Abstract

In this paper we report on the numerical analysis of a time-resolved terahertz (THz) spectroscopy experiment using a modified finite-difference time-domain method. Using this method, we show that ultrafast carrier dynamics can be extracted with a time resolution smaller than the duration of the THz probe pulse and can be determined solely by the pump pulse duration. Our method is found to reproduce complicated two-dimensional transient conductivity maps exceedingly well, demonstrating the power of the time-domain numerical method for extracting ultrafast and dynamic transport parameters from time-resolved THz spectroscopy experiments. The numerical implementation is available online.

© 2011 Optical Society of America

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  1. J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag, 1999).
  2. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
    [CrossRef]
  3. G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
    [CrossRef]
  4. R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
    [CrossRef] [PubMed]
  5. R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
    [CrossRef] [PubMed]
  6. C. Schmuttenmaer, “Exploring dynamics in the far-infrared with terahertz spectroscopy,” Chem. Rev. 104, 1759–1780 (2004).
    [CrossRef] [PubMed]
  7. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
    [CrossRef]
  8. H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
    [CrossRef]
  9. M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
    [CrossRef]
  10. J. Kindt and C. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589–8596 (1999).
    [CrossRef]
  11. M. Beard and C. Schmuttenmaer, “Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments,” J. Chem. Phys. 114, 2903–2909(2001).
    [CrossRef]
  12. M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
    [CrossRef]
  13. H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
    [CrossRef] [PubMed]
  14. G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212(1999).
    [CrossRef]
  15. D. Côté, J. Sipe, and H. van Driel, “Simple method for calculating the propagation of terahertz radiation in experimental geometries,” J. Opt. Soc. Am. B 20, 1374–1385 (2003).
    [CrossRef]
  16. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005).
  17. W. Pernice, F. Payne, and D. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightwave Technol. 25, 2306–2314 (2007).
    [CrossRef]
  18. M. Dressel and M. Scheffler, “Verifying the Drude response,” Ann. Phys. 15, 535–544 (2006).
    [CrossRef]
  19. C. Stanton and D. Bailey, “Rate equations for the study of femtosecond intervalley scattering in compound semiconductors,” Phys. Rev. B 45, 8369–8377 (1992).
    [CrossRef]
  20. K. Iwaszczuk, D. G. Cooke, M. Fujiwara, H. Hashimoto, and P. Jepsen, “Simultaneous reference and differential waveform acquisition in time-resolved terahertz spectroscopy,” Opt. Express 17, 21969–21976 (2009).
    [CrossRef] [PubMed]
  21. C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

2011 (2)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[CrossRef]

C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

2009 (1)

2007 (1)

2006 (1)

M. Dressel and M. Scheffler, “Verifying the Drude response,” Ann. Phys. 15, 535–544 (2006).
[CrossRef]

2005 (2)

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005).

2004 (1)

C. Schmuttenmaer, “Exploring dynamics in the far-infrared with terahertz spectroscopy,” Chem. Rev. 104, 1759–1780 (2004).
[CrossRef] [PubMed]

2003 (2)

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

D. Côté, J. Sipe, and H. van Driel, “Simple method for calculating the propagation of terahertz radiation in experimental geometries,” J. Opt. Soc. Am. B 20, 1374–1385 (2003).
[CrossRef]

2002 (1)

H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
[CrossRef]

2001 (3)

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

M. Beard and C. Schmuttenmaer, “Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments,” J. Chem. Phys. 114, 2903–2909(2001).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
[CrossRef]

2000 (1)

M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
[CrossRef]

1999 (4)

J. Kindt and C. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589–8596 (1999).
[CrossRef]

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag, 1999).

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212(1999).
[CrossRef]

1998 (1)

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[CrossRef]

1992 (1)

C. Stanton and D. Bailey, “Rate equations for the study of femtosecond intervalley scattering in compound semiconductors,” Phys. Rev. B 45, 8369–8377 (1992).
[CrossRef]

Abstreiter, G.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Bailey, D.

C. Stanton and D. Bailey, “Rate equations for the study of femtosecond intervalley scattering in compound semiconductors,” Phys. Rev. B 45, 8369–8377 (1992).
[CrossRef]

Beard, M.

