effect of group velocity dispersion on supercontinuum generation and filamentation in transparent...

Effect of group velocity dispersion on supercontinuum generationand filamentation in transparent solids

Jayashree A. Dharmadhikari • Rucha A. Deshpande •

Arpita Nath • Krithika Dota • Deepak Mathur •

Aditya K. Dharmadhikari

Received: 1 November 2013 / Accepted: 3 May 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract We experimentally investigate the spectral

extent and spectral profile of the supercontinuum (SC)

generated in transparent solids: barium fluoride, calcium

fluoride, and fused silica upon irradiation by intense fem-

tosecond-long pulses of 800, 1,380, and 2,200 nm light.

These wavelengths correspond to the normal and anoma-

lous group velocity dispersion (GVD) regimes in fused

silica calcium fluoride and barium fluoride. We observe an

isolated (anti-Stokes) wing on the blue side most promi-

nently in fused silica but also in CaF2. The SC conversion

efficiency is measured for the long wavelengths used in our

experiments. We also present results on filamentation in

BaF2 in the anomalous GVD regime, including visualiza-

tion of focusing–refocusing events within the crystal; the

size of a single filament is also determined. The 15-photon

absorption cross section in BaF2 is deduced to be

6.5 9 10-190 cm30 W-15 s-1.

1 Introduction

Propagation of ultrashort laser pulses through transparent

media continues to be a subject of contemporary interest

[1, 2]. The high peak powers associated with ultrashort

pulses give rise to a host of nonlinear optical effects such as

supercontinuum (SC) generation and filamentation in such

media. Supercontinuum generation has been a source of

coherent broadband light for a variety of applications, such

as pump–probe spectroscopy, white light microscopy,

carrier-envelope phase stabilization, optical frequency

comb [3], and as a seed for optical parametric amplification

of ultrashort pulses. Considerable work has been reported

in recent years on SC generation in solid media as a

function of laser energy, polarization, pulse duration, and

focusing conditions, but almost all experimental work has

been carried out at 800 nm central wavelength and under

conditions of normal group velocity dispersion (GVD) [1].

In the normal dispersion regime, it has been shown that it is

the ratio of the material’s band gap to the incident photon

energy that determines the extent of anti-Stokes broadening

[4]. However, there are few reports that have investigated

filamentation and supercontinuum generation at longer

wavelengths [5–11]. Some authors have also performed

numerical simulations of the propagation of longer wave-

length laser pulses in fused silica (SiO2) [11–13]. Mea-

surements in fused silica have shown that the threshold

for filamentation increases with wavelength in the

1,200–2,400 nm range, in agreement with the fact that

critical power increases with the square of the wavelength

[7]. A broadband emission is formed, which can lead to

efficient pulse self-compression [5, 14, 15]. In bulk media,

detailed investigation of supercontinuum generation in

several host crystals has been carried out for various

experimental parameters: in sapphire, a tunable source

(1,100–1,600 nm) has been used to produce a broader

spectrum compared to that obtained with 800 nm pump-

ing [16]. Recent reports on supercontinuum emission

from filaments at longer wavelengths confirm this obser-

vation while also showing that filamentation in the

anomalous dispersion regime gives rise to a broader con-

tinuum, spanning more than 3 octaves that could ideally

J. A. Dharmadhikari � K. Dota � D. Mathur

Centre for Atomic and Molecular Physics, Manipal University,

Manipal 576 104, India

R. A. Deshpande � A. Nath � K. Dota � D. Mathur �A. K. Dharmadhikari (&)

Tata Institute of Fundamental Research, 1 Homi Bhabha Road,

Mumbai 400 005, India

e-mail: [email protected]

123

Appl. Phys. B

DOI 10.1007/s00340-014-5857-3

self-compress to a single-cycle pulse [7, 17–20]. Pulse self-

compression in both normal dispersion regime [21] and

anomalous dispersion regime using 3.1 lm wavelength is

reported [22].

