Slowing down of solitons by intrapulse Raman
scattering in fibers with frequency cutoff
Alexey V. Yulin and Dmitry V. Skryabin
Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, UK
Received May 15, 2006; accepted August 10, 2006;
posted August 22, 2006 (Doc. ID 70955); published October 11, 2006
A method for transforming fast solitons into slow ones in bandgap fibers is proposed. The approach is based
on the deceleration of the solitons by intrapulse Raman scattering, which can be achieved for fiber modes
having a cutoff frequency. We develop a comprehensive theory for the soliton slowdown and elucidate how
the fiber losses introduce a fundamental minimum for the soliton velocity. © 2006 Optical Society of America
OCIS code: 060.5530.
Methods of reduction of the group velocity of light
pulses have recently attracted significant theoretical
and experimental efforts.1Slow light regimes can be
achieved via use of the intrinsic material resonances
as is done, e.g., in the electromagnetically induced
transparency1or Raman and Brillouin scattering
based schemes.2,3Light can also be slowed down in
specially designed waveguides, which in a certain
spectral region support guiding modes with small
group velocities. A classical optical system of this
type is the fiber Bragg grating.4,5More recently, slow
light has been demonstrated in planar photonic crys-
tal waveguides6,7and predicted in photonic crystal fi-
bers and waveguides with metal cladding.8–10
Instead of working with dispersive pulses one can
consider the use of solitons in slow-light settings.
Solitons are expected to bring an obvious advantage
of suppressed group-velocity dispersion, which allows
one to work with narrower pulses and thereby in-
creases the system bandwidth. Slow solitons have
been considered, e.g., in multilevel atomic systems11
and in fiber Bragg gratings.4,5The concept of zero-
velocity cutoff solitons in axially uniform photonic
bandgap fibers, directly relevant to what we are con-
sidering below, has also been introduced.10The cutoff
solitons are similar to the gap solitons existing in the
fiber Bragg gratings,4in the sense that the frequen-
cies below the cutoff are forbidden for linear propaga-
tion but are accessible for nonlinear pulses. A large
impedance mismatch between the slow modes and
fast external radiation is a common problem prevent-
ing achievement of acceptable coupling efficiency to
slow-light regimes. In particular, no practical way of
excitation of the cutoff solitons has been suggested so
In planar photonic crystal waveguides the excita-
tion problem has been addressed by specially de-
signed termination of the photonic crystal at the cou-
pling interface.6Another approach that has been put
forward in the context of fiber gratings is to pump the
waveguide at the fast frequency and then use Raman
effect to generate the slow or zero-velocity Stokes
signal.12A similar scheme has been recently realized
with four-wave mixing in fibers.13The methods of
Refs. 12 and 13 are applicable for relatively long
pulses. Pulses in the picosecond to femtosecond range
in silica fibers start to experience intrapulse Raman
scattering, which does not lead to generation of new
spectral components but amplifies the red part of the
pulse spectrum with simultaneous depletion of its
blue part. In the soliton regime this process gradu-
ally shifts the spectral maximum of the pulse toward
the redder frequencies without destroying it.14
The central idea of this work is that the fast soli-
tons can, by means of the Raman soliton self-
frequency shift, be gradually transformed to the cut-
off solitons having small group velocities. The cutoff
frequency typically corresponds to the frequency
minimum,10therefore a shift of the soliton frequency
beyond the cutoff is not possible and the soliton
freezes in the spectral proximity of the point of the
zero group velocity. Below we provide theoretical
foundations for this effect and confirm its feasibility
To describe the evolution of the field in the axially
uniform photonic bandgap fiber in the spectral range
including the cutoff frequency, we use the slow-
varying amplitude approach and write the equation
for the complex amplitude E of the field E??,??e−i?0?,
where ?0is the cutoff frequency corresponding to the
i??E = − ?? ˜2??
2+ ? ˜4??
4+ ... ?E − ???1 − ???E?2+ Q˜?E
Note that in our case it is not convenient to use
propagation coordinate ? as the evolution variable be-
cause near the cutoff frequency the waveguide sup-
ports waves with both positive and negative group
and phase velocities. Here ? ˜nare the dispersion coef-
ficients, ? is the nonlinearity parameter, ?ais the lin-
ear loss in the waveguide, and Q˜is the amplitude of
the Raman oscillator. Without nonlinear and dissipa-
tive terms Eq. (1) has the solution exp?−i?˜?+ik˜??, and
?−1?n+1? ˜2nk˜2n. The dispersion operator includes only
even derivatives because forward and backward di-
rections must be equivalent, i.e., ?˜?k˜?=?˜?−k˜?. Typical
OPTICS LETTERS / Vol. 31, No. 21 / November 1, 2006
0146-9592/06/213092-3/$15.00© 2006 Optical Society of America
dependence of ?˜versus k˜is of the parabolic type with
velocity dispersion, giving bright solitons for self-
focusing nonlinearity ???0?. The group velocity ?k˜?˜is
zero for k˜=0.8–10
Q˜is governed by the equation for the classical os-
cillator driven by the intensity of the optical field:
2?˜?0, which corresponds to the anomalous group-
?Rand ?Rare the decay rate and frequency of the Ra-
man oscillator. ? is the dimensionless Raman cou-
pling and the factor ?1−?? is introduced into Eq. (1)
for normalization convenience.
