Abstract

Solid-state high-harmonic generation (HHG) by an intense infra-red (IR) laser field offers a new route to generate coherent attosecond light pulses in the extreme ultraviolet regime. The propagation of the IR driving field in the dense solid medium is accompanied by non-linear processes which shape the generating waveform. In this work, we introduce a monolithic scheme in which we both exploit the non-linear propagation to manipulate a two color driving field, as well as generate high harmonics within a single crystal. We show that the resulting non-commensurate, bi-chromatic, generating field provides precise control over the periodicity of the HHG process. This control enables us to manipulate the spectral positions of the discrete harmonic peaks. Our method advances solid-state HHG spectroscopy, and offers a simple route towards tunable, robust XUV sources.

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

Full Article  |  PDF Article
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References

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    [Crossref]
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2018 (1)

R. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, “High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems,” Nat. Photonics 12(5), 266–270 (2018).
[Crossref]

2017 (5)

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Phys. 13(3), 262–265 (2017).
[Crossref]

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017).
[Crossref]

Y. S. You, Y. Yin, Y. Wu, A. Chew, X. Ren, F. Zhuang, S. Gholam-Mirzaei, M. Chini, Z. Chang, and S. Ghimire, “High-harmonic generation in amorphous solids,” Nat. Commun. 8(1), 724 (2017).
[Crossref]

Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017).
[Crossref]

T. Hammond, D. Villeneuve, and P. Corkum, “Producing and controlling half-cycle near-infrared electric-field transients,” Optica 4(7), 826–830 (2017).
[Crossref]

2016 (4)

M. Garg, M. Zhan, T. T. Luu, H. Lakhotia, T. Klostermann, A. Guggenmos, and E. Goulielmakis, “Multi-petahertz electronic metrology,” Nature 538(7625), 359–363 (2016).
[Crossref]

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref]

2015 (7)

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref]

P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015).
[Crossref]

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523(7562), 572–575 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
[Crossref]

2014 (1)

G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

2013 (1)

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
[Crossref]

2011 (2)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

J. Dahlström, A. L’Huillier, and J. Mauritsson, “Quantum mechanical approach to probing the birth of attosecond pulses using a two-colour field,” J. Phys. B: At., Mol. Opt. Phys. 44(9), 095602 (2011).
[Crossref]

2003 (1)

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[Crossref]

2002 (1)

B. Shan, A. Cavalieri, and Z. Chang, “Tunable high harmonic generation with an optical parametric amplifier,” Appl. Phys. B: Lasers Opt. 74(S1), s23–s26 (2002).
[Crossref]

2000 (1)

M. Bellini, “Generation of widely tunable harmonic pulses in the UV and VUV from a NIR optical parametric amplifier,” Appl. Phys. B: Lasers Opt. 70(6), 773–776 (2000).
[Crossref]

1999 (1)

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
[Crossref]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref]

1986 (1)

M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

1967 (1)

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Altucci, C.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
[Crossref]

Ammosov, M.

M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Argenti, L.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

Arnold, C.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

Arusi-Parpar, T.

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

Baltuška, A.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[Crossref]

Barreau, L.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

Bellini, M.

M. Bellini, “Generation of widely tunable harmonic pulses in the UV and VUV from a NIR optical parametric amplifier,” Appl. Phys. B: Lasers Opt. 70(6), 773–776 (2000).
[Crossref]

Bengtsson, S.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

Blanchet, V.

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Blinov, I. V.

R. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, “High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems,” Nat. Photonics 12(5), 266–270 (2018).
[Crossref]

Boscolo, A.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
[Crossref]

Brabec, T.

G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Brambila, D.

O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
[Crossref]

Browne, D. A.

Bruner, B.

O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
[Crossref]

Bruner, B. D.

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Bruzzese, R.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
[Crossref]

Caillat, J.

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A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
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Kroon, D.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
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A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

L’Huillier, A.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

J. Dahlström, A. L’Huillier, and J. Mauritsson, “Quantum mechanical approach to probing the birth of attosecond pulses using a two-colour field,” J. Phys. B: At., Mol. Opt. Phys. 44(9), 095602 (2011).
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M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
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Lakhotia, H.

M. Garg, M. Zhan, T. T. Luu, H. Lakhotia, T. Klostermann, A. Guggenmos, and E. Goulielmakis, “Multi-petahertz electronic metrology,” Nature 538(7625), 359–363 (2016).
[Crossref]

Langer, F.

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523(7562), 572–575 (2015).
[Crossref]

Larsen, E.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

Laux, M.

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
[Crossref]

Légaré, F.

