SUPPLMNTARY INFORMATION Shoufeng Lan 1, Lei Kang 2, David T. Schoen 3, Sean P. Rodrigues 1,2, Yonghao Cui 2, Mark L. Brongersma 3, Wenshan Cai 1,2* 1 School of lectrical and Computer ngineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA 2 School of Materials Science and ngineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Backward phase-matching for nonlinear optical generation in negative-index materials 3 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA * Correspondence to: wcai@gatech.edu This PDF file includes: 1. Locating the Operating Point 2. Symmetric Breaking in Waveguide 3. Numerical Simulation 4. xperimental Setup 5. Control xperiment with Uniform Dielectric Core 6. stimation of the Conversion fficiency Supplementary Figures S1-S5 Supplementary References 1-11 1. Locating the Operating Point The goal of the design strategy is to identify an operating point in the plasmonic waveguide, such that the backward phase matching condition of = and = 2 is satisfied. For this purpose we exploit the H-asymmetric plasmonic mode at 2 with a negative mode refractive index, which can be phase-matched to the H-symmetric mode at the fundamental NATUR MATRIALS www.nature.com/naturematerials 1
SUPPLMNTARY INFORMATION frequency of with a positive mode index of equal magnitude. The dispersive refractive index for each of these modes supported in a metal-dielectric-metal (MIM) waveguide can be calculated by solving Maxwell s equations with the appropriate boundary conditions. 1 The optimal choice of the operating condition, with a 30 nm dielectric spacer of =2 sandwiched between two silver films, was determined by considering a number of factors. These factors include the loss tangent of the H-asymmetric mode, the feasibility to balance the magnitudes of the two mode indices, the stability of the operating point in case of fabrication uncertainties, and the tuning range of the ultrafast laser available in the lab. Figure S1. Dispersion of the mode refractive indices in the plasmonic waveguide. The dielectric spacer has a thickness of 40 nm and a material index of 2.0. Both the real (solid) and imaginary (dashed) parts of the mode refractive index are plotted. The surface plasmon frequency ( = ) is at 3.17 ev. The operating point indicated by the crossed dash lines is located at = ev (λ = 786 nm) and 2 = ev (λ 2 = 393 nm). 2 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION For comparison purposes, in Fig. S1 we show the operating point of the system when the thickness of the spacer of the Ag-Si 3 N 4 -Ag waveguide is increased to 40 nm. The permittivity of silver is adopted from the tabulated data 2 and a refractive index of =2 is used for the dielectric layer. As the gap size increases, the mode index of the typical, H-symmetric mode decreases and approaches the material index of the dielectric spacer, which leads to an increase in the loss tangent of the corresponding H-asymmetric mode at 2. More critically, compared to the dispersion curves in Figure 1b of the manuscript with = 0 nm, the operating point ( 2 = ev) with a 40 nm spacer would inevitably approach the surface plasmon frequency ( = 7 ev) of the metal-dielectric interface. As shown in Fig. S1, the index of the H- asymmetric mode exhibits drastic dispersion near the cutoff frequency of p, with both its real and imaginary parts changing rapidly with wavelength. This represents an undesired instability of the operating point that we would try to avoid in the experiment. A similar instability occurs when a low-index dielectric material is used as the spacer, because in this case the frequency range (between =0 and = ) for the H-asymmetric mode would be very narrow. Given all these considerations, in the experimental work we used a thin layer of dielectric ( = 0 nm) with a relatively large index ( =2) to achieve a phase-matched operating point ( 2 = 27 ev) away from p with a reasonably small loss tangent. 2. Symmetric Breaking in Waveguide The second harmonic intensity created in a nonlinear waveguide of length L due to wavemixing from mode i at the fundamental wavelength λ to mode j at the harmonic wavelength λ 2 can be expressed as NATUR MATRIALS www.nature.com/naturematerials 3
SUPPLMNTARY INFORMATION I 2 P 2 2 2 2 sin( kl 2 ) ( Si, j ) L kl 2 2 ( 2) 2 ( χ ) I eff Here superscripts and 2 correspond to the fundamental and harmonic waves, respectively, I is the intensity, P is the nonlinear polarization, 2 k = k 2 k represents the phase mismatch, χ ( 2) eff denotes the effective second-order susceptibility, and the nonlinear overlap factor S i, j describes the normalized spatial coupling coefficient defined as 3 : S i, j = S i χ ( 2) jii i ds 2 j i i ds 2 2 ( j ) ds The phase matching condition ( k = 0 ) in this work is fulfilled by matching the mode refractive indices for λ and λ 2 in the plasmonic waveguide. The lacking of an efficient second-order susceptibility in centro-symmetric materials can be overcome by introducing a voltage-induced ( 2) χ eff as described in the main text. Since the nonlinear generation efficiency is critically