View
219
Download
3
Category
Preview:
Citation preview
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: aditya@tifr.res.in
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
J. A. Dharmadhikari et al.
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
Effect of group velocity dispersion
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
J. A. Dharmadhikari et al.
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
Effect of group velocity dispersion
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)
J. A. Dharmadhikari et al.
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.
References
1. A. Couairon, A. Mysyrowicz, Phys. Rep. 441, 47 (2007)
2. L. Berge, S. Skupin, R. Nuter, J. Kasparian, J.P. Wolf, Rep. Prog.
Phys. 70, 1633 (2007)
3. T. Brabec, F. Krauz, Rev. Mod. Phys. 72, 545 (2000)
4. C. Nagura, A. Suda, H. Kawano, M. Obara, K. Midorikawa,
Appl. Opt. 41, 3735 (2002)
5. A. Saliminia, S.L. Chin, R. Vallee, Opt. Express 13, 5731 (2005)
6. M.L. Naudeau, R.J. Law, T.S. Luk, T.R. Nelson, S.M. Cameron,
Opt. Express 14, 6194 (2006)
7. M. Durand, K. Lim, V. Jukna, E. McKee, M. Baudelet, A.
Houard, M. Richardson, A. Mysyrowicz, A. Couairon, Phys. Rev.
A 87, 043820 (2013)
8. J. Darginavicius, D. Majus, V. Jukna, N. Garejev, G. Valiulis, A.
Couairon, A. Dubietis, Opt. Express 21, 25210 (2013)
9. V.P. Kandidov, E.O. Smetanina, A.E. Dormidonov, V.O. Ko-
mpanets, S.V. Chekalin, JETP 113, 422 (2011)
10. S.V. Chekalin, V.O. Kompanets, E.O. Smetanina, V.P. Kandidov,
Quantum Electron. 43, 326 (2013)
11. K.D. Moll, A.L. Gaeta, Opt. Lett. 29, 995 (2004)
12. J. Liu, R. Li, Z. Xu, Phys. Rev. A 74, 043801 (2006)
13. L. Berge, S. Skupin, Phys. Rev. E 71, 065601 (2005)
14. D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas,
A. Matijosius, F. Bragheri, M. Porras, A. Piskarskas, P. Di Tra-
pani, Phys. Rev. E 74, 047603 (2006)
15. A. Mysyrowicz, A. Couairon, U. Keller, New J. Phys. 10, 025023
(2008)
16. M. Bradler, P. Baum, E. Riedle, Appl. Phys. B 97, 561 (2009)
17. E.O. Smetanina, V.O. Kompanets, S.V. Chekalin, A.E. Dormi-
donov, V.P. Kandidov, Opt. Lett. 38, 16 (2013)
18. E.O. Smetanina, V.O. Kompanets, S.V. Chekalin, V.P. Kandidov,
Quantum Electron. 42, 913 (2012)
19. E.O. Smetanina, V.O. Kompanets, S.V. Chekalin, V.P. Kandidov,
Quantum Electron. 42, 920 (2012)
20. F. Silva, D.R. Austin, A. Thai, M. Baudisch, M. Hemmer, D.
Faccio, A. Couairon, J. Biegert, Nat. Commun. 3, 807 (2012)
21. A. Couairon, J. Biegert, C.P. Hauri, W. Kornelis, F.W. Helbing,
U. Keller, A. Mysyrowicz, J. Mod. Opt. 53, 75 (2006)
22. M. Hemmer, M. Baudisch, A. Thai, A. Couairon, J. Biegert, Opt.
Express 21, 28095 (2013)
23. A.K. Dharmadhikari, F.A. Rajgara, D. Mathur, Appl. Phy. B 80,
61 (2005)
24. A.K. Dharmadhikari, F.A. Rajgara, D. Mathur, Appl. Phys. B 82,
575 (2006)
25. A.K. Dharmadhikari, K.M. Alti, J.A. Dharmadhikari, D. Mathur,
Phys. Rev. A 76, 033811 (2007)
26. A.K. Dharmadhikari, J.A. Dharmadhikari, D. Mathur, Appl.
Phys. B 94, 259 (2009)
27. S. Polyakov, F. Yoshino, G. Stegeman, J. Opt. Soc. Am. B 18,
1891 (2001)
28. A. Dubietis, E. Gaizauskas, G. Tamosauskas, P. Di Trapani, Phys.
Rev. Lett. 92, 253903 (2004)
29. S. Skupin, R. Nuter, L. Berge, Phys. Rev. A 74, 043813 (2006)
30. A.K. Dharmadhikari, F.A. Rajgara, N.C.S. Reddy, A.S. Sandhu,
D. Mathur, Opt. Express 12, 695 (2004)
31. Y.H. Chen, S. Varma, T.M. Antonsen, H.M. Milchberg, Phys.
Rev. Lett. 105, 215005 (2010)
32. F. Wise, P. Di Trapani, Opt. Photonics News 13, 28 (2002)
33. B. Shim, S.E. Schrauth, A.L. Gaeta, Opt. Express 19, 9118 (2011)
34. F.A. Jenkins, H.E. White, in Fundamentals of Optics, 4th Edition
(McGraw Hill, New York 2001)
35. M.J. Weber, Handbook of Optical Materials (CRC Press, Roca
Baton, 2003)
36. K. Dota, J.A. Dharmadhikari, D. Mathur, A.K. Dharmadhikari,
Appl. Phys B. 107, 703 (2012)
37. D. Faccio, A. Averchi, A. Lotti, M. Kolesik, J.V. Moloney, A.
Couairon, P. Di Trapani, Phys. Rev. A 78, 033825 (2008)
38. M. Kolesik, E.M. Wright, J.V. Moloney, Opt. Express 13, 10729
(2005)
39. E.O. Smetanina, V.O. Kompanets, A.E. Dormidonov, S.V. Che-
kalin, V.P. Kandidov, Laser Phys. Lett. 10, 105401 (2013)
40. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, G. Mourou, Opt.
Lett. 20, 73 (1995)
41. M.R. Junnarkar, Opt. Commun. 195, 273 (2001)
Effect of group velocity dispersion
123
Recommended