Abstract

Due to strong absorption of above-bandgap high harmonics generated from solids, the surface should play an important role. However, surface-induced contributions have yet to be identified. Here we show that the presence of the interface between a Si crystal and a native SiO2 layer is encoded in the appearance of weak even-order high harmonics. The even harmonics arise from a surface-induced static space charge that perturbs the laser-driven motion of electrons and holes in the bulk of the material. Our demonstration extends surface-induced nonlinear optical effects to the non-perturbative regime, characterized by unbound large-amplitude motion of charge carriers, and therefore paves the way for new methodologies to probe surfaces and their chemistry.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In recent years, crystal nonlinear optics have been extended to the non-perturbative regime with the generation of strong-field-induced currents [1] and the conversion of the driving laser photon energy to its high-order harmonics [210]. Due to strong absorption of above-bandgap radiation, it is commonly understood that only an order of an absorption length from the output surface contributes to the detected high-harmonic emission [11]. Therefore, the strongest contribution must come from regions near the surface, but its role has, so far, eluded scrutiny. Surface effects have been detected in perturbative second-harmonic and sum-frequency generation in centrosymmetric media, where the broken inversion symmetry at the surface leads to a finite second-order susceptibility χ(2) [1215]. At relativistic intensities, strongly asymmetric plasma dynamics at a surface result in equally bright even- and odd-order high harmonics [16,17]. At the intermediate intensities considered here (0.11TW/cm2), however, emission of even-order harmonics from two-dimensional crystals [5,8] has been interpreted as arising from a nonlinear polarization in the plane of the material where the inversion symmetry is broken, rather than as an interfacial effect. Here we show surface-induced effects on high-order harmonic generation from a centrosymmetric crystal, silicon. Even-order high harmonics are generated when the Si/SiO2/vacuum interface is illuminated at off-normal incidence with infrared laser pulses having a component of the polarization normal to the surface. A slow buildup time of the even harmonics and their saturation for increasing intensity of the driving infrared laser further indicate that the static space charge that builds up at the Si-SiO2 interface perturbs high-harmonic emission from the Si crystal. The power of the even harmonics is a few percent of the nearest odd-order harmonics, a ratio which is uncommonly large for static-field-induced effects, possibly because the short emission depth of bulk harmonics (10nm) magnifies the influence of the interface. Our demonstration extends surface-induced nonlinear optical effects to harmonics higher than the second harmonic, from the perturbative to the non-perturbative regime, with enhanced sensitivity over the symmetry breaking. Just as this regime of light–matter interaction has provided novel means to probe gas-phase atoms and molecules and bulk solids [18,19], we expect our demonstration will result in more sensitive methods to probe surfaces and their chemistry.

In the experiment, we generate high harmonics of a 2.2 μm laser in a 500 nm thick Si single crystal (intrinsic, resistivity ρ>100Ωcm), (001) oriented, epitaxially grown on a sapphire substrate. The 2.2 μm source is the idler beam of an optical parametric amplifier pumped by 2 mJ, 50 fs pulses from a Ti:sapphire regenerative amplifier operating at a 1 kHz repetition rate. We have examined four different experiment configurations [Figs. 1(a)1(c)]: with the Si film facing the incident beam, detected in reflection (panel a) and transmission (panel b), and with the sapphire substrate facing the incident beam, detected in transmission, for p-polarized (panel c, blue line) and s-polarized (panel c, red line) infrared excitation. The angle of incidence is 70°. We find that even harmonics are generated together with the odd-order harmonics only when the Si film faces the detector for p-polarized infrared excitation and for large angles of incidence. The power of the even harmonics can reach 3% of the nearest odd-order harmonics, whereas the second harmonic is undetectable (<0.5% of the third harmonic, panel c, blue line). Because the Si film is thicker than the short absorption length of radiation above the direct gap (10nm in Si), high harmonics can discriminate which interface is responsible for the emission of the even orders (sapphire is transparent to the harmonics, so the emission from the Si–sapphire interface can still be detected). The generation of even-order high harmonics is observed only at the Si–air interface, when the electric field has a component normal to the interface. Finally, the incident intensity is too weak for observable high-harmonic generation in sapphire.

 figure: Fig. 1.

