## Duration of ultrashort pulses in the presence of spatio-temporal coupling |

Optics Express, Vol. 19, Issue 18, pp. 17357-17371 (2011)

http://dx.doi.org/10.1364/OE.19.017357

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### 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

## 1. Introduction

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]

2. 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]

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

4. 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]

10. 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]

11. C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A **27**, 1395–1403 (2010). [CrossRef]

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

13. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**, 093001 (2010). [CrossRef]

9. 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]

10. 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]

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

13. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**, 093001 (2010). [CrossRef]

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

14. P. Gabolde, D. Lee, S. Akturk, and R. Trebino, “Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters,” Opt. Express **15**, 242–252 (2007). [CrossRef] [PubMed]

## 2. Theoretical Study

*E*solution of the propagation equation, that is Eq. (1), where

*z*stands for the position along the pulse propagation axis,

*x*and

*y*are the transverse coordinates and

*t*is the time variable:

*z*

_{0}, one typically assumes that the spatio-temporal electric field (resp. the spatio-spectral electric field) can be written [3

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

*g*(

*x*,

*y*) with a time-dependent function

*f*(

*t*) (resp. a frequency-dependent function

*f̃*(

*ω*), where

*f̃*stands for the Fourier Transform of

*f*), see Eq. (2):

*f*|

^{2}. One possible way to get such a width is to calculate the RMS duration Δ

*t*defined by Eq. (3): where 〈〉 stands for the usual mean operator weighted by |

*f*|

^{2}, so that 〈

*t*〉 is equal to

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

*GD*is the

*Group Delay*and is equal to

*dφ/dω*, with

*φ*the spectral phase of the pulse. According to Eq. (4), the duration Δ

*t*depends on two parameters:

- Δ
*t*is the_{FT}*Fourier Transform limited*duration. It corresponds to the duration of a pulse, the GD of which is constant with respect to frequency. It is the shortest RMS duration attainable with a given spectrum. - Δ
*GD*is the RMS spectral variation of the GD. It quantifies the temporal synchronization of the spectral components.

*GD*will stretch an ultrashort pulse, according to Eq. (4).

9. 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]

10. 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]

*Global Pulse I*(

_{G}*t*), would be the signal detected by an imaginary photodiode with a suitable temporal resolution but no spatial resolution. Moreover, the detection plane of such a sensor would be orthogonal to the pulse propagation axis, see Eq. (5):

*Global Spectrum*as the spatially integrated spectrum

*S*(

_{G}*ω*), and the

*Global Group Delay*as the spatially averaged group delay

*GD*(

_{G}*ω*), see Eqs. (6) and (7): where 〈〉

_{(}

_{x,y}_{)}is the mean weighted by the spatial intensity. Hereafter, it will be convenient to also use the spectral weighted mean 〈〉

_{(}

_{ω}_{)}. Moreover, the combination of the spatial mean with the spectral mean will be summarized into the operator 〈〉

_{(}

_{x,y,}_{ω}_{)}.

*I*(

_{G}*t*), we get the

*global pulse duration*Δ

*t*, as named in [12

_{G}**13**, 8642–8661 (2005). [CrossRef] [PubMed]

13. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**, 093001 (2010). [CrossRef]

*t*involves four parts:

_{G}- Δ
*t*is the_{FTG}*Global Fourier Transform limited duration*. It is the shortest RMS global duration attainable with the involved spectral components. It corresponds to the duration of a temporal pulse*I*(_{FTG}*t*), the spectrum of which is the global spectrum*S*(_{G}*ω*), and the spectral phase of which is zero. This fundamental limit is reached if there is no spectro-spatial coupling and the spectral phase is linear, see Fig. 1(a). - Δ
*GD*is the_{G}*Dispersion of the Global Group Delay*. It represents the global synchronization of the spectral components for a spatio-temporal pulse, as shown on Fig. 1(b). This term stands for phenomena such as temporal chirp or higher order dispersion. When there is no space-time coupling, the*GD*does not depend on (*x,y*) anymore, so Δ*GD*reduces to Δ_{G}*GD*, which was previously defined in Eq. (4). *τ*represents the temporal stretch due to a coupling of the spatio-spectral amplitude. Hereafter, such a coupling will be referred to as an_{AC}*Amplitude Coupling*and*τ*as the_{AC}*Amplitude Coupling Duration*, see Fig. 1(c). This duration is zero if every spectral component is equally distributed spatially, that is if the local Fourier-Transform limited duration Δ*t*(_{FT}*x*,*y*) is equal to Δ*t*in every point in space, see Eq. (9)._{FTG}*τ*includes the influences of phenomena such as spatial chirp._{AC}*τ*stands for the_{PC}*Phase Coupling Duration*, that is the temporal stretch induced by a coupling on the spatio-spectral phase. It quantifies the spatial synchronization of every spectral component, see Eq. (10) and Fig. 1(d). This has an influence when there is a spatio-spectral coupling in the phase,*i.e.*when the phase cannot be written as a*sum*of a frequency-dependent function with a space-dependent function. It summarizes the effects of phenomena such as spatially varying time delay.

