Abstract

A spatio-temporal analysis of terahertz generation by optical rectification of tilted pulse fronts is presented. Closed-form expressions of terahertz transients and spectra in two spatial dimensions are furnished in the undepleted limit. Importantly, the analysis incorporates spatio-temporal distortions of the optical pump pulse such as angular dispersion, group velocity dispersion due to angular dispersion, spatial and temporal chirp, as well as beam curvature. The influence of the radius of curvature on the tilt angle is shown. Furthermore, the impact of group velocity dispersion due to angular dispersion on terahertz frequency, conversion efficiency and peak field is revealed. In particular, the deterioration of terahertz frequency, efficiency and field at large pump bandwidths and beam sizes by group velocity dispersion due to angular dispersion is expressed analytically.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
  2. A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
    [Crossref]
  3. L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense thz pulse,” Phys. Rev. ST Accel. Beams 17, 031301 (2014).
    [Crossref]
  4. D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
    [Crossref]
  5. C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
    [Crossref]
  6. W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
    [Crossref]
  7. C. Ropers, “Electrons catch a terahertz wave,” Science 352, 410–411 (2016).
    [Crossref]
  8. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
    [Crossref]
  9. T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
    [Crossref]
  10. G. Gallerano and S. Biedron, “Overview of terahertz radiation sources,” in Proceedings of the 2004 FEL Conference, (2004), 1, pp. 216–221.
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    [Crossref]
  12. L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kärtner, “250 w average power, 100 khz repetition rate cryogenic yb:yag amplifier for opcpa pumping,” Opt. Lett. 41, 492–495 (2016).
    [Crossref]
  13. W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
    [Crossref]
  14. C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
    [Crossref]
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    [Crossref]
  17. J. A. Fülöp, Z. Ollmann, C. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, “Efficient generation of thz pulses with 0.4 mj energy,” Opt. Express 22, 20155–20163 (2014).
    [Crossref]
  18. M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
    [Crossref]
  19. J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010).
    [Crossref]
  20. S. C. Zhong, Z. H. Zhai, J. Li, L. G. Zhu, J. Li, K. Meng, Q. Liu, L.-H. Du, J.-H. Zhao, and Z. R. Li, “Optimization of terahertz generation from linbo 3 under intense laser excitation with the effect of three-photon absorption,” Opt. Express 23, 31313–31323 (2015).
    [Crossref]
  21. K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
    [Crossref]
  22. A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
    [Crossref]
  23. M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus cherenkov effect,” Journal of Applied Physics 104, 073105 (2008).
    [Crossref]
  24. M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
    [Crossref]
  25. M. I. Bakunov and S. B. Bodrov, “Terahertz generation with tilted-front laser pulses in a contact-grating scheme,” Journal of Optical Society of America B 31, 2549–2557 (2014).
    [Crossref]
  26. J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
    [Crossref]
  27. Y. J. Ding, “Quasi-single-cycle terahertz pulses based on broadband-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,” IEEE Journal of Selected Topics in Quantum Electronics 10, 1171–1179 (2004).
    [Crossref]
  28. A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
    [Crossref] [PubMed]
  29. K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
    [Crossref]
  30. J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Optical and Quantum Electronics 28, 1759–1763 (1996).
    [Crossref]
  31. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
    [Crossref]
  32. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005).
    [Crossref] [PubMed]
  33. L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
    [Crossref]
  34. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE Journal of Quantum Electronics 24, 2530–2536 (1988).
    [Crossref]
  35. 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,” Applied Physics Letters 98, 091106 (2011).
    [Crossref]

2018 (1)

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

2016 (5)

C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

C. Ropers, “Electrons catch a terahertz wave,” Science 352, 410–411 (2016).
[Crossref]

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
[Crossref]

L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kärtner, “250 w average power, 100 khz repetition rate cryogenic yb:yag amplifier for opcpa pumping,” Opt. Lett. 41, 492–495 (2016).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

2015 (4)

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

S. C. Zhong, Z. H. Zhai, J. Li, L. G. Zhu, J. Li, K. Meng, Q. Liu, L.-H. Du, J.-H. Zhao, and Z. R. Li, “Optimization of terahertz generation from linbo 3 under intense laser excitation with the effect of three-photon absorption,” Opt. Express 23, 31313–31323 (2015).
[Crossref]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

