Abstract

The superposition of signal and idler pulses in dual-chirped optical parametric amplification is proposed for the efficient generation of intensity-modulated pulses with periodic modulation. Both the duration and the modulation period are easily and independently adjustable. Numerical simulations for a three-stage optical parametric amplifier system predicted an efficiency as high as ~50% for about 40 mJ of output pulse energy at a wavelength of 2 µm. Sources of such intensity-modulated pulses near 1.6 µm or 2 µm wavelength, pumped by Ti:sapphire or Yb-doped lasers, can be ideally suited for intense multicycle THz pulse generation with tunable frequency and bandwidth by optical rectification for example in organic, semiconductor, or lithium niobate materials.

© 2017 Optical Society of America

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References

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2017 (5)

F. Ahr, S. W. Jolly, N. H. Matlis, S. Carbajo, T. Kroh, K. Ravi, D. N. Schimpf, J. Schulte, H. Ishizuki, T. Taira, A. R. Maier, and F. X. Kärtner, “Narrowband terahertz generation with chirped-and-delayed laser pulses in periodically poled lithium niobate,” Opt. Lett. 42(11), 2118–2121 (2017).
[PubMed]

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

Y. Fu, E. J. Takahashi, and K. Midorikawa, “Energy scaling of infrared femtosecond pulses by dual-chirped optical parametric amplification,” IEEE Photonics J. 9(3), 1503108 (2017).

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

2016 (8)

J. A. Fülöp, G. Polónyi, B. Monoszlai, G. Andriukaitis, T. Balciunas, A. Pugzlys, G. Arthur, A. Baltuska, and J. Hebling, “Highly efficient scalable monolithic semiconductor terahertz pulse source,” Optica 3(10), 1075–1078 (2016).

G. Polónyi, B. Monoszlai, G. Gäumann, E. J. Rohwer, G. Andriukaitis, T. Balciunas, A. Pugzlys, A. Baltuska, T. Feurer, J. Hebling, and J. A. Fülöp, “High-energy terahertz pulses from semiconductors pumped beyond the three-photon absorption edge,” Opt. Express 24(21), 23872–23882 (2016).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

E. Kaksis, G. Almási, J. A. Fülöp, A. Pugžlys, A. Baltuška, and G. Andriukaitis, “110-mJ 225-fs cryogenically cooled Yb:CaF2 multipass amplifier,” Opt. Express 24(25), 28915–28922 (2016).
[PubMed]

K. Ravi, D. N. Schimpf, and F. X. Kärtner, “Pulse sequences for efficient multi-cycle terahertz generation in periodically poled lithium niobate,” Opt. Express 24(22), 25582–25607 (2016).
[PubMed]

A. Sharma, Z. Tibai, and J. Hebling, “Intense tera-hertz laser driven proton acceleration in plasmas,” Phys. Plasmas 23(6), 063111 (2016).

D. Nicoletti and A. Cavalleri, “Nonlinear light-matter interaction at terahertz frequencies,” Adv. Opt. Photonics 8(3), 401–464 (2016).

2015 (5)

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

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[PubMed]

J. Lu, H. Y. Hwang, X. Li, S.-H. Lee, O.-P. Kwon, and K. A. Nelson, “Tunable multi-cycle THz generation in organic crystal HMQ-TMS,” Opt. Express 23(17), 22723–22729 (2015).
[PubMed]

S. Carbajo, J. Schulte, X. Wu, K. Ravi, D. N. Schimpf, and F. X. Kärtner, “Efficient narrowband terahertz generation in cryogenically cooled periodically poled lithium niobate,” Opt. Lett. 40(24), 5762–5765 (2015).
[PubMed]

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

2014 (7)

K. Ravi, W. R. Huang, S. Carbajo, X. Wu, and F. Kärtner, “Limitations to THz generation by optical rectification using tilted pulse fronts,” Opt. Express 22(17), 20239–20251 (2014).
[PubMed]

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

L. Pálfalvi, J. A. Fülöp, Gy. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).

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,” Opt. Lett. 39(23), 6632–6635 (2014).
[PubMed]

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(17), 20155–20163 (2014).
[PubMed]

T. I. Oh, Y. J. Yoo, and Y. S. You, “andk K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105, 041103 (2014).

Z. Hong, Q. Zhang, P. Lan, and P. Lu, “Generation of few-cycle infrared pulses from a degenerate dual-pump OPCPA,” Opt. Express 22(5), 5544–5557 (2014).
[PubMed]

2013 (4)

2012 (1)

2011 (4)

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19(8), 7190–7212 (2011).
[PubMed]

M. C. Hoffmann and J. A. Fülöp, “Intense ultrashort terahertz pulses: generation and applications,” J. Phys. D Appl. Phys. 44, 083001 (2011).

