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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 3 — Feb. 9, 2004
  • pp: 473–482
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Pulse trains produced by phase-modulation of ultrashort optical pulses: tailoring and characterization

M. Renard, R. Chaux, B. Lavorel, and O. Faucher  »View Author Affiliations


Optics Express, Vol. 12, Issue 3, pp. 473-482 (2004)
http://dx.doi.org/10.1364/OPEX.12.000473


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Abstract

In this paper, creation of pulse doublets and pulse trains by spectral phase modulation of ultrashort optical pulses is investigated. Pulse doublets with specific features are generated through step-like and triangular spectral phase modulation, whereas sequences of pulses with controllable delay and amplitude are produced via sinusoidal phase modulations. A temporal analysis of this type of tailored pulses is exposed and a complete characterization with the SPIDER technique (Spectral Phase Interferometry for Direct Electric-field Reconstruction) is presented.

© 2004 Optical Society of America

1. Introduction

2. Experimental setup

Our pulse shaping apparatus relies on a usual zero-dispersion line. A pair of 1200 lines/mm gratings and 200-mm focal length cylindrical mirrors is used in a 4f-arrangement. A programmable one-dimensional LC-SLM array (SLM-128, CRI, Inc.) is inserted in the Fourier plane, midway between the cylindrical mirrors. The LC-SLM allows for independent control of the phase for each of its 128 pixels. The pixel spacing is 100 µm center to center with 3 µm gaps between pixel, and the active area is 2 mm high. In order to attribute one pixel to a spectral wavelength, we have measured the frequency spatial distribution on the Fourier plane with the following method. A knife was translated across the Fourier plane, perpendicularly to the beam axis. For each position of the knife, the transmitted spectrum was measured and a cutting wavelength was deduced from each spectrum. By assuming a linear distribution, the frequency spatial distribution is deduced by a linear fit of the cutting wavelength λ versus the knife position x. We obtain a value of δλ/δx=2.85 nm/mm, leading to a spectral sampling of γ=0.842 (1012 rad.s-1/pixel). The LC-SLM of 12.8 mm total aperture covers in the Fourier plane a spectral bandwidth of 36.5 nm, which is three times larger than spectrum of the laser used in the present work (FWHM~11 nm). By analyzing a truncated spectrum around the cutting wavelength, we have deduced the transverse radius of the focused beam for a single frequency component. The spot waist (130 µm) is nearly equal to the pixel width (100 µm) allowing a good spectral resolution with smooth pixelation effects [1

1. A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]

].

For the complete characterization of shaped pulses, a home-made SPIDER apparatus was employed. The SPIDER technique [15

15. C. Iaconis and I.A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998). [CrossRef]

] uses spectral shearing interferometry to retrieve the spectral phase of tailored pulses. This type of interferometry measures the interference between two pulses separated in time, which are identical except for their respective central frequency (frequency sheared pulses). In our setup [16

16. S. Xu, B. Lavorel, O. Faucher, and R. Chaux, “Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,” J. Opt. Soc. Am. B 19, 165–168 (2002). [CrossRef]

], the spectrally sheared pulse pair was generated by up-converting a pair of time delayed identical pulses with a highly chirped pulse in a nonlinear crystal (type I BBO). The interference fringes were analyzed by a commercial spectrometer. The recorded signals were acquired with a personal computer. A fast data inversion algorithm permitted to display the measured quantities (spectral intensity and phase) in real time. The temporal intensity and phase were determined by an inverse Fourier transformation. In the experiments performed in the present work, a 100 fs chirped pulse amplified Ti:sapphire laser operating at 800 nm at a repetition rate of 20 Hz was used. The pulse energy measured after the pulse shaping apparatus was about 100 µJ/pulse. In our setup, the maximum energy limited by the SLM damage threshold was estimated around 1.5 mJ/pulse.

3. Pulse trains generation

3.1 Step phase modulation

In what follows, the complex spectral electric field is written E(ω)=|E(ω)|exp[iφ(ω)], where |E(ω)| and φ(ω) are respectively the spectral amplitude and phase. The spectral amplitude, defined by our laser system, is fixed and is assumed to be gaussian |E(ω)|=E0exp[-(ω-ω0)22)], with ω0 the carrier frequency and σ the half width at 1/e.

