Frequency resolved cross-correlation between optical and terahertz pulses: application to ultrashort laser pulse characterization

M. Cornet; J. Degert; E. Abraham; E. Freysz

doi:10.1364/OE.24.003003

Abstract

We have analyzed both theoretically and experimentally the spectrum of the optical pulses produced by the interaction of optical and THz pulses in a ZnTe crystal. Recorded as a function of the delay between the two pulses, the resulting spectrogram can be viewed as a frequency resolved cross-correlation between the optical and THz pulses making it possible to characterize the optical pulse.

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References

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Year
Author
Publication

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques,” J. Opt. Soc. Am. B 19(11), 2692–2703 (2002).
[Crossref]
L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19(11), 2704–2715 (2002).
[Crossref]
G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16(8), 1204–1212 (1999).
[Crossref]
Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]
Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]
M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]
S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]
M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]
P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).
G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]
D. T. F. Marple, “Refractive index of ZnSe, ZnTe, and CdTe,” J. Appl. Phys. 35(3), 539–542 (1964).
[Crossref]
Q. Wu and X.-C. Zhang, “Design and characterization of traveling-wave electrooptic terahertz sensors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 693–700 (1996).
[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(16), 2321–2323 (1996).
[Crossref]
M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]
J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006).

2014 (1)

M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]

2008 (1)

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

2007 (2)

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

2002 (2)

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques,” J. Opt. Soc. Am. B 19(11), 2692–2703 (2002).
[Crossref]

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19(11), 2704–2715 (2002).
[Crossref]

1999 (1)

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16(8), 1204–1212 (1999).
[Crossref]

1998 (1)

S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]

1996 (2)

Q. Wu and X.-C. Zhang, “Design and characterization of traveling-wave electrooptic terahertz sensors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 693–700 (1996).
[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(16), 2321–2323 (1996).
[Crossref]

1973 (1)

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

1964 (1)

D. T. F. Marple, “Refractive index of ZnSe, ZnTe, and CdTe,” J. Appl. Phys. 35(3), 539–542 (1964).
[Crossref]

Abraham, E.

M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]

Arena, D. A.

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]

Boyd, G. D.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).

Carr, G. L.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Cornet, M.

M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).

Coutaz, J.-L.

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques,” J. Opt. Soc. Am. B 19(11), 2692–2703 (2002).
[Crossref]

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19(11), 2704–2715 (2002).
[Crossref]

Degert, J.

M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]

Diels, J.-C.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006).

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]

Grischkowsky, D.

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16(8), 1204–1212 (1999).
[Crossref]

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(16), 2321–2323 (1996).
[Crossref]

Kao, C.-C.

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Kreß, M.

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]

Löffler, T.

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Marple, D. T. F.

D. T. F. Marple, “Refractive index of ZnSe, ZnTe, and CdTe,” J. Appl. Phys. 35(3), 539–542 (1964).
[Crossref]

Murphy, J. B.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

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(16), 2321–2323 (1996).
[Crossref]

Pollack, M. A.

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

Rialland, S.

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques,” J. Opt. Soc. Am. B 19(11), 2692–2703 (2002).
[Crossref]

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19(11), 2704–2715 (2002).
[Crossref]

Roskos, H. G.

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Rudolph, W.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006).

Shen, Y.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Thomson, M. D.

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Tsang, T. Y.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Tsang, T.Y.

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Wang, X.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Wang, X. J.

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Watanabe, T.

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

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(16), 2321–2323 (1996).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]

Wu, Q.

Q. Wu and X.-C. Zhang, “Design and characterization of traveling-wave electrooptic terahertz sensors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 693–700 (1996).
[Crossref]

Yang, X.

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Zhang, X.-C.

