Abstract

We describe the use of mixing linearly chirped optical pulses in biased photoconductors to generate tunable narrow-band terahertz (THz) radiation with enhanced spectral brightness. The increase in conversion efficiency from optical to THz radiation at a given THz frequency arises from the improved saturation characteristics of the photoconductor for chirped-pulse mixing compared with the usual case of excitation by an ultrafast optical pulse. In the weak saturation limit, the enhancement in the saturation fluence scales with the ratio of the duration of the chirped optical pulse to the photocurrent relaxation time in the emitter and is essentially independent of the beat frequency generated by the chirped-pulse mixing technique. This dependence allows for substantial enhancements in the saturation fluence and, hence, in the THz spectral brightness. We demonstrate enhanced saturation fluences experimentally for dipole emitters fabricated on radiation-damaged Si on sapphire.

© 1999 Optical Society of America

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    [CrossRef]

1997

1996

1995

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

R. A. Cheville and D. R. Grischkowsky, “Time-domain terahertz impulse ranging studies,” Appl. Phys. Lett. 67, 1960 (1995).
[CrossRef]

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

1994

P. K. Benicewicz, J. P. Roberts, and A. J. Taylor, “Scaling of terahertz radiation from large-aperture biased photoconductors,” J. Opt. Soc. Am. B 11, 2533 (1994).
[CrossRef]

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

1993

A. J. Taylor, P. K. Benicewicz, and S. M. Roberts, “Modeling of femtosecond electromagnetic pulses from large-aperture photoconductors,” Opt. Lett. 18, 1340 (1993).
[CrossRef] [PubMed]

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

1992

J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, “Saturation properties of large-aperture photoconducting antennas,” IEEE J. Quantum Electron. 28, 1607 (1992).
[CrossRef]

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

1990

M. Van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

B. B. Hu, X.-C. Zhang, and D. H. Auston, “Free space radiation from electro-optic crystals,” Appl. Phys. Lett. 56, 506 (1990).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

1988

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

1987

F. Doany, D. Grischkowsky, and C.-C. Chi, “Carrier lifetime versus ion-implantation dose in silicon-on-sapphire,” Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

1986

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics: I & II,” Phys. Rev. B 33, 6954 (1986).
[CrossRef]

Auston, D. H.

A. S. Weling and D. H. Auston, “Novel sources and detectors for coherent tunable narrowband terahertz radiation in free space,” J. Opt. Soc. Am. B 13, 2783 (1996).
[CrossRef]

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, “Saturation properties of large-aperture photoconducting antennas,” IEEE J. Quantum Electron. 28, 1607 (1992).
[CrossRef]

B. B. Hu, X.-C. Zhang, and D. H. Auston, “Free space radiation from electro-optic crystals,” Appl. Phys. Lett. 56, 506 (1990).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Benicewicz, P. K.

Birkelund, K.

R. H. Jacobsen, K. Birkelund, T. Holst, P. Uhd Jepsen, and S. R. Keiding, “Interpretation of photocurrent correlation measurements used for ultrafast photoconductive switch characterization,” J. Appl. Phys. 79, 2649 (1996).
[CrossRef]

Brown, E. R.

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

Bucksbaum, P. H.

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

Cheville, R. A.

R. A. Cheville and D. R. Grischkowsky, “Time-domain terahertz impulse ranging studies,” Appl. Phys. Lett. 67, 1960 (1995).
[CrossRef]

Chi, C.-C.

F. Doany, D. Grischkowsky, and C.-C. Chi, “Carrier lifetime versus ion-implantation dose in silicon-on-sapphire,” Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Chuang, S. L.

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

Darrow, J. T.

J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, “Saturation properties of large-aperture photoconducting antennas,” IEEE J. Quantum Electron. 28, 1607 (1992).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

Dennis, C. L.

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

Doany, F.

F. Doany, D. Grischkowsky, and C.-C. Chi, “Carrier lifetime versus ion-implantation dose in silicon-on-sapphire,” Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Dykaar, D. R.

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

Froberg, N. M.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

Greene, B. I.

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

Grischkowsky, D.

M. Van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

F. Doany, D. Grischkowsky, and C.-C. Chi, “Carrier lifetime versus ion-implantation dose in silicon-on-sapphire,” Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Grischkowsky, D. R.

R. A. Cheville and D. R. Grischkowsky, “Time-domain terahertz impulse ranging studies,” Appl. Phys. Lett. 67, 1960 (1995).
[CrossRef]

Holst, T.