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

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

2009 (2)

K. L. Vodopyanov, “Terahertz-wave generation with periodically inverted gallium arsenide,” Laser Phys. 19(2), 305–321 (2009).

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95, 103304 (2009).

2008 (2)

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

A. Gaydardzhiev, I. Nikolov, I. Buchvarov, V. Petrov, and F. Noack, “Ultrabroadband operation of a femtosecond optical parametric generator based on BiB3O6 in the near-IR,” Opt. Express 16(4), 2363–2373 (2008).
[PubMed]

2006 (1)

T. Plettner, P. P. Lu, and R. L. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel. Beams 9, 111301 (2006).

2005 (1)

K. Wynne and J. J. Carey, “An integrated description of terahertz generation through optical rectification, charge transfer, and current surge,” Opt. Commun. 256(4–6), 400–413 (2005).

2004 (1)

2003 (1)

T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal Coherent Control of Lattice Vibrational Waves,” Science 299(5605), 374–377 (2003).
[PubMed]

2002 (2)

2000 (2)

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

1999 (1)

1998 (1)

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998).

1997 (3)

1996 (1)

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

1994 (1)

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

1992 (1)

D. A. Roberts, “Simplified Characterization of Uniaxial and Biaxial Nonlinear Optical Crystals: A Plea for Standardization of Nomenclature and Conventions,” IEEE J. Quantum Electron. 28(10), 2057–2074 (1992).

1986 (1)

Agranat, M. B.

Ahr, F.

F. Ahr, S. W. Jolly, N. H. Matlis, S. Carbajo, T. Kroh, K. Ravi, D. N. Schimpf, J. Schulte, H. Ishizuki, T. Taira, A. R. Maier, and F. X. Kärtner, “Narrowband terahertz generation with chirped-and-delayed laser pulses in periodically poled lithium niobate,” Opt. Lett. 42(11), 2118–2121 (2017).
[PubMed]

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Ališauskas, S.

Almási, G.

Andriukaitis, G.

Arthur, G.

Ashitkov, S. I.

Auston, D. H.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

Backus, S.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998).

Balciunas, T.

Baltuska, A.

Baltuška, A.

Biegert, J.

Blanchard, F.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

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

Branning, D.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Broeuf, N.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Buchvarov, I.

Byer, R. L.

T. Plettner, P. P. Lu, and R. L. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel. Beams 9, 111301 (2006).

Calendron, A.-L.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Cankaya, H.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Carbajo, S.

Carey, J. J.

K. Wynne and J. J. Carey, “An integrated description of terahertz generation through optical rectification, charge transfer, and current surge,” Opt. Commun. 256(4–6), 400–413 (2005).

Cavalleri, A.

D. Nicoletti and A. Cavalleri, “Nonlinear light-matter interaction at terahertz frequencies,” Adv. Opt. Photonics 8(3), 401–464 (2016).

Chang, Z.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Chaperot, I.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Chen, Z.

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

Chew, A.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Chua, C. F.

Cirmi, G.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Cooke, D. G.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Cousin, S. L.

Dauler, E.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Deng, Y.

DeSalvo, R.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

Doi, A.

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

Durfee, C. G.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998).

Ebrahimzadeh, M.

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO3, KNbO3, β-BaB2O4, and LiB3O5,” Opt. Commun. 142, 294–298 (1997).

Engelbrecht, M.

Fallahi, A.

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

L. J. Wong, A. Fallahi, and F. X. Kärtner, “Compact electron acceleration and bunch compression in THz waveguides,” Opt. Express 21(8), 9792–9806 (2013).
[PubMed]

Fattahi, H.

Fernández, A.

Feurer, T.

Flöry, T.

Fortov, V. E.

Froberg, N. M.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

Fu, Y.

Y. Fu, E. J. Takahashi, and K. Midorikawa, “Energy scaling of infrared femtosecond pulses by dual-chirped optical parametric amplification,” IEEE Photonics J. 9(3), 1503108 (2017).

Fülöp, J. A.

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

J. A. Fülöp, G. Polónyi, B. Monoszlai, G. Andriukaitis, T. Balciunas, A. Pugzlys, G. Arthur, A. Baltuska, and J. Hebling, “Highly efficient scalable monolithic semiconductor terahertz pulse source,” Optica 3(10), 1075–1078 (2016).

