Frequency chirp of long electron quantum paths in high-order harmonic
generation

doi:10.1364/OE.14.002242

Frequency chirp of long electron quantum paths in high-order harmonic generation

Enrico Benedetti, Jean-Pascal Caumes, Giuseppe Sansone, Salvatore Stagira, Caterina Vozzi, and Mauro Nisoli »View Author Affiliations

Optics Express, Vol. 14, Issue 6, pp. 2242-2249 (2006)
http://dx.doi.org/10.1364/OE.14.002242

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– Abstract
1. Introduction
2. Experimental results and discussion
3. Conclusion
– References and links

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Abstract

We report on the first experimental measurement of the spectral broadening of the harmonic emission associated with only the long electron quantum paths as a function of the driving pulse intensity. For such measurements we have estimated the chirp coefficient associated with the long quantum paths, within the strong-field approximation. This coefficient describes how the harmonic phase depends on the intensity of the driving laser field. The dependence of the chirp coefficient on the driving pulse intensity and on the harmonic order, which strongly influence the characteristics of the harmonic radiation, has been experimentally investigated. The experimental values turn out to be in excellent agreement with the results of numerical simulations based on the use of the nonadiabatic saddle-point method.

1. Introduction

The production of extreme ultraviolet (XUV) radiation by high-order harmonic generation (HHG) in noble gases is a well developed research topic due to its applications in various fields of physics and XUV technology [1

E.A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I.P. Christov, A. Aquila, E.M. Gullikson, D. T. Attwood, M.M. Murnane, and H.C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science 302, 95–98 (2003). [CrossRef] [PubMed]

, 2

A. L’Huillier, D. Descamps, A. Johansson, J. Norin, J. Mauritsson, and C.-G. WahlstrÖm, “Applications of high-order harmonics,” Eur. Phys. J. D 26, 91–98 (2003). [CrossRef]

], including the emerging field of attosecond science. In particular, in the last five years dramatic progress in the field of attosecond physics has been obtained thanks to the introduction of new techniques for the characterization of the attosecond pulses [3

P. Agostini and L.F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855 (2004). [CrossRef]

]. In the case of isolated attosecond pulses [4

I.P. Christov, M.M. Murnane, and H. Kapteyn, “High-harmonic generation of attosecond pulses in the “single-cycle” regime,” Phys. Rev. Lett. 78, 1251–1254 (1997). [CrossRef]

], generated by phase-stabilized few-optical-cycle light pulses, the attosecond-streak camera method [5

J. Itatani, F. Quéré, G.L. Yudin, M.Yu. Ivanov, F. Krausz, and P.B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903-1-4 (2002). [CrossRef] [PubMed]

] allowed the measurement of 250-as isolated pulses [6

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, Th. West-erwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427, 817–821 (2004). [CrossRef]

] and the first direct measurement of the electric field of a few-cycle light pulse [7

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltusška, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004). [CrossRef] [PubMed]

]. In the case of multiple-cycle pulses, trains of attosecond pulses have been generated and their average temporal intensity profile has been determined using the reconstruction of attosecond beating by interference of two-photon transition (RABITT) method [8

P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Aug>, Ph. Balcou, H.G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001). [CrossRef] [PubMed]

, 9

H.G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, S17–S21 (2002). [CrossRef]

, 10

V. Véniard, R. Taïeb, and A. Maquet,“Phase dependence of (N + 1)-color (N > 1) ir-uv photoionization of atoms with higher harmonics,” Phys. Rev. A 54, 721–728 (1996). [CrossRef] [PubMed]

] and by the measurement of the second-order auto-correlation of the train of attosecond pulses [11

P. Tzallas, D. Charalambidis, N.A. Papadogiannis, K. Witte, and G.D. Tsakiris, “Direct observation of attosecond light bunching,” Nature (London) 426, 267–271 (2003). [CrossRef]

The physical processes leading to HHG and to the formation of attosecond pulses can be clearly understood in the framework of a quantum-mechanical theory using the concept of Feynman’s path integrals [12

P. Saliéres, B. Carré, L.Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D.B. Milošsević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001). [CrossRef] [PubMed]

, 13

M. Lewenstein, Ph. Balcou, M.Y. Ivanov, A. L’Huillier, and P.B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]

], which correspond to the complex trajectories (quantum paths) followed by the electrons from the ionization instant to the recombination with the parent ion, after acceleration in the infrared driving field [13

, 14

M. Lewenstein, P. Saliéres, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52, 4747–4754 (1995). [CrossRef] [PubMed]

]. Two classes of electron quantum paths give the most relevant contribution to the harmonic generation process: the so-called short and long paths, characterized by electron travel times in the continuum of the order of one-half and one optical period, respectively [15

P. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using highorder harmonics,”Phys. Rev. Lett. 77,, 1234–1237 (1996). [CrossRef] [PubMed]

, 16

M.B. Gaarde and K.J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. 89, 213901 (2002). [CrossRef] [PubMed]

