1 October 2001

Optics Communications 197 (2001) 475±480

www.elsevier.com/locate/optcom

Characteristics of the oscillatory spectrum due to only
induced-phase modulation in an argon-®lled hollow
waveguide accompanied with intense self-phase modulation
Naoki Karasawa a,*, Ryuji Morita a, Hidemi Shigekawa b, Mikio Yamashita aaDepartment of Applied Physics, Hokkaido University, Core Research for Evolutional Science and Technology, Japan Scienceand Technology Corporation, Kita-13, Nishi-8, Kita-Ku, Sapporo 060-8628, Japanb
Institute of Applied Physics, University of Tsukuba, Core Research for Evolutional Science and Technology, Japan Scienceand Technology Corporation, 1-1-1 Tennodai, Tsukuba 205-8573, Japan
Received 18 December 2000; received in revised form 18 July 2001; accepted 19 July 2001

Abstract
We experimentally observed the periodic oscillatory structure in the second-harmonic spectrum when we simultaneously propagated intense fundamental and second-harmonic pulses from a Ti:sapphire laser-ampli®er system in an
argon-®lled capillary ®ber. The periodic oscillatory structure in the second-harmonic spectrum is only due to the induced-phase modulation (IPM) which, in general, is hidden because of strong self-phase modulation. It is found that
this structure can be well explained by the interference of the electric ®eld components in the second-harmonic pulse.
These electric ®eld components are generated based on IPM by the fundamental pulse. The theoretical analysis of
the nonlinear pulse propagation agrees well with experimental results. In addition, the measured dependence of themodulated spectral behavior on the delay time, including the spectral feature, the oscillatory period and the range of thedelay time between two pulses, indicates the excellent agreement with the numerical analysis which includes the medium
and waveguide dispersion and the steepening eect. Ó 2001 Elsevier Science B.V. All rights reserved.
Keywords: Induced-phase modulation (IPM); Self-phase modulation (SPM); Capillary ®ber; Nonlinear pulse propagation

1. Introduction
When an intense optical pulse is propagated in
an optically nonlinear medium, self-phase modulation (SPM) occurs. The phase of the pulse is*Corresponding author. Present address: Department ofApplied Photonics System Technology, Chitose Institute of
Science and Technology, 758-65 Bibi, Chitose 066-8655, Japan.
Fax: +81-123-27-6063.
E-mail address: n-karasa@photon.chitose.ac.jp (N. Karasawa).

modulated because of the intensity-dependentchange of the refractive index of the medium, andthe spectrum of the pulse is broadened by chirp
arising from this temporal change of the phase. It is
well known that the periodic oscillatory structurein the spectrum arises from SPM [1]. This periodic
structure can be explained from the interference
between the intra-pulse electric ®elds with theidentical frequencies separated in time [2]. The
number of peaks in the structure gives the estimateof the phase shift induced by the pulse intensity. Recently, we have applied the induced-phase0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 5 6 - 0

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N. Karasawa et al. / Optics Communications 197 (2001) 475±480

modulation (IPM) eect [3,4] to generate ultrabroadband optical pulses in a fused-silica ®ber [5±7] and in a capillary ®ber ®lled with argon [8]. This
IPM is also known as cross-phase modulation(XPM) [1]. (However in Ref. [3], XPM is speci®cally used to represent the phase modulation andchange of emission spectrum caused by the interference between stimulated Raman scattering andSPM.) In the capillary ®ber experiments, both intense fundamental and second-harmonic pulses
from a Ti:sapphire laser-ampli®er system were copropagated. The intensity of both pulses was about
the same and the phase modulation of one pulse
was induced by the other pulse (IPM) in addition tothat due to its own pulse (SPM). As a result, thecoherent pulse with the bandwidth exceeding one
octave (300±1000 nm) has been generated [8],which is essential for the successful achievement of
monocycle pulses to be generated after chirp
compensation covering the entire frequency range.
However, the detailed mechanism of the complicated structure in the modulated spectrum in thecase that the IPM and SPM eects coexist in thealmost same magnitude has not been revealed, andits clari®cation is signi®cantly important for complete chirp compensation.
In this paper, we clarify through theoretical andexperimental observation the origin of the complicated oscillatory structure in the spectrum underintense induced- and self-phase modulations in an
argon-®lled capillary ®ber by controlling precisely
the delay time between two pulses. Furthermore, it
is shown that the observed IPM-induced spectral
behavior excellently agrees with the numerical results of the theoretical analysis.
2. Experiment
The experimental setup is shown in Fig. 1,
which is similar to the apparatus used by Karasawa et al. [8]. The output beam of a Ti:sapphirelaser-ampli®er system (Femtopower 1000; center
wavelength  790 nm; pulse width 32 fs; repetition
rate, 1 kHz; pulse energy, 1 mJ) was separated by abeam splitter (BS), which re¯ected 30% of pulse
energy. The transmitted beam was passed througha 0.5-mm-thick b-barium borate (BBO) crystal.

