Abstract

We present and demonstrate experimentally two simple and distortion-free methods to sample terahertz pulse in zinc-blende-type crystals like ZnTe or GaP. They are based on an optical heterodyne detection scheme which makes it possible to measure the terahertz field by detecting the optical probe beam or its second harmonic. Both are compared to conventional electro-optic sampling and give results in good agreement with the latter.

© 2016 Optical Society of America

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References

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    [Crossref]
  2. 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, 2704–2715 (2002).
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    [Crossref]
  4. Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
    [Crossref]
  5. A. Schneider and P. Günter, “Measurement of the terahertz induced phase shift in electro-optic sampling for an arbitrary biasing phase,” Appl. Opt. 45, 6598–6601 (2006).
    [Crossref]
  6. F. D. J. Brunner, J. A. Johnson, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of terahertz signals measured by electrooptic sampling. I. Theory,” J. Opt. Soc. Am. B 31, 904–910 (2014).
    [Crossref]
  7. J. A. Johnson, F. D. J. Brunner, S. Grübel, A. Ferrer, S. L. Johnson, and T. Feurer, “Distortion-free enhancement of terahertz signals measured by electrooptic sampling. II. Experiment,” J. Opt. Soc. Am. B 31, 1035–1040 (2014).
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    [Crossref]
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    [Crossref]
  14. C. Vicario, M. Jazbinsek, A. V. Ovchinnikov, O. V. Chefonov, S. I. Ashitkov, M. B. Agranat, and C. P. Hauri, “High efficiency THz generation in DSTMS, DAST and OH1 pumped by Cr:forsterite laser,” Opt. Express 23, 4573–4580 (2015).
    [Crossref]
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    [Crossref]
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2016 (1)

2015 (1)

2014 (3)

2008 (1)

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

2007 (3)

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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

2006 (1)

2004 (1)

2002 (1)

1999 (1)

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

1996 (1)

Q. Wu and X.-C. Zhang, “Ultrafast electro-optical field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[Crossref]

Abraham, E.

Agranat, M. B.

Aravazhi, S.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Ashitkov, S. I.

Bliss, D.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Brunner, F. D. J.

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990).

Chefonov, O. V.

Chen, Q.

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

Chen, Y.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Cornet, M.

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990).

Coutaz, J.-L.

Dai, J.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Degert, J.

Duvillaret, L.

Fejer, M. M.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Ferrer, A.

Feurer, T.

Fletcher, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Dover, 1989).

Freysz, E.

Gramlich, V.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Grübel, S.

Günter, P.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

A. Schneider and P. Günter, “Measurement of the terahertz induced phase shift in electro-optic sampling for an arbitrary biasing phase,” Appl. Opt. 45, 6598–6601 (2006).
[Crossref]

Harris, J. S.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Hauri, C. P.

Jazbinsek, M.

C. Vicario, M. Jazbinsek, A. V. Ovchinnikov, O. V. Chefonov, S. I. Ashitkov, M. B. Agranat, and C. P. Hauri, “High efficiency THz generation in DSTMS, DAST and OH1 pumped by Cr:forsterite laser,” Opt. Express 23, 4573–4580 (2015).
[Crossref]

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Jiang, Z. P.

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

Johnson, J. A.

Johnson, K.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Johnson, S. L.

Karpowicz, N.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Kozlov, V. G.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

Kuo, P. S.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Lesimple, A.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

Lu, X.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Lynch, C.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Mamer, O.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Mutter, L.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Ovchinnikov, A. V.

Planken, P. C. M.

Price-Gallagher, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Rialland, S.

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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

Ruiz, B.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Schaar, J. E.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Schneider, A.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

A. Schneider and P. Günter, “Measurement of the terahertz induced phase shift in electro-optic sampling for an arbitrary biasing phase,” Appl. Opt. 45, 6598–6601 (2006).
[Crossref]

Stillhart, M.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Sun, F. G.

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

van der Valk, N. C. J.

Vicario, C.

Vodopyanov, K. L.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Wecnkebach, T.

Wu, Q.

Q. Wu and X.-C. Zhang, “Ultrafast electro-optical field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[Crossref]

Yamaguchi, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Yang, Z.

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, 1989).

Yu, X.

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Zhang, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Zhang, L.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Zhang, X.-C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

Q. Wu and X.-C. Zhang, “Ultrafast electro-optical field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[Crossref]

Zhao, H.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

Adv. Funct. Mater. (1)

Z. Yang, L. Mutter, M. Stillhart, B. Ruiz, S. Aravazhi, M. Jazbinsek, A. Schneider, V. Gramlich, and P. Günter, “Large-size bulk and thin-film stilbazolium-salt single crystals for nonlinear optics and THz generation,” Adv. Funct. Mater. 17, 2018–2023 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

Q. Wu and X.-C. Zhang, “Ultrafast electro-optical field sensors,” Appl. Phys. Lett. 68, 1604–1606 (1996).
[Crossref]

Z. P. Jiang, F. G. Sun, Q. Chen, and X.-C. Zhang, “Electro-optic sampling near zero optical transmission point,” Appl. Phys. Lett. 74, 1191–1193 (1999).
[Crossref]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”,” Appl. Phys. Lett. 92, 011131 (2008).
[Crossref]

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

Laser Photon. 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 Photon. Rev. 1, 349–368 (2007).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (1)

K. L. Vodopyanov, J. E. Schaar, P. S. Kuo, M. M. Fejer, X. Yu, J. S. Harris, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz wave generation in orientation-patterned GaAs using resonantly enhanced scheme,” Proc. SPIE 6455, 645509 (2007).
[Crossref]

Other (3)

