great potential of all-dielectric resonant nanostructured surfaces for nonlinear nanophotonics and metaoptics.
n contrast to conventional optics, metasurfaces are composed of subwavelength elements (“meta-atoms’) that can manipulate light−matter interaction by employing a very compact platform. Metasurfaces with spatially varying metaatoms allow the control of the polarization, phase, and amplitude of light.1−4 By now, many phenomena have been demonstrated successfully in linear optics; however, nonlinear metasurfaces are expected to meet the growing demand for tailored nonlinearities5−7 and realize nonlinear optical chirality, nonlinear geometric Berry phase, and nonlinear wavefront engineering. Several recent pioneering demonstrations of nonlinear metasurfaces for wavefront control8−17 were based on plasmonic designs. One of the main limitation of plasmonic designs for the nonlinear metasurfaces is the nonlinear conversion efficiency. Different design strategies were implemented to increase the nonlinear performance of plasmonic and hybrid metasurfaces18−20 based on the resonant coupling of meta-atoms. However, the overall efficiency of the nonlinear frequency conversion in such planar plasmonic nanostructures remained very small, being of the order of ∼10−10. Despite extremely high intrinsic nonlinearities of plasmonic materials, the experimental efficiencies were limited by Ohmic losses, small mode volumes, and low laser-damage thresholds. In addition, all previous approaches do not provide a direct © 2018 American Chemical Society
Received: April 11, 2018 Revised: May 9, 2018 Published: May 11, 2018 3978
DOI: 10.1021/acs.nanolett.8b01460 Nano Lett. 2018, 18, 3978−3984
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higher-order multipoles as their optical volume increases. This allows an analytical treatment for the intensity of the third harmonic in both the forward (F) and backward (B) directions in terms of multipoles having even or odd parity with respect to the vertical reflection symmetry:
Here, we suggest a new concept for embedding any functionality into a nonlinear metasurface and demonstrate our approach for silicon-based metasurfaces that generate the third-harmonic (TH) field. Constituent elements of our metasurfaces are sets of different elliptical nanopillar resonators, each of them providing similar nonlinear conversion efficiency but accumulating a different phase for the third-harmonic signal covering the full 0−2π range. The resonant-phase accumulation of the third harmonic field in the dielectric metasurface occurs in the regime when both electric and magnetic Mie multipoles are excited simultaneously with comparable amplitudes. The simultaneous generation of several Mie multipoles allows the control of the directionality of the third-harmonic signal in each individual nanopillar and, in particular, to achieve a high forward-to-backward ratio of the TH signal. When assembled into a planar structure, these dielectric resonators allow the creation of smooth phase gradients in the generated thirdharmonic field. Here, we exemplarily present two demonstrations of such metasurfaces. The first metasurface acts as a nonlinear beam deflector that generates a third-harmonic beam at a selected angle with respect to the direction of the pump beam (as visualized in Figure 1a). The second metasurface generates a nonlinear focusing vortex beam with orbital angular momentum 1 from a Gaussian pump (as shown in Figure 1b).
(3ω) (3ω) 2 B = |Eeven − Eodd |
(1)
(3ω) (3ω) 2 F = |Eeven + Eodd |
(2)
where E(3ω) even(odd) is the TH electric field generated by all even (odd) multipoles. Eq 1 suggests that the necessary condition for forward directionality requires the balance between the even and odd multipoles: (3ω) (3ω) Eeven = Eodd = |E|e iφ
(3)
Therefore, the total field of the third harmonic in the forward direction E(3ω) is: (3ω) (3ω) E(3ω) = Eeven + Eodd = 2|E|e iφ
(4)
By shaping the pillar geometry, a full 2π phase coverage with a uniform amplitude can be achieved keeping high forward directionality at the same time. Importantly, as the nanocylinders are optically resonant at both the pump wavelength and the TH wavelength, the multipolar resonances at the TH are inevitably of higher-order multipoles, making the decomposition go beyond the dipole and quadrupole approximations. The balance of multipoles with opposite parities required by eq 3, leading to the unidirectional pattern, represents the generalized Huygens’ condition4,30,31 extended to the case of nonlinear harmonic generation. To find a specific set of geometries, we perform full-wave nonlinear simulations of amorphous silicon particles on glass substrate using the finite element method solver of COMSOL multiphysics. We choose a specific excitation wavelength of 1615 nm and an initial pillar geometry that supports electric and magnetic dipole modes at the excitation wavelength. We note that calculations of the third harmonic are computerintensive because they require at least a 33 times finer mesh compared to linear calculations. Moreover, our calculations are performed over a four-dimensional (4D) parameter space (composed of pillar height, two elliptical axes, and the unit cell size), resulting in a large total volume of the 4D parameter space. To minimize the demand for computing power, we implemented the gradient descent method in the COMSOL calculations. This method analyses the calculated data sets, learns to find the quickest route to optimization, and is commonly used in machine intelligence (in particular, as a subset of the deep-learning method). The optimization was set to find the parameters that provide the exact desired values of TH phase near the maxima of the TH amplitude and the forward-to-backward ratio. Details of the numerical simulations are provided in the Methods section. The resulting design provides a set of particles with a height of 617 nm in a square array with 550 nm period and elliptical cross-section with the axes values ranging between 320 and 535 nm (see Figure 2 for the details of the nanopillar geometry). This range of parameters allows a full 2π phase coverage in the TH to be achieved while maintaining similar amplitudes. All of the nanopillars have a high forward-to-backward ratio of TH: overall, over 90% of the third harmonic is generated in forward direction, and the TH generation in the backward direction is suppressed (see Figure 2 for details on individual nanopillars).
