(PDF) Backward phase-matching for nonlinear optical generation in negative-index materials

ARTICLES PUBLISHED ONLINE: 15 JUNE 2015 | DOI: 10.1038/NMAT4324 Backward phase-matching for nonlinear optical generation in negative-index materials Shoufeng Lan1, Lei Kang2, David T. Schoen3, Sean P. Rodrigues1,2, Yonghao Cui2, Mark L. Brongersma3 and Wenshan Cai1,2* Metamaterials have enabled the realization of unconventional electromagnetic properties not found in nature, which provokes us to rethink the established rules of optics in both the linear and nonlinear regimes. One of the most intriguing phenomena in nonlinear metamaterials is ‘backward phase-matching’, which describes counter-propagating fundamental and harmonic waves in a negative-index medium. Predicted nearly a decade ago, this process is still awaiting a definitive experimental confirmation at optical frequencies. Here, we report optical measurements showing backward phase-matching by exploiting two distinct modes in a nonlinear plasmonic waveguide, where the real parts of the mode refractive indices are 3.4 and −3.4 for the fundamental and the harmonic waves respectively. The observed peak conversion efficiency at the excitation wavelength of ∼780 nm indicates the fulfilment of the phase-matching condition of k2ω = 2kω and n2ω = −nω , where the coherent harmonic wave emerges along a direction opposite to that of the incoming fundamental light. T he phase-matching condition, resulting from the conserva- to each other, the harmonic output will travel towards the source tion of photon momentum, is among the most vital aspects to of the fundamental wave20–23 . This new type of phase-matching consider when multiple frequencies are mixed in bulk non- condition, known as ‘backward phase-matching’ or ‘nonlinear linear media. Strictly speaking, all materials are dispersive and the mirror’ in negative-index media, has been predicted for years, but conversion efficiency of a nonlinear process is critically dependent an experimental verification at optical frequencies is still lacking. on the relationship among the wavevectors involved. Taking second- The difficulty in experimentally validating this condition arises in harmonic generation (SHG) as an example, phase-matching implies the production of a bulk optical negative-index medium of many a wavevector relation of k2ω = 2kω or a refractive index relation of wavelengths in length, together with perfect, simultaneous tailoring n2ω = nω . To achieve these parameters the orientation or temperature of the refractive indices at both ω and 2ω frequencies to meet the of a nonlinear crystal must be fine tuned, or an approximation can aforementioned conditions. be reached by flipping the crystal axis in a periodic manner. The In this work, we report the experimental demonstration of specific scheme to fulfil the phase-matching requirement is critically backward phase-matching for nonlinear generation of light in a dependent on the optical properties of the nonlinear medium, such negative-index material (NIM). As in NIMs a negative index can as the material’s chromatic dispersion, crystal anisotropy, waveguid- occur only within a rather limited frequency range, here we set the ing characteristics, and thermal coefficient of the refractive index1,2 . refractive index for the fundamental wave (ω) to be positive and The exotic electromagnetic parameters made possible by set the harmonic frequency (2ω) to be within the negative-index metamaterials provoke us to re-evaluate the established rules band. In Fig. 1, we illustrate the scenarios of frequency doubling of phase-matching for nonlinear optical interactions. As one in different nonlinear media. A phase mismatch generally leads of the most exciting frontiers in optics and materials science, to a poor conversion efficiency, because constructive SHG occurs metamaterials have enabled unprecedented flexibility in producing only within the coherence length of lc = π/|k2ω − 2kω | (Fig. 1a). unconventional optical properties that have not been observed in The general condition for phase-matching in SHG is k2ω = 2kω , as the past3–10 . The linear responses of metamaterials have substantially governed by the law of momentum conservation1,2 . Phase-matching