Abstract

To circumvent a velocity mismatch between optical pump and terahertz waves in electro-optic crystals, we propose to use dual-wavelength optical beams tilted with respect to their planes of equal amplitude. The tilt is achieved by transmission of a dual-wavelength laser beam through a diffraction grating placed on the crystal boundary. The proposed technique extends optical rectification of tilted-front femtosecond laser pulses to difference-frequency generation with longer (nanosecond) pulses. Our analysis of the technique for LiNbO3 pumped at 1.3 μm and GaAs pumped at 1.55 μm shows its efficiency.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
    [CrossRef]
  39. K. L. Vodopyanov, “Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: conversion efficiency and optimal laser pulse format,” Opt. Express14, 2263–2276 (2006).
    [CrossRef] [PubMed]
  40. M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
    [CrossRef]

2012 (1)

2011 (2)

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbibg crystals,” J. Opt. Soc. Am. B28, 1724–1734 (2011).
[CrossRef]

2010 (2)

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010).
[CrossRef] [PubMed]

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

2009 (2)

2008 (5)

2007 (2)

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
[CrossRef]

2006 (2)

K. L. Vodopyanov, “Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: conversion efficiency and optimal laser pulse format,” Opt. Express14, 2263–2276 (2006).
[CrossRef] [PubMed]

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

2005 (3)

2004 (7)

H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett.29, 1751–1753 (2004).
[CrossRef] [PubMed]

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19, 964–970 (2004).
[CrossRef]

T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys.95, 7588–7591 (2004).
[CrossRef]

W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett.84, 1635–11637 (2004).
[CrossRef]

N. S. Stoyanov, T. Feurer, D. W. Ward, E. R. Statz, and K. A. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express12, 2387–2396 (2004).
[CrossRef] [PubMed]

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

2003 (2)

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett.82, 4435–4437 (2003).
[CrossRef]

2002 (3)

2001 (2)

Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B18, 823–831 (2001).
[CrossRef]

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001).
[CrossRef]

1998 (1)

Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun.148, 105–109 (1998).
[CrossRef]

1990 (1)

1983 (1)

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).

1974 (1)

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech.22, 995–1000 (1974).
[CrossRef]

1969 (2)

T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron.5, 140–146 (1969).
[CrossRef]

R. H. Stolen, “Far-infrared absorption in high resistivity GaAs,” Appl. Phys. Lett.15, 74–75 (1969).
[CrossRef]

1968 (1)

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R1 and R2 laser lines,” Phys. Lett. A26, 281–282 (1968).
[CrossRef]

Akiba, T.

Almási, G.

J. A. Fülöp, L. Pálfalvi, S. Klingebiel, G. Almási, F. Krausz, S. Karsch, and J. Hebling, “Generation of sub-mJ terahertz pulses by optical rectification,” Opt. Lett.37, 557–559 (2012).
[CrossRef] [PubMed]

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010).
[CrossRef] [PubMed]

L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “Novel setups for extremely high power single-cycle terahertz pulse generation by optical rectification,” Appl. Phys. Lett.92, 171107 (2008).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express10, 1161–1166 (2002).
[CrossRef] [PubMed]

Avetisyan, Y.

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] [PubMed]

Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO3 crystal,” Appl. Phys. Lett.81, 3323–3325 (2002).
[CrossRef]

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001).
[CrossRef]

Bagdasaryan, D. A.

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).

Bakunov, M. I.

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbibg crystals,” J. Opt. Soc. Am. B28, 1724–1734 (2011).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
[CrossRef]

Bartal, B.

J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B25, B6–B19 (2008).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

Becouarn, L.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Berger, V.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19, 964–970 (2004).
[CrossRef]

Bessho, T.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Blanchard, F.

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

Bodrov, S. B.

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbibg crystals,” J. Opt. Soc. Am. B28, 1724–1734 (2011).
[CrossRef]

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
[CrossRef]

Bonacina, L.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett.33, 2497–2499 (2008).
[CrossRef] [PubMed]

Cao, H.

Cao, Z. S.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Chekalin, S. V.

Chen, Q.

Chen, W. D.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Chen, Y. H.

Chiang, A. C.