Q. Wu and X.-C. Zhang, “Design and characterization of traveling-wave electrooptic terahertz sensors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 693–700 (1996).
[Crossref]

Appl. Phys. Lett. (1)

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(16), 2321–2323 (1996).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

Q. Wu and X.-C. Zhang, “Design and characterization of traveling-wave electrooptic terahertz sensors,” IEEE J. Sel. Top. Quantum Electron. 2(3), 693–700 (1996).
[Crossref]

J. Appl. Phys. (1)

D. T. F. Marple, “Refractive index of ZnSe, ZnTe, and CdTe,” J. Appl. Phys. 35(3), 539–542 (1964).
[Crossref]

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

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques,” J. Opt. Soc. Am. B 19(11), 2692–2703 (2002).
[Crossref]

L. Duvillaret, S. Rialland, and J.-L. Coutaz, “Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation,” J. Opt. Soc. Am. B 19(11), 2704–2715 (2002).
[Crossref]

G. Gallot and D. Grischkowsky, “Electro-optic detection of terahertz radiation,” J. Opt. Soc. Am. B 16(8), 1204–1212 (1999).
[Crossref]

M. Cornet, J. Degert, E. Abraham, and E. Freysz, “Terahertz Kerr effect in gallium phosphide crystal,” J. Opt. Soc. Am. B 31(7), 1648–1652 (2014).
[Crossref]

Laser Photonics Rev. (1)

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007).
[Crossref]

Phys. Rev. A (1)

Y. Shen, G. L. Carr, J. B. Murphy, T. Y. Tsang, X. Wang, and X. Yang, “Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses,” Phys. Rev. A 78, 043813 (2008).
[Crossref]

Phys. Rev. B (1)

G. D. Boyd and M. A. Pollack, “Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities,” Phys. Rev. B 7(12), 5345–5359 (1973).
[Crossref]

Phys. Rev. Lett. (1)

Y. Shen, T. Watanabe, D. A. Arena, C.-C. Kao, J. B. Murphy, T.Y. Tsang, X. J. Wang, and G. L. Carr, “Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses,” Phys. Rev. Lett. 99, 043901 (2007).
[Crossref] [PubMed]

Phys. Stat. Sol.(b) (1)

S. Linden, H. Giessen, and J. Kuhl, “XFROG – A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Stat. Sol.(b) 206(1), 119–124 (1998).
[Crossref]

Other (3)

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991).

M. Born and E. Wolf, Principles of optics (Cambridge University, 1999).
[Crossref]

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 2006).

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Fig. 1 (a) Experimental setup. BS: beamsplitter; HDPE: high density polyethylene. (b) Geometry of the experiment.

(\hat{X}, \hat{Y}, \hat{Z})

: Cartesian frame of the crystal;

(\hat{x}, \hat{y}, \hat{z})

: Cartesian frame of the laboratory.

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Fig. 2 (a) S₁: intensity profile of the Pockels-induced optical pulse; S_EO: intensity profile of the THz wave measured by conventional EO sampling. (b) Peak value of the Pockels-induced optical pulse with respect to the THz electric field strength E_THz.

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Fig. 3 (a) Spectrogram of the Pockels-induced optical pulse obtained with a pulse compression optimizing the peak of the THz pulse. (b) Spectrogram of the Pockels-induced optical pulse recorded when a 3mm thick window of ZnSe is inserted on the path of the incident optical pulse. The dashed white lines indicate the overall slope of the spectrogram in the time delay – angular frequency plane.

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Equations (17)

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

E (t, x) = \frac{1}{2} \hat{E} (t, x) \exp [i (k_{pr} x - ω_{pr} t)] + c .c.

P^{(2)} (t, x) = \frac{1}{2} {\hat{P}}^{(2)} (t, x) \exp [i (k_{pr} x - ω_{pr} t)] + c .c.

\frac{\partial {\hat{E}}_{μ} (t^{'}, x)}{\partial x} = i \frac{π}{λ_{pr} n_{pr} ε_{0}} {\hat{P}}_{μ}^{(2)} (t^{'}, x),

{\hat{P}}^{(2)} (t^{'}, x) = 2 ε_{0} χ^{(2)} : \hat{E} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x),

{\hat{P}}_{μ}^{(2)} (t^{'}, x) = ε_{0} n_{pr}^{4} r_{41} {\hat{E}}_{z} (t^{^{'}}, x) E_{THz} (t^{^{'}} + \frac{x}{v_{g}}, x) δ_{μ, y} (μ = x, y, z) .

