Propagation of short laser pulses through water is animportant research area that has recently been sur-rounded by a controversy concerning possible viola-tions of the exponential attenuation law of Bouguer,Lambert, and Beer (BLB) that have been attributedto formation of femtosecond optical precursors. Opti-cal precursors, theoretically predicted by Brillouinand Sommerfeld roughly a century ago [1], have re-mained largely elusive in experimental observations.
As a notable exception, in recent decades precursorshave been studied in media with strong and narrowabsorption lines, i.e., semiconductors [2] and atomicgases [3]. Recently, optical precursors were observedwhile studying propagation of entangled photonsthrough rubidium vapor [4]. However, observationof optical precursors in bulk liquids, such as water,has been a subject of substantial debate and dis-agreement. Choi and Österberg [5] have reported ameasurement of femtosecond precursors in deionizedwater; in that paper, as well as in their follow-upwork [6,7], Österberg and coworkers have stated thatthe occurrence of these precursors is associated witha violation of the exponential absorption law (BLB
law). Apparently, the subsequent controversy re-sulted from amisunderstanding of the connection be-tween the existence of optical precursors and theseeming violation of the BLB law even when thislaw was applied to each individual spectral compo-nent separately. Follow-up studies by other groups(including ours) [8–10] have experimentally shownthat at low light intensities there is no noticeable de-viation from the (spectral-domain) BLB law, while aspectrally varying absorption, even in the linear re-gime, may indeed result in nonexponential decay ofthe total laser pulse energy integrated over a finitespectral band.The original claims of BLB law violations, as well
as the follow-up studies that tended to disprove thoseclaims, focused on orders-of-magnitude deviationsand discrepancies and took special care to make surethe light field was sufficiently weak so as to avoid anypossible sources of nonlinearity. Since this subjectarea is of substantial fundamental significance andhas potential implications to such important applica-tions as underwater communications and biomedicalimaging, it warrants in our view further investi-gation. The scope of our present study is twofold:(1) We perform a balanced side-by-side comparisonof water absorption for weak laser pulses of differentduration, aiming to detect small (fraction of a percentlevel) duration-dependent differences. We employ aspectral hole filling technique developed by Warrenand coworkers [11], where a short pulse and a longpulse (of the same spectral intensity at the same cen-ter wavelength but with opposite phase) are addedcoherently to produce a spectral distribution witha narrow gap (hole). In this situation, any pulse-duration dependent absorption is expected to resultin “filling” of this hole. (2) We quantify the transitionfrom linear to nonlinear pulse propagation in water.We proceed by first reviewing the basics of our
technique, and then we describe experimental re-sults for pulses with peak intensities varied overmany orders of magnitude. Electric fields of twocoherent laser pulses (a short and a long one) ofthe same spectral intensity at the same center fre-quency ωc, but with opposite phase, add up to producea spectrum with a hole [Fig. 1(a)], since at ωc the twocontributions exactly cancel:
EðωÞ ¼ A
�exp
�−
ðω − ωcÞ28 lnð2Þ τ21
�− exp
�−
ðω − ωcÞ28 lnð2Þ τ22
��;
ð1aÞ
EðtÞ ¼ B
�exp
�−2 lnð2Þ
�tτ1
�2�
−
τ1τ2
exp�−2 lnð2Þ
�tτ2
�2��
cosðωctÞ; ð1bÞ
where EðωÞ is the spectral amplitude, EðtÞ is the am-plitude of the electric field [I ¼ jEðtÞj2 is shown sche-matically in Fig. 1(b)], and τ1 and τ2 are the intensity
full width at half-maximum (FWHM) pulse dura-tions. When the resultant waveform is sent througha linear absorbing or scattering medium, this bal-ance will persist if the absorption is independentof pulse duration. Any absorption that would affectthe longer and shorter pulses differently would inevi-tably destroy the perfect destructive interference atthe center wavelength and result in the filling of thespectral hole.
Figure 1(c) shows a typical setup for a spectral-hole-filling experiment, where a pulse shaper is usedto obtain the required waveforms, and the spectratransmitted through the sample are measured bya spectrometer.
