662

J. Opt. Soc. Am. B / Vol. 16, No. 4 / April 1999

Karasawa et al.

Theory of ultrabroadband optical pulse generation
by induced phase modulation
in a gas-filled hollow waveguide
Naoki Karasawa, Ryuji Morita, and Lin Xu
Department of Applied Physics, Hokkaido University, and Core Research for Evolutional Science and Technology,
Japan Science and Technology Corporation, Kita-13, Nishi-8, Kita-Ku, Sapporo, 060-8628 Japan

Hidemi Shigekawa
Institute of Materials Science, and Center for Tsukuba Advanced Research Alliance, University of Tsukuba,
and Core Research for Evolutional Science and Technology, Japan Science and Technology Corporation,
1-1-1 Tennodai, Tsukuba, 305-8573 Japan

Mikio Yamashita
Department of Applied Physics, Hokkaido University, and Core Research for Evolutional Science and Technology,
Japan Science and Technology Corporation, Kita-13, Nishi-8, Kita-Ku, Sapporo, 060-8628 Japan
Received September 21, 1998; revised manuscript received December 24, 1998
Generation of an ultrabroadband optical pulse with a fluent frequency dependency of the phase is importantfor creating a monocyclelike optical pulse and for shaping multiwavelength optical pulses. A previously proposed method that uses induced phase modulation between a femtosecond fundamental wave v 1 and its
second-harmonic wave v 2 5 2 v 1 in a fused-silica fiber is applied to a capillary fiber filled with noble gas.
Analytic results of chirps without dispersion but with loss in the fiber are shown, and the optimum conditions
relating to a delay time between two pulses and to input peak powers are found for fully covering the spectrum
between v 1 and v 2 . Furthermore, numerical calculations, including dispersion effects of the fundamental
and the second-harmonic waves from a Ti:sapphire laser-amplifier system with experimentally realizable parameters, are presented. These calculations show that it is possible to generate an ultrabroadband opticalpulse whose spectrum ranges from 300 to 900 THz (330 to 1000 nm) with quasi-linear chirp by this method.
© 1999 Optical Society of America [S0740-3224(99)01604-5]
OCIS codes: 060.4370, 060.7140, 320.5520, 320.5540, 320.7140.

1. INTRODUCTION
Recently, generation of an ultrabroadband optical pulsethat uses induced phase modulation (IPM) in an opticalglass fiber was proposed.1,2 In this method two or three
optical pulses, one fundamental and others generated bysecond-harmonic generation or optical parametric amplification, were propagated inside a single-mode silica fiberat the same time; IPM, as well as self-phase modulation(SPM), among the pulses was used to broaden the spectrum of each pulse. Subsequently the phase of the combined pulses with quasi-linear chirp was compensated forby a spatial phase modulator. It was predicted that amonocyclelike optical pulse could be generated with thismethod.
Fused-silica fibers have conventionally been used tobroaden the spectrum of an optical pulse through dispersive SPM, and one of the shortest pulses to date was generated by this method.3 Optical propagation loss of afused-silica fiber is negligibly small for small distances.
However, the damage threshold of fused-silica fiber isrelatively small, which makes applications requiring
0740-3224/99/040662-07$15.00

large energies difficult. The maximum energy that canbe allowed in this method is ;40 nJ for a 13-fs input
pulse.4
In 1995 a high-powered, ultrabroadband continuum optical pulse was generated by multichannel propagation ofa terawatt femtosecond near-infrared pulse in a long multimode capillary fiber filled with noble gas.5 However,the phases between frequency components of the generated optical wave were not constant in this method, sincemultichannel and multimode propagation was employed.
More recently, single-mode capillary fibers filled with
noble gases were used to broaden the spectrum of opticalpulses through dispersive SPM in single-channel
propagation,6,7 and sub-5-fs, high-powered pulses weregenerated with this technique.7 Because of the chemical
stability of the noble gases, the damage threshold is relatively high, and the maximum energy that can be allowed
is ;10 mJ. However, since the linear refractive index of
the noble gas (.1.0) is smaller than that of the glass usedin the capillary wall, the optical field is not completely
confined in the fiber during propagation. Hence loss al-© 1999 Optical Society of America

Karasawa et al.

