Abstract

Frequency-domain optical parametric amplification (FOPA) is a new scheme that enables extremely broadband amplification of ultraintense pulses. The spatiotemporal coupling property of signal pulses can make the coherent noise of FOPA sharply different from that of conventional OPCPA. This paper presents a first theoretical study on the coherent noise produced in a FOPA system. We reveal that the coherent noise acquires the spatiotemporal coupling, and thus distinguishes the compressed signal pulse not only in time but also in space, which allows the suppression of coherent noise via optical manipulations in the spatial domain. The quantitative impacts of spatiotemporal coherent noise originated from the imperfections in either pump laser or crystal surfaces, are numerically studied. The result provides a new perspective on improving the coherent contrast of ultraintense lasers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

2016 (2)

2015 (3)

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

2014 (2)

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

M. Chyla, T. Miura, M. Smrz, H. Jelinkova, A. Endo, and T. Mocek, “Optimization of beam quality and optical-to-optical efficiency of Yb:YAG thin-disk regenerative amplifier by pulsed pumping,” Opt. Lett. 39(6), 1441–1444 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2009 (2)

J. Moses, C. Manzoni, S. W. Huang, G. Cerullo, and F. X. Kärtner, “Temporal optimization of ultrabroadband high-energy OPCPA,” Opt. Express 17(7), 5540–5555 (2009).
[Crossref] [PubMed]

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

2008 (1)

2007 (4)

C. Dorrer, “Analysis of pump-induced temporal contrast degradation in optical parametric chirped-pulse amplification,” J. Opt. Soc. Am. B 24(12), 3048–3057 (2007).
[Crossref]

I. Ross, G. New, and P. Bates, “Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system,” Opt. Commun. 273(2), 510–514 (2007).
[Crossref]

E. Gerstner, “Laser physics: extreme light,” Nature 446(7131), 16–18 (2007).
[Crossref] [PubMed]

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

2006 (2)

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

2005 (4)

1986 (1)

Akturk, S.

Albert, O.

Albright, B. J.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Antonucci, L.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Arissian, L.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

Audebert, P.

Augé-Rochereau, F.

Bates, P.

I. Ross, G. New, and P. Bates, “Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system,” Opt. Commun. 273(2), 510–514 (2007).
[Crossref]

Bionta, M. R.

Boivin, M.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Brambrink, E.

Bromage, J.

Burgy, F.

Canova, L.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Cerullo, G.

Chaker, M.

Chambaret, J. P.

Chériaux, G.

Chyla, M.

Cobble, J.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Corkum, P. B.

Cotel, A.

Dai, Y.

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

Daniels, J.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Danson, C.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

Di Mauro, L.

Divall, E.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Dorrer, C.

Druon, F.

Durfee, C. G.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Endo, A.

Ernotte, G.

V. Gruson, G. Ernotte, P. Lassonde, A. Laramée, M. R. Bionta, M. Chaker, L. Di Mauro, P. B. Corkum, H. Ibrahim, B. E. Schmidt, and F. Legaré, “2.5 TW, two-cycle IR laser pulses via frequency domain optical parametric amplification,” Opt. Express 25(22), 27706–27714 (2017).
[Crossref] [PubMed]

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

Etchepare, J.

Falcoz, F.

Fan, D.

Fernández, J. C.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Flippo, K.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Forget, N.

Gabolde, P.

Genevrier, K.

Georges, P.

Gerstner, E.

E. Gerstner, “Laser physics: extreme light,” Nature 446(7131), 16–18 (2007).
[Crossref] [PubMed]

Gonsalves, A.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Gontier, E.

Gordon, D. F.

Gruson, V.

Gu, X.

Hafizi, B.

Hamoniaux, G.

Hegelich, B. M.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Hein, J.

Helle, M. H.

Hellwing, M.

Hillier, D.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

Hopps, N.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

Hornung, M.

Huang, S. W.

Ibrahim, H.