Fig. 1. Measured high-harmonic spectra from a Si film facing the incident beam in (a) transmission and (b) reflection for a p-polarized driver and for (c) the sapphire substrate facing the incident beam in transmission for a p-polarized (blue line) and s-polarized (red line) driver. Weak even harmonics are only detected for p polarization when the Si film faces the detector. The powers of the second and third harmonics are unrelated to those of the higher harmonics because they are measured with a different spectrometer with non-overlapping spectra. (d) The power of the even harmonics (colored lines) saturates with the intensity of the driver, contrary to the odd harmonics (gray lines). (e) Real-time acquisition of the seventh and eighth harmonics for alternate illumination of two adjacent regions of the Si film (one region is marked by the blue dot and the other by the red dot). The sample is laterally translated between these two regions at the times marked by the colored dots. The fundamental beam is blocked when the power is zero. The even-harmonic power shows a transient behavior developing over 1sec, contrary to the odd harmonics. Data in panels (d) and (e) is acquired in the configuration of panel (c) with p-polarized light.

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The power of the even harmonics saturates for sufficiently high intensity of the infrared laser [Fig. 1(d), colored lines], in contrast to the odd harmonics (gray lines), whose powers continue to increase as I5 (and therefore in a non-perturbative fashion). Furthermore, as shown in Fig. 1(e), the even harmonic power slowly increases upon sudden illumination of the sample (at 5 s and 70 s) and features an abrupt suppression followed by the same slow recovery upon a sudden translation (at 30, 35, and 80 s).

The slow buildup suggest that a long-lived static space charge generated by a sequence of intense infrared pulses breaks the inversion symmetry of the bulk, as previously demonstrated for perturbative second-harmonic generation at Si–oxide interfaces [2022]. Since our experiments are conducted under ambient atmospheric conditions, an amorphous native oxide layer 5nmthick should be present on the Si crystal [22,23]. The process is depicted schematically in Fig. 2(a). The space-charge field (FSC, blue arrows) is generated by electrons or holes that migrate from the Si to the oxide layer and are trapped near the oxide–air interface. For trapping to be effective, thin interfacial layers are necessary [20]. It is likely that the sapphire substrate is too thick for sufficient charge trapping to occur, explaining why even harmonics are only detected at the Si–oxide–air interface. Saturation of the even-harmonic power is likely due to a reduction of the space-charge field upon tunneling and trapping of the minority oppositely charged carriers [22]. Within the framework of perturbative nonlinear optics, the space-charge field breaks the inversion symmetry of the bulk and results in weak asymmetric motion of the bound nonlinear charge. In the strong-field regime that we explore here, the space-charge field acts upon unbound electrons and holes that are accelerated apart from each other up to nanometer distances [9,24]. For example, the calculated trajectories of electrons and holes responsible for the emission of the eighth and 13th harmonic are shown in Fig. 2(b). Typical unperturbed electron-hole trajectories are sketched with dashed black lines in panel a, and those perturbed by the space-charge field are sketched with solid black lines. Only the component of the space-charge field parallel to the laser polarization [FIR, red arrow in Fig. 2(a)] is effective at perturbing the nonlinear dipole. If this were not the case, the s polarization would also generate even harmonics—contrary to our findings.

 figure: Fig. 2.

Fig. 2. (a) Sketch of how the space-charge field perturbs bulk high-harmonic emission. High harmonics (purple beam, co-propagating with the red driver) are emitted within one absorption length from the Si–oxide interface. The infrared field (FIR) launches electrons (e) and holes (h+) on trajectories in Si (dashed black lines) along the driver’s polarization (red arrow). The component of the space-charge field (blue arrows, FSC) along the driver’s polarization renders the trajectories asymmetric (solid black lines), resulting in the emission of even-order high harmonics. (b) Electrons’ (solid lines) and holes’ (dashed lines) trajectories responsible for emission of the eighth harmonic (blue lines) and the thirteenth harmonic (red lines, corresponding to the cutoff harmonic). The trajectories are calculated according to the model developed in Ref. [24] with the following parameters: λ0=2.2μm (driving laser wavelength), FIR=0.18V/Å, a=5.43Å (lattice constant of Si). The electron and the hole propagate in the first conduction and split-off valence bands, respectively.

