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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 17357–17371
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Duration of ultrashort pulses in the presence of spatio-temporal coupling

C. Bourassin-Bouchet, M. Stephens, S. de Rossi, F. Delmotte, and P. Chavel  »View Author Affiliations


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

The generation, manipulation and characterization of ultrashort light pulses have become an area of considerable interest in the past few decades. Such pulses allow probing ultrafast phenomena such as chemical reactions [1

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]

] using femtosecond pulses, or the electronic motion in molecules [2

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]

] thanks to the decrease of the pulse duration down to the attosecond time scale. One major assumption in ultrafast optics is that time and space are uncoupled variables, which means that an ultrashort pulse can be described as the product of a time dependent pulse shape with a space dependent beam profile [3

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

]. It considerably simplifies the way to describe a pulse since the pulse shape is supposed to be same in every point in space.

However, since the beginning of ultrafast science, it has been shown that this assumption is often invalid. For example, misaligned prisms or gratings compressors, or aberrated optical systems can spatio-temporally distort a pulse [4

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

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]

], which causes a coupling between time and space variables. Moreover, as shorter and shorter pulses become accessible to experiment, this coupling has become more and more difficult to avoid. For example, it has been shown that attosecond extreme ultraviolet (XUV) pulses are very sensitive to optical aberrations of typical focusing mirrors [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]

]. So there is an increasing need for theoretical tools to describe the coupling phenomenon.

Especially, associating a duration to a pulse is not straightforward anymore. Indeed, since coupling makes the pulse change in space, its duration has to be locally defined [12

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

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

]. Nevertheless, it remains important to be able to summarize the full spatio-temporal pulse into a single duration. To do so, it has been proposed to consider the duration of the spatially integrated pulse, namely the global pulse duration [9

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

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

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

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

]. Some theoretical models studying the global duration already exist [12

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

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]

] for both gaussian temporal and spatial profiles and for first order coupling.

In this article, we propose a more general analysis of the global duration of arbitrary pulses. More precisely, we report on a way to decompose the root mean square (RMS) global pulse duration of a spatio-temporal pulse into four basic durations based on the behavior of the complex spectrum versus frequency and space. Each of these durations has a simple physical explanation. Firstly, we describe the principle of the duration decomposition. As an illustration, we single out each of them in a simple situation that may arise in the femtosecond regime, and we proceed with a example in the attosecond regime. Finally, we show that the contributions of three of the four elementary durations to the global pulse duration are measurable without any spectral phase characterization.

2. Theoretical Study

A linearly polarized light pulse propagating in vacuum can be considered as a scalar electric field 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:
2E(x,y,z,t)1c22E(x,y,z,t)t2=0
(1)

If considering the problem at a given position 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]

] as a product of a space-dependent function g(x,y) with a time-dependent function f (t) (resp. a frequency-dependent function (ω), where stands for the Fourier Transform of f), see Eq. (2):
E(x,y,z0,t)=g(x,y)f(t)
(2)

Defining a duration for such a pulse is straightforward since it suffices to estimate the characteristic width of the intensity function |f|2. One possible way to get such a width is to calculate the RMS duration Δt defined by Eq. (3):
Δt2=t2t2
(3)
where 〈〉 stands for the usual mean operator weighted by |f|2, so that 〈t〉 is equal to +|f(t)|2tdt/+|f(t)|2dt.

When developing Eq. (3), we get Eq. (4) which is well-established [3

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

]:
Δt2=ΔtFT2+ΔGD2
(4)

In this equation, 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 Δt depends on two parameters:
  1. ΔtFT is the 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.
  2. ΔGD is the RMS spectral variation of the GD. It quantifies the temporal synchronization of the spectral components.

To summarize, any spectral variations of the GD will stretch an ultrashort pulse, according to Eq. (4).