V. Gruson, G. Ernotte, P. Lassonde, A. Laramée, M. R. Bionta, M. Chaker, L. Di Mauro, P. B. Corkum, H. Ibrahim, B. E. Schmidt, and F. Legaré, “2.5 TW, two-cycle IR laser pulses via frequency domain optical parametric amplification,” Opt. Express 25(22), 27706–27714 (2017).
[Crossref] [PubMed]

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Jang, Y. H.

Jelinkova, H.

Jullien, A.

Jungquist, R. K.

Kaganovich, D.

Kalashnikov, M.

Kalashnikov, M. P.

Kaluza, M. C.

H. Liebetrau, M. Hornung, S. Keppler, M. Hellwing, A. Kessler, F. Schorcht, J. Hein, and M. C. Kaluza, “High contrast, 86 fs, 35 mJ pulses from a diode-pumped, Yb:glass, double-chirped-pulse amplification laser system,” Opt. Lett. 41(13), 3006–3009 (2016).
[Crossref] [PubMed]

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Kamperidis, C.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Kärtner, F. X.

Keppler, S.

Kessler, A.

Khodakovskiy, N.

Krushelnick, K.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Lancaster, K. L.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Laramee, A.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

Laramée, A.

Lassonde, P.

V. Gruson, G. Ernotte, P. Lassonde, A. Laramée, M. R. Bionta, M. Chaker, L. Di Mauro, P. B. Corkum, H. Ibrahim, B. E. Schmidt, and F. Legaré, “2.5 TW, two-cycle IR laser pulses via frequency domain optical parametric amplification,” Opt. Express 25(22), 27706–27714 (2017).
[Crossref] [PubMed]

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

Le Blanc, C.

Lebas, N.

Lebrun, G.

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Lee, C. W.

Lee, H. W.

Lee, S. K.

Leemans, W.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Legare, F.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

Legaré, F.

Légaré, F.

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Letzring, S.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Li, Z.

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

Liebetrau, H.

Lindau, F.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Lopez-Martens, R.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Lundh, O.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Ma, J.

J. Wang, J. Ma, P. Yuan, D. Tang, B. Zhou, G. Xie, and L. Qian, “Nonlinear beat noise in optical parametric chirped-pulse amplification,” Opt. Express 25(24), 29769–29777 (2017).
[Crossref] [PubMed]

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

Magna, A.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Mangles, S. P. D.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Manzoni, C.

Mao, H.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Martin, L.

Martinez, O. E.

Mathieu, F.

Mercier, B.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Minkovski, N.

Mittelberger, D.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Miura, T.

Mocek, T.

Monot, P.

Moses, J.

Murphy, C. D.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Najmudin, Z.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Nakamura, K.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Nam, C. H.

Neely, D.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

New, G.

I. Ross, G. New, and P. Bates, “Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system,” Opt. Commun. 273(2), 510–514 (2007).
[Crossref]

Ozaki, T.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Paffett, M.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Papadopoulos, D. N.

Paul, P. M.

Pellegrina, A.

Peñano, J. R.

Persson, A.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Poitras, F.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Qian, L.

Ramirez, P.

Ranc, L.

Risse, E.

Ross, I.

I. Ross, G. New, and P. Bates, “Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system,” Opt. Commun. 273(2), 510–514 (2007).
[Crossref]

Rousseau, J. P.

Rousseau, J.-P.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Ruhl, H.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Saltiel, S. M.

Sandner, W.

Schmidt, B. E.

V. Gruson, G. Ernotte, P. Lassonde, A. Laramée, M. R. Bionta, M. Chaker, L. Di Mauro, P. B. Corkum, H. Ibrahim, B. E. Schmidt, and F. Legaré, “2.5 TW, two-cycle IR laser pulses via frequency domain optical parametric amplification,” Opt. Express 25(22), 27706–27714 (2017).
[Crossref] [PubMed]

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Schönnagel, H.

Schorcht, F.

Schreiber, J.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Schulze, R. K.

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Smrz, M.

Son, Y. J.

Sung, J. H.

