spie proceedings [spie spie optical engineering + applications - san diego, california (sunday 21...

Supercontinuum generation in filamentation of femtosecond laser pulse in fused silica

Smetanina E.O., Dormidonov A.E., Kandidov V.P. Moscow Lomonosov State University, Physics Department, Moscow, Russia

[email protected] ABSTRACT The Supercontinuum (SC) generation during femtosecond laser pulse filamentation with various central wavelengths in fused silica is investigated by numerical solution of nonlinear Schroedinger equation. Material dispersion of the medium is considered due to Sellmeier formula. Nonmonotonic dependence of spectral intensity on wavelength in anti-Stokes wing for anomalous group velocity dispersion (GVD) region (λ0 = 1900 nm) was found. There is a local minimum in SC spectrum from 800 nm to 1200 nm, and there is also a local maximum in SC spectrum from 400 nm to 700 nm. We suppose such modification of pulse spectrum during filamentation process to be caused by interference modulation of SC spectrum in presence of anomalous GVD. nonlinear optics, filamentation, supercontinuum generation, self-phase modulation, Kerr nonlinearity 1. INTRODUCTION

Spatial and temporal transformation of femtosecond laser pulse during filamentation leads to transformation

of frequency spectrum of pulse and supercontinuum (SC) generation1,2. In the first experiments on SC generation the

530 nm 4 ps 5 GW laser pulses were focused in a number of transparent dielectrics such as calcite, quartz, sodium

chloride, and glasses3. The asymmetrical spectral super-broadening was observed in all materials listed above. Anti-

Stokes wing was wider then the Stokes wing of SC spectrum. Maximal broadening was about 120 nm and 290 nm

for Stokes and Anti-Stokes wings correspondingly for BK-7 glass. In multi filamentation of 1540 nm 300 fs 100 GW

laser pulse in fused silica4 the spectral broadening was also asymmetric, and in the blue side of spectrum the broad

emission feature centered near 700 nm and null from 950 to 1150 nm was found4. In experiments on white-light

continuum generation during self-focusing in extended transparent media using 796 nm 140 fs laser pulses5,6, a band-

gap threshold was discovered above which the width of the continuum tends to increase with increasing band gap

and below which there is no continuum generation. More general conclusion was given in ref.7. In this work7 the

generation of white-light continuum by femtosecond laser pulses in transparent condensed media was investigated

with 262, 393, and 785 nm laser pulses. It was found that the ratio of the medium’s bandgap energy iU to the photon

energy 0ωh of the pulse central wavelength determines the amount of anti-Stokes broadening, independently of the

central wavelength and the medium’s bandgap. The threshold for SC generation is 20 =ωhiU . Anti-Stokes

broadening is greater for the longer central wavelength. The theoretical investigation8 shows that the space-time

focusing and self-steepening during filamentation give rise to broadening of pulse spectrum. The coherent length of

SC spectral components is close to the value of coherent length of the input laser pulse9. At present, SC sources

generated by filamentation in transparent condensed media1, are used in nonlinear spectroscopy10, in cavity ring-

down absorption spectrography11. Broadband SC produced by filamentation of ultrashort intense laser pulses in a

femtosecond terawatt lidar system "Teramobile" 12 was used for probing the atmosphere and for aerosol studies. The

overviews of remote sensing of the atmosphere using femtosecond laser filamentation are presented in ref.13,14. SC

sources working on the basis of the nonlinear-optical transformation of laser pulses in photonic crystal fibers2 are

Lidar Remote Sensing for Environmental Monitoring XII, edited by Upendra N. Singh, Proc. of SPIE Vol. 8159, 81590L · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.893108

Proc. of SPIE Vol. 8159 81590L-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/16/2013 Terms of Use: http://spiedl.org/terms

used as an instrument for measuring small concentrations of constituents on the distances from 10 m to 10 km in the

atmosphere15, optical coherent tomography16, and other applications.

2. MATHEMATICAL MODEL

For numerical analysis of SC generation in femtosecond laser pulse filamentation in fused silica we use the

mathematical model17 that includes diffraction, Kerr nonlinearity, plasma nonlinearity, multi-photon absorption and

linear losses, self-steeping and full dispersion according to Sellmeier formula18.