We first study a spectral phase of the form

φ(ω)=αH(ωω0),
(1)

were H is the step function and α is the adjustable amplitude factor. The complex spectral electric field is converted to the time domain by an inverse Fourier transformation. After integration, we obtain the expression of the complex temporal electric field

E(t)=(E0σ2π)cos(α2)exp[(σt2)2][1+tg(α2)erfi(σt2)]exp(iω0t)
=A(t)exp(iω0t),
(2)

with the antisymmetric erfi function defined by

erfi(x)=2π120xexp(y2)dy,
(3)

and A(t) the time dependent envelope. For α=0 equation (2) leads to the well-known expression of the electric-field for a Fourier transform-limited gaussian pulse. For α≠0, A(t) is expressed as a sum of two real terms, a gaussian function and an antisymmetric function weighted by tg(α/2), resulting in a double hump of opposite signs. As a result, Eq. (2) represents a double pulse field with a constant π-step temporal phase centred in between the two pulses where the envelope is zero. An example of SPIDER-characterization of a phase-locked double-pulse, created by our pulse shaper apparatus, is shown in Fig. 1.

Fig. 1. Electric-field characteristics of a pulse shaped through step spectral phase modulation of amplitude α=-π/2 rad. : (a) spectral intensity, (b) spectral phase, (c) temporal intensity and (d) temporal phase. Measurements are represented with black lines and calculations with red lines. See text for details.

All the data presented in this paper have been averaged over 20 laser shots. An appropriate voltage was applied to the 128 pixels, in order to produce a spectral phase step with α=-π/2. The characteristics of the electric-field are presented respectively in spectral (Figs. 1(a) and (b)) and temporal (Figs. 1(c) and (d)) domain. The red lines in Figs.1(a), (b), (c) and (d) represent respectively the gaussian fit of the measured spectrum, the target spectral phase, the time dependent intensity and the time dependent phase. The two last ones are obtained by a Fourier transformation of the spectral (red lines) characteristics. The black lines represent the SPIDER measurements. By adjusting the magnitude of the spectral phase jump α, i.e. the weight of the antisymmetric function appearing in Eq. (3), different intensity ratios between the two pulses have been achieved. For -π<α< 0 the intensity of the pulse arising at short times increases with respect to the one arising at long times, whereas it decreases for 0<α<π. Pulse doublets generated by the pulse shaper apparatus, applying phase step α of -π/2, -2π/3, and -3π/4, are presented respectively in Fig. 2(a)(i–iii).

Fig. 2. Creation of pulse doublets with adjustable maxima intensity ratio through spectral phase step modulation. (a) Examples of temporal intensity of pulse doublets generated through step modulation of amplitude α and characterised by a SPIDER apparatus : (i) α=-π/2 rad., (ii) α=-2π/3 rad. and (iii) α=-3π/4 rad.. Measurements (black lines) and calculated results (red lines). (b) Evolution of the maxima intensity ratio versus modulation magnitude α. Measurements (blue circles) and calculated results (solid line). See text for details.

The measurements (black lines) are in good agreement with the numerical simulations (red lines). The ability to precisely produce pulse doublets, with a large range of maxima ratio is shown in Fig. 2(b). A ratio as low as ~1% was measured for α=-π/4. The time delay between the two pulses is mainly due to σ, the spectral bandwidth of the input spectrum. The variation of this time delay with respect to α is rather small. Given in σ-1 unit, it varies from 3.7, for α=-π, to 4.3, for α=-π/4. Hence, the use of spectral phase step filter does not allow an extensive control of the time delay between the two pulses.

3.2 Triangular phase modulation

Independent control of the pulse ratio and the time delay can be achieved by the use of a triangular spectral phase modulation defined by

φ(ω)=sgn[ω(ω0+δω)]Δτ[ω(ω0+δω)],
(4)

where Δτ is the spectral phase slope and (ω0+δω) is the spectral phase breakpoint frequency.

