Abstract
Highharmonic generation in isolated atoms and molecules has been widely utilized in extreme ultraviolet photonics and attosecond pulse metrology. Recently, highharmonic generation has been observed in solids, which could lead to important applications such as alloptical methods to image valance charge density and reconstruct electronic band structures, as well as compact extreme ultraviolet light sources. So far these studies are confined to crystalline solids; therefore, decoupling the respective roles of longrange periodicity and high density has been challenging. Here we report the observation of highharmonic generation from amorphous fused silica. We decouple the role of longrange periodicity by comparing harmonics generated from fused silica and crystalline quartz, which contain the same atomic constituents but differ in longrange periodicity. Our results advance current understanding of the strongfield processes leading to highharmonic generation in solids with implications for the development of robust and compact extreme ultraviolet light sources.
Introduction
Highharmonic generation (HHG) in isolated atoms and molecules^{1, 2} has been the foundation of attosecond pulse metrology^{3}, extreme ultraviolet (XUV) photonics^{4}, and molecular orbital tomography^{5}. The observation of nonperturbative high harmonics generation from strongly driven bulk crystals^{6,7,8,9,10,11,12,13,14} has motivated new research aiming to probe the electronic structure of solids including normally unoccupied conduction bands^{9, 15} and to overcome the drawbacks of conventional XUV sources. Gas phase XUV sources suffer from low efficiency and therefore do not provide sufficient flux desired for many applications such as metrology and imaging^{16}. Solidstate HHG has the potential for high efficiency and high stability because of the use of highdensity and rigid target as interaction medium. Since the original discovery in singlecrystal ZnO, several crystalline solids, such as GaSe, SiO_{2}, Ar, Kr, MgO, and MoS_{2}, have been used for HHG^{6,7,8,9,10,11,12,13,14}. The important findings, such as highenergy cutoff scaling with the field^{6}, emergence of a secondary plateau^{11, 17}, and novel ellipticity dependence^{14}, indicate that the underlying electron dynamics are markedly different from the threestep recollision model, which is widely accepted for atomic and molecular HHG^{18}. These fundamental differences are attributed to the high density and periodicity present in bulk crystals as the fielddriven electron is always in the proximity of the Coulomb potential^{6, 19}. In order to incorporate the fundamental solidstate response, two major mechanisms are being considered, which are based on the emission from nonlinear interband polarization and intraband current^{19,20,21,22,23}. While this topic is of intense debate, both mechanisms rely on the basic assumption of the Bloch theorem^{24}—the energy eigenstates of the electron moving in a periodical potential are Bloch waves.
Here we ask one of the most fundamental questions: can strongly driven amorphous solids, where the electron can no longer be described by Bloch wave, produce high harmonics? The answer would help to elucidate the role of longrange order in HHG, since amorphous systems do not exhibit longrange periodicity. We report the observation of nonperturbative HHG from amorphous silicon dioxide (fused silica) subjected to intense fewcycle laser pulses of field strength ~2 VÅ^{−1} without damage. The highharmonic spectrum shows characteristics of both spatial and temporal coherence and the photon energy extends up to ~25 eV. In order to understand the role of longrange periodicity, we perform similar measurements in singlecrystal SiO_{2} (quartz). We compare their generation efficiency and the dependence of the spectra on both the input field strength and the fewcycle field waveform through the carrierenvelope phase (CEP) setting of the driving laser. While in both cases the harmonic spectra cover similar spectral ranges, up to ~25 eV, the harmonics from crystalline quartz show a minimum separating two plateau structures. We reproduce our experimental observation by performing quantum calculations of driven multilevel systems, where the energy levels correspond to the respective electronic band structures.
