Abstract

We theoretically investigate the propagation of broadband single-cycle terahertz (THz) pulses through a medium with a nonlinear optical response. Our model takes into account non-paraxial effects, self-focusing and diffraction, as well as dispersion, in both the linear and nonlinear optical regimes. We investigate the contribution of non-instantaneous Kerr-type nonlinearity to the overall instantaneous and delayed Kerr effect at the THz frequencies. We show how increasing the nonlinear relaxation time and its dispersion modifies the THz pulse after the propagation through a transparent medium. We also discuss the effect of linear dispersion on self-action during the pulse propagation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2018 (2)

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
[Crossref]

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

2015 (2)

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92(2), 023809 (2015).
[Crossref]

A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17(4), 043002 (2015).
[Crossref]

2014 (4)

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m Single-Cycle Terahertz Fields from a Laser-Driven Large-Size Partitioned Organic Crystal,” Phys. Rev. Lett. 112(21), 213901 (2014).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

L. M. Kovachev, “The light filament as vector solitary wave,” AIP Conf. Proc. 1629(1), 167–171 (2014).
[Crossref]

2013 (2)

H. Y. Hwang, N. C. Brandt, H. Farhat, A. L. Hsu, J. Kong, and K. A. Nelson, “Nonlinear thz conductivity dynamics in p-type cvd-grown graphene,” J. Phys. Chem. B 117(49), 15819–15824 (2013).
[Crossref]

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

2012 (3)

S. Tani, F. Blanchard, and K. Tanaka, “Ultrafast Carrier Dynamics in Graphene under a High Electric Field,” Phys. Rev. Lett. 109(16), 166603 (2012).
[Crossref]

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85(20), 201304 (2012).
[Crossref]

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E 86(3), 036706 (2012).
[Crossref]

2011 (1)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

2009 (1)

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

2008 (2)

J. Hebling, K. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-Power THz Generation, THz Nonlinear Optics, and THz Nonlinear Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14(2), 345–353 (2008).
[Crossref]

J. Hebling, K. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25(7), B6–B19 (2008).
[Crossref]

2006 (1)

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96(18), 187402 (2006).
[Crossref]

2004 (2)

2002 (1)

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89(28), 283902 (2002).
[Crossref]

2001 (1)

1997 (1)

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

1993 (1)

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70(9), 1236–1239 (1993).
[Crossref]

1991 (1)

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of znse in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989).
[Crossref]

1988 (1)

1986 (1)

K. P. Cheung and D. H. Auston, “A novel technique for measuring far-infrared absorption and dispersion,” Infrared Phys. 26(1), 23–27 (1986).
[Crossref]

1985 (1)

K. P. Cheung and D. H. Auston, “Excitation of Coherent Phonon Polaritons with Femtosecond Optical Pulses,” Phys. Rev. Lett. 55(20), 2152–2155 (1985).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Andreasen, J.

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E 86(3), 036706 (2012).
[Crossref]

Auston, D. H.

K. P. Cheung and D. H. Auston, “A novel technique for measuring far-infrared absorption and dispersion,” Infrared Phys. 26(1), 23–27 (1986).
[Crossref]

K. P. Cheung and D. H. Auston, “Excitation of Coherent Phonon Polaritons with Femtosecond Optical Pulses,” Phys. Rev. Lett. 55(20), 2152–2155 (1985).
[Crossref]

Awari, N.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Ayesheshim, A.

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

Baierl, S.

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

Bandulet, H.-C.

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

Bartal, B.

Blanchard, F.

S. Tani, F. Blanchard, and K. Tanaka, “Ultrafast Carrier Dynamics in Graphene under a High Electric Field,” Phys. Rev. Lett. 109(16), 166603 (2012).
[Crossref]

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989).
[Crossref]

Bonn, M.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Bougeard, D.

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

Boyd, R. W.

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92(2), 023809 (2015).
[Crossref]

R. W. Boyd, Nonlinear Optics (Elsevier, 2003).

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

Brandt, N. C.

H. Y. Hwang, N. C. Brandt, H. Farhat, A. L. Hsu, J. Kong, and K. A. Nelson, “Nonlinear thz conductivity dynamics in p-type cvd-grown graphene,” J. Phys. Chem. B 117(49), 15819–15824 (2013).
[Crossref]

Brömmel, D.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

Bucksbaum, P. H.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70(9), 1236–1239 (1993).
[Crossref]

Chai, X.