]. Macroscopic phase-matching conditions and proper spatial filtering allows one to isolate a single class of electron quantum paths. Each quantum path contributes to the harmonic dipole moment with a term characterized by a phase, Φ_i, which is given by the action along the considered electron trajectory and turns out to be proportional to the intensity, I, of the driving laser field: Φ_i(t) = -α_iI(t). The intensity dependent phase gives rise to spectral broadening and frequency modulation (chirp) of each harmonic. The chirp coefficient, α_i , is different for the short and long paths and it is closely related to the time spent by the electron in the continuum and to the coherence properties of the generated harmonic radiation [17

C. Lyngå, M.B. Gaarde, C. Delfin, M. Bellini, T.W. Hänsch, A. L’Huillier, and C.-G. WahlstrÖm, “Temporal coherence of high-order harmonics,” Phys. Rev. A 60, 4823–4830 (1999). [CrossRef]

]. The intensity and harmonic order dependence of the chirp coefficient strongly influences the characteristics of the XUV radiation. Such coefficients have been calculated both for the short and long quantum paths [18

M.B. Gaarde and K.J. Schafer, “Quantum path distributions for high-order harmonics in rare gas atoms,” Phys. Rev. A 65, 031406(R)-1-4 (2002). [CrossRef]

], but a few experimental data have been reported only in the case of the short quantum paths [19

Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Saliéres, and B. Carré, “High harmonic XUV spectral phase interferometry for direct electric-field reconstruction,” Phys. Rev. Lett. 94, 173903-1-4 (2005). [CrossRef]

In this work we report, for the first time at our knowledge, on the experimental measurement of the intensity dependent spectral broadening of harmonic radiation produced by only the long electron quantum paths, from which we estimate the corresponding chirp coefficient within the Lewenstein model [13

]. The intensity and harmonic order dependence of such parameter has been measured using the intensity dependent spectral broadening of the harmonic peaks. The experimental data have been analyzed in the framework of the strong-field approximation [13

] using the saddle-point method generalized to account for nonadiabatic effects [13

, 20

G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411-1-10 (2004). [CrossRef]

2. Experimental results and discussion

Harmonic radiation has been produced by focusing 30-fs light pulses, produced by a Ti:sapphire laser system (800-nm central wavelength, 1-kHz repetition rate), on an Argon jet placed around the position of the laser focus. We have then blocked the central part of the harmonic beam with a circular beam stop centered on the XUV beam. In this way it was possible to select the contribution of the long quantum paths. Indeed, when the gas jet is located around the position of the laser focus the contribution of the long paths increases due to phase-matching processes [12

]. Moreover, the XUV emission due to the long paths is characterized by a larger divergence with respect to that due to the short paths [21

M. Bellini, C. Lyngå, A. Tozzi, M.B. Gaarde, T.W. Hänsch, A. L’Huillier, and C.-G. WahlstrÖm, “Temporal coherence of ultrashort high-order harmonic pulses,” Phys. Rev. Lett. 81, 297–300 (1998). [CrossRef]

], therefore it is possible to isolate the contributions of the long quantum paths simply by spatial filtering. The harmonic radiation was sent to a XUV spectrometer by a grazing-incidence toroidal mirror. The spectrometer is composed by one toroidal mirror mounted at grazing incidence for stigmatic imaging with almost unitary magnification, and one variable-line-spaced grating, mounted after the mirror [22

L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, “Instrumentation for analysis and utilization of extreme-ultraviolet and soft x-ray high-order harmonics,” Rev. Sci. Instrum. 75, 4413–4418 (2004). [CrossRef]

]. The resulting spectrum is stigmatic and almost flat even in a wide spectral region. The detector is a multichannel-plate intensifier with phosphor screen optically coupled to a low-noise fast-readout charge-coupled device (CCD) camera.

Fig. 1. Measured evolution of the spectral characteristics of the 19-th harmonic generated in Argon by 30-fs pulses as a function of the driving pulse intensity.

Figure 1 shows the evolution of the spectral characteristics of the 19-th harmonic as a function of the peak intensity, I ₀, of the driving pulses, ranging from ~ 1.2 × 10¹⁴ W/cm² to ~ 1.8×10¹⁴ W/cm²; similar results have been observed in correspondence of all the harmonic orders. Upon increasing the excitation intensity the harmonic peak continuously broadens; at I ₀ > 2×10¹⁴ W/cm² the spectral broadening becomes larger than the frequency separation between two consecutive harmonics. In this case (not shown in Fig. 1) interference fringes can be observed in the region between consecutive harmonics using driving pulses with stable carrier-envelope phase [23

G. Sansone, E. Benedetti, J-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S.De Silvestri, and M. Nisoli, “Measurement of Harmonic Phase Differences by Interference of Attosecond Light Pulses,” Phys. Rev. Lett. 94, 193903-1-4 (2005). [CrossRef] [PubMed]

]. We note that, while the bandwidth of the harmonic peaks generated by the long paths is strongly influenced by the driving pulse intensity, the spectral characteristics of the harmonic radiation associated to the short paths are much less affected by intensity [18

M.B. Gaarde and K.J. Schafer, “Quantum path distributions for high-order harmonics in rare gas atoms,” Phys. Rev. A 65, 031406(R)-1-4 (2002). [CrossRef]

]. This is clearly demonstrated in Figs. 2(a) and (b), which show the spectral profiles of the 19-th harmonic generated by the long and short paths, respectively, at two different pulse intensities. The short paths have been selected by placing the gas jet after the laser focus [12

] and by removing the beam stop placed on the XUV beam. As expected, in the case of the short quantum paths the measured bandwidth is barely affected by the driving pulse intensity.