However, these results are based on the major assumption that the pulse is not distorted by any space-time coupling, see Eq. (2), which is a strong restriction. In order to establish a more general formula for the duration of an arbitrary spatio-temporal pulse, we first have to find a way to convert a space-time signal into a temporal one, whose duration will be easier to estimate. As proposed in [9

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

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]

], the simplest way to do so is to spatially integrate the pulse. The resulting temporal signal, hereafter referred to as the Global Pulse IG (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):
IG(t)=+|E(x,y,t)|2dxdy
(5)

By analogy, one can define global quantities in the spectral domain, such as the Global Spectrum as the spatially integrated spectrum SG (ω), and the Global Group Delay as the spatially averaged group delay GDG (ω), see Eqs. (6) and (7):
SG(ω)=+|E˜(x,y,ω)|2dxdy
(6)
GDG(ω)=GD(x,y)=+|E˜(x,y,ω)|2GDdxdy/+|E˜(x,y,ω)|2dxdy
(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 ).

Applying Eq. (3) to the temporal signal IG (t), we get the global pulse duration ΔtG, as named in [12

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

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

], that is the RMS duration of the spatially integrated pulse. This duration can be written the following way:
ΔtG2=ΔtFTG2+ΔGDG2+τAC2+τPC2
(8)

The complete demonstration of such a decomposition is given in Appendix A. According to Eq. (8), the duration ΔtG involves four parts:
  1. ΔtFTG is the 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 IFTG (t), the spectrum of which is the global spectrum SG (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).
  2. ΔGDG is the 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 ΔGDG reduces to ΔGD, which was previously defined in Eq. (4).
  3. τAC represents the temporal stretch due to a coupling of the spatio-spectral amplitude. Hereafter, such a coupling will be referred to as an Amplitude Coupling and τAC as the 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 ΔtFT (x, y) is equal to ΔtFTG in every point in space, see Eq. (9). τAC includes the influences of phenomena such as spatial chirp.
    τAC=ΔtFT2(x,y)ΔtFTG2(x,y)1/2
    (9)
  4. τPC stands for the 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.
    τPC=[GD(x,y,ω)GDG(ω)]2(x,y,ω)1/2
    (10)

Equation (4) highlighted that a pulse without space-time coupling can be described as a Fourier-transform limited pulse stretched by any non linear variations of its spectral phase. Equation (8) extends this analysis to spatio-temporal pulses. It shows that any space-time coupling can be seen as 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.

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.

3. Examples of Decomposition of the Duration

3.1. Basic Examples

First, let us consider a pulse that is distorted by a time delay which linearly varies in space with a slope 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):
GD(x)=γ(xx0)
(11)

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).

In this equation, x 0 is the central radial position of the pulse, which is equal to 0μ m. Since the group delay does not depend on ω, all the spectral components remains synchronized at a given point in space, so that GDG is zero. On the other hand, a spatially varying GD indicates a phase coupling. Indeed, using Eq. (10) gives immediately Eq. (12):
τPC=|γ|Δx
(12)

Given that Δx is equal to 1μ m, Eq. (12) predicts a pulse stretch equal to 20 fs, as shown on Fig. 2(a).

Now, consider the situation where the pulse is distorted by both spatial and temporal chirps. The spatial chirp is characterized by a spatial shift ζ of the spectral components of −0.118μm/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]

], which is visible on Fig. 2(b). According to our duration decomposition, it appears that the global pulse is equally stretched by ΔGDG 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 ΔGDG or τPC:
GD(ω)=GDD(ωω0)
(13)
where 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 GDG (ω) in every point, so that τPC is zero whereas ΔGDG is given by Eq. (14):
ΔGDG=|GDD|Δω
(14)

This expression stands for the usual stretch term of an uncoupled pulse distorted by a constant GDD [3

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

]. As for spatial chirp, it spatially spreads the spectral components, so that the spectrum becomes narrower than SG (ω) at a given point in space, as depicted on Fig. 1(c). Consequently, τAC cannot be null insofar as the local Fourier Transform limited duration ΔtFT (x) becomes greater than ΔtFTG. Indeed, it was shown [6