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The relative intensity of the even- and odd-order harmonics encodes the magnitude of the space-charge field—and therefore of the trapped electron density. Employing a model developed for high-harmonic generation [4,24], one finds that the ratio between the even and odd harmonics is

I2n/2n±1=tan2σ,
where σ=ttΔv[k+A(τ)A(t)][A2(τ)A2(t)]dτ; t and t are the times of creation of the electron-hole pair and of collision of the electron with the hole, respectively; Δv=vcvv is the difference in the velocities of the conduction and valence band electrons; k is the crystal momentum; and A(t) and A2(t)are the vector potentials of the infrared laser field and of the perturbing field, respectively. The latter is the projection of the static space-charge field on the polarization direction [therefore, A2=FSC,p(tτ)]. The integral σ is numerically calculated by solving a set of trajectories for 0<t<T0/4 (T0 is the period of the infrared laser). To match the experimentally measured I8/(7,9)3%, and taking into account the 15° angle formed by the laser polarization and the space-charge field inside Si, we find FSC=0.04V/Å. Assuming a uniform sheet of charge, the electron density is Σ=(2ϵ0FSC)/e=4.6×1012e/cm2, which is comparable to that measured in the perturbative regime [21] with resonant near-infrared pumping at a peak intensity of only 1GW/cm2.

By perturbing the high-harmonic generation process with a weak second-harmonic field polarized parallel to the driver, we control the degree of asymmetry induced by the surface, a methodology first introduced by Glauber [25] and which developed in the field of optical poling [26]. As observed in centrosymmetric gas molecules [27] and solids [4], varying the phase delay between the second harmonic and the fundamental fields induces modulation of the power of the odd and even harmonics, with two oscillations per cycle of the second harmonic. This is also what we measure at the SiSiO2 interface for the ninth harmonic [black line in Fig. 3(a)]. Contrary to centrosymmetric media, however, the intrinsic broken symmetry induced by the surface charge results in the addition of a strong modulation of the even harmonics with half the periodicity of the odd harmonics, as shown in Fig. 3(a) for the eighth harmonic (red and orange lines). Manipulating the electron-hole trajectories by coherently adding space-charge and second-harmonic fields can also restore near-complete centrosymmetry, as exemplified by the suppression of the even harmonic signal to 50% of the intrinsic value, for an appropriate delay of the second harmonic, as shown in Fig. 3(a). The residual even-harmonic signal possibly originates from the outskirts of the beam, where the shorter asymmetric electron-hole trajectories resulting from the lower intensity are not balanced at the same second-harmonic delay.

 figure: Fig. 3.

Fig. 3. Controlling symmetry breaking with a bichromatic laser field (fundamental and second harmonic). (a) While the power of the ninth harmonic is modulated twice per cycle of the second harmonic, the eighth harmonic develops an additional strong modulation with half the period. For some optimum delays, destructive interference between space-charge and second-harmonic fields suppresses the power of the eighth harmonic to below that obtained without the second-harmonic field (labeled “intrinsic” in the figure). (b) The amplitude of the bulk (2/cycle) and surface-induced (1/cycle) modulations of the eighth harmonic are analyzed as a function of the second-harmonic power. The surface contribution reaches a maximum as a result of optimum interference with the surface charge. (c) The surface contribution remains strong even for shallow (<20°) angles of incidence (measured from normal to the surface), where intrinsic even harmonics are too weak to be detected.