Fig. 1. Experimental setup. See text for de®nitions.

The delay of the generated second-harmonic pulse
was adjusted by a retrore¯ector (RR) and separated by a harmonic separator (HS). The polarization direction of the fundamental beam wasrotated by 90° using two periscopes (PS) to match
it with that of the second-harmonic beam. Neutral
density ®lters (ND) were used to adjust the intensities of both beams. Two beams were combinedby a dichroic mirror (DM) and focused at the input end of the ®ber (A), which was positioned
inside a chamber ®lled with an argon gas with apressure of 2.8 atm. The chamber had two 1-mmthick sapphire windows (S1 and S2 ). The capillary®ber was made of fused silica, and its length andinner diameters were 34 cm and 0.1 mm, respectively. To determine the delay time of the secondharmonic pulse relative to the fundamental pulse(Td2 ), the combined beam was redirected by a ¯ipmirror (B), passed through 1-mm-thick sapphire
plate (S3 ), and focused at a 10-lm-thick BBO
crystal (C) to generate the sum-frequency signal.
The optical distance between the ¯ip mirror (B)and the BBO crystal (C), and that between the ¯ipmirror (B) and the input end of the ®ber (A), were
set to be equal by inserting an extra sapphire plate(S3 ) and by taking into account of the dispersionof the argon and air [8]. The delay time that gave
the maximum sum-frequency signal corresponded
to the case for which both pulses met at the ®ber
input end (A) (Td2  0). When it was negative, thesecond-harmonic pulse entered the ®ber before thefundamental pulse did. Input pulse energies were
40.8 lJ for fundamental and 27 lJ for secondharmonic pulses and output pulse energies were
15.5 lJ for fundamental and 10.5 lJ for second-

N. Karasawa et al. / Optics Communications 197 (2001) 475±480

harmonic pulses, respectively. The temporal
widths of the input pulses were measured by atransient-grating frequency-resolved optical gating
apparatus [9]. They were found to be 32 and 72 fs
for fundamental and second-harmonic pulses, respectively. The peak powers for the pulses were1.28 GW for fundamental and 0.377 GW for second harmonic, and the center wavelengths were
790 nm for fundamental and 405 nm for secondharmonic pulses.
The spectra observed for the second-harmonic
pulse are shown in Fig. 2. In Fig. 2(a), the delay
time (Td2  207 fs) was selected such that bothpulses did not overlap inside a ®ber. In this case,
the IPM spectrum (solid curve) was almost identical to the SPM spectrum (dotted curve, measured
for the propagation of only the second-harmonic477

pulse). This showed the oscillatory structure with aperiod of 20 THz in the high frequency region.
When the delay time Td2 was increased (Fig. 2(b)±(d)), pulses began to overlap inside a ®ber, and a
new oscillatory structure with the shorter period of
10 THz in the low frequency region was observed
as indicated by arrows. As the overlap increased,
this new structure became more pronounced. For
the even larger delay time Td2 (Fig. 2(e)), the overlap between fundamental and second-harmonic
pulses became large such that the original SPM
spectrum shape was modi®ed considerably. Although the oscillatory structure was still present
in this case, the strong modi®cation of the main
spectrum by SPM made the identi®cation of IPM
oscillation dicult.

3. Comparison with theory
Since the dispersion length of a capillary ®ber ismuch longer than the nonlinear length in this case,the dispersion eect can be ignored in the ®rst
approximation. We have used the exact solution of
chirps for Gaussian input pulses for a dispersionless waveguide with a loss [10]. In Fig. 3, the instantaneous frequency dm2 ( dx2 =2p: nonlinear
chirp) for the second-harmonic pulse (at zl 
34 cm) is shown with the relative position of two
pulses for dierent delay times used in the experiments. The nonlinear chirp at the position zl ,dx2 zl ; T2  is given by
dx2 zl ; T2  
2n2 x2 zl z2eff 0s
sd2 exp s02 sd2 2 P2
cA T02 zl 2p 
1
exp 2g1 s01  g21 P1 pg1 erf s01  g1  d1 
d2
1
erf s01  g1 
exp 2g1 s01  g21 P1
d2
n
o
2
2
0
0
 exp s1  g1  d1   exp s1  g1   :Fig. 2. Experimental spectra where both fundamental and second-harmonic pulses were co-propagated in an argon-®lledcapillary ®ber with dierent delay times (a±e). Oscillatory
structure due to only IPM is indicated by arrows. In (a), thespectrum where only second-harmonic pulses were propagatedis shown by a dotted line.