R. H. Jacobsen, K. Birkelund, T. Holst, P. Uhd Jepsen, and S. R. Keiding, “Interpretation of photocurrent correlation measurements used for ultrafast photoconductive switch characterization,” J. Appl. Phys. 79, 2649 (1996).
[CrossRef]

Hu, B. B.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

B. B. Hu, X.-C. Zhang, and D. H. Auston, “Free space radiation from electro-optic crystals,” Appl. Phys. Lett. 56, 506 (1990).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

Hvam, J. M.

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Jacobsen, R. H.

R. H. Jacobsen, K. Birkelund, T. Holst, P. Uhd Jepsen, and S. R. Keiding, “Interpretation of photocurrent correlation measurements used for ultrafast photoconductive switch characterization,” J. Appl. Phys. 79, 2649 (1996).
[CrossRef]

Jones, R. R.

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

Keiding, S. R.

R. H. Jacobsen, K. Birkelund, T. Holst, P. Uhd Jepsen, and S. R. Keiding, “Interpretation of photocurrent correlation measurements used for ultrafast photoconductive switch characterization,” J. Appl. Phys. 79, 2649 (1996).
[CrossRef]

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Keilmann, F.

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics: I & II,” Phys. Rev. B 33, 6954 (1986).
[CrossRef]

Lindelof, P. E.

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Lyssenko, V. G.

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Macintosh, K. A.

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

Mayer, A.

A. Mayer and F. Keilmann, “Far-infrared nonlinear optics: I & II,” Phys. Rev. B 33, 6954 (1986).
[CrossRef]

Morse, J. D.

J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, “Saturation properties of large-aperture photoconducting antennas,” IEEE J. Quantum Electron. 28, 1607 (1992).
[CrossRef]

Nichols, K. B.

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

Nuss, M. C.

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Pedersen, J. E.

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Raman, C.

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

Roberts, J. P.

Roberts, S. M.

Rodrigues, G.

Rodriguez, G.

Saeta, P. N.

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

Schmitt-Rink, S.

B. I. Greene, P. N. Saeta, D. R. Dykaar, S. Schmitt-Rink, and S. L. Chuang, “Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,” IEEE J. Quantum Electron. 28, 2302 (1992).
[CrossRef]

Smith, P. R.

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Some, D.

Sorensen, C. B.

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Taylor, A. J.

Tielking, N. E.

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

Uhd Jepsen, P.

R. H. Jacobsen, K. Birkelund, T. Holst, P. Uhd Jepsen, and S. R. Keiding, “Interpretation of photocurrent correlation measurements used for ultrafast photoconductive switch characterization,” J. Appl. Phys. 79, 2649 (1996).
[CrossRef]

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

Van Exter, M.

M. Van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684 (1990).
[CrossRef]

Weling, A. S.

A. S. Weling and D. H. Auston, “Novel sources and detectors for coherent tunable narrowband terahertz radiation in free space,” J. Opt. Soc. Am. B 13, 2783 (1996).
[CrossRef]

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

You, D.

R. R. Jones, N. E. Tielking, D. You, C. Raman, and P. H. Bucksbaum, “Ionization of oriented Rydberg states by sub-picosecond half-cycle electromagnetic pulses,” Phys. Rev. A 51, R2687 (1995).
[CrossRef]

Zhang, X.-C.

J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, “Saturation properties of large-aperture photoconducting antennas,” IEEE J. Quantum Electron. 28, 1607 (1992).
[CrossRef]

B. B. Hu, X.-C. Zhang, and D. H. Auston, “Free space radiation from electro-optic crystals,” Appl. Phys. Lett. 56, 506 (1990).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

Appl. Phys. Lett.

B. B. Hu, X.-C. Zhang, and D. H. Auston, “Free space radiation from electro-optic crystals,” Appl. Phys. Lett. 56, 506 (1990).
[CrossRef]

B. B. Hu, J. T. Darrow, X.-C. Zhang, D. H. Auston, and P. R. Smith, “Optically steerable photoconducting antennas,” Appl. Phys. Lett. 56, 886 (1990).
[CrossRef]

R. A. Cheville and D. R. Grischkowsky, “Time-domain terahertz impulse ranging studies,” Appl. Phys. Lett. 67, 1960 (1995).
[CrossRef]

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrowband THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
[CrossRef]