augmented the linear properties available from naturally occurring implies an infinite coherence length, such that as the fundamental materials. Similarly, the studies of nonlinear metamaterials may wave is gradually depleted and converted to its second-harmonic have a revolutionary impact on the entire field of nonlinear signal, the source polarization and the generated harmonic field optics11–19 . A particularly fundamental topic to reconsider is the remain in phase across the length of the nonlinear medium (Fig. 1b). set of novel relations necessary to achieve constructive nonlinear In a negative-index medium with nω > 0 and n2ω < 0, however, the frequency conversion when one of the mixed waves has a negative phase-matching relation implies that n2ω = −nω . Thus, as governed index of refraction. For example, when a metamaterial possesses by the Poynting vector, the energy flow in a negative-index medium opposite signs in its indices of refraction at the fundamental and the is directed against the phase propagation direction described by the second-harmonic frequencies, phase-matched frequency doubling wavevector k. Consequently, when the backward phase-matching requires that the indices for the fundamental and harmonic waves occurs, the harmonic output grows towards the source of the have the same magnitude but opposite signs. As the Poynting vector fundamental light, as depicted in Fig. 1c. As a side note, the and the wavevector in a negative-index medium are antiparallel counter-propagation of the fundamental and harmonic waves can 1 School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. 2 School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. 3 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA. *e-mail:

NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials 1 © 2015 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE MATERIALS DOI: 10.1038/NMAT4324 a kω Sω a n2ω ≠ nω k2ω ≠ 2kω 3.5 k2ω S2ω λ 2ω = 380 nm εm = −ε d b kω Sω 3.0 n2ω = nω Photon energy (eV) k2ω = 2kω n2ω = −3.4 + 0. 46i k2ω S2ω nω = 3.4 + 0.01i 2.5 c kω Sω n2ω = −nω k2ω = 2kω 2.0 k2ω S2ω λω = 760 nm 1.5 Figure 1 | Phase-matching conditions for second-harmonic generation in nonlinear optical media. a, Low-efficiency frequency doubling of light in a −5 0 5 positive-index medium without phase-matching (k2ω 6 = 2kω , n2ω 6 = nω ). Mode refractive index b, Conventional phase-matching for SHG (k2ω = 2kω ), where the fundamental and harmonic waves possess the same index of refraction and b co-propagate along the same direction. c, Backward phase-matching (k2ω = 2kω , n2ω = −nω ) in a negative-index material, where the frequency-doubled signal is directed towards the source of the fundamental wave. The red- and blue-coloured lines in the schematics represent the fundamental and second-harmonic waves, respectively. be realized based on an entirely different scheme24 , using quasi- Figure 2 | Operating point for backward phase-matching in a plasmonic phase-matching instead of a negative index of refraction. waveguide. a, Dispersion relations in a silver–dielectric–silver waveguide To circumvent the difficulty in engineering the effective param- (nd = 2, td = 30 nm). The H-symmetric and H-asymmetric modes are eters in a bulk photonic metamaterial across multiple wavelength represented by the red and blue curves, respectively. Both the real (solid) bands, we realize the index-matching relation of n2ω = −nω by and imaginary (dashed) parts of the mode refractive index are plotted. The using two distinct modes supported in a plasmonic waveguide. condition for backward phase-matching is represented by the marker lines, When nonlinear wavemixing occurs in a waveguide instead of a where nω = 3.4 + 0.01i at λω = 760 nm and n2ω = −3.4 + 0.46i at bulk medium, the phase-matching condition for SHG requires the λ2ω = 380 nm. b, Field mapping for the H-asymmetric mode for λ2ω (left) propagation constants of the fundamental and harmonic waves to and the H-symmetric mode for λω (right). The two modes exhibit opposite be identical, which can be satisfied by exploiting the dispersion field profiles, which leads to a vanishing mode-overlapping factor for characteristics of different modes in the waveguide25,26 . A