Vol. 16, No. 4 / April 1999 / J. Opt. Soc. Am. B

ways occurs in this method. In addition the nonlinear refractive index and second-order dispersion of the noble
gas are smaller than those of fused silica by a factor ofthousands.
In this paper the analytic equations describing thechirp of each pulse in the case of two-pulse propagation ina single-mode capillary fiber are derived when dispersion
is neglected. By use of these equations, effects of a delay
time between two pulses (a femtosecond fundamental
wave and its second-harmonic wave) for spectrum broadening are examined, and the optimum delay time and the
input peak powers for fully covering the spectrum between two pulses are obtained. Furthermore, numericalcalculations including second- and third-order dispersion
terms are performed under experimentally feasible conditions. Since the second-harmonic wave as well as thefundamental wave are generated from one common femtosecond pulse, the carrier-phase difference between theirwaves is constant.8–11 This uniformity enables us to synthesize constructively the two spectrally broadened waves
at the fiber output. A compressed pulse width is evaluated after its phase is compensated for by use of a spatial
phase modulator.

2. CHIRPS WITHOUT DISPERSIONEFFECTS
Nonlinear chirps are considered in the case in which dispersion effects can be neglected. For a capillary fiberwith a 50-mm radius filled with argon at 3.3 atm (2.5
3 103 torr) and at 300 K, the dispersion length L d
5 T 0 2 / b 2 is 35.3 m for a fundamental wave (l 1 5 790
nm, pulse 1) of a Ti:sapphire laser-amplifier system when
its full width at half-maximum (FWHM, T FWHM) is 30 fs@ T 0 5 T FWHM /(2 Aln 2) for a Gaussian pulse]. For asecond-harmonic wave (l 2 5 395 nm, T FWHM 5 30 fs,
pulse 2), it is 2.46 m. As will be shown below, we consider mainly a short fiber length of ;0.3 m. Thus, forthis fiber length, it can be estimated that the dispersion
effect is small and not very significant. Also, since theexperimentally observed spectra6,7 of an argon-filled fiber
taken under conditions similar to those in this paper donot show significant asymmetry owing to self-steepening
effects, we neglect these terms in the following analysis.
We consider a case in which two optical pulses with different center frequencies and widths are copropagating ina capillary fiber with a length z l whose axis is parallel tothe z direction. For each pulse i (i 5 1, 2) the electricfield (linearly polarized in the x direction) can be written
as follows:

the inverse of the group velocity of pulse i. By use of aslowly varying envelope approximation and if dispersion
terms are neglected, A i (z, T i ) satisfies the following
equations12,13:]A1
a1
n 2v 15 2 A1 1 i@ f 11u A 1 u 2 1 2 f 12u A 2 u 2 # A 1 ,
]z
2
c
(2)

]A2
a2
n 2v 25 2 A2 1 i@ f 22u A 2 u 2 1 2 f 12u A 1 u 2 # A 2 ,
]z
2
c
(3)
where a i represents the loss, n 2 is the nonlinear refractive index, and c is the speed of light. f ij is the mode
overlap integral between transverse modes of pulses i and
j. Below these values are all assumed to be the inverse of
the core area of the fiber. The relative phase difference
between pulses ( f 01 2 f 02) does not enter into Eqs. (2)and (3). Thus the spectrum broadening due to IPM does
not depend on phase difference. However, it has some effects when two pulses are combined for pulse compression
as discussed in Section 6. With

S D

a iz
A i 5 AP i exp 2
U i ~ z, T i ! ,2where P i is the peak power of pulse i, the solutions of Eqs.
(2) and (3) are given by
U i ~ z, T i ! 5 exp@ i f i ~ z, T i !# U i ~ 0, T i ! .
Here the nonlinearly time-dependent phases f i are givenbyf 1~ z l , T 1 ! 5Fn 2v 1
f 11P 1 u U 1 u 2 z 1 eff 1 2 f 12P 2
c
3

E

zl

G

exp~ 2a 2 x ! u U 2 ~ 0, T 1 2 xd ! u 2 dx ,0(4)

f 2~ z l , T 2 ! 5Fn 2v 2
f 22P 2 u U 2 u 2 z 2 eff 1 2 f 12P 1
c
3

E

zl

G

exp~ 2a 1 x ! u U 1 ~ 0, T 2 1 xd ! u 2 dx ,0(5)where d is the difference between the inverses of thegroup velocities,
d 5 b 12 2 b 11 .