Tang, D.

Thiré, N.

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

Thomas, A. G. R.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Ting, A.

Toth, C.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Trebino, R.

Trisorio, A.

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

Vincenti, H.

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

Wahlström, C.-G.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Wang, J.

J. Wang, J. Ma, P. Yuan, D. Tang, B. Zhou, G. Xie, and L. Qian, “Nonlinear beat noise in optical parametric chirped-pulse amplification,” Opt. Express 25(24), 29769–29777 (2017).
[Crossref] [PubMed]

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

Wang, T.

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

Wang, Y.

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

Wei, X.

Xie, G.

J. Wang, J. Ma, P. Yuan, D. Tang, B. Zhou, G. Xie, and L. Qian, “Nonlinear beat noise in optical parametric chirped-pulse amplification,” Opt. Express 25(24), 29769–29777 (2017).
[Crossref] [PubMed]

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

Xu, G.

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

Yang, J. M.

Yoo, J. Y.

Yoon, J. W.

Yuan, P.

Zhou, B.

Zhu, H.

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

X. Wei, L. Qian, P. Yuan, H. Zhu, and D. Fan, “Optical parametric amplification pumped by a phase-aberrated beam,” Opt. Express 16(12), 8904–8915 (2008).
[Crossref] [PubMed]

Zou, J. P.

Appl. Phys. B (1)

A. Jullien, C. G. Durfee, A. Trisorio, L. Canova, J.-P. Rousseau, B. Mercier, L. Antonucci, G. Chériaux, O. Albert, and R. Lopez-Martens, “Nonlinear spectral cleaning of few-cycle pulses via cross-polarized wave (XPW) generation,” Appl. Phys. B 96(2–3), 293–299 (2009).
[Crossref]

High Power Laser Sci. Eng. (1)

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3(1), 5–18 (2015).

High-Power Lasers and Applications IV (1)

Z. Li, Y. Dai, T. Wang, and G. Xu, “Influence of spectral clipping in chirped pulse amplification laser system on pulse temporal profile,” High-Power Lasers and Applications IV 6823(1), 682315 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Nakamura, H. Mao, A. Gonsalves, H. Vincenti, D. Mittelberger, J. Daniels, A. Magna, C. Toth, and W. Leemans, “Diagnostics, control and performance parameters for the BELLA high repetition rate petawatt class laser,” IEEE J. Quantum Electron. 53(4), 1–21 (2017).
[Crossref]

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

P. Lassonde, N. Thiré, L. Arissian, G. Ernotte, F. Poitras, T. Ozaki, A. Laramee, M. Boivin, H. Ibrahim, F. Legare, and B. E. Schmidt, “High gain frequency domain optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–10 (2015).
[Crossref]

J. Opt. Soc. Am. B (3)

Nat. Commun. (2)

B. E. Schmidt, N. Thiré, M. Boivin, A. Laramée, F. Poitras, G. Lebrun, T. Ozaki, H. Ibrahim, and F. Légaré, “Frequency domain optical parametric amplification,” Nat. Commun. 5(6183), 3643 (2014).
[PubMed]

J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. 6(1), 6192 (2015).
[Crossref] [PubMed]

Nature (2)

E. Gerstner, “Laser physics: extreme light,” Nature 446(7131), 16–18 (2007).
[Crossref] [PubMed]

B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, “Laser acceleration of quasi-monoenergetic MeV ion beams,” Nature 439(7075), 441–444 (2006).
[Crossref] [PubMed]

Opt. Commun. (1)

I. Ross, G. New, and P. Bates, “Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system,” Opt. Commun. 273(2), 510–514 (2007).
[Crossref]

Opt. Express (5)

Opt. Lett. (9)