1Here, the input envelope for thefundamentalpulsep
2(i  1) is given by A1 0; T1   P1 exp T12 = 2T01
and that of the second-harmonic
pulse
(i

2)
is
p
2
given by A2 0; T2   P2 exp T2 Td2 2 = 2T02


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N. Karasawa et al. / Optics Communications 197 (2001) 475±480

Fig. 3. Calculated instantaneous frequencies m t (solid line) andthe intensities I2x t (dotted line) of the second-harmonic pulse
and the intensities Ix at the ®ber output at dierent delay times.
Thin horizontal lines show the minimum instantaneous frequency created by SPM. The instantaneous frequencies and theintensities of second-harmonic pulses in the temporal region
where the instantaneous frequencies are created twice due toonly IPM are hatched by thin lines and thick lines, respectively.
Gaussian input pulses are assumed and the dispersion and thesteepening eects are neglected. See text for input parameters.

in the time coordinates Ti moving at group velocities vgi , where Pi is the peak power, T0i is the pulse
width (full-width-at-half-maximum-width divided
by 1.665), and Td2 is the delay time of the secondharmonic pulse relative to the fundamental pulse.
The parameters n2 , x2 , c, and A are the nonlinear
index of refraction (2:744  10 23 m2 /W [8]), thecenter angular frequency of the second-harmonic
pulse (2pc= 405 nm), the speed of light, andthe cross-section of the capillary ®ber (pa2 , a 
50 lm). z2eff  1 exp a2 zl =a2 is the eective

length of the ®ber for the second-harmonic pulse
with the loss a2  0:568 m 1 . Also, the normalized
parameters are de®ned as s0i  T2 =T0i , sd2  Td2 =T02 , di  zl d=T0i , and g1  a1 T01 =2d with a1  2:16
m 1 . d is the dierence of the inverse group velocities of pulses, and is given by d  1=vg2
1=vg1  150 fs/m. The ®rst term of Eq. (1) shows
the chirp arising from SPM. The second and thirdterms arise from IPM. The second term arises
from the phase change of the second-harmonic
pulse induced by the shape change of the fundamental pulse due to its loss during propagation.
The third term arises mainly from the imperfectoverlap of both pulses at ®ber ends [10]. The second term is insigni®cant for the present experimental condition because the loss is small.
As shown in Fig. 3, the spectral components forthe second-harmonic pulse whose frequency islower than the center frequency (740 THz) arecreated by both SPM and IPM. However, it isfound that the spectral components less than theminimum frequency created by SPM (720 THz,
indicated by thin horizontal lines in Fig. 3) arecreated only by IPM from the intense fundamental
pulse (which is described by the third term of Eq.
(1)) and these frequencies are created twice at different times (hatched by thin lines in Fig. 3). The
novel oscillatory structure in the spectrum appears
because of the interference between these twoelectric ®eld components with the identical instantaneous frequencies by IPM. When the delay issuciently negative (Fig. 3(a)), the intensity of thesecond-harmonic pulse I2x t in this region is essentially zero, so that the interference is not observed. When the delay times become less negative(Fig. 3(b)±(d)), the intensity in this region (hatchedby thick lines) becomes stronger and the interference patterns can be observed. When the delay
becomes even less negative (Fig. 3(e)), the chirps
due to SPM and IPM in the region from 675 to 775
THz are becoming similar in magnitude because of
the large intensity of the pulse and the interference
pattern shown here becomes dicult to identify.
In Fig. 4, the spectra calculated by numericallysolving the nonlinear ®ber propagation equation
at the same delay times as in the experiment are
shown. In these calculations, the dispersion of thegas medium and the waveguide (including loss

N. Karasawa et al. / Optics Communications 197 (2001) 475±480479pulse shapes. In calculations, Gaussian input pulses were assumed. On the other hand, small deviations from Gaussian pulses were observed inexperiments. Moreover, small side pulses wereobserved for fundamental pulses. These should be
responsible for the dierences of detailed shapes ofsecond-harmonic spectra between calculations andexperiments.
4. Conditions for observing IPM-only oscillation

Fig. 4. Numerically calculated spectra at the same delay time
for the same input parameters as shown in Fig. 3. In calculations, the gas as well as the waveguide dispersions and thesteepening eect were taken into account.