2014 (6)

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense thz pulse,” Phys. Rev. ST Accel. Beams 17, 031301 (2014).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
[Crossref]

M. I. Bakunov and S. B. Bodrov, “Terahertz generation with tilted-front laser pulses in a contact-grating scheme,” Journal of Optical Society of America B 31, 2549–2557 (2014).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
[Crossref]

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

J. A. Fülöp, Z. Ollmann, C. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, “Efficient generation of thz pulses with 0.4 mj energy,” Opt. Express 22, 20155–20163 (2014).
[Crossref]

2013 (2)

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
[Crossref]

2011 (3)

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
[Crossref]

J. A. Fülöp, L. Pálfalvi, M. C. Hoffmann, and J. Hebling, “Towards generation of mj-level ultrashort thz pulses by optical rectification,” Opt. Express 19, 15090–15097 (2011).
[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,” Applied Physics Letters 98, 091106 (2011).
[Crossref]

2010 (1)

2009 (1)

M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
[Crossref]

2008 (1)

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus cherenkov effect,” Journal of Applied Physics 104, 073105 (2008).
[Crossref]

2007 (2)

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
[Crossref]

2005 (2)

L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
[Crossref]

S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005).
[Crossref] [PubMed]

2004 (2)

Y. J. Ding, “Quasi-single-cycle terahertz pulses based on broadband-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,” IEEE Journal of Selected Topics in Quantum Electronics 10, 1171–1179 (2004).
[Crossref]

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
[Crossref]

2002 (1)

1997 (1)

S. H. Gold and G. S. Nusinovich, “Review of high-power microwave source research,” Review of Scientific Instruments 68, 3945–3974 (1997).
[Crossref]

1996 (1)

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Optical and Quantum Electronics 28, 1759–1763 (1996).
[Crossref]

1988 (1)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE Journal of Quantum Electronics 24, 2530–2536 (1988).
[Crossref]

Agranat, M. B.

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

Akturk, S.

S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005).
[Crossref] [PubMed]

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
[Crossref]

Almási, G.

Arrieta, M.

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
[Crossref]

Ashitkov, S. I.

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

Avetisyan, Y.

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

Bakunov, M. I.

M. I. Bakunov and S. B. Bodrov, “Terahertz generation with tilted-front laser pulses in a contact-grating scheme,” Journal of Optical Society of America B 31, 2549–2557 (2014).
[Crossref]

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
[Crossref]

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus cherenkov effect,” Journal of Applied Physics 104, 073105 (2008).
[Crossref]

Baum, P.

C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

Beigang, R.

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

Biedron, S.

G. Gallerano and S. Biedron, “Overview of terahertz radiation sources,” in Proceedings of the 2004 FEL Conference, (2004), 1, pp. 216–221.

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,” Applied Physics Letters 98, 091106 (2011).
[Crossref]

Bodrov, S. B.

M. I. Bakunov and S. B. Bodrov, “Terahertz generation with tilted-front laser pulses in a contact-grating scheme,” Journal of Optical Society of America B 31, 2549–2557 (2014).
[Crossref]

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
[Crossref]

M. I. Bakunov, S. B. Bodrov, and M. V. Tsarev, “Terahertz emission from a laser pulse with tilted front: Phase-matching versus cherenkov effect,” Journal of Applied Physics 104, 073105 (2008).
[Crossref]

Busacca, A.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

Calendron, A.-L.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

Cankaya, H.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

Carbajo, S.

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
[Crossref]

Chirkin, A. S.

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
[Crossref]

Cino, A.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

Ding, Y. J.

Y. J. Ding, “Quasi-single-cycle terahertz pulses based on broadband-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,” IEEE Journal of Selected Topics in Quantum Electronics 10, 1171–1179 (2004).
[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,” Applied Physics Letters 98, 091106 (2011).
[Crossref]

Du, L.-H.

Dwayne Miller, R. J.

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

Ehberger, D.