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
[Crossref]

Chen, M.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Chen, Z.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Cheung, K. P.

K. P. Cheung and D. H. Auston, “A novel technique for measuring far-infrared absorption and dispersion,” Infrared Phys. 26(1), 23–27 (1986).
[Crossref]

K. P. Cheung and D. H. Auston, “Excitation of Coherent Phonon Polaritons with Femtosecond Optical Pulses,” Phys. Rev. Lett. 55(20), 2152–2155 (1985).
[Crossref]

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

Couairon, A.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

Dai, J.

de J. Ramírez-Góngora, O.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

Deinert, J.-C.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Deneuville, A.

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of znse in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Dillner, U.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

Dolgaleva, K.

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92(2), 023809 (2015).
[Crossref]

Edwards, E. R. J.

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

Elsaesser, T.

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H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
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A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
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M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70(3), 036604 (2004).
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M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89(28), 283902 (2002).
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C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m Single-Cycle Terahertz Fields from a Laser-Driven Large-Size Partitioned Organic Crystal,” Phys. Rev. Lett. 112(21), 213901 (2014).
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H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
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H. Y. Hwang, N. C. Brandt, H. Farhat, A. L. Hsu, J. Kong, and K. A. Nelson, “Nonlinear thz conductivity dynamics in p-type cvd-grown graphene,” J. Phys. Chem. B 117(49), 15819–15824 (2013).
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P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96(18), 187402 (2006).
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X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
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L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
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Reid, M.

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
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L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
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P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96(18), 187402 (2006).
[Crossref]

Reinhard, A.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

Ropagnol, X.

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
[Crossref]

Saari, P.

Safavi-Naeini, S.

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
[Crossref]

Schmidt, A.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

Schubert, O.

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Sharma, G.

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

Singh, P.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

Strikwerda, A. C.

A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17(4), 043002 (2015).
[Crossref]

Su, F. H.

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

Tanaka, K.

S. Tani, F. Blanchard, and K. Tanaka, “Ultrafast Carrier Dynamics in Graphene under a High Electric Field,” Phys. Rev. Lett. 109(16), 166603 (2012).
[Crossref]

Tani, S.

S. Tani, F. Blanchard, and K. Tanaka, “Ultrafast Carrier Dynamics in Graphene under a High Electric Field,” Phys. Rev. Lett. 109(16), 166603 (2012).
[Crossref]

Tanner, D.

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of znse in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Tarekegne, A. T.

A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17(4), 043002 (2015).
[Crossref]

Teichert, J.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Tielrooij, K.-J.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Tóth, G.

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz spm,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds. (Springer, Berlin, Heidelberg, 2009), pp. 651–653.

Turchinovich, D.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85(20), 201304 (2012).
[Crossref]

Urbanek, B.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Vicario, C.

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m Single-Cycle Terahertz Fields from a Laser-Driven Large-Size Partitioned Organic Crystal,” Phys. Rev. Lett. 112(21), 213901 (2014).
[Crossref]

Wang, Z.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

Woerner, M.

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96(18), 187402 (2006).
[Crossref]

Woltersdorf, G.

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989).
[Crossref]

Yeh, K.

J. Hebling, K. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-Power THz Generation, THz Nonlinear Optics, and THz Nonlinear Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14(2), 345–353 (2008).
[Crossref]

J. Hebling, K. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25(7), B6–B19 (2008).
[Crossref]

Yeh, K.-L.

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz spm,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds. (Springer, Berlin, Heidelberg, 2009), pp. 651–653.

You, D.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70(9), 1236–1239 (1993).
[Crossref]

Zalkovskij, M.

A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17(4), 043002 (2015).
[Crossref]

Zhang, J.

Zhang, W.

Ziegler, W.