*a deviation of local properties from global properties*(see Eqs. (9), and (10)), and that any coupling stretches a pulse. Moreover, Eq. (8) is a general formula which is accurate for

*arbitrarily*complex spatio-temporal pulses.

## 3. Examples of Decomposition of the Duration

### 3.1. Basic Examples

*nm*gaussian pulse with a 10

*fs*RMS duration. The pulse spatial profile is Gaussian and its RMS width Δ

*x*is equal to 1

*μm*. Then we add various distortions on the pulse, and numerically calculate the duration decomposition in each case. To simplify the understanding of the described pulses, we here consider only (

*x,t*) dependent pulses, so the previous equations remain valuable if removing the

*y*coordinate.

*γ*equal to 20

*fs*/

*μm*. We obtain the pulse reported on Fig. 2(a), the pulse front of which is tilted. The numerically obtained duration decomposition highlights that the pulse stretch is due to phase coupling. Indeed, there is no amplitude coupling since the spectrum remains the same in every point in space. This distortion only affects the group delay, the latter being given by Eq. (11):

*x*

_{0}is the central radial position of the pulse, which is equal to 0

*μ*

*m*. Since the group delay does not depend on

*ω*, all the spectral components remains synchronized at a given point in space, so that

*GD*is zero. On the other hand, a spatially varying GD indicates a phase coupling. Indeed, using Eq. (10) gives immediately Eq. (12):

_{G}*x*is equal to 1

*μ*

*m*, Eq. (12) predicts a pulse stretch equal to 20

*fs*, as shown on Fig. 2(a).

*ζ*of the spectral components of −0.118

*μm/nm*, and the

*Group Delay Dispersion*(GDD) inducing the temporal chirp is equal to 400

*fs*

^{2}/

*rad*. It is well established that adding temporal and spatial chirp creates pulse front tilt [15

15. 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]

*GD*and amplitude coupling. Indeed, the GDD modifies the spectral phase while leaving the spectrum unchanged, as shown by Eq. (13), so it can only modify Δ

_{G}*GD*or

_{G}*τ*: where

_{PC}*ω*

_{0}equals 800

*nm*and represents the central angular frequency of the spectrum. Since the group delay is constant in space,

*GD*(

*x,y,ω*) equals

*GD*(

_{G}*ω*) in every point, so that

*τ*is zero whereas Δ

_{PC}*GD*is given by Eq. (14):

_{G}**72**, 1–29 (2001). [CrossRef]

*S*(

_{G}*ω*) at a given point in space, as depicted on Fig. 1(c). Consequently,

*τ*cannot be null insofar as the local Fourier Transform limited duration Δ

_{AC}*t*(

_{FT}*x*) becomes greater than Δ

*t*. Indeed, it was shown [6

_{FTG}6. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. **242**, 599–604 (2004). [CrossRef]

*ω*′ (

*x*) decrease, see Eq. (15): where Δ

*ω*is the spectrum width without spatial chirp. Moreover, it is well-known that the quantity Δ

*t*· Δ

_{FT}*ω*′ is equal to 1/2 for gaussian pulses. This allows to deduce Δ

*t*and the local Fourier Transform limited duration Δ

_{FTG}*t*(

_{FT}*x*), see Eqs. (16) and (17):

*GD*and

_{G}*τ*have to be equal to 20

_{AC}*fs*. Finally, by comparing Figs. 2(a) and 2(b), it appears that pulse front tilt, or more generally distortion of the shape of the pulse front, can be obtained either by adjusting the phase coupling, or by choosing a combination of amplitude coupling with a nonzero global group delay.