Ei ~ ri , t ! 5 1 / 2 x̂$ F i ~ x, y ! A i ~ z, t !
3 exp@ i ~ b 0i z 2 v i t 1 f 0i !# 1 c.c.% ,

663

(1)

where b 0i , v i , and f 0i are the wave number of the propagation mode, the center angular frequency, and the constant phase for a pulse i and c.c. represents the complex
conjugate. F i (x, y) represents the transverse distribution of the mode, and A i (z, t) represents the slowly varying envelope part of the field. For convenience two timecoordinates that move with the speed of a group velocity
for each pulse are defined as T i 5 t 2 b 1i z, where b 1i is

The first terms on the right-hand sides of Eqs. (4) and(5) show the time-dependent phase that is due to SPM.
Because of the loss, fiber length z l is reduced to effective
length z i eff :z i eff 5 @ 1 2 exp~ 2a i z l !# / a i .
The nonlinear chirp, which represents the instantaneous
frequency shift from the carrier frequency that is due tothe nonlinearly time-dependent phase, can be calculated
to be

664

J. Opt. Soc. Am. B / Vol. 16, No. 4 / April 1999

Karasawa et al.

]fi
dvi 5 2,]Ti
where the minus signs arise from the definition of theelectric field in Eq. (1). These quantities can be calculated if the pulse shapes are specified. In this section, toobtain the analytic formula, we assume the following
Gaussian input-pulses:

for v 2 [Fig. 1(b)] the pulse center is at time t d2
5 22.32. Also, T 01 5 T 02 is assumed for this figure.
In Eqs. (6) and (7) the first terms on the right-handsides are due to SPM, and the second and third terms are2
A 1 ~ 0, T 1 ! 5 AP 1 exp@ 2T 12 / ~ 2T 01!# ,2
A 2 ~ 0, T 2 ! 5 AP 2 exp@ 2~ T 2 2 T d2 ! 2 / ~ 2T 02!# ;T 0i is the pulse width, where the intensity becomes 1/e,
and T d2 is the delay time of pulse 2 with respect to pulse
1. The chirps are calculated to be

d v 1~ z l , T 1 !
5X

2n 2 v 1 z l z 1 eff
f 12
f 11t 1 exp~ 2t 12 ! P 1 1c
T 01 z l
d1
3 exp@ 22 h 2 ~ t 2 2 t d2 ! 1 h 2 2 # P 2 Ap h 2
3 $ erf~ t 2 2 t d2 2 h 2 ! 2 erf~ t 2 2 t d2 2 h 2 2 d 2 ! %
1

f 12

d1

exp@ 22 h 2 ~ t 2 2 t d2 ! 1 h 2 2 # P 2

3 $ exp@ 2~ t 2 2 t d2 2 h 2 2 d 2 ! 2 #C

2 exp@ 2~ t 2 2 t d2 2 h 2 ! 2 # % ,d v 2~ z l , T 2 !
5

(6)

X

2n 2 v 2 z l z 2 eff
f ~ t 8 2 t d2 ! exp@ 2~ t 28 2 t d2 ! 2 # P 2
c
T 02 z l 22 2
2

f 12

d2

exp~ 2 h 1 t 18 1 h 1 2 ! P 1 Ap h 1

3 $ erf~ t 18 1 h 1 1 d 1 ! 2 erf~ t 18 1 h 1 ! %
2

f 12

d2

exp~ 2 h 1 t 18 1 h 1 2 ! P 1 $ exp@ 2~ t 18 1 h 1 1 d 1 ! 2 #C

2 exp@ 2~ t 18 1 h 1 ! 2 # % .