Results
Highharmonic spectrum from amorphous solids
In the experiments, we focus twocycle laser pulses obtained from a highefficiency optical parametric chirpedpulse amplification (OPCPA) system^{25} with a center wavelength of ~1700 nm (0.73 eV) into a 100 μm thick sample of amorphous fused silica (see “Methods” section). The estimated maximum peak field strength without damage is ~2 VÅ^{−1}. The relatively high damage threshold is achieved with the use of ultrashortpulse duration and long wavelength. For comparison, we use crystalline quartz sample of similar thickness in place of the fused silica. Figure 1 shows representative highharmonic spectra from fused silica and crystalline quartz. In both cases, the spectrum extends to ~25 eV, corresponding to the 33rd harmonic. The harmonic spectrum from fused silica consists of oddorder harmonic peaks, although the peak position may or may not correspond to the exact harmonic order depending on the CEP setting (see “Methods” section). The spectrum from crystalline quartz has more peaks because of the inclusion of additional evenorder harmonics, thus forming a more continuous pattern. The presence of even harmonics in crystalline quartz is consistent with the fact that the crystal structure lacks inversion symmetry^{26}. Fused silica, on the other hand, is isotropic and, therefore, does not produce evenorder harmonics. A unique feature in the spectrum from quartz is the presence of a prominent minimum in the range from 17 to 18 eV, which separates the spectrum into two plateaus. As in rare gas solids^{11}, the origin of multiple plateaus can be attributed to the important role of highlying conduction bands.
Scaling of high harmonics with laser peak fields
We study how the generation from periodic and nonperiodic medium depends on the laser field. The harmonic spectra resulting from laser fields with amplitude ranging from 1.4 to 2.0 VÅ^{−1} are shown in Fig. 2a, b. We find that the spectral minimum of crystalline quartz does not shift as a function of the laser field. At moderate field strengths, the harmonic yield from fused silica and crystalline quartz are similar. In both cases, the harmonic yield increases with the input laser field strength; however, the yields scale differently such that at the highest peak field crystalline quartz is about four times more efficient (Fig. 2c). Crystalline quartz exhibits longrange order in contrast to the random network structure present in amorphous fused silica. Therefore, in the limit that electron excursion distances are much larger than the size of a unit cell, coherent collisions in several unit cells could contribute significantly to the HHG. To verify this, we change electron excursion distances via the laser wavelength (see Supplementary Discussion). At 800 nm, the estimated excursion distances are about the size of the unit cell. In that case, we find that both amorphous and crystalline SiO_{2} show similar nonlinear scaling (see Supplementary Fig. 1b). We note that Luu et al.^{9} have also reported similar nonlinear scaling in crystalline quartz when compared with conventional gas phase atomic targets.
CEP dependence of highharmonic spectrum
To gain insights into the underlying electron dynamics, we perform timedomain measurements through the CEP dependence^{17}. Figure 3a, b shows the measured CEP scans for fused silica and crystalline quartz, respectively. It is seen that in fused silica the photon energies of the harmonic peaks shift in energy (indicated by the dashed line) with a CEP slope ~3 eVπ^{−1} and the harmonic spectrum repeats with π periodicity (horizontal axis). However, the harmonic spectra from crystalline quartz show a strong modulation of the harmonic yield with a 2π periodicity. Due to the random interatomic structure in fused silica on a scale much smaller than the laser wavelength, both positive and negative polarity (half cycle) experience the same average response consistent with the observed π periodicity in the CEP. However, crystalline quartz exhibits broken inversion symmetry, and as a result the positive and negative half cycles experience different collective response, resulting in a 2π periodicity. The origin of shifts in the harmonic energies is an indication of attochirp, which we discuss below.
Simulation
To model our results, we solve timedependent Schrödinger equations in a multilevel system^{22} using the density matrix approach (see “Methods” section). For fused silica, we consider a simple twolevel system, where the energy separation corresponds to the experimental band gap of ~9 eV^{27}. Figure 3c shows the simulation results, which reproduce the experimentally observed CEP dependences with appropriate periodicities. In order to explain the observed minima in harmonic spectrum from crystalline quartz we use a simple threelevel system similar to previous results^{17}. Here the energy separation between the first and second level corresponds to the primary experimental bandgap (~9 eV) and the separation between second and third level corresponds to the second band gap (~4 eV). Also, in order to account for the noncentrosymmetric structure of crystalline quartz we include a permanent dipole moment term (see “Methods” section). The simulation results are shown in Fig. 3d. We reproduce the dominant monotonic 2π periodicity along with the spectral minima (at around 18 eV) that is independent of the laser parameters. Our CEP measurements are relative, however it can be seen that in our scale the spectrum is maximized when CEP = π, 3π, and 5π, which means that the permanent dipole moment is periodically aligned/antialigned to the peak of the laser field. The limitation of this simulation is that we consider multilevel coupling only at the zone center, where the dominant contribution comes from the minimum band gap. This could be improved by considering the entire Brillouin zone.