Before proceeding further, we need to make a noteon the rapid dispersion of short laser pulses in water;a detailed discussion of this phenomenon is includedin Appendix A. Although a plethora of virtues of ul-trashort laser pulses have been discovered, theirduration depends on the medium dispersion, theirspectral bandwidth, and the propagation distance.Because of the large spectral width of these pulses,we had to investigate the effect of the glass windowof our water cell on the shape and duration of a pulseas it emerges from the cell window and enters thewater. Our simulations (see Appendix A) show thatan initial 7 fs pulse broadens significantly faster thana 30 fs pulse and becomes longer than the latter evenfor a 2mm thick glass window. Therefore, in order toobtain a 7 fs pulse at the entrance of our water cell,the pulse has to be prechirped to compensate for thedispersion in the glass window. Note that a similareffect of fast pulse spreading due to dispersion alsotakes place in the water itself. Therefore, eventhough our femtosecond oscillator was capable of pro-ducing pulses as short as 7 fs, in our experiments wechose to limit the total pulse bandwidth and workwith pulses of not less than 25 fs duration.
2. Experiments with Femtosecond Oscillator Pulses
To study spectral hole filling experimentally, weuse an ultrabroadband mode-locked femtosecond
Fig. 1. (Color online) Spectral-hole-filling experiment: (a) Fre-quency domain and (b) time domain schematics. (c) Experimentalsetup (femtosecond laser oscillator not shown). Initial 7 fs laserpulses pass through the pulse shaper, which produces a spectralhole and reduces the total spectral bandwidth, and enter the watercell. The transmitted spectra are measured by the spectrometer.
Ti:sapphire oscillator (Rainbow, FemtoLasers) fol-lowed by an acousto-optic programmable pulse sha-per (Dazzler, FastLite) to obtain a broad, smoothspectrum with a spectral hole whose position andwidth can be precisely controlled. The laser oscillatorproduces pulses of 7 fs duration with a spectrumthat extends from 660 to 980nm; its average outputpower is 340mW, and the repetition rate is 78MHz,so that the energy per pulse is about 4nJ. We set theDazzler to reduce the full spectral bandwidth to80nm. The width of the hole is chosen to be 10nm,and for three measurements its center is positionedat three different wavelengths, i.e., at 767nm,800nm, and 827nm [example spectra can be seenin Figs. 2(a)–2(c)]. Note that the pulse shaper iscapable of compensating additional dispersion (in-cluding its own dispersion) and therefore allowssynthesis of transform-limited waveforms. Assuminga constant spectral phase, we calculate pulse shapes(not shown), which indeed contain long- and short-duration components (200 fs and 25 fs, respectively),as drawn in Fig. 1(b).The resultant shaped pulses are directed to a
tilled water. Laser light emerging from the water cellis focused by a lens and scatters off a white screenvibrating at a 50Hz rate; then a fraction of it couplesinto a fiber and is detected by a spectrometer (USB2000, Ocean Optics), as shown in Fig. 1(c). Thisconfiguration is used in order to make the measure-ment insensitive to small variations of laser beamalignment; the screen vibration reduces the possibleeffects of intensity speckles and diffraction thatresult in fluctuating modulations of the measuredspectrum. We check that the reshaping of the spectra[water cell input, Figs. 2(a)–2(c), versus output,Figs. 2(d)–2(f)] is mostly consistent with linear ab-sorption in water (which is strongly wavelengthdependent), although these present experimentsare not optimized for measuring the wavelength-dependent absorption as reliably as some of ourearlier experiments were [10].
We vary laser power at the input of the water cellfrom 1mW to 12mW in 1mW increments using aneutral density filter while recording the output spec-tra [Figs. 2(d)–2(f)]. To compare the shapes of the spec-tra for different input powers, we first take the dataand divide each measured spectral curve by the
Fig. 2. (Color online) Results from the spectral hole filling experiment with low-power (laser oscillator) shaped pulses. Parts (a)–(c) showinput spectra with the spectral hole centered at (a) 767nm, (b) 800nm, and (c) 827nm wavelengths. Parts (d)–(f) show the correspondingoutput spectra after propagation through 1:15m of distilled water; insets show zoomed-in spectral hole regions.