In earlier work from our laboratory, we have demon-

strated highly efficient white light generation in barium

fluoride (BaF2) and have reported the results of systematic

measurements of the spectral extent of the supercontinuum

under different experimental conditions, such as laser

energy, polarization, pulse duration, and external focusing

in normal dispersion regime [23]. At high-incident powers

(*3,000 Pcr), we estimated the time-varying change in

electron densities [24]. Some measure of control on the

onset of filamentation within a large BaF2 crystal has been

demonstrated [25], and the six-photon absorption cross

section has been reported [26].

Filamentation is a visual manifestation of the propaga-

tion of ultrashort, high-intensity pulses through matter.

Such propagation causes self-focusing because of the

optical Kerr effect that, in turn, enables beams to propagate

over distances much larger than the normal Rayleigh range.

However, the self-focusing cannot continue indefinitely,

and various mechanisms set into arrest the self-focusing

action. In condensed media, such effects include diffrac-

tion, group velocity dispersion, self-phase modulation,

pulse self-steepening, higher-order nonlinear effects (v(5)

effects), and defocusing induced by plasma formation; all

these factors contribute to the overall propagation dynam-

ics. An alternate mechanism for the arrest of self-focusing

has also been proposed that is based on multiphoton

absorption [27].

Propagation over long distances occurs because of a

chain of focusing–refocusing events, a consequence of the

dynamic competition between the optical Kerr effect and

multiphoton ionization. Following this idea, Dubietis

et al. [28] have proposed a model that does not require

plasma generation, without ruling out the possibility that

electrons are generated. Recently, Skupin et al. [29] have

also theoretically investigated the self-guiding effect in

condensed media. At very high laser powers, multiple

filamentation occurs, which scales with laser power [30].

The factors that affect the filament length and plasma

density are the initial pulse intensity, initial pulse dura-

tion, and beam convergence [31]. It has been shown that

both plasma- and GVD-induced pulse splitting lead to

pulse self-shortening, with a single compressed pulse

component emerging incidentally at some distance, fol-

lowed by rapid deterioration [6]. As a result, it becomes

difficult to achieve a controlled and stable filamentation

regime with laser pulse features preserved over extended

distances. On the other hand, in the anomalous dispersion

regime, one expects the self-guided pulses not to spread

in time and space over long distances [32]. Some

theoretical and experimental efforts have recently begun

to investigate this regime. Filamentation in air at 3.1 lm

(negative GVD regime) has been simulated by Shim et al.

[33] showing that air is not a suitable medium in this

regime. In condensed medium, using a 30-cm-long BK7

glass filamentation in anomalous GVD regime was

investigated [11].

Here, we report on experimental investigations of su-

percontinuum generation at three different incident wave-

lengths (in the infrared region) in transparent solids:

barium fluoride, calcium fluoride, and fused silica. These

wavelengths correspond to normal and anomalous group

velocity dispersion (GVD) regimes. The SC conversion

efficiency is measured for the long wavelengths used in our

experiments. We also investigate filamentation in BaF2 in

the anomalous GVD regime.

2 Experimental details

The experimental setup used by us to study the filamen-

tation is shown schematically in Fig. 1. We used a Ti–

sapphire laser (800 nm wavelength, 1 kHz repetition rate),

with beam diameter 1 cm, 4 mJ energy, and 35 fs pulse

duration in our experiments. The beam from this laser was

used to pump an optical parametric amplifier (OPA) that

generates wavelengths over the range 1.1–2.5 lm.

Care was taken to block residual 800 nm light by using

an RG850 filter. A pair of dielectric mirrors enabled sep-

aration of signal and idler wavelengths. We measured the

pulse duration of these IR beams using a homemade

autocorrelator. The pulse duration at 1.3 lm was *56 fs

and that at 2.2 lm was *64 fs. The laser beam from our

OPA was focused using a 30-cm lens onto the sample:

fused silica, CaF2, and BaF2. Each sample was 15 mm in

length. The generated supercontinuum (SC) was then

characterized using three different spectrometers. Note

here that we have collected only the axial part of the su-

percontinuum spectrum.