Attochirp analysis
As shown by You et al.^{17} originally and briefly here in the Methods section below, the photon energy shift of the harmonic peaks with the CEP corresponds to variation of emission time of harmonics emitted at subsequent half cycles of the laser field. In the gas phase, the subcycle delay between harmonics, which is known as attochirp were found largely independent of the target atom^{28}. It means that the CEP slope in the gas phase also would not depend on the target atom. In contrast, we find that the measured CEP slope of plateau harmonics in fused silica is about two–three times larger than that in the case of MgO under similar laser parameters^{17}. Also, in the case of crystalline quartz harmonics are emitted predominantly at specific CEP settings (π, 3π, and 5π). This monotonic CEP dependence (without slope) is consistent with chirpfree harmonics, recently demonstrated through timedomain measurements^{12}. In our simulation, such monotonic dependence comes from the noncentrosymmetric structure of quartz, where harmonics are strongly enhanced when the laser field is parallel to the permanent dipole moment. Therefore, our analysis indicates that the intrinsic attochirp in solidstate harmonics is materials dependent.
Discussion
We present the generation of nonperturbative highorder harmonics from nonperiodic transparent solids subjected to strong laser fields. The harmonic spectrum exhibits a broad plateau structure, which extends to ~25 eV limited by the damage threshold at ~2 VÅ^{−1}. The strong CEP dependence of the HHG spectrum confirms that the harmonics are locked in phase with the driving laser field. The measured CEP slope in fused silica indicates that harmonics are delayed with respect to each other on the subcycle timescale. The observation of HHG from amorphous materials means that periodicity of atomic arrangement inside solids is not a requirement for generating coherent XUV radiation. When combined with the modest requirements in the peak intensity (~10^{13} W cm^{−2}), the solidstate HHG technique becomes an attractive candidate for future highrepetition rate compact XUV light sources^{29, 30}. Amorphous optical materials are readily available and can be incorporated relatively easily in photonics design, such as in intracavity XUV frequency combs, which currently rely on gas targets^{31}. Another advantage of using solid targets is that relatively large beam size can be utilized to further enhance the efficiency. Finally, the ability to generate high harmonics in amorphous optical medium could open up new series of possibilities in nanophotonics and XUV waveguides^{32}.
Methods
Experimental setup
We focus twocycle laser pulses produced from a highefficiency OPCPA laser system^{25} into 100 μm thick samples placed inside the vacuum chamber. The laser spectrum is centered ~1700 nm and the pulse duration is ~11 fs, measured by frequencyresolved optical gating. The samples withstand the peak intensity ~10^{14} W cm^{−2} (~2.1 VÅ^{−1}) at 1 kHz repetition rate. Such a relatively high damage threshold is reached due to the combination of relatively large band gap (9 eV), small photon energy (0.73 eV), and the ultrashort pulse duration. The CEP settings are adjusted using an acoustooptical modulator, which provides relative values. We record the harmonic spectra with an imaging spectrometer consisting of a flatfield variable groove density grating and microchannel plates. The spectral range is from 13 to 25 eV, limited by the collection angle of the XUV spectrometer. The spectrum is not corrected for the sensitivity of the grating.
Longrange order
To discuss the role of periodicity for the microscopic HHG process, we compare the maximum excursion distance of semiclassical electron with the grain size of the sample. Based on Scherrer particle size equation^{33}, the measured width of Xray diffraction ring (~10°) corresponds to a grain size of about 9 Å, which is an upper bound (resolution limited).The maximum excursion distance of semiclassical electron is r _{max} = eEλ ^{2}/4π ^{2} mc ^{2}, where E is the electric field strength and λ is the wavelength. For E = 2 VÅ^{−1} and λ = 1700 nm, this corresponds to ~30 Å. Clearly, at these laser parameters the local correlation lengths in fused silica are much shorter than the estimated electron excursion distances. In contrast, the correlation length in crystalline quartz is infinitely long compared to the estimated excursion distances. Therefore, we expect that amorphous and crystalline medium behave differently. However, at moderate fields, such as at E = 0.5 VÅ^{−1}, the excursion length would be around 8 Å, approaching the measured correlation length in fused silica. Therefore, at moderate fields we do not expect significantly different harmonic efficiency between fused silica and crystalline quartz, consistent to the experimental results. In order to perform a more systematic study, we extend our measurements to other laser wavelengths since the maximum electron excursion distances are expected to depend quadratically on the laser wavelength (see Supplementary Fig. 1b, c).