In our simulations we obtained the distribution of complex amplitude ),,( ztrA and pulse spectrum ),( zS λ ,

where z is the pulse propagation coordinate. The power spectral density ),( zS ω of laser pulse at a distance z is

rdrzrSconstzS ∫∞

ω⋅=ω0

1 ),,(),( , (1)

where

2

2 ),,(),,( ∫∞

∞−

Ω−⋅=ω dtetzrAconstzrS ti is intensity distribution of spectral components in the cross section,

0ω−ω=Ω is the frequency offset of SC from the central frequency 0ω . The distribution of power spectral density

as a function of wavelength ),( zS λ is

2)(2)),((),(

λλπ

λω=λcnzSzS . (2)

We call ),( zS λ as the spectrum of the pulse at a distance z .

3. SPECTRAL COMPONENTS OF SC IN THE FILAMENTATION OF FS LASER PULSES WITH DIFFERENT CENTRAL WAVELENGTHES

In this work we perform numerical experiments for different laser pulses with the same similarity parameters

that determine the initial stage of filamentation caused by diffraction and Kerr self-focusing. These parameters are

the diffraction length 20kaLd = , and the nonlinearity parameter nlR , which is equal to the ratio of pulse peak power

P to the critical power of self-focusing crP . For all pulses considered the diffraction length was taken 4≈dL cm

and the beam radius was tens of microns. These values are close to parameters of the beams used in experiments19.

Nonlinearity parameter is equal to 10=nlR , that corresponds to the single filament in the beam cross section and is

enough for several pulse refocusings. At a certain beam radius the nonlinearity parameter determines the peak

intensity 0I and, consequently, the nonlinear length02

00

2 InnaLnl = . The dispersion length 2

2 kL FWHMdis τ= ,

where 0

2

2

2ω=ωω∂

∂=

kk , was taken about 1.64 cm for pulses with central wavelengths from normal and anomalous

dispersion regions. Complete set of parameters is given in Table1.



Table 1. Pulses and media parameters.

Parameter Normal GVD Zero GVD Anomalous GVD

Wavelength λ0, nm 400 800 1300 1900

2k , fs2 cm-1 975 360 –23 –800

Pulse half duration 0τ (e-1 level), fs 40 34 40 36

Dispersion length disL , cm 1.64 1.64 70.12 1.64

Beam waist 0a , μm 40 57 73 88

Peak intensity 0I , W/m2 9.2×1014 18×1014 3×1015 4.4×1015

Pulse energy, μJ 0.33 1.1 3.5 6.8 Critical self-focusing power crP , MW 0.46 1.87 4.94 10.6

Order of multiphoton process K 4 7 11 15

Thus, the conditions of pulse propagation in the media before the nonlinear focus, and before the plasma

formation are similar for all wavelengths studied. The order of multiphoton process ⎡ ⎤0ω= hiUK , determines the

rate of plasma generation. Plasma defocusing limits the growth of the intensity under the self-focusing that

determines the steepness of the pulse fronts and, consequently, the spectral broadening. SC spectra from one

extended emitting region in filament at distance z before pulse refocusing for different pulse central wavelengths

are shown in fig.1.The length of emitting region in filament in each case was about 1 mm.

In a single-filament regime in area of normal dispersion we worked with wavelengths 4000 =λ nm and

8000 =λ nm. So for the wavelength 4000 =λ nm the order of multiphoton process 3=K and the spectrum width

110=λΔ nm (fig.1 a), whereas for 8000 =λ nm the ratio is: 6=K and the spectral width 790=λΔ nm (fig.1 b).

The broadening is asymmetric, anti-Stokes wing is larger than the Stokes one. These results are in agreement with

the statement that above the bandgap threshold for continuum generation the continuum width increases with

growing the pulse central wavelength7. The spectral width λΔ was found for level: 7))(/)(lg( 0 −=λλ SS , where

)(λS is an intensity of spectral component, 0λ is the pulse central wavelength. In zero dispersion region in fused

silica spectrum broadening for 13000 =λ nm is 1600=λΔ nm (fig.1 c) and the order of multiphoton process

11=K . In anomalous dispersion region the broadening for pulse spectrum with central wavelength 19000 =λ nm is