metal– second-harmonic generation. insulator–metal (MIM) plasmonic waveguide is typically operated below the surface plasmon frequency (ω < ωsp ), where the disper- have located an operational point at which the H-symmetric funda- sion curve ω(k) has a positive slope below the light line of the mental wave (nω = 3.4 + 0.01i) at λω = 760 nm is phase-matched to dielectric medium in the gap. This regular plasmon mode exhibits the generated H-asymmetric harmonic signal (n2ω = −3.4 + 0.46i) a symmetric profile in its magnetic field (‘H-symmetric’) along at λ2ω = 380 nm. with a relatively low loss factor. It has been demonstrated that if Two immediate challenges stand in the way of realizing phase- the working frequency falls between the surface and bulk plasmon matched SHG using the plasmonic waveguide. First, the thin frequencies (ωsp < ω < ωp ), the magnetic-field component of the dielectric layer made by typical deposition methods has an amor- dominant propagating mode possesses an asymmetric distribu- phous nature and a vanishing χ (2) response. Second, the two modes tion (‘H-asymmetric’). More intriguingly, the dispersion relation involved in the proposed SHG process have opposing symmetry, for this mode is characterized by a negative slope, which leads to as indicated by the magnetic-field mapping of the fundamental an antiparallel relation between the phase velocity ω/k and the and harmonic modes at the operational point illustrated in Fig. 2b group velocity dω/dk—and, consequently, a negative index for the (details in Supplementary Section 3). This mode asymmetry along H-asymmetric mode27–33 . For all propagating waveguide modes, the the transverse direction prevents the nonlinear coupling between imaginary parts of the wavevector k and the mode index n must the two modes R from happening because the nonlinear overlapping be non-negative, as required by the basic laws of causality and factor Si,j ∝ S χjii(2) Ej2ω Eiω Eiω dS, which is defined as an integral of the energy conservation. The H-asymmetric mode extends well into product of the susceptibility χ (2) , the SHG mode, and the square the metal and therefore is relatively lossy compared to the regular, of fundamental mode over the entire cross-section35 , is precisely H-symmetric mode in a MIM waveguide. The dispersion curves zero. These obstacles are overcome in our experiment by exploit- for the plasmonic waveguide used in our experiment are shown ing the electrically induced χ (2) effect in the dual-layered dielec- in Fig. 2a, where a sufficiently thin dielectric spacer of relatively tric spacer in a planar plasmonic waveguide (fabrication details high refractive index (nd = 2, td = 30 nm) is sandwiched between in Methods), as illustrated in Fig. 3a. Instead of the regular χ (2) two flat silver films. A large dielectric constant in the gap enables response intrinsic to non-centrosymmetric crystals and interfaces, a pronounced separation between the surface plasmon (εm = −εd ) here the effective second-order nonlinear effect is produced by the and the bulk plasmon (εm = 0) frequencies, while a narrow gap interplay between the ubiquitous third-order nonlinear suscepti- (2) pushes the operating point away from surface plasmon frequency bility χ (3) and the applied control field (χ (3) EC → χeff ; refs 36–38). ωp and helps to balance the magnitudes of the refractive indices of Moreover, the 30-nm-thick dielectric spacer is split into two halves, the two modes (see Supplementary Section 1). A similar analysis for 15 nm each, made of Si3 N4 and HfO2 (Fig. 3b). These two materials matching the indices of two distinct modes in a plasmonic waveg- have similar linear (nd ≈ 2), but distinct nonlinear behaviour and uide has been numerically proposed before34 . By optimizing the electrical conductivity39 . As a result, the electrically induced χ (2) material and geometry parameters of the plasmonic waveguide, we effect is effectively disabled in one half of the dielectric channel. 