Coleman, P. D.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech.22, 995–1000 (1974).
[CrossRef]

Deng, L. H.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Ding, Y. J.

W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett.84, 1635–11637 (2004).
[CrossRef]

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett.82, 4435–4437 (2003).
[CrossRef]

W. Shi, Y. J. Ding, N. Fernelius, and K. Vodopyanov, “Efficient, tunable, and coherent 0.18–5.27-THz source based on GaSe crystal,” Opt. Lett.27, 1454–1456 (2002).
[CrossRef]

Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun.148, 105–109 (1998).
[CrossRef]

Doi, A.

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

Eyres, L. A.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Fan, S.

Fattinger, Ch.

Fejer, M. M.

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17, 13502–13515 (2009).
[CrossRef] [PubMed]

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Fernelius, N.

Feurer, T.

Fülöp, J. A.

Gao, X. M.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Gerard, B.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Gong, Z. B.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Grischkowsky, D.

Harris, J. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Hebling, J.

J. A. Fülöp, L. Pálfalvi, S. Klingebiel, G. Almási, F. Krausz, S. Karsch, and J. Hebling, “Generation of sub-mJ terahertz pulses by optical rectification,” Opt. Lett.37, 557–559 (2012).
[CrossRef] [PubMed]

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010).
[CrossRef] [PubMed]

L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “Novel setups for extremely high power single-cycle terahertz pulse generation by optical rectification,” Appl. Phys. Lett.92, 171107 (2008).
[CrossRef]

J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B25, B6–B19 (2008).
[CrossRef]

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007).
[CrossRef]

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express10, 1161–1166 (2002).
[CrossRef] [PubMed]

Henin, S.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

Hirori, H.

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

Hirosumi, T.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Hoffman, M. C.

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007).
[CrossRef]

Hoffmann, M. C.

Huang, Y. C.

Inoue, K.

T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron.5, 140–146 (1969).
[CrossRef]

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R1 and R2 laser lines,” Phys. Lett. A26, 281–282 (1968).
[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] [PubMed]

Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO3 crystal,” Appl. Phys. Lett.81, 3323–3325 (2002).
[CrossRef]

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001).
[CrossRef]

Jiang, Z.

Karsch, S.

Kasparian, J.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

Kawase, K.

Keiding, S.

Khurgin, J. B.

Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun.148, 105–109 (1998).
[CrossRef]

Klingebiel, S.

Koketsu, K.

Kozma, I. Z.

Krausz, F.

Kuech, T.

Kuhl, J.

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express10, 1161–1166 (2002).
[CrossRef] [PubMed]

Kuo, P. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Lallier, E.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Lee, H. H.

Levi, O.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Lin, S. T.

Lin, Y. Y.

Linke, R. A.

Makaryan, A. D.

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).

Mashkovich, E. A.

Maslov, A. V.

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
[CrossRef]

McCaughan, L.

Nagai, M.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Nahata, A.

Nakanishi, H.

T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys.95, 7588–7591 (2004).
[CrossRef]

Nelson, K. A.

Ohtake, H.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Otani, C.

Pálfalvi, L.

J. A. Fülöp, L. Pálfalvi, S. Klingebiel, G. Almási, F. Krausz, S. Karsch, and J. Hebling, “Generation of sub-mJ terahertz pulses by optical rectification,” Opt. Lett.37, 557–559 (2012).
[CrossRef] [PubMed]

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010).
[CrossRef] [PubMed]

L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “Novel setups for extremely high power single-cycle terahertz pulse generation by optical rectification,” Appl. Phys. Lett.92, 171107 (2008).
[CrossRef]

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

Péter, Á.

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

Petit, Y.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

Pinguet, T. J.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Pogosyan, P. S.

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).

Polgár, K.

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

Ruan, Z.

Sasaki, Y.

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] [PubMed]

Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO3 crystal,” Appl. Phys. Lett.81, 3323–3325 (2002).
[CrossRef]

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001).
[CrossRef]

Shi, W.