The associated temporal complex electric field is given by

E(t)=(E0σ4π)exp(iω0t){exp[(σ(t+Δτ)2)2][1erf(iσ(t+Δτ)2+δωσ)]
+exp[(σ(tΔτ)2)2][1erf(iσ(tΔτ)2δωσ)]},
(5)

with the erf function defined by

erf(x)=2π120xexp(y2)dy.
(6)

A clear understanding of the temporal electric field structure can be achieved if one focused on the particular case of a triangular spectral phase with a phase breakpoint centred in ω0 (i.e. δω=0). The total temporal complex electric field is decomposed in a sum of two temporal complex electric field, 1 2 E(t)=E(t)+E(t), with

E1,2(t)=(E0σ4π)exp(iω0t)exp[(σ(t±Δτ)2)2][1ierfi(σ(t±Δτ)2)].
(7)

Temporal amplitudes A 1,2(t) and temporal phases φ 1,2(t) of E 1,2(t) deduced from eq.(7) are respectively

A1,2(t)=(E0σ4π)exp[(σ(t±Δτ)2)2][1+{erfi(σ(t±Δτ)2)}2]12,
(8)
φ1,2(t)=tg1[erfi(σ(t±Δτ)2)].
(9)

From expression (8), we notice that both envelopes have exactly the same temporal shape, a gaussian form slightly distorted by the erfi function, and are respectively centered at t=-Δτ and t=Δτ. For Δτ larger than the initial pulse duration (Fourier transform-limited pulse), the time delay between the two pulses is equal to 2Δτ. Each pulse is globally frequency shifted relatively to the central frequency ω0 (see Eq.9). Indeed, around t=-Δτ, the temporal phase decreased with time from π/2 to -π/2, leading to a red shift. Symmetrically, the temporal phase around t=Δτ increased from -π/2 to π/2, leading to a blue shift. A complete SPIDER-characterization of a phase-locked double-pulse, created by a triangular spectral phase, is depicted in Fig. 3. A spectral phase slope of Δτ=193 fs, and a phase breakpoint, centred in ω0, i.e. δω=0, was programmed. The characteristics of the electric field are presented in spectral (Figs. 3(a) and (b)) and temporal (Figs. 3(c) and (d)) domain. The numerical simulations are represented with red lines as in Fig. 1. The agreement between theory and experiment is excellent. The time-delay between the two pulses could be precisely adjusted by changing the spectral phase slope Δτ. In Fig. 4(a), three pulse-doublets with different time delay are depicted. Fig. 4(a)(i–iii) represents the temporal intensity obtained by applying a centred (δω=0) triangular spectral phase with slope of respectively 193 fs, 258 fs, and 396 fs. In agreement with Eq. (8), the time delay between the two pulses is 2 Δτ. The control of the ratio between the intensity maxima of the pulse doublet is achieved by shifting the triangular spectral phase, i.e. δω≠0. For δω<0 the intensity of the pulse arising at short times increases with respect to the one arising at long times, whereas it decreases for δω>0.

Fig. 3. Same as fig.1 with a centered triangular spectral phase modulation (δω=0) of spectral phase slope Δτ=193 fs.
Fig. 4. Tailoring of pulse doublets through triangular spectral phase modulation. (a) Temporal intensity of pulse doublets generated through triangular spectral phase modulation with spectral phase slope Δτ : (i) Δτ=193 fs, (ii) Δτ=258 fs and (iii) Δτ=396 fs. (b) Temporal intensity of pulse doublets generated through triangular spectral phase modulation with relative phase breakpoint position δω/Δω. The spectral phase slope Δτ is fixed to 193 fs. (i) δω/Δω=-0.3, (ii) δω/Δω=-0.2, (iii) δω/Δω=-0.1. (c) Evolution of maxima intensity ratio versus relative phase breakpoint position δω/Δω (Δτ=193fs). (a–b) Experimental (black lines) and calculated results (red lines). (c) Experimental (blue circles) and calculated results (black lines).

3.3 Sinusoidal phase modulation

By applying a sinusoidal spectral phase modulation,

φ(ω)=βcos[Δτ(ωω0)],
(10)

where β is the modulation amplitude and Δτ the spectral modulation frequency, we obtain in the temporal domain a pulse train with a temporal separation Δτ between subsequent pulse maxima

E(t)n=+Jn(β)exp[i(nπ2)]exp[(tnΔτ)2σ24],
(11)

where Jn are Bessel functions of the first kind.

The maxima intensity and the temporal phase of the nth pulse, i.e., centred in (t-nΔτ), are respectively Jn2(β) and |n|π/2.