2 NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2015 Macmillan Publishers Limited. All rights reserved NATURE MATERIALS DOI: 10.1038/NMAT4324 ARTICLES a λ ω , fundamental wave b background due to multi-photon luminescence is negligibly low within the wavelength range of interest. We note that the observed harmonic signal in Fig. 4a may arise from multiple sources, λ 2 , ha including the exposed front metallic surface, the focused ion beam ω rmonic sign al milled nanoslit, and the internal metal–dielectric interfaces inside the waveguide. Only the latter can be ascribed to the nonlinear 5 μm conversion between the aforementioned plasmonic modes. We are able, however, to produce and analyse harmonic signals exclusively stemming from the waveguide core by applying a voltage across Si3 N Sil 15 nmver (2) χ 1,eff ≠ χ (2) 4, the dielectric spacer. This electrically induced frequency doubling, V+ 2,eff V− Silve HfO , superimposed onto the regular, static SHG signal (Fig. 4b), is r 2 15 n m expected to follow the mode index and phase-matching analysis c 10 1.0 laid out in Figs 2a and 3c. The dependences of the static and d 1 electrically induced SHG levels on the excitation wavelength λω Phase mismatching Δk/k0,ω are illustrated in Fig. 5a and b, respectively. Provided a constant fundamental intensity, a maximum SHG conversion efficiency can be identified at the fundamental wavelength of λω = 780 nm, I2ω (a.u.) 5 0.5 0 indicating the existence of backward phase-matching in the plasmonic waveguide. In particular, the measured efficiency curve for the purely voltage-induced SHG signal (Fig. 5b) agrees very well with the theoretical prediction in Fig. 3c, where the peak SHG 0 0.0 −1 conversion efficiency corresponds to the backward phase-matching 720 740 760 780 condition of k2ω = 2kω and n2ω = −nω . The incident pump light can λω (nm) be efficiently coupled into the waveguide only when its polarization is perpendicular to the nanoslit, as evidenced by the inset polar Figure 3 | Experimental design for backward phase-matching in a diagrams in Fig. 5a,b. In fact, the dependence of the static SHG waveguide with a dual-layered dielectric core. a, Schematic of intensity on the polarization of the fundamental light (inset of experimental set-up and structure of sample. The dielectric spacer of the Fig. 5a) exhibits an extinction ratio around 5:1, indicating that waveguide consists of two ultra-smooth layers with similar linear, but a small portion of SHG stems from the metallic surface outside distinct nonlinear behaviours to enable strong harmonic generation. of the slit. In contrast, the level of electrically induced SHG is b, Scanning electron microscope image of the fabricated structure. Inset is reduced to zero when the fundamental wave is polarized along a cross-sectional view illustrating the metallic and dielectric layers. the nanoslit (inset of Fig. 5b), which serves as an additional c, Simulated intensity of the backward second-harmonic signal (red) and validity check that the voltage-enabled SHG is generated inside the degree of phase mismatch (1k/k0,ω , blue) at different wavelengths, the designed waveguide because light can couple only when where 1k = |k2ω − 2kω | and k0,ω represents the wavevector at the polarized perpendicularly. Thus the signal is generated from an fundamental wavelength in the air. d, Magnetic-field mapping for the intentionally induced symmetric break with an applied voltage second-harmonic wave in the plasmonic waveguide with (top) and without bias and demonstrates a second-harmonic conversion efficiency (bottom) intentionally induced symmetry breaking. The implementation of peaking at λω = 780, where the backward phase-matching occurs. symmetric breaking in the nonlinear property is essential for the efficient The voltage-induced frequency doubling from the plasmonic generation of the H-asymmetric SHG mode. waveguide features a few characteristics typical to the electrically induced SHG process, such as a linear dependence of the observed This induced symmetry breaking gives rise to a non-zero nonlinear SHG on the level of applied bias voltage (Fig. 5c) and a relation overlap factor between the H-symmetric fundamental wave and the between the fundamental and second-harmonic intensities given H-asymmetric SHG signal, and therefore enables a strong nonlin- as I2ω ∝ (Iω )2 (Fig. 5d). We note that a number of assumptions ear conversion between the two modes (details in Supplementary have been made in the preceding