Supplementary Figure 1 Schematics of an optical pulse in a nonlinear medium. A Gaussian optical pulse propagates along z-axis in a nonlinear medium

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1 Supplementary Figure 1 Schematics of an optical pulse in a nonlinear medium. A Gaussian optical pulse propagates along z-axis in a nonlinear medium with thickness L.

2 Supplementary Figure Measurement of Zne refractive index in Hz range. a, Hz time domain signals transmitted through free space (purple) and a. mm Zne crystal (red). b, he measured refractive index of the Zne crystal as a function of frequency.

3 Supplementary Figure 3 Calculation of second-order nonlinear susceptibility of SRRs. heoretical calculation showing the Hz fields generated from a single layer of SRR emitter (red) and a. mm Zne emitter (purple). he nonlinear second-order susceptibility of a single layer of SRR is extracted by comparing the peak-to-peak amplitudes of SRR and Zne emitters. 3

4 Supplementary Figure 4 Current distributions. Distributions of the linear current (blue), nonlinear polarization (red) for a linear cut-wire (a) as well as the electric resonance (b) and electrically excited magnetic resonance (c) of the SRR. he + and indicate charge accumulation. Only the magnetic resonance of the SRR (c) exhibits an appreciable Hz via difference frequency generation (DFG) due to the broken symmetry. 4

5 Supplementary Note 1. he calculation of the second-order nonlinear susceptibility of SRRs We describe the procedure to extract the second-order nonlinear susceptibility of a single layer SRR emitter. We start with the wave equation in a nonlinear medium propagating in z-axis as 8 : () () E ( z, n E ( z, 1 P ) ( z, Eo ( z, t z c t c t c t (1) where E ( z, is the generated Hz field, n the refractive index of Hz pulses in the nonlinear () medium, the free space permittivity, c the speed of light, P ( z, the second-order polarization of () the nonlinear medium due to the optical pump beam E ( z,, and the second-order nonlinear susceptibility of the nonlinear medium. As shown in Supplementary Fig. 1, we define a Gaussian optical pump pulse propagating in a nonlinear medium, and at o z z' the optical pulse field amplitude can be expressed as E o ( z, E e a( tz'/ v o ), where a ln / and is the optical pulse duration, which is 14 fs in our case, and v o is the group velocity of the optical pump pulse in the medium. Hz radiation generated from an infinitesimal thin layer of the nonlinear medium z z' can be expressed as E () ax ( z', A (1 4ax ) e, where z' L z' x t v v o, A is a constant, L the thickness of the medium, and v the phase velocity of Hz pulses in the medium. Consequently, the Hz field generated from a single sheet of SRRs can be obtained as E SRR () at ( A SRR(1 4at ) e () In a thick nonlinear medium, the Hz field generated can be obtained by taking the integration of the sheet result over the thickness. For a. mm thick Zne crystal, we have: 5

6 E Zne ( () Zne L.mm E ( z', dz' L.mm () Ac t1.89ps Zne ax (1 4 ) d.1ps ax e x n n t- o 4 A (9.11 m/ps)[( t 1.89ps) e A () Zne a( t1.89ps) (1 4ax ) e dz' ( t.1ps) e ax a( t.1ps) ] (3) In the calculation, the group refractive index of optical pump beam at 15 nm is n o = n gr (15 nm)=.8 (ref 35). he central frequency of the Hz spectrum ~ Hz is used to determine the refractive index of Hz in Zne, i.e., n =n ( Hz). Because we found that our calculation is very sensitive to the indices which depend on crystal specifications, so in order to get as accurate calculation as possible, we measured it experimentally by comparing the time domain Hz transmission through free space and the. mm Zne crystal, from where the refractive index is obtained. he results are shown in Supplementary Fig., which indicates that n = 3.15 at Hz. Knowing the measured peak-to-peak amplitudes E SRR. E Zne and r 8 1 m/v, () 1 Zne 41 where 1 r m/v () (ref 36,37) is the electro-optic coefficient of Zne, we can extract SRR from calculated Hz electric fields by comparing Supplementary Equations () and (3). he result is shown in () 16 Supplementary Fig. 3, where.81 m /V reproduces the experimental peak-to-peak ratio. SRR herefore, we reveal a gigantic resonant sheet nonlinear susceptibility of SRRs, which is three orders of magnitude higher than the typical surface and sheet values. For example, the typical surface values are in the range of 1 - m /V ~ 1-1 m /V for fused silica or BK7 glass, ~1-19 m /V for liquid crystals, inorganic and organic thin films Note the above calculations do not take into account the detector response function since Hz waves from both SRRs and Zne are measured by the same detector. 6