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

] that adding a linear spatial chirp to a pulse, the beam and pulse profile of which are Gaussian, makes the local width of the spectrum Δω′ (x) decrease, see Eq. (15):
Δω=(1Δω2+ζ2Δx2)1/2
(15)
where Δω is the spectrum width without spatial chirp. Moreover, it is well-known that the quantity ΔtFT · Δω′ is equal to 1/2 for gaussian pulses. This allows to deduce ΔtFTG and the local Fourier Transform limited duration ΔtFT (x), see Eqs. (16) and (17):
ΔtFTG2=14Δω2
(16)
ΔtFT2=ΔtFT2(x)=14Δω2=14Δω2+ζ24Δx2
(17)

Thus, using the definition of the amplitude coupling duration, we obtain τAC:
τAC=|ζ|2Δx
(18)

According to Eq. (14) and (18), both ΔGDG and τAC have to be equal to 20 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.

As a third example, let us consider a radially varying GDD characterized by a parameter 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 ΔGDG remains null. Indeed, since the GDD is alternatively negative and positive in different points in space, GDG (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):
GD(x,ω)=ξ(ωω0)(xx0)
(19)
τPC=|ξ|ΔxΔω
(20)

The latter gives a value of τPC equal to 25 fs, as shown on Fig. 2(c).

3.2. Complex Example

To go one step further with this analysis of the duration decomposition, we now consider more complex spatio-temporal pulses. Optical aberrations are known to be a typical source of spatio-temporal distortions of ultrashort pulses [7

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

9

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]

]. In particular, it has been shown that attosecond XUV pulses are very sensitive to aberrations induced by most focusing mirrors [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]

]. Moreover, such pulses are chirped due to the generation process [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]

]. The latter phenomenon, called the atto-chirp, is unavoidable. So the attosecond pulses can be stretched in a highly complex way due to a combination of atto-chirp and optical aberrations. Now we will investigate the influence of these various phenomena by decomposing the global duration of such distorted attosecond pulses.

To do so, we simulate an attosecond pulse reflected off a grazing incidence ellipsoidal mirror, and we compare the aberration-free case with a configuration that is aberrated due to misalignment. The focal length of the mirror is equal to 750mm and its optimal grazing angle is equal to 11.5°, leading to sagittal and tangential radii of curvature equal to 1500mm and 299mm, 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).

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.

The attosecond pulse generated at the source is chosen not to be distorted by space-time coupling. Its XUV spectrum has a gaussian envelope, centered at 75eV with a full width at half maximum (FWHM) of 30eV. The intrinsic GDD has a typical value of 6000as 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]

], and the divergence of the beam is between 1 and 2mrad 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]

]. As shown in Fig. 3(b), the full (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

17. 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.

], 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.

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).

We now consider the evolution of the decomposition while changing the grazing angle from 11.5° down to 11.4°, i. e. while adding astigmatism to the attosecond pulse. ΔtFTG remains constant since geometric aberrations do not change the involved spectral components, that is SG (ω). As for ΔGDG, it appears that the global group delay GDG (ω) does not vary either, since the atto-chirp remains constant whatever the strength of aberrations. Moreover, the amplitude coupling duration remains null whereas τPC 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.

These numerical simulations confirm that the duration decomposition is suitable to describe various, realistic and potentially highly complex spatio-temporal pulses.

4. Experimental Considerations and Discussion

For now, the duration decomposition has only been used to theoretically describe spatio-temporal pulses. But it could be interesting to know how to measure these durations experimentally. Of course, if we use techniques able to characterize the full spatio-temporal electric field of a pulse [18

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

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]

], the four durations can be easily extracted. But some other methods, which are easier to implement, may in fact give access to other measures of duration. Moreover, since the elementary durations correspond to four independent phenomena, these techniques are not necessarily the same for all the durations:
  1. ΔtFTG is given by the spatially integrated spectrum SG (ω). So a typical spectrometer without spatial resolution is sufficient to get IFTG (t) and its duration.
  2. τAC can be obtained by measuring the spatially resolved spectrum with an imaging spectrometer [6