The spectral extent of the supercontinuum emerging

from each sample was measured using a combination of a

lens and a spectrometer: we used three spectrometers to

cover the entire spectral range of interest in the present

experiments: spectrometer I (Ocean Optics USB 4000;

range 350–1,100 nm), spectrometer II (Aventes NIR256-

2.5; range 1,100–2,500 nm), and spectrometer III (Ocean

Optics USB 2000; range 200–870 nm). For spectrometers I

and III, the spectra were averaged over 50 shots and in case

of spectrometer II, it was averaged over 200 shots. We

adjusted the incident laser energy at each wavelength to

ensure that all our measurements of the supercontinuum

were taken in the single-filament regime and close to the

threshold for SC generation. Supercontinuum conversion

J. A. Dharmadhikari et al.

123

efficiency was estimated by measuring the ratio of SC

energy (in the range 400–1,100 nm) to the incident energy.

In order to study filamentation in detail, we used a BaF2

crystal. This enabled us to visualize the filaments by the

multiphoton absorption-induced emission in the blue

region, as has been described in earlier work from our

laboratory [24]. A digital camera (Camera 2: Nikon D200

equipped with a 28–200 mm AF Nikon lens) enabled us to

capture the focusing–defocusing events within 15-mm-long

crystal. The camera was positioned transverse to the laser

propagation direction. To measure the size of filaments,

another CCD camera (Camera 1—JVC model TK-

C1480E) was used, which was positioned along the

direction of laser beam propagation, coupled with a long-

working-distance objective lens (Mitutoyo, M Plan Apo

20). Neutral density filters and color glass filters (VG 6)

were used to avoid saturation of the CCD camera due to

laser light. The pixel size was calibrated with reference to a

slit whose width was 250 lm; this enabled us to quantify

the sizes of filament images.

3 Results and discussion

3.1 Group velocity dispersion (GVD) calculations

In order to carry out the experiments in different GVD

regimes, we require prior knowledge of GVD values for

each sample at specific wavelengths. In the following, we

describe how we obtained estimates of the GVD values for

various samples. GVD is defined by the following

equation:

GVD ¼ k3=ð2pc2Þ d2n

dk2

� �ð1Þ

The form of the dispersion curve is a material property

and is readily deduced using the Sellmeier equation valid in

the transparency region. The general form of this equation

is [34]:

n2 kð Þ ¼ 1þ B1k2

k2 � C1

þ B2k2

k2 � C2

þ B3k2

k2 � C3

þ � � � ; ð2Þ

where the coefficients for BaF2, CaF2, and fused silica are

obtained from [35].

In Fig. 2, we plot the GVD values calculated using the

above formula for fused silica, BaF2, and CaF2; the

numerical values are tabulated in Table 1 for specific

wavelengths used in our measurements. In case of fused

silica, the wavelength at which the GVD value is zero is

1.27 lm, whereas for BaF2, this wavelength is 1.92 lm

Fig. 1 Schematic

representation of the

experimental setup. M1 and M2

are the steering and reflecting

mirrors, respectively. NDF is

the neutral density filter. L1 and

L2 are the focusing and

collimating lenses. Integrating

sphere (IS) measures the energy

of white light. The three

different spectrometers are

described in the text

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-100

-80

-60

-40

-20

0

20

40

60

80

100

GV

D (f

s2 /mm

)

Wavelength (µm)

BaF2

CaF2

Fused SiO2

Fig. 2 Group velocity dispersion curve as a function of wavelength

for fused silica, BaF2, and CaF2

Effect of group velocity dispersion

123

and for CaF2, it is 1.54 lm. We have also calculated the Pcr

values for different wavelengths [36].

3.2 Supercontinuum generation in fused silica

Earlier work in fused silica has shown an isolated wing in

the visible part of the supercontinuum. This wing is

observed to be blueshifted as the wavelength of the inci-

dent pulse is increased [7, 17]. In Fig. 3, we show how the

supercontinuum spectrum varies with pump wavelength.

