Abstract

We report on a simple method allowing one to decompose the duration of arbitrary ultrashort light pulses, potentially distorted by space-time coupling, into four elementary durations. Such a decomposition shows that, in linear optics, a spatio-temporal pulse can be stretched with respect to its Fourier limit by only three independent phenomena: nonlinear frequency dependence of the spectral phase over the whole spatial extent of the pulse, spectral amplitude inhomogeneities in space, and spectral phase inhomogeneities in space. We illustrate such a decomposition using numerical simulations of complex spatio-temporal femtosecond and attosecond pulses. Finally we show that the contribution of two of these three effects to the pulse duration is measurable without any spectral phase characterization.

© 2011 OSA

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2010 (3)

2009 (1)

2008 (2)

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
[CrossRef] [PubMed]

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (2)

2004 (4)

T. A. Planchon, S. Ferré, G. Hamoniaux, G. Chériaux, and J.-P. Chambaret, “Experimental evidence of 25-fs laser pulse distortion in singlet beam expanders,” Opt. Lett. 29, 2300–2302 (2004).
[CrossRef] [PubMed]

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242, 599–604 (2004).
[CrossRef]

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

2003 (2)

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

P. Mercère, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (2)

1994 (1)

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

1992 (2)

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

1989 (1)

Agostini, P.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Akturk, S.

Alonso, B.

Blanchot, N.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Bor, Z.

Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119–121 (1989).
[CrossRef] [PubMed]

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” in International Conference on Ultrafast Phenomena (UP), 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper ThD16.

Bourassin-Bouchet, C.

Boutu, W.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Bowlan, P.

Breger, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Bucourt, S.

Caillat, J.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Carré, B.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Chambaret, J.-P.

Chavel, P.

Chériaux, G.

Csatari, M.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

de Bohan, A.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

de Rossi, S.

Delmotte, F.

Dinu, L. C.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Dorrer, C.

Douillet, D.

Dovillaire, G.

Ferré, S.

Fiorini, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Frasinski, L. J.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Fuchs, U.

Gabolde, P.

Gallmann, L.

Goldberg, K. A.

Gu, X.

Haessler, S.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Hamoniaux, G.

Heiner, Z.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

Herek, J. L.

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

Hernández-Toro, J.

Horváth, Z. L.

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” in International Conference on Ultrafast Phenomena (UP), 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper ThD16.

Iaconis, C.

Idir, M.

Keller, U.

Kempe, M.

Klebniczki, J.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

Kosik, E. M.

Kovacev, M.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Kovács, A. P.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” in International Conference on Ultrafast Phenomena (UP), 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper ThD16.

Kurdi, G.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

Le Pape, S.

Lee, D.

Levecq, X.

Liu, Q.

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

Mairesse, Y.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Maquet, A.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Méndez, C.

Mercère, P.

Merdji, H.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Migus, A.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Monchicourt, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Muller, H. G.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Naulleau, P. P.

Osvay, K.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Quantum Electron. 10, 213–220 (2004).
[CrossRef]

Pedersen, S.

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

Planchon, T. A.

Potter, E. D.

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

Rekawa, S.

Roso, L.

Rouyer, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Rudolph, W.

Rupp, T.

Salières, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

San Román, J.

Sauteret, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Seznec, S.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30, 1662–1670 (1994).
[CrossRef]

Sola, I. J.

Stamm, U.

Stankiewicz, M.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Steinmeyer, G.

Sutter, D. H.

Taïeb, R.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Trebino, R.

Tünnermann, A.

Varela, O.

Walmsley, I.

I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009).
[CrossRef]

I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. 72, 1–29 (2001).
[CrossRef]

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J. Opt. Soc. Am. A (1)

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W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Nature (1)

E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature 355, 66–68 (1992).
[CrossRef]

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Opt. Lett. (6)

Rev. Sci. Instrum. (1)

I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. 72, 1–29 (2001).
[CrossRef]

Science (1)

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302, 1540–1543 (2003).
[CrossRef] [PubMed]

Other (1)

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” in International Conference on Ultrafast Phenomena (UP), 2006 OSA Technical Digest Series (Optical Society of America, 2006), paper ThD16.