C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

Fakhari, M.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
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A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
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W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
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E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
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L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense thz pulse,” Phys. Rev. ST Accel. Beams 17, 031301 (2014).
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J. A. Fülöp, Z. Ollmann, C. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, “Efficient generation of thz pulses with 0.4 mj energy,” Opt. Express 22, 20155–20163 (2014).
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D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
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L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kärtner, “250 w average power, 100 khz repetition rate cryogenic yb:yag amplifier for opcpa pumping,” Opt. Lett. 41, 492–495 (2016).
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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,” Applied Physics Letters 98, 091106 (2011).
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Hohenleutner, M.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Hong, K.-H.

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
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D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

Huang, S.-W.

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

Huang, W. R.

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
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Huber, R.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Huttner, U.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Jewariya, M.

M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
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Kampfrath, T.

T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
[Crossref]

Karsch, S.

Kärtner, F.

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
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Kärtner, F. X.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
[Crossref]

L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kärtner, “250 w average power, 100 khz repetition rate cryogenic yb:yag amplifier for opcpa pumping,” Opt. Lett. 41, 492–495 (2016).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
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C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
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Kira, M.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Koch, S. W.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Kozma, I. Z.

Krausz, F.

C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
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J. A. Fülöp, Z. Ollmann, C. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, “Efficient generation of thz pulses with 0.4 mj energy,” Opt. Express 22, 20155–20163 (2014).
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L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
[Crossref]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area thz pulse generation,” Opt. Express 10, 1161–1166 (2002).
[Crossref]

L’huillier, J.

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
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M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
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Matlis, N. H.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
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O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
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Meng, K.

Morandotti, R.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
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Moriena, G.

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

Nagai, M.

M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
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Nanni, E. A.

Nazarov, M. M.

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
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Nelson, K. A.

T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
[Crossref]

Nusinovich, G. S.

S. H. Gold and G. S. Nusinovich, “Review of high-power microwave source research,” Review of Scientific Instruments 68, 3945–3974 (1997).
[Crossref]

Ollmann, Z.

Ovchinnikov, A. V.

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

Pálfalvi, L.

Parisi, A.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
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Peccianti, M.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

Peter, A.

L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
[Crossref]

Polgár, K.

L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
[Crossref]

Ravi, K.

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
[Crossref]

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C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

Schimpf, D. N.

Schneider, W.

C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

Schubert, O.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
[Crossref]

Shkurinov, A. P.

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
[Crossref]

Shuvaev, A. V.

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
[Crossref]

Skrobol, C.

Stivala, S.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

Tanaka, K.

T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
[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,” Applied Physics Letters 98, 091106 (2011).
[Crossref]

M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
[Crossref]

Theuer, M.

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

Tomasino, A.

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
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J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

Tóth, G.

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense thz pulse,” Phys. Rev. ST Accel. Beams 17, 031301 (2014).
[Crossref]

Trebino, R.

S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005).
[Crossref] [PubMed]

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
[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,” Journal of Applied Physics 104, 073105 (2008).
[Crossref]

Urbanek, B.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
[Crossref]

Vicario, C.

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

Wu, X.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
[Crossref]

Yahaghi, A.

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
[Crossref]

Zapata, L. E.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kärtner, “250 w average power, 100 khz repetition rate cryogenic yb:yag amplifier for opcpa pumping,” Opt. Lett. 41, 492–495 (2016).
[Crossref]

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

Zeek, E.

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
[Crossref]

Zhai, Z. H.

Zhang, D.

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

W. R. Huang, A. Fallahi, X. Wu, H. Cankaya, A.-L. Calendron, K. Ravi, D. Zhang, E. A. Nanni, K.-H. Hong, and F. X. Kärtner, “Terahertz-driven, all-optical electron gun,” Optica 3, 1209–1212 (2016).
[Crossref]

Zhao, J.-H.

Zhong, S. C.

Zhu, L. G.