G. Polónyi, B. Monoszlai, G. Gäumann, E. J. Rohwer, G. Andriukaitis, T. Balciunas, A. Pugzlys, A. Baltuska, T. Feurer, J. Hebling, and J. A. Fülöp, “High-energy terahertz pulses from semiconductors pumped beyond the three-photon absorption edge,” Opt. Express 24(21), 23872–23882 (2016).
[PubMed]

E. Kaksis, G. Almási, J. A. Fülöp, A. Pugžlys, A. Baltuška, and G. Andriukaitis, “110-mJ 225-fs cryogenically cooled Yb:CaF2 multipass amplifier,” Opt. Express 24(25), 28915–28922 (2016).
[PubMed]

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

L. Pálfalvi, J. A. Fülöp, Gy. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).

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(17), 20155–20163 (2014).
[PubMed]

M. C. Hoffmann and J. A. Fülöp, “Intense ultrashort terahertz pulses: generation and applications,” J. Phys. D Appl. Phys. 44, 083001 (2011).

Z. Tibai, L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “THz-pulse-driven particle accelerators,” in 4th EOS Topical Meeting on Terahertz Science & Technilogy (Camogli, Italy, 2014).

Gaižaiskas, E.

Gäumann, G.

Gaydardzhiev, A.

Granados, E.

Grigonis, R.

Gu, X.

Guérin, S.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Hagan, D. J.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

Hauri, C. P.

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[PubMed]

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,” Opt. Lett. 39(23), 6632–6635 (2014).
[PubMed]

Hebling, J.

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

A. Sharma, Z. Tibai, and J. Hebling, “Intense tera-hertz laser driven proton acceleration in plasmas,” Phys. Plasmas 23(6), 063111 (2016).

J. A. Fülöp, G. Polónyi, B. Monoszlai, G. Andriukaitis, T. Balciunas, A. Pugzlys, G. Arthur, A. Baltuska, and J. Hebling, “Highly efficient scalable monolithic semiconductor terahertz pulse source,” Optica 3(10), 1075–1078 (2016).

G. Polónyi, B. Monoszlai, G. Gäumann, E. J. Rohwer, G. Andriukaitis, T. Balciunas, A. Pugzlys, A. Baltuska, T. Feurer, J. Hebling, and J. A. Fülöp, “High-energy terahertz pulses from semiconductors pumped beyond the three-photon absorption edge,” Opt. Express 24(21), 23872–23882 (2016).
[PubMed]

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

L. Pálfalvi, J. A. Fülöp, Gy. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).

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(17), 20155–20163 (2014).
[PubMed]

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

A. Stepanov, J. Hebling, and J. Kuhl, “Generation, tuning, and shaping of narrow-band, picosecond THz pulses by two-beam excitation,” Opt. Express 12(19), 4650–4658 (2004).
[PubMed]

J. Hebling, G. Almási, I. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large area THz-pulse generation,” Opt. Express 10(21), 1161–1166 (2002).
[PubMed]

Z. Tibai, L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “THz-pulse-driven particle accelerators,” in 4th EOS Topical Meeting on Terahertz Science & Technilogy (Camogli, Italy, 2014).

Hemmer, M.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

H. Hoogland, A. Thai, D. Sánchez, S. L. Cousin, M. Hemmer, M. Engelbrecht, J. Biegert, and R. Holzwarth, “All-PM coherent 2.05 µm Thulium/Holmium fiber frequency comb source at 100 MHz with up to 0.5 W average power and pulse duration down to 135 fs,” Opt. Express 21(25), 31390–31394 (2013).
[PubMed]

Hirori, H.

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

Hoffmann, M. C.

M. C. Hoffmann and J. A. Fülöp, “Intense ultrashort terahertz pulses: generation and applications,” J. Phys. D Appl. Phys. 44, 083001 (2011).

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

Holzwarth, R.

Hong, K.-H.

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

S.-W. Huang, E. Granados, W. R. Huang, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “High conversion efficiency, high energy terahertz pulses by optical rectification in cryogenically cooled lithium niobate,” Opt. Lett. 38(5), 796–798 (2013).
[PubMed]

Hong, Z.

Hoogland, H.

Hu, B. B.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

Huang, S.-W.

Huang, W. R.

Hwang, H. Y.

Ishizuki, H.

Ito, R.

Jaeger, G.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Jolly, S. W.

Jundt, D.

Kaksis, E.

Kapteyn, H. C.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998).

Karpowicz, N.

Karsch, S.

Kärtner, F.