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Figures (4)

Fig. 1
Fig. 1

Description of the decomposition of the global duration of an arbitrary pulse into four elementary durations. The scales are in arbitrary units. For each case, the instantaneous frequency of the spatio-temporal pulse (central part) and its corresponding global pulse (on top) are depicted, along with the local spectrum (gray shaded lines) and group delay (black dashed line) for three positions (on the right part). (a) The shortest pulse duration attainable ΔtFTG is reached when every spectral component is synchronized both in space and time and is equally spread in space. The global pulse will be stretched by (b) any temporal desynchronization of the spectral components, due to a temporal chirp, for example, (c) any inhomogeneity in their spatial spread, induced by phenomena such as spatial chirp, or (d) any spatial desynchronization of these spectral components, caused by spatially varying time delay, for example.

Fig. 2
Fig. 2

Three examples of the duration decomposition of spatio-temporal pulses. The spatio-temporal pulses and the global pulses (red shaded curves) are reported on the upper panels. The associated duration decompositions are reported on the lower panels. (a) Pulse front tilt caused by a time delay linearly varying in space. (b) Pulse front tilt caused by both spatial and temporal chirps. (c) Pulse distorted by a space dependent Group Delay Dispersion (negative for negative positions, and positive for positive ones). The Global Fourier Transform limit IFTG (t) of the three distorted pulses is reported in a) (blue shaded curve).

Fig. 3
Fig. 3

Simulation of the full spatio-temporal electric field distribution of an attosecond pulse distorted by optical aberrations. (a) The attosecond pulse is focused by a grazing incidence ellipsoidal mirror, the optimal grazing angle of which is equal to 11.5°. The mirror is set with a grazing angle of 11.4° to see the impact of astigmatism on the pulse. (b) The spatio-temporal intensity distribution of the pulse at the paraxial focus is represented. The projections on the (x,t) and (y,t) sides of the box stand for two slices of the 3D pulse located on the dashed lines. The projection on the (x,y) side represents the temporally integrated pulse, that is the image that should be obtained if using a CCD sensor. The pulses IG (t) (red shaded curve) and IFTG (t) (blue shaded curves) associated to the spatio-temporal pulse are represented above the box. (c) (resp. (d)) The instantaneous frequency of the (x,t) (resp. (y,t)) projection is plotted. Astigmatism is responsible for the inverted curvatures of the pulse front on the two projections, whereas the atto-chirp makes the instantaneous frequency vary linearly throughout the pulse envelope.

Fig. 4
Fig. 4

Evolution of the duration decomposition of a refocused attosecond pulse with respect to the strength of astigmatism, that is with respect to the deviation from the optimal grazing angle of 11.5° to 11.4°. (a) Simulation of the spatio-temporal pulses (lower part) and their global pulses (upper part) with respect to the grazing angle. (b) Evolution of the global duration of the pulse (black diamonds) and its decomposition into the four parameters: ΔtFTG (blue circles), ΔGDG (yellow squares), τAC (orange up triangles), τPC (red down triangles), and their corresponding fits (dashed lines).

Equations (36)