Fig. 2. Normalized spectral profiles of 19-th harmonic generated in Argon by the long (a) and short (b) quantum paths for two intensity values: I≈1.2 × 10¹⁴ W/cm² (solid lines), I≈1.8 × 10¹⁴ W/cm² (dashed lines).

It is well known that the observed spectral broadening of the harmonic peaks is related to the intensity dependence of the dipole phase, Φ_i(t), associated to each quantum path contributing to the harmonic generation, given by Φ_i(t) = -α_iI(t), where: i=1,2 refers to the short (i = 1) and long (i = 2) electron quantum paths contributing to the harmonic emission [12

,24

P. Saliéres, A. L’Huillier, and M. Lewenstein, “Coherence Control of High-Order Harmonics,” Phys. Rev. Lett. 74, 3776–3779 (1995). [CrossRef] [PubMed]

, 25

M.B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K.J. Schafer, K.C. Kulander, and A. L’Huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A 59, 1367–1373 (1999). [CrossRef]

, 18

M.B. Gaarde and K.J. Schafer, “Quantum path distributions for high-order harmonics in rare gas atoms,” Phys. Rev. A 65, 031406(R)-1-4 (2002). [CrossRef]

]; I(t) is the intensity profile of the infrared pulse; the coefficients ai are related to the time spent by the electron in the continuum and are different for the short and long quantum paths. Such intensity dependence of the dipole phase determines a variation of the instantaneous frequency (chirp) of the single harmonic δω_i (t) = -∂Φ_i,/∂t/, which gives rise to the spectral broadening. Therefore such spectral broadening depends on the α_i coefficient and can be used to estimate it. We have then calculated the bandwidths of the harmonic peaks as standard deviation of the measured harmonic spectra. Figure 3 shows the rms bandwidth of the 19-th harmonic normalized to the fundamental angular frequency, Δω/(ω ₀, as a function of the laser peak intensity. In Fig. 3 we have also reported, as dashed line, the rms bandwidth calculated using the nonadiabatic saddle-point (NASP) method [13

, 20

], which allows one to consider separately the contribution of the various electron quantum paths. The calculated Δω values turn out to be in excellent agreement with the experimental results. Due to our experimental conditions, the Fourier transform of the dipole moment can be calculated as coherent superposition of the contributions of the long quantum paths as follows:

Fig. 3. rms bandwidth of the 19-th harmonic normalized to the fundamental angular frequency (ω ₀ as a function of the peak intensity of the driving pulses. The dots with error bars are the experimental values. The dashed curve was calculated using the nonadiabatic saddle-point method. Pulse duration T ₀ = 30 fs.

x (ω) = \sum_{s \in long} ∣ x_{s} (ω) ∣ \exp [i Ψ_{s} (ω)],

(1)

where Ψ_s(ω) is the phase of the complex function x_s (ω). It is worth to point out that the calculated spectral characteristics of the harmonic radiation reproduce the experimental results remarkably well.

Using the measured spectral broadening we have then evaluated the intensity dependence of the chirp coefficient α. For the calculation we have assumed that the electric field of the q-th harmonic is given by:

E_{q} (t) = E_{q} 0 \exp (- \frac{t^{2}}{τ_{q}^{2}}) \exp (- iα I_{0} e^{\frac{- 2 t^{2}}{τ_{0}^{2}}}),

(2)

Fig. 4. Chirp coefficients α associated to the long and short quantum paths contributing to the 19-th harmonic, as a function of the pulse peak intensity. The symbols with error bars have been obtained using the data reported in Fig. 3 and Eq. 4. The triangles have been calculated assuming a constant duration of the harmonic pulses, τ_q = τ ₀/2; the dots have been obtained using the harmonic duration calculated using the NASP method. The dashed curve has been calculated using the NASP method, as described in the text. Pulse duration T ₀ = 30 fs.

where τ_q is related to the full-width-at-half-maximum duration, T_q , of the q-th harmonic pulse by

T_{q} = τ_{q} \sqrt{2 \log 2}

and I = I ₀ e^-2t²/

τ_{0}^{2}

is the temporal intensity profile of the fundamental pulse. As reported in Ref. [26