Applied Physics B (1)

J. L’huillier, G. Torosyan, M. Theuer, Y. Avetisyan, and R. Beigang, “Generation of thz radiation using bulk, periodically and aperiodically poled lithium niobate–part 1: Theory,” Applied Physics B 86, 185–196 (2007).
[Crossref]

Applied Physics Letters (1)

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,” Applied Physics Letters 98, 091106 (2011).
[Crossref]

IEEE Journal of Quantum Electronics (1)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE Journal of Quantum Electronics 24, 2530–2536 (1988).
[Crossref]

IEEE Journal of Selected Topics in Quantum Electronics (1)

Y. J. Ding, “Quasi-single-cycle terahertz pulses based on broadband-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,” IEEE Journal of Selected Topics in Quantum Electronics 10, 1171–1179 (2004).
[Crossref]

Journal of Applied Physics (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,” Journal of Applied Physics 104, 073105 (2008).
[Crossref]

L. Pálfalvi, J. Hebling, J. Kuhl, A. Peter, and K. Polgár, “Temperature dependence of the absorption and refraction of mg-doped congruent and stoichiometric linbo 3 in the thz range,” Journal of Applied Physics 97, 123505 (2005).
[Crossref]

Journal of Modern Optics (1)

W. R. Huang, S.-W. Huang, E. Granados, K. Ravi, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “Highly efficient terahertz pulse generation by optical rectification in stoichiometric and cryo-cooled congruent lithium niobate,” Journal of Modern Optics 62, 1486–1493 (2015).
[Crossref]

Journal of Optical Society of America B (3)

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbing crystals,” Journal of Optical Society of America B 28, 1724–1734 (2011).
[Crossref]

M. I. Bakunov and S. B. Bodrov, “Terahertz generation with tilted-front laser pulses in a contact-grating scheme,” Journal of Optical Society of America B 31, 2549–2557 (2014).
[Crossref]

M. Jewariya, M. Nagai, and K. Tanaka, “Enhancement of terahertz wave generation by cascaded χ (2) processes in linbo3,” Journal of Optical Society of America B 26, A101–A106 (2009).
[Crossref]

Nature Communications (1)

E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne Miller, and F. X. Kärtner, “Terahertz-driven linear electron acceleration,” Nature Communications 6, 8486 (2015).
[Crossref] [PubMed]

Nature Photonics (3)

D. Zhang, A. Fallahi, M. Hemmer, X. Wu, M. Fakhari, Y. Hua, H. Cankaya, A.-L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kärtner, “Segmented terahertz electron accelerator and manipulator (steam),” Nature Photonics 12, 336–342 (2018).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photonics 8, 119 (2014).
[Crossref]

T. Kampfrath, K. Tanaka, and K. A. Nelson, “Resonant and nonresonant control over matter and light by intense terahertz transients,” Nature Photonics 7, 680–690 (2013).
[Crossref]

Opt. Express (7)

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area thz pulse generation,” Opt. Express 10, 1161–1166 (2002).
[Crossref]

S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005).
[Crossref] [PubMed]

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010).
[Crossref]

J. A. Fülöp, L. Pálfalvi, M. C. Hoffmann, and J. Hebling, “Towards generation of mj-level ultrashort thz pulses by optical rectification,” Opt. Express 19, 15090–15097 (2011).
[Crossref]

J. A. Fülöp, Z. Ollmann, C. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, “Efficient generation of thz pulses with 0.4 mj energy,” Opt. Express 22, 20155–20163 (2014).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, E. A. Nanni, D. N. Schimpf, E. P. Ippen, and F. X. Kärtner, “Theory of terahertz generation by optical rectification using tilted-pulse-fronts,” Opt. Express 23, 5253–5276 (2015).
[Crossref]

S. C. Zhong, Z. H. Zhai, J. Li, L. G. Zhu, J. Li, K. Meng, Q. Liu, L.-H. Du, J.-H. Zhao, and Z. R. Li, “Optimization of terahertz generation from linbo 3 under intense laser excitation with the effect of three-photon absorption,” Opt. Express 23, 31313–31323 (2015).
[Crossref]

Opt. Lett. (1)

Optica (1)

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J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Optical and Quantum Electronics 28, 1759–1763 (1996).
[Crossref]

Optics Express (2)

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express 12, 4399–4410 (2004).
[Crossref]

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to thz generation by optical rectification using tilted pulse fronts,” Optics Express 22, 20239–20251 (2014).
[Crossref]

Optics Letters (1)

C. Vicario, A. V. Ovchinnikov, S. I. Ashitkov, M. B. Agranat, V. E. Fortov, and C. P. Hauri, “Generation of 0.9-mj thz pulses in dstms pumped by a cr:mg2sio4 laser,” Optics Letters 39, 6632–6635 (2014).
[Crossref]