At 800 nm pump wavelength, the GVD value for fused

silica is positive: the extent of the SC in this case covers the

range 410–2,080 nm, spanning more than 2 octaves. When

the incident wavelength is increased to 1,380 nm, the GVD

value becomes -10 fs2 mm-1 and the SC extent

(410–2,080 nm) is nearly the same as that for 800 nm. As

the incident wavelength is further increased to 2,200 nm,

the GVD value becomes more negative (GVD =

-107 fs2 mm-1) and we see a clear extension of the SC

spectrum. The enhancement extends the SC to cover the

range 370–2,300 nm. We observe a single isolated wing

(370–650 nm) with a peak at 530 nm. Our results are in

consonance with those reported very recently [7, 17].

We have also measured the SC conversion efficiency

over the wavelength range 400–1,100 nm using a cali-

brated Si photodiode connected to an integrating sphere

(3A-IS-V1-ROHS from OPHIR). The efficiency in case of

1,380 nm incident wavelength is determined by us to be

25 %; it becomes 8 % for 2,200 nm incident wavelength,

both values being measured at an input energy of 160 lJ.

Thus, our observations indicate that the supercontinuum

efficiency reduces as the incident wavelength is increased.

3.3 Supercontinuum generation in BaF2

Supercontinuum generation was also explored in BaF2 at

longer wavelengths. In Fig. 4, we show the variation of the

SC with incident wavelength. For both 800 nm and

Table 1 GVD (fs2 mm-1) and Pcr (MW) for different materials at

three wavelengths

Wavelength (nm) Fused silica BaF2 CaF2

GVD Pcr GVD Pcr GVD Pcr

800 36 2.5 37 3 27 4

1,380 -10 7.4 16 9 6 12

2,200 -107 19 -10 23 -33 30

1E-3

0.01

0.1

1

400 500 600 700 800 900 10001E-3

0.01

0.1

1

1E-3

0.01

0.1

1

1200 1400 1600 1800 2000 2200 2400

Inte

nsity

(ar

b. u

nits

)

Wavelength (nm)

800 nm incident

4 1380 nm incident

2200 nm incident

Spectrometer 2

2200 nm

14 Spectrometer 1

1380 nm

4

800 nm

µJ

µJ

µJ

Fig. 3 Supercontinuum generation in fused silica with three different incident laser wavelengths: 40 Pcr at 800 nm; 10 Pcr at 1,380 nm; and 10

Pcr at 2,200 nm. All the incident spectra are attenuated


123

1,380 nm incident wavelengths, the GVD value for BaF2 is

positive: the extent of the SC in case of 800 nm pump

spans the range 320–1,980 nm (obtained using spectrom-

eters II and III), while that for 1,380 nm incident wave-

length is 330–2,240 nm, spanning more than 2 octaves. It

may be noted that there is a dip in SC generation from

750–900 nm. This dip is due to the presence of a highly

reflecting dielectric mirror at 800 nm. This allows us to

observe the weak visible radiation. As the pump wave-

length changes to 2,200 nm, we access the negative GVD

region (GVD is -10 fs2 mm-1) and we see a marked

extension in the SC spectrum. The SC spectrum now

extends up to 2,350 nm. Note here that since GVD value is

less negative, we do not observe an isolated anti-Stoke

shifted wing on the blue side, as was observed in the case

of fused silica.

The SC conversion efficiency is measured to be 32 % at

1,380 nm incident wavelength and 13 % for 2,200 nm

incident wavelength, with both values being determined at

an input energy of 160 lJ.

3.4 Supercontinuum generation in CaF2

Figure 5 shows the SC spectra as a function of different

pump wavelengths for CaF2. For both 800 and 1,380 nm

pump wavelengths, the GVD value for CaF2 is positive: the

extent of the SC (measured using spectrometers II and III)

in these cases covers the range 300–2,000 nm (in the case

of 800 nm pumping) and 320–2,200 nm (for 1,380 nm

pumping); both spectra span more than 2 octaves. It may be

noted that there is a dip in SC generation from

750–900 nm. This dip is due to the presence of a highly

reflecting dielectric mirror at 800 nm. This allows us to

observe the weak visible radiation. As the pump wave-

length changes to 2,200 nm, we access the negative GVD

region (GVD is -33 fs2 mm-1) and we see a marked

extension of the SC spectrum. It now covers the range

340–2,450 nm. Note that even though the GVD value is -

33 fs2 mm-1, we still observe a shift in the axial compo-

nent of conical emission toward shorter wavelength, as was

noted for fused silica. But the extent in the visible part of

the spectrum is broader (370–850 nm) compared to that

observed in fused silica (370–650 nm).

