Abstract

A theory of Cherenkov phase-matched monochromatic terahertz (THz)-wave generation via difference frequency generation in a nonlinear crystal is developed. An experimental situation (LiNbO3 pumped by dual-wavelength near-infrared nanosecond pulses) is considered. This theory accounts for the finite size of pump beam and allows us to explore the generation of transverse THz wave vector. The output characteristic of this THz source is analyzed based on the analytical expression, including radiation pattern, conversion efficiency, and tuning range. Calculated tuning curves are presented, which reasonably agree with previous experimental results. The influence of divergence of the focused pump beam on total radiated energy is studied in detail. Optimal pump beam size that maximizing generated THz energy is obtained.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005).
    [CrossRef]
  5. J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
    [CrossRef]
  6. J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
    [CrossRef]
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    [CrossRef]
  10. A. G. Stepanov, J. Hebling, and J. Kuhl, “THz generation via optical rectification with ultrashort laser pulse focused to a line,” Appl. Phys. B 81, 23–26 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express 17, 6676–6681 (2009).
    [CrossRef]
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  16. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003), Chap. 1.
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    [CrossRef]
  24. M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus Cherenkov effect,” J. Appl. Phys. 104, 073105(2008).
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    [CrossRef]
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    [CrossRef]

2011 (1)

P. X. Liu, D. G. Xu, C. M. Liu, D. Lv, Y. J. Lv, P. Wang, and J. Q. Yao, “P-polarized Cherenkov THz wave radiation generated by optical rectification for a Brewster-cut LiNbO3 crystal,” J. Opt. 13, 085202 (2011).
[CrossRef]

2010 (2)

T. Shibuya, K. Suizu, and K. Kawase, “Widely tunable monochromatic Cherenkov phase-matched Terahertz wave generation from bulk lithium niobate,” Appl. Phys. Express 3, 082201 (2010).
[CrossRef]

K. Suizu, T. Shibuya, H. Uchida, and K. Kawase, “Prism-coupled Cherenkov phase-matched terahertz wave generation using a DAST crystal,” Opt. Express 18, 3338–3344 (2010).
[CrossRef]

2009 (2)

K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express 17, 6676–6681 (2009).
[CrossRef]

T. Shibuya, T. Tsutsui, K. Suizu, T. Akiba, and K. Kawase, “Efficient Cherenkov-type phase-matched widely tunable THz-wave generation via an optimized pump beam shape,” Appl. Phys. Express 2, 032302 (2009).
[CrossRef]

2008 (2)

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus Cherenkov effect,” J. Appl. Phys. 104, 073105(2008).
[CrossRef]

K. Suizu, T. Tutui, T. Shibuya, T. Akiba, and K. Kawase, “Cherenkov phase-matched monochromatic THz-wave generation using difference frequency generation with lithium niobate crystal,” Opt. Express 16, 7493–7498 (2008).
[CrossRef]

2007 (4)

K. Reimann, “Table-top sources of ultrashort THz pulses,” Rep. Prog. Phys. 70, 1597–1632 (2007).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photon. 1, 97–105 (2007).
[CrossRef]

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76, 085346 (2007).
[CrossRef]

2006 (1)

2005 (3)

Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005).
[CrossRef]

A. G. Stepanov, J. Hebling, and J. Kuhl, “THz generation via optical rectification with ultrashort laser pulse focused to a line,” Appl. Phys. B 81, 23–26 (2005).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Cherenkov radiation of THz surface plasmon polaritons from a superluminal optical spot,” Phys. Rev. B 72, 195336 (2005).
[CrossRef]

2004 (1)

2003 (1)

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

1997 (1)

1984 (1)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

1977 (1)

J. R. Morris and Y. R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–1156 (1977).
[CrossRef]

1970 (1)

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cherenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450(1970).
[CrossRef]

1968 (1)

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

Akiba, T.

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Avetisyan, Y.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005).
[CrossRef]

Bakunov, M. I.

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus Cherenkov effect,” J. Appl. Phys. 104, 073105(2008).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76, 085346 (2007).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Cherenkov radiation of THz surface plasmon polaritons from a superluminal optical spot,” Phys. Rev. B 72, 195336 (2005).
[CrossRef]

Beigang, R.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Bodrov, S. B.

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus Cherenkov effect,” J. Appl. Phys. 104, 073105(2008).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76, 085346 (2007).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Cherenkov radiation of THz surface plasmon polaritons from a superluminal optical spot,” Phys. Rev. B 72, 195336 (2005).
[CrossRef]

Boyd, G. D.

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

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Ding, Y. J.

Fernelius, N.

Feurer, T.