M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
[CrossRef]

M. Beard and C. Schmuttenmaer, “Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments,” J. Chem. Phys. 114, 2903–2909(2001).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
[CrossRef]

Bichler, M.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Brodschelm, A.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Carnahan, M. A.

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Chemla, D. S.

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Cooke, D. G.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[CrossRef]

C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

K. Iwaszczuk, D. G. Cooke, M. Fujiwara, H. Hashimoto, and P. Jepsen, “Simultaneous reference and differential waveform acquisition in time-resolved terahertz spectroscopy,” Opt. Express 17, 21969–21976 (2009).
[CrossRef] [PubMed]

Côté, D.

Dressel, M.

M. Dressel and M. Scheffler, “Verifying the Drude response,” Ann. Phys. 15, 535–544 (2006).
[CrossRef]

Fujiwara, M.

Gallagher, D.

Gallot, G.

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212(1999).
[CrossRef]

Grischkowsky, D.

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16, 1204–1212(1999).
[CrossRef]

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

Hagele, D.

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005).

Hashimoto, H.

Huber, R.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Iwaszczuk, K.

Jeon, T.

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

Jepsen, P.

Jepsen, P. U.

C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[CrossRef]

Jungwirth, P.

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

Kadlec, F.

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
[CrossRef]

Kaindl, R. A.

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Kindt, J.

J. Kindt and C. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589–8596 (1999).
[CrossRef]

Koch, M.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[CrossRef]

Kužel, P.

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
[CrossRef]

Larsen, C.

C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

Leitenstorfer, A.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Lovenich, R.

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

McGowan, R.

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

Nemec, H.

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
[CrossRef]

Othonos, A.

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[CrossRef]

Payne, F.

Pernice, W.

Scheffler, M.

M. Dressel and M. Scheffler, “Verifying the Drude response,” Ann. Phys. 15, 535–544 (2006).
[CrossRef]

Schmuttenmaer, C.

C. Schmuttenmaer, “Exploring dynamics in the far-infrared with terahertz spectroscopy,” Chem. Rev. 104, 1759–1780 (2004).
[CrossRef] [PubMed]

M. Beard and C. Schmuttenmaer, “Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments,” J. Chem. Phys. 114, 2903–2909(2001).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
[CrossRef]

J. Kindt and C. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589–8596 (1999).
[CrossRef]

Shah, J.

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag, 1999).

Sipe, J.

Stanton, C.

C. Stanton and D. Bailey, “Rate equations for the study of femtosecond intervalley scattering in compound semiconductors,” Phys. Rev. B 45, 8369–8377 (1992).
[CrossRef]

Surendran, S.

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005).

Tauser, F.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Turner, G.

M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
[CrossRef]

van Driel, H.

Zhang, J.

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

Ann. Phys. (1)

M. Dressel and M. Scheffler, “Verifying the Drude response,” Ann. Phys. 15, 535–544 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

G. Gallot, J. Zhang, R. McGowan, T. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74, 3450–3452 (1999).
[CrossRef]

Chem. Rev. (1)

C. Schmuttenmaer, “Exploring dynamics in the far-infrared with terahertz spectroscopy,” Chem. Rev. 104, 1759–1780 (2004).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[CrossRef]

M. Beard, G. Turner, and C. Schmuttenmaer, “Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy,” J. Appl. Phys. 90, 5915–5923 (2001).
[CrossRef]

J. Chem. Phys. (4)

H. Němec, F. Kadlec, S. Surendran, P. Kužel, and P. Jungwirth, “Ultrafast far-infrared dynamics probed by terahertz pulses: a frequency domain approach. I. model systems,” J. Chem. Phys. 122, 104503 (2005).
[CrossRef] [PubMed]

J. Kindt and C. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589–8596 (1999).
[CrossRef]

M. Beard and C. Schmuttenmaer, “Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments,” J. Chem. Phys. 114, 2903–2909(2001).
[CrossRef]

H. Němec, F. Kadlec, and P. Kužel, “Methodology of an optical pump-terahertz probe experiment: an analytical frequency-domain approach,” J. Chem. Phys. 117, 8454–8466 (2002).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photon. Rev. 5, 124–166 (2011).
[CrossRef]