Kärtner, F. X.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

F. Ahr, S. W. Jolly, N. H. Matlis, S. Carbajo, T. Kroh, K. Ravi, D. N. Schimpf, J. Schulte, H. Ishizuki, T. Taira, A. R. Maier, and F. X. Kärtner, “Narrowband terahertz generation with chirped-and-delayed laser pulses in periodically poled lithium niobate,” Opt. Lett. 42(11), 2118–2121 (2017).
[PubMed]

K. Ravi, D. N. Schimpf, and F. X. Kärtner, “Pulse sequences for efficient multi-cycle terahertz generation in periodically poled lithium niobate,” Opt. Express 24(22), 25582–25607 (2016).
[PubMed]

S. Carbajo, J. Schulte, X. Wu, K. Ravi, D. N. Schimpf, and F. X. Kärtner, “Efficient narrowband terahertz generation in cryogenically cooled periodically poled lithium niobate,” Opt. Lett. 40(24), 5762–5765 (2015).
[PubMed]

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

S.-W. Huang, E. Granados, W. R. Huang, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “High conversion efficiency, high energy terahertz pulses by optical rectification in cryogenically cooled lithium niobate,” Opt. Lett. 38(5), 796–798 (2013).
[PubMed]

L. J. Wong, A. Fallahi, and F. X. Kärtner, “Compact electron acceleration and bunch compression in THz waveguides,” Opt. Express 21(8), 9792–9806 (2013).
[PubMed]

Kato, K.

Kienberger, R.

Kitamoto, A.

Klingebiel, S.

Kobayashi, T.

Kondo, T.

Kozma, I.

Krausz, F.

Kroh, T.

Kuhl, J.

Kwon, O.-P.

Lan, P.

Lee, S.-H.

Lee, Y.-S.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

Légaré, F.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Li, J.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Li, X.

Lin, K.-H.

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95, 103304 (2009).

Lombosi, C.

Lombosi, Cs.

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

Lu, J.

Lu, P.

Lu, P. P.

T. Plettner, P. P. Lu, and R. L. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel. Beams 9, 111301 (2006).

Maier, A. R.

Malevich, P.

Marcus, G.

Martinez, O. E.

Matlis, N. H.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

F. Ahr, S. W. Jolly, N. H. Matlis, S. Carbajo, T. Kroh, K. Ravi, D. N. Schimpf, J. Schulte, H. Ishizuki, T. Taira, A. R. Maier, and F. X. Kärtner, “Narrowband terahertz generation with chirped-and-delayed laser pulses in periodically poled lithium niobate,” Opt. Lett. 42(11), 2118–2121 (2017).
[PubMed]

Meade, T.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

Mechler, M.

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

Mechler, M. I.

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

Metzger, T.

Midorikawa, K.

Y. Fu, E. J. Takahashi, and K. Midorikawa, “Energy scaling of infrared femtosecond pulses by dual-chirped optical parametric amplification,” IEEE Photonics J. 9(3), 1503108 (2017).

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19(8), 7190–7212 (2011).
[PubMed]

Migdall, A.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Miller, R. J. D.

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

Monoszlai, B.

Morandotti, R.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Moriena, G.

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

Mücke, O. D.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19(8), 7190–7212 (2011).
[PubMed]

Muller, A.

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Murnane, M. M.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69(3), 1207–1223 (1998).

Nanni, E. A.

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

Nelson, K. A.

J. Lu, H. Y. Hwang, X. Li, S.-H. Lee, O.-P. Kwon, and K. A. Nelson, “Tunable multi-cycle THz generation in organic crystal HMQ-TMS,” Opt. Express 23(17), 22723–22729 (2015).
[PubMed]

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95, 103304 (2009).

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal Coherent Control of Lattice Vibrational Waves,” Science 299(5605), 374–377 (2003).
[PubMed]

Nicoletti, D.

D. Nicoletti and A. Cavalleri, “Nonlinear light-matter interaction at terahertz frequencies,” Adv. Opt. Photonics 8(3), 401–464 (2016).

Nikolov, I.

Noack, F.

Norris, T. B.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

Oh, T. I.

T. I. Oh, Y. J. Yoo, and Y. S. You, “andk K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105, 041103 (2014).

Ollmann, Z.

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

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(17), 20155–20163 (2014).
[PubMed]

Ossiander, M.

Ovchinnikov, A. V.

Ozaki, T.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Pálfalvi, L.

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

L. Pálfalvi, J. A. Fülöp, Gy. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).

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(17), 20155–20163 (2014).
[PubMed]

Z. Tibai, L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “THz-pulse-driven particle accelerators,” in 4th EOS Topical Meeting on Terahertz Science & Technilogy (Camogli, Italy, 2014).

Perlin, V.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

Pervak, V.

Petrov, V.

Phua, P. B.

Plettner, T.

T. Plettner, P. P. Lu, and R. L. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” Phys. Rev. Spec. Top. Accel. Beams 9, 111301 (2006).

Polónyi, G.

Polónyi, Gy.

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

Pugzlys, A.

Pugžlys, A.

Ravi, K.

Reichert, F.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

Ren, X.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Roberts, D. A.

D. A. Roberts, “Simplified Characterization of Uniaxial and Biaxial Nonlinear Optical Crystals: A Plea for Standardization of Nomenclature and Conventions,” IEEE J. Quantum Electron. 28(10), 2057–2074 (1992).

Rohwer, E. J.