2400=λΔ nm (fig.1 d) and respectively the order of multiphoton process is 15=K . It was found for this

wavelength, that the spectral intensity decrease in anti-Stokes wing is nonmonotonic. There is a local minimum in

spectrum from 800 nm to 1200 nm, and then there is a local maximum from 400 nm to 700 nm. Such a spectrum was

observed experimentally for 15400 =λ nm in fused silica4. This nonmonotonic superbroadening can't be explained

by 3-rd harmonic generation (THG), since it isn’t included in our numeric model. We suppose such modification of



pulse spectrum during filamentation process to be caused by interference modulation of SC spectrum in presence of

anomalous GVD. Fig. 1. Spectrum of SC from one emitting region generated in the filamentation of fs laser pulses with different central wavelengths a) λ0 = 400 nm; б) λ0 = 800 nm; c) λ0 = 1300 nm; d) λ0= 1900 nm. Solid line - the distribution of spectral components of the SC in a logarithmic scale )),(/),(lg( 0 zSzS λλ ; dotted line - the initial pulse spectrum in a logarithmic scale ))0,(/)0,(lg( 0 =λ=λ zSzS .

4. SPECTRUM TRANSFORMATION IN FS LASER PULSE FILAMENTATION IN FUSED SILICA

4.1. Normal GVD In region of strong normal GVD in fused silica we considered filamentation of laser pulse at 4000 =λ nm

( 122 cmfs975 −⋅=k ). Fig. 2 shows the transformation of pulse spectrum in logarithmic scale ( )zSzS 0),(ln λ

(a) and

the transformation of the on-axis pulse time profile 0/),0,( IzrI =τ in the frame moving with group velocity of input

pulse gv , and the dependence of the electron density )(zNe in self-induced laser plasma at the beam axis on distance

z (b).

The nonlinear focus is formed at a distance cm8.0=fz , the pulse intensity reaches the value 100 I0, that

gives start to the ionization. Self-induced laser plasma defocuses the pulse tail, so the pulse tail becomes steeper. The

pulse spectrum rapidly broadens because of SPM and self-steeping of the pulse tail. Strong normal GVD in presence

of SPM stretches pulse in time and leads to the pulse intensity clamping. So pulse splits into two subpulses. They are

presented in fig.2 b). The first area of high intensity 0/),( IzI τ splits into two "wings" coming angularly to the

propagation axis z. Durations of both subpulses (FWHM) are about 40 fs. Intensity of both subpulses is sufficient to

form extended emitting region in filament. Length of the first emitting region is about 1mm. The spectrum from the

first emitting region at distance 9.0=z cm is shown in fig.1 а). The spectrum is broadened slightly, its width is about

110 nm. A chain of two coaxial separate plasma channels and corresponding emitting regions is formed in the

filament after pulse refocusing ( 3.1=z cm). As the pulse refocuses spectrum broadens again. The is a modulation of

the SC spectrum. The period of modulation in SC spectrum from the first emitting region is greater, than the period



of spectrum modulation that appears after pulse refocusing. The modulation in spectrum before pulse refocusing is a

result of interference of radiation from first extended emitting region, and the additional spectrum modulation that

appears after pulse refocusing is a result of interference of radiation from two emitting regions in the filament17,19.

a) Fig. 2. Filamentation of 400 nm laser pulse in fused silica. (а) Transformation of pulse spectrum in logarithmic scale

( )zSzS 0),(ln λ , )),(max(0 zSS z λ= ; (b) Transformation of the on-axis pulse time profile

0/),0,( IzrI =τ , ))0,,(max(0 =τ= zrII

in the frame moving with group velocity of input pulse and the dependence of the electron density )(zNe in self-induced laser plasma at the beam axis on propagation distance z .

b)

The dispersion stretching for 800 nm pulse is smaller than for 400 nm pulse, so the start of the filament

occurs on the smaller distance from the entrance of radiation in media: cm65.0=fz , and the “wings” angle is

smaller for subpulses (fig.3). The duration of each subpulse is about 8-10 fs, and the maximum intensity that pulse

reaches under self-focusing is 200 I0. The plasma density is higher than in 400 nm pulse filamentation. In areas of

high intensity, that correspond to the self-focusing and following refocusings of pulse, the spectrum rapidly broadens

(fig.3). The spectrum of radiation provided by the first emitting region at distance 8.0=z cm is shown in fig.1 b).

The spectral broadening is 790 nm. Also it is seen an interference modulation of SC spectrum appears from the first

extended emitting region and additional modulation from the sequence of emitting regions in filament appears after

pulse refocusing (fig.3).



