## 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

Over the last decade, much attention has been paid to the development of techniques for the generation of shaped femtosecond optical pulses [1]. The most common technique to shape ultrashort optical pulses into target waveforms is based on intensity and phase filtering of the spectrally dispersed original pulse in the Fourier plane. In principle, almost any arbitrary pulse shapes can be produced with such spectral phase and amplitude modulation setup. However, phase-only modulations can already provide a large variety of useful output waveforms. For instance, periodic spectral phase modulation leads to the generation of pulse trains, highly involved in coherent quantum control of atomic [2,3] and molecular [4–6] systems. Spectral phase step modulation leads to pulse doublets also used in coherent control [7] as well as in dark soliton generation in optical fibers [8]. Considering the implication of pulse trains in the context of coherent control, one must distinguish between two categories of processes in which pulse trains can be involved. In the first one, only the temporal envelope of the field matters, whereas the temporal phase of field plays no role upon the process to control. It is for instance the case for systems to be controlled through nonresonant impulsive stimulated Raman processes, for which the coupling intensity depends on the field parameters only through the temporal shape [6]. The unimportance of the temporal phase is also generally verified in ionization and harmonic generation processes occurring in a nonresonant multiphoton regime or in a strong field regime (field ionization) with multi-cycle pulses. The second category encompasses excitation processes that not only depend on the field envelope but also on the temporal phase, meaning that the quantum system response is sensitive the temporal change of the laser phase during the pulse. In general, it applies to systems resonantly driven through single- or multi-photon absorption [2,9]. In the present work, we are interested in producing phase-locked pulse doublets and pulse trains of adjustable relative amplitudes that could be employed in both categories, only by using pure phase filtering with a standard liquid-crystal spatial light modulator (LC-SLM).

The task of shaping ultrashort pulses is strongly related to the capability of characterizing such tailored pulses. The field of ultrashort pulse shape characterization has undergone remarkable progress during last years. Various characterization techniques have been used to control the fidelity of shaped pulses. Among them, intensity cross correlation was the most commonly used [10–12]. It provides a good approximation about the tailored intensity profile, but no phase information is available. Complete characterization (envelope and phase) of shaped pulses has been performed with the well-known method of frequency-resolved optical gating (FROG) [13] or the more recent SPIDER technique [14]. We use the last one [15] for characterization of the tailored ultra-short optical pulses produced in this work. The generation of specific pulse-doublets using suitable spectral phase step or triangular spectral phase is investigated. Control of time separation and intensity ratio between the two generated pulses is explored. High fidelity pulse trains using sinusoidal spectral phase are as well tailored and characterized.

## 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 (10^{12} 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].

In order to achieve gray-level phase control, a calibration of the modulator phase response as a function of applied voltage is needed. We have obtained this calibration by using the mask as an amplitude modulator. A He-Ne laser was linearly polarized, with its polarization rotated 45° relative to the plane containing the alignment direction of the liquid crystal, and focused onto the calibration pixel of the modulator. The phase calibration curve was obtained from the transmission T versus voltage V by measuring the transmission T(V) through an analyzer and using the relation T(V)=sin^{2}[ϕ(V)/2] [11]. An analytical function of the voltage versus phase, V=G(ϕ), was deduced from a polynomial fit of the experimental curve. In order to produce any desired spectral phase ϕ=φ(ω), the knowledge of G (voltage versus phase) and γ (angular frequency versus pixel) was used to program the voltage pattern [V_{1},V_{2},…,V_{128}] via the following function, V_{i}=G [φ(γ(i-i_{0})+ω_{0})], where V_{i} is the voltage applied to the pixel i, and i_{0} the pixel related to the carrier frequency.

For the complete characterization of shaped pulses, a home-made SPIDER apparatus was employed. The SPIDER technique [15] 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], 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)|=E_{0}exp[-(ω-ω_{0})^{2}/σ^{2})], with ω_{0} the carrier frequency and σ the half width at 1/e.

We first study a spectral phase of the form

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

$$=A\left(t\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\phantom{\rule{.2em}{0ex}}\left(-i{\omega}_{0}t\right),$$

with the antisymmetric erfi function defined by

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.

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).

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

where Δτ is the spectral phase slope and (ω_{0}+δω) is the spectral phase breakpoint frequency.

The associated temporal complex electric field is given by

$$+\mathrm{exp}\text{}\left[-{\left(\frac{\sigma \left(t-\Delta \tau \right)}{2}\right)}^{2}\right]\phantom{\rule{.3em}{0ex}}\left[1-\mathit{erf}\left(i\frac{\sigma \left(t-\Delta \tau \right)}{2}-\frac{\delta \omega}{\sigma}\right)\right]\},$$

with the erf function defined by

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 ω_{0} (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

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

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 ω_{0} (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.

Pulse doublets with different maxima intensity ratio, programmed with a slope of 193 fs, are illustrated in Fig. 4(b). The values of relative phase breakpoint frequency (δω/Δω) are respectively -0.3, -0.2, and -0.1 in Fig. 4(b) (i–iii), with Δω=(2ln2)^{1/2}σ. The ability to produce a large range of maxima ratio, [0.01 to 1], is illustrated in Fig. 4(c). A good agreement between theoretical and experimental results reveals the ability of the pulse shaper to produce pulse doublets of adjustable ratio and delay. It is pointed out that the use of a triangular spectral phase allows to accurately determine the pixel i_{0}, related to the central frequency. Indeed, a significant feedback is given by the SPIDER when δω=0 : two pulses of same intensity are measured.

#### 3.3 Sinusoidal phase modulation

By applying a sinusoidal spectral phase modulation,

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

where J_{n} are Bessel functions of the first kind.

The maxima intensity and the temporal phase of the n^{th} pulse, i.e., centred in (t-nΔτ), are respectively ${{\mathrm{J}}_{\mathrm{n}}}^{2}$(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.

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<β_{1}~1.43 rad., the largest pulse is the central one (n=0); for β_{1}<β<β_{2}~2.63 rad., largest ones are the first pulses (n=1, -1), with β_{1} and β_{2} defined respectively by J_{0}(β_{1})=J_{1}(β_{1}) and J_{1}(β_{2})=J_{2}(β_{2}). 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.

## 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]). 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] 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]. 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]