discussions to simplify our Information). In the experiment a nanoslit cut through the top analysis. In the actual sample, the dielectric spacer, consisting of metallic layer serves as the in-coupling entrance for the fundamental Si3 N4 and HfO2 sublayers, may have thickness fluctuation along light as well as the out-coupling port for the generated SHG signal. the waveguide, which would result in undesired non-uniformity Figure 3c shows the full-wave simulation result for the conversion along the propagation direction in both the mode refractive index efficiency for the electrically induced SHG as a function of the and the effective second-order nonlinear susceptibility induced fundamental wavelength. The frequency-doubled output peaks at by the applied voltage. In addition, any difference in the linear λω = 760 nm, precisely where the backward phase-matching condi- refractive indices of the two distinct dielectric media may further tion of 1k = |k2ω − 2kω | = 0 or n2ω = −nω is satisfied, as determined complicate the situation. Moreover, an extra loss factor included in Fig. 2a. The importance of the induced symmetry breaking by in the permittivity of nanostructured metals compared to that of using a dual-layered spacer is further illustrated in Fig. 3d, which bulk metals will cause a shorter propagation distance for the surface shows that the amount of generated second-harmonic signal will be plasmon waves, especially the H-asymmetric mode at the second- reduced by nearly two orders of magnitude if a uniform dielectric harmonic frequency. Nevertheless, the ultra-smooth dielectric films channel is employed. used in the experiment, fabricated by low-pressure chemical vapour A comprehensive set of measurements has been carried deposition and atomic layer deposition, respectively, ensure a device out to experimentally confirm the phenomenon of backward performance that agrees remarkably well with the prediction from phase-matching in the proposed plasmonic waveguide (details in the theoretical analysis and numerical simulations. The measured Supplementary Sections 4 and 5). Figure 4a shows the collected values of the spectral location (λω = 780 nm) and bandwidth nonlinear spectra under a constant pump intensity at a series of (1λω ≈ 30 nm) for the backward phase-matching in the plasmonic excitation wavelengths λω without an externally applied voltage. The waveguide fits nicely with the theoretical quantities of λω = 760 nm sharp and frequency-doubled peak at each excitation wavelength and 1λω ≈ 25 nm as illustrated in Fig. 3c. Just as momentum designates the second-harmonic signal, meanwhile broadband conservation leads to the requirement of phase-matching for NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials 3 © 2015 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE MATERIALS DOI: 10.1038/NMAT4324 a b 1 1.0 I2ω (a.u.) 0.5 0.0 820 340 800 360 780 380 400 760 nm) λ2ω (nm) 420 440 740 λω ( 0 Figure 4 | Frequency-doubled signals emerging from the plasmonic waveguide. a, Nonlinear spectra from the waveguide at a series of fundamental wavelengths λω without an externally applied voltage. The intensity of the pump light is maintained at a constant level. At each excitation wavelength, the output signal features a distinct peak centred at λω /2. b, Images of second-harmonic light spots with (bottom) and without (top) an applied voltage at λω = 780 nm. The electrically induced second-harmonic signal, which corresponds to the difference between the two observed spots, is expected to meet the backward phase-matching condition at the prescribed operating point. a λ 2ω (nm) b λ 2ω (nm) 370 380 390 400 410 370 380 390 400 410 1.0 No voltage 1.0 Voltage-induced 0.8 0.8 0.6 0.6 I2ω (a.u.) I2ω (a.u.) 90° 90° 0.4 0.4 0.2 0.2 0.0 180° 0° 0.0 180° 0° 740 760 780 800 820 740 760 780 800 820 λω (nm) λω (nm) c d 1.0 1.0 Voltage-induced Voltage-induced 0.8 0.8 0.6 0.6 I2ω (a.u.) I2ω (a.u.) 