dependent on the nonlinear overlap 2 2 factor, I ), in the following section we describe our approach to optimize in this ( S i, j work. Assuming a uniform susceptibility ( 2 ) χ exists in the core of the plasmonic waveguide, the 2 nonlinear overlap factor S i, j is determined by the field integral j i i ds within the cross- S section of the waveguide. In the proposed phase-matching scheme, which involves two modes of opposite symmetry, this integral is reduced to zero and no SHG should be expected. The scenario is better visualized in Figure S2, which shows the profiles of all field components of the H- symmetric mode at λ and the H-asymmetric mode at λ 2. S i, j 4 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION Figure S2. Nonlinear overlap factor and symmetric breaking in the plasmonic waveguide. The profiles of different field components are shown for (a) the fundamental wave at λ = 760 nm and (b) the second harmonic wave at λ 2 = 380 nm. (c) The field integral within the cross-section. 2 dx Figure S2c illustrates how the integral 2 dx evolves along a direction perpendicular to the metal-dielectric layers. Due to the opposing symmetry between the two modes involved, the integral and consequently the nonlinear overlap factor S i, j, reach the maximum value half-way into the waveguide core and then are reduced to zero when the entire waveguide cross-section is accounted. In order to circumvent this cancellation effect and enable NATUR MATRIALS www.nature.com/naturematerials 5
SUPPLMNTARY INFORMATION efficient SHG in the waveguide, we purposely introduce a symmetric breaking mechanism by splitting the middle dielectric layer into two sub-layers with identical linear but distinct nonlinear properties. This way the effective susceptibility ( 2) χ eff of one of the sub-layer dominates over that of the other, and efficient wave-mixing is facilitated between the two modes of opposite symmetry. In this work, Si 3 N 4 and HfO 2 layers with the same thickness of 15 nm were used in our sample. Both Si 3 N 4 and HfO 2 have a similar linear refractive index of approximately 2. Specifically, the values of the refractive index are n,si3n4 = 2.0; n,hfo2 = 2.09 ; n 2,Si3N4 = 2.07, and n 2,HfO2 = 2.16 at the operating points (λ = 780 nm and λ 2 = 390 nm). 4,5 The third order nonlinear susceptibility χ (3) is approximately 3 10 21 m 2 /V 2 for Si 3 N 4. 6 The χ (3) of HfO 2 is not readily available in the literature and can be deduced from the magnitude and dispersion of the linear refractive index. 7 The value of χ (3) for HfO 2 is hence estimated to be 8 10 22 m 2 /V 2. The resistivity for the two dielectrics are ρ Si3N4 ~10 16 Ω cm and ρ HfO2 ~ 10 12 Ω cm. 8,9 Therefore the value of χ (3) ρ for Si 3 N 4 is four orders of magnitude larger than that of HfO 2. Consequently, the voltage-induced ( ) is mainly in the Si 3 N 4 sublayer, dramatically breaking down the symmetry in the waveguide. We note that these values in the literature serve only as a reference, because the exact value of the nonlinear susceptibility and electrical resistivity of thin films are critically sensitive to the fabrication process. Nevertheless, the use of two distinct dielectric materials enables the asymmetry of the FISH generation, which will primarily exist within one of the two sublayers. In fact, any nontrivial difference in χ (3) ρ between the two dielectrics will lead to a notable difference in the voltage-induced ( ), thereby resulting in a distinct contribution to the frequency-double output ( ( ) ). 6 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION 3. Numerical Simulation The simulations of the linear and nonlinear responses of the structure were performed in the frequency domain using a commercial finite-element solver (COMSOL). The mode refractive indices of the metal-insulator-metal waveguide can be computed either analytically by solving Maxwell s equations or numerically using the mode analysis module of the finite-element solver. Perfect agreement was achieved between the mode indices obtained by these two methods. The dispersive, complex indices of the two modes in the plasmonic waveguide were plotted in Fig. 1b and Fig. S1. The mode refractive indices were further used to deduce the degree of phase mismatch (, ) in Figure 2c, where a perfectly phase matched point (, =0, which indicates the backward phase-matching condition of = and = 2 ) is obtained at λ = 760 nm. To simulate the nonlinear response in the plasmonic waveguide, the internal equation system of the COMSOL solver was modified to include a source term related to the nonlinear polarization. Two models, one at the frequency of and the other at 2, were solved interactively, with mode coupling between the fundamental wave and the frequency-doubled signal via the nonlinear medium taken into account. 10 To simplify the modeling process while capturing the essentials of the physics, we assume that all the materials in the structure are homogeneous and exhibit bulk properties. As mentioned in the previous section, the voltageinduced SHG mainly arises from one of the two 15-nm dielectric spacers due to the contrast in χ (3) ρ. Therefore in the nonlinear model, the effective second order susceptibility ( ) = / 2 0 m/v was assigned to only one of the sublayers when a voltage of 10 V is applied across the device. NATUR MATRIALS www.nature.com/naturematerials 7