Hangyo, M.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76, 085346 (2007).
[CrossRef]

Hebling, J.

A. G. Stepanov, J. Hebling, and J. Kuhl, “THz generation via optical rectification with ultrashort laser pulse focused to a line,” Appl. Phys. B 81, 23–26 (2005).
[CrossRef]

Ito, H.

Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005).
[CrossRef]

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

Jundt, D.

Karino, K.

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

Kawase, K.

K. Suizu, T. Shibuya, H. Uchida, and K. Kawase, “Prism-coupled Cherenkov phase-matched terahertz wave generation using a DAST crystal,” Opt. Express 18, 3338–3344 (2010).
[CrossRef]

T. Shibuya, K. Suizu, and K. Kawase, “Widely tunable monochromatic Cherenkov phase-matched Terahertz wave generation from bulk lithium niobate,” Appl. Phys. Express 3, 082201 (2010).
[CrossRef]

K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express 17, 6676–6681 (2009).
[CrossRef]

T. Shibuya, T. Tsutsui, K. Suizu, T. Akiba, and K. Kawase, “Efficient Cherenkov-type phase-matched widely tunable THz-wave generation via an optimized pump beam shape,” Appl. Phys. Express 2, 032302 (2009).
[CrossRef]

K. Suizu, T. Tutui, T. Shibuya, T. Akiba, and K. Kawase, “Cherenkov phase-matched monochromatic THz-wave generation using difference frequency generation with lithium niobate crystal,” Opt. Express 16, 7493–7498 (2008).
[CrossRef]

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

Kimura, T.

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

Kitamura, K.

N. E. Yu, K.-S. Lee, D.-K. Ko, S. Takekawa, and K. Kitamura, “Fine tuning terahertz generation in fanned-out periodically poled stoichiometric lithium tantalate crystal,” in Proceedings of IEEE Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz) (IEEE, 2010).

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

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

Ko, D.-K.

N. E. Yu, K.-S. Lee, D.-K. Ko, S. Takekawa, and K. Kitamura, “Fine tuning terahertz generation in fanned-out periodically poled stoichiometric lithium tantalate crystal,” in Proceedings of IEEE Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz) (IEEE, 2010).

Koketsu, K.

Kuhl, J.

A. G. Stepanov, J. Hebling, and J. Kuhl, “THz generation via optical rectification with ultrashort laser pulse focused to a line,” Appl. Phys. B 81, 23–26 (2005).
[CrossRef]

L’Huillier, J. A.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Lee, K.-S.

N. E. Yu, K.-S. Lee, D.-K. Ko, S. Takekawa, and K. Kitamura, “Fine tuning terahertz generation in fanned-out periodically poled stoichiometric lithium tantalate crystal,” in Proceedings of IEEE Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz) (IEEE, 2010).

Liu, C. M.

P. X. Liu, D. G. Xu, C. M. Liu, D. Lv, Y. J. Lv, P. Wang, and J. Q. Yao, “P-polarized Cherenkov THz wave radiation generated by optical rectification for a Brewster-cut LiNbO3 crystal,” J. Opt. 13, 085202 (2011).
[CrossRef]

Liu, P. X.

P. X. Liu, D. G. Xu, C. M. Liu, D. Lv, Y. J. Lv, P. Wang, and J. Q. Yao, “P-polarized Cherenkov THz wave radiation generated by optical rectification for a Brewster-cut LiNbO3 crystal,” J. Opt. 13, 085202 (2011).
[CrossRef]

Lv, D.

P. X. Liu, D. G. Xu, C. M. Liu, D. Lv, Y. J. Lv, P. Wang, and J. Q. Yao, “P-polarized Cherenkov THz wave radiation generated by optical rectification for a Brewster-cut LiNbO3 crystal,” J. Opt. 13, 085202 (2011).
[CrossRef]

Lv, Y. J.

P. X. Liu, D. G. Xu, C. M. Liu, D. Lv, Y. J. Lv, P. Wang, and J. Q. Yao, “P-polarized Cherenkov THz wave radiation generated by optical rectification for a Brewster-cut LiNbO3 crystal,” J. Opt. 13, 085202 (2011).
[CrossRef]

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cherenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450(1970).
[CrossRef]

Maslov, A. V.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B 76, 085346 (2007).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Cherenkov radiation of THz surface plasmon polaritons from a superluminal optical spot,” Phys. Rev. B 72, 195336 (2005).
[CrossRef]

Morris, J. R.

J. R. Morris and Y. R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–1156 (1977).
[CrossRef]

Nelson, K. A.

Nishizawa, J.

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

Rau, C.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Reimann, K.