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

AIP Conf. Proc. (1)

L. M. Kovachev, “The light filament as vector solitary wave,” AIP Conf. Proc. 1629(1), 167–171 (2014).
[Crossref]

Eur. Phys. J.: Spec. Top. (1)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J.: Spec. Top. 199(1), 5–76 (2011).
[Crossref]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hebling, K. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-Power THz Generation, THz Nonlinear Optics, and THz Nonlinear Spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14(2), 345–353 (2008).
[Crossref]

Infrared Phys. (1)

K. P. Cheung and D. H. Auston, “A novel technique for measuring far-infrared absorption and dispersion,” Infrared Phys. 26(1), 23–27 (1986).
[Crossref]

J. Opt. Soc. Am. B (3)

J. Phys. Chem. B (1)

H. Y. Hwang, N. C. Brandt, H. Farhat, A. L. Hsu, J. Kong, and K. A. Nelson, “Nonlinear thz conductivity dynamics in p-type cvd-grown graphene,” J. Phys. Chem. B 117(49), 15819–15824 (2013).
[Crossref]

Nat. Photonics (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, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Nature (1)

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot dirac fermions,” Nature 561(7724), 507–511 (2018).
[Crossref]

New J. Phys. (1)

A. T. Tarekegne, K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Impact ionization in high resistivity silicon induced by an intense terahertz field enhanced by an antenna array,” New J. Phys. 17(4), 043002 (2015).
[Crossref]

Opt. Express (1)

Phys. Rev. A (1)

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92(2), 023809 (2015).
[Crossref]

Phys. Rev. B (3)

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85(20), 201304 (2012).
[Crossref]

L. Razzari, F. H. Su, G. Sharma, F. Blanchard, A. Ayesheshim, H.-C. Bandulet, R. Morandotti, J.-C. Kieffer, T. Ozaki, M. Reid, and F. A. Hegmann, “Nonlinear ultrafast modulation of the optical absorption of intense few-cycle terahertz pulses in n-doped semiconductors,” Phys. Rev. B 79(19), 193204 (2009).
[Crossref]

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of znse in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Phys. Rev. E (2)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70(3), 036604 (2004).
[Crossref]

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E 86(3), 036706 (2012).
[Crossref]

Phys. Rev. Lett. (10)

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89(28), 283902 (2002).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

S. Tani, F. Blanchard, and K. Tanaka, “Ultrafast Carrier Dynamics in Graphene under a High Electric Field,” Phys. Rev. Lett. 109(16), 166603 (2012).
[Crossref]

C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, “Extremely nonperturbative nonlinearities in gaas driven by atomically strong terahertz fields in gold metamaterials,” Phys. Rev. Lett. 113(22), 227401 (2014).
[Crossref]

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96(18), 187402 (2006).
[Crossref]

X. Chai, X. Ropagnol, S. M. Raeis-Zadeh, M. Reid, S. Safavi-Naeini, and T. Ozaki, “Subcycle terahertz nonlinear optics,” Phys. Rev. Lett. 121(14), 143901 (2018).
[Crossref]

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70(9), 1236–1239 (1993).
[Crossref]

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m Single-Cycle Terahertz Fields from a Laser-Driven Large-Size Partitioned Organic Crystal,” Phys. Rev. Lett. 112(21), 213901 (2014).
[Crossref]

A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H-G. Meyer, and G. G. Paulus, “Observation of Gigawatt-Class THz Pulses from a Compact Laser-Driven Particle Accelerator,” Phys. Rev. Lett. 111(7), 074802 (2013).
[Crossref]

K. P. Cheung and D. H. Auston, “Excitation of Coherent Phonon Polaritons with Femtosecond Optical Pulses,” Phys. Rev. Lett. 55(20), 2152–2155 (1985).
[Crossref]

Other (4)

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz spm,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds. (Springer, Berlin, Heidelberg, 2009), pp. 651–653.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

M. Kolesik, “gUPPEcore simulation framework” (2019). http://acms.arizona.edu/FemtoTheory/MK_personal/guppelab/

R. W. Boyd, Nonlinear Optics (Elsevier, 2003).

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Figures (6)