Ropagnol, X.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Said, A. A.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

Sánchez, D.

Schimpf, D. N.

Schmidt, B. E.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Schulte, J.

Schwarz, A.

Shalaby, M.

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[PubMed]

Sharma, A.

A. Sharma, Z. Tibai, and J. Hebling, “Intense tera-hertz laser driven proton acceleration in plasmas,” Phys. Plasmas 23(6), 063111 (2016).

Sheik-Bahae, M.

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO3, KNbO3, β-BaB2O4, and LiB3O5,” Opt. Commun. 142, 294–298 (1997).

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

Shirane, M.

Shoji, I.

Sirutkaitis, V.

Skrobol, C.

Small, D. L.

Stepanov, A.

Taira, T.

Takahashi, E. J.

Y. Fu, E. J. Takahashi, and K. Midorikawa, “Energy scaling of infrared femtosecond pulses by dual-chirped optical parametric amplification,” IEEE Photonics J. 9(3), 1503108 (2017).

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19(8), 7190–7212 (2011).
[PubMed]

Tan, L. H.

Tanaka, K.

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

Thai, A.

Thiré, N.

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

Tibai, Z.

A. Sharma, Z. Tibai, and J. Hebling, “Intense tera-hertz laser driven proton acceleration in plasmas,” Phys. Plasmas 23(6), 063111 (2016).

Z. Tibai, L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “THz-pulse-driven particle accelerators,” in 4th EOS Topical Meeting on Terahertz Science & Technilogy (Camogli, Italy, 2014).

Tokodi, L.

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

Tóth, Gy.

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

L. Pálfalvi, J. A. Fülöp, Gy. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).

Ueffing, M.

Umemura, N.

Van Stryland, E. W.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

Vaughan, J. C.

T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal Coherent Control of Lattice Vibrational Waves,” Science 299(5605), 374–377 (2003).
[PubMed]

Verhoef, A. J.

Vicario, C.

Vodopyanov, K. L.

K. L. Vodopyanov, “Terahertz-wave generation with periodically inverted gallium arsenide,” Laser Phys. 19(2), 305–321 (2009).

Wang, Y.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Weling, A. S.

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

Werley, C. A.

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95, 103304 (2009).

Winful, H.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

Wong, L. J.

Wu, X.

Wu, Y.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Wynne, K.

K. Wynne and J. J. Carey, “An integrated description of terahertz generation through optical rectification, charge transfer, and current surge,” Opt. Commun. 256(4–6), 400–413 (2005).

Yeh, K.-L.

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

Yin, Y.

Y. Yin, A. Chew, X. Ren, J. Li, Y. Wang, Y. Wu, and Z. Chang, “Towards Terawatt Sub-Cycle Long-Wave Infrared Pulses via Chirped Optical Parametric Amplification and Indirect Pulse Shaping,” Sci. Rep. 8, 45794 (2017).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

Yoo, Y. J.

T. I. Oh, Y. J. Yoo, and Y. S. You, “andk K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105, 041103 (2014).

Yoshida, K.

You, Y. S.

T. I. Oh, Y. J. Yoo, and Y. S. You, “andk K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105, 041103 (2014).

Zapata, L. E.

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

S.-W. Huang, E. Granados, W. R. Huang, K.-H. Hong, L. E. Zapata, and F. X. Kärtner, “High conversion efficiency, high energy terahertz pulses by optical rectification in cryogenically cooled lithium niobate,” Opt. Lett. 38(5), 796–798 (2013).
[PubMed]

Zelmon, D. E.

Zhang, Q.

Zhou, X.

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

Adv. Opt. Photonics (1)

D. Nicoletti and A. Cavalleri, “Nonlinear light-matter interaction at terahertz frequencies,” Adv. Opt. Photonics 8(3), 401–464 (2016).

Appl. Opt. (1)

Appl. Phys. Lett. (7)

T. I. Oh, Y. J. Yoo, and Y. S. You, “andk K. Y. Kim, “Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling,” Appl. Phys. Lett. 105, 041103 (2014).

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

Z. Chen, X. Zhou, C. A. Werley, and K. A. Nelson, “Generation of high power tunable multicycle terahertz pulses,” Appl. Phys. Lett. 99, 071102 (2011).

A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64(2), 137–139 (1994).

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, and T. B. Norris, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505 (2000).

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle terahertz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95, 103304 (2009).

F. Blanchard, B. E. Schmidt, X. Ropagnol, N. Thiré, T. Ozaki, R. Morandotti, D. G. Cooke, and F. Légaré, “Terahertz pulse generation from bulk GaAs by a tilted-pulse-front excitation at 1.8 μm,” Appl. Phys. Lett. 105(24), 241106 (2014).