F. Doany, D. Grischkowsky, and C.-C. Chi, “Carrier lifetime versus ion-implantation dose in silicon-on-sapphire,” Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

J. E. Pedersen, V. G. Lyssenko, J. M. Hvam, P. Uhd Jepsen, S. R. Keiding, C. B. Sorensen, and P. E. Lindelof, “Ultrafast local field dynamics in photoconductive THz antennas,” Appl. Phys. Lett. 62, 1265 (1993).
[CrossRef]

E. R. Brown, K. A. Macintosh, K. B. Nichols, and C. L. Dennis, “Photomixing up to 3.8 THz in low-temperature-grown GaAs,” Appl. Phys. Lett. 66, 285 (1995).
[CrossRef]

IEEE J. Quantum Electron.

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

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[CrossRef]

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Y. Liu, S.-G. Paek, and A. M. Weiner, “Enhancement of narrow-band terahertz radiation from photoconducting antennas by optical pulse shaping,” Opt. Lett. 21, 1762 (1996); Y. Liu, S.-G. Paek, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709 (1996).
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D. You, R. R. Jones, P. H. Bucksbaum, and D. R. Dykaar, “Generation of high-power sub-single-cycle 500-fs electromagnetic pulses,” Opt. Lett. 18, 290 (1993); E. Budiarto, J. Margolies, S. Jeong, J. Son, and J. Bokor, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. 32, 1839 (1996).
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Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of THz beams,” Appl. Phys. Lett. 67, 3523 (1995); A. Nahata, D. H. Auston, T. F. Heinz, and C. Wu, “Coherent detection of freely propagating terahertz radiation by electro-optic sampling in a poled polymer,” Appl. Phys. Lett. 68, 150 (1996); P. Uhd Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E PLEEE8 53, R3052 (1996).
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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of the enhancement in THz spectral irradiance from an enhanced saturation fluence for the case of narrow-band excitation. The THz output is illustrated for (a) the low-fluence case in which chirped-pulse mixing is equivalent to spectral filtering of the broadband THz output and (b) the high-fluence case in which broadband output is saturated while narrow-band saturation is just incipient. The saturation fluences for broadband (Fbbsat) and narrow-band (Fnbsat) excitation are discussed in Subsection 3.B of the text.

Fig. 2
Fig. 2

Results of numerical calculations of the variation of THz spectral irradiance S(Ω) with absorbed optical fluence for three frequencies Ω within the emission spectrum of a large-aperture RDSOS photoconducting emitter for broadband (dotted curves) and narrow-band (solid curves) excitation. The broadband excitation pulses are Gaussian with Tbb=100 fs (FWHM), and the narrow-band excitation pulses are obtained by mixing of linearly chirped Gaussian pulses with Tnb=100 ps (FWHM). For these calculations, based on the saturation model described in the text, the following parameters were assumed: τr=0.27 ps, τd=0.6 ps, μdc=30 cm2/V s.

Fig. 3
Fig. 3

Results of numerical calculations for narrow-band saturation fluence Fsatnb (fluence at 50% deviation from ideal behavior): (a) Dependence of Fsatnb on photocurrent relaxation time τd for mixing of 100-ps-long (FWHM) chirped Gaussian pulses at beat frequencies Ω of 100 and 500 GHz and 1.0 THz. (b) Dependence of Fsatnb on THz beat frequency Ω for chirped Gaussian pulses with values of τd and T shown. (c) Dependence of Fsatnb on the duration (FWHM) of the chirped Gaussian pulses for τd=0.6 ps at the three THz beat frequencies given in (a).

Fig. 4
Fig. 4

Calculated enhancement in spectral irradiance S(Ω) at the three THz beat frequencies of Fig. 3(a) of the narrow-band THz output obtained by mixing of 100-ps-long chirped Gaussian pulses compared with the linearly filtered broadband THz output obtained from a single 100-fs Gaussian pulse. This enhancement is determined for a pump fluence yielding a 50% deviation from ideal behavior in the narrow-band case (Fsatnb) and is plotted as a function of photocurrent relaxation time τd. The prediction of the simple analytical theory presented in Subsection 3.B is also shown.

Fig. 5
Fig. 5

Experimental results of the variation of the THz spectral irradiance S(Ω) with absorbed optical fluence for (a) broadband (∼110-fs FWHM pulses) and (b) narrow-band (obtained by mixing of two 100-ps FWHM chirped pulses with a variable delay) excitation at 800 nm of RDSOS dipole emitters. Dashed curves, values of S(Ω) calculated numerically from the model for saturation described in text. The scale of the numerically calculated curves has been adjusted to match the experimental results at low fluence.