*ξ*, which equals 500

*fs*

^{2}/

*rad*/

*μm*. To be more specific, the pulse depicted on Fig. 2(c) is distorted by a positive (resp. negative) GDD for positive (resp. negative) positions and zero on the propagation axis. The resulting global pulse is clearly stretched by these distortions. But according to the duration decomposition, the global pulse is not chirped since Δ

*GD*remains null. Indeed, since the GDD is alternatively negative and positive in different points in space,

_{G}*GD*(

_{G}*ω*) is zero. In other words, the spectral components remain temporally synchronized on average. Finally, the real source of pulse stretching is phase coupling, insofar as the GD depends on space. More precisely, the group delay is given by Eq. (19), and the corresponding phase coupling duration by Eq. (20):

### 3.2. Complex Example

*mm*and its optimal grazing angle is equal to 11.5°, leading to sagittal and tangential radii of curvature equal to 1500

*mm*and 299

*mm*, respectively. To add some aberrations to the pulse, the mirror is set in the focus-focus configuration, but with a grazing angle reduced to 11.4°, see Fig. 3(a).

*eV*with a full width at half maximum (FWHM) of 30

*eV*. The intrinsic GDD has a typical value of 6000

*as*

^{2}/rad [16

16. 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]

*mrad*over the whole spectrum. Since we simulate a grazing incidence mirror, we can consider that its reflectivity is constant and its spectral phase is linear over the whole spectrum, which is the case for gold or platinum made mirrors. The theoretical model for the simulations is described in [11

11. C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A **27**, 1395–1403 (2010). [CrossRef]

*x,y,t*) pulse is simulated. This becomes necessary to completely see the influence of aberrations that clearly depend both on

*x*and

*y*. To be more specific, Fig. 3(b) reports on the evolution in space and time of the intensity of the attosecond pulse at the paraxial focus of the ellipsoidal mirror. The obtained spatio-temporal pulse is clearly distorted by astigmatism [17], leading to a stretch of the global pulse. In addition to astigmatism, the atto-chirp increases the pulse duration too. As shown on Figs. 3(c) and 3(d), this causes the instantaneous frequency to vary throughout the envelope of the pulse.

*i. e.*where the only possible stretch phenomenon is the atto-chirp, to the previous aberrated case. We first consider that the mirror is set at its optimal grazing angle of 11.50°, leading to a diffraction limited pulse, see Fig. 4(a). At this angle, the duration of the obtained global pulse is equal to 118

*as*(278

*as*FWHM). The decomposition shows that Δ

*t*is equal to 26

_{FTG}*as*, see Fig. 4(c). This duration is also the Fourier Transform limited pulse duration given by Eq. (16) using the initial gaussian spectrum. Indeed the global spectrum does not change after the reflection off the mirror since its reflectivity was supposed to be constant over the whole spectrum. Moreover, the spectral phase of the grazing incidence mirror was assumed to be linear due to the total reflection phenomenon. So the obtained Δ

*GD*of 115

_{G}*as*also corresponds to the temporal stretch of a pulse, the spectrum and GDD of which are the initial gaussian spectrum and the initial GDD, as confirmed by Eq. (14). Finally it appears that the two coupling durations are zero, which is consistent since a diffraction limited pulse at its focus is not distorted by any space-time coupling.

*i. e.*while adding astigmatism to the attosecond pulse. Δ

*t*remains constant since geometric aberrations do not change the involved spectral components, that is

_{FTG}*S*(

_{G}*ω*). As for Δ

*GD*, it appears that the global group delay

_{G}*GD*(

_{G}*ω*) does not vary either, since the atto-chirp remains constant whatever the strength of aberrations. Moreover, the amplitude coupling duration remains null whereas

*τ*increases linearly with respect to the grazing angle. Indeed, as it was in the case of Fig. 3(c) and 3(d), the pulse is composed of the same spectral components in every point in space, so the spectrum does not vary in space. But the major consequence of astigmatism is to curve the pulse front by applying a time delay depending on space to the diffraction limited pulse. This phenomenon was already described on Fig. 1(d) and Fig. 2(a), and is known to be pure phase coupling. So the stronger the astigmatism, the greater the phase coupling duration. Moreover, it appears that a 0.1° misalignment of the mirror is sufficient to almost double the global pulse duration, which confirms the high sensitivity of attosecond pulses to optical aberrations.