dispersion) was rigorously included, and the steepening eect was taken into account [11]. As shownin Fig. 4, the clear oscillatory structure in the low
frequency region, as observed in the experimentfor cases (b)±(d), was obtained. The spectral feature, the oscillatory period (10 THz) and thedelay-time range show the excellent agreementwith the experimental results. This clearly demonstrates that Eq. (1) is valid in the current experimental situation. At the largest negative delay
time (a), the oscillatory structure as in (b)±(d) was
not observed. At the smallest negative delay time(e), even though the oscillatory structure was observed, the original SPM structure was stronglymodi®ed, as clearly indicated by the suppressionof the peak at 765 THz. The detailed shapes ofspectra did not agree quantitatively with experimental ones mainly due to dierences of initial

From the previous results of comparisons between experiment and theory, the conditions toobserve IPM-only-induced spectral oscillations
can be determined. First, to observe the visible
fringes, we need to have sucient intensity for thesecond-harmonic pulse at which chirp from IPM ispresent. This limits the upper bound of the temporal separation between the two pulses. For cases
treated here, the loss is small (gi ' 0:1), thus thesecond term in Eq. (1) can be neglected. The IPM
negative chirp is given by the last term in Eq.
(1). This term becomes most negative near s01 
d1  0 or equivalently, s02  d2 . The intensityof the second-harmonic pulse is proportional to2exp s02 sd2  . By requiring its intensity at
s02  d2 to be more than 10 4 of its peak value toobserve the fringe, we obtain the condition, sd2 >4:22 or Td2 > 181 fs. This implies that the
separation between two pulses should be less than
181 fs to observe the fringe. This agrees well withthe experiments where the fringes were observed at
Td2 > 160 fs. Second, to observe the clear oscillation due to only IPM, the SPM chirp should not
be in¯uenced by the IPM chirp too much. The
chirp arising from SPM is given by the ®rst term in
Eq. (1), which is positive for s02 > sd2 . When theseparation of both pulses becomes small, this positive chirp from SPM is partly canceled by the negative chirp from IPM, which is again given by thelast term in Eq. (1). For the small loss, z2eff ' zl
and the ®rst term in the bracket in Eq. (1) becomes2
P2 s02 sd2  exp s02 sd2  . This term hasp themaximum value of 0:429P2 at s02 sd2  1= 2 andhas the half-maximum value of 0:214P2 at s02
sd2  1:36. If we consider the case where the positive SPM chirp is canceled by the negative IPM

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N. Karasawa et al. / Optics Communications 197 (2001) 475±480

chirp at this half-maximum time, we can determine sd2 from the condition 0:214P2  P1 =d2  
2exp s01  d1  . The last relationship is obtained
from the ®rst and last terms in Eq. (1) where s01 
s02 T02 =T01 . In the present case, using s02  1:36  sd2
in the above equation, we obtain sd2  3:26 or
Td2  140 fs. Thus, if the temporal separation
between two pulses becomes less than 140 fs, theSPM chirp is strongly in¯uenced by the IPM, andhence the SPM spectrum will be strongly modi®ed,
which makes the identi®cation of the interference
pattern due to only IPM dicult. In the experimental results, when the separation time was lessthan about 130 fs, the original SPM spectrum wasmodi®ed so strongly that clear fringes by IPM
were not observed. This agrees well with the above
estimate.
Thus from the theoretical estimate, the fringe
pattern due to only from IPM can be clearly identi®ed when 3:26 > sd2 > 4:22 or 140 > Td2 >181 fs, which agrees well with the experimental
results.

5. Conclusion
The novel oscillatory structure in second-harmonic spectrum due to only IPM was observed
when we co-propagated fundamental and secondharmonic pulses in an argon-®lled single mode
capillary ®ber. The analytic solutions without dispersion indicate de®nitely that it is indeed arisingfrom the interference between the identical frequency components created only by IPM, and it
was con®rmed by the numerical calculations including the rigorous dispersion and the steepeningeects. By using the simple analyses, we can esti-

mate the range of the delay time between two
pulses, in which we can clearly identify the interference pattern due to only IPM. Conversely, it
is expected that by measuring in this range, thequantitative information regarding the pulse shape,
the phase dispersion and the strength of the IPM
eect in the nonlinear medium can be extracted.
Moreover, these results suggest that the capillary®ber is an ideal nonlinear medium to clarify thecomplicated dynamic mechanism of the IPM chirp
because it provides the small dispersion, the small
nonlinear refractive index, the small loss and thewell-controlled beam diameters.
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