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 100 Applied voltage (V) Fundamental intensity (%) Figure 5 | Backward phase-matched second-harmonic generation. a, Intensity of the static SHG at different excitation wavelengths. b, Intensity of the electrically induced SHG at a series of excitation wavelengths. The realization of backward phase-matching is evidenced by the sharp peak in the conversion efficiency at λω = 780 nm. The polar diagrams (insets) in a and b illustrate the output signal as a function of the polarization angle of the fundamental light, which is defined as 0◦ (90◦ ) when the input polarization is parallel (perpendicular) to the slit. c, Electrically induced SHG as a function of an externally applied voltage. Data are collected at the fundamental wavelength of λω = 780 nm. d, Dependence of the electrically induced SHG intensity on the intensity of the fundamental light. The dashed curve in d depicts a quadratic function I2ω ∝ (Iω )2 whereas in a–c the dashed curves are a guide to the eye. Error bars in all figures indicate the system uncertainty in the respective measurement and have been calculated by taking the standard deviations from five measurements. efficient nonlinear wavemixing, the law of energy conservation |Aω |2 − |A2ω |2 in our negative-index medium, where Aω and A2ω necessitates the Manley–Rowe relation, a fundamental rule in represent the electric field of the fundamental and harmonic waves, nonlinear optics that governs the rates of creation and annihilation respectively21 . This would be very different from the scenario in of photons. For this reason, we anticipate that the total energy a conventional SHG process, where the Manley–Rowe relation flux in a Manley–Rowe relation takes the unusual form of describes the evolution of the sum, rather than the difference, 4 NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2015 Macmillan Publishers Limited. All rights reserved NATURE MATERIALS DOI: 10.1038/NMAT4324 ARTICLES of the squared amplitudes along the propagation direction of the 22. Shadrivov, I. V., Zharov, A. A. & Kivshar, Y. S. Second-harmonic generation in fundamental wave2 . nonlinear left-handed metamaterials. J. Opt. Soc. Am. B 23, 529–534 (2006). In conclusion, we have experimentally verified the backward 23. Rose, A., Huang, D. & Smith, D. R. Controlling the second harmonic in a phase-matched negative-index metamaterial. Phys. Rev. Lett. 107, phase-matching condition for frequency doubling of light in optical 063902 (2011). negative-index materials. 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Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification. Appl. Competing financial interests Phys. B 84, 131–137 (2006). The authors declare no competing financial interests. NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials 5 © 2015 Macmillan Publishers Limited. All rights reserved ARTICLES NATURE MATERIALS DOI: 10.1038/NMAT4324 Methods complete the core of the waveguide. Subsequently, a 140-nm-thick silver layer was The fabrication of the device starts with an atomic layer deposition of HfO2 onto an evaporated on the other side of the membrane using e-beam evaporation (CVC). A ultra-smooth Si3 N4 membrane, followed by a metallization process to generate the focused ion beam system (FEI Nova Nanolab 200) was used to define a nanoslit silver layers, a focused ion beam (FIB) nanopatterning step to create the 20 µm in length through the silver layer of the Si3 N4 side with a width of inlet/outlet-coupling nanoslit, and additional steps for electrical connections. The approximately 100 nm. The dielectric layers were not cut through during FIB frame-supported nitride membrane, grown on a silicon wafer by low-pressure milling so the two sides of the dielectric spacer remain electrically insulated. When chemical vapour deposition and followed by etching away a window in the silicon the nanoslit was ready, a second silver film of thickness 140 nm was evaporated on substrate, has a thickness of 15 nm and a surface roughness of 0.65 nm. A HfO2 top of the HfO2 layer using e-beam evaporation to complete the layer 15 nm in thickness was coated onto the nitride membrane by an atomic layer metal–dielectric–metal waveguide. Further steps were carried out to connect the deposition system (Cambridge NanoTech) to form the second sublayer and two metallic layers to opposite electrodes for electrical connection. NATURE MATERIALS | www.nature.com/naturematerials View publication stats © 2015 Macmillan Publishers Limited. All rights reserved