Abstract

We explore optical-to-terahertz conversion efficiencies which can be achieved with femto- and picosecond optical pulses in electro-optic crystals with periodically inverted sign of second-order susceptibility. Optimal crystal lengths, pulse durations, pulse formats and focusing are regarded. We show that for sufficiently short (femtosecond) optical pulses, with a pulsewidth much shorter than the inverse terahertz frequency, conversion efficiency does not depend on pulse duration. We also show that by mixing two picosecond pulses (bandwidth-limited or chirped), one can achieve conversion efficiency, which is the same as in the case of femtosecond pulse with the same pulse energy. Additionally, when the group velocity dispersion of optical pulses is small, one can substantially exceed Manley‒Rowe conversion limit due to cascaded processes.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970).
    [Crossref]
  2. K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
    [Crossref]
  3. L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992)
    [Crossref]
  4. B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002).
    [Crossref]
  5. A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
    [Crossref]
  6. R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
    [Crossref]
  7. Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002).
    [Crossref]
  8. T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
    [Crossref]
  9. A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
    [Crossref] [PubMed]
  10. Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
    [Crossref]
  11. K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.
  12. A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.
  13. A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
    [Crossref]
  14. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).
  15. W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996).
    [Crossref]
  16. D. Grischkovsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [Crossref]
  17. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
    [Crossref]
  18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [Crossref]
  19. R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137.
    [Crossref]
  20. A. S. Weling, B. B. Hu, N.M. Froberg, and D.H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994).
    [Crossref]
  21. A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Ch.9.
  22. J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977).
    [Crossref]
  23. Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976)
    [Crossref]
  24. S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998).
    [Crossref]
  25. M. Cronin-Golomb, “Cascaded nonlinear difference-frequency generation of enhanced terahertz wave production,” Opt. Lett. 29, 2046–48 (2004).
    [Crossref] [PubMed]
  26. T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
    [Crossref]
  27. J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
    [Crossref]
  28. Y.J. Ding, “Efficient generation of high-power quasi-single-cycle terahertz pulses from a single infrared beam in a second-order nonlinear medium,” Opt. Lett. 29, 2650–52 (2004).
    [Crossref] [PubMed]
  29. A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
    [Crossref]
  30. B.S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984).
    [Crossref]
  31. M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
    [Crossref]

2005 (1)

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (1)

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

2002 (3)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002).
[Crossref]

Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002).
[Crossref]

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

2000 (1)

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

1999 (1)

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

1998 (1)

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998).
[Crossref]

1996 (2)

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
[Crossref]

W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996).
[Crossref]

1995 (2)

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

1994 (1)

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

1992 (2)

1991 (1)

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

1990 (1)

1984 (1)

1977 (1)

J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977).
[Crossref]

1976 (1)

Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976)
[Crossref]

1971 (1)

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

1970 (2)

T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970).
[Crossref]

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

1968 (1)

G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Almási, G.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Auston, D. H.

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992)
[Crossref]

Auston, D.H.

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

Bonvalet, A.

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

Boyd, G.D.

G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Byer, R.L.

R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137.
[Crossref]

Carrig, T. J.

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

Caumes, J.-P.

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

Chiao, R.Y.

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

Clement, T. S.

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

Cronin-Golomb, M.

Ding, Y.J.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

Eickemeyer, F.

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

Elsaesser, T.

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

Exter, M. van

Eyres, L. A.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Fattinger, Ch.

Fejer, M. M.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

Ferguson, B.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002).
[Crossref]

Freysz, E.

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

Froberg, N.M.

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

Galvanauskas, A.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

Grischkovsky, D.

Guha, S.

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998).
[Crossref]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

Gustafson, T.K.

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

Hagan, D.J.

A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

Harris, J. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Hebling, J.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Heinz, T. F.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
[Crossref]

Herbst, R.L.

R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137.
[Crossref]

Holm, R.T.

W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996).
[Crossref]

Hu, B. B.

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

Hutchings, D.C.

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

Joffre, M.

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

Kaindl, R. A.

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

Keiding, S.

Kelley, P.L.