The SC conversion efficiency in the range 400–1,100 nm

is measured to be 32 % at 1,380 nm incident wavelength and

14 % for 2,200 nm incident wavelength, both at an input

energy of 160 lJ. Recent work in CaF2 has shown spectral

broadening from 450 to 2,500 nm in a 6-mm-long crystal

using 2 lm incident wavelength [8].

We note that of all the three samples we have studied,

the extent of the SC spectrum is largest in the case of CaF2.

Moreover, this is accompanied by high conversion effi-

ciency in the visible region for both 1,380 and 2,200 nm

incident wavelengths. As discussed earlier, in normal dis-

persion regime, it has been shown by Midorikawa and

coworkers [4] that it is the ratio of material’s bandgap

energy to the incident photon energy that determines the

extent of anti-Stokes broadening that is obtained.

400 500 600 700 800 900 10001E-3

0.01

0.1

11E-3

0.01

0.1

11E-3

0.01

0.1

1

1200 1400 1600 1800 2000 2200 2400

5 J

2200 nm

1380 nmIn

tens

ity (

arb.

uni

ts)

Wavelength (nm)

800 nm

(a)

(b)

Spectrometer 2 14 J(c) Spectrometer 1

4 J

µ

µ

µ

Fig. 4 Supercontinuum

generation in BaF2 with incident

laser wavelengths: 33 Pcr at

800 nm; 8 Pcr at 1,380 nm; and

10 Pcr at 2,200 nm


123

In Table 2, we compare the ratio of material’s band gap

to the incident laser photon energy with the anti-Stokes

width (measured at full width at one tenth maximum) in the

anomalous dispersion regime. From the table, it is seen that

for GVD values less than -33 fs2 mm-1, the anti-Stoke

width increases with bandgap energy as well as with the

ratio of band gap to photon energy. Note here that as the

GVD value decreases from -10 to -107 fs2 mm-1, we see

a reduction in the anti-Stokes width in fused silica, in

agreement with earlier measurements [7].

Earlier measurements in fused silica in the anomalous

GVD regime (at 1.5 lm central wavelength) showed a

broad maximum around 600 nm, with the spectral extent

covering the range 400–950 nm [5, 6]. Filamentation in

fused silica in the anomalous dispersion regime is observed

to give rise to an extreme blueshifted continuum peak in

the visible region, even though the filament is formed by

near-IR pulses [36]. The blue-side peak has been identified

as an axial component of the conical emission, as indi-

cated by a three-wave mixing model [7]. Furthermore,

anomalous dispersion in SC generation in fused silica has

been shown to lead to the formation of an isolated anti-

Stokes wing (ASW) that is located in the visible region of

the SC; this isolated ASW is formed by the interference of

the SC light field encountering anomalous group velocity

dispersion [17].

The highly asymmetric features of SC spectra are due to

odd-order dispersion terms and are interpreted in terms of

spontaneous formation of stationary conical waves in a

dispersive medium [14]. If the group velocity of the

X-wave is significantly different from the group velocity of

the incident laser pulse, a strongly blueshifted peak would

be expected in the pulse spectrum [37]. The scattering of

input pulse by the material waves constituting a nonlinear

response gives rise to SC generation [38]. In this case, the

dispersion properties of the medium are crucial ingredients

in the SC formation process.

3.5 Visualization of filamentation in BaF2

The band gap of BaF2 is 9 eV. Irradiation by intense

800 nm laser light gives rise to six-photon absorption-

induced emission [24]. We have shown earlier that the

fluorescence peaks at 330 nm and extends toward 450 nm.

Also, the blue fluorescence is a direct mapping of the

intensity within the filament [26] and, thus, enables the

visualization of the propagating beam undergoing focus-

ing–refocusing cycles when the power exceeds the critical

power for self-focusing, Pcr. The number of cycles that are

visualized depends on the peak power of the input pulse.