S.C. Pinault and M.J. Potasek, “Frequency broadening by self-phase modulation in optical fibers,” J. Opt. Soc. Am. B 2, 1318–1319 (1985). [CrossRef]

] the variance of the spectrum of the q-th harmonic can be evaluated in closed form as follows:

{(Δω)}^{2} = \frac{\int {∣ E_{q}^{'} (t) ∣}^{2} dt}{\int {∣ E_{q} (t) ∣}^{2} dt} + {[\frac{\int E_{q}^{'} (t) E_{q}^{*} (t) dt}{\int {∣ E_{q} (t) ∣}^{2} dt}]}^{2},

(3)

where E′_q(t) is the temporal derivative of the electric field of the q-th harmonic and

E_{q}^{*}

(t) is its complex conjugate. We thus obtain:

{(Δ ω)}^{2} = \frac{1}{τ_{q}^{2}} [1 + \frac{4 α^{2} I_{0}^{2}}{δ {(δ^{2} + 2)}^{\frac{3}{2}}}],

(4)

where δ = τ ₀/τ_q . Therefore, in order to calculate the α coefficient from the measured band-widths it is required to know the duration of the harmonic pulses. As a first approximation we will assume τ_q = τ ₀/2; such assumption is supported by the results of nonadiabatic saddle-point simulations, as will be discussed in the following. Using Eq. (4) and the experimental results shown in Fig. 3, we have calculated the intensity dependence of the coefficient α. The results are reported as triangles in Fig. 4. The chirp coefficient α is nearly constant with intensity, α ≈ 21 × 10^-14 cm²/W. We have then evaluated the dependence of the chirp coefficient on the harmonic order at a fixed driving pulse intensity. The results obtained assuming τ_q = τ ₀/2 are shown as triangles in Fig. 5. As predicted by theory [18

M.B. Gaarde and K.J. Schafer, “Quantum path distributions for high-order harmonics in rare gas atoms,” Phys. Rev. A 65, 031406(R)-1-4 (2002). [CrossRef]

], the chirp coefficient of the long paths slightly decreases with order.

Fig. 5. Chirp coefficients α associated to the long and short quantum paths, as a function of the harmonic order. The symbols with error bars have been obtained using the measured harmonic bandwidths and Eq. 4. The triangles have been calculated assuming a constant duration of the harmonic pulses, τ_q = τ ₀/2; the dots have been obtained using the harmonic duration calculated using the NASP method. The squares and open circles are the chirp coefficients for the long and short paths, respectively, calculated using the NASP method, as described in the text. Pulse duration T ₀ = 30 fs; peak intensity of the driving pulse I ₀ = 1.54 × 10¹⁴ W/cm².

As previously noted, in order to have a better estimation of the chirp coefficients, the duration of the harmonic pulses is required [see Eq. (4)]. We have then calculated the duration of the harmonic pulses generated by the contribution of the long quantum paths using the NASP simulations. The calculated pulse duration slightly changes in the intensity range from 1.2 to 2 × 10¹⁴ W/cm², with a mean value T ₁₉ ≈ 13.7 fs in the case of the 19-th harmonic, which is close to the T ₀/2 value used as a first approximation. Moreover, at a fixed value of the driving pulse peak intensity the harmonic pulse duration slightly increases with the harmonic order: at I ₀ = 1.54 × 10¹⁴ W/cm² T_q ranges from 13.6 fs, in the case of the 13-th harmonic, to 14.8 fs, in the case of the 23-rd harmonic. We have then used the calculated τ_q values in Eq. (4): the results are shown as dots with error bars in Figs. 4 and 5. The chirp coefficients calculated with the correct τ_q values are very close to the values obtained considering a constant duration of the harmonic pulses (τ_q = τ ₀/2). Using the NASP simulations we have then directly calculated the chirp coefficients as α_q = -∂Φ_q/∂_I , where Φ_q is the phase of the q-th harmonic. The results of the calculations are shown in Figs. 4 and 5, for both short and long quantum paths. The chirp coefficients evaluated from the experimental data turn out to be in excellent agreement with the predicted values: this is a further confirmation of the validity of our experimental approach.

Using the measured values of the chirp coefficients as a function of intensity and harmonic order we have calculated the corresponding chirp rate, b_q = -∂²Φ_q/∂t ², which, in the case of a Gaussian pulse, as given by Eq. (2), can be written as:

b_{q} = \frac{- 4 α_{q} I_{0}}{τ_{0}^{2}} .

(5)

Upon increasing the driving pulse intensity from 1.2 to 2 × 10¹⁴ W/cm² the chirp rate decreases from -0.16 fs^-2 to -0.28 fs^-2 in the case of the 19-th harmonic. At a fixed value of the driving pulse peak intensity the chirp rate slightly increases with the harmonic order: at I ₀ = 1.54 × 10¹⁴ W/cm² bq ranges from -0.23 fs^-2, in the case of the 15-th harmonic, to -0.20 fs^-2, in the case of the 21-st harmonic.