SUPPLMNTARY INFORMATION The far-field coupling of the fundamental wave from the free space to the core of the plasmonic waveguide was also simulated using the finite-element package. Figure S3a illustrates the field profile of the H z component (out of the plane) for the fundamental wave at λ = 760 nm. Slit-coupling to metal-insulator-metal waveguides from free space, while possessing a rather low efficiency (~14% from our simulations), is a standard technique and has been used extensively in plasmonics. With the H-symmetric plasmonic mode coupled into the dielectric core via the nanoslit, Figures S3b demonstrates the necessity of broken symmetry in the plasmonic waveguide, where frequency-doubled signals stem from the voltage induced ( ) response. Figure S3. Far field coupling and FISH generation in the nonlinear plasmonic structure. H z component (out of the plane) of the magnetic field is mapped for (a) the fundamental wave at λ = 760 nm and (b) the frequency-doubled light at λ 2 = 380 nm. In (b) the metal-insulatormetal waveguide with a single uniform core ( = 0 nm) was used in the left half, and the right half has a dual-layered spacer consisting of two distinct sublayers ( = nm, = nm). White dashed lines indicate the geometric boundaries in the structure. With purposely introduced symmetry breaking (the right half of Fig. S3b, = nm, = nm), the amplitude of the magnetic field at 2 is 10 times (10 2 times in terms of 8 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION intensity) larger than that with a uniform dielectric core of 30 nm thick (the left half of Fig. S3b). A similar result was provided in the main manuscript (Fig. 2d) without depicting the far-field coupling effect. The finite SHG from the symmetric dielectric core (Fig. S3b, left) is attributed to non-propagating modes locally generated in the waveguide. The nonlinear polarization at 2 serves as a local dipole source, which will couple to all modes supported in the waveguide. All but one (the H-asymmetric mode) of these modes at 2 are not propagating, but they do exist locally and contribute to the non-zero field profile in Figure 2d (bottom) of the manuscript. 4. xperimental Setup The experimental setup for this study is illustrated in Figure S4. The fundamental light was from a Ti:Sapphire ultrafast oscillator (Spectra-Physics, Mai Tai HP) with a pulse duration of 100 fs, a repetition rate of 80 MHz, and a tuning range of 690-1040 nm. The output beam from the source first passes through a long-pass filter to prevent any high-frequency residue from entering later stages of the system. A set of Glan polarizers and half wave plates is employed to control the power level and polarization state of the excitation pulses. After passing through a beam splitter, the fundamental beam is delivered to the nonlinear plasmonic device mounted under an inverted optical microscope (Zeiss, Axio Observer D1m) with a 20 objective (NA = 0.5), which results in a spot size of approximately 50 m on the sample. The generated signal from the sample is collected by the same objective and sent back to the beam splitter. A bandpass filter is placed in front of the spectrometer to eliminate the high intensity fundamental waves. The detection system includes a monochromator (Princeton Instruments, IsoPlane) followed by a CCD camera (Princeton Instruments, Pixis). Nonlinear signals with and without an NATUR MATRIALS www.nature.com/naturematerials 9
SUPPLMNTARY INFORMATION applied voltage were collected separately and the purely voltage-induced SHG component was extracted by further data processing. Figure S4. xperimental setup for the backward phase-matching measurements. Abbreviations for optical components: LPF long-pass filter; HWP half wave plate; GP Glan polarizer; M mirror; BS beam splitter; BPF band-pass filter. 5. Control xperiment with Uniform Dielectric Core To demonstrate the necessity of purposely induced broken symmetry, in this part we report the results from a control experiment, where the core of the plasmonic waveguide is formed by a uniform layer of Si 3 N 4. The fabrication steps for this control sample are largely the same as the procedure described in the Methods section, except a thickness of 30 nm is used for the initial Si 3 N 4 membrane in place of the asymmetric Si 3 N 4 /HfO 2 dual layers. Figure S5 shows the experimentally collected excitation spectrum for the second harmonic generation from the control sample. The intensity of the fundamental wave is kept at a constant level and the 10 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION wavelength of the fundamental light is varied from 740 nm to 820 nm at 10 nm step size. Compared to the results using a dual-layered waveguide core ( = nm, = nm, plotted in Fig. 3c and repeated in Fig. S5 for comparison), the