corresponding input power. This procedure by itselfproduces curves deviating from each other by nomorethan the relative uncertainty of our power measure-ments (with the absolute experimental uncertainty of0:1mW, the relative uncertainty varies from 10% for1mW of input power to 0.8% for 12mW). To avoidthese variations due to uncertainty in the powermea-surements, we normalize each curve to its peak value,in order to be able to look at possible small (well lessthan 1%) relative variations in the resultant spectralshapes. The insets shown in Figs. 2(d)–2(f) illustratean essential absence of a mismatch among differentcurves at the spectral hole wavelengths. This mis-match is less than0.1%of the peakvalue and iswithinthe spectrometer noise.The above results, to within our precision of 0.1%
showing no “hole filling,” indicate that absorptionaffects a 25 fs pulse exactly the same way it affectsa 200 fs pulse. For a linear regime this is as expected,since new frequency components can only appear asa result of nonlinear generation. Simple estimatesshow that in the experiments with the oscillatorpulses, we are well below intensity levels requiredfor any nonlinear effects (i.e., self-focusing or self-phase modulation) to occur.
3. Experiments with Amplified Pulses: Transition tothe Nonlinear Regime
To investigate the propagation of laser pulses in abroad range of intensities and to quantify the transi-tion to nonlinear behavior, we used amplified laserpulses. Figure 3 shows a schematic of our experimen-tal setup: after amplification (Femtopower multipasssystem from Femtolasers) a spectral hole with thecentral wavelength of 794nm is produced in the pulsespectrum by placing an obstruction that blocks a nar-row spectral interval in the compressor (see Fig. 3).The resultant spectrum is centered at 800nm, andhas a total bandwidth of about 70nm and a spectralhole width of 20nm (corresponding to 30 fs and 100 fspulse durations, respectively). The pulse repetitionrate is 5kHz. The laser pulses propagate through avariable neutral density filter and then through a
telescope, which reduces the beam size by a factorof 3 (to a waist of 2:1mm). The collimated laser beampasses through theglasswindowandenters thewatersample. After propagating the single-pass length of1:15m in distilled water, the pulses are scatteredoff of a vibrating white screen and measured with afiber-coupled spectrometer, similar to the above-described experiment with laser oscillator pulses.The input power is measured by redirecting the laserbeam to a powermeter (Nova II, Ophir) with a remo-vable mirror just before the water cell.
Figure 4(a) shows the dependence of the trans-mitted pulse spectrum on the input power. Eachcurve has been normalized to the peak value of thespectrum obtained at 340mW of input power. Thecurves are vertically displaced, with greater inputpowers corresponding to greater upward vertical dis-placement. It can be seen that, as input power in-creases, the hole begins to fill in and also shiftssomewhat from the initial wavelength of 794nmtoward smaller wavelengths at higher input powers.
Fig. 4. (Color online) Transformation of transmitted spectra withincreasing input laser power: (a) Measured spectra. To aid in vi-sualization, each spectrum has been vertically displaced, with lar-ger upward vertical displacements corresponding to larger inputpowers. The input powers for the shown spectra are, from bottomto top: 5mW, 15mW, 25mW, 40mW, 60mW, 80mW, 100mW,120mW, 140mW, 180mW, 250mW, and 340mW. (b) Changesin the spectrum, relative to spectra expected for linear transmis-sion and calculated by scaling the output spectrum obtained for20mW input power. These curves are obtained by subtractingthe expected linear-transmission spectra from the actual mea-sured spectra, dividing by the input power, and scaling relativeto the peak value obtained in the 200mW curve. The input powersshown, with increasing spectral transformation, are 40mW,80mW, 120mW, 160mW, and 200mW. In both (a) and (b), dottedvertical lines correspond to wavelengths of 794nm (spectral holecenter at low power), and two wavelengths at the wings of the spec-trum, 758 and 836nm.
Fig. 3. (Color online) Experimental setup for measuring propaga-tion of amplified laser pulses through water. An obstruction isplaced in the compressor to produce a hole in the laser spectrum.Then the pulses propagate through the water cell, and the trans-mitted spectra are measured by the spectrometer.