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The amount of surface and bulk contribution to the modulated high-harmonic signal is obtained by fitting with the linear addition of two cosine waves: one with a period of half that of the second-harmonic, characteristic of bulk emission (measured from the odd-harmonic modulation), and one with twice that period, characteristic of the surface contribution. The free parameters of the fit are the amplitude and phase of each component. As the second-harmonic power is increased [Fig. 3(b)], the modulation of the eighth harmonic acquires increasing bulk character (red line). The surface character, instead, reaches a maximum, corresponding to optimum interference with the intrinsic even harmonic (blue line). The maximum surface character can, in principle, be used to determine the space-charge field strength, knowing the second-harmonic power. Finally, we find that the addition of the second harmonic increases the sensitivity of the high harmonics to the surface. Figure 3(c) shows the amplitude of the bulk- and surface-induced oscillations on the eighth harmonic as a function of the angle of incidence. The symmetry breaking induced by the surface remains very apparent even at shallower angles of incidence (<20°), where no intrinsic even harmonics are measured without a second-harmonic field.

To conclude, we have found that charge accumulated in the native oxide layer of a Si crystal generates a space-charge field that breaks the symmetry of high-harmonic generation from the bulk of Si. This surface-mediated contribution to high-harmonic generation from solids bridges perturbative surface nonlinearities—characterized by bound non-centrosymmetric nonlinear oscillations—to the nonperturbative regime—characterized by large-scale motion of electron-hole pairs. The high degree of symmetry breaking, exemplified by the significant even-harmonic power relative to that of the odd-order harmonics, suggests that the large amplitude oscillations magnify the effect of the surface compared to perturbative nonlinearities, enabling a new class of studies of surfaces and interfaces. Our demonstration is a first step towards new ways of probing surfaces and adsorbates in the strong-field regime, where techniques developed for gas-phase experiments can be adapted. Finally, direct surface effects hold the potential for engineering strong-field phenomena by modifying the electron and hole propagation across interfaces.

Funding

Air Force Office of Scientific Research (AFOSR) (FA9550-14-1-0108); U.S. Department of Energy (DOE) (DE-AC02-76SF00515).

REFERENCES

1. A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013). [CrossRef]  

2. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Nat. Phys. 7, 138 (2011). [CrossRef]  

3. T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015). [CrossRef]  

4. G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, Nature 522, 462 (2015). [CrossRef]  

5. M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015). [CrossRef]  

6. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014). [CrossRef]  