3. Conclusion

We have measured the intensity dependent spectral broadening of the harmonic radiation generated by the long electron quantum paths. The chirp coefficients,α, and chirp rates, b, associated to the long quantum paths have been evaluated from the measured harmonic bandwidths in the framework of the strong-field approximation. The dependence of such parameters on the peak intensity of the driving field and on the harmonic order has been experimentally investigated. The experimental data turn out to be in excellent agreement with the results of numerical simulations based on the use of the nonadiabatic saddle-point method.

4. Acknowledgements

This work was partially supported by European Community under project MRTN-CT-2003-505138 (XTRA), and by MIUR under project “Processi fisici nel dominio degli attosecondi”. We thank L. Poletto and P. Villoresi for the development of the XUV spectrometer.

References and links

1.	E.A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I.P. Christov, A. Aquila, E.M. Gullikson, D. T. Attwood, M.M. Murnane, and H.C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science 302, 95–98 (2003). [CrossRef] [PubMed]
2.	A. L’Huillier, D. Descamps, A. Johansson, J. Norin, J. Mauritsson, and C.-G. WahlstrÖm, “Applications of high-order harmonics,” Eur. Phys. J. D 26, 91–98 (2003). [CrossRef]
3.	P. Agostini and L.F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855 (2004). [CrossRef]
4.	I.P. Christov, M.M. Murnane, and H. Kapteyn, “High-harmonic generation of attosecond pulses in the “single-cycle” regime,” Phys. Rev. Lett. 78, 1251–1254 (1997). [CrossRef]
5.	J. Itatani, F. Quéré, G.L. Yudin, M.Yu. Ivanov, F. Krausz, and P.B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903-1-4 (2002). [CrossRef] [PubMed]
6.	R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, Th. West-erwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427, 817–821 (2004). [CrossRef]
7.	E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltusška, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science 305, 1267–1269 (2004). [CrossRef] [PubMed]
8.	P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Aug>, Ph. Balcou, H.G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001). [CrossRef] [PubMed]
9.	H.G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, S17–S21 (2002). [CrossRef]
10.	V. Véniard, R. Taïeb, and A. Maquet,“Phase dependence of (N + 1)-color (N > 1) ir-uv photoionization of atoms with higher harmonics,” Phys. Rev. A 54, 721–728 (1996). [CrossRef] [PubMed]
11.	P. Tzallas, D. Charalambidis, N.A. Papadogiannis, K. Witte, and G.D. Tsakiris, “Direct observation of attosecond light bunching,” Nature (London) 426, 267–271 (2003). [CrossRef]
12.	P. Saliéres, B. Carré, L.Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D.B. Milošsević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001). [CrossRef] [PubMed]
13.	M. Lewenstein, Ph. Balcou, M.Y. Ivanov, A. L’Huillier, and P.B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]
14.	M. Lewenstein, P. Saliéres, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52, 4747–4754 (1995). [CrossRef] [PubMed]
15.	P. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using highorder harmonics,”Phys. Rev. Lett. 77,, 1234–1237 (1996). [CrossRef] [PubMed]
16.	M.B. Gaarde and K.J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. 89, 213901 (2002). [CrossRef] [PubMed]
17.	C. Lyngå, M.B. Gaarde, C. Delfin, M. Bellini, T.W. Hänsch, A. L’Huillier, and C.-G. WahlstrÖm, “Temporal coherence of high-order harmonics,” Phys. Rev. A 60, 4823–4830 (1999). [CrossRef]
18.	M.B. Gaarde and K.J. Schafer, “Quantum path distributions for high-order harmonics in rare gas atoms,” Phys. Rev. A 65, 031406(R)-1-4 (2002). [CrossRef]
19.	Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Saliéres, and B. Carré, “High harmonic XUV spectral phase interferometry for direct electric-field reconstruction,” Phys. Rev. Lett. 94, 173903-1-4 (2005). [CrossRef]
20.	G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411-1-10 (2004). [CrossRef]
21.	M. Bellini, C. Lyngå, A. Tozzi, M.B. Gaarde, T.W. Hänsch, A. L’Huillier, and C.-G. WahlstrÖm, “Temporal coherence of ultrashort high-order harmonic pulses,” Phys. Rev. Lett. 81, 297–300 (1998). [CrossRef]
22.	L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, “Instrumentation for analysis and utilization of extreme-ultraviolet and soft x-ray high-order harmonics,” Rev. Sci. Instrum. 75, 4413–4418 (2004). [CrossRef]
23.	G. Sansone, E. Benedetti, J-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S.De Silvestri, and M. Nisoli, “Measurement of Harmonic Phase Differences by Interference of Attosecond Light Pulses,” Phys. Rev. Lett. 94, 193903-1-4 (2005). [CrossRef] [PubMed]
24.	P. Saliéres, A. L’Huillier, and M. Lewenstein, “Coherence Control of High-Order Harmonics,” Phys. Rev. Lett. 74, 3776–3779 (1995). [CrossRef] [PubMed]
25.	M.B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K.J. Schafer, K.C. Kulander, and A. L’Huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A 59, 1367–1373 (1999). [CrossRef]
26.	S.C. Pinault and M.J. Potasek, “Frequency broadening by self-phase modulation in optical fibers,” J. Opt. Soc. Am. B 2, 1318–1319 (1985). [CrossRef]