Equations (41)

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PNL(Ω)=χ(2)(Ω):-dωEin(ω+Ω)Ein*(ω).
PNL(Ω)=χ(2)(Ω):-dωEin(ω+Ω)×exp[ j(ω-ω0+Ω)2/2µ]Ein*(ω)×exp[-j(ω-ω0)2/2µ-jτ(ω-ω0)].
S(Ω)=R(cε0/2)|E(Ω)|2.
N(t)=βω-tdtIin(t)exp-(t-t)τd,
E(t)=-η01+nJ(t)=-ηJ(t).
J(t)=σ(t)Ebσ(t)η+1.
σ(t)=e-tdtμ(t-t)N(t),
σ(t)=K-tdt exp[-(t-t)/τr]τr-tdt×exp[-(t-t)/τd]I(t).
F(t)=F0(t)+Re[FΩ(t)exp(jΩt)],
I(t)=I0(t)+Re[IΩ(t)exp(jΩt)].
I(t)=I0(t)+I0(t+τ)+2[I0(t)I0(t+τ)]1/2 cos Ωt.
I0(t)=I0(t)+I0(t+τ),
IΩ(t)=2[I0(t)I0(t+τ)]1/2.
σ(t)KI0(t)τd+Kτd ReIΩ(t)exp( jΩt)(1+jΩτd)(1+jΩτr)=σ0(t)+Re[σΩ(t)exp( jΩt)].
τdΩτd(1+jΩτr)(1+jΩτd).
θSi(Ω)-S(Ω)Si(Ω),
S=R(cε0/2)-|EΩ(t)|2dt,
θSi(Ω)-S(Ω)Si(Ω)Si-SSiΩ=-dt[|EiΩ(t)|2-|EΩ(t)|2]-dt|EiΩ(t)|2.
J(t)=Ji(t)+ΔJ(t),
ΔJ(t)-Ebη[σ(t)]2=-Ebη{σ0(t)+Re[σΩ(t)exp(jΩt)]}2.
ΔJΩ(t)-2ησΩ(t)σ0(t)Eb=-2ηK2τdτdΩI0(t)IΩ(t)Eb
JiΩ(t)=σΩ(t)Eb=KIΩ(t)τdΩEb.
θ=-dt[|JiΩ(t)|2-|JΩ(t)|2]-dt|JiΩ(t)|22 -dt Re{[JiΩ(t)]*ΔJΩ(t)}-dt|JiΩ(t)|2.
θ=4Kτdη -I0(t)|IΩ(t)|2dt-|IΩ(t)|2dt.
Fsatbb=(2-1)Kη=(2-1)1+nη0 ωeμdc.
Fnb=-dtI(t)=-dtI0(t).
θ=12FnbFsatbbτdTeff.
Teff=(2+1)-|IΩ(t)|2dt-I0(t)dt8-I0(t)|IΩ(t)|2dt.
Teff=(2+1)-I0(t)I0(t+τ)dt-I0(t)dt4-[I02(t)I0(t+τ)+I02(t+τ)I0(t)]dt.
Fsatnb=Fsatbb(Teff/τd).
αFsatnb/Fsatbb=Teff/τd,
I(t)=I0 exp(-t2/T2)(1+cos Ωt)=I0 exp(-t2/T2)+Re[I0 exp(-t2/T2)exp(jΩt)].
Teff=(2+1)83π2T0.655T.
SsatbbSbb|F=Fsatbb=(1/2)Sibb|F=Fsatbb.
SsatnbSnb|F=Fsatnb=(1/2)Sinb|F=Fsatnb.
Ssatnb/Ssatbb=(Fsatnb/Fsatbb)2=α2=(Teff/τd)2.
ηsatnbηsatbbSsatnb/FsatnbSsatbb/Fsatbb=α=Teffτd.
Sbb|F=FnbsatSmaxbb=Ssatbb22-1211.66Ssatbb.
SnbSbbF=Fsatnb=SsatnbSsatbbSsatbbSbb|F=Fsatnbα211.66.
Finc=--dxdy[F(x, y)]2F(x, y)--dxdy[F(x, y)]2,
Finc=(2/3)F0=(2/3)(Einc/πρ2),

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