7. G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016). [CrossRef]  

8. H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017). [CrossRef]  

9. Y. S. You, D. A. Reis, and S. Ghimire, Nat. Phys. 13, 345 (2017). [CrossRef]  

10. M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017). [CrossRef]  

11. S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Phys. Rev. A 85, 043836 (2012). [CrossRef]  

12. T. F. Heinz, Modern Problems in Condensed Matter Sciences (Elsevier, 1991), vol. 29, pp. 353–416.

13. N. Bloembergen, R. K. Chang, S. Jha, and C. Lee, Phys. Rev. 174, 813 (1968). [CrossRef]  

14. N. Bloembergen, Appl. Phys. B 68, 289 (1999). [CrossRef]  

15. Y. Shen, Nature 337, 519 (1989). [CrossRef]  

16. C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007). [CrossRef]  

17. B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006). [CrossRef]  

18. G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015). [CrossRef]  

19. T. T. Luu and H. J. Wörner, Nat. Commun. 9, 916 (2018). [CrossRef]  

20. J. Bloch, J. Mihaychuk, and H. Van Driel, Phys. Rev. Lett. 77, 920(1996). [CrossRef]  

21. J. Mihaychuk, J. Bloch, Y. Liu, and H. Van Driel, Opt. Lett. 20, 2063 (1995). [CrossRef]  

22. T. Scheidt, E. Rohwer, H. Von Bergmann, and H. Stafast, Phys. Rev. B 69, 165314 (2004). [CrossRef]  

23. M. Morita, T. Ohmi, E. Hasegawa, M. Kawakami, and M. Ohwada, J. Appl. Phys. 68, 1272 (1990). [CrossRef]  

24. G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, Phys. Rev. B 91, 064302 (2015). [CrossRef]  

25. R. Glauber, Quantum Optics (Academic, 1969), p. 15.

26. S. Brasselet and J. Zyss, J. Opt. Soc. Am. B 15, 257 (1998). [CrossRef]  

27. N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006). [CrossRef]  

References

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  1. A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
    [Crossref]
  2. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Nat. Phys. 7, 138 (2011).
    [Crossref]
  3. T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
    [Crossref]
  4. G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, Nature 522, 462 (2015).
    [Crossref]
  5. M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
    [Crossref]
  6. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
    [Crossref]
  7. G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016).
    [Crossref]
  8. H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017).
    [Crossref]
  9. Y. S. You, D. A. Reis, and S. Ghimire, Nat. Phys. 13, 345 (2017).
    [Crossref]
  10. M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017).
    [Crossref]
  11. S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Phys. Rev. A 85, 043836 (2012).
    [Crossref]
  12. T. F. Heinz, Modern Problems in Condensed Matter Sciences (Elsevier, 1991), vol. 29, pp. 353–416.
  13. N. Bloembergen, R. K. Chang, S. Jha, and C. Lee, Phys. Rev. 174, 813 (1968).
    [Crossref]
  14. N. Bloembergen, Appl. Phys. B 68, 289 (1999).
    [Crossref]
  15. Y. Shen, Nature 337, 519 (1989).
    [Crossref]
  16. C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).
    [Crossref]
  17. B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
    [Crossref]
  18. G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015).
    [Crossref]
  19. T. T. Luu and H. J. Wörner, Nat. Commun. 9, 916 (2018).
    [Crossref]
  20. J. Bloch, J. Mihaychuk, and H. Van Driel, Phys. Rev. Lett. 77, 920(1996).
    [Crossref]
  21. J. Mihaychuk, J. Bloch, Y. Liu, and H. Van Driel, Opt. Lett. 20, 2063 (1995).
    [Crossref]
  22. T. Scheidt, E. Rohwer, H. Von Bergmann, and H. Stafast, Phys. Rev. B 69, 165314 (2004).
    [Crossref]
  23. M. Morita, T. Ohmi, E. Hasegawa, M. Kawakami, and M. Ohwada, J. Appl. Phys. 68, 1272 (1990).
    [Crossref]
  24. G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, Phys. Rev. B 91, 064302 (2015).
    [Crossref]
  25. R. Glauber, Quantum Optics (Academic, 1969), p. 15.
  26. S. Brasselet and J. Zyss, J. Opt. Soc. Am. B 15, 257 (1998).
    [Crossref]
  27. N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006).
    [Crossref]

2018 (1)

T. T. Luu and H. J. Wörner, Nat. Commun. 9, 916 (2018).
[Crossref]

2017 (3)

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017).
[Crossref]

Y. S. You, D. A. Reis, and S. Ghimire, Nat. Phys. 13, 345 (2017).
[Crossref]

M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017).
[Crossref]

2016 (1)

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016).
[Crossref]

2015 (5)

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015).
[Crossref]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, Nature 522, 462 (2015).
[Crossref]

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, Phys. Rev. B 91, 064302 (2015).
[Crossref]

2014 (1)

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
[Crossref]

2013 (1)

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
[Crossref]

2012 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Phys. Rev. A 85, 043836 (2012).
[Crossref]

2011 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Nat. Phys. 7, 138 (2011).
[Crossref]

2007 (1)

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).
[Crossref]

2006 (2)

B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
[Crossref]

N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006).
[Crossref]

2004 (1)

T. Scheidt, E. Rohwer, H. Von Bergmann, and H. Stafast, Phys. Rev. B 69, 165314 (2004).
[Crossref]

1999 (1)

N. Bloembergen, Appl. Phys. B 68, 289 (1999).
[Crossref]

1998 (1)

1996 (1)

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M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017).
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S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Phys. Rev. A 85, 043836 (2012).
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S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Nat. Phys. 7, 138 (2011).
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M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017).
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G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016).
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G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016).
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S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Phys. Rev. A 85, 043836 (2012).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
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N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006).
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N. Bloembergen, R. K. Chang, S. Jha, and C. Lee, Phys. Rev. 174, 813 (1968).
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A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
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M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
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G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015).
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M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
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M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
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Krausz, F.

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
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T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
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Krushelnick, K.

B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
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Langer, F.