Phys. Rev. Accel. Beams (1)

A. Fallahi, M. Fakhari, A. Yahaghi, M. Arrieta, and F. X. Kärtner, “Short electron bunch generation using single-cycle ultrafast electron guns,” Phys. Rev. Accel. Beams 19, 081302 (2016).
[Crossref]

Phys. Rev. ST Accel. Beams (1)

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense thz pulse,” Phys. Rev. ST Accel. Beams 17, 031301 (2014).
[Crossref]

Radiophysics and Quantum Electronics (1)

A. V. Shuvaev, M. M. Nazarov, A. P. Shkurinov, and A. S. Chirkin, “Čerenkov radiation excited by an ultrashort laser pulse with oblique amplitude front,” Radiophysics and Quantum Electronics 50, 922–928 (2007).
[Crossref]

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C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F. Krausz, and P. Baum, “All-optical control and metrology of electron pulses,” Science 352, 429–433 (2016).
[Crossref]

C. Ropers, “Electrons catch a terahertz wave,” Science 352, 410–411 (2016).
[Crossref]

Scientific Reports (1)

A. Tomasino, A. Parisi, S. Stivala, P. Livreri, A. Cino, A. Busacca, M. Peccianti, and R. Morandotti, “Wideband thz time domain spectroscopy based on optical rectification and electro-optic sampling,” Scientific Reports 3, 3116 (2013).
[Crossref] [PubMed]

Other (1)

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Figures (12)