0.01

0.1

1

400 500 600 700 800 900 1000

0.01

0.1

1

1200 1400 1600 1800 2000 2200 2400

1E-3

0.01

0.1

1

(b)

(a)

Inte

nsity

(ar

b. u

nits

)

Wavelength (nm)

5

10

Spectrometer 1(c)

2200 nm

1380 nm

800 nm

14 Spectrometer 2 µJ

µJ

µJ

Fig. 5 Supercontinuum

generation in CaF2 at incident

laser wavelengths: 30 Pcr at

800 nm; 15 Pcr at 1,380 nm;

and 7 Pcr at 2,200 nm

Table 2 Variation of the anti-Stokes width with material properties

Band gap

(eV)

Ratio of bandgap/

photon energy

GVD fs2

mm-1Anti-Stokes

width (nm)

7.5 8.3 -10 213

7.5 13.3 -107 104

9.1 16 -10 252

10.2 18 -33 284


123

Other effects, such as diffraction, group velocity disper-

sion, self-phase modulation, and pulse self-steepening, also

contribute to the self-guiding process. In the present series

of experiments, we probe the effect of GVD by directly

visualizing the filaments in the BaF2 crystal at longer

wavelengths.

With 1,380 nm wavelength incident light, one requires

11-photon absorption for the blue fluorescence to be

observed. Note here that at this wavelength, the GVD value

is still positive (normal dispersion regime). In Fig. 6, the

filament image shows the focusing–refocusing cycles at

different values of incident laser power with corresponding

intensities of the blue fluorescence. Our observation is that

there is no significant difference in the filament focusing–

refocusing cycles compared to what we observed in our

earlier measurements with 800 nm light. At this incident

energy, the SC conversion efficiency is measured to be

10 %. The choice of energy values in these measurements

is such that only a single filament is visible in the propa-

gation direction even though we observe focusing–refo-

cusing events.

By changing the incident wavelength from 1.3 to 2.2 lm

(Fig. 7), we now access the anomalous GVD regime and we

observe focusing–refocusing cycles in top two panels,

whereas with increase in energy from 45 to 90 lJ, refocus-

ing events merge followed by extended fluorescence, clearly

demonstrating the role of GVD in the dynamics. In the case

of 2.2 lm light, there is 16-photon absorption that gives rise

to the corresponding blue fluorescence shown in Fig. 7.

Earlier measurements in BK7 glass have shown that

length of filament is larger in anomalous GVD (-25 fs2 -

mm-1) regime compared to the case of normal GVD

regime [11]. Also, they had shown that the separation

between refocusing events is significantly larger in anom-

alous GVD regime. The mechanism responsible for these

observations was attributed to transfer of energy into the

collapse region even after plasma arresting the collapse

resulting in the formation of extended filaments before the

beam defocuses. Theoretical and experimental work on

spectral transformation and spatiotemporal distribution of

ultrashort laser pulses during filamentation in fused silica

has been reported previously [10, 13, 39]. The formation of

light bullets was first observed in a femtosecond laser pulse

in the anomalous group velocity dispersion regime at a

wavelength of 1,800 nm. The filament start distance is

smaller for a fixed ratio of P/Pcr in the anomalous GVD

regime, than in the normal GVD regime in fused silica.

3.6 Estimation of multiphoton absorption cross section

in BaF2

By imaging the filament in the transverse direction, we

have measured the filament radius (Lmin) inside the BaF2

crystal at 2.0 lm wavelength. As noted above, this wave-

length accesses the anomalous GVD regime. We measure

the filament radius to be 8.5 lm, a value that is almost

double the value obtained at 800 nm [26]. We obtain the

estimates of peak intensities (Imax) and electron densities

(ne) by considering diffraction, the Kerr effect, and ioni-

zation responses, as described below.