Nature (2)

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

R. A. Kaindl, M. A. Carnahan, D. Hagele, R. Lovenich, and D. S. Chemla, “Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas,” Nature 423, 734–738 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (2)

M. Beard, G. Turner, and C. Schmuttenmaer, “Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy,” Phys. Rev. B 62, 15764–15777(2000).
[CrossRef]

C. Stanton and D. Bailey, “Rate equations for the study of femtosecond intervalley scattering in compound semiconductors,” Phys. Rev. B 45, 8369–8377 (1992).
[CrossRef]

Other (3)

C. Larsen, D. G. Cooke, and P. U. Jepsen, “A numerical implementation in MATLAB,” available online at http://www.physics.mcgill.ca/~cooke/ and http://www.terahertz.dk (2011).

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005).

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag, 1999).

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Figures (7)

Fig. 1
Fig. 1

Time lines of the three pulses involved in TRTS: pump, probe, and sampling pulses. Each pulse has been designated an absolute time τ, and the definition of relative times t are shown in the bottom. The optical pump pulse of less than 50 fs excites a carrier density in the sample that decays over time, which is illustrated at the pump time line. The THz probe pulse is shown with a full-width at half-maximum of 300 fs of the central peak in the electric field.

Fig. 2
Fig. 2

Illustration of the numerical setup. A source injects the THz probe pulse, a pump excites carriers in the sample, and detectors measure reflection and transmission. PML are the absorbing perfectly matched layer boundary conditions.

Fig. 3
Fig. 3

Amplitude and phase of the total response function | f det f apr | and atan ( { f det f apr } / { f det f apr } ) , respectively. The amplitude of the individual response functions of the aperture and detector using Eqs. (35, 37) is also shown. The THz refractive index of ZnTe is measured, and the nonlinearity is modeled as χ ( 2 ) n ( Ω ) 2 1 [14]. The aperture radius or sampling beam waist is 200 μm .

Fig. 4
Fig. 4

Measured and simulated electric field of the THz probe pulses before ( t p = 2 ps ) and after ( t p = 3 ps ) optical pumping. The GaAs sample is excited with 800 nm , 7 μJ / cm 2 , 45 fs pulses.

Fig. 5
Fig. 5

Real ( σ 1 ) and imaginary ( σ 2 ) transient conductivity of the experiment and the simulation. Before extracting the experimental conductivity in (a) and (b), the response functions have been deconvoluted in order to be directly comparable to the simulation results in (c) and (d). The GaAs sample is excited with 800 nm , 7 μJ / cm 2 , 45 fs pulses. The simulation parameters are given in Section 4.

Fig. 6
Fig. 6

Real part of conductivity σ 1 for (a) experiment, (b) simulation convoluted with the detector and aperture response functions before reconstruction and Fourier transformation, (c) as (b) convoluted with the detector function only, and (d) as (b) convoluted with the aperture function only. The GaAs sample is excited with 800 nm , 7 μJ / cm 2 , 45 fs pulses. The simulation parameters are given in Section 4.

Fig. 7
Fig. 7

Imaginary part of conductivity σ 2 for (a) experiment, (b) simulation convoluted with the detector and aperture response functions before reconstruction and Fourier transformation, (c) as (b) convoluted with the detector function only, and (d) as (b) convoluted with the aperture function only. The GaAs sample is excited with 800 nm , 7 μJ / cm 2 , 45 fs pulses. The simulation parameters are given in Section 4.

Tables (1)

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Table 1 Corrective Terms to Be Used in Eqs. (30, 31) a

Equations (37)