D. Kaganovich, J. R. Peñano, M. H. Helle, D. F. Gordon, B. Hafizi, and A. Ting, “Origin and control of the subpicosecond pedestal in femtosecond laser systems,” Opt. Lett. 38(18), 3635–3638 (2013).
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M. Chyla, T. Miura, M. Smrz, H. Jelinkova, A. Endo, and T. Mocek, “Optimization of beam quality and optical-to-optical efficiency of Yb:YAG thin-disk regenerative amplifier by pulsed pumping,” Opt. Lett. 39(6), 1441–1444 (2014).
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H. Liebetrau, M. Hornung, S. Keppler, M. Hellwing, A. Kessler, F. Schorcht, J. Hein, and M. C. Kaluza, “High contrast, 86 fs, 35 mJ pulses from a diode-pumped, Yb:glass, double-chirped-pulse amplification laser system,” Opt. Lett. 41(13), 3006–3009 (2016).
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N. Khodakovskiy, M. Kalashnikov, E. Gontier, F. Falcoz, and P. M. Paul, “Degradation of picosecond temporal contrast of Ti:sapphire lasers with coherent pedestals,” Opt. Lett. 41(19), 4441–4444 (2016).
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J. H. Sung, H. W. Lee, J. Y. Yoo, J. W. Yoon, C. W. Lee, J. M. Yang, Y. J. Son, Y. H. Jang, S. K. Lee, and C. H. Nam, “4.2 PW, 20 fs Ti:Sapphire laser at 0.1 Hz,” Opt. Lett. 42(11), 2058–2061 (2017).
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D. N. Papadopoulos, P. Ramirez, K. Genevrier, L. Ranc, N. Lebas, A. Pellegrina, C. Le Blanc, P. Monot, L. Martin, J. P. Zou, F. Mathieu, P. Audebert, P. Georges, and F. Druon, “High-contrast 10 fs OPCPA-based front end for multi-PW laser chains,” Opt. Lett. 42(18), 3530–3533 (2017).
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N. Forget, A. Cotel, E. Brambrink, P. Audebert, C. Le Blanc, A. Jullien, O. Albert, and G. Chériaux, “Pump-noise transfer in optical parametric chirped-pulse amplification,” Opt. Lett. 30(21), 2921–2923 (2005).
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A. Jullien, O. Albert, F. Burgy, G. Hamoniaux, J. P. Rousseau, J. P. Chambaret, F. Augé-Rochereau, G. Chériaux, J. Etchepare, N. Minkovski, and S. M. Saltiel, “10-10 temporal contrast for femtosecond ultraintense lasers by cross-polarized wave generation,” Opt. Lett. 30(8), 920–922 (2005).
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Plasma Phys. Contr. Fusion (1)

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Contr. Fusion 48(12B), B83–B90 (2006).
[Crossref]

Other (1)

L. Veisz, “Contrast improvement of relativistic few-cycle light pulses,” Coherence and Ultrashort Pulse Laser Emission. InTech, (2010).

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Figures (5)