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

Kleinman, D.A.

G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Kozlov, V.G.

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

Kozma, I. Z.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Kuhl, J.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Kuo, P. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Lee, Y.-S.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

Levi, O.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Martin, J.-L.

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

Meade, T.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

Migus, A.

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

Moore, W.J.

W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996).
[Crossref]

Morris, J.R.

J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977).
[Crossref]

Nahata, A.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
[Crossref]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

Norris, T. B.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

Perlin, V.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

Pinguet, T. J.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Richards, P.L.

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

Riedle, E.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Rodriguez, G.

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

Said, A.A.

Sheik-Bahae, M.

A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

Shen, Y.R.

J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977).
[Crossref]

Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976)
[Crossref]

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

Siegel, Peter H.

Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Ch.9.

Simanovskii, D. M.

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

Skauli, T.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

Stepanov, A. G.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Stryland, E.W. Van

A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

Takeuchi, N.

T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970).
[Crossref]

Taran, J.-P.E.

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

Taylor, A. J.

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

Touyez, C.

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

Videau, L.

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

Vodopyanov, K. L.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

Wang, J.

Wei, T.H.

Weling, A. S.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
[Crossref]

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

Wherrett, B.S.

Winful, H.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

Woerner, M.

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

Xu, L.

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992)
[Crossref]

Yajima, T.

T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970).
[Crossref]

Yang, K.H.

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

Yariv, A.

A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.

Young, J.

Zhang, X.-C.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002).
[Crossref]

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992)
[Crossref]

Appl. Phys. B (1)

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998).
[Crossref]

Appl. Phys. Lett. (8)

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

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992)
[Crossref]

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995).
[Crossref]

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999).
[Crossref]

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995).
[Crossref]

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000).
[Crossref]

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996).
[Crossref]

IEEE J. of Quant. Electron. (1)

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991).
[Crossref]

IEEE Transactions on Microwave Theory and Techniques (1)

Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002).
[Crossref]

J. Appl. Phys. (2)

W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996).
[Crossref]

G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

J. of Appl. Phys. (1)

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003).
[Crossref]

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

Jpn. J. Appl. Phys. (1)

T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970).
[Crossref]

Nature Materials (1)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002).
[Crossref]

Opt. Commun. (1)

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[Crossref]

Opt. Lett. (2)

Optics Express (1)

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005).
[Crossref] [PubMed]

Phys. Rev. A (1)

J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977).
[Crossref]

Phys. Rev. Letters (1)

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002).
[Crossref]

Prog. Quant. Electr. (1)

Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976)
[Crossref]

Other (5)

A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Ch.9.

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.

R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137.
[Crossref]

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

Supplementary Material (1)

» Media 1: AVI (1391 KB)     

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Illustration of optical rectification in a media with periodically inverted χ (2) sign, using femtosecond pump pulses. The static picture shows optical and THz electric fields, while the animation shows the nonlinear driving polarization and the THz electric field (779kB).

Fig. 2.
Fig. 2.

Reduction factor g 1 as a function of vTHzT for the case of femtosecond pump pulses (νTHz=Ω/2π).

Fig. 3.
Fig. 3.

Reduction factor g 2 as a function of l w/L for the case of picosecond pump pulses.

Fig. 4.
Fig. 4.

Scheme to generate tunable THZ radiation from the overlap of two linearly chirped pulses.

Fig. 5.
Fig. 5.

Enhancement factor h as a function of the focusing parameter ξ. Solid curve is based on ref. 22. Dashed curve ‒ plane-wave approximation. Dots represent our calculations based on the Green’s function method. Inset: far field THz intensity profiles at different ξ for a 1-cm-long GaAs.

Fig. 6.
Fig. 6.

Number of THz cascading cycles as a function of THz frequency and pump wavelength for GaAs, L=1cm.

Fig. 7.
Fig. 7.

(a) The optical pulse intensity profile, (b) rectified field profile, (c) phase profile across the optical pulse after traveling the crystal, and (d) frequency shift of the optical pulse.