A. Yariv, Quantum Electronics (Wiley, 1989).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990).

G. R. Fowles, Introduction to Modern Optics (Dover, 1989).

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Figures (5)

Fig. 1.
Fig. 1. (a) Experimental setup. BS, beam splitter; nBS, nonpolarizing beam splitter; P, polarizer; PD, photodiode. (b) Geometry of the experiment. ( X ^ , Y ^ , Z ^ ) , Cartesian frame of the crystal; ( x ^ , y ^ , z ^ ) , Cartesian frame of the laboratory. k THz (resp. k pr ) is the wave vector of the THz (resp. probe) pulse.
Fig. 2.
Fig. 2. EO sampling by optical heterodyne detection performed in ZnTe [(a) and (b)] and GaP [(c) and (d)]. (a),(c) Normalized THz electric field, for different β angles, versus the time delay τ compared to the field measured with a CEOS setup (dotted line). (c),(d) Corresponding spectra.
Fig. 3.
Fig. 3. EO sampling by optical heterodyne detection performed in ZnTe [(a) and (b)] and GaP [(c) and (d)]. (a),(c) THz electric field, for different β angles, versus the time delay τ . (c),(d) Evolution of the amplitude of the signal displayed in (a) and (c) with respect to the heterodyning angle β . The solid line curve corresponds to a fit of the data with the function sin ( 2 β ) .
Fig. 4.
Fig. 4. Experimental setup. PMT, photomultiplier tube; ( x ^ , y ^ , z ^ ) , Cartesian frame of the laboratory.
Fig. 5.
Fig. 5. (a) PISH signals for two opposite heterodyning angles: β = 10 ° and β = 10 ° . (b) EO signal measured with a CEOS setup (dashed red curve) and heterodyned PISH signal difference, Δ I 2 ω , for β = 10 ° (black curve). All the signals are plotted as a function of the time delay, τ , between the probe and THz pulses. (c) Corresponding spectra.

Equations (30)

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E = E 0 [ 0 1 ] .
E = E 0 [ i sin ( Γ / 2 ) cos ( Γ / 2 ) ] ,
Γ = 2 π L λ n ω 3 r 41 E THz
I 1 ( Γ , β ) = I 0 2 [ sin 2 ( Γ 2 ) cos 2 ( 2 β ) + 1 2 sin 2 ( 2 β ) + 1 2 sin ( 2 β ) sin Γ ]
I LO = I 1 ( 0 , β ) = I 0 4 sin 2 ( 2 β ) ,
I sig = I 1 ( Γ , 0 ) = I 0 2 sin 2 ( Γ 2 ) .
M ( Γ , β ) = g Γ + d ( g Γ ) 2 + O ( Γ 3 ) .
g = 1 sin ( 2 β ) ,
d = 1 2 cos 2 ( 2 β ) .
Δ I ( Γ , β ) = I 0 2 sin ( 2 β ) sin Γ ,
E ( t , x ) = 1 2 E ω ( x ) exp [ i ( ω t k ω x ) ] + c.c. ( = y , z )
E THz ( t , x ) = 1 2 E THz exp [ i ( Ω t k Ω x ) ] y ^ + c.c. ,
[ E y ω ( β , 0 ) E z ω ( β , 0 ) ] = E 0 [ 1 + i 2 sin ( 2 β ) ( 1 + i ) sin 2 β i ] .
E X ω ( β , x ) = E Y ω ( β , x ) ,
E Y ω ( β , x ) = E 0 2 2 [ f β e i Γ x / 2 + g β e i Γ x / 2 ] ,
E Z ω ( β , x ) = E 0 2 [ f β e i Γ x / 2 g β e i Γ x / 2 ] ,
f β = 1 + i 2 sin ( 2 β ) + ( 1 + i ) sin 2 β i ,
g β = 1 + i 2 sin ( 2 β ) + i ( 1 + i ) sin 2 β ,
P i 2 ω = 1 2 ϵ 0 χ i j k ( 2 ) E j ω E k ω ,
P x 2 ω ( β , x ) = 0 ,
P y 2 ω ( β , x ) = ϵ 0 χ ( 2 ) E 0 2 8 [ f β 2 e i Γ x g β 2 e i Γ x ] ,
P z 2 ω ( β , x ) = ϵ 0 χ ( 2 ) E 0 2 16 [ f β e i Γ x / 2 + g β e i Γ x / 2 ] 2 ,
( x + α 2 ω ) E 2 ω = i ω c n 2 ω ϵ 0 P 2 ω exp ( i x Δ k ) ( = y , z ) ,
E y 2 ω ( β , L ) = i A 4 [ f β 2 e i Γ g β 2 e i Γ ] ,
E z 2 ω ( β , L ) = i A 8 [ f β e i Γ / 2 + g β e i Γ / 2 ] 2 ,
I 2 ω ( Γ , β ) | A | 2 4 [ 2 sin 2 ( 2 β ) + sin 2 Γ + sin ( 2 β ) sin ( 2 Γ ) 3 2 sin 2 ( 2 β ) sin 2 Γ ] .
I LO 2 ω = I 2 ω ( 0 , β ) | A | 2 2 sin 2 ( 2 β ) ,
I sig 2 ω = I 2 ω ( Γ , 0 ) | A | 2 4 sin 2 Γ .
M ( Γ , β ) = g Γ + d ( g Γ ) 2 + O ( Γ 3 ) ,
Δ I 2 ω ( Γ , β ) = I 2 ω ( Γ , β ) I 2 ω ( Γ , β ) , | A | 2 2 sin ( 2 β ) sin ( 2 Γ ) , 2 L λ β [ χ ( 2 ) r 41 ] 2 I ω 2 E THz .

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