W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett.84, 1635–11637 (2004).
[CrossRef]

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett.82, 4435–4437 (2003).
[CrossRef]

W. Shi, Y. J. Ding, N. Fernelius, and K. Vodopyanov, “Efficient, tunable, and coherent 0.18–5.27-THz source based on GaSe crystal,” Opt. Lett.27, 1454–1456 (2002).
[CrossRef]

Shibuya, T.

Sirtori, C.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19, 964–970 (2004).
[CrossRef]

Skauli, T.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

Statz, E. R.

Staus, C.

Stepanov, A. G.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett.33, 2497–2499 (2008).
[CrossRef] [PubMed]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

Stolen, R. H.

R. H. Stolen, “Far-infrared absorption in high resistivity GaAs,” Appl. Phys. Lett.15, 74–75 (1969).
[CrossRef]

Stoyanov, N. S.

Sugiura, T.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Suizu, K.

Tanaka, K.

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Tani, M.

Taniuchi, T.

T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys.95, 7588–7591 (2004).
[CrossRef]

Thompson, D. E.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech.22, 995–1000 (1974).
[CrossRef]

Tsutsui, T.

Tutui, T.

van Exter, M.

Veronis, G.

Vodopyanov, K.

Vodopyanov, K. L.

Wang, T. D.

Ward, D. W.

Wolf, J.-P.

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett.33, 2497–2499 (2008).
[CrossRef] [PubMed]

Yajima, T.

T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron.5, 140–146 (1969).
[CrossRef]

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R1 and R2 laser lines,” Phys. Lett. A26, 281–282 (1968).
[CrossRef]

Yeh, K. L.

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007).
[CrossRef]

Yeh, K.-L.

Yokoyama, H.

Yoshida, M.

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

Yuan, Y. Q.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Zhang, W. J.

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Zhang, X.-C.

Appl. Phys. B (3)

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001).
[CrossRef]

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B101, 11–14 (2010).
[CrossRef]

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B78, 593–599 (2004).
[CrossRef]

Appl. Phys. Lett. (8)

L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “Novel setups for extremely high power single-cycle terahertz pulse generation by optical rectification,” Appl. Phys. Lett.92, 171107 (2008).
[CrossRef]

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett.85, 3974–3976 (2004).
[CrossRef]

R. H. Stolen, “Far-infrared absorption in high resistivity GaAs,” Appl. Phys. Lett.15, 74–75 (1969).
[CrossRef]

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007).
[CrossRef]

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett.98, 091106 (2011).
[CrossRef]

Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO3 crystal,” Appl. Phys. Lett.81, 3323–3325 (2002).
[CrossRef]

W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett.84, 1635–11637 (2004).
[CrossRef]

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett.82, 4435–4437 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron.5, 140–146 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech.22, 995–1000 (1974).
[CrossRef]

J. Appl. Phys. (3)

T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys.95, 7588–7591 (2004).
[CrossRef]

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO3 in the THz range,” J. Appl. Phys.97, 123505 (2005).
[CrossRef]

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys.94, 6447–6455 (2003).
[CrossRef]

J. Opt. Soc. Am. B (4)

JETP Lett. (1)

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).

Opt. Commun. (2)

Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun.148, 105–109 (1998).
[CrossRef]

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO3 crystal using mid-infrared difference-frequency generation,” Opt. Commun.268, 110–114 (2006).
[CrossRef]

Opt. Express (8)

N. S. Stoyanov, T. Feurer, D. W. Ward, E. R. Statz, and K. A. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express12, 2387–2396 (2004).
[CrossRef] [PubMed]

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010).
[CrossRef] [PubMed]

K. Suizu, T. Shibuya, T. Akiba, T. Tutui, C. Otani, and K. Kawase, “Cherenkov phase-matched monochromatic THz wave generation using difference frequency generation with a lithium niobate crystal,” Opt. Express16, 7493–7498 (2008).
[CrossRef] [PubMed]

K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express17, 6676–6681 (2009).
[CrossRef] [PubMed]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express10, 1161–1166 (2002).
[CrossRef] [PubMed]

C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded waveguide structure supporting only fundamental modes,” Opt. Express16, 13296–13303 (2008).
[CrossRef] [PubMed]

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17, 13502–13515 (2009).
[CrossRef] [PubMed]

K. L. Vodopyanov, “Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: conversion efficiency and optimal laser pulse format,” Opt. Express14, 2263–2276 (2006).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Lett. A (1)

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R1 and R2 laser lines,” Phys. Lett. A26, 281–282 (1968).
[CrossRef]

Phys. Rev. Lett. (1)

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett.99, 203904 (2007).
[CrossRef]

Semicond. Sci. Technol. (1)

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19, 964–970 (2004).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Generation scheme.