(7)Here erf(x) 5 (2/Ap ) * 0x exp(2t2)dt is an error function,and the following quantities are defined as

t i 5 T 1 /T 0i ,t i8 5 T 2 /T 0i ,d i 5 z l d/T 0i ,h i 5 a i T 0i /2d.

t d2 5 T d2 /T 02 ,

(8)
(9)

By defining the walk-off length L wi as L wi
5 T 0i / u d u , 12,13 we can write d i 5 sgn(d)zl /Lwi and h i
5 sgn(d)aiLwi/2, where sgn(d) is 61 according to the signof d. Thus h i shows the estimate of the loss during the
propagation of pulse i for the walk-off length L wi .
In Fig. 1 chirps are shown as functions of normalized
time for the case in which v 1 , v 2 and the group velocityof pulse 1 is greater than that of pulse 2 (d . 0, in thenormal dispersion region). For Fig. 1 the delay time T d2
is chosen such that two pulses meet near the fiber exit
end. For v 1 [Fig. 1(a)] the pulse center is at time 0, andFig. 1. Calculated chirps versus normalized time for (a) d v 1 and(b) d v 2 when dispersion terms are neglected. Solid curves show
chirps from SPM, long- and short-dashed curves show chirps
from IPM, and dotted curves show the total chirps. Pulse centers are at 0 for (a) and at 22.32 for (b). In calculations v 1 is thefundamental wave and v 2 is the second-harmonic wave of a
Ti:sapphire laser system (l 1 5 790 nm). These are propagatedin an argon-filled (300 K, 3.3 atm) capillary fiber with a radius of
50 mm and a length of 29.2 cm. Also, P 1 5 P 2 5 1.29 GW, T 01
5 T 02 5 30/(2 Aln 2) fs (t 1 5 t 2 , t 18 5 t 28 ), n 2 5 3.234 3 10223
m2/W, and T d2 5 241.8 fs (T d2 . 2z l d) are used.

Karasawa et al.

due to IPM. Significantly, the second terms appear because of the loss during propagation.
Physically, these terms can be interpreted as follows.
The intense electric field of the pulse causes an increasein the refractive index of the medium owing to the thirdorder optical nonlinearity, and its refractive-index change
is proportional to the instantaneous intensity of the pulse.
The refractive-index change causes nonlinearly timedependent change in the phase of the pulse. In the caseof SPM, the temporal profile of this phase is proportional
to the pulse shape. Chirp is the negative of the time derivative of this phase, which yields a lower frequency shift
for the pulse leading edge and a higher frequency shift forthe pulse trailing edge [solid curves (SPM) in Fig. 1]. In
the case of IPM, the phase and the chirp depend on thegroup-velocity difference between pulses and on the fiber
length. If a delay time is chosen such that 2z l d , T d2
, 0 (pulse 2 leads), the center of pulse 1 passes the center of pulse 2 inside the fiber. For pulse 1, the effect ofpulse 2 on the phase is larger at the leading edge thanthat at the trailing edge as pulse 1 passes pulse 2, since
the intensity of pulse 2 decreases, owing to the loss during
propagation. This effect causes the higher frequency
shift [long-dashed curve (IPM 1) in Fig. 1(a)]. On theother hand, for pulse 2, the effect of pulse 1 on the phasebecomes larger at the trailing edge than that at the leading edge as pulse 2 is passed by pulse 1, owing to the lossof pulse 1. This effect causes the lower frequency shift[long-dashed curve (IPM 1) in Fig. 1(b)]. The third termsof Eqs. (6) and (7) are caused mainly by the imperfectoverlap between two pulses at one end of the fiber. For
T d2 . 0 (when the pulses meet near the entrance of thefiber), the trailing edge of pulse 1 interacts with the leading edge of pulse 2. This interaction causes an increaseof the phase only at the trailing edge of pulse 1 (the leading edge of pulse 2). Thus for pulse 1 (pulse 2), the phaseis an increasing (decreasing) function with respect totime. Thus pulse 1 (pulse 2) is frequency-shifted lower(higher) near the center of the pulse. For T d2 . 2z l d(when the pulses meet near the exit of the fiber), the leading edge of pulse 1 interacts with the trailing edge ofpulse 2, and pulse 1 is frequency-shifted higher near thecenter of the pulse [short-dashed curve (IPM 2) in Fig.
1(a)]. Also, pulse 2 is frequency-shifted lower near thecenter of the pulse [short-dashed curve (IPM 2) in Fig.
1(b)]. The total chirp for each wave is given by the sumof three terms in Eqs. (6) or (7), and its shape can be controlled by changing the delay time T d2 , as is shown Section 3.