CEP dependence of high harmonics spectrum
The CEP dependence is modeled by considering the interference between adjacent XUV pulses as shown originally by You et al.^{17}. The XUV bursts generated in each half cycle have different amplitudes and phases that depend on the instantaneous intensity and the CEP, which can lead to constructive or destructive interference. Consider the timedependent dipole of two adjacent bursts^{34}:
where d(t) is the dipole from a single attosecond burst, T is the laser cycle period, and θ _{12} is the phase difference between these two attosecond bursts. We ignore the amplitude change and consider only the phase difference, which can be written as:
where t _{i} and t _{ri} are the time of tunneling and the time when frequency ω is generated, respectively. ε(t) is the timedependent energy difference between the first and second instantaneous eigenstates of the twolevel system. So instead of peaking at the odd harmonics, the constructed interference requires that the harmonics peak at the frequency given by:
where ω _{ L } is the laser frequency, n is integer, and θ _{12} is the energydependent phase difference between the two bursts. If θ _{12} = 0, the harmonic photon energies are equal to the odd harmonics of laser photon energy. When the CEP is varied, θ _{12} is modulated and, therefore, the harmonic photon energies shift to other frequencies, thus forming slopes as seen in Fig. 3a.
Quantum calculations
To simulate high harmonics generation in solids, we solve the timedependent Schrödinger equations in multilevel systems^{22} using the density matrix approach. For fused silica, we use a system of two levels, where the separation corresponds to the experimental band gap ~9 eV (see inset of Fig. 3c). The Hamiltonian of the system can be expressed as (in atomic unit (a.u.))
where ω _{0} is the level separation, μ _{0} is the dipole moment between two levels, and E(t) is the electric field. We use μ _{0} = 10 a.u. and peak field of 2 VÅ^{−1}. For crystalline quartz, we add one more energy level, which is separated from the second level by ~4 eV corresponding to the spacing of the second conduction band. Also, we include a permanent dipole moment to account for broken inversion symmetry. Therefore, the Hamiltonian becomes:
where ω _{1} is the separation between first and third level (see Fig. 3d), μ _{p}is the permanent dipole moment in the valence band, and μ _{1} is the dipole moment between second and third level. For quartz we use, μ _{0} = 5 a.u., μ _{1} = 6.6 a.u., and μ _{p} = 2.5 a.u. Initially the first level is fully occupied and upper levels are empty. The dephasing time is set to 2.8 fs similar to Vampa et al.^{20}. The harmonic spectrum is obtained by Fourier transform of the timedependent current. We note that our quantum calculations do not predict absolute efficiency that can be compared directly to the experiments. For amorphous fused silica, simulation based on timedependent density functional matrix could provide further insight into the microscopic electronic response^{35}. For crystalline quartz, adding the contribution from highlying conduction bands^{36} would be beneficial to study the origin of spectral features.
Data availability
The data relating to this work are available on request from the corresponding author.
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Acknowledgements
At Stanford/SLAC, this work is supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the Early Career Research Program. At UCF, this work is supported by the Air Force Office of Scientific Research under award number FA95501510037 and FA95501610149, the Army Research Office W911NF1410383 and W911NF1510336, the DARPA PULSE program by a grant from AMRDEC W31P4Q1310017, and the National Science Foundation 1506345. S.G. and Y.S.Y. thank David Reis, Mengxi Wu, Mette Gaarde, Kenneth Schafer, Jerry Hastings, and Kelly Gaffney for fruitful discussions.
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Y.S.Y. and S.G. conceived the experiments. Y.S.Y., Y.Y., A.C., X.R., M.C., S.G.M., Y.W., F.Z., and S.G. collected and analyzed data. All authors contributed to the interpretation of the results and writing of the manuscript.
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You, Y., Yin, Y., Wu, Y. et al. Highharmonic generation in amorphous solids. Nat Commun 8, 724 (2017). https://doi.org/10.1038/s41467017009894
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