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t p = τ sampling τ pump = t p p t ,
Δ P ( t , t p ) = E ( t t ) χ ˜ ( t , t p ) d t ,
χ ( Ω , t p ) = Δ P ( Ω , t p ) / E ( Ω ) ,
Δ P * ( t , t p p ) = E ( t t ) χ ˜ ( t , t p p t ) d t ,
E z = μ 0 H t ,
H z = ϵ 0 ϵ E t + P Ω t ,
P Ω t = e p ( x p N p t + N p x p t ) ,
2 x p ( t ) t 2 + γ p ( t ) x p ( t ) t + ω 0 p ( t ) 2 x p ( t ) = e m p * E ( t ) ,
N p t = y p I γ p b N p + z { D p N p z } + q p P ( γ p q N q γ q p N p ) ,
I ( z , t ) = N i δ λ p π Δ t p exp { z δ λ p } × exp { ( z c 0 / n g ( t t p p ) ) 2 ( c 0 Δ t p / n g ) 2 } ,
E z | i n E i + 1 / 2 n E i 1 / 2 n h ,
Q = c 0 δ t / h 1 ,
2 δ t D h 2 < 1 , δ t s 2 h < 1 .
P Ω t | i n + 1 / 2 = e δ t ( N p i n + 1 x p i n + 1 N p i n x p i n ) .
x p i n + 1 = α p i n x p i n + β p i n x p i n 1 θ p i n E i n ,
α p i n = 4 2 ( ω 0 p i n ) 2 δ t 2 γ p i n δ t + 2 ,
β p i n = γ p i n δ t 2 γ p i n δ t + 2 ,
θ p i n = 2 e δ t 2 m p * ( γ p i n δ t + 2 ) .
N p i n + 1 = N p i n + δ t N p t | i n
= δ t { N p i n ( 1 δ t γ p i b 2 D p i n h 2 ) + q p [ γ p q N q i n γ q p N p i n ] + y p I i n + D p i n h 2 ( N p i + 1 n + N p i 1 n ) } .
I i n = 2 π F λ p ( 1 R ) c 0 δ λ p 2 π Δ t p exp { i h z 0 δ λ p } × exp { [ i h z 0 c 0 / n g ( n δ t + τ pump τ probe ) ] 2 ( c 0 Δ t p / n g ) 2 } ,
N p i L n + 1 = δ t { N p i L n ( 1 δ t s p n h γ p i L b D p i L n h 2 ) + q p [ γ p q N q i L n γ q p N p i L n ] + y p I i L n + D p i L n h 2 N p i L + 1 n } ,
E z = μ 0 H t + σ * H + j H ,
H z = ϵ 0 ϵ E t + ϵ 0 P Ω t + σ E + j E ,
A i = σ i δ t 2 ϵ i = σ i * δ t 2 μ i = A max ( i A h ) A m .
H ˜ i + 1 / 2 n + 1 / 2 ϵ 0 / μ 0 H i n .
H i n + 1 = 1 A i + 1 / 2 1 + A i + 1 / 2 { H i n Q ( E i + 1 n + 1 E i n + 1 ) 1 A i + 1 / 2 } + Q j E i n ,
E i n + 1 = 1 A i 1 + A i { E i n Q ϵ i H i n H i 1 n 1 A i } + e ϵ 0 ϵ i p ( N p i n + 1 x p i n + 1 N p i n x p i n ) + j E i n ,
x ˜ p i n = N p i n x p i n + 0 · I i n δ t N p i n + I i n δ t .
x ˜ p i n = N p i n x p i n + ϕ Δ N ϕ i n x ϕ i n N p i n + 1 ,
x ˜ p i n 1 = x p i n 1 + ϕ Δ N ϕ i n { x ϕ i n ( 1 m ϕ * m p * ) + m ϕ * m p * x ϕ i n 1 } N p i n + 1 .
T ( Ω , t p ) = E pump ( Ω , t p ) E ref ( Ω , t p ) = | T ( Ω , t p ) | e i Φ ( Ω , t p ) ,
σ 1 ( Ω ) = n ( Ω ) + 1 Z 0 δ λ p ( cos Φ | T | 1 ) ,
σ 2 ( Ω ) = n ( Ω ) + 1 Z 0 δ λ p sin Φ | T | ,
f d e t ( Ω ) χ ( 2 ) ( ω 0 , Ω ) e i Δ k ( ω 0 , Ω ) d 1 i Δ k ( ω 0 , Ω ) 2 1 + n ( Ω ) × A * ( ω ) A ( ω Ω ) d ω ,
Δ k ( ω 0 , Ω ) Ω { n g ( ω 0 ) n ( Ω ) } c 0 ,
f apr ( Ω ) = erf 2 ( Ω R 2 c 0 ) ,

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