_{PC}## 4. Experimental Considerations and Discussion

18. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Direct space-time characterization of the electric fields of ultrashort optical pulses,” Opt. Lett. **27**, 548–550 (2002). [CrossRef]

21. B. Alonso, I. J. Sola, O. Varela, J. Hernández-Toro, C. Méndez, J. San Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by Fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B **27**, 933–940 (2010). [CrossRef]

- Δ
*t*is given by the spatially integrated spectrum_{FTG}*S*(_{G}*ω*). So a typical spectrometer without spatial resolution is sufficient to get*I*(_{FTG}*t*) and its duration. *τ*can be obtained by measuring the spatially resolved spectrum with an imaging spectrometer [6_{AC}].6. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun.

**242**, 599–604 (2004). [CrossRef]- Δ
*GD*can be determined applying interferometric spectral phase characterization techniques such as Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER). Indeed, when using such techniques, the retrieved group delay is usually known up to a constant [22_{G}, 2322. L. Gallmann, G. Steinmeyer, D. H. Sutter, T. Rupp, C. Iaconis, I. A. Walmsley, and U. Keller, “Spatially resolved amplitude and phase characterization of femtosecond optical pulses,” Opt. Lett.

**26**, 96–98 (2001). [CrossRef]] which can depend on position in presence of phase coupling. Consequently, performing independent measurements in different points of the pulse does not allow to reconstruct the full spatio-temporal pulse. Nevertheless, it does not prevent from determining Δ23. I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.

**1**, 308–437 (2009). [CrossRef]*GD*, since this space dependent constant GD does not play any role in the global group delay. More precisely, the experimental group delay_{G}*GD*can be considered as a centered group delay_{xp}*GD*(*x*_{0},*y*_{0},*ω*) – 〈*GD*〉_{(}_{ω}_{)}(*x*_{0},*y*_{0}) where the absolute GD at the point (*x*_{0},*y*_{0}) has been lost. According to Eq. (21), Δ*GD*depends only on_{G}*GD*. So knowing the latter is sufficient to determine Δ_{xp}*GD*. Nevertheless, it should be noted that the mean of the GD has to be weighted by the spatially-resolved spectrum, meaning that the spatio-spectral intensity has to be extracted from the measurements._{G} *τ*can be measured using a simple wavefront sensor. More precisely, a Shack-Hartmann-like wavefront sensor can measure the shape of monochromatic wavefronts but not the phase relation between these wavefronts, that is the spectral phase. However, to do a similar analysis as in the case of Δ_{PC}*GD*, the measured wavefront_{G}*φ*(_{xp}*x*,*y,**ω*_{0}) at the*ω*_{0}frequency can be seen as a centered wavefront*φ*(*x,y,**ω*_{0}) − 〈*φ*〉_{(}_{x,y}_{)}(*ω*_{0}) without any spectral phase information. Thus it becomes simple to get the phase coupling duration by performing spectrally resolved wavefront measurements, see Eq. (22).

*i. e.*to measure Δ

*t*,

_{FTG}*τ*and

_{AC}*τ*at the same time. Moreover, Hartmann wavefront sensors exist for the XUV range [24

_{PC}24. 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 *λ _{EUV}*/120 accuracy,” Opt. Lett.

**28**, 1534–1536 (2003). [CrossRef] [PubMed]

25. C. Dorrer and I. A. Walmsley, “Simple linear technique for the measurement of space-time coupling in ultrashort optical pulses,” Opt. Lett. **27**, 1947–1949 (2002). [CrossRef]

*t*,

_{FTG}*τ*and

_{AC}*τ*. These simple setups could be a good way to measure and minimize both amplitude and phase coupling without the need for complete spatio-temporal characterization techniques.

_{PC}## 5. Conclusion

*arbitrary*spatio-temporal light pulses into four elementary durations, namely one fundamental limit and three extra stretch terms. These three terms highlight the influence of three independent stretch phenomena on spatio-temporal pulses, named the dispersion of the global group delay, the amplitude coupling and the phase coupling. We illustrated this decomposition using numerical simulations of femtosecond and attosecond pulses distorted by various phenomena, such as pulse front tilt or optical aberrations. Moreover, we discussed a possible way to experimentally measure these durations, and considered simple setups to retrieve three of these terms without spectral phase characterization. We conclude that this duration decomposition appears to have the potential to become a useful tool for describing and characterizing ultrashort light pulses.