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

    ].
  3. ΔGDG 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

    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. 26, 96–98 (2001). [CrossRef]

    , 23

    23. I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [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 ΔGDG, since this space dependent constant GD does not play any role in the global group delay. More precisely, the experimental group delay GDxp can be considered as a centered group delay 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), ΔGDG depends only on GDxp. So knowing the latter is sufficient to determine ΔGDG. 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.
    ΔGDG=(GD(x,y)GD(x,y,ω))2(ω)1/2=GDxp(x,y)2(ω)1/2
    (21)
  4. τPC 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 ΔGDG, the measured wavefront φxp (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).
    τPC=(GDGD(x,y))2(x,y,ω)1/2=(ϕxpω)2(x,y,ω)1/2
    (22)

Therefore, it appears that the four durations can be measured using independent experimental techniques and various devices, such as an imaging spectrometer, a SPIDER-like system or a wavefront sensor. In particular, the latter is usually able to measure both the intensity and the wavefront of a beam. So if the sensor is coupled with a tunable filter for the spectral resolution, it would allow to also access the spatially resolved spectrum, i. e. to measure ΔtFTG, τAC and τPC at the same time. Moreover, Hartmann wavefront sensors exist for the XUV range [24

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 λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003). [CrossRef] [PubMed]

], so such a setup would be potentially usable for XUV attosecond pulses if coupled with selective XUV mirrors to ensure the spectral resolution. Another way to access the space-time coupling without spectral phase measurements was given in [25

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]

]. The proposed solution is based on two spatially sheared and phase-shifted replicas of a pulse sent into an imaging spectrometer. By doing so, one can measure the spatially resolved spectrum along with the variations of the spectral phase across the beam, and thus get ΔtFTG, τAC and τPC. 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.

5. Conclusion

We described a way to summarize the complexity of 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

We consider the RMS duration ΔtG of the spatially integrated pulse, namely the Global pulse IG (t):
ΔtG2=t2t2=K1+IG(t)t2dt(K1+IG(t)tdt)2
(23)
where K stands for a normalization constant and is equal to +IG(t)dt.

6.1. Calculation of 〈t〉2

In order to develop Eq. (23), we first calculate 〈t2, which leads to Eq. (24):
t=+IG(t)tdt+IG(t)dt=K1+|E(x,y,t)|2tdtdxdy=K1+E*(x,y,t)E(x,y,t)tdtdxdy
(24)
where E * (x,y,t) corresponds to the complex conjugate of E (x,y,t). Using Parseval’s theorem, which gives Eq. (25):
t=iK1+E˜*(x,y,ω)E˜(x,y,ω)ωdxdydω
(25)
where (x,y,ω) represents the Fourier transform of E (x,y,t) and is equal to |(x,y,ω)|exp( (x,y,ω)). Hereafter |(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):
t=iK1+|E˜|exp(iϕ)[|E˜|ω+i|E˜|ϕω]exp(iϕ)dxdydω=0+K1+|E˜|2ϕωdxdydω
(26)

If noticing that K is equal to +|E˜|2dωdxdy, we get Eq. (27):
t=(+|E˜|2dω)GD(ω)dxdy+(+|E˜|2dω)dxdy=GD(ω)(x,y)=GD(x,y,ω)
(27)

Moreover, we can decompose 〈t2, leading to Eq. (28):
t2=GD(x,y,ω)2=GD(x,y)(ω)2+GD(x,y)2(ω)GD(x,y)2(ω)=GD(x,y)2(ω)ΔGDG2
(28)

6.2. Calculation of 〈t〉2

We still have to calculate 〈t2 in order to get ΔtG. Using Parseval’s theorem leads to Eq. (29):
t2=K1+IG(t)t2dt=K1+[(|E˜|2ω)2+|E˜|2(ϕω)2]dxdydω=K1+(|E˜|ω)2dxdydω+GD2(x,y,ω)
(29)