OCIS Codes
(190.4180) Nonlinear optics : Multiphoton processes
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 3, 2006
Revised Manuscript: March 13, 2006
Manuscript Accepted: March 14, 2006
Published: March 20, 2006

Citation
Enrico Benedetti, Jean-Pascal Caumes, Giuseppe Sansone, Salvatore Stagira, Caterina Vozzi, and Mauro Nisoli, "Frequency chirp of long electron quantum paths in high-order harmonic generation," Opt. Express 14, 2242-2249 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2242

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References

E.A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I.P. Christov, A. Aquila, E.M. Gullikson, D. T. Attwood, M.M. Murnane, and H.C. Kapteyn, "Coherent soft X-ray generation in the water window with quasi-phase matching," Science 302,95-98 (2003). [CrossRef] [PubMed]
A. L’Huillier, D. Descamps, A. Johansson, J. Norin, J. Mauritsson, and C.-G. Wahlström, "Applications of highorder harmonics," Eur. Phys. J. D 26,91-98 (2003). [CrossRef]
P. Agostini and L.F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67,813-855 (2004). [CrossRef]
I.P. Christov, M.M. Murnane and H. Kapteyn, "High-harmonic generation of attosecond pulses in the "singlecycle" regime," Phys. Rev. Lett. 78,1251-1254 (1997). [CrossRef]
J. Itatani, F. Quéré, G.L. Yudin, M.Yu. Ivanov, F. Krausz, and P.B. Corkum, "Attosecond streak camera," Phys. Rev. Lett. 88,17390314 (2002). [CrossRef] [PubMed]
R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature (London) 427,817-821 (2004). [CrossRef]
E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher and F. Krausz, "Direct measurement of light waves," Science 305,1267-1269 (2004). [CrossRef] [PubMed]
P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, and P. Agostini, "Observation of a train of attosecond pulses from high harmonic generation," Science 292,1689-1692 (2001). [CrossRef] [PubMed]
H.G. Muller, "Reconstruction of attosecond harmonic beating by interference of two-photon transitions," Appl. Phys. B 74,S17-S21 (2002). [CrossRef]
V. V´eniard, R. Taieb, and A. Maquet,"Phase dependence of (N+1)-color (N > 1) ir-uv photoionization of atoms with higher harmonics," Phys. Rev. A 54,721-728 (1996). [CrossRef] [PubMed]
P. Tzallas, D. Charalambidis, N.A. Papadogiannis, K. Witte, and G.D. Tsakiris, "Direct observation of attosecond light bunching," Nature (London) 426, 267-271 (2003). [CrossRef]
P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D.B. Milo¡sevi´c, A. Sanpera, and M. Lewenstein, "Feynman’s path-integral approach for intense-laser-atom interactions," Science 292,902-905 (2001). [CrossRef] [PubMed]
M. Lewenstein, Ph. Balcou, M.Y. Ivanov, A. L’Huillier, and P.B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49,2117-2132 (1994). [CrossRef] [PubMed]
M. Lewenstein, P. Salières, and A. L’Huillier, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52,4747-4754 (1995). [CrossRef] [PubMed]
P. Antoine, A. L’Huillier, and M. Lewenstein, "Attosecond pulse trains using highorder harmonics,"Phys. Rev. Lett. 77,1234-1237 (1996). [CrossRef] [PubMed]
M.B. Gaarde and K.J. Schafer, "Space-time considerations in the phase locking of high harmonics," Phys. Rev. Lett. 89,213901 (2002). [CrossRef] [PubMed]
C. Lynga, M.B. Gaarde, C. Delfin, M. Bellini, T.W. H¨ansch, A. L’Huillier, and C.-G. Wahlstr¨om, "Temporal coherence of high-order harmonics," Phys. Rev. A 60,4823-4830 (1999). [CrossRef]
M.B. Gaarde and K.J. Schafer, "Quantum path distributions for high-order harmonics in rare gas atoms," Phys. Rev. A 65,03140614 (2002). [CrossRef]
Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Salières, and B. Carre, "High harmonic XUV spectral phase interferometry for direct electric-field reconstruction," Phys. Rev. Lett. 94,17390314 (2005). [CrossRef]
G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, "Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths," Phys. Rev. A 70,013411110 (2004). [CrossRef]
M. Bellini, C. Lynga, A. Tozzi, M.B. Gaarde, T.W. H¨ansch, A. L’Huillier, and C.-G. Wahlström, "Temporal coherence of ultrashort high-order harmonic pulses," Phys. Rev. Lett. 81,297-300 (1998). [CrossRef]
L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, "Instrumentation for analysis and utilization of extremeultraviolet and soft x-ray high-order harmonics," Rev. Sci. Instrum. 75,4413-4418 (2004). [CrossRef]
G. Sansone, E. Benedetti, J-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, and M. Nisoli, "Measurement of Harmonic Phase Differences by Interference of Attosecond Light Pulses," Phys. Rev. Lett. 94, 193903-1-4 (2005). [CrossRef] [PubMed]
P. Salières, A. L’Huillier and M. Lewenstein, "Coherence Control of High-Order Harmonics," Phys. Rev. Lett. 74,3776-3779 (1995). [CrossRef] [PubMed]
M.B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K.J. Schafer, K.C. Kulander, and A. L’Huillier, "Spatiotemporal separation of high harmonic radiation into two quantum path components," Phys. Rev. A 59,1367-1373 (1999). [CrossRef]
S.C. Pinault and M.J. Potasek, "Frequency broadening by self-phase modulation in optical fibers," J. Opt. Soc. Am. B 2,1318-1319 (1985). [CrossRef]