0/),0,( IzrI =τ , ))0,,(max(0 =τ= zrII


b)



The wavelength 800 nm lies in the normal GVD region, but close to the region of zero dispersion in fused silica ( 12

2 cmfs360 −⋅≈k ). So the spectrum of the front subpulse, that moves in the medium with a velocity gvv >1 , is

shifted in the Stokes region from 0λ . Spectrum of this subpulse is formed like spectrum of pulse in medium with zero GVD17. So appearance of the weak Stokes components of the SC spectrum near 1500 nm is due to front subpulse spectrum.

4.2. Zero GVD Let’s consider the formation of the SC in filamentation of pulse with central wavelength =λ0 1300 nm, for

this wavelength the group-velocity dispersion in fused silica is close to zero ( 122 cmfs23 −⋅−=k ). The start of the

filament with parameters of the pulse given occurs at a smaller distance than in presence of normal GVD:

cm56.0=fz (fig. 4). Rapid broadening of the spectrum occurs in the nonlinear focus.



0/),0,( IzrI =τ , ))0,,(max(0 =τ= zrII


b)

The sequence of energy localization regions appears on the trailing edge of the pulse (pulse tail corresponds to

positive values of time τ ) after the nonlinear focus (fig.4 b). The formation of multi-peak structure of the pulse in

time is not caused by the dispersion stretching, but occurs as a result of Kerr self-focusing of the pulse tail, that was

defocused in self-induced laser plasma. Each peak reaches the intensity about 150I0 and its duration is 5-10 fs. As a

result of such spatio-temporal transformation almost uniformly broadened SC is generated over 0.7 cm along the

axis of pulse propagation (fig.4 a). Components of SC become weaker only when a series of pulse tail refocusings

stops ( 5.1=z cm). The spectrum of radiation provided by the first emitting region at distance 7.0=z cm is shown in

fig. 1 c). The spectral width is 1540 nm. This value exceeds broadening of the spectrum in filamentation of 800 nm

pulse. The interference modulation of the spectrum from the extended non-uniform emitting region also appears.

4.3. Anomalous GVD

We now consider the SC formation in filamentation of the pulse with central wavelength =λ0 1900 nm, when the

group-velocity dispersion in fused silica is anomalous ( 122 cmfs23 −⋅−=k ). In presence of anomalous GVD the

filament starts at the distance smallest among GVD regimes considered: cm45.0=fz (fig.5). As a result of nonlinear

interaction of laser pulse with the medium during its propagation a sequence of so-called "light bullets" is formed.



The peak intensity in “light bullet” reaches ~120 I0, and its duration ~10 fs. Because of spatio-temporal compression

of the pulse a strong broadening of pulse spectrum (from 400 nm to 3000 nm) occurs. Further propagation of the

"light bullet" causes its peak intensity decrease, and it moves to the pulse tail, while in the central time layers a new

self-focusing begins. So this cycle repeats until there is enough power in the central time layers while the power flux

from pulse front and tail is provided by the anomalous dispersion. It can be seen that in quasi-periodic regime of

"light bullets" formation (fig. 5, z ≈ 1÷2 cm) each of self-focused bullet gives an almost uniformly broadened SC in

its nonlinear focus. Local minimum appears in anti-Stokes wing of the SC after the nonlinear focus. The local

minimum appears in area of 1100 nm and its width is about 500 nm. It can be explained as a result of an interference

modulation of pulse SC spectrum in presence of anomalous GVD.



0/),0,( IzrI =τ , ))0,,(max(0 =τ= zrII


b)

5. Conclusion Increase of central wavelength of pulse on one side determines the increase of the ratio of bandgap energy to the

energy of photon. It means, that in the self-focusing process the peak intensity of pulse with longer wavelength

reaches larger value before plasma generation and start of pulse tail defocusing. As a result of steeper pulse tail

formation more intense anti-Stokes broadening of the spectrum occurs. On the other side, when the central

wavelength of pulse increases, the GVD changes from strong normal GVD, that stretches pulse in presence of SPM,

and therefore decreases the steepness of pulse fronts, to small normal GVD, when pulse stretching is not so hard,

pulse fronts are steeper, and pulse spectrum becomes more broadened. When GVD is zero the power pulse tail, that

was defocused in laser plasma, undergoes the self-focusing and forms a multi-peak structure, every self-focusing

event of this structure is a source of SC. In anomalous GVD region “light bullets” compressed in space and time are

sources of SC, but during the propagation of such “light bullet” the dip appears in anti-Stokes wing of SC from

800 nm to 1200 nm. This local minimum in spectra is formed as a result of the interference of SC radiation in

presence of anomalous GVD.