{\hat{P}}^{' (2)} (t^{'}, x) = 2 ε_{0} χ^{(2)} : {\hat{E}}_{y} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x),

\hat{P}'_{μ}^{(2)} (t^{'}, x) = ε_{0} n_{pr}^{4} r_{41} {\hat{E}}_{y} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x) δ_{μ, z} (μ = x, y, z) .

\frac{\partial {\hat{E}}_{y} (t^{'}, x)}{\partial x} = i κ {\hat{E}}_{z} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x),

\frac{\partial {\hat{E}}_{z} (t^{'}, x)}{\partial x} = i κ {\hat{E}}_{y} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x),

\frac{\partial {\hat{E}}_{\pm} (t^{'}, x)}{\partial x} = \pm i κ {\hat{E}}_{\pm} (t^{'}, x) E_{THz} (t^{'} + \frac{x}{v_{g}}, x),

\begin{array}{l} {\hat{E}}_{y} (t^{'}, L) = i {\hat{E}}_{z} (t^{'}, 0) \sin [κ \int_{0}^{L} E_{THz} (t^{'} + \frac{x}{v_{g}} + τ, x) d x], \\ {\hat{E}}_{z} (t^{'}, L) = {\hat{E}}_{z} (t^{'}, 0) \cos [κ \int_{0}^{L} E_{THz} (t^{'} + \frac{x}{v_{g}} + τ, x) d x], \end{array}

{\hat{E}}_{y} (t^{'}, L) ≃ i κ {\hat{E}}_{z} (t^{'}, 0) \int_{0}^{L} E_{THz} (t^{'} + \frac{x}{v_{g}} + τ, x) d x .

E_{THz} (t^{'} + \frac{x}{v_{g}} + τ, x) = \int_{- \infty}^{+ \infty} \frac{d Ω}{2 π} e^{- i Ω t^{'}} E_{THz} (Ω, 0) e^{i Ω [\tilde{n} (Ω) - n_{g}] x / c} e^{- i Ω τ},

\begin{array}{l} {\hat{E}}_{y} (ω, L) = \int_{- \infty}^{+ \infty} d t^{'} e^{i ω t^{'}} {\hat{E}}_{y} (t^{'}, L) \\ = i κ \int_{0}^{L} d x \int_{- \infty}^{+ \infty} \frac{d Ω}{2 π} {\hat{E}}_{z} (ω - Ω, 0) E_{THz} (Ω, 0) e^{i \frac{Ω x}{c} [\tilde{n} (Ω) - n_{g}] - i Ω τ} . \end{array}

{\hat{E}}_{y} (ω, L) = i κ L \int_{- \infty}^{+ \infty} \frac{d Ω}{2 π} sinc (\frac{Δ k^{'} L}{2}) {\hat{E}}_{z} (ω - Ω, 0) E_{THz} (Ω, 0) e^{i [\frac{1}{2} ϕ_{in}^{(2)} {(ω - Ω)}^{2} - Ω τ + \frac{1}{2} Δ k^{'} L]},

{\hat{E}}_{y} (ω, L) = i κ L \int_{- \infty}^{+ \infty} \frac{d Ω}{2 π} \sin c (\frac{Δ k^{'} L}{2}) {\hat{E}}_{z} (ω - Ω, 0) E_{THz} (Ω, 0) e^{- \frac{1}{2 c} κ (Ω) Ω L} \times e^{i φ (ω, Ω)},

φ (ω, Ω) = \frac{1}{2} ϕ_{in}^{(2)} {(ω - Ω)}^{2} - Ω τ + \frac{1}{4} ϕ_{ZnTe}^{(2)} {(ω - Ω)}^{2} + \frac{1}{2} Δ k (Ω) L .

Abstract

References

2014 (1)

2008 (1)

2007 (2)

2002 (2)

1999 (1)

1998 (1)

1996 (2)

1973 (1)

1964 (1)

Abraham, E.

Arena, D. A.

Born, M.

Boyd, G. D.

Butcher, P. N.

Carr, G. L.

Cornet, M.

Cotter, D.

Coutaz, J.-L.

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Tsang, T.Y.

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