Supporting information for Metamaterials with tailored nonlinear optical response

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1 Supporting information for Metamaterials with tailored nonlinear optical response Hannu Husu, Roope Siikanen, Jouni Mäkitalo, Joonas Lehtolahti, Janne Laukkanen, Markku Kuittinen and Martti Kauranen Nonlinear response tensor The local-field distribution in nanostructures can differ significantly from the incident field used to illuminate the structure, in particular when plasmon resonances are excited. The electromagnetic energy is typically concentrated to small nanoscale locations and the local field can contain polarization components not present in the incident field. Furthermore, the optical response functions, including the nonlinear susceptibilities, exhibit strong nanoscale variations. In order to decsribe the nonlinear response of a nanostructure in detail, one should therefore account for the local variations in electromagnetic fields, nonlinear susceptibility tensors, and the generated nonlinear sources as well as the coupling of the incoming and outgoing radiation fields to the local fields. For the present, this is a challenging task even computationally. The nonlinear response tensor (NRT) A jkl is defined in terms of the incoming and outgoing radiation fields as 30 E ( 2ω) A E ( ω) E ( ω) =. (S1) j jkl k l k, l NRT therefore avoids all the complications related to the local nanoscale effects. Its main disadvantage is that it is more related to the experimental setup than the sample itself. On the other hand, it allows the macroscopic nonlinear response to be described using electric-dipole-type selection rules, although all multipole effects are included implicitly. The NRT is therefore essentially equivalent to the macroscopic nonlinear susceptibility of the sample in the effective medium limit, an appropriate approach from the metamaterials point of view. The samples investigated in the present work have a mirror plane, which limits the number of nonvanishing tensor components. In short, if the sample has a mirror plane in y direction, the tensor components with an odd number of index x, which is perpendicular to y, are forbidden. The allowed 1

2 tensor components for the Standard sample are thus yyy, yxx, xxy, xyx, where the last two must be equal for SHG. Similarly, the allowed tensor components for Samples A and B are vvv, vuu, uvu=uuv. Sample quality issues The data of the present work was fitted by assuming that the forbidden tensor components of the samples vanish. Furthermore, we also assumed the allowed tensor components to be real valued, although generally they are complex. Both simplifications, neglecting the forbidden components and assuming real valued components, were justified by the good fits to the measurement data. However, our earlier work and the work of others have shown that the ideal symmetry of the samples can be broken by defects 20-23,27,28. In some cases, this has led to forbidden SHG signals that are even larger than the allowed ones 22. In order to verify that this is not a problem in the present work, we have performed two additional tests. The results for all the samples were fitted by taking the forbidden components as nonvanishing. The results are shown in Table S1, indicating that the magnitudes of the forbidden components are always much smaller than the allowed components. This result is a direct consequence of the high quality of the present samples. Table S1 SHG values for certain NRT components taken from the fit to the measured data. The components forbidden by the symmetry are shown with grey background. The values are normalized to yyy-component of the Standard array. xxx xyy yxx yyy Standard uuu uvv vuu vvv Sample A Sample B xxx xyy yxx yyy Sample C As a second test, our samples also included Sample C (Fig. S1a), which has four-fold symmetry about the surface normal. Such a sample is expected to have isotropic linear response, which was verified by measuring identical extinction spectra for x and y polarizations (Fig. S1b). Furthermore, 2

3 Figure S1 a) Layout of Sample C and the coordinate system. b) The extinction spectra for the eigenpolarizations. Sample C looks centrosymmetric at normal incidence and should not yield SHG signals. We measured a number of SHG tensor components for Sample C and found that all of them are close to the noise level (Table S1). Detailed analysis of resonance factors When the nonlinear response is enhanced by a resonance at the fundamental frequency, the Lorentz model suggests that the NRT should depend inversely on the square of the resonance denominator ( ω) ω ω 2ωγ 2ω( ω ω γ) 2ω( γ) D = i i = i, (S2) 0 0 where ω 0 is the resonance frequency, ω is the laser frequency, γ is the half width of the resonance line, and is the detuning of the laser frequency from exact resonance. The approximate results are valid when the laser frequency ω is tuned close to the resonance frequency ω 0. In the following all other quantities are normalized to the half width of the Standard array. Table S2 Predicted resonance enhancement factors. The enhancement factor is the inverse of the square of the denominator scaled to the factor for Standard. Polarization Half width Detuning iγ Factor Standard y γ γ 1.41γ 1 Sample A u 3γ γ 3.16γ 0.20 Sample A v 1.5γ (4 / 3)γ 2.01γ 0.49 Sample B u, v (2 / 3)γ (1/ 5)γ 0.70γ 4.06 The predicted enhancement factors are shown in Table S2 and are seen to provide a reasonable estimate of the differences in the nonlinear responses of the samples as shown in Table 1. However, this approach does not account for the more subtle details that may arise from the influence of the metamolecular ordering on the strength of the various polarization-dependent resonances and on the 3

4 Figure S2 The distribution of the in-plane local field components for Standard array, Sample A and Sample B for the eigenpolarization excitations at the fundamental frequency. The fields are simulated in the middle of the particle along the vertical direction. The z components are very weak and thus not shown. details of local-field distribution for the various resonances. Local electric field distributions The enhanced nonlinear responses in metal nanostructures arise from the strong local fields in the structures. For second-harmonic generation, the local-field strength at the fundamental frequency plays a particularly important role, because the nonlinear response scales with the second power of the fundamental field. In order to address the differences in the local fields in our structures, we have performed electromagnetic simulations. We emphasize, however, that these results can only provide a rough estimate for the differences in the nonlinear response, because the second-order response depends also on the symmetry of the local-field distribution26. The local electric fields were simulated by solving the Maxwell s equations using the finitedifference time-domain (FDTD) method. The simulations were performed for 3D-structures, where the array structure, substrate and protective glass layer were taken into account. The results in Fig. S2 represent the local field strengths at the fundamental frequency in the middle of the particles along the vertical (z) direction and are shown for the eigenpolarizations of each sample (x and y or u and v). The z component is not shown because it was found to be significantly weaker than the in-plane components. We have also calculated the local fields 2 nm below and above the metal layer with similar conclusions. 4

5 We note that even qualitative comparison between the Standard sample and the other two samples (A and B) is difficult because of their different eigenpolarizations. However, the strengths of the local fields in the modified structures A and B are in general similar to those that correspond to y polarization for the Standard sample, which is resonant. Beyond this, the strongest local fields around each particle for Samples A and B are at different locations compared to the Standard sample due to the different overall symmetry of the samples. We will therefore focus on comparing Samples A and B, which have the same eigenpolarizations. The incident fundamental field interacts with the structures through the local susceptibility tensor of the particle surface. In addition, the dominant contribution to the local nonlinear response can be assumed to arise from the tensor component where both the local fundamental and second-harmonic fields are along the local surface normal (Ref. S1). Thus, for the v-polarized SH output the nonlinear sources are driven by the v-polarized local field at the fundamental frequency. For both input polarizations the v-polarized fields for Sample B are about 1.44 times as strong as the fields for Sample A, which furthermore corresponds to a factor of 2.1 in terms of the second-order response tensors. In Table 1, the enhancement between Samples A and B is 2.8 for vuu component and 2.6 for vvv component, which are both already close to the factor of 2.1. Furthermore, the local field distribution within an individual particle is more asymmetric for Sample B than for Sample A, which further enhances the factor from 2.1. The third allowed tensor component uuv depends on both components of the fundamental field, thus requiring a mixed-polarization input. This component can therefore not be directly compared to the local-field calculations, because the local fields calculated with a mixed input would contribute also to other tensor components. Supporting references S1. Wang, F. X.; Rodríquez, F. J.; Albers, W. M.; Ahorinta, R.; Sipe, J. E.; Kauranen, M. Phys. Rev. B 2009, 80,