Vol. 16, No. 4 / April 1999 / J. Opt. Soc. Am. B

become less than 1/e owing to the loss after the field
propagates over the length Ap L w2 , we may ignore thesecond term. By use of t 2 5 T 01t 1 /T 02 5 T 01 /( A2T 02) inthe last equation and Eqs. (8) and (9), the optimum delay
is given by
T d2 5 T 01 / A2 2 T 02h 2 2 z l d.

We set v 1 as the fundamental wave and v 2 5 2 v 1 as thesecond-harmonic wave. To cover the spectrum fully between v 1 and v 2 , we desire to maximize d v 1 and 2d v 2( d v 2 , 0). From Eq. (6) and Fig. 1 it can be seen thatthe first (SPM) term becomes maximum at t 1 5 1/A2.
For not very small d 2 , the second term becomes maximum approximately at t 2 5 t d2 1 h 2 1 d 2 /2, and thethird term becomes maximum approximately at t 2 5 t d2
1 h 2 1 d 2 . If Ap h 2 , 1, i.e., the optical field does not(10)

From Eq. (7), similar considerations give, for the condition of maximizing 2d v 2 ,T d2 5 T 02 / A2 2 T 01h 1 2 z l d.

(11)

A comparison of Eqs. (10) and (11) shows that, for small

h 1 and h 2 , the two conditions above can be satisfied if

both pulses have the same widths (T 01 5 T 02). In particular, if there are no losses ( h 1 5 h 2 5 0), these conditions can be satisfied exactly. The condition
T d2 5 T 02 / A2 2 z l d(12)specifies that both pulses meet almost at the fiber exitend.

4. REQUIRED PEAK POWERS FOR
SPECTRUM BROADENING
Here we evaluate the pulse peak powers that can broaden
the spectrum of each pulse such that spectra of these twopulses overlap. For this purpose we use the results ofSection 3; i.e., the widths of both pulses are assumed to beequal, and the optimal delay time is given by Eq. (10).
Also, loss terms in the chirp equations [the second termson the right-hand sides in Eqs. (6) and (7)] are assumed tobe small and can be ignored. However, since these terms
give chirps in the favored directions (a higher frequency
shift for d v 1 and a lower frequency shift for d v 2 ), the required peak powers obtained will be slightly larger thanthe practical experimental powers. Moreover, inspectionof Eqs. (6) and (7) shows that the second and third terms
become smaller than the first terms if d 1 or d 2 becomeslarger than ;3. This characteristic implies that the fiber
length must be smaller than the same value to have theeffect of IPM comparable with that of SPM. Here we set
the values of d i to be 3, i.e., the fiber length z l is equal to3L w1 ( 5 3L w2 ). Under these assumptions, the maximum chirps are given as follows:~ d v 1 ! max 5

2n 2 v 1 3
cpa
1

3. DETERMINATION OF A DELAY TIMEFOR SPECTRUM BROADENING

665~ 2d v 2 ! max 5

2

1
3

d

1

2

1
3

0.4289z 1 eff
zl

P1

J

(13)

J

(14)

exp@ 2h 2 ~ h 2 1 6 !# P 2 ,4n 2 v 1 3
cpa

H

d

H

0.4289z 2 effzl

P2

exp@ 2h 1 ~ h 1 1 6 !# P 1 .

Here 0.4289 is the maximum of the SPM termt i exp(2ti2), and f ij is replaced by 1/p a 2 (a is the capillaryradius). For fully covering the spectrum between v 1 andv 2 , we set the sum of Eqs. (13) and (14) to be v 1 . This
modification gives

666

J. Opt. Soc. Am. B / Vol. 16, No. 4 / April 1999Hz 1 eff
2
1 exp@ 2h 1 ~ h 1 1 6 !# P 1
zl
3

0.4289Karasawa et al.

Jn 2 5 3.234 3 10223 m2/W,

H

l 1 5 790 nm,

Jz 2 eff
1
1 0.8578
1 exp@ 2h 2 ~ h 2 1 6 !# P 2zl3

a 1 /2 5 1.0807 m ,h 1 5 0.1051,

cpa2 d.
52n 2 3(15)

5. ARGON CASE
It is assumed that we employ an amplified Ti:sapphire laser system with a fundamental wavelength of 790 nm anda pulse width of 30 fs. The pulse width of the secondharmonic wave is assumed to be 30 fs also. To calculate
dispersion and loss characteristics of the fiber propagation mode in an argon-filled capillary fiber, we assume
that the dominant propagation mode is EH 11 , whose loss
is the smallest. Then the phase constant b and the fieldattenuation constant a/2 are given by14

a
2

5

S

DG

2.405 2 l 2

n2 1 1F

2p

S

n 0 2 2 1 pT 0

2a 3 ~ n 2 2 1 ! 1/2

n 02 1 2 p 0T

DS
1/2

11

12

.