## 6. Appendix A: Duration of an Arbitrary Spatio-Temporal Pulse

*t*of the spatially integrated pulse, namely the

_{G}*Global pulse I*(

_{G}*t*): where

*K*stands for a normalization constant and is equal to

## 6.1. Calculation of 〈t〉^{2}

*t*〉

^{2}, which leads to Eq. (24):

*E*

^{*}(

*x,y,t*) corresponds to the complex conjugate of

*E*(

*x,y,t*). Using Parseval’s theorem, which gives Eq. (25): where

*Ẽ*(

*x*,

*y*,

*ω*) represents the Fourier transform of

*E*(

*x*,

*y*,

*t*) and is equal to |

*Ẽ*(

*x*,

*y*,

*ω*)|exp(

*iφ*(

*x*,

*y*,

*ω*)). Hereafter |

*Ẽ*(

*x*,

*y*,

*ω*)| (resp.

*φ*(

*x*,

*y*,

*ω*)) will be written |

*Ẽ|*(resp.

*φ*). Moreover, since 〈

*t*〉 is a real quantity, we obtain Eq. (26):

## 6.2. Calculation of 〈t〉^{2}

*t*〉

^{2}in order to get Δ

*t*. Using Parseval’s theorem leads to Eq. (29):

_{G}*Amplitude Coupling Duration*

*τ*which is a positive or null quantity homogenous to a duration:

_{AC}*τ*is null if |

_{AC}*Ẽ*| can be written as a product of a spectral function with a space-dependent function, that is if there is no coupling on the spatio-spectral amplitude. Moreover: where

*I*(

_{FTG}*t*) is the

*Global Fourier Transform limited pulse*which is, by definition, obtained by calculating the Fourier transform of the amplitude of

*S*(

_{G}*ω*) for a zero spectral phase. If noticing that the squared Fourier transform limited pulse duration at a given point

_{AC}is equal to

*τ*is equal to

_{PC}*GD*does not depend on space,

*τ*is null whatever the spectral phase.

_{PC}## Acknowledgments

## References and links

1. | E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, “Femtosecond laser control of a chemical reaction,” Nature |

2. | 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. |

3. | I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. |

4. | 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. |

5. | 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. |

6. | X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. |

7. | Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. |

8. | U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express |

9. | 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 |

10. | 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. |

11. | C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A |

12. | S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams,” Opt. Express |

13. | S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. |

14. | P. Gabolde, D. Lee, S. Akturk, and R. Trebino, “Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters,” Opt. Express |

15. | S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express |

16. | 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 |

17. | Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” in |

18. | C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Direct space-time characterization of the electric fields of ultrashort optical pulses,” Opt. Lett. |

19. | P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multispectral digital holography,” Opt. Express |

20. | 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 |

21. | B. Alonso, I. J. Sola, O. Varela, J. Hernández-Toro, C. Méndez, J. San Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by Fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B |

22. | L. Gallmann, G. Steinmeyer, D. H. Sutter, T. Rupp, C. Iaconis, I. A. Walmsley, and U. Keller, “Spatially resolved amplitude and phase characterization of femtosecond optical pulses,” Opt. Lett. |

23. | I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. |

24. | 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 28, 1534–1536 (2003). [CrossRef] [PubMed] |

25. | C. Dorrer and I. A. Walmsley, “Simple linear technique for the measurement of space-time coupling in ultrashort optical pulses,” Opt. Lett. |

**OCIS Codes**

(320.5520) Ultrafast optics : Pulse compression

(320.5550) Ultrafast optics : Pulses

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 28, 2011

Revised Manuscript: July 25, 2011

Manuscript Accepted: July 25, 2011

Published: August 18, 2011

**Citation**

C. Bourassin-Bouchet, M. Stephens, S. de Rossi, F. Delmotte, and P. Chavel, "Duration of ultrashort pulses in the presence of spatio-temporal coupling," Opt. Express **19**, 17357-17371 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17357

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### References

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