Agostini, P.
Antoine, P.
Aquila, A.
Attwood, D. T.
Augé, F.
Backus, S.
Balcou, Ph.
Baltuska, A.
Bammer, F.
Becker, W.
Bellini, M.
Bonora, S.
Breger, P.
Carre, B.
Carré, B.
Charalambidis, D.
Christov, I.P.
Constant, E.
Corkum, P.B.
Delfin, C.
Descamps, D.
DiMauro, L.F.
Drescher, M.
Gaarde, M.B.
Gaudiosi, D.
Gibson, E.A.
Gobert, O.
Goulielmakis, E.
Grasbon, F.
Gullikson, E.M.
Heinzmann, U.
Itatani, J.
Ivanov, M.Y.
Ivanov, M.Yu.
Johansson, A.
Kapteyn, H.
Kapteyn, H.C.
Kienberger, R.
Kleineberg, U.
Kopold, R.
Krausz, F.
Kulander, K.C.
L’Huillier, A.
Le Déroff, L.
Lewenstein, M.
Lynga, C.
Mairesse, Y.
Maquet, A.
Mauritsson, J.
Merdji, H.
Meynadier, P.
Monchicourt, P.
Muller, H.G.
Mullot, G.
Murnane, M.M.
Nisoli, M.
Norin, J.
Papadogiannis, N.A.
Pascolini, M.
Paul, A.
Paul, P.M.
Paulus, G. G.
Perdrix, M.
Pinault, S.C.
Poletto, L.
Potasek, M.J.
Quéré, F.
Salières, P.
Salin, F.
Sansone, G.
Schafer, K.J.
Scrinzi, A.
Stagira, S.
Taieb, R.
Tobey, R.
Toma, E.S.
Tozzi, A.
Tsakiris, G.D.
Tzallas, P.
Uiberacker, M.
V´eniard, V.
Villoresi, P.
Vozzi, C.
Wagner, N.
Wahlström, C.-G.
Walther, H.
Westerwalbesloh, Th.
Witte, K.
Yakovlev, V.
Yudin, G.L.

Appl. Phys. B

H.G. Muller, "Reconstruction of attosecond harmonic beating by interference of two-photon transitions," Appl. Phys. B 74,S17-S21 (2002). [CrossRef]

Eur. Phys. J. D

A. L’Huillier, D. Descamps, A. Johansson, J. Norin, J. Mauritsson, and C.-G. Wahlström, "Applications of highorder harmonics," Eur. Phys. J. D 26,91-98 (2003). [CrossRef]

J. Opt. Soc. Am. B

S.C. Pinault and M.J. Potasek, "Frequency broadening by self-phase modulation in optical fibers," J. Opt. Soc. Am. B 2,1318-1319 (1985). [CrossRef]

Nature (London)

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature (London) 427,817-821 (2004). [CrossRef]
P. Tzallas, D. Charalambidis, N.A. Papadogiannis, K. Witte, and G.D. Tsakiris, "Direct observation of attosecond light bunching," Nature (London) 426, 267-271 (2003). [CrossRef]

Phys. Rev. A

C. Lynga, M.B. Gaarde, C. Delfin, M. Bellini, T.W. H¨ansch, A. L’Huillier, and C.-G. Wahlstr¨om, "Temporal coherence of high-order harmonics," Phys. Rev. A 60,4823-4830 (1999). [CrossRef]
M.B. Gaarde and K.J. Schafer, "Quantum path distributions for high-order harmonics in rare gas atoms," Phys. Rev. A 65,03140614 (2002). [CrossRef]
M. Lewenstein, Ph. Balcou, M.Y. Ivanov, A. L’Huillier, and P.B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49,2117-2132 (1994). [CrossRef] [PubMed]
M. Lewenstein, P. Salières, and A. L’Huillier, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52,4747-4754 (1995). [CrossRef] [PubMed]
V. V´eniard, R. Taieb, and A. Maquet,"Phase dependence of (N+1)-color (N > 1) ir-uv photoionization of atoms with higher harmonics," Phys. Rev. A 54,721-728 (1996). [CrossRef] [PubMed]
M.B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K.J. Schafer, K.C. Kulander, and A. L’Huillier, "Spatiotemporal separation of high harmonic radiation into two quantum path components," Phys. Rev. A 59,1367-1373 (1999). [CrossRef]
G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, "Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths," Phys. Rev. A 70,013411110 (2004). [CrossRef]

Phys. Rev. Lett.