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2 E ( x , y , z , t ) 1 c 2 2 E ( x , y , z , t ) t 2 = 0
E ( x , y , z 0 , t ) = g ( x , y ) f ( t )
Δ t 2 = t 2 t 2
Δ t 2 = Δ t F T 2 + Δ G D 2
I G ( t ) = + | E ( x , y , t ) | 2 d x d y
S G ( ω ) = + | E ˜ ( x , y , ω ) | 2 d x d y
G D G ( ω ) = G D ( x , y ) = + | E ˜ ( x , y , ω ) | 2 G D d x d y / + | E ˜ ( x , y , ω ) | 2 d x d y
Δ t G 2 = Δ t F T G 2 + Δ G D G 2 + τ A C 2 + τ P C 2
τ A C = Δ t F T 2 ( x , y ) Δ t F T G 2 ( x , y ) 1 / 2
τ P C = [ G D ( x , y , ω ) G D G ( ω ) ] 2 ( x , y , ω ) 1 / 2
G D ( x ) = γ ( x x 0 )
τ P C = | γ | Δ x
G D ( ω ) = G D D ( ω ω 0 )
Δ G D G = | G D D | Δ ω
Δ ω = ( 1 Δ ω 2 + ζ 2 Δ x 2 ) 1 / 2
Δ t F T G 2 = 1 4 Δ ω 2
Δ t F T 2 = Δ t F T 2 ( x ) = 1 4 Δ ω 2 = 1 4 Δ ω 2 + ζ 2 4 Δ x 2
τ A C = | ζ | 2 Δ x
G D ( x , ω ) = ξ ( ω ω 0 ) ( x x 0 )
τ P C = | ξ | Δ x Δ ω
Δ G D G = ( G D ( x , y ) G D ( x , y , ω ) ) 2 ( ω ) 1 / 2 = G D x p ( x , y ) 2 ( ω ) 1 / 2
τ P C = ( G D G D ( x , y ) ) 2 ( x , y , ω ) 1 / 2 = ( ϕ x p ω ) 2 ( x , y , ω ) 1 / 2
Δ t G 2 = t 2 t 2 = K 1 + I G ( t ) t 2 d t ( K 1 + I G ( t ) t d t ) 2
t = + I G ( t ) t d t + I G ( t ) d t = K 1 + | E ( x , y , t ) | 2 t d t d x d y = K 1 + E * ( x , y , t ) E ( x , y , t ) t d t d x d y
t = i K 1 + E ˜ * ( x , y , ω ) E ˜ ( x , y , ω ) ω d x d y d ω
t = i K 1 + | E ˜ | exp ( i ϕ ) [ | E ˜ | ω + i | E ˜ | ϕ ω ] exp ( i ϕ ) d x d y d ω = 0 + K 1 + | E ˜ | 2 ϕ ω d x d y d ω
t = ( + | E ˜ | 2 d ω ) G D ( ω ) d x d y + ( + | E ˜ | 2 d ω ) d x d y = G D ( ω ) ( x , y ) = G D ( x , y , ω )
t 2 = G D ( x , y , ω ) 2 = G D ( x , y ) ( ω ) 2 + G D ( x , y ) 2 ( ω ) G D ( x , y ) 2 ( ω ) = G D ( x , y ) 2 ( ω ) Δ G D G 2
t 2 = K 1 + I G ( t ) t 2 d t = K 1 + [ ( | E ˜ | 2 ω ) 2 + | E ˜ | 2 ( ϕ ω ) 2 ] d x d y d ω = K 1 + ( | E ˜ | ω ) 2 d x d y d ω + G D 2 ( x , y , ω )
( ( S G ( ω ) ) 1 / 2 ω ) 2 = 1 4 S G ( ω ) ( S G ( ω ) ω ) 2 = 1 4 S G ( ω ) ( + | E ˜ | 2 ω d x d y ) 2 = 1 S G ( ω ) ( + | E ˜ | | E ˜ | ω d x d y ) 2
( + | E ˜ | | E ˜ | ω d x d y ) 2 + | E ˜ | 2 d x d y + ( | E ˜ | ω ) 2 d x d y ( ( S G ( ω ) ) 1 / 2 ω ) 2 + ( | E ˜ | ω ) 2 d x d y
K 1 + ( S G ( ω ) 1 / 2 ω ) 2 d ω + τ A C 2 = K 1 + ( | E ˜ | ω ) 2 d x d y d ω
K 1 + ( S G ( ω ) 1 / 2 ω ) 2 d ω = K 1 + I F T G ( t ) t 2 d t = Δ t F T G 2
K 1 + ( | E ˜ | ω ) 2 d x d y d ω = + ( + | E | 2 d t ) Δ t F T 2 ( x , y ) d x d y + ( + | E | 2 d t ) d x d y = Δ t F T 2 ( x , y ) ( x , y )
t 2 = Δ t F T G 2 + τ A C 2 + G D 2 ( x , y , ω )
Δ t G 2 = Δ t F T G 2 + τ A C 2 + G D 2 ( x , y ) ( ω ) G D ( x , y ) 2 ( ω ) + Δ G D G 2 = Δ t F T G 2 + Δ G D G 2 + τ A C 2 + τ P C 2

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