Using the definition of the global spectrum SG (ω), see Eq. (6), one can establish Eq. (30):
((SG(ω))1/2ω)2=14SG(ω)(SG(ω)ω)2=14SG(ω)(+|E˜|2ωdxdy)2=1SG(ω)(+|E˜||E˜|ωdxdy)2
(30)

Moreover, using the Cauchy-Schwarz inequality allows to get inequality [Eq. (31)]:
(+|E˜||E˜|ωdxdy)2+|E˜|2dxdy+(|E˜|ω)2dxdy((SG(ω))1/2ω)2+(|E˜|ω)2dxdy
(31)

We can introduce the Amplitude Coupling Duration τAC which is a positive or null quantity homogenous to a duration:
K1+(SG(ω)1/2ω)2dω+τAC2=K1+(|E˜|ω)2dxdydω
(32)

It is easily checkable that τAC is null if || 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:
K1+(SG(ω)1/2ω)2dω=K1+IFTG(t)t2dt=ΔtFTG2
(33)
where IFTG (t) is the Global Fourier Transform limited pulse which is, by definition, obtained by calculating the Fourier transform of the amplitude of SG (ω) for a zero spectral phase. If noticing that the squared Fourier transform limited pulse duration at a given point ΔtFT2(x,y) is equal to +(|E˜|ω)2dω/+|E˜|2dω, we get Eq. (34):
K1+(|E˜|ω)2dxdydω=+(+|E|2dt)ΔtFT2(x,y)dxdy+(+|E|2dt)dxdy=ΔtFT2(x,y)(x,y)
(34)

Using Eqs. (29), (32), (33) and (34) leads to Eq. (35):
t2=ΔtFTG2+τAC2+GD2(x,y,ω)
(35)
where τAC is equal to ΔtFT2(x,y)ΔtFTG2(x,y)1/2.

Finally, using Eqs. (23), (28) and (35), one can get the RMS global pulse duration, see Eq. (36):
ΔtG2=ΔtFTG2+τAC2+GD2(x,y)(ω)GD(x,y)2(ω)+ΔGDG2=ΔtFTG2+ΔGDG2+τAC2+τPC2
(36)
where τPC is equal to GD2(x,y)GD(x,y)2(ω)1/2, which can be rewritten (GDGDG)2(x,y,ω)1/2. It should be noticed that if there is no coupling on the spatio-spectral phase, that is if GD does not depend on space, τPC is null whatever the spectral phase.

Acknowledgments

We would like to acknowledge support through NSF grant “Modern Optics in the City of Light” PHY 0851614, and the University of Michigan’s Center for Ultrafast Optical Science. We would like to thank Alexander G. Anderson for his contribution to the final manuscript.

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 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]

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. 10, 213–220 (2004). [CrossRef]

6.

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

7.

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

8.

U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express 13, 3852–3861 (2005). [CrossRef] [PubMed]

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]

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]

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]

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]

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]

17.

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.

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]

19.

P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multispectral digital holography,” Opt. Express 14, 11460–11467 (2006). [CrossRef] [PubMed]

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 16, 13663–13675 (2008). [CrossRef] [PubMed]

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]

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. 26, 96–98 (2001). [CrossRef]

23.

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

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 λEUV/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]

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

  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]
  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. 10, 213–220 (2004). [CrossRef]
  6. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242, 599–604 (2004). [CrossRef]
  7. Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119–121 (1989). [CrossRef] [PubMed]
  8. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express 13, 3852–3861 (2005). [CrossRef] [PubMed]
  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]
  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]
  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]
  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]
  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]
  17. 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.
  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]
  19. P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multispectral digital holography,” Opt. Express 14, 11460–11467 (2006). [CrossRef] [PubMed]
  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 16, 13663–13675 (2008). [CrossRef] [PubMed]
  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]
  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. 26, 96–98 (2001). [CrossRef]
  23. I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [CrossRef]
  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 λEUV/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]

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