Origin of strong-field-induced low-order harmonic generation in amorphous quartz


Kerr-type nonlinearities form the basis for our physical understanding of nonlinear optical phenomena in condensed matter, such as self-focusing, solitary waves and wave mixing1,2,3. In strong fields, they are complemented by higher-order nonlinearities that enable high-harmonic generation, which is currently understood as the interplay of light-driven intraband charge dynamics and interband recombination4,5,6. Remarkably, the nonlinear response emerging from the subcycle injection dynamics of electrons into the conduction band, that is from ionization, has been almost completely overlooked in solids and only partially considered in the gas phase7,8,9,10. Here, we reveal this strong-field-induced nonlinearity in a-SiO2 as a typical wide-bandgap dielectric by means of time-resolved, low-order wave-mixing experiments, and show that, close to the material damage threshold, the so far unexplored injection current provides the leading contribution. The sensitivity of the harmonic emission to the subcycle ionization dynamics offers an original approach to characterize the evolution of laser-induced plasma formation in optical microprocessing.

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Fig. 1: Schematic description of harmonic generation in solids.
Fig. 2: Time-resolved measurement of low-order two-colour harmonics.
Fig. 3: Experiments in c-SiO2.
Fig. 4: Characterization of m underlying the harmonic emission (for n = 1) as a function of the pump intensity in a-SiO2.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We wish to thank M. Jupé (Laser Zentrum Hannover) for measuring the bandgap of the samples used in this study. A.M.-B. acknowledges financial support from the Deutsche Forschungsgemeinschaft (IDs DFG ME4427/1-1 and DFG ME4427/1-2). T.F. acknowledges financial support from the Deutsche Forschungsgemeinschaft via a Heisenberg Grant (ID 398382624) and via SPP1840 (ID 281272685), from the Bundesministerium für Bildung und Forschung (BMBF, ID 05K16HRB) and from the European Social Fund (ID ESF/14-BM-A55-0007/19) and the Ministry of Education, Science and Culture of Mecklenburg-Vorpommern, Germany, via the project ‘NEISS’. Computing time was provided by the North German Supercomputing Alliance (HLRN, ID mvp00013).

Author information




A.M.-B., P.J. and M.J.J.V. conceived, performed and analysed the experiment, with the help of D.E., T.W. and B.L. B.K., C.P., A.H., M.I. and T.F. developed the theoretical model. A.M.-B. and T.F. wrote the manuscript, with input from all authors.

Corresponding authors

Correspondence to T. Fennel or A. Mermillod-Blondin.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Ruifeng Lu, Alexander Pukhov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Extended Data Fig. 1 Schematic experimental setup.

A strong mid-IR pump laser pulse is temporally and spatially overlapped with a NIR probe laser pulse in the volume of a SiO2 sample. After recollimation by a CaF2 lens (L1) the signals emerging from the sample are separately analyzed in the two parts of the detection unit. L1 - L4: Lenses, BS: Beam splitters, DM: Dielectric mirror (high reflectivity at 0.8 μm), PD: Photodiode, BD: Beam dump, P1, P2: prisms.

Extended Data Fig. 2 Probe intensity scans in a-SiO2.

Harmonic yield of the first four harmonic orders depending on the intensity of the probe laser pulse together with linear fits to the experimental results. The pump and probe beams have parallel polarizations. The linear relationship between harmonic yield and probe intensity indicates that only one probe photon is involved in the nonlinear wave mixing. The probe intensity used in the experiments is Iprobe = 0.015 TW/cm2.

Extended Data Fig. 3 Plasma diagnostics in a-SiO2.

Loss of probe transmission in the irradiated region for a pump intensity of ≈12 TW/cm2. The first maximum of the signal is attributed to probe depletion due to reversible two-beam coupling processes during the pump–probe overlap (symbolized as an orange region). These contributions can be safely excluded when the pump beam has propagated through the focal region, that is for τ > 200 fs. The remaining signal is then only due to absorption in the undercritical electron–hole plasma. The probe absorption decays with a characteristic time on the order of ≈150 fs, corresponding to the carrier trapping constant usually reported for fused silica31.

Extended Data Fig. 4 Determination of the pump–probe overlap.

a, Variation of the central probe wavelength as a function of the number of pump–probe delay increments (one increment represents a delay of ≈6.7 fs). The shift of the central wavelength is due to cross-phase modulation and shows a maximum when pump and probe overlap in the focal region. b, projection of the n = 1 harmonic on the delay axis. The grey dotted curve was obtained by direct integration of the data shown in a.

Extended Data Fig. 5 Effective non-linearity as function of the bandgap.

Orange dashed line: the sum of Kerr-type and Brunel nonlinearities are considered. Orange solid line: the contribution of the injection is also taken into account. The region shaded in violet shows the effective order of nonlinearity obtained experimentally for an intensity of 12 TW/cm2 when measurement uncertainties are included. The squares correspond to the bandgap values (that is 7.5, 7.7, and 8.0 eV) chosen to illustrate the robustness of our model versus bandgap variations in Fig. 4 of the main manuscript.

Supplementary information

Supplementary Information

Supplementary Figs. 1–4 and discussion.

Source data

Source Data Fig. 2

Raw experimental data for Fig. 2b.

Source Data Fig. 3

Raw experimental data for Fig. 3.

Source Data Fig. 4

Raw experimental data, statistical data (error bars) and numerical simulation results for Fig. 4.

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Jürgens, P., Liewehr, B., Kruse, B. et al. Origin of strong-field-induced low-order harmonic generation in amorphous quartz. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0943-4

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