Fig. 1
Fig. 1 A tilted-pulse-front setup comprised of a diffraction grating and an imaging system. The angularly dispersed pulse produces a tilted pulse front (red ellipse) with tilt angle γ, resulting in terahertz radiation propagating at an angle γ with respect to the direction of pump pulse propagation.
Fig. 2
Fig. 2 Explaining the finite radius of curvature produced by a finite value of group-velocity dispersion due to angular dispersion (GVD-AD) kT. (a) kT = 0 results in the wave vectors of various pump frequency components being spaced by roughly equal angles, which results in all terahertz vectors produced by each pair k ( ω + m Ω ) , k ( ω + ( m 1 ) Ω ) being parallel to each other. This results in a flat terahertz phase-front or R 1 = 0. (b) When | k T | > 0, different pump frequency components (dotted lines) are displaced from their original angles (solid lines), producing terahertz vectors pointing in slightly different directions. This results in a curved phase front.
Fig. 3
Fig. 3 Understanding the spatial distribution of terahertz energy in tilted-pulse-front setups. (a) Distribution of terahertz fluence in the xz plane of the crystal. The boundary of the crystal is represented by the white-dotted line. The asymmetry in the energy distribution is due to an interplay between the geometry and the effects of absorption and walk-off. (b) The distribution of E T H z ( Ω , x ) in x is the difference of two profiles centered at x = 0 (blue) and x = z tan γ (red) evident from Eq. (15a). At z=0, the two terms cancel out to result in null intensity (black-dotted). (c) For larger z = 2 cos γ / α, the displaced and attenuated second term (red) produces an asymmetric total transverse profile of larger absolute value (black-dotted). This is the case for z < 1 mm in (a). (d) For large enough z, the second term in Eq. (11a) is attenuated sufficiently and displaced faraway from x = 0 to result in the | E T H z ( Ω , x , z ) | reaching a maximum value. This is the case for z > 2mm in (a).
Fig. 4
Fig. 4 Comparison of conversion efficiencies η calculated from expressions for spectra in Eq. (11a) and Eq. (8a) for a lithium niobate crystal. (a) At T=300 K, Δ η 20 % if constraints in Eq. (10a) (Black-dotted) and Eq. (10b) (Blue-dotted) are satisfied. (b) At T=100 K, relative error is large predominantly due to violation of constraint in Eq. (10a) due to reduced absorption at cryogenic temperatures.
Fig. 5
Fig. 5 Spatio-temporal evolution of terahertz transients for a τ 0 = 500 fs, w 0 = 2 mm beam radius in lithium niobate at T=300 K. The propagation of the terahertz transient at an angle γ 63 is evident in panels (a)-(d). A continuous growth in intensity is seen. Due to the large duration of the pump pulse, spatial inhomogeneities are relatively low.
Fig. 6
Fig. 6 Spatio-temporal evolution of terahertz transients for a τ = 50 fs, w 0 = 2 mm beam radius in lithium niobate at T=300 K. Due to the short duration of the pump pulse, spatial in homogeneities are obviously present in (d).
Fig. 7
Fig. 7 (a) Relative error of peak field calculations between Eqs. (15a) and (8c). Consistent with Fig. 4, the relative error is 20 % at T=300 K upon satisfying the constraints in Eqs. (10a) (black-dotted) and (10b) (blue-dotted). (b) At T=100 K, the relative error is large, analogous to Fig. 4(b) due to violation of Eq. (10a) resulting from small absorption at cryogenic temperatures.
Fig. 8
Fig. 8 Variation of pulse-front tilt angle γ just inside a lithium niobate crystal with lens-to-crystal distance variations Δ s 2. Angular dispersion (blue) does not change with s2 as expected. Variations of pulse-front tilt due to spatial-chirp ζ ( z ) and radius of curvature R ( z ) (third term, Eq. (6)) however produces significant variations in tilt-angle. The variations are more severe for short focal length of f = 7.5 cm (red) compared to longer focal length of f = 15 cm (green). Variation of tilt-angle due to spatial-chirp and radius of curvature inside the crystal (black-dotted) is negligible due to reduction in growth of Δ ζ ( z ) due to reduced angular dispersion (reduces by a factor n ( ω 0 ) due to Snell’s law) as well as a slower rate of change of radius of curvature.
Fig. 9
Fig. 9 Terahertz spectra obtained from Eqs. (8a)-(8b) in lithium niobate at T=300 K. (a) Spatial distribution of the terahertz spectrum for a pump pulse duration of τ 0 = 500 fs and w 0 = 2 mm shows relatively homogeneous properties consistent with transients in Fig. 5. (b) Spatial distribution of the terahertz spectrum for a pump pulse duration of τ 0 = 50 fs and w 0 = 2 mm shows significant asymmetry analogous to Fig. 6. (c) Average spectra | E T H z ( Ω , x , z ) | 2 d x for varying beam sizes shows the reduction in frequency bandwidth for larger beam radii due to the effects of group-velocity dispersion due to angular dispersion (GVD-AD). (d) Average spectra for τ 0 = 50 fs shows evengreater reduction of frequency due to enhancement of GVD-AD effects at short durations (see Eq. (14)).
Fig. 10
Fig. 10 Central terahertz frequencies as a function of τ 0 , w 0 calculated using Eq. (8a) in lithium niobate for: (a) No GVD-AD, i.e. k T = 0 , T = 300 K shows a monotonic increase with shorter durations and no w0 dependence, consistent with Eq. (14). (b) For k T = 1.3 × 10 23 s 2 / m,T=300 K, the reduction of terahertz frequency for large beam sizes is evident. (c) For k T = 0 , T = 100 K, an increase in average frequency compared to (a) due to reduced absorption is seen. (d) For non-zero k T , T = 100 K, the frequency is closer to that in (b), indicating relative importance of GVD-AD effects.
Fig. 11
Fig. 11 Conversion efficiency η as a function of τ 0 , w 0 based on Eq. (8a) in lithium niobate. (a) For k T = 0 , T = 300 K, η saturates beyond a critical beam radius and shows an optimum value at 250 fs. (b) For k T = 1.3 × 10 23 s 2 / m and T=300 K, effects of GVD-AD increase optimal duration as alluded to by Eq. (13a). (c) A reduction rather than saturation at large w0 is seen For k T = 0 , T = 100 K, saturation occurs at much larger w0 due to increased interaction lengths by virtue of reduced absorption. (d) For | k T | > 0, peak efficiency occurs at larger τ compared to (c), consistent with (b).
Fig. 12
Fig. 12 Peak electric field obtained from Eqs. (8a)(8c) for (a) k T = 0 , T = 300 K. (b) k T = 1.3 × 10 23 s 2 / m , T = 300 K. (c) k T = 0 , T = 100 K. (d) | k T | > 0 , T = 100 K. The results are essentially proportional to the product of η and f m a x. As a result peak fields are obtained at lower pump durations for T = 100 K whereas they are relatively flat for T = 300 K. The values are well described by Eq. (16) for T=300 K due to satisfaction of constraints in Eqs. (10a)(10b).

