September 15, 2007 / Vol. 32, No. 18 / OPTICS LETTERS 2753

Conditional femtosecond pulse collapse forwhite-light and plasma delivery to a

controlled distance

M. Kolesik,* D. E. Roskey, and J. V. MoloneyCollege of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson,

Arizona 85721-0094, USA*Corresponding author: kolesik@acms.arizona.edu

Received June 28, 2007; revised August 6, 2007; accepted August 21, 2007;posted August 21, 2007 (Doc. ID 84617); published September 14, 2007

Collisions between ultrashort pulses with different wavelengths are studied numerically. The relative delay,wavelength difference, focusing geometry, and chirp are used to accurately control the distance at whichpulses undergo conditional collapse and generate plasma and white light. A wide supercontinuum spectrumis achievable even with pulses that by themselves do not have sufficient power for filament formation.© 2007 Optical Society of America

OCIS codes: 320.2250, 320.5550, 190.5940, 190.7110, 190.4380, 190.2620.

A great deal of effort has been going into control offilamentation and white-light generation in high-power femtosecond pulses in gases, condensed media,and fibers alike. For remote sensing applications(see, e.g., [1,2]), the control of the filamentation-onsetdistance is of paramount importance. Currently,there are several methods available.

One approach is based on the notion of temporal fo-cusing. A suitable prechirp is imposed on the pulse tostretch it such that the ultrashort, high-peak-powerpulse is only temporally refocused at the desired dis-tance. This was demonstrated, e.g., in [3–5]. An adap-tive chirp technique can be also used to optimize theproperties of the resulting continuum [6].

Another method uses spatial focusing. While asingle-lens technique can be sufficient for relativelyshort propagation distances and lower pulse powers,more sophisticated telescope designs are used forlong distances and high powers [7,8]. Accurate con-trol can be achieved by using a deformable mirror [9]to adjust the beam divergence angle.

Filamentation also can be affected by using mul-tiple pulses. Twin-pulse (same wavelength) interac-tions have been used in gases to extend the filamentlength by concatenation [10–12]. Two-pulse interac-tions in white-light generation have been studied forsome time in fibers. It was shown that dual-wavelength pumping can produce stronger and widersupercontinuum spectra [13–15]. Faccio et al. inves-tigated pump–seed interactions in water and ethanolin the presence of resonant interactions [16] andshowed that strong temporal and spatial reshaping ofthe probe pulse occurs in the double-pulse filament.This indicates the potential of dual- and multiple-pulse supercontinuum excitation in bulk media. Re-cently, different-wavelength pulses were used in gen-erating tunable, visible-wavelength ultrashort pulses[17]. However, supercontinuum generation inmultiple-pulse interaction in gases over long dis-tances has not been studied so far.

In this Letter, we explore, by means of numerical

simulation, a new approach that combines temporal

0146-9592/07/182753-3/$15.00 ©

and spatial focus control, with dual-pulse excitation.It takes the idea of temporal focusing to the extremeby utilizing two pulses with sufficiently differentwavelengths. This makes it possible to tune the dis-tance at which filament and supercontinuum are cre-ated and, to some degree, to shape the supercon-tinuum spectrum at the same time. These are veryappealing capabilities for remote sensing; one couldprobe a target at a precisely given distance and si-multaneously optimize the spectrum of the excita-tion.

Although the idea of this paper applies to high-power pulses and possibly to regular filament arraysas well, we concentrate on using relatively low powerto ensure that accurate control of the filamentationdistance is achieved by linear means. Since such in-dividual pulses do not have sufficient power, theircollapse, and subsequent filament formation, is con-ditional on their overlap and fulfillment of certainconditions that we derive next.

We consider two (similar) individual pulses cen-tered at wavelengths �1,2. Each pulse power PI (sub-script I denotes values achieved in the interactionzone) is below or around the critical power, Pc, forself-focusing in air. We launch them with an appro-priate delay ��, chirp, and focusing such that thetemporally and spatially focused pulses will overlapin the desired interaction region at distance zI. An es-timate for the required delay is given by

�� � zIk���, �1�

where ��=2�c�1/�1−1/�2� and k� is the group-velocity dispersion. Clearly, once the pulses overlap,a necessary condition for the subsequent collapse isthat the composite pulse exhibits higher than criticalpower at least in some of its temporal slices. How-ever, this condition is still not sufficient. What weneed to achieve is that the collapse occurs before thedifferent group velocities cause the two pulses towalk-off from each other. For the pulse duration �I,

the walk-off distance can be estimated as 2�I / �k����.

2007 Optical Society of America

2754 OPTICS LETTERS / Vol. 32, No. 18 / September 15, 2007

On the other hand, the collapse distance depends onthe beam waist, shape, and power [18]. For thepresent purpose, a simplified formula gives us the re-quirement that the walk-off distance be larger thanthe self-focusing distance:

2�I

��k��

kwI2

2�4PI/Pc − 1. �2�

Together with Eq. (1) this gives us a rough guidanceto achieve the conditional collapse at a given dis-tance.