7 Supplementary Note. he theory he observed Hz emission from the SRR sample is due to optical rectification (OR) by the second-order nonlinear electric response arising from the electrons in the metal that makes up the SRR. As shown in previous literature, the electron gas in the metal can be described by a hydrodynamic model -4 known as Maxwell-Vlassov theory, which goes beyond the usual linear Drude model for the metallic response and contains second-order nonlinearities: t p ne j m 1 5 E j m e m 5/3 ( n) (3 ) /3 n 1 ( E dive j B) ( j grad) j ne e m grad p, (4) he first two terms comprise the linear response as described by the conventional Drude model, the following terms are the nonlinear response of the electrons and represent Lorentz force, convective acceleration and Fermi pressure, respectively. All the nonlinear terms are second-order in the Maxwell fields, giving rise to a non-zero second-order electric polarizability () harmonic generation as well as for the Hz generation via OR considered here., which is responsible for nd As was shown in Ref. 4, the main nonlinear contributions are coming from the E dive and ( j grad ) terms, which do not contribute in the bulk but do contribute on surfaces. Qualitatively, both terms behave like j, current times accumulated density on the surface: As a consequence the nonlinear current is parallel or antiparallel to the linear current induced by the external excitation in regions of increasing or decreasing surface charge, respectively. his behaviour is illustrated in Supplementary Fig. 4: In the case of a simple straight nanorod, shown in Supplementary Fig. 4(a), the radiation caused by the nonlinear current contribution (red) in the two regions of surface charge accumulation (i.e. both ends of the nanorod) is out of phase, interferes destructively, and is not observed in the far field. Similarly, the j 7

8 symmetry of the current distribution at the electric dipole resonance of the SRR shown in Supplementary Fig. 4(b), i.e. with the external electric field along the two parallel arms of the SRR, leads to nonlinear polarization that is parallel to the linear currents at the top ends of the SRR where (positive) charge is accumulating and anti-parallel at the bottom where (positive) charge is depleting. Hence, again, no radiation due to the nonlinear current contributions is observed in the far field. If, however, we bend the nanorod into an U-shaped SRR and excite the resonant mode that has a continuous current (without nodes) around the SRR ring as shown in Supplementary Fig. 4(c), i.e. from the tip of one arm to the tip of the other, the nonlinear current in both arm are now parallel, their radiated fields interfere constructively and are observed in the far field. his mode is the magnetic resonance of the SRR, to which we couple electrically by the incident optical pulse (i.e. via the bi-anisotropic electric dipole moment across the gap), such that the resonance can be excited for normal incidence to the SRR plane. It should be noted that, although the mechanism also holds for second harmonic generation reported before,3, the current phaseresolved Hz results allow us to solve the completely different pressing challenges in Hz optoelectronics and gain new insights in nonlinear optics of metamaterials, i.e., quantitatively reveal the second-order susceptibility of the SRRs and phase reversal of Hz emission, which have not been obtained in prior the SHG experiment. In first approximation we can describe the temporal optical input signal as a linearly chirped Gaussian pulse, (5) he linear chirp / is just a redefinition 1 i and can be obtained from the difference in spectral width of the measured power spectrum of the pump pulse ( 9 Hz) and the temporal width of the pump pulse derived from a cross-correlation measurement ( Hz). is the center frequency of the pump pulse and a phase shift between carrier and envelope. 8 o

9 (6) with the second-order nonlinear response being proportional to (7) he OR is the first term in the spectrum above; the other two summands represent the SHG, which is removed from the signal by the eflon filter in the Hz signal path and by the detector crystals acting as an effective low-pass filter. Note that the chirp drops out and the OR signal only depends on the temporal envelope of the pump pulse. We can write the radiated Hz field in terms of a () polarizability of the SRR sample: (8) he time derivative in the radiated fields suppresses zero-frequency components such that the Hz spectrum has a peak at finite frequency, ( nl) () 4 ( nl) () t rad ( ) ~ e Erad ( ~ (1 t ) e E. (9) he Hz spectrum peaks at and has a bandwidth FWHM. 31. Strictly, the radiated nonlinear fields are given by the currents in the SRR, which are proportional to the reflection amplitude of the electric sheet given by the SRR metasurface. So in the formulas above we ( in) have G ( ) ~ R( ) G ( ). From the linear transmittance measurements of the SRR we get R( ) Z ( ) /[ Z ( )] with ( ) i /( i) where 384, 199 e e Z e 9

10 Hz, and 15.4 Hz. hus, since the SRR resonance is very wide compared to the bandwidth of the optical pump pulse envelope ( Hz) the effect of R () on the Hz pulse shape is negligible: In our experiments, the achievable Hz bandwidth is limited by the duration of the optical pump pulse, not the SRR response. A second, more severe problem is that the bandwidth and, in particular, the upper cut-off frequency (lowpass, between and 3 Hz depending on the used crystal) of the detectors is much smaller than the expected Hz signal. As a consequence, the spectrum of the observed Hz signal in our experiments is not limited by the OR but essentially given by the bandwidth of the detectors. he so predicted Hz waveform agrees well with the experimental observation. 1