Tables (2)

Tables Icon

Table 1 List of parameters used in calculations.

Tables Icon

Table 2 List of variables

Equations (38)

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2 E T H z ( Ω , x , z ) + k 2 ( Ω ) E T H z ( Ω , x , z ) = j k ( Ω ) α ( Ω ) E T H z ( Ω , x , z ) Ω 2 ε 0 c 2 P T H z ( Ω , x , z )
P T H z ( Ω , x , z ) = ε 0 χ ( 2 ) 0 E o p ( ω + Ω , x , z ) E o p * ( ω , x , z ) d ω
2 E T H z ( Ω , k x , z ) z 2 + k z 2 ( Ω , k x ) E T H z ( Ω , k x , z ) = j k ( Ω ) α ( Ω ) E T H z ( Ω , k x , z ) Ω 2 P T H z ( Ω , k x , z ) e j Ω v g 1 z ε 0 c 2
E T H z ( Ω , k x , z ) = e α ( Ω ) z 2 cos γ 0 z j Ω 2 2 ε 0 k z c 2 P T H z ( Ω , k x , z ) e j Δ k ( Ω , k x ) z + α ( Ω ) z 2 cos γ d z
Δ k ( Ω , k x ) = k z Ω n g c 1
E o p ( ω , x , z ) = E 0 e Δ ω 2 τ 0 2 4 e ( x ζ ( z ) Δ ω ) 2 w 0 2 e j ϕ 0 Δ ω 2 e j ω n ( ω ) ( x ζ ( z ) Δ ω ) 2 2 c R 0 ( z ) e j [ k 0 + v g 1 Δ ω + k m Δ ω 2 ] z e j [ k 0 β 0 Δ ω + k T Δ ω 2 ] x
P T H z ( Ω , x , z ) = 2 π τ [ ε 0 E 0 2 χ ( 2 ) e Ω 2 τ 2 8 e 2 [ ϕ 0 + k T x + k m z ] 2 Ω 2 τ 2 e 2 x 2 w 0 2 ] e j [ n g 2 c R 0 ( z ) + 8 ζ ( z ) k T τ 2 w 0 2 ] Ω x 2 × e j v g 1 Ω z e j ( k 0 β 0 + 2 ζ ( z ) ϕ 0 τ 0 2 w 0 2 / 4 + ζ ( z ) 2 ω 0 n ( ω 0 ) ζ ( z ) c R 0 ( z ) ) Ω x
τ = ( τ 0 2 + 4 ζ ( z ) 2 w 0 2 ) 1 2
tan γ = k 0 β 0 v g + 2 ζ ( z ) ϕ 0 v g τ 0 2 w 0 2 4 + ζ ( z ) 2 ω 0 n ( ω 0 ) ζ ( z ) v g c R 0 ( z )
P T H z ( Ω , k x , z ) = ε 0 E 0 2 χ ( 2 ) w 2 τ e Ω 2 τ 2 8 e 2 ϕ 2 Ω 2 τ 2 + w 0 2 k T 2 Ω 2 e ( k x k 0 β Ω ) 2 [ w 2 2 j Ω w 2 8 c R ( z ) ] 4 e j ( k x k 0 β Ω ) x 0
ϕ = ϕ 0 + k m z
R 1 ( z ) = n ( ω 0 ) R 0 1 ( z ) + 16 c ζ ( z ) k T τ 2 w 0 2
w 2 = w 0 2 + k T 2 Ω 2 τ 2
β = k 0 β 0 n ( ω 0 ) ω 0 ζ ( z ) c R 0 ( z ) + 2 ζ ( z ) ϕ 0 τ 0 2 w 0 2 4 + ζ ( z ) 2
x 0 = ϕ k T k T 2 Ω 2 w 2 τ 2
E T H z ( Ω , k x , z ) = j Ω 2 2 ε 0 c 2 k z P T H z ( Ω , k x , z ) [ e j Ω v g 1 z e α ( Ω ) z 2 cos γ e j k z z α ( Ω ) 2 cos γ + j Δ k ( Ω , k x ) ] l e t h z k x c
E T H z ( Ω , x , z ) = F x 1 { E T H z ( Ω , k x , z ) }