K. Reimann, “Table-top sources of ultrashort THz pulses,” Rep. Prog. Phys. 70, 1597–1632 (2007).
[CrossRef]

Saito, K.

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

Sasaki, Y.

Shen, Y. R.

J. R. Morris and Y. R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–1156 (1977).
[CrossRef]

Shi, W.

Shibuya, T.

Shikata, J.

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

Small, D. L.

Statz, E. R.

Stepanov, A. G.

A. G. Stepanov, J. Hebling, and J. Kuhl, “THz generation via optical rectification with ultrashort laser pulse focused to a line,” Appl. Phys. B 81, 23–26 (2005).
[CrossRef]

Stoyanov, N. S.

Suizu, K.

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003), Chap. 2.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003), Chap. 1.

Suto, K.

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

Takekawa, S.

N. E. Yu, K.-S. Lee, D.-K. Ko, S. Takekawa, and K. Kitamura, “Fine tuning terahertz generation in fanned-out periodically poled stoichiometric lithium tantalate crystal,” in Proceedings of IEEE Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz) (IEEE, 2010).

Tanabe, T.

T. Tanabe, K. Suto, J. Nishizawa, K. Saito, and T. Kimura, “Tunable terahertz wave generation in the 3- to 7 THz region from GaP,” Appl. Phys. Lett. 83, 237–239 (2003).
[CrossRef]

Taniuchi, T.

J. Shikata, K. Kawase, K. Karino, T. Taniuchi, and H. Ito, “Tunable terahertz-wave parametric oscillators using LiNbO3 and MgO:LiNbO3 crystals,” IEEE Trans. Microwave Theory Tech. 48, 653–661 (2000).
[CrossRef]

Theuer, M.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Tien, P. K.

P. K. Tien, R. Ulrich, and R. J. Martin, “Optical second harmonic generation in form of coherent Cherenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450(1970).
[CrossRef]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photon. 1, 97–105 (2007).
[CrossRef]

Torosyan, G.

J. A. L’Huillier, G. Torosyan, M. Theuer, C. Rau, Y. Avetisyan, and R. Beigang, “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate—Part 2: Experiments,” Appl. Phys. B 86, 197–208 (2007).
[CrossRef]

Tsarev, M. V.

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus Cherenkov effect,” J. Appl. Phys. 104, 073105(2008).
[CrossRef]

Tsutsui, T.

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N. E. Yu, K.-S. Lee, D.-K. Ko, S. Takekawa, and K. Kitamura, “Fine tuning terahertz generation in fanned-out periodically poled stoichiometric lithium tantalate crystal,” in Proceedings of IEEE Conference on Infrared Millimeter and Terahertz Waves (IRMMW-THz) (IEEE, 2010).

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003), Chap. 1.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 2003), Chap. 2.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Geometry of Cherenkov THz radiation and (b) wave vector diagram.

Fig. 2.
Fig. 2.

Spatial distribution of THz electric field via Cherenkov-type DFG.

Fig. 3.
Fig. 3.

Tuning curves for different pump beam width. Triangles and squares are experimental data taken from [12] when the beam widths were 35 μm and 46 μm, respectively.

Fig. 4.
Fig. 4.

Distribution of THz intensity in z direction with different beam width for (a) 1 THz and (b) 1.5 THz generation (given pump intensity).

Fig. 5.
Fig. 5.

Curves of optimal pump beam width (FWHM) varying with output frequency for given pump energy (solid) and pump intensity (dashed).

Equations (10)

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Em(x,z,t)=E0exp(x2/r2)exp[i(kmzωmt)].
2ET(x,z,t)εc2ωT2ET(x,z,t)=μ0ωT2PNL(x,z,t),
ET(x,z,t)=exp(iωTt)×E˜T(g,κ)exp[i(gx+κz)]dg,
PNL(x,z,t)=2ε0d33E02exp[i(kgzωTt)]×G(g)exp(igx)dg,
exp(iωTt)×E˜T(g,κ)(g2+κ2εc2ωT2)exp[i(gx+κz)]dg=2μ0ε0d33E02ωT2exp[i(kgzωTt)]×G(g)exp(igx)dg.
E˜T(g)=2d33E02ωT2G(g)/c2g2+kg2εωT2/c2.
ET(x,z,t)=2d33E02G(g*)ωT2c22πg*sin(kgz+g*|x|ωTt).
IT=πε1/2r2μ0c3(εng2)d332E04ωT2exp(r2g*2/4).
Em(x,z,t)=E0[r0r(z)]1/2exp[x2r2(z)]exp[i(kmzωmt)].
IT(z)r02E04ωT2exp(r02g*24)×exp[λ2g*24π2r02nopt2(zz0)2].

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