11 Supplementary Note 3. Comparison with other Hz nano-plasmonic structures We now make a thorough comparison between our approach and the prior Hz generation work in silver nanoparticles as shown in Ref. 9 in the main text. his clearly shows not only the distinctly different nonlinear mechanisms involved between the two Hz generation processes, but also the exceedingly new findings in our work over the prior work. 1. Our work demonstrates broadband Hz generation, ranging from about.1 to 4 Hz, free from quasi-phase-matching and phonon limitation. Instead, this Hz bandwidth is limited mostly by the excitation pulse duration ~ 14 fs (spectral width at FWHM ~13 mev or 3. Hz) and have potential to produce broadband and gapless spectrum covering the entire Hz range up to 15 Hz with shorter pulses. In strong contrast, the silver nanoparticles work only demonstrates less than 1.5 Hz bandwidth even with shorter excitation pulse of 1 fs (spectral width at FWHM ~18 mev or 4.4 Hz). his shows that the scheme there still suffers from those conventional limitations unlike the new demonstration in our work.. Regarding the Hz emission from the various emitters (silver nanoparticles, Zne, SRRs), it is most straightforward to compare the raw data of Hz emission which are shown in Fig. 3 of our work and Fig. 7 of ref 9 in the main text. In our SRR emitter, the peak-to-peak amplitude of Hz electric field versus pump power shows that the emitted Hz field is ~1/5 of the maximum from the optimized Zne emitter for the same pumping energy at pumping power ~ 1. GW/cm. In the silver nanoparticles work, the Hz intensity (E field square) is only ~1-6 to 1-7 of the Zne emitter performance at the same pumping power ~1. GW/cm, and increase to ~1-4 of the intensity from Zne (1 1 vs. 1 5 ) at much higher pump power close to the saturation condition. he Hz output intensity of our SRR emitter exceeds the silver nanoparticles under the same pumping. 3. he distinctly different power dependence of the Hz emission between our SRR emitter ( nd order for intensity) and silver nanoparticles (5 th order) up to pumping power ~ 1. GW/cm, shown in Fig. 3 of our work and Fig. 7 of ref 9 in the main text, clearly indicates the completely 11

12 different optical nonlinear mechanism used. Our work also reveals much more aspects of nonlinear optical phenomena ranging from distinct electric vs. magnetic dipole excitation dependence, polarization control to resonant enhancement and pump photon energy selectivity that are rarely addressed in any other platform, far beyond the silver nano-plasmonic structures and gas phase work cited. 4. Our SRR scheme and Hz results allow a thorough understanding regarding the Hz physics of nonlinear metamaterials that is beyond the silver nanoparticles work. here are at least three strategic advantages of our scheme: (1) our Hz setup allows for tuning the pump photon energy to study a single sample and to show the resonant enhancement effect (Fig. in the main tex. he silver nanoparticles work, with fixed pump wavelength, has to use various samples with different unit cell sizes which are susceptible to many other external factors, e.g., inhomogeneous broadening, collective effects, detail fabrication conditions () Our SRR sample exhibits distinct polarization dependence (Fig. 4a) and electric vs. magnetic dipole excitation physics (Fig. 1), which are missed in any other work and add another layer of novel functionality; (3) our results identify the nonlinear mechanism in our SRR scheme that arises from exciting the magnetic-dipole resonance of SRRs and second-order optical rectification effect (Fig. 1). However, limitations in the silver nanoparticles work, although very interesting, don t allow any transparent picture for nonlinear physics yet. Particularly, the model discussed cannot predict the right nonlinear orders and has many key assumptions in their model used, e.g., the exact nature of the photo-current, uniform pondermotive potential, only peak power used without considering excitation intensity change during optical pulse, only a single frequency used for the incident pump light without considering spectral width of ~ 18 mev 1

13 Supplementary References 35. Nahata, A., Weling, A. S. & Heinz,. F. A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling. Appl. Phys. Lett. 69, (1996). 36. Faust, W. L. & Henry, C. H. Mixing of Visible and Near-Resonance Infrared Light in GaP. Phys. Rev. Lett. 17, (1966). 37. Leitenstorfer, A., Hunsche, S., Shah, J., Nuss, M. C. & Knox, W. H. Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory. Appl. Phys. Lett.74, (1999). 13

Multi-cycle THz pulse generation in poled lithium niobate crystals

Laser Focus World April 2005 issue (pp. 67-72). Multi-cycle THz pulse generation in poled lithium niobate crystals Yun-Shik Lee and Theodore B. Norris Yun-Shik Lee is an assistant professor of physics