6. Acknowledgements

This work was supported by Russian Fund for Basic Research, grants № 11-02-00556-а, № 11-02-90421-Укр_ф_а.



7. References

[1] Kandidov V.P., Kosareva O.G., I.S. Golubtsov., W. Liu , S.L. Chin, "Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical madia (or supercontinuum generation)", Appl. Phys. B 77,149-165 (2003). [2] Zheltikov A. M., "Let there be white light: Supercontinuum generation by ultrashort laser pulses," Phys. Uspekhi 49, 605-628 (2006). [3] Alfano R.R., Shapiro L.S., "Observation of self-phase modulation and small-scale filaments in cristals and glasses", Phys. Rev. Lett. 24(11), 592-594 (1970). [4] Naudeau M.L. et al, "Observation of nonlinear optical phenomena in air and fused silica using a 100 GW , 1.54 μm source", Optics Express 14(13), 6194-6200 (2006). [5] Brodeur A., Chin S. L., "Band–gap dependence of the ultrafast white–light continuum", Physical Review Letters 1998, 80(20), 4406 (1998). [6] Brodeur A., Chin S. L. "Ultrafast white–light continuum generation and self–focusing in transparent condensed media", JOSA B 16(4), 637,(1999). [7] Naguro Ch.,Suda A., Kawano H., "Generation and characterization of ultrafast white-light continuum in condensed media", Appl. Opt. 41(18), 3735 (2002). [8] Gaeta A. L., "Catastrophic Collapse of Ultrashort Pulses", Phys. Rev. Lett. 84(16), 3582-3585 (2000). [9] Chin S.L., Brodeur A., Petit S.,et al., "Filamentation and supercontinuum generation during the propagation of powerful ultrashort laser pulses in optical media (white light laser)", J. Nonlinear Opt. Phys. Mater 8, 121-146 (1999). [10] Melnikov A.A., Misochko O.V., Kompanets V.O., Dobryakov A.L., Chekalin S.V.," Investigation of ultrafast processes in photoexcited bismuth by broadband probing in the wavelength range 0.4–0.9 μm ", JETP 111(3), 431-439 (2010). [11] Stelmaszczyk K., Rohwetter P., Fechner M., Queißer M., CzyŜewski A., Stacewicz T., Wöste L., "Cavity Ring-Down Absorption Spectrography based on filament-generated supercontinuum light", Optics Express 17(5), 3673 (2009). [12] Mejean G., Kasparian J., Yu J., Frey S., Salmon E., Ackermann R., Wolf J.P., Berge L., Skupin S., "UV-Supercontinuum generated by femtosecond pulse lamentation in air: Meter-range experiments versus numerical simulations", Appl. Phys. B 83,341-345 (2006). [13] Xu H.L., Chin S.L.“Femtosecond Laser Filamentation for Atmospheric Sensing”, Sensors11, 32 (2011). [14] Kasparian J., Wolf J.P. "Physics and applications of atmospheric nonlinear optics and filamentation", Optics Express 16, 466 (2008). [15] Brown D., Philbrick R., "Long-path supercontinuum absorption spectroscopy for measurement of atmospheric constituents", Optics Express 16(12), 8457 (2008) [16] Hartl I., Li X. D., Chudoba C., Ghanta R. K., Ko T. H., Ranka J. K., Windeler R. S., Fujimoto J. G., "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber", Optics Letters 26(9), 608-610 (2001). [17] Dormidonov A. E., Kandidov V. P., "Interference Model of Femtosecond Laser Pulse Conical Emission in Dispersive Medium",. Laser Physics 19(10), 1993(2009). [18] Govind P. Agrawal, [Nonlinear Fiber Optics 3rd Edition], New York: Academic Press, (2007). [19] Smetanina E.O., Dormidonov A.E., Kompanets V. O., " Supercontinuum conical emission accompanying filamentation of a femtosecond laser pulse in fused quartz", Journal of Optical Technology 77(7), 463-464 (2010).



spie proceedings [spie spie optical engineering + applications - san diego, california (sunday 21...

Documents