Following the treatment detailed in [1, 2, 26, 40], we use

the following equations for estimating the peak intensity,

electron density, and the 15-photon absorption cross sec-

tion from our experimental observation of filament radius:

Imax ¼qmax

2qcn0n2

þ ð1:22kÞ2

8pn0n2L2min

; ð3Þ

qmax ffi tpqntWðImaxÞ; ð4Þ

Lmin ¼ 2n0

k20n2Imax

� �1=2

; ð5Þ

where k is the incident wavelength, tp is the pulse duration,

n0 is the refractive index, n2 is the nonlinear index

0.0 0.4 0.8 1.20

40

80

120

160

Inte

nsity

(ar

b. u

nits

)

Distance (cm)

18

0.0 0.4 0.8 1.2 1.60

40

80

120

160

20030

Inte

nsity

(ar

b. u

nits

)

Distance (cm)

Laser propagation direction

µJ

µJ

Fig. 6 Visualization of filamentation in BaF2 at 1.3 lm (GVD value

of 16 fs2 mm-1) incident wavelength showing focusing–refocusing

cycles at different values of incident laser power. The right panel

shows the image and the left panel shows the corresponding

intensities of blue fluorescence along the length of the crystal. The

upper panel shows the images for 18 lJ (35 Pcr) incident energy and

the lower panel for 30 lJ (60 Pcr) energy. The laser beam is incident

from the right side of the images


123

coefficient of BaF2 [41], qc is the critical plasma density,

and W(Imax) is the photoionization rate in the multiphoton

(MPI) regime such that W(Imax) = rkImaxk , where rk is the

15-photon absorption cross section. The expression for Lmin

assumes a Gaussian beam profile [1].

Thus, by using the above expressions along with the

measured size of the single filament (radius 8.5 lm in our

experiments), we obtain the estimates of peak intensity

(Imax) and electron density (qmax) within the crystal to be

2.7 9 1013 W cm-2 and 1.7 9 1020 cm-3, respectively.

Using these values, we estimate the 15-photon absorption

cross section to be 6.5 9 10-190 cm30 W-15 s-1. We also

estimate the cross section, using Keldysh’s theory in dense

media [2], to be 7.2 9 10-187 cm30 W-15 s-1. These val-

ues are higher than those computed using our experimental

measurements of filament radius. We note that in case of

fused silica, the 6-photon absorption cross section com-

puted using Keldysh’s theory was found to be four orders

higher than experimentally deduced values.

4 Summary

We have experimentally investigated the spectral extent as

well as the spectral profile of SC in transparent solids such as

fused silica, barium fluoride, and calcium fluoride at three

different wavelengths: 800, 1,380, and 2,200 nm. These

wavelengths correspond to normal and anomalous group

velocity dispersion (GVD) regimes in fused silica, calcium

fluoride, and barium fluoride. The SC spectral profile is

markedly different in the negative GVD regime compared to

normal and zero GVD regimes. A distinct isolated blue-side

continuum is produced whose width narrows as the GVD

value becomes more negative. We also observed a reduction

in SC efficiency in the region 400–1,100 nm when the

incident wavelength is increased. We probed, by direct

visualization, filamentation in BaF2 in the anomalous GVD

regime. The 15-photon absorption cross sections have been

estimated. We believe that these values will be of utility in

numerically simulating the experimental results.

0.0 0.5 1.00

100

200

0.0 0.5 1.00

100

200

0.0 0.5 1.00

100

200

26 J

Distance (cm)

90 J

Inte

nsity

(ar

b.un

its)

45 J

Laser propagation

direction

µ

µ

µ

Fig. 7 Visualization of

filamentation in BaF2 at 2.2 lm

(GVD value of -10 fs2 mm-1)

incident wavelength. The image

and the corresponding cross

section of the intensity profile

for 26 lJ (18 Pcr) incident

energy is shown in the upper

panel; 45 lJ (30 Pcr) incident

energy is shown in the middle

panel and for 90 lJ (60 Pcr)

energy is shown in the lower

panel (0 cm corresponds to the

exit face of the crystal)


123

Acknowledgments The Department of Science and Technology is

thanked for assistance to JAD under the Women Scientists Scheme

and to DM for the J C Bose National Fellowship. Rucha Deshpande

was a project student from Fergusson College, Pune.

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effect of group velocity dispersion on supercontinuum generation and filamentation in transparent...

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