D

n 0 2 2 1 pT 0 21/2
n 02 1 2 p 0T

S

5.15 3 105

4.09 3 1017

4.32 3 1023

1

l 06

1

l 02
l 08

5 1.85 3 10213 s/m.
To satisfy the condition d i 5 z l d/T 0i 5 3, we need tohave z l 5 29.2 cm; for this fiber length the loss for thefundamental wave is 47% and that of the secondharmonic wave is 15%. Also, z 1 eff 5 21.7 cm, and z 2 eff
5 27.0 cm.
The optimal delay time from Eq. (10) is given by
T d2 /T 01 5 1/A2 2 d i 2 h 2 5 22.3192,
T d2 5 241.8 fs.
From Eq. (15) the minimum powers required for fully
covering the spectrum between the fundamental and thesecond-harmonic waves are given by

(16)

,

where n 0 is the linear refractive index under standard
conditions (p 0 5 1 atm, T 0 5 273.15 K) and is given by
the dispersion formula16
n 0 2 2 1 5 5.547 3 1024 1 1

d 5 b 12 2 b 11

where P i is measured in gigawatts. From this equation,
if we set P 1 5 P 2 , the required minimum powers are P 1
5 P 2 5 1.29 GW.

Here l is the wavelength in the gas medium; n is the ratioof the linear refractive indices of the external medium tothat of the internal media and is assumed to be 1.45. To
obtain the group velocity, the group-velocity dispersion,and the third-order dispersion, it is necessary to calculate
the derivatives of the phase constant with respect to the
angular frequency. To include the effect of medium dispersion, pressure, and temperature, l in Eq. (16) is replaced by 2 p c/( v n), where the linear refractive index ofgas n depends on pressure p (atm; 1 atm5735.6 torr) andtemperature T (K) as follows:15
n5 2

h 2 5 0.0263,

0.298P 1 1 0.480P 2 5 1,

2p
1 2.405l 2
12
,
l
2 2pa

S D

a 2 /2 5 0.2702 m21,

5 3.338737 3 10209 s/m 2 3.338552 3 10209 s/m

In Section 5 the practical values of P 1 and P 2 will be calculated for a capillary fiber filled with argon.

b5

l 2 5 395 nm,
21

1

D

4.19 3 1011
l 04,

where l 0 (in angstroms) is the vacuum wavelength.
For argon at T 5 300 K, p 5 3.3 atm, and a
5 50 m m, we obtain the following parameters (n 2 from
Ref. 6):

6. NUMERICAL CALCULATIONS
To verify the above results with the dispersion terms, numerical calculations are performed. Here the programused in Ref. 1 is slightly modified to include the secondand the third-order dispersion terms of argon. The calculated values of the second- and the third-order dispersion terms (b 2i and b 3i ) are as follows:b 21 5 9.18 fs2/m,b 22 5 132 fs2/m,

b 31 5 91.9 fs3/m,

b 32 5 44.1 fs3/m.

Note that the second-order dispersion term for the fundamental wave (9.18 fs2/m) is much smaller than thevalue (40.3 fs2/m) for a 5 80 m m with the same pressure
and temperature (cf. Ref. 7).
In the numerical calculations all the parameters shownin Section 5 are used except for peak powers and a delay
time. Figure 2 shows the calculated spectrum with T d2
5 241.8 fs and P 1 5 P 2 5 1.29 GW. Figure 3 shows thecalculated spectrum with T d2 5 0 fs and P 1 5 P 2
5 1.29 GW. In these calculations the constant carrier
phases are set to be f 01 5 0 and f 02 5 2p /2. It was
found that these values do not influence the spectrum except for the small interference part in which the highestfrequency edge of fundamental-wave spectrum and thelowest-frequency edge of the second-harmonic-wave spectrum overlap.2 Figure 4 shows the compressed pulse
whose spectrum is shown in Fig. 2. This compression is
performed by compensating for the phase of the fiberoutput composed pulse by the 256-channel spatial phasemodulator according to Ref. 1. The calculated pulse du-Karasawa et al.