Fig. 1.
Fig. 1. Group velocity dispersion curves for (a) silicon and (b) ZnSe materials calculated from [31] and [32], respectively. The inset shows the GVD in the frequency range of interest. The GVD value of ZnSe is 200 times larger compared to the GVD in Si at that frequency range.
Fig. 2.
Fig. 2. Propagation of the THz pulse in (a) Si. The linear (solid light blue curve) arrives the first. The THz pulse propagated in the Kerr medium (dashed green line) arrives with a time delay of 3.4 ps. Finally, the THz pulses propagated in the Kerr+Raman medium (dotted blue and solid violet lines) arrive one after another with a time delay of $\approx$ 5 ps. (b) shows the pulse propagation in ZnSe. Again, the linear (solid light blue curve) arrives the first. The THz pulse propagated in the Kerr medium (dashed green line) arrives with a time delay of 240 fs. Finally, the THz pulse propagated in the Kerr+Raman medium (dash-dotted violet line) arrives with a time delay of $\approx$ 500 fs. The inset in Fig. 2(a) shows the initial THz electric field.
Fig. 3.
Fig. 3. Intensity distribution of the THz field at different propagation distances (left to right) in Si. Figures 3(a)–3(c) assume a linear propagation, Figs. 3(d)–3(f) show the results of Kerr effect (no Raman effect), and Figs. 3(g)–3(i) give the intensity distribution for the combined Kerr$+$Raman effect ($\alpha =0.9$) during the pulse propagation. The pulse is delayed by $\Delta t_1=3.4~\textrm {ps}$ due to the Kerr effect. This amount of time shift is a result of an overall refractive index change of at least $\Delta n=n_2I=0.01$. Kerr+Raman contribution introduces a larger time shift of $\Delta t_2=5~\textrm {ps}$.
Fig. 4.
Fig. 4. Intensity distribution of the THz field at different propagation distances (left to right) in ZnSe. Figures 4(a)–4(c) assume a linear propagation, Figs. 4(d)–4(f) show the results of Kerr effect (no Raman effect), and Figs. 4(g)–4(i) gives the intensity distribution for the combined Kerr$+$Raman effect during the pulse propagation. The amount of time shift introduced by Kerr+Raman effect is twice as large as the time shift introduced by pure Kerr effect.
Fig. 5.
Fig. 5. Intensity distribution (logarithmic units) of THz spectral power at different propagation distances (left to right) in Si. Spatio-spectral plots visualize the spectral broadening/shrinking of the pulse as well as the diffraction effects. First row (Figs. 5(a)–5(c)) assumes a linear propagation. Second row (Figs. 5(d)–5(f)) shows the results of Kerr effect (no Raman effect), and third row (Figs. 5(g)–5(i)) gives the results for the combined Kerr$+$Raman effects. No significant changes are observed when replacing 10$\%$ of the Kerr contribution by a Raman effect.
Fig. 6.
Fig. 6. Intensity distribution (logarithmic units) of THz spectral power at different propagation distances (left to right) in ZnSe. Spatio-spectral plots visualize the spectral broadening/shrinking of the pulse as well as the diffraction effects. First row (Figs. 6(a)–6(c)) assumes a linear propagation. Unlike Si (results shown in Fig. 5), linear absorption strongly suppresses the higher frequency portion of the spectrum. Second row (Figs. 6(d)–6(f)) shows the results of Kerr effect (no Raman effect), and third row (Figs. 6(g)–6(i)) gives the results for the combined Kerr$+$Raman effects. No significant changes are observed when replacing 10$\%$ of the Kerr contribution by a Raman effect.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

2 E ( E ) + ω 2 n 2 ( ω ) c 2 E = μ 0 ( i ω J ω 2 P ) .
( z 2 + 2 ) E + k 2 ( ω ) E = μ 0 ω 2 P ,
[ z + i k ( ω ) ] [ z i k ( ω ) ] E = 2 E μ 0 ω 2 P .
E = E + exp ( i k ( ω ) z ) + E exp ( i k ( ω ) z ) .
E z = i [ k ( ω ) k 2 2 k ( ω ) ] E + i μ 0 ω 2 2 k ( ω ) P .
{ z 2 + [ k 2 ( ω ) k 2 ] } E = μ 0 ω 2 P .
E z = i k 2 ( ω ) k 2 E + i μ 0 ω 2 2 k 2 ( ω ) k 2 P .
P ( t ) = ϵ 0 χ 0 ( 3 ) E ( t ) g ( t t ) E 2 ( t ) d t .
g ( t ) = α δ ( t ) + ( 1 α ) g R ( t ) .
g R ( ω ) = ω R 2 ω R 2 + 2 i ω δ R ω 2 .
α = g ( t ) g R ( t ) δ ( t ) g R ( t ) .
L D = τ 2 | β 2 | .
L NL = 1 γ P ,
γ = n 2 ω 0 c A eff .

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