Equations (51)

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

E ( t ) = Re { E 0 exp ( t 2 τ 2 ) exp [ i ( ω 0 t ) ] } = 1 2 { E 0 exp ( t 2 τ 2 ) exp [ i ( ω 0 t ) ] + c . c . } ,
f ( t ) = f ( ω ) exp ( iωt )
f ( ω ) = 1 2 π f ( t ) exp ( iωt ) dt ,
E ( ω ) = E 0 τ 2 π exp ( τ 2 ( ω ω 0 ) 2 4 )
E ω z = E ( ω ) exp [ ik ( ω ) z ] ,
dE Ω z dz = i μ 0 Ω c 2 n 1 P NL ( Ω ) exp ( i Δ kz ) ,
P NL ( Ω ) = ε 0 χ ( 2 ) E ( ω + Ω ) E * ( ω )
P NL ( Ω ) = ε 0 χ ( 2 ) E 0 2 τ 2 2 π exp ( τ 2 Ω 2 8 )
dE Ω z dz = i Ω χ ( 2 ) E 0 2 τ 4 2 π c n 1 exp ( τ 2 Ω 2 8 ) exp ( i Δ kz ) .
Δ k = k ( Ω ) + k ( ω ) k ( ω + Ω ) 2 π ʌ .
Δ k = Ω n THz c ( dk ) opt Ω 2 π ʌ = Ω c ( n THz n opt gr ) 2 π ʌ ,
E Ω L 2 = Ω 2 d eff 2 E 0 4 τ 2 8 π c 2 n 1 2 L 2 exp ( τ 2 Ω 2 4 ) sin c 2 ( Δ kL 2 ) .
Ω 0 = 2 πc Λ Δ n , λ THz = Λ Δn ,
Δ Ω accept = c Δ n Δ k accept = 2 πc L Δ n
l c = πc ΩΔ n ,
Ω c ( n THz + n opt gr ) 2 π Λ = 0
η THz PW = Fluence ( THz ) Fluence ( pump ) .
F pu = c ε 0 n 2 2 E t 0 2 dt = π 2 c ε 0 n 2 2 E 0 2 τ ,
F THz = c ε 0 n 1 2 E THz t L 2 dt = c ε 0 n 1 2 2 π 0 E Ω L 2 d Ω ,
η THz PW = Ω 2 d eff 2 E 0 2 τ 2 2 π c 2 n 1 n 2 L 2 0 exp ( τ 2 Ω 2 4 ) sinc 2 [ Δ n ( Ω Ω 0 ) L 2 c ] d Ω
η THz PW = g 1 2 Ω 0 3 d eff 2 π ε 0 c 3 n 1 n 2 2 Ll c F pu = g 1 2 Ω 0 3 d eff 2 L ε 0 c 3 n 1 n 2 2 Δ n F pu
g 1 = exp ( ( τ Ω 0 2 ) 2 ) = exp ( ( π ν THz τ ) 2 )
E i ( t ) = Re { E i exp ( t 2 τ 2 ) exp [ i ( ω i t ) ] } ,
E i ( ω ) = E i τ 2 π exp ( τ 2 ( ω ω i ) 2 4 ) .
P NL ( Ω ) = ε 0 χ ( 2 ) 0 E 3 ( ω + Ω ) E 2 * ( ω ) = ε 0 χ ( 2 ) E 2 E 3 τ 2 2 π exp ( τ 2 ( Ω Ω 0 ) 2 8 ) .
E Ω L 2 = Ω 2 d eff 2 E 2 2 E 3 2 τ 2 8 π c 2 n 1 2 L 2 exp ( τ 2 ( Ω Ω 0 ) 2 4 ) sinc 2 ( Δ kL 2 ) ,
η THz PW = Fluence ( THz ) Fluence ( ω 2 ) = Ω 2 d eff 2 E 3 2 τ 2 2 π c 2 n 1 n 2 L 2 0 exp ( τ 2 Ω' 2 4 ) sin c 2 [ Δ n Ω' L 2 c ] d Ω'