E T H z ( t , x , z ) = 1 2 F t 1 [ E T H z ( Ω , x , z ) Θ ( Ω ) + E T H z ( Ω , x , z ) Θ ( Ω ) ]
η ( z ) = 2 π c ε 0 F p u m p π w 0 n T H z ( Ω ) | E T H z ( Ω , x , z ) | 2 d x d Ω
E T H z ( Ω , k x , z ) = j Ω 2 2 ε 0 c 2 k z P T H z ( Ω , k x , z ) sinc ( Δ k z 2 )
w 0 > 4 2 sin γ α
4 w 0 2 k T 2 τ 4 1
E THz (Ω,x,z)= jΩ n THz (Ω)α(Ω)c χ (2) E 0 2 2π τ e jΩ v g 1 z e jk(Ω)sinγx [ e Ω 2 τ 1 2 8 e x 2 (2 w 0 2 + jΩ 2cR(z) ) e α(Ω)z 2cosγ e Ω 2 τ 2 2 8 e (xztanγ) 2 (2 w 0 2 + jΩ 2cR(z) ) ]
τ 1 = τ [ 1 + 16 x 2 k T 2 τ 4 + 16 ϕ 2 τ 4 ] 1 / 2
τ 2 = τ [ 1 + 16 ( x z tan γ ) 2 k T 2 τ 4 + 16 ϕ 2 τ 4 + 4 β T τ 2 ] 1 / 2
η ( z ) = F p u m p χ ( 2 )   2 2 π ε 0 n T H z n I R 2 c 3 F ( Ω ) [ ( 1 e α z 2 cos γ ) 2 + 2 e α z 2 cos γ ( 1 e z 2 tan 2 γ w 0 2 ) ]
F ( Ω ) = [ 0 Ω 2 e Ω 2 τ 2 4 α ( Ω ) 2 1 + k T 2 Ω 2 w 0 2 τ 2 d Ω ]
η F p u m p χ ( 2 )   2 π ε 0 n T H z n I R 2 c 3 τ 0 3 [ 1 3 k T 2 w 0 2 τ 0 4 ]
τ o p t 1.62 ( | k T | w 0 ) 1 / 2
Ω m a x 2 τ [ 1 2 k T 2 w 0 2 τ 0 4 ]
E T H z ( t , x , z ) = 8 π χ ( 2 ) E 0 2 n T H z α ( 2 / τ 0 ) c τ [ 1 τ 1 3 e 2 x 2 w 0 2 t e 2 t 2 τ 1 2 1 τ 2 3 e α 0 z 2 cos γ e 2 ( x z tan γ ) 2 w 0 2 t " e 2 t " 2 τ 2 2 ]
t = t x sin γ v T H z 1 z cos γ v T H z 1 + x 2 2 c R ( z )
t " = t x sin γ v T H z 1 z cos γ v T H z 1 + ( x z tan γ ) 2 2 c R ( z )
| E T H z , m a x ( τ 0 ) | 4 π χ ( 2 ) E 0 2 e 1 2 n T H z α ( 2 / τ 0 ) c τ 0 3
f ( ω ) = F t { f ( t ) } = 1 2 π f ( t ) e j ω t d t f ( t ) = F t 1 { f ( ω ) } = f ( ω ) e j ω t d ω
g ( k x ) = F x { g ( x ) } = 1 2 π g ( x ) e j k x . x d x g ( x ) = F x 1 { g ( k x ) } = g ( k x ) e j k x x d k x
| f ( x ) | 2 d x = 2 π | f ( k x ) | 2 d k x
0 z P T H z ( Ω , k x , z ) e j Δ k z + α 2 z d z = P T H z ( Ω , k x , z ) e j Δ k z + α 2 z α / 2 + j Δ k P T H z   ' ( Ω , k x , z ) e j Δ k z + α 2 z ( α / 2 + j Δ k ) 2 + P T H z " ( Ω , k x , z ) e j Δ k z + α 2 z ( α / 2 + j Δ k ) 3 ..

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