We define dimensionless time t as t=??Rand di-
=1/??R/?2? ˜2?. We also normalize ?E?2and Q˜on ?/?R,
so that ?A?2=?E?2?/?Rand Q=Q˜?/?R. The fibers pro-
posed in Refs. 8–10 are yet to be fabricated, and
quantitative estimates for the physical scales derived
and used below should be viewed as a rough guide-
line only. For silica glass ?R
For example, choosing the cutoff wavelength to be
1.5 ?m and using the parameters of the bandgap fi-
ber discussed in Ref. 10, we find that the core diam-
eter should be D?0.5 ?m, ? ˜2?25 m2/s and the nor-
nonlinear parameter ? we use the fact that the cutoff
mode is the nonpropagating mode bouncing between
the lateral walls of the cladding and therefore its
resonant frequency can be estimated as ?0??c/nD,
where n is the core refractive index. This estimate
works well for the fibers discussed in Refs. 8 and 10.
If n=n0+n2??E?2, then the nonlinear shift of the cut-
off frequency is given by −?c2n2??E?2/Dn0
tions discussed below vary from propagating to
standing waves, therefore it is convenient for us to
measure ?E?2in the units of the linear energy density,
i.e., J/m. This, together with n2measured in m2/W,14
determines the expression for the scaling coefficient:
?=c/n01/S, where S is the modal area. Thus we have
?1015m/J/s and scaling for the energy L?R/?
The resulting dimensionless equations used below
−1?16 fs and ?R
2. The solu-
i?tA = − ??2?x
4+ ... ?A − ?1 − ???A?2A − QA − i?A,
+ Q = ??A?2,
We assume that the time derivatives of Q can be
treated perturbatively and solve Eq. (4) by iterations.
In the leading order we find Q=??A?2+..., and in the
Q = ??A?2+ ???t?A?2+ ... .
An approximate soliton solution of Eq. (3) detuned
arbitrarily far from the cutoff frequency is given by
A = F?
U?vst − x?
F??? = U sech???.
Equation (6) satisfies Eq. (3) with Q=??A?2, ?=0, pro-
vided that all the derivatives of F higher than the
second are neglected. Here U is the soliton parameter
defining its amplitude, width, and the nonlinear fre-
quency shift. ?sis the soliton wavenumber. Defining
v=?k? and dispersion d=?k
=v?ks?, and ds=d?ks??0.
We will describe soliton dynamics using adiabatic
equations for the soliton parameters, which are most
easily derivable from the laws of evolution of the en-
ergy W and momentum M
N?−1?n+1?2nk2n, we have the group velocity
2?, so that ?s=??ks?, vs
?A?xA*− A*?xA?dx. ?7?
Losses destroy conservation of W,
?tW = − 2?W,
and the Raman effect violates conservation of M,
?tM = −?
Using Eqs. (5), (6), and (9) and assuming adiabatic
dependence of kson t, we demonstrate that
Solving Eq. (8), we have W=W0e−2?t, where W0
=2U?dsis the initial soliton energy; dsand vsare
functions of ks, and hence Eq. (10) is the nonlinear
equation for ks, which can always be solved numeri-
cally. In the simplest case of the parabolic dispersion
?=k2/2, which is a good approximation for the disper-
sion curves calculated in Ref. 10, we have ks=vsand
ds=1, giving us a fully explicit solution for ks: ks
=ks0exp???e−8?t−1?/?8???, where ks0 is the initial
wavenumber. Using vs=ks, we have for the velocity
Thus the minimal group velocity, vmin, that can be
reached by the soliton in infinite time is vmin
=vs0exp?−?/?8???, where vs0is the initial soliton ve-
locity. In the idealized limit of a lossless fiber, ?=0,
the minimal possible velocity is zero. Physically,
8??1 − e−8?t??,
November 1, 2006 / Vol. 31, No. 21 / OPTICS LETTERS
??0 broadens the soliton and therefore slows down Download full-text
and finally cancels the Raman-induced soliton self-
frequency shift, thereby preventing the soliton from
reaching the cutoff frequency.
lim?→0vs=vs0exp?−?t?, and by integrating the latter
equation into t we find the time evolution of the soli-
ton coordinate: xs=vs0?1−e−?t?/?. Thus, for ?=0, there
is a distance
beyond which the soliton cannot penetrate.
To verify our analytical results we performed a se-
ries of numerