Second Harmonic Generation from Critically Coupled Surface Phonon Polaritons

TL;DR

This paper demonstrates how critically coupled surface phonon polaritons at a 6H-SiC-air interface can significantly enhance second harmonic generation in the mid-infrared, with precise control and detailed analysis of the process.

Contribution

It provides the first experimental demonstration of second harmonic generation from critically coupled surface phonon polaritons with tunable excitation conditions and detailed theoretical modeling.

Findings

High-accuracy match between experimental and theoretical spectra

Quantification of optical field enhancement

Identification of low out-coupling efficiency mechanism

Abstract

Mid-infrared nanophotonics can be realized using sub-diffractional light localization and field enhancement with surface phonon polaritons in polar dielectric materials. We experimentally demonstrate second harmonic generation due to the optical field enhancement from critically coupled surface phonon polaritons at the 6H-SiC-air interface, employing an infrared free-electron laser for intense, tunable, and narrowband mid-infrared excitation. Critical coupling to the surface polaritons is achieved using a prism in the Otto geometry with adjustable width of the air gap, providing full control over the excitation conditions along the polariton dispersion. The calculated reflectivity and second harmonic spectra reproduce the full experimental data set with high accuracy, allowing for a quantification of the optical field enhancement. We also reveal the mechanism for low out-coupling efficiency of the second harmonic light in the Otto geometry. Perspectives on surface phonon polariton-based nonlinear sensing and nonlinear waveguide coupling are discussed.

Equations

$$L=\frac{\lambda }{2\pi \sqrt{k_{\parallel }^2{c}^2/{\omega }^2-1}},$$

$$I(2\omega )$$

Full text

1 Introduction

Surface polaritons are the key building block of nanophotonics since these excitations allow for extreme light localization accompanied with significant enhancement of the local optical fields. A large body of research has focused on surface plasmon polaritons (SPPs) at noble metal surfaces,1 which has led to a number of applications ranging from optical near-field microscopy to nonlinear plasmonic nanosensors.2, 3, 4, 5, 6, 7 Recently, an alternative approach was introduced employing surface phonon polaritons (SPhPs) which can be excited in the mid-infrared (mid-IR) at the surface of polar dielectrics.8, 9, 10 In these materials, optical phonon resonances in the dielectric response result in a negative permittivity in the reststrahl range between the transverse optical (TO) and longitudinal optical (LO) phonon frequencies and, in consequence, the existence of a highly dispersive surface polariton.11 Notably, the much reduced optical losses of SPhPs as compared to SPPs have been argued to potentially solve the loss problem that was identified as the key limitation for wide-spread implementation of plasmonic devices.12, 13 Recent pioneering experiments have employed SPhPs for optical switching,14 as well as sub-diffractional light confinement15, 16, 17 and strongly enhanced nonlinear response in sub-wavelength nanostructures.18

A systematic study of the linear and nonlinear-optical response of surface polaritons is enabled by prism coupling either in Kretschmann-Raether19 or Otto20 configuration. In both cases, the high refractive index of the prism operated in a regime of total internal reflection provides the large momenta required to excite the surface waves.19 Whilst mostly employed for studies of SPPs,21, 22, 23, 24, 25 a few works also investigated SPhPs with prism coupling in the mid-IR.8, 26, 10, 27 Specifically in the Otto configuration, the surface polariton is excited across an air gap of adjustable width, providing extrinsic tunability of the excitation efficiency which can lead to critical coupling conditions.10 Strong optical field enhancement at critical coupling was predicted but could not be experimentally confirmed using linear optical techniques. Instead, nonlinear-optical approaches like second harmonic generation (SHG) are highly sensitive to the localized electromagnetic fields.28, 29, 22, 30, 18, 25

In this Letter, we experimentally demonstrate the first SHG from critically coupled SPhPs, allowing us to accurately determine the associated optical field enhancement. We employ the Otto geometry for prism coupling to SPhPs at the 6H-SiC-air interface across a variable air gap, and detect reflectivity and SHG output spectroscopically at various positions in the SPhP dispersion, using an infrared free-electron laser (FEL) for tunable narrowband excitation. By varying the air gap width between the prism and the sample, we demonstrate critical coupling behavior of the SPhP excitation efficiency. The calculated linear and nonlinear response reproduces the full dataset of reflectivity and SHG spectra with high accuracy. We extract the optical field enhancement, analyze the out-coupling of the SHG intensity in the Otto geometry, and discuss several potential applications of the nonlinear response from SPhPs.

2 Results and Discussion

Excitation of propagating SPhPs and detection of their reflectivity and SHG response is realized experimentally as schematically shown in Fig. 1 (a). We implement the Otto arrangement20, 8, 10 by placing a triangular prism (KRS5, $n_{\rm prism}\approx 2.4$, Korth) operated in the regime of total internal reflection onto a motorized mount in front of the sample, allowing for continuous tuning of the air gap width $d$. Rotation of the prism-sample assembly by angle $\Delta\theta_{ext}\approx n_{\rm prism}\Delta\theta$ changes the in-plane momentum $k_{\parallel}=\frac{\omega}{c}\,n_{\rm prism}\sin\theta$ of the incoming wave, allowing to excite SPhPs at different points along the dispersion.8 Here, $\theta$ is the incidence angle inside the prism of refractive index $n_{\rm prism}$, and $\omega$ the frequency of the incoming mid-IR beam. We use an infrared FEL31 as tunable, narrow band p-polarized excitation source, and the reflected fundamental and SHG beams are detected after a dichroic beam splitter. As a sample, we use a semi-insulating 6H-SiC c-cut single crystal, see Suppl. Mat. for further details on the experiment.

Notably, the efficiency of coupling the incoming light to the SPhPs in this geometry sensitively depends on the air gap width $d$.10 The decay length $L$ of the evanescent waves into the air gap for both, the totally reflected incoming light and the SPhP, strongly varies with the in-plane momentum $k_{\parallel}$:19

[TABLE]

where $\lambda$ is the wavelength and $c$ the speed of light in vacuum. For small gaps $d\ll L$ with large overlap of the two evanescent waves, the strong radiative coupling of the SPhP back into the prism is a significant loss channel and prevents efficient excitation, while for large gaps $d\gg L$ the small overlap between the two evanescent waves inhibits an efficient energy transfer. There is, however, a critical coupling gap width $d_{\rm crit}$ where the radiative and intrinsic losses of the SPhP exactly balance each other, and critical coupling to SPhPs is achieved.10 This is illustrated with the red and green shaded areas in Fig. 1 (a) for prism-side and SPhP waves, respectively. Since $k_{\parallel}$ strongly varies along the SPhP dispersion, both $L$ and the critical gap width $d_{\rm crit}$ also vary from less than 1 $\mu$m for large momenta to tens of $\mu$m at small momenta when approaching the light line.

In consequence, the Otto arrangement allows for a high-precision, direct measurement of the full surface polariton dispersion if the critical coupling conditions are adapted appropriately. This mapping of the SPhP dispersion is demonstrated in Fig. 1 (b) and (c) where we show experimental reflectivity and SHG spectra, respectively, at four different incidence angles, each taken near the respective $d_{\rm crit}$. Under these conditions, a reflectivity dip of $\approx 80\%$ indicates efficient excitation of the SPhP. The spectral position of the dip follows the SPhP dispersion, see Fig. 1 (d), as the incidence angle is changed. At the same time, the field enhancement associated with the efficiently excited SPhP results in significant increase of the SHG yield, as evidenced by the single, narrow peak in each of the SHG spectra (c).

For comparison, we show calculated reflectivity and SHG maps at critical coupling conditions in Fig. 1 (d) and (e), respectively, plotted as a function of fundamental frequency $\omega$ and in-plane momentum $k_{\parallel}$. The reflectivity is calculated using the transfer matrix approach,32, 33, 34 see Suppl. Mat. for details, while the SHG intensity is computed using:

[TABLE]

Here, $\vec{E}_{\rm SiC}(\omega)$ is the local optical field on the SiC side of the SiC-air interface which we obtain from the transfer matrix approach, $\chi^{(2)}$ the nonlinear susceptibility of 6H-SiC,30 while $\Delta k$ accounts for the wave vector mismatch for SHG in reflection.35, 30 Additionally, we explicitly account for the field coupling of the nonlinear polarization by projecting it onto $T_{b}\vec{E}_{\rm SiC}(2\omega)$, with $\vec{E}_{\rm SiC}(2\omega)$ the local field of the respective mode at $2\omega$ and $2k_{\parallel}$ propagating from SiC back into the prism, and $T_{b}$ the transmission coefficient back to the prism, see Suppl. Mat. for details on the SHG calculations.

To demonstrate the critical behavior of the response, we acquired reflectivity spectra for multiple values of the air gap width $d$ for selected incidence angles, exemplified for $\theta=28.8^{\circ}$ and $\theta=35.1^{\circ}$ in Fig. 2 (a) and (b), respectively. At the smallest gap (black lines), a shallow broad dip in the reflectivity at $\omega\approx 900$ cm*-1* reports on excitation of highly lossy SPhPs with pronounced radiative coupling. As $d$ is increased, the reflectivity dip increases in amplitude and narrows down while simultaneously blue-shifting. A maximum dip depth of $\approx 80\%$ (blue lines) indicates that the critical coupling conditions are reached. For even larger gaps (orange lines), the amplitude of the dip drops again, while the narrowing and blue-shifting of the resonance converge, approaching the intrinsic line width and frequency of the uncoupled surface polariton, cf. Fig. 1 (d). The qualitative behavior is identical between the two incidence angles shown here. However, the respective air gaps are clearly different, see legends in (a,b), where $d$ essentially scales with the evanescent length $L$ in Eq. 1. The calculated reflectivity perfectly reproduces the experimental data, including subtle features due to the 6H-SiC crystal anisotropy, such as reflectivity dips at the zone-folded weak modes at $\approx 885$ cm*-1* and the axial LO phonon at $964$ cm*-1*.36, 37, 30

The simultaneously acquired SHG spectra are shown in Fig. 2 (c,d). At small gaps, the data exhibit some additional spectral features,35, 30 however, a pronounced SPhP resonance can be clearly distinguished (please note the logarithmic scale), with SPhP peak positions and line widths qualitatively following those observed in the reflectivity as $d$ is increased. However, the amplitude behavior is clearly different for the SHG; we observe a steady decrease of the SHG amplitude at the SPhP resonance with increasing $d$. The calculated SHG spectra also shown in Fig. 2 (c,d, dash-dotted lines) not only reproduce all the features in the spectra with high accuracy. They also predict the steady decrease of SPhP resonance amplitude of the SHG with increasing width $d$ of the air gap. This is very surprising, since the maximum field enhancement and, in consequence, most efficient nonlinear signal generation is expected at critical coupling.19, 10

We summarize the SPhP resonance behavior for the full experimental data set in Fig. 3. Lorentzian fits of the SPhP resonance in the experimental (symbols) and theoretical (lines) reflectivity (a,c,e) and SHG (b,d,f) spectra yield the amplitude (a,b), Q-factor (c,d), and position (e,f) of the reflectivity dip and SHG peak, respectively. The high Q-factors, i.e., the ratios between center frequency and line width, mark the high quality of the SPhP resonance. Notably, the SHG resonances are intrinsically narrower as compared to the reflectivity dips due to the second-order nature of the interaction, leading to the consistently higher Q-factors. Remarkable agreement between experiment and theory is observed throughout the full data set. We note here that these calculations use a single set of parameters globally extracted from the full reflectivity data set, see Supp. Mat. for details.

The direct comparison of the SPhP resonance amplitudes of the reflectivity dip and the SHG peak, see Fig. 3 (a) and (b), respectively, for the different incident angles suggest a correlation between the critical coupling gap marking the strongest modulation of the reflectivity and the apparent decay length of the SHG amplitude. To understand these observations, we need to consider two mechanisms determining the detectable SHG signal in the far field: (i) efficiency of SHG due to the local field enhancement provided by the SPhPs, and (ii) the out-coupling of that nonlinear signal across the air gap into the prism, and into the far field. In the case of low losses, the local field enhancement associated with surface polariton excitations is typically expected to follow the magnitude of the reflectivity dip,19, 10 i.e., the maximum enhancement should be achieved at the critical coupling condition.

It is, however, important to realize that the SHG is generated with large in-plane momentum $k_{\rm\parallel,SHG}=2k_{\rm SPhP}>k_{\rm 0,SHG}$, where $k_{\rm 0,SHG}$ is the wavenumber of the SHG light propagating in air. This corresponds to the condition of total internal reflection for the SHG at the SiC-air interface, and only an evanescent wave is expected to leak into the air gap. Evaluating Eq. 1 for this wave reveals exactly half the evanescent length for the SHG as compared to the fundamental radiation on both sides of the air gap, resulting in an extremely poor overlap and largely suppressed far field coupling of the SHG at critical coupling conditions. In our SHG calculations, this effect is explicitly accounted for by the gap width $d$-dependence of $T_{b}\vec{E}_{\rm SiC}(2\omega)$ in Eq. 2, whose amplitude rapidly decays with $d$. In fact, we find that at critical coupling only $\sim 0.1\%$ of the generated nonlinear signal is harvested into the far-field intensity.

Therefore, we can recover the second harmonic intensity generated in SiC by correcting the experimental signals for the out-coupling efficiency $|T_{b}\vec{E}_{\rm SiC}(2\omega)|^{2}$, as shown in Fig. 4 (a). As described by the second factor in Eq. 2, these signals now follow fourth power of the resonantly enhanced fundamental optical fields $|E_{\rm SiC}(\omega)|$, which are also plotted there. With this information, we can confidently extract the magnitude of the field enhancement in SiC at critical coupling, which is plotted in Fig. 4 (b) along the SPhP dispersion. We plot both in-plane and out-of-plane fields, $E_{x}$ and $E_{z}$, respectively, since due to the symmetry of the $\chi^{(2)}$ tensor, both components contribute significantly to the SHG output.30 Notably, the small dip in these curves at $\theta\approx 27.5^{\circ}$ marks the resonant interaction of the SPhP with the zone-folded phonon modes in 6H-SiC.18 For illustration, we also show an example of the spatio-spectral distribution of field enhancement along the normal-to-surface direction $z$ in Fig. 4 (c).

The excellent agreement between the experimental and calculated SHG response corroborates our observation of the extremely efficient nonlinear-optical conversion inside the SiC crystal, in particular in comparison to SPPs. From our data, we estimate an exceptional nonlinear conversion efficiency exceeding $\sim 10^{-6}$, which is a result of (i) the large field enhancement due to the high Q-factor of the SPhP resonance, and (ii) the broken inversion symmetry as well as proximity to an ionic resonance of the nonlinear susceptibility38, 30 in the SPhP host.

The latter two features, broken inversion and ionic resonance, are unique fingerprints of SPhP materials as opposed to noble metals used in plasmonics.1 Our approach is applicable to all non-centrosymmetric, polar dielectrics, such as $\alpha$-quartz, ZnO, AlN, InP, GaAs, ZnTe, to name a few, known to exhibit large nonlinear coefficients, as well as artificially designed hybrid materials39 with yet unknown SPhP dispersion relation. Additionally, the strong dispersion of SPhPs provides spectral tunability of the field enhancement resonance with extraordinarily high Q-factors.

The combination of these features suggests several appealing scenarios for future applications of nonlinear nanophononics.18 Previous linear sensing schemes40, 27 could be extended into the nonlinear domain with larger contrasts and higher Q-factors. Taking advantage of the large resonant, high-Q nonlinear signals from the broken-inversion SPhP host, these approaches could straight-forwardly be implemented using narrow-bandwidth quantum-cascade lasers. Similarly, our approach could be extended to four-wave mixing experiments,21, 41 sensitive to either the SPhP host or a liquid phase analyte in the gap, cf. Fig. 4 (c) for the relevant length scales of field enhancement inside the gap. This could, for instance, lead to a drastic resonant enhancement of time-resolved vibrational spectroscopy signals from molecules in solution.42 Additionally, we also note that SPhP materials are typically largely transparent above the reststrahl spectral range. Together with the generation of nonlinear signal under conditions of total internal reflection, this suggests that the Otto geometry could be used for high-contrast nonlinear waveguide coupling.43

3 Conclusion

In conclusion, we demonstrated the first SHG from critically coupled SPhPs, enabling high-precision measurement of the associated optical field enhancement, here shown for 6H-SiC as a model system. We use the critical coupling behavior of the Otto geometry to maximize the SPhP excitation efficiency. Despite poor far-field coupling of the SHG signals, which is shown to be a generic feature of the Otto arrangement, the large bulk nonlinearity of the crystal facilitates exceptional nonlinear-optical conversion from SPhPs, owing to the broken inversion symmetry of the host material and the high quality of the SPhP resonance. Being applicable to a wide range of polar dielectrics supporting SPhPs, our approach could be employed to extract the unknown polariton dispersion and field enhancement in artificially designed SPhP hybrids, and opens the path to several new and appealing applications of nonlinear nanophononics.

4 Acknowledgments

The authors thank K. Horn (FHI Berlin) for providing the SiC sample.