SHG from the control experiment is only ~ 20% of the previous value at a fundamental wavelength of ~ 780 nm. This remaining SHG largely stems from the flat metal surface and the in-coupling slit, instead of the dielectric core of the MIM waveguide. More importantly, the resonance behavior in the SHG excitation spectrum, which serves as the fingerprint of the phase matching condition, disappeared in the control experiment. There is a very small, bump-like feature in the excitation spectrum for the control sample, which we believe is due to the residue symmetry-breaking in the 30 nm Si 3 N 4 channel. This is because the two interfaces of the Si 3 N 4 membrane have experienced different deposition and etching conditions during fabrication. The new control experiment evidences the need of purposely induced broken symmetry for the work. Figure S5. Control experiment in absence of introduced symmetry breaking. The blue markers show the SHG from the control sample of the uniform dielectric core as a function of the fundamental frequency. The excitation spectrum of the main sample with dual-layered spacer (green, NATUR MATRIALS www.nature.com/naturematerials 11
SUPPLMNTARY INFORMATION which is identical to the curve in Fig. 3c) is also included for comparison purposes. The data (green and blue) are collected under the same experimental conditions. The dashed lines are a guide to the eye. 6. stimation of the Conversion fficiency Here we roughly estimate the conversion efficiency of second harmonic generation from the plasmonic structure. Under the excitation of an ultrafast laser at the wavelength of λ = 780 nm with a time-averaged power of 168.5 mw, approximately 3000 photons per second of the frequency-doubled wave were detected by the silicon CCD camera. Considering a laser spot size of ~ 80 µm and the slit dimension of 100 nm 20 µm, photons at the fundamental frequency on the level of 7 0 /second were carried by the H-symmetric mode in the waveguide and drove the nonlinear light-matter interaction. Other factors to be accounted for include the finite collection cone of the 20 objective lens, the optical loss associated with the filters and other optical elements, and the efficiency of the grating in the monochromator. Taken all these experimental factors into account, the overall conversion efficiency for the static SHG was estimated to be 2 0. The conversion efficiency for the voltage-induced SHG signal, which is approximately 22% of the static SHG at the externally applied voltage of 3V, is estimated to be 4 4 0. The major reason for this limited conversion efficiency is that while the H- symmetric fundamental wave is able to propagate for tens of microns in the waveguide, only the SHG generated in a sub-micrometer section can make its way to the outcoupling slit because of the lossy nature of the H-asymmetric harmonic mode. Nevertheless, the benefit from the backward phase-matching is evidenced by the peak value of the conversion efficiency when the excitation wavelength approaches λ = 780. Moreover, thanks to the phase matching condition, the conversion efficiency in this structure is over two orders larger than that of the previous work 12 NATUR MATRIALS www.nature.com/naturematerials
SUPPLMNTARY INFORMATION under similar experimental conditions without phase matching. 10,11 We note again the goal of this research is to provide the first experimental evidence (i.e., the conversion peak at ~ 780 nm observed in the experiment) of a decade-long prediction instead of producing the best configuration for the most efficient harmonic generation. Supplementary References 1 Maier, S. A. Plasmonics: Fundamentals and Applications. (Springer, New York, 2007). 2 Johnson, P. B. & Christy, R. W. Optical-constants of noble-metals. Phys. Rev. B 6, 4370-4379 (1972). 3 Stegeman, G. I. & Seaton, C. T. Nonlinear integrated-optics. J. Appl. Phys. 58, R57-R78 (1985). 4 Philipp, H. R. Optical properties of silicon-nitride. J. lectrochem. Soc. 120, 295-300 (1973). 5 Wood, D. L., Nassau, K., Kometani, T. Y. & Nash, D. L. Optical-properties of cubic hafnia stabilized with Yttria. Appl. Opt. 29, 604-607 (1990). 6 Ikeda, K., Saperstein, R.., Alic, N. & Fainman, Y. Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/silicon dioxide waveguides. Opt. xpress 16, 12987-12994 (2008). 7 Boling, N. L., Glass, A. J. & Owyoung, A. mpirical relationships for predicting nonlinear refractive-index changes in optical solids. I J. Quantum lectron. 14, 601-608 (1978). 8 Nishi, Y. & Doering, R. Handbook of Semiconductor Manufacturing Technology, 2 nd ed., (CRC Press, Boca Raton, 2007). 9 Hildebrandt,. et al. Controlled oxygen vacancy induced p-type conductivity in HfO2-x thin films. Appl. Phys. Lett. 99, 112902 (2011). 10 Cai, W. S., Vasudev, A. P. & Brongersma, M. L. lectrically controlled nonlinear generation of light with plasmonics. Science 333, 1720-1723 (2011). 11 Klein, M. W., nkrich, C., Wegener, M. & Linden, S. Second-harmonic generation from magnetic metamaterials. Science 313, 502-504 (2006). NATUR MATRIALS www.nature.com/naturematerials 13