IEEE J. Quantum Electron. (2)

D. A. Roberts, “Simplified Characterization of Uniaxial and Biaxial Nonlinear Optical Crystals: A Plea for Standardization of Nomenclature and Conventions,” IEEE J. Quantum Electron. 28(10), 2057–2074 (1992).

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids,” IEEE J. Quantum Electron. 32(8), 1324–1333 (1996).

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

Gy. Polónyi, M. I. Mechler, J. Hebling, and J. A. Fülöp, “Prospects of Semiconductor Terahertz Pulse Sources,” IEEE J. Sel. Top. Quantum Electron. 23(4), 8501208 (2017).

IEEE Photonics J. (1)

Y. Fu, E. J. Takahashi, and K. Midorikawa, “Energy scaling of infrared femtosecond pulses by dual-chirped optical parametric amplification,” IEEE Photonics J. 9(3), 1503108 (2017).

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

J. Phys. At. Mol. Opt. Phys. (1)

G. Cirmi, M. Hemmer, K. Ravi, F. Reichert, L. E. Zapata, A.-L. Calendron, H. Cankaya, F. Ahr, O. D. Mücke, N. H. Matlis, and F. X. Kärtner, “Cascaded second-order processes for the efficient generation of narrowband terahertz radiation,” J. Phys. At. Mol. Opt. Phys. 50, 088002 (2017).

J. Phys. D Appl. Phys. (1)

M. C. Hoffmann and J. A. Fülöp, “Intense ultrashort terahertz pulses: generation and applications,” J. Phys. D Appl. Phys. 44, 083001 (2011).

Laser Phys. (1)

K. L. Vodopyanov, “Terahertz-wave generation with periodically inverted gallium arsenide,” Laser Phys. 19(2), 305–321 (2009).

Nat. Commun. (2)

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[PubMed]

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

New J. Phys. (1)

Cs. Lombosi, Gy. Polónyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fülöp, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).

Opt. Commun. (4)

K.-L. Yeh, J. Hebling, M. C. Hoffmann, and K. A. Nelson, “Generation of high average power 1 kHz shaped THz pulses via optical rectification,” Opt. Commun. 281(13), 3567–3570 (2008).

K. Wynne and J. J. Carey, “An integrated description of terahertz generation through optical rectification, charge transfer, and current surge,” Opt. Commun. 256(4–6), 400–413 (2005).

Gy. Tóth, L. Pálfalvi, L. Tokodi, J. Hebling, and J. A. Fülöp, “Scalable broadband OPCPA in Lithium Niobate with signal angular dispersion,” Opt. Commun. 370, 250–255 (2016).

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO3, KNbO3, β-BaB2O4, and LiB3O5,” Opt. Commun. 142, 294–298 (1997).

Opt. Eng. (1)

N. Broeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals,” Opt. Eng. 39(4), 1016–1024 (2000).

Opt. Express (14)

A. Gaydardzhiev, I. Nikolov, I. Buchvarov, V. Petrov, and F. Noack, “Ultrabroadband operation of a femtosecond optical parametric generator based on BiB3O6 in the near-IR,” Opt. Express 16(4), 2363–2373 (2008).
[PubMed]

E. Kaksis, G. Almási, J. A. Fülöp, A. Pugžlys, A. Baltuška, and G. Andriukaitis, “110-mJ 225-fs cryogenically cooled Yb:CaF2 multipass amplifier,” Opt. Express 24(25), 28915–28922 (2016).
[PubMed]

G. Polónyi, B. Monoszlai, G. Gäumann, E. J. Rohwer, G. Andriukaitis, T. Balciunas, A. Pugzlys, A. Baltuska, T. Feurer, J. Hebling, and J. A. Fülöp, “High-energy terahertz pulses from semiconductors pumped beyond the three-photon absorption edge,” Opt. Express 24(21), 23872–23882 (2016).
[PubMed]

H. Hoogland, A. Thai, D. Sánchez, S. L. Cousin, M. Hemmer, M. Engelbrecht, J. Biegert, and R. Holzwarth, “All-PM coherent 2.05 µm Thulium/Holmium fiber frequency comb source at 100 MHz with up to 0.5 W average power and pulse duration down to 135 fs,” Opt. Express 21(25), 31390–31394 (2013).
[PubMed]

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19(8), 7190–7212 (2011).
[PubMed]

Z. Hong, Q. Zhang, P. Lan, and P. Lu, “Generation of few-cycle infrared pulses from a degenerate dual-pump OPCPA,” Opt. Express 22(5), 5544–5557 (2014).
[PubMed]

Y. Yin, J. Li, X. Ren, Y. Wang, A. Chew, and Z. Chang, “High-energy two-cycle pulses at 3.2 μm by a broadband-pumped dual-chirped optical parametric amplification,” Opt. Express 24(22), 24989–24998 (2016).
[PubMed]