Figure 1. Concept images of functional nonlinear metasurfaces. (a) A nonlinear beam deflector. (b) A nonlinear vortex beam generator. The metasurfaces are assembled from a set of different silicon nanopillars, generating third harmonics with different phases.
Theoretical Approach. We employ a set of silicon nanopillars as building blocks for our nonlinear metasurfaces. We require each nanopillar to support third-harmonic generation with forward directionality. We further require that the different pillars provide similar amplitudes of the thirdharmonic signal but have different phases ranging from 0 to 2π. This could be achieved via the interplay between the Mie resonances of a single nanopillar at both the pump wavelength and the TH wavelength. Multipolar analysis based on Mie theory proved to be a powerful tool to understand the light scattering from resonant nanoscale objects. In the past, it yielded important insights into various fields of nanophotonics. In the past, this framework was employed for describes the scattering intensities and radiation pattern of dominant excited multipole modes that yielded important insights into various fields of nanophotonics.21,27−29 We extend this approach for the wavefront control in nonlinear regime (see the Supporting Information). We consider pillars that at the pump wavelength support electric and magnetic dipole Mie resonances. Correspondingly, at the TH wavelength, they support several 3979
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Figure 2. Geometries, nonlinear phases, and TH amplitudes of nanopillar meta-atoms. Shown are the sizes of the nanopillars and corresponding analytical and numerical results for the phase, amplitude (in the forward direction), and directionality (the forward-to-backward ratio of intensity) of the third-harmonic field. The results are for the optimal pump wavelength of 1615 nm and linear polarization of the pump along the a axis.
offset that we associate with the effect of the substrate (see details in the Supporting Information). Our multipolar analysis shows that at the pump wavelength the forward-scattering of all the pillars is dominated by the first-order multipoles (electric and magnetic dipoles), and correspondingly, the thirdharmonic field is dominated by several low-order multipoles: electric and magnetic dipoles, quadrupoles, and octupoles (see details in the Supporting Information). Importantly, the sum of contributions of all even multipoles to the TH signal is similar to the sum of contributions of all odd multipoles (in terms of both amplitudes and phases; see the details in the Supporting Information). Thus, our system satisfies eqs 3 and 4 and achieves the generalized Huygens’ condition in the nonlinear regime, which provides full 2π phase coverage combined with the similar TH amplitude in the forward direction and suppressed reflection in backward direction for all nanopillars (see Figure 2). Full-phase coverage combined with nearly the
The optimization is performed for a linearly polarized pump along the a axis of the pillars (as marked in Figure 2). We note the possibility of independent phase and amplitude control for different polarizations. We numerically estimate the TH generation efficiency of the pillars. For the average pump power of 200 mW, a pulse length of 300 fs, a repetition rate 20 MHz, and a diffraction-limited focusing on a single nanopillar, the calculated TH average power is ∼300 nW. Thus, the overall conversion efficiency −6 pump PTH ave /Pave = 10 , which is similar to previous reports on TH efficiency in Si nanostructures in the vicinity of a magnetic dipole modes and composite resonances.32,33 The peak powerpump 3 −4 independent conversion efficiency PTH W−2. ave /(Pave ) = 10 We compare our numerical results with the multipolar analysis (see Figure 2) and find that calculations of phases of the TH are in a perfect agreement. Calculations of the amplitudes and the forward-to-backward ratios have a small 3980
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Figure 3. Nonlinear beam-deflector metasurface. (a) Phase profile of the third-harmonic field encoded into the metasurface. (b) Electron microscope image of the silicon metasurface. (c) Directionality diagram (back-focal plane image) of the forward TH field. A total of 92% of TH is directed into the designed angle θ = 5.6°, where kx/k0 = −0.098. (d) Cross-section of experimental back-focal plane image normalized to the TH maximum.
Figure 4. Spectral performance of a nonlinear metasurface. (a) Theoretically calculated and (b) experimentally measured directionality diagrams resolved spectrally. Images show the true colors of the TH field for different wavelengths. (c) The diffraction efficiency spectrum defined as the percentage of the forward-generated third harmonic directed into the −1st order. Shown are theoretical (solid) and experimental (dots) results.
same intensity of forward-generated TH signal allows for an efficient arbitrary complex wavefront shaping in the nonlinear regime. We finally perform the multipolar analysis over the spectral range of pump wavelengths from 1300 to 1800 nm with the details provided in the Supporting Information. Experimental Demonstrations. To demonstrate the capabilities for nonlinear wavefront control, we fabricate two metasurfaces: a nonlinear beam deflector, a metasurface that generates a beam of light at the third-harmonic wavelength
propagating at an angle to the k-vector of the pump; and a nonlinear focusing vortex beam generator, a metasurface that generates a focusing TH beam with an orbital angular momentum m = 1. The first metasurface acting as a nonlinear beam deflector serves as a canonical example of wavefront control similar to common demonstrations in linear metaoptics.34−39 The second metasurface provides an example of a more-complex wavefront control revealing the potential of our general approach. 3981
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Nano Letters For the beam-deflector metasurface, we arrange the 10-pillar set from Figure 2 into a supercell that creates a linear-phase gradient at the TH wavelength. The supercell is then arranged into a periodic array (see Figure 3a). Such a metasurface generates the TH mainly in the direction determined by the condition |kx/k0| = λTH/(10 × period) = 0.098, where k0 and kx are the full length and in-plane projection of the TH wavevector and λTH is the TH wavelength. We fabricate the nonlinear beam deflector from amorphous silicon on a glass substrate using electron beam lithography (for details, see the Methods section). An electron-microscope image of the fabricated structure is shown in Figure 3b. To generate the TH signal, we pump the metasurface at a wavelength of 1615 nm with femtosecond pulses from an optical parametric amplifier pumped by a mode-locked Ytterbium laser. To observe the directionality of the TH and to measure the beam deflection we image the back-focal plane onto a camera (see the details in the Methods section). Nearly perfect TH beam deflection is observed at an angle of θ = 5.6° (where sin[θ] = |kx/k0|) relative to the direction of pump with 92% diffraction efficiency in forward direction. Importantly, virtually zero TH is generated along the direction of pump (i.e., into the zero-diffraction order). Considering the high forwardto-backward ratio of TH generation, we conclude that over 80% of the overall third-harmonic energy is generated into the desired channel. The experimentally measured TH conversion −4 pump 3 efficiency PTH W−2. ave /(Pave ) = 1.4 × 10 We then studied the performance of the metasurface as the pump wavelength is tuned from 1330 to 1750 nm, the range available from our optical parametric amplifier. Directionality curves, similar to Figure 3b, are experimentally retrieved as the pump wavelength was varied and then combined into the diagram in Figure 4b. The color space (HSL color-map) is used to represent the TH directionality and wavelength information, in which the hue (H) shows the true color of TH for given wavelength, saturation (S) is 1, and lightness (L) corresponds to the normalized TH intensity at each wavelength. The complementary theoretical calculation is shown in Figure 4a. The calculation took both the amplitudes and phases of the TH into account across the spectral range. We can see that around the optimal wavelength of 1615 nm, the only diffraction order clearly visible is the −1st order. As the pump wavelength deviated from its optimal value, other diffraction orders appear in the directionality diagram. From this, we could determine the variation of the diffraction efficiency with wavelength (see Figure 4c). We observe good correspondence between theoretically calculated and experimentally measured diffraction efficiencies of the wavefront control. For the vortex metasurface, we design a phase profile shown in Figure 5a. We assemble the metasurface from an increased set of pillars found numerically (see Figure 5b for the optical image of the fabricated sample). We pump the metasurface with a 1615 nm wavelength and scan the formed TH beam along the optical axis. Figure 5c shows the cross-section of a donut-shape vortex TH beam taken perpendicular to the optical axis at 25 μm distance from the focus. Figure 5c shows the cross-section along the optical axis of the focusing and defocusing TH beam. The nonlinear vortex beam generation illustrates the immediate applicability of our approach for engineering the complex phase fronts of parametric waves. In summary, we have presented a general approach and versatile experimental platform for nonlinear wavefront control with highly efficient nonlinear dielectric metasurfaces. Our
Figure 5. Nonlinear vortex beam generator. (a) Phase profile of the third-harmonic field encoded into the metasurface. (b) Optical microscope image of the fabricated metasurface. (c) Cross-section of a generated donut-shape vortex beam taken perpendicular to the optical axis at distance z = 25 μm from the focus. (d) Cross-section along the optical axis of the focusing and defocusing TH beam.