Fig. 1
Fig. 1 (a) FOPA setup with nonlinear crystals placed in the spectral Fourier-plane. G1 and G2, diffraction gratings; M1 and M2, concave mirrors with a focal length f; x and x' represents the transverse spatial coordinate in the spectral Fourier-plane and that in output near-field of FOPA, respectively, which links each other via a spatial-domain Fourier transformation; t and ω denotes time and frequency, respectively. (b) Diagram of the noise mechanism of FOPA. Spatiotemporal coherent noise is produced via three steps: (1) in the amplification stage, the pump beam modulation is nonlinearly imparted onto the signal beam and also spectrum; (2) in the pulse compression stage, the induced spectral modulation on the amplified signal gets transformed into pre- and post-pulses; (3) in the spatial Fourier-transform stage from the nonlinear crystals to grating G2, the induced spatial modulation on the amplified signal beam gets transformed into beam side-lobes. Owing to the interdependence between x and ω introduced in the first step, the second and third steps jointly make the output noise exhibit spatiotemporal coupling.
Fig. 2
Fig. 2 Spatiotemporal (the first row) and spatial-sp0ectral (the second row) intensity profiles of the signal at the planes of G1 input (a), G1 output (b), M1 input (c), spectral Fourier-plane (d), M2 input (e), G2 input (f) and G2 output (g), respectively. Dashed lines in the first row highlight the slope of spatiotemporal coupling.
Fig. 3
Fig. 3 Spatiotemporal characterization of the FOPA noise originated from temporal pump-intensity modulation. (a) Intensity profile of the pump laser with a temporal sinusoidal modulation of Ωt = 6.28 THz and rt = 0.1. (b), (c), (d) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal in the spectral Fourier-plane, the output near-field and output far-field, respectively. Red dotted line highlights the spatial-spectral coupling property of the induced noise. The FOPA is assumed in the small-signal amplification regime with G0 = ~104 and pump depletion ηp = 0.5%. (e) Spatial profile (temporally integrated) of the amplified signal beam in the far-field of FOPA output, calculated at a trivial pump-depletion ηp = 0.5% (black line) in comparison with that in a significant pump-depletion of ηp = 38% (blue line). (f) Temporal profiles of the amplified signal for the case of ηp = 0.5% (black line) and ηp = 38% (red line). Inset plots the signal spectrum before (black) and after amplification (red).
Fig. 4
Fig. 4 Characterization of the spatiotemporal coherent noise originated from spatial pump-intensity modulation. (a) Intensity profile of the pump laser with a sinusoidal spatial modulation of Ωx = 7.85 × 103 m−1 and rx = 0.01. (b), (c), (d) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal and noise in the spectral Fourier-plane, the output near-field and output far-field, respectively. (e) Temporal profiles of the amplified signal pulse on the axis of FOPA output near-field (solid line) and far-field (dotted line), respectively. (f) The intensity of the temporal spike at t = −2.5ps (blue) and t = −5ps (red) versus the amplification parameters of parametric gain G0 and pump-depletion ratio ηp. The seed-to-pump intensity ratio is fixed at 1.8 × 10−6. (g) The temporal spike intensities at t = −2.5ps (blue triangles) and t = −5ps (red triangles) versus the seed-to-pump intensity ratio.
Fig. 5
Fig. 5 Characterization of the spatiotemporal coherent noise originated from the crystal surface roughness. (a) Crystal surface profile characterized with a RMS height of 0.3 nm. (b) Temporal profile of the amplified signal pulse profiles with a clean pump laser and the amplification parameters the same with Fig. 3. (c), (d), (e) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal in the spectral Fourier-plane, the output near-field and output far-field, respectively. (f) Output coherent contrast at t = −5 ps (red line) and t = −10 ps (blue line) versus parametric gain G0 and pump-depletion ratio ηp. (g) Coherent contrast values at t = −5 ps (red line) and t = −10 ps (blue line) versus the RMS height of crystal surface roughness, under the condition of a fixed amplification condition at G0 = 104 and pump-depletion ratio ηp = 0.5%.

Equations (32)