Vol. 16, No. 4 / April 1999 / J. Opt. Soc. Am. B

667

ration is 1.53 fs. In this method17 each spectral component of the output pulse from the fiber is projected to adifferent channel of the spatial phase modulator by a first
spectral–spatial converter and a lens. Then the phase ateach channel is adjusted independently to make it essentially the same at every channel for pulse compression.
After this adjustment these spectral components are recombined by a second spectral–spatial converter and alens to form a final pulse. Gratings have been used forspectral–spatial converters. However, these may notwork for ultrabroadband pulses treated here owing to theoverlap of different diffraction orders.18 Prisms can beused here, since added nonlinear material dispersion canbe compensated for by the spatial phase modulator.19

Fig. 4. Calculated intensity versus time of the compressed pulse
after its phase is compensated for by the spatial phase modulator. Other parameters are the same as those of Fig. 2.

7. DISCUSSIONS

Fig. 2. Calculated intensity, phase, and group delay versus frequency of the fundamental and second-harmonic waves after
propagation through an argon-filled capillary fiber under thecondition P 1 5 P 2 5 1.29 GW, T d2 5 241.8 fs.

Figure 2 shows that the spectra of fundamental andsecond-harmonic waves are just overlapping at peak powers of 1.29 GW each, as is expected from the evaluation of
the required peak powers in Section 5. By comparing
Figs. 2 and 3, we can see the effect of a delay time on the
spectrum broadening. For Fig. 2 the delay time is chosensuch that both pulses meet near the fiber exit. In thiscase the IPM effect tends to cover the spectrum fully between fundamental and second-harmonic waves. For
Fig. 3, on the other hand, the delay is chosen such thatboth pulses meet at the fiber entrance. In this case theIPM effect tends to broaden the spectrum in opposite directions, and not much power is left around the center.
The maximum power that does not damage the medium may be determined by use of the self-focusing
effect.7 For bulk argon, the maximum power may beevaluated to be
P c 5 l 2 /2p n 2 5

H

3.1 GW~ l 5 790 nm!

0.77 GW~ l 5 395 nm!

,

which is smaller than the 1 GW of power employed for thesecond-harmonic wave. However, a recent theoretical estimation has shown that for a hollow waveguide the damage threshold is ;5 times larger than the bulk medium.20
Thus it is likely that pulses with 1.29-GW of peak power
can be used without damage problems.

8. CONCLUSION

Fig. 3. Calculated intensity, phase, and group delay versus frequency of the fundamental and second-harmonic waves after
propagation through an argon-filled capillary fiber under thecondition P 1 5 P 2 5 1.29 GW, T d2 5 0 fs.

Ultrabroadband optical pulse generation that uses SPM
and IPM in a gas-filled hollow waveguide has been analyzed. Analytic results of chirps without dispersion but
with loss in the fiber have been derived for the Gaussian
input pulses. By use of these equations, the optimum
conditions relating to a delay time between two pulses
and to input peak powers have been found for fully cover-668

J. Opt. Soc. Am. B / Vol. 16, No. 4 / April 1999

ing the spectrum between v 1 and v 2 . As an example, forthe experimentally realizable parameters that use thefundamental and the second-harmonic waves from a
Ti:sapphire laser-amplifier system in an argon-filled capillary fiber, the optimum delay time and the required
pulse peak powers have been obtained. These results
have been verified by numerical calculations, including
dispersion effects. These calculations show that by thismethod it is in fact possible to generate an ultrabroadband quasi-linear chirped optical pulse that fully covers
the spectrum between these two waves.
This technique for the generation of quasi-linearchirped, white-spectral pulses can be applied to opticalpulse monocyclization as well as to the realization of independently synthesized, synchronized multicolor femtosecond beams as a new optical source, such as a femtosecond photon factory. Experimental verification of thistheory is under way in our laboratory.

Karasawa et al.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

ACKNOWLEDGMENTS
The authors thank Hiroyasu Sone of Muroran Institute ofTechnology for providing the nonlinear fiber propagation
program that is used in this study with some modifications.

14.
15.
16.
17.

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