An example of SPIDER-characterization of a shaped pulse, with β=0.5. rad. and Δτ=330 fs is depicted in Fig. 5. In agreement with Eq. (11), the temporal phase exhibits jump of π/2 magnitude between subsequent pulses.

Fig. 5. Same as in fig.1 with a sinusoidal spectral phase modulation. The modulation amplitude β and the spectral modulation frequency Δτ are respectively fixed to 0.5 rad. and 330 fs.

The theoretical evolution of maxima intensity versus modulation amplitude is illustrated in Fig. 6. The intensity of each pulse has been normalized to the maxima intensity of the highest pulse(s). For 0<β<β1~1.43 rad., the largest pulse is the central one (n=0); for β1<β<β2~2.63 rad., largest ones are the first pulses (n=1, -1), with β1 and β2 defined respectively by J01)=J11) and J12)=J22). Figure 6(a) reveals the impossibility to produce trains with more than three pulses of equal intensity with sinusoidal phase pattern.

Experimental pulse train resulting from different values of β, with a spectral modulation frequency Δτ fixed to 211 fs, are displayed in Fig. 6(b). The intensity profile represented in Fig. 6(b)(i–v), corresponds respectively to modulation amplitude of 0.5, 1, 1.2, 1.5, and 2 radians. The experimental results (black lines) are in very good agreement with the theoretical curve (red lines). The pulse positions are independent of the value of modulation amplitude β. The intensity maxima ratio of each pulse in the train depicted in Fig. 6(b)(v) can be deduced from the intersection of the vertical line with the different curves in Fig. 6(a).

Examples of pulse trains shaped with similar modulation amplitude of 0.5 radian and spectral modulation frequency of 211 fs, 310 fs, 405 fs are shown respectively in Fig. 6(c)(i–iii). In agreement with Eq. (2), experimental intensity profiles exhibit a constant intensity ratio between pulses in the three pulse train.

Fig. 6. Pulse trains generation through sinusoidal spectral phase modulation. (a) Calculated evolution of maxima intensity versus modulation amplitude β. (b) Temporal intensity of pulse train generated through sinusoidal spectral phase modulation for different values of modulation amplitude β. The spectral modulation frequency Δτ is 211 fs. (i) β=0.5 rad., (ii) β=1 rad., (iii) β=1.2 rad., (iv) β=1.5 rad. and (v) β=2 rad.. (c) Temporal intensity of pulse train generated through sinusoidal spectral phase modulation for different values of spectral modulation frequency Δτ. The modulation amplitude β is fixed to 0.5 rad.. (i) Δτ=211 fs, (ii) Δτ=310 fs and (iii) Δτ=405 fs. See text for details.

4. Conclusion

In conclusion, we have investigated the generation of phase-locked pulse doublets and pulse trains by ultrashort optical pulse phase modulation. Doublets of adjustable maxima intensity ratio were generated through step phase modulation. The use of triangular spectral phase modulation to control the time delay and the maxima intensity ratio of pulse doublets was experimentally demonstrated. Pulse trains with variable repetition rates were created through sinusoidal phase modulation. All tailored pulses have been characterized with a SPIDER apparatus and a complete time domain analysis is supplied for the different spectral phase modulations investigated. As mentioned in the introduction, pulse doublets and pulse trains of adjustable relative amplitudes are of particular interest in the field of coherent control (see for instance [17

17. S. Vajda, A. Bartelt, E. Kaposta, T. Leisner, C. Lupulescu, S. Minemoto, P. Rosendo-Francisco, and L. Wöste “Feedback optimization of shaped femtosecond laser pulses for controlling the wavepacket dynamics and reactivity of mixed alkaline clusters,” Chem. Phys. 267, 231–239 (2001). [CrossRef]

]). The advantage of synthesizing pulses with a spectral phase modulator, instead of using standard interferometers, is twofold. The first one is practical. A spatial light modulator is a simple optical device that avoids the user constraining optical alignments, as encountered when using two- or multi-beam interferometers delivering pulses of adjustable relative intensities. The second one is the already mentioned possibility of producing train of phase-locked pulses. Although pulses exhibiting phase jumps are achievable with standard optics, in practical it requires interferometers stabilized to much better than half of the optical wavelength. However, a triangular spectral phase like the one presented in Fig.3 can only be produced by use of a modulator. It provides two separated pulses having a spectrum of about same bandwidth, but frequency shifted with respect to each other. Such pulse sequence is relevant for manipulation of three-level systems, for instance. More generally, it offers the opportunity of achieving “two-color” pump-probe experiments with a single laser beam. Finally, it should be mentioned that deformable mirrors [18