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a in ( x,ω )= a 0 exp( x 2 2 D 2 )exp[ ω 2 2 ( Δω ) 2 ].
a FP ( x,ω )=exp( ik 2f x 2 )exp( ik β 2 ω 2 f 2 )exp[ ω 2 2 ( Δω ) 2 ]exp( ikβωx )exp[ ( x+βfω ) 2 2 σ 2 ],
ζ= dx dω =βf.
A FP ( x,t )=exp[ x 2 2 ( ζΔω ) 2 ]exp[ t 2 2 ( Δ T FP ) 2 ]exp( ixt ζ ),
Δ T FP = 1 Δ ω FP =kβD.
I p ( x,t )= I p0 [ 1+ r t cos( Ω t t ) ]= I p0 [ 1+ r t 2 exp( ±i Ω t t ) ],
A FP amp ( x,t )= A FP ( x,t )×cosh[ Γz I p0 [ 1+ r t cos( Ω t t ) ] ] A FP ( x,t )× G 0 ×( 1+ m [ ln( 4 G 0 ) 8 ] m r t m m! e ±im Ω t t ),
a FP amp ( x,ω )= a s ( xζω,ω )+ m [ ln( 4 G 0 ) 8 ] m r t m m! a s ( xζωζm Ω t ,ω±m Ω t ) ,
a out NF ( x,ω )= a NF ( x,ω )+ m [ ln( 4 G 0 ) 8 ] m r t m m! a NF ( x,ω±m Ω t )exp( ikβm Ω t x ) .
a out FF ( x,ω )= a FF ( x,ω )+ m [ ln( 4 G 0 ) 8 ] m r t m m! a FF ( xmβf Ω t ,ω±m Ω t ) .
Δx( m Ω t )=mβf Ω t =m× kβD 1 Ω t ×σ.
A out FF ( x,t )= A FF ( x,t )+ m [ ln( 4 G 0 ) 8 ] m m t m m! A FF ( xmβf Ω t ,t )exp( im Ω t t ) .
I p ( x,t )= I p0 ( x,t )[ 1+ r x 2 exp( ±i Ω x x ) ],
a FP amp ( x,ω )= a FP ( xζω,ω )×{ 1+ m [ ln( 4 G 0 ) 8 ] m r x m m! exp( im Ω x x ) exp( ±i m Ω x ζ ω ) },
A out NF ( x,t )= A NF ( x,t )+ m [ ln( 4 G 0 ) 8 ] m r x m m! A NF ( x±m f Ω x k ,t±ζm Ω x ) ,
Δt( m Ω x )=m Ω x ×ζ.
Δx( m Ω x )=m Ω x × f k .
A out FF ( x,t )= A FF ( x,t )+ m [ ln( 4 G 0 ) 8 ] m r x m m! A FF ( x,t±ζm Ω x )exp[ i f k m Ω x x ] ,
I spike (1) I signal = [ ln( 4 G 0 ) 8 × r x ] 2 , I spike (2) I signal = { [ ln( 4 G 0 ) 8 × r x ] 2 × 1 2 } 2 .
a G 1 ( x,ω )= a in ( x,ω )exp( ikβωx ),
β= dθ dω = λ 3 2πcdcosθ ,
A G 1 ( x,t )= a G 1 ( x,ω )exp( iωt )dω= a 1 exp( x 2 2 D 2 )exp[ ( t+kβx ) 2 2 τ 0 2 ].
ξ= dt dx =kβ.
a z ( x,ω,z )= i λz a G 1 ( x 0 ,ω)exp[ ik 2z ( x x 0 ) 2 ] d x 0 exp[ ω 2 2 ( Δω ) 2 ]exp( ikβωx )exp{ ( x+βωz ) 2 2 D 2 }exp( ik 2 β 2 ω 2 z ).
u( z )= βΔωz D ,C( z )=k β 2 ( Δω ) 2 z.
I z ( x,t,z )= | a z ( x,ω,z ) e iωt dω | 2 =exp{ [ tξ( z )x ] 2 (1+ u 2 ) 2 + C 2 1+ u 2 τ 0 2 }exp[ x 2 ( 1+ u 2 ) D 2 ],
ξ( z )=kβ× 1 1+ u 2 .
T local ( z )= ( 1+ u 2 ) 2 + C 2 1+ u 2 τ 0 .
a FP ( x,ω )= i λf a z ( x 0 ,ω,z=f )exp( ik 2f x 0 2 )exp[ ik 2f ( x x 0 ) 2 ] d x 0 =exp[ ω 2 2 ( Δω ) 2 ]exp[ ( x+βfω ) 2 2 σ 2 ]exp( ik 2f x 2 )exp( ik β 2 ω 2 f 2 )exp( ikβωx ).
ζ= dx dω =βf.
a M 2 ( x,ω )=exp[ ω 2 2 ( Δω ) 2 ]exp{ ( x+βωf ) 2 2 D 2 }exp( ikβωx )exp( ik 2 β 2 ω 2 f ),
a G 2 ( x,ω )=exp[ ω 2 2 ( Δω ) 2 ]exp( x 2 2 D 2 ).