K. Ravi, D. N. Schimpf, and F. X. Kärtner, “Pulse sequences for efficient multi-cycle terahertz generation in periodically poled lithium niobate,” Opt. Express 24(22), 25582–25607 (2016).
[PubMed]

A. Stepanov, J. Hebling, and J. Kuhl, “Generation, tuning, and shaping of narrow-band, picosecond THz pulses by two-beam excitation,” Opt. Express 12(19), 4650–4658 (2004).
[PubMed]

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Figures (6)

Fig. 1
Fig. 1 Spectrograms of pump, signal, and idler pulses in case of FL (a) and chirped (b) pulses. The central frequencies of pump, signal, and idler are denoted by ω p 0 , ω s 0 , and ω i 0 , respectively. (c) and (d) show the temporal intensities for the superstition of signal and idler.
Fig. 2
Fig. 2 Relative deviation in the intensity modulation frequency caused by a 40-fs jitter between pump and signal as function of the FL pump pulse duration and the pump stretching factor. The jitter-free modulation frequency was 1 THz.
Fig. 3
Fig. 3 The stretched pulse duration ratio M as function of the pump stretching factor N and the ratio μ = Δ t s / Δ t p of the FL pulse durations.
Fig. 4
Fig. 4 Schematics of the simulated DC-OPA setup.
Fig. 5
Fig. 5 Phase mismatch as function of the signal wavelength in LN and KN for collinear ( α = 0 ° ) and noncollinear ( α = 0.5 ° ) geometries. The shaded range indicates the bandwidth of the seed laser.
Fig. 6
Fig. 6 Temporal and spatial intensity profiles of intensity-modulated pulses obtained by the superposition of signal and idler pulses in the DC-OPA for 5-ps pump pulses and 1 THz modulation frequency (a), 10-ps pump pulses and 1 THz modulation frequency (b), and 10-ps pump pulses and 0.5 THz modulation frequency (c). The lower panels show the temporal intensity shapes at the beam center.

Tables (2)

Tables Icon

Table 1 Parameters of the two noncollinear OPA preamplifier stages. Wp: input pump energy,Ws0: input signal energy,ws,p: FWHM beam diameter of signal (s) and pump (p), L: length of the LN nonlinear crystal, Ws: output amplified signal pulse energy.

Tables Icon

Table 2 Parameters of the DC-OPA. η : efficiency of the DC-OPA stage, η Σ efficiency of the entire system including also the two preamplifier stages, Ws + i: energy of the intensity-modulated output pulse (superposition of signal and idler).

Equations (29)