18. E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murname, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493–495 (1999). [CrossRef]

] offer an alternative to liquid-crystal spatial light modulators for producing only smoothly-varying phase modulations. They share the practicality and generality of the latter, with convenience of supporting larger energy over a broader bandwidth [19

19. It is noticed that the energy transmittable through the LC-SLM has been recently improved with the last commercialized generation, see for instance: G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72, 627–630 (2001). [CrossRef]

]. At the moment, the limitation with deformable mirrors has to do with their relative low resolution, incompatible with some applications.

Acknowledgments

The authors would like to thank Edouard Hertz for useful discussions. This work was supported in part by the Conseil Régional de Bourgogne and by the ACI “photonique” from the French Minestery of Research. Financial support from the CNRS is also acknowledged.

References

1.

A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]

2.

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239 (1998). [CrossRef]

3.

T. Hornung, R. Meier, D. Zeidler, L.-L. Kompa, D Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000). [CrossRef]

4.

T. Hornung, R. Meier, and M. Motzkus, “Optimal control of molecular states in a learning loop with a parametrization in frequency and time domain,” Chem. Phys. Lett. 326, 445–453 (2000). [CrossRef]

5.

T. Hornung, R. Meier, and M. Motzkus, “Coherent control of the molecular four-wave mixing response by phase and amplitude shaped pulses,” Chem. Phys. 267, 261–276 (2001). [CrossRef]

6.

M. Renard, E. Hertz, B. Lavorel, and O. Faucher, “Controlling ground-state rotational dynamics of molecule by shaped femtosecond laser pulses,” Phys. Rev. A (to be published).

7.

D. Meshulach and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,” Phys. Rev A 60, 1287–1292 (1999). [CrossRef]

8.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Kirschner, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988). [CrossRef] [PubMed]

9.

Y.V. Yakovlev, C. J. Bardeen, J. Che, J. Cao, and K. R. Wilson, “Chirped pulse enhancement of multiphoton absorption in molecular iodine,” J. Chem. Phys. 108(6), 2309–2313 (1998). [CrossRef]

10.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563 (1988). [CrossRef]

11.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326 (1990). [CrossRef] [PubMed]

12.

D. H. Reitze, A.M. Weiner, and D.E Leiard, “Shaping of wide bandwidth 20 femtosecond optical pulses,” Appl. Phys. Lett. 61 (11), 1260, (1992). [CrossRef]

13.

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. R. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483 (1995). [CrossRef] [PubMed]

14.

P. Baum, S. Lochbrunner, L. Gallmann, G. Steinmeyer, U. Keller, and E. Riedle, “Real-time characterization and optimal phase control of tunable visible pulses with a flexible compressor,” Appl. Phys. B 74, 219–224 (2002). [CrossRef]

15.

C. Iaconis and I.A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998). [CrossRef]

16.

S. Xu, B. Lavorel, O. Faucher, and R. Chaux, “Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,” J. Opt. Soc. Am. B 19, 165–168 (2002). [CrossRef]

17.

S. Vajda, A. Bartelt, E. Kaposta, T. Leisner, C. Lupulescu, S. Minemoto, P. Rosendo-Francisco, and L. Wöste “Feedback optimization of shaped femtosecond laser pulses for controlling the wavepacket dynamics and reactivity of mixed alkaline clusters,” Chem. Phys. 267, 231–239 (2001). [CrossRef]

18.

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murname, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493–495 (1999). [CrossRef]

19.