In addition, the wings of the spectrum experi-ence broadening. Figure 4(b) shows the differentialchanges of the spectrum at different power levels.Each of the shown curves was obtained with the fol-lowing procedure: a spectrum, calculated assuminglinear behavior, is subtracted from the measuredspectrum and divided by the input power; each resul-tant curve is then scaled relative to the obtainedpeak value of the curve corresponding to an inputpower of 200mW. As can be seen in the figure, theenergy from the intense spectral components istransferred to the hole and wings (i.e., to spectralcomponents of low intensity) as the input power in-creases. We note that this behavior is reminiscentof diffusion, with a spectral diffusion coefficient in-creasing with the laser power.Figure 5 shows the measured transmitted spectral
intensities at three wavelengths (marked in Fig. 4 bydotted vertical lines) as a function of the input power.The three wavelengths correspond to the low-powerspectral hole center (794nm) and two wavelengthsat the spectral wings (758 and 836nm) having, atlow input powers, the same spectral intensity as thehole center. At low input powers (below about40mW), we observe linear growth of the spectralintensities, turning to nonlinear behavior at higherinput powers.This behavior can be seen especially clearly on the
logarithmic plot of the transmitted spectral intensityat the hole central wavelength of 794nmas a functionof the input power (see Fig. 6). A significant (1% com-pared to the peak spectral intensity) deviation fromthe linear behavior starts at around 40mW power,which correspond to a peak value of 0:12mJ=cm2 ofinput laser energy fluence and, given the pulse dura-tions used, to a peak intensity of about 1010 W=cm2.Above input powers of 40mW, the spectral intensityat 794nm exhibits quadratic growth. A new drasticchange in the behavior happens at around 250mW(which corresponds, for the parameters of our experi-ment, to a peak input fluence of 0:73mJ=cm2).
The quadratic power dependencemay result from athird-order nonlinear mixing process (please notethat second-order nonlinearities are forbidden in cen-trosymmetric media such as liquid water). As a likelypossibility, we consider two-photon processes such asstimulated Raman scattering (SRS). At powers above100mW we find that the spectral hole position (thelocation of the minimum in the output spectrum)shifts toward shorter wavelengths; this behavior isalso consistent with SRS, which converts shorter-wavelength photons to longer-wavelength ones. Itshould be pointed out that even though SRSis a likely process to come into play first, as the laserintensity is gradually increased, many other phe-nomena such as thermal lensing, self-focusing, self-phasemodulation, andmedium ionization all becomesignificant at somewhat higher intensities [12–14]. Inour experiment we observe an interplay of theseeffects at powers above 250mW; however, detailedinvestigation of these phenomena is beyond the scopeof our present work.
4. Conclusion
We have investigated propagation of femtosecondlaser pulses with the central wavelength around800nm in water by employing a spectral hole fillingtechnique. We showed experimentally and withprecision of better than 0.1% that there is no pulse-duration dependent absorption for ultrashort pulsesof duration around 30 fs in the linear light–matter in-teraction regime.We then quantified the transition ofsuch pulse behavior from the linear to the nonlinearinteraction regime, and we found that this transitionoccurs at around 0:12mJ=cm2. Above this value aquadratic growth of the intensity at the spectral holewas observedup to about0:73mJ=cm2. Also,we foundthat as the light–matter interaction becomes increas-ingly nonlinear, the location of the minimum in the
Fig. 5. (Color online) Transmitted intensity for three wave-lengths as a function of input laser power. The three wavelengthscorrespond to the low-power spectral hole center (794nm) and twowavelengths at the spectral wings (758 and 836nm) having, at lowinput powers, the same spectral intensity as the hole center.
Fig. 6. (Color online) Plot on a log-log scale of transmitted spec-tral intensity at 794nm (initial spectral hole center) as a functionof the input power. The blue circles show the measured data. Thered reference line has a slope of 1 and fits well to the data points inthe low-power region, indicating a linear dependence. The greenreference line has a slope of 2 and fits the data points in the regionof intermediate power up to about 250mW, indicating a quadraticdependence. The change from the linear dependence to quadraticdependence can be clearly seen at around 40mW (where a dottedvertical line is drawn).
output spectrum shifts toward shorter wavelengths.We suggest stimulated Raman scattering as a possi-ble mechanism for this spectral hole filling and shift-ing effect. However, it should be pointed out thatmany other phenomena, such as thermal lensing,self-focusing, self-phase modulation, and medium io-nization, can affect propagation of short and intenselaser pulses in the nonlinear regime, and the detailedinvestigation of these mechanisms is certainly out ofthe scope of this paper.Themain conclusion is that noviolations of theBLB
law in the linear light–matter interaction regime wasregistered in this work. It may still be a matter ofdebate whether pulses of any duration, most signifi-cantly durations that are a small fraction of the vibra-tional periods of water molecules, will induce aviolation of the BLB law; however, we can now statethat none have yet been documented. As we haveshown,due todispersion, ultrashortpulsesbecome in-creasingly fragile as their duration decreases, so thatthey canmaintain their pulseduration on correspond-ingly decreasing distances (less than a millimeter forsub-10 fs pulses), unless a special compensation of thedispersion spreading is implemented.