l w = π Δ n ,
η THz PW = Ω 2 d eff 2 E 3 2 τ 2 2 π c 2 n 1 n 2 L 2 0 exp ( τ 2 Ω' 2 4 ) sin c 2 [ π 2 ( L l w ) τ Ω' ] d Ω'
η THz = Ω 2 d eff 2 E 3 2 L 2 2 c 2 n 1 n 2 = 2 Ω 2 d eff 2 L 2 ε 0 c 2 n 1 n 2 n 3 I 3 2 ,
η THz = Ω 2 d eff 2 E 3 2 L l w 2 c 2 n 1 n 2 = 2 Ω 2 d eff 2 L l w ε 0 c 2 n 1 n 2 n 3 I 3 2 ,
η THz = 2 Ω 3 d eff 2 L l c π ε 0 c 3 n 1 n 2 n 3 F 3 ,
η THz = g 2 2 Ω 3 d eff 2 L l c π ε 0 c 3 n 1 n 2 n 3 F 3 = g 2 2 Ω 2 d eff 2 L ε 0 c 2 n 1 n 2 n 3 Δ n F 3 ,
g 2 ( x ) = 1 π 0 exp ( x 2 μ 2 π ) sin c 2 ( μ ) .
E 2 ( t ) = Re { E 0 exp ( t 2 τ 2 ) exp [ i ( ω 0 t + b t 2 ) ] }
E 3 ( t ) = E 2 ( t + Δ t ) = Re { E 0 exp ( ( t + Δ t ) 2 τ 2 ) exp [ i ( ω 0 ( t + Δ t ) + b ( t + Δ t ) 2 ) ] }
P NL ( t ) = Re { ε 0 χ ( 2 ) ( E 2 ( t ) + E 3 ( t ) ) 2 ω = 0 }
= Re { ε 0 χ ( 2 ) E 0 2 exp [ 2 τ 2 ( t 2 + Δ tt + Δ t 2 2 ) + i ( 2 b Δ tt + b Δ t 2 ) ] }
P NL ( Ω ) = ε 0 χ ( 2 ) E 0 2 2 π exp [ 2 τ 2 ( t 2 + Δ tt + Δ t 2 2 ) + i ( 2 b Δ tt + b Δ t 2 Ω t ) ] dt
= ε 0 χ ( 2 ) E 0 2 τ 2 2 π exp ( τ 2 Ω' 2 8 1 2 Δ t 2 τ 2 ) exp ( i φ 1 )
η THz PW = Ω 2 d eff 2 E 0 2 τ 2 2 π c 2 n 1 n 2 L 2 0 exp ( τ 2 Ω' 2 4 Δ t 2 τ 2 ) sin c 2 [ Δ n ( Ω' + Ω" ) L 2 c ] d Ω'
η THz = g 2 exp ( Δ t 2 τ 2 ) 2 Ω 2 d eff 2 Ll c π ε 0 c 3 n 1 n 2 n 3 F 3 ,
η THz = g 1 g 2 2 Ω 2 d eff 2 Ll c π ε 0 c 3 n 1 n 2 n 3 F 3 .
η THz ( L ) 1 α [ 1 exp ( αL ) ] = L eff ,
η THz ( L ) g 3 L ; g 3 = 1 αL [ 1 exp ( αL ) ]
η THz = U THz U pu = g 1 g 3 2 Ω 2 d eff 2 L ε 0 c 3 n 1 n 2 2 Δ n U pu π w 2 ,
η THz = U THz U pu = g 1 g 3 2 Ω 3 d eff 2 π ε 0 c 3 n 1 n 2 2 Δ n U pu h ( ξ ) ,
η THz = U THz U 2 = g 2 g 3 2 Ω 3 d eff 2 π ε 0 c 3 n 2 n 3 Δ n U 3 h ( ξ ) ,
Δ k z = k 1 z ( k 3 k 2 ) = k 1 cos θ ( k 3 k 2 ) Δ k 0 k 1 2 θ 2 = 0
ξ = λ 1 L 2 π n 1 w 2 < π .
Δ ω pump accept = 2 πc L ω pump Ω ( λ d n gr ) 1

Metrics