It is noticed that the energy transmittable through the LC-SLM has been recently improved with the last commercialized generation, see for instance: G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72, 627–630 (2001). [CrossRef]

OCIS Codes
(320.0320) Ultrafast optics : Ultrafast optics
(320.5540) Ultrafast optics : Pulse shaping
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Research Papers

History
Original Manuscript: December 10, 2003
Revised Manuscript: January 29, 2004
Published: February 9, 2004

Citation
Mathias Renard, R. Chaux, B. Lavorel, and O. Faucher, "Pulse trains produced by phase-modulation of ultrashort optical pulses: tailoring and characterization," Opt. Express 12, 473-482 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-473


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References

  1. A.M. Weiner, �??Femtosecond pulse shaping using spatial light modulators,�?? Rev. Sci. Instrum. 71, 1929-1960 (2000). [CrossRef]
  2. D. Meshulach, Y. Silberberg, �??Coherent quantum control of two-photon transitions by a femtosecond laser pulse,�?? Nature 396, 239 (1998). [CrossRef]
  3. T. Hornung, R. Meier, D. Zeidler, L.-L. Kompa, D Proch, M. Motzkus, �??Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,�?? Appl. Phys. B 71, 277-284 (2000). [CrossRef]
  4. T. Hornung, R. Meier, M. Motzkus, �??Optimal control of molecular states in a learning loop with a parametrization in frequency and time domain,�?? Chem. Phys. Lett. 326, 445-453 (2000). [CrossRef]
  5. T. Hornung, R. Meier, M. Motzkus, �??Coherent control of the molecular four-wave mixing response by phase and amplitude shaped pulses,�?? Chem. Phys. 267, 261-276 (2001). [CrossRef]
  6. M. Renard, E. Hertz, B. Lavorel, and O. Faucher, �??Controlling ground-state rotational dynamics of molecule by shaped femtosecond laser pulses,�?? Phys. Rev. A (to be published).
  7. D. Meshulach, Y. Silberberg, �??Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses,�?? Phys. Rev A 60, 1287-1292 (1999). [CrossRef]
  8. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Kirschner, �??Experimental observation of the fundamental dark soliton in optical fibers,�?? Phys. Rev. Lett. 61, 2445-2448 (1988). [CrossRef] [PubMed]
  9. Y.V. Yakovlev, C. J. Bardeen, J. Che, J. Cao and K. R. Wilson, �??Chirped pulse enhancement of multiphoton absorption in molecular iodine,�?? J. Chem. Phys. 108(6), 2309-2313 (1998). [CrossRef]
  10. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, �?? High-resolution femtosecond pulse shaping, �?? J. Opt. Soc. Am. B 5, 1563 (1988). [CrossRef]
  11. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, �?? Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator, �?? Opt. Lett. 15, 326 (1990). [CrossRef] [PubMed]
  12. D. H. Reitze, A.M. Weiner, and D.E Leiard, �?? Shaping of wide bandwidth 20 femtosecond optical pulses,�?? Appl. Phys. Lett. 61 (11), 1260, (1992). [CrossRef]
  13. B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. R. DeLong, and R. Trebino, �??Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,�?? Opt. Lett. 20, 483 (1995). [CrossRef] [PubMed]
  14. P. Baum, S. Lochbrunner, L. Gallmann, G. Steinmeyer, U. Keller, and E. Riedle, �??Real-time characterization and optimal phase control of tunable visible pulses with a flexible compressor,�?? Appl. Phys. B 74, 219-224 (2002). [CrossRef]
  15. C. Iaconis, and I.A. Walmsley, �??Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,�?? Opt. Lett. 23, 792-794 (1998). [CrossRef]
  16. S. Xu, B. Lavorel, O. Faucher, and R. Chaux, �??Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,�?? J. Opt. Soc. Am. B 19, 165-168 (2002). [CrossRef]
  17. S. Vajda, A. Bartelt, E. Kaposta, T. Leisner, C. Lupulescu, S. Minemoto, P. Rosendo-Francisco and L. Wöste �??Feedback optimization of shaped femtosecond laser pulses for controlling the wavepacket dynamics and reactivity of mixed alkaline clusters,�?? Chem. Phys. 267, 231-239 (2001). [CrossRef]
  18. E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murname, G. Mourou, H. Kapteyn and G. Vdovin, �??Pulse compression by use of deformable mirrors,�?? Opt. Lett. 24, 493-495 (1999). [CrossRef]
  19. It is noticed that the energy transmittable through the LC-SLM has been recently improved with the last commercialized generation, see for instance: G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus and F. Reichel, �??A new high-resolution femtosecond pulse shaper,�?? Appl. Phys. B 72, 627-630 (2001). [CrossRef]

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