Fig. 2
Fig. 2

Tilt angle α vs. phase-matched frequency Ω for LN pumped at 1.3 μm and GaAs pumped at 1.55 μm.

Fig. 3
Fig. 3

Snapshots of |Ez(x,y,t)| in LN for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the blazed grating scheme. Optical beams are shown by dashed lines. The pump power is 4 ×106 W/cm, Ω/(2π) = 1 THz, and α = 63.8°.

Fig. 4
Fig. 4

Efficiency as a function of the LN crystal thickness L for several values of aFWHM (labeled next to the corresponding curves). The pump parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Snapshots of |Ez(x,y,t)| in LN for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles Ez(y) are shown in the air at x = 2 mm (solid) or x = 0.5 mm (dotted). The pump parameters are the same as in Fig. 3.

Fig. 6
Fig. 6

Snapshots of |Ez(x,y,t)| in GaAs for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the blazed grating scheme. Optical beams are shown by dashed lines. The pump power is 4 ×106 W/cm, Ω/(2π) = 1 THz, and α = 12.2°.

Fig. 7
Fig. 7

Efficiency as a function of the GaAs crystal thickness L for several values of aFWHM (labeled next to the corresponding curves). The pump parameters are the same as in Fig. 6.

Fig. 8
Fig. 8

Snapshots of |Ez(x,y,t)| in GaAs for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles Ez(y) are shown in the air at x = 2.4 cm (a) and x = 0.8 cm (b). The pump parameters are the same as in Fig. 6.

Equations (14)

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E ( 0 , y , t ) = m = 1 2 E 0 exp ( i ω m t y 2 2 a 2 ) .
E ± 1 ( x , y , t ) = m = 1 2 a ( 2 π ) 1 / 2 T ± 1 E 0 e i ω m t i G y d γ e i γ y i h m x ( γ a ) 2 / 2 ,
E ± 1 ( x , y , t ) 2 T ± 1 E 0 cos ( Ω ξ / 2 ) e η ± 2 / ( 2 a 2 ) + i ω ¯ t i G y i G x cot α ,
P NL = p e i Ω ξ η + 2 / a 2 ,
P NL = p e i Ω ξ [ e η + 2 / a 2 + e η 2 / a 2 + 2 cos ( 2 G y ) e ( y 2 + x 2 tan 2 α ) / a 2 ] .
ε = ε + ( ε 0 ε ) ω TO 2 ω TO 2 Ω 2 + i ν Ω .
2 E ˜ z x 2 + κ 2 E ˜ z = 4 π Ω 2 c 2 P ˜ NL ,
P ˜ NL ( x , g , t ) = p a 2 π 1 / 2 e i Ω ξ g 2 a 2 / 4 + i g x tan α
P ˜ NL ( x , g , t ) = p a 2 π 1 / 2 e i Ω ξ g 2 a 2 / 4 × [ e i g x tan α + e i g x tan α + e ( x / a ) 2 tan 2 α G 2 a 2 ( e g G a 2 + e g G a 2 ) ] .
E ˜ z ( x , g , t ) = 2 π Ω 2 i κ c 2 0 x d x P ˜ NL ( x , g , t ) e i κ ( x x ) .
E z ( x , y , t ) = d g E ˜ t ( x , g , y ) e i g y .
E z ( x , t ) = 2 π p Ω x c ε sinc [ Ω x 2 c ( n g cos α ε ) ] sin [ Ω t Ω x 2 c ( n g cos α + ε ) ] .
cos α = n g / Re ε ( Ω ) .
E z ( x , t ) 2 π p | ε | Im ε ( 1 e Ω x c Im ε ) sin ( Ω t Ω x 2 c Re ε )

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