approach is based on the generalized Huygens’ principle extended to nonlinear optics. This allows the creation of arbitrary phase gradients and wavefronts in nonlinear optics via multipolar nanophotonics by the excitation of electric and magnetic Mie multipoles. The multipolar generation also allows the achievement of a high forward-to-backward ratio of nonlinear signals. Based on our concept, we have designed and demonstrated experimentally the first nonlinear alldielectric metasurface that generates the third harmonic signal in the forward direction with 92% precision in its wavefront control. Methods. Sample Fabrication. The silicon metasurfaces are fabricated on a 4 in. fused silica wafer that was 500 μm thick. First, a 617 nm polycrystalline silicon layer is deposited onto the substrate by low-pressure chemical vapor deposition (LPCVD). Subsequently, a thin layer of an electron-resist PMMA A4 950 is spin-coated onto the sample, followed by electron-beam lithography (JEOL 100 eV) and development. After this, a 20 nm Cr film is evaporated onto the sample, followed by the lift-off process to generate a Cr mask. Reactiveion etching (RIE) is used to transfer the Cr mask pattern into the silicon film. Finally, the residual Cr mask is removed via wet Cr etching, and the sample is cleaned with oxygen plasma. Nonlinear Optics Setup. For nonlinear optical measurements, we pump the metasurfaces with an optical parametric amplifier (MIROPA-fs-M from Hotlight Systems) tunable across the range from 1330 to 1750 nm and generating 300 fs duration pulses at a repetition rate of 21 MHz and average power at the laser output ranging from 100 to 300 mW. The power is then attenuated to 9 mW. The metasurface is mounted on a three-dimensional stage and is placed at normal incidence in the focal region of two confocal Olympus Plan N objective lenses. A low-NA lens is used to focus the pump (NA = 0.1, 4×). The TH is collected using a lens with NA = 0.4. The beam is focused from the front side of the sample. To detect the TH radiation, we use a cooled CCD camera (Starlight Xpress Ltd., SXVR-H9). The TH signal was separated from the pump using short-pass filter Thorlabs FESH0650 for spectra measurements. A pair of confocal lenses are used to transfer the 3982
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Zhang, T. Zentgraf, and A. Zayats and also thank D. Smirnova for a help with the numerical code and advice for numerical simulations.
back-focal plane image of the TH radiation onto the camera. The spectrum of the pump beam is cleared up by a dichroic mirror and long-pass infrared filters (Thorlabs FEL 1000 and FELH1150) before the first objective. Numerical Calculations. For numerical simulations of the nonlinear amplitudes and phases of the forward-scattered TH wave, we use the finite-element-method solver in COMSOL multiphysics in the frequency domain. All calculations are realized for a single nanopillar of a specific size placed on a semi-infinite substrate with periodic boundary conditions mimicking a square array. Thus, the simulation takes into account the effect of coupling between identical neighbor particles but neglects the difference in coupling strengths between non-identical near-neighbors. We employ the approach based on an undepleted pump approximation using two steps to calculate the radiated nonlinear emission. First, we simulate the linear scattering at the fundamental wavelength and then obtain the nonlinear polarization induced inside the nanopillar. Then, we employ this as a source for the electromagnetic simulation at the harmonic wavelength to obtain the generated TH field. The nonlinear susceptibility tensor χ(3) was considered as a constant scalar value of 2.45 × 10−19 m2/V2.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.8b01460. Details of the analytical approach. Figures showing the convergence of multipoles, spectra of amplitudes and phase differences, and a multipolar analysis. (PDF)
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Sergey Kruk: 0000-0003-0624-4033 Yuri Kivshar: 0000-0002-3410-812X Author Contributions
L.W. and S.K. conceived the research. L.W and K.K. performed numerical simulations and theoretical calculations. S.K. and I.K. fabricated the samples. L.W., S.K., and B.L.D. provided experimental data. S.K., K.K., and Y.K. wrote the manuscript based on the input from all authors. Y.K. supervised the project. All authors contributed to the editing of the manuscript. Funding
The authors acknowledge a financial support from the Ministry of Education and Science of the Russian Federation (grant no. 3.1668.2017), the Australian Research Council and the Australian National University. Part of this research was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. K.K. acknowledges FASIE (grant no. 10864GU/2016) for the valuable support. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge useful discussions with H. Atwater, M. Brongersma, D. Christodoulides, A. Kuznetsov, W. Liu, S. 3983
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