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ω p ( t ) = ω s ( t ) + ω i ( t )
Δ ω = 2 π Δ ν = ω s ( t ) ω i ( t ) = c o n s t a n t
E ( t ) = E 0 e 2 ln ( 2 ) t 2 Δ t 2 e j ω 0 t
E ˜ ( ω ) = F { E ( t ) } = 1 2 π E ( t ) e j ω t d t = E 0 Δ t ln ( 2 ) e Δ t 2 8 ln ( 2 ) ( ω ω 0 ) 2
E ( t ) = 1 2 π E ˜ ( ω ) e j ( G D ( ω ω 0 ) + G D D 2 ( ω ω 0 ) 2 ) e j ω t d ω = E 0 Δ t τ e 2 ln ( 2 ) ( t G D ) 2 τ 2 e j [ ω 0 t + β ( t G D ) 2 1 2 arc tan ( 4 ln ( 2 ) G D D Δ t 2 ) ]
τ = Δ t 1 + ( 4 ln ( 2 ) G D D Δ t 2 ) 2 ,
β = ( 2 G D D + Δ t 4 8 ln 2 ( 2 ) G D D ) 1 .
ω ( t ) = ω 0 + 2 β ( t G D ) .
β s = β p / 2.
G D D s = G D D p + Δ t p 4 16 ln 2 ( 2 ) G D D p + ( G D D p + Δ t p 4 16 ln 2 ( 2 ) G D D p ) 2 Δ t s 4 4 ln 2 ( 2 ) .
ω p ( t ) = ω p 0 + 2 β p t ,
ω s ( t ) = ω s0 + 2 β s ( t G D sp ) = ω p 0 + Δ ω 0 2 + β p ( t G D sp ) ,
ω i ( t ) = ω p ( t ) ω s ( t ) = ω p 0 Δ ω 0 2 + β p ( t + G D sp ) .
Δ ω = ω s ( t ) ω i ( t ) = Δ ω 0 2 β p G D s p .
Δ ω = Δ ω 0 4 ln ( 2 ) N 2 1 Δ t p 2 N 2 G D sp ,
M = τ s τ p = μ N 1 + ( N 2 + N 4 4 μ 4 ( N 2 1 ) ) 2 μ 4 ( N 2 1 ) .
E ( r , t ) = 1 2 l = p , s , i A l ( r , t ) e j ( k l r ω l t ) + c . c . ,
z A p ( ω , z , ρ ) = j ( k p ( ω ) e z ω v s ) A p ( ω , z , ρ ) + j ω p d e f f n p c F { A s ( τ , z , ρ ) A i ( τ , z , ρ ) e j Δ k e z z } +   j ω p ε 0 n 2 n p 2 F { A p ( τ , z , ρ ) ( 2 | A s ( τ , z , ρ ) | 2 + 2 | A i ( τ , z , ρ ) | 2 + | A p ( τ , z , ρ ) | 2 ) } ,
z A s ( ω , z , ρ ) = j ( k s ( ω ) e z ω v s ) A s ( ω , z , ρ ) + j ω s d e f f n s c F { A p ( τ , z , ρ ) A i * ( τ , z , ρ ) e j Δ k e z z } + j ω s ε 0 n 2 n s 2 F { A s ( τ , z , ρ ) ( | A s ( τ , z , ρ ) | 2 + 2 | A i ( τ , z , ρ ) | 2 + 2 | A p ( τ , z , ρ ) | 2 ) } ,
z A i ( ω , z , ρ ) = j ( k i ( ω ) e z ω v s ) A i ( ω , z , ρ ) + j ω i d e f f n i c F { A p ( τ , z , ρ ) A s * ( τ , z , ρ ) e j Δ k e z z } + j ω i ε 0 n 2 n i 2 F { A i ( τ , z , ρ ) ( 2 | A s ( τ , z , ρ ) | 2 + | A i ( τ , z , ρ ) | 2 + 2 | A p ( τ , z , ρ ) | 2 ) } .
A s ( t , z = 0 , ρ ) = 4 ln 3 4 ( 2 ) ε 0 c n s π 3 2 Δ t s w s W s e 2 ln ( 2 ) t 2 Δ t s 2 e 2 ln ( 2 ) ρ 2 w s 2 ,
A p ( t , z = 0 , ρ ) = 4 2 ln ( 2 ) ε 0 c n p Δ t p π w p W p e 2 ln ( 2 ) t 2 Δ t p 2 e 8 ln ( 2 ) ρ 4 w p 4 .
G D D ( λ , d , γ , L f ) = λ 3 L g π c 2 d 2 ( 1 ( λ d sin γ ) 2 ) 3 2 ,
τ s = Δ t s 1 + ( 4 ln ( 2 ) G D D s Δ t s 2 ) 2
τ p = Δ t p 1 + ( 4 ln ( 2 ) G D D p Δ t p 2 ) 2 = Δ t p N .
G D D p = Δ t p 2 N 2 1 4 ln ( 2 ) .
τ s = Δ t s 1 + ( 4 ln ( 2 ) [ G D D p + Δ t p 4 16 ln 2 ( 2 ) G D D p + ( G D D p + Δ t p 4 16 ln 2 ( 2 ) G D D p ) 2 Δ t s 4 4 ln 2 ( 2 ) ] Δ t s 2 ) 2 = Δ t s 1 + ( 4 ln ( 2 ) [ Δ t p 2 N 2 1 4 ln ( 2 ) + Δ t p 2 4 ln ( 2 ) N 2 1 + ( Δ t p 2 N 2 1 4 ln ( 2 ) + Δ t p 2 4 ln ( 2 ) N 2 1 ) 2 Δ t s 4 4 ln 2 ( 2 ) ] Δ t s 2 ) 2 = Δ t s 1 + ( ( Δ t p Δ t s ) 2 N 2 1 + ( Δ t p Δ t s ) 2 1 N 2 1 + ( Δ t p Δ t s ) 4 ( N 2 1 + 1 N 2 1 ) 2 4 ) 2 .
τ s = Δ t s 1 + ( ( 1 μ ) 2 N 2 1 + ( 1 μ ) 2 1 N 2 1 + ( 1 μ ) 4 ( N 2 1 + 1 N 2 1 ) 2 4 ) 2 = Δ t s 1 + ( N 2 μ 2 N 2 1 + 1 μ 2 N 4 N 2 1 4 μ 4 ) 2 = Δ t s 1 + ( N 2 + N 4 4 μ 4 ( N 2 1 ) μ 2 N 2 1 ) 2 .
M = τ s τ p = Δ t s 1 + ( N 2 + N 4 4 μ 4 ( N 2 1 ) μ 2 N 2 1 ) 2 Δ t p N = μ N 1 + ( N 2 + N 4 4 μ 4 ( N 2 1 ) ) 2 μ 4 ( N 2 1 ) .

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