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
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V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

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M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

Li, Y.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Phys. 13(3), 262–265 (2017).
[Crossref]

Lindroth, E.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
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Liu, H.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Phys. 13(3), 262–265 (2017).
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M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

Luu, T. T.

M. Garg, M. Zhan, T. T. Luu, H. Lakhotia, T. Klostermann, A. Guggenmos, and E. Goulielmakis, “Multi-petahertz electronic metrology,” Nature 538(7625), 359–363 (2016).
[Crossref]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref]

Maquet, A.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

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M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

Mauritsson, J.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
[Crossref]

J. Dahlström, A. L’Huillier, and J. Mauritsson, “Quantum mechanical approach to probing the birth of attosecond pulses using a two-colour field,” J. Phys. B: At., Mol. Opt. Phys. 44(9), 095602 (2011).
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McDonald, C.

G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Meyer, K.

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
[Crossref]

Miranda, M.

M. Kotur, D. Guénot, Á. Jiménez-Galán, D. Kroon, E. Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mauritsson, C. Arnold, S. E. Canton, M. Gisselbrecht, T. Carette, J. M. Dahlström, E. Lindroth, A. Maquet, L. Argenti, F. Martín, and A. L’Huillier, “Spectral phase measurement of a fano resonance using tunable attosecond pulses,” Nat. Commun. 7(1), 10566 (2016).
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O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
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G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
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C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
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O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Orlando, G.

G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
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Ott, C.

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
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O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Pfeifer, T.

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
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Poletto, L.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
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Popov, V.

M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Raith, P.

C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, “Lorentz meets fano in spectral line shapes: a universal phase and its laser control,” Science 340(6133), 716–720 (2013).
[Crossref]

Reis, D. A.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Phys. 13(3), 262–265 (2017).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref]

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
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Ren, X.

Y. S. You, Y. Yin, Y. Wu, A. Chew, X. Ren, F. Zhuang, S. Gholam-Mirzaei, M. Chini, Z. Chang, and S. Ghimire, “High-harmonic generation in amorphous solids,” Nat. Commun. 8(1), 724 (2017).
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Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017).
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Risoud, F.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

Roessler, D.

Rubtsov, A. N.

R. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, “High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems,” Nat. Photonics 12(5), 266–270 (2018).
[Crossref]

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Ruchon, T.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
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Salières, P.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
[Crossref]

Schafer, K. J.

Schmidt, B.

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
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Schubert, O.

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. Koch, M. Kira, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523(7562), 572–575 (2015).
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Scrinzi, A.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
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Serbinenko, V.

O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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B. Shan, A. Cavalieri, and Z. Chang, “Tunable high harmonic generation with an optical parametric amplifier,” Appl. Phys. B: Lasers Opt. 74(S1), s23–s26 (2002).
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Silva, R.

R. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, “High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems,” Nat. Photonics 12(5), 266–270 (2018).
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Silva, R. E.

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Sistrunk, E.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
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Smirnova, O.

R. Silva, I. V. Blinov, A. N. Rubtsov, O. Smirnova, and M. Ivanov, “High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems,” Nat. Photonics 12(5), 266–270 (2018).
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O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

Soifer, H.

O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. Bruner, A. Uzan, D. Brambila, A. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, “Attosecond tunnelling interferometry,” Nat. Phys. 11(10), 815–819 (2015).
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Stagira, S.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
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Svelto, O.

C. Altucci, R. Bruzzese, C. De Lisio, M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, A. Boscolo, P. Ceccherini, L. Poletto, G. Tondello, and P. Villoresi, “Tunable soft-x-ray radiation by high-order harmonic generation,” Phys. Rev. A 61(2), 021801 (1999).
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Taïeb, R.

V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J.-F. Hergott, T. Ruchon, L. Argenti, R. Taïeb, F. Martín, and P. Salières, “Attosecond dynamics through a fano resonance: Monitoring the birth of a photoelectron,” Science 354(6313), 734–738 (2016).
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Tamaya, T.

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017).
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Tanaka, K.

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017).
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A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

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G. Vampa, C. McDonald, G. Orlando, P. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, and P. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref]

G. Vampa, T. Hammond, N. Thiré, B. Schmidt, F. Légaré, C. McDonald, T. Brabec, D. Klug, and P. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref]

G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

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M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

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Y. S. You, Y. Yin, Y. Wu, A. Chew, X. Ren, F. Zhuang, S. Gholam-Mirzaei, M. Chini, Z. Chang, and S. Ghimire, “High-harmonic generation in amorphous solids,” Nat. Commun. 8(1), 724 (2017).
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Nat. Photonics (1)

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[Crossref]

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G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
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Science (3)

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Sov. Phys. JETP (1)

M. Ammosov, N. Delone, V. Krainov, A. Perelomov, V. Popov, M. Terent’ev, G. L. Yudin, and M. Y. Ivanov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electric field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Other (1)

A. J. Uzan, G. Orenstein, Á. Jiménez-Galán, C. McDonald, R. E. Silva, B. D. Bruner, N. D. Klimkin, V. Blanchet, T. Arusi-Parpar, M. Krüger, A. N. Rubtsov, O. Smirnova, M. Ivanov, B. Yan, T. Brabec, and N. Dudovich, “Multi-band petahertz currents resolved via high harmonic generation spectroscopy,” arXiv preprint arXiv:1812.02498 (2018).