In our numerical experiment, we launched twopulses centered at 800 and 600 nm, with respectiveenergies of 0.25 and 0.14 mJ, and focused them to acommon focal point at a distance of 25 m (distancechosen small enough to accommodate both pulses inthe same computational domain during the wholepropagation). Since the pulse energies are too low forcreating a well-developed filament and supercon-tinuum in each pulse separately, we can choose therelative delay between the two pulses and their tar-get beam waist such that we control if and where thefilament and, consequently, white light are created.According to the qualitative argument outlinedabove, for the chosen distance and wavelengths, fila-mentation should occur for relative pulse delays ofabout 400 fs and a target beam waist of roughly lessthan 1 mm. The pulse chirp plays a smaller role forthis small distance, but for longer distances the tem-poral focusing becomes important. We have used thescalar, z-propagated UPPE (for “unidirectional pulsepropagation equation”) [19] simulator to perform ourcalculations.

We first examine the dependence on the relativepulse delay. The results shown were obtained for atarget beam waist wI=0.4 mm. Figure 1 demon-strates that the collapse is conditional on the properoverlap in the focal region and that it is possible toswitch filament creation on and off at a predeter-mined distance. The figure also shows that the pre-cise location of the filament (defined by maximumplasma generation) can be further fine-tuned bychoosing an appropriate delay, while the total num-

Fig. 1. (Color online) Total number of electrons generatedin the filament (circles), and filament location (squares) asfunctions of the delay between the pulses. The plasma fila-ment is created only for optimal delays that ensure condi-

tional dual-pulse collapse.

ber of electrons generated in the filament remainsroughly the same. Thus, with the two pulses, it is inprinciple possible to place the filament formation ac-curately on the desired target.

Figure 2 shows a few examples of the linearplasma density profiles as generated in filaments fordifferent relative pulse delays. We can see that thelength of the filament stays roughly around 25 cm,while maximal plasma densities vary a little. It hasto be stressed that the modulations seen in theplasma density profiles do not correspond to refocus-ing cycles as one could expect based on the spatial re-plenishment scenario [20]. Rather, in this case themodulation is brought about by pulse synthesis, orinterference between the two major color componentsof the composite waveform. The plasma densityvariations reflect the evolution of the interferencepattern with the propagation distance.

The effect of the conditional collapse is most evi-dent in the supercontinuum generation as illustratedin Fig. 3. For small ��400 fs� or large (�600 fs, notshown) delay, the spectral broadening is identical tothat in individual pulses, because the mutual inter-action becomes negligible. However, there is an inter-

Fig. 2. (Color online) Linear plasma density along thepropagation distance for different delays. The length of thefilament is about 25 cm, largely independent of the delay.

Fig. 3. (Color online) Supercontinuum generation in a con-ditional femtosecond pulse collapse. If and only if the twopulse overlap such that the resulting collapse distance isshorter than their walk-off distance, strong supercontinua

are generated.

September 15, 2007 / Vol. 32, No. 18 / OPTICS LETTERS 2755

val of delays that ensure that the overlap occurs justwhen the pulses are about to reach their maximalpeak power due to temporal and spatial focusing.Then and only then are very wide and intense super-continua generated.

Figure 3 also shows that the spectral power at afixed frequency can depend significantly on the rela-tive pulse delay. As a consequence, should a remotelyprobed target excitation depend on the spectralpower in a certain frequency range, it could be opti-mized by choosing an appropriate delay.

Let us look next on the role of the focusing geom-etry. Figure 4 shows several spectra for a fixed delaybetween the pulses �420 fs�, but for varying targetbeam waists. As the beam waist achieved in the in-teraction region increases, so does the collapse dis-tance of the resulting dual-color beam. Beyond thewaist size of 1–1.5 mm [compare with the �1 mm es-timate from Eq. (2)] the walk-off distance of the twocolors becomes shorter than the self-focusing dis-tance. Consequently, collapse is avoided, and littlespectral broadening and plasma are generated.

In this study, we used similar pulses; i.e., the pulseduration, power, and focusing geometry were similarin both pulses. Naturally, one can also consider caseswith dissimilar pulses. For example a longer-duration pulse may be tighter focused in the targetdistance to serve as a catalyst to trigger the collapsein the probe pulse.

In conclusion, we have demonstrated that spatialand temporal focusing can be effectively combinedwith dual-pulse excitation to achieve conditional col-lapse and, consequently, filament formation at a pre-determined distance. The method thus allows one tocontrol accurately if and at what distance plasmaand white light are produced. These results should beuseful for both remote sensing and control of filamen-tation and supercontinuum generation in general.

Fig. 4. (Color online) Spectral broadening for different tar-get beam waists. To achieve supercontinuum generation,the beam waist must be small enough for the self-focusingdistance to be smaller than the walk-off distance.

The authors acknowledge support from the US AirForce Office for Scientific Research under grantFA9550-07-1-0010. J. V. Moloney acknowledges sup-port from the Alexander von Humboldt Foundation.

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