M. Bellini, C. Lynga, A. Tozzi, M.B. Gaarde, T.W. H¨ansch, A. L’Huillier, and C.-G. Wahlström, "Temporal coherence of ultrashort high-order harmonic pulses," Phys. Rev. Lett. 81,297-300 (1998). [CrossRef]
P. Salières, A. L’Huillier and M. Lewenstein, "Coherence Control of High-Order Harmonics," Phys. Rev. Lett. 74,3776-3779 (1995). [CrossRef] [PubMed]
I.P. Christov, M.M. Murnane and H. Kapteyn, "High-harmonic generation of attosecond pulses in the "singlecycle" regime," Phys. Rev. Lett. 78,1251-1254 (1997). [CrossRef]
J. Itatani, F. Quéré, G.L. Yudin, M.Yu. Ivanov, F. Krausz, and P.B. Corkum, "Attosecond streak camera," Phys. Rev. Lett. 88,17390314 (2002). [CrossRef] [PubMed]
P. Antoine, A. L’Huillier, and M. Lewenstein, "Attosecond pulse trains using highorder harmonics,"Phys. Rev. Lett. 77,1234-1237 (1996). [CrossRef] [PubMed]
M.B. Gaarde and K.J. Schafer, "Space-time considerations in the phase locking of high harmonics," Phys. Rev. Lett. 89,213901 (2002). [CrossRef] [PubMed]
Y. Mairesse, O. Gobert, P. Breger, H. Merdji, P. Meynadier, P. Monchicourt, M. Perdrix, P. Salières, and B. Carre, "High harmonic XUV spectral phase interferometry for direct electric-field reconstruction," Phys. Rev. Lett. 94,17390314 (2005). [CrossRef]

Rep. Prog. Phys.

P. Agostini and L.F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67,813-855 (2004). [CrossRef]

Rev. Sci. Instrum.

L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, "Instrumentation for analysis and utilization of extremeultraviolet and soft x-ray high-order harmonics," Rev. Sci. Instrum. 75,4413-4418 (2004). [CrossRef]

Science

E.A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I.P. Christov, A. Aquila, E.M. Gullikson, D. T. Attwood, M.M. Murnane, and H.C. Kapteyn, "Coherent soft X-ray generation in the water window with quasi-phase matching," Science 302,95-98 (2003). [CrossRef] [PubMed]
E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher and F. Krausz, "Direct measurement of light waves," Science 305,1267-1269 (2004). [CrossRef] [PubMed]
P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, and P. Agostini, "Observation of a train of attosecond pulses from high harmonic generation," Science 292,1689-1692 (2001). [CrossRef] [PubMed]
P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D.B. Milo¡sevi´c, A. Sanpera, and M. Lewenstein, "Feynman’s path-integral approach for intense-laser-atom interactions," Science 292,902-905 (2001). [CrossRef] [PubMed]

Other

G. Sansone, E. Benedetti, J-P. Caumes, S. Stagira, C. Vozzi, M. Pascolini, L. Poletto, P. Villoresi, S. De Silvestri, and M. Nisoli, "Measurement of Harmonic Phase Differences by Interference of Attosecond Light Pulses," Phys. Rev. Lett. 94, 193903-1-4 (2005). [CrossRef] [PubMed]

2005, Mairesse, Phys. Rev. Lett.
2004, Sansone, Phys. Rev. A
2004, Poletto, Rev. Sci. Instrum.
2004, Agostini, Rep. Prog. Phys.
2004, Kienberger, Nature (London)
2004, Goulielmakis, Science
2003, Gibson, Science
2003, L’Huillier, Eur. Phys. J. D
2003, Tzallas, Nature (London)
2002, Itatani, Phys. Rev. Lett.
2002, Gaarde, Phys. Rev. Lett.
2002, Gaarde, Phys. Rev. A
2002, Muller, Appl. Phys. B
2001, Paul, Science
2001, Salières, Science
1999, Lynga, Phys. Rev. A
1999, Gaarde, Phys. Rev. A
1998, Bellini, Phys. Rev. Lett.
1997, Christov, Phys. Rev. Lett.
1996, V´eniard, Phys. Rev. A
1996, Antoine, Phys. Rev. Lett.
1995, Lewenstein, Phys. Rev. A
1995, Salières, Phys. Rev. Lett.
1994, Lewenstein, Phys. Rev. A
1985, Pinault, J. Opt. Soc. Am. B

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