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Figures (5)

Fig. 1.
Fig. 1. Experimental setup. A two color HHG driving field is produced by frequency doubling a $\lambda$=1300 nm laser in a type-I phase matched BBO crystal. The SH and fundamental fields propagate colinearly. The first neutral density filter (NDF) controls the conversion ratio of the intensity dependent SH generation and the second adjusts the total intensity focused into the crystal. HHG emission is measured with a flat field XUV spectrometer. The XUV grating can be retracted to enable spectral measurements of the fundamental and SH fields by an IR and visible spectrometer. See text for further details.
Fig. 2.
Fig. 2. Non-commensurate bi-chromatic HHG in a crystal. a Schematic description of the non-linear propagation of the driving fields and the HHG emission region inside the MgO lattice. The red arrow indicates the propagation direction of the driving fields. The red and blue lines are snapshots of the generating pulse, showing the temporal profiles of the fundamental and SH fields, respectively. When the two fields enter the crystal (right snapshot) their frequencies are commensurate. The non-linear propagation through the crystal detunes the relative frequency between the driving fields (left snapshot). The HHG signal is emitted near the output surface of the crystal (purple region) only, due to absorption of the harmonics. b Temporal snapshot of HHG driven by a non-commensurate bi-chromatic field. The growing phase mismatch, caused by the detuning of the relative frequency between the fundamental (red line) and SH (blue line) fields, is presented by the vertical black dotted lines. The black arrows schematically illustrate the subcycle electronic trajectories for zero detuning. The trajectories for non-zero detuning are shown in magenta and green lines for positive and negative half cycles, respectively, exhibiting opposite trends.
Fig. 3.
Fig. 3. Fundamental field and SH spectra. The spectra of the fundamental (dark blue line) and SH (red line) fields after propagating through the MgO crystal. The wavelength of SH spectrum was multiplied by two in order to visualize the spectral detuning between the fields. The fundamental spectrum before the crystal is described by the dashed green line. All spectra were normalized by their maximum value, respectively. a Propagation through a 100 $\mu m$ thick crystal. The SH spectrum shows no significant detuning from the fundamental spectrum. b Propagation through a 200 $\mu m$ thick crystal. The SH shows significant detuning from the fundamental spectrum.
Fig. 4.
Fig. 4. HHG spectral tuning. Experimentally measured HHG spectrum (horizontal axis) versus the two-color delay (vertical axis) in logarithmic scale. a Propagation through a 100 $\mu m$ thick MgO crystal. The harmonics exhibit a periodic modulation at $4\omega _0$. b propagation through a 200 $\mu m$ thick MgO crystal. Scanning the two-color delay leads to a dramatic shift of the central frequency associated with each harmonic.
Fig. 5.
Fig. 5. Modeling the control mechanism. Simulation of the $18^{th}$ harmonic (17.1 eV) generated by a non-commensurate bi-chromatic field. a Purely real $\sigma$. The two opposite temporal slopes split the harmonic spectrum into two equal branches. b Ccomplex $\sigma$. The positive and negative half cycles obtain different amplitudes, resulting in a suppression of one of the branches presented in a. c Experimentally measured spectrum of the $18^{th}$ harmonic. This spectrum exhibits a broad harmonic which shifts periodically with the two-color delay, showing a good agreement with the simulated results presented in b.

Equations (17)

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σ ( τ , δ , n ) σ 0 ( τ , δ , 0 ) + κ ( τ ) n δ for positive half cycles σ ( τ , δ , n + 1 / 2 ) σ 0 ( τ , δ , 1 / 2 ) + κ ( τ ) n δ for negative half cycles
I ( ω , τ ) | e i S 0 ( e i σ 0 ( τ , δ , 0 ) n exp ( i ω n T 0 i κ ( τ ) n δ ) e i σ 0 ( τ , δ , 1 / 2 ) + i ω T 0 / 2 n exp ( i ω n T 0 + i κ ( τ ) n δ ) ) | 2
n exp ( i ω n T 0 i κ ( τ ) n δ ) = ω 0 n δ ( ω κ ( τ ) δ T 0 + n ω 0 )
I ( ω , τ ) = a | e i σ 0 ( τ , δ , 0 ) n sech 2 ( n w e f f ) exp ( i ω n T 0 i κ ( τ ) n δ ) e i σ 0 ( τ , δ , 1 / 2 ) + i ω T 0 / 2 n sech 2 ( n w e f f ) exp ( i ω n T 0 + i κ ( τ ) n δ ) | 2
c + I ( ω , τ ) σ 0 ( τ , δ , 0 ) = σ R cos ( 2 ω 0 τ + θ 1 ) + i σ I cos ( 2 ω 0 τ + θ 2 ) κ ( τ ) = κ R cos ( 2 ω 0 τ + θ 3 ) + i κ I cos ( 2 ω 0 τ + θ 4 )
j e r ( ω , τ ) = ω B . Z . d 3 k g ( k , t , t ) e i S ( k , t , t ) + i ω t d t d t S ( k , t , t ) = t t ε g ( k A ( t ) + A ( t ~ ) ) d t ~ ε g ( k ) = E c ( k ) E v ( k )
j e r ( ω , τ ) = ω stationary solutions g ~ ( st ) e i S ( st ) + i ω t s t st = ( k s t , t s t , t s t )
A F ( t ) = F 0 ω 0 sin ( ω 0 t ) n ^ F A S H ( t ) = ϵ F 0 ( 2 + δ ) ω 0 sin ( ( 2 + δ ) ω 0 ( t τ ) ) n ^ F
S ( st ) S 0 ( st ) + S ϵ ( st ) ϵ + S ϵ δ ( st ) ϵ δ
S 0 ( st ) = S ( st ) | ϵ , δ = 0 S ϵ ( st ) = S ( st ) ϵ | ϵ , δ = 0
S ϵ δ ( st ) = 2 S ( st ) ϵ δ | ϵ , δ = 0 = F 0 4 ω 0 t s t t s t { k ε g ( k s t A F ( t s t ) + A F ( t ~ ) ) n ^ F × [ sin ( 2 ω 0 ( t s t τ ) ) sin ( 2 ω 0 ( t ~ τ ) ) + 2 ω 0 τ cos ( 2 ω 0 ( t s t τ ) ) 2 ω 0 τ cos ( 2 ω 0 ( t ~ τ ) ) 2 cos ( 2 ω 0 ( t ~ τ ) ) ω 0 t ~ 2 cos ( 2 ω 0 ( t s t τ ) ) ω 0 t s t ] } d t ~  
st = ( k ¯ s t , t ¯ s t + n T 0 , t ¯ s t + n T 0 ) for positive half cycles
st = ( k ¯ s t , t ¯ s t + ( n + 1 / 2 ) T 0 , t ¯ s t + ( n + 1 / 2 ) T 0 ) for negative half cycles
κ ( τ ) = ϵ F 0 2 ω 0 t ¯ s t t ¯ s t { k ε g ( k ¯ s t A F ( t ¯ s t ) + A F ( t ^ ) ) n ^ F × [ cos ( 2 ω 0 ( t ^ τ ) ) cos ( 2 ω 0 ( t ¯ s t τ ) ) ] ω 0 T 0 } d t ^
for positive half cycles σ ( τ , δ , n ) = ϵ S ( st ) ϵ | ϵ = 0 σ 0 ( τ , δ , 0 ) = S ϵ ( st ) ϵ + S ϵ δ ( st ) ϵ δ κ ( τ ) n δ } st = ( k ¯ s t , t ¯ s t , t ¯ s t ) S ( st ) S 0 + σ 0 ( τ , δ , 0 ) + κ ( τ ) n δ
for negative half cycles σ ( τ , δ , n + 1 / 2 ) = ϵ S ( st ) ϵ | ϵ = 0 σ 0 ( τ , δ , 1 / 2 ) = S ϵ ( st ) ϵ S ϵ δ ( st ) ϵ δ + κ ( τ ) n δ } st = ( k ¯ s t , t ¯ s t + T 0 / 2 , t ¯ s t + T 0 / 2 ) S ( st ) S 0 σ 0 ( τ , δ , 1 / 2 ) κ ( τ ) n δ
j e r ( ω , τ ) = ω g ~ ( st ) e i S 0 + i ω t ¯ s t ( e i σ 0 ( τ , δ , 0 ) n exp ( i ω n T 0 i κ ( τ ) n δ ) e i σ 0 ( τ , δ , 1 / 2 ) + i ω T 0 / 2 n exp ( i ω n T 0 + i κ ( τ ) n δ ) )

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