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, Nature 523, 572 (2015).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nat. Photonics 8, 119 (2014).
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N. Bloembergen, R. K. Chang, S. Jha, and C. Lee, Phys. Rev. 174, 813 (1968).
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M. Taucer, T. Hammond, P. Corkum, G. Vampa, C. Couture, N. Thiré, B. Schmidt, F. Légaré, H. Selvi, N. Unsuree, B. Hamilton, T. J. Echtermeyer, and M. A. Denecke, Phys. Rev. B 96, 195420 (2017).
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G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015).
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G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, Nature 522, 462 (2015).
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N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006).
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Levy, A.

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).
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Li, Y.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017).
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Liu, H.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017).
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Luu, T. T.

T. T. Luu and H. J. Wörner, Nat. Commun. 9, 916 (2018).
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T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
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Mairesse, Y.

N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, Nat. Phys. 2, 781 (2006).
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Marjoribanks, R.

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).
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Martin, P.

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).
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McDonald, C.

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, Phys. Rev. Lett. 115, 193603(2015).
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G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, Nature 522, 462 (2015).
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G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, Phys. Rev. B 91, 064302 (2015).
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B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nat. Phys. 2, 456 (2006).
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Figures (3)

Fig. 1.
Fig. 1. Measured high-harmonic spectra from a Si film facing the incident beam in (a) transmission and (b) reflection for a p -polarized driver and for (c) the sapphire substrate facing the incident beam in transmission for a p -polarized (blue line) and s -polarized (red line) driver. Weak even harmonics are only detected for p polarization when the Si film faces the detector. The powers of the second and third harmonics are unrelated to those of the higher harmonics because they are measured with a different spectrometer with non-overlapping spectra. (d) The power of the even harmonics (colored lines) saturates with the intensity of the driver, contrary to the odd harmonics (gray lines). (e) Real-time acquisition of the seventh and eighth harmonics for alternate illumination of two adjacent regions of the Si film (one region is marked by the blue dot and the other by the red dot). The sample is laterally translated between these two regions at the times marked by the colored dots. The fundamental beam is blocked when the power is zero. The even-harmonic power shows a transient behavior developing over 1 sec , contrary to the odd harmonics. Data in panels (d) and (e) is acquired in the configuration of panel (c) with p -polarized light.
Fig. 2.
Fig. 2. (a) Sketch of how the space-charge field perturbs bulk high-harmonic emission. High harmonics (purple beam, co-propagating with the red driver) are emitted within one absorption length from the Si–oxide interface. The infrared field ( F IR ) launches electrons ( e ) and holes ( h + ) on trajectories in Si (dashed black lines) along the driver’s polarization (red arrow). The component of the space-charge field (blue arrows, F SC ) along the driver’s polarization renders the trajectories asymmetric (solid black lines), resulting in the emission of even-order high harmonics. (b) Electrons’ (solid lines) and holes’ (dashed lines) trajectories responsible for emission of the eighth harmonic (blue lines) and the thirteenth harmonic (red lines, corresponding to the cutoff harmonic). The trajectories are calculated according to the model developed in Ref. [24] with the following parameters: λ 0 = 2.2 μm (driving laser wavelength), F IR = 0.18 V / Å , a = 5.43 Å (lattice constant of Si). The electron and the hole propagate in the first conduction and split-off valence bands, respectively.
Fig. 3.
Fig. 3. Controlling symmetry breaking with a bichromatic laser field (fundamental and second harmonic). (a) While the power of the ninth harmonic is modulated twice per cycle of the second harmonic, the eighth harmonic develops an additional strong modulation with half the period. For some optimum delays, destructive interference between space-charge and second-harmonic fields suppresses the power of the eighth harmonic to below that obtained without the second-harmonic field (labeled “intrinsic” in the figure). (b) The amplitude of the bulk (2/cycle) and surface-induced (1/cycle) modulations of the eighth harmonic are analyzed as a function of the second-harmonic power. The surface contribution reaches a maximum as a result of optimum interference with the surface charge. (c) The surface contribution remains strong even for shallow ( < 20 ° ) angles of incidence (measured from normal to the surface), where intrinsic even harmonics are too weak to be detected.

Equations (1)

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I 2 n / 2 n ± 1 = tan 2 σ ,

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