Appendix A
Ultrashort femtosecond pulse broadening is investi-gated here.We consider sufficiently low incident laserintensities to involve only linear response of the med-ia under study. The electric field of a pulse resultsfrom a superposition of its Fourier components:
Eðz; tÞ ¼ 12π
Z∞
−∞
EðωÞ exp½iðkðωÞz − ωtÞ�dω; ðA1Þ
where we assume that the pulse is propagating in thepositive z direction and regard its divergence due todiffraction as negligible, Eðz; tÞ is the electric field,ω is the angular frequency, z is the propagation dis-tance in the medium, t is time, EðωÞ is the Fouriercomponent (or complex spectral amplitude) of the in-cident pulse at the input interface of the considereddielectric medium (at z ¼ 0), and kðωÞ is the complexwave vector in an absorptive medium such as water.Usually, the obtainable data characteristic of themedium are the refractive index nðωÞ and the absorp-tion coefficient αðωÞ instead of the wave vectorkðωÞ. The relations between kðωÞ, the refractive indexnðωÞ, and the absorption coefficient αðωÞ areRe½kðωÞ� ¼ ½nðωÞω�=c and Im½kðωÞ� ¼ αðωÞ=2.We consider the broadening of pulse duration of an
ultrashort pulse passing through a glass window anda water cell. In our simulation, we use documentedrefractive indices of BK7 glass and water. The refrac-tive index nðλÞ of BK 7 glass follows the Sellmeierequation [15]:
where λ is the wavelength of light,B1 ¼ 1:03961212 μm−2, B2 ¼ 2:31792344 × 0:1 μm−2,B3 ¼ 1:01046945 μm−2,C1 ¼ 6:00069867 × 0:001 μm2,C2 ¼ 2:00179144 × 0:01 μm2,and C3 ¼ 1:03560653 × 100 μm2.
The refractive index of water is given by Quan andFry [16]:
nðλÞ ¼ 1:31279þ 15:762λ−1 − 4328λ−2
þ 1:1455 × 106λ−3; ðA3Þ
where λ is in nanometers. This formula is verified bythe experimental data of [17] in a range of 200 to1100nm. The absorption coefficient is obtained fromthe data of Kou, et al. [18] and Pope and Fry [19],which in combination cover the range from 380to 2500nm.
The pulse duration is defined as the full width athalf-maximum (FWHM) of the pulse intensity:
Iðz; tÞ ¼ Eðz; tÞEðz; tÞ�: ðA4Þ
In our simulation, we assume a pulse with a durationτ0 and a carrier frequency ωc in the form
Eð0; tÞ ¼ 1ffiffiffiffiffiffi2π
p exp�−
2 lnð2Þ tτ20
2�cosðωctÞ: ðA5Þ
The spectral amplitude EðωÞ is obtained by the Four-ier transformation of Eq. (A5):
EðωÞ ¼ exp�−
ðω − ωcÞ2τ208 lnð2Þ
�
¼ exp�−γðω − ωcÞ2
�: ðA6Þ
We define γ ¼ τ20=½8 lnð2Þ�. After propagating a cer-tain distance z, the phase delays between differentspectral components determine the pulse broaden-ing. Using the carrier frequency ωc as a reference,the phase difference of ikðωÞz − iωt can be expressedas a Taylor expansion:
ðizb1 − itÞðω − ωcÞ þ�izb22
�ðω − ωcÞ2 þ…; ðA7Þ
where bj ¼ djkðωÞ=dωjjωcand j ¼ 1; 2; 3…. The group
velocity dispersion (GVD) approximation keeps theexpansion to the second order and is given byGVD ¼ d2kðωÞ=dω2jωc
. Therefore, the value of GVDequals b2. By substituting the GVD approximationinto Eq. (A1), an analytic solution of the intensityenvelope is obtained:
Shown in Fig. 7 are the simulation results obtainedfor pulses with initial durations of 7 and 30 fs, bothcentered at 800nm.
This work was supported by the Office of Naval Re-search (ONR) under contract N00014-08-1-0037 andthe Defense University Research Initiative Program(DURIP) under contract N00014-08-1-0804, and inpart by National Science Foundation (NSF) grants0722800 and 354897, Texas Advanced ResearchProgram grant 010366-0001-2007, Army ResearchOffice grant W911NF-07-1-0475, and Robert A.Welch Foundation grant A1547. We also wish tothank Matthew Springer for his help with the manu-script.
References1. L. Brillouin, Wave Propagation and Group Velocity
(Academic, 1960).
2. J. Aaviksoo, J. Kuhl, and K. Ploog, “Observation of optical pre-cursors at pulse propagation in GaAs,” Phys. Rev. A 44,R5353–R5356 (1991).
3. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observa-tion of optical precursors in a region of anomalous dispersion,”Phys. Rev. Lett. 96, 143901 (2006).
4. S. Du, C. Belthangady, P. Kolchin, G. Y. Yin, and S. E.Harris, “Observation of optical precursors at the biphotonlevel,” Opt. Lett. 33, 2149–2151 (2008).
5. S. H. Choi and U. Österberg, “Observation of optical precur-sors in water,” Phys. Rev. Lett. 92, 193903 (2004).
6. U. J. Gibson and U. L. Osterberg, “Optical precursors andBeer’s law violations: nonexponential propagation losses inwater,” Opt. Express 13, 2105–2110 (2005).
7. A. E. Fox and U. Osterberg, “Observation of nonexponentialabsorption of ultra-fast pulses in water,” Opt. Express 14,3688–3693 (2006).
8. J. C. Li, D. R. Alexander, H. F. Zhang, U. P. Parali, D. W. Doerr,J. C. Bruce III, and H. Wang, “Propagation of ultrashort laserpulses through water,” Opt. Express 15, 1939–1945 (2007).
9. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A.L. Gaeta, “Absorption of ultrashort optical pulses in water,” J.Opt. Soc. Am. A 24, 3343–3347 (2007).
10. L. M. Naveira, B. D. Strycker, J. Wang, G. O. Ariunbold, A. V.Sokolov, and G. W. Kattawar, “Propagation of femtosecondlaser pulses through water in the linear absorption regime,”Appl. Opt. 48, 1828–1836 (2009).
11. M. C. Fischer, H. C. Liu, I. R. Piletic, and W. S. Warren, “Si-multaneous self-phase modulation and two-photon absorptionmeasurement by a spectral homodyne Z-scan method” Opt.Express 16, 4192–4205 (2008).
12. N. Tcherniega, A. Sokolovskaia, A. D. Kudriavtseva, R.Barille, and G. Rivoire, “Backward stimulated Raman scatter-ing in water,” Opt. Commun. 181, 197–205 (2000).
13. A. Brodeur and S. L. Chin, “Band-gap dependence of the ultra-fast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409(1998).
14. M. Koselik, G. Katona, J. V. Moloney, and E. M. Wright,“Theory and simulation of supercontinuum generation intransparent bulk media,” Appl. Phys. B 77, 185–195 (2003).
15. “Optical glass data sheets,” Schott, Inc., http://www.us.schott.com/advanced_optics/english/download/datasheet_all_us.pdf
16. X. Quan and E. S. Fry, “Empirical equation for the index ofrefraction of seawater,” Appl. Opt. 34, 3477–3480 (1995).
17. P. D. T. Huibers, “Models for the wavelength dependence of theindex of refraction of water,” Appl. Opt. 36, 3785–3787 (1997).
18. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of waterand ice in the 0.65 to 2:5 μm spectral range,” Appl. Opt. 32,3531–3540 (1993).
19. R. M. Pope and E. S. Fry, “Absorption spectrum (380–700nm)of pure water. II. Integrating cavity measurements,” Appl.Opt. 36, 8710–8723 (1997).
Fig. 7. (Color online) Calculated pulse duration versus propaga-tion distance (a) in K7 glass and (b) in water for two transform-limited input pulses of 7 fs (blue lines and circles) and 30 fs (blacklines and stars) initial duration (both at an 800nm center wave-length). In both figures, the data points show the results calculatedwith the actual wavelength-dependent refractive indices, whilethe lines are obtained with the group velocity dispersion (GVD)approximation.
![Generation of short pulses 2.7 fs. Ultrashort pulse generation 15 fs pulse Time [fs] Wavelength [m] Single cycle pulse.](https://static.documents.pub/doc/80x56/56649d2b5503460f94a00e2a/generation-of-short-pulses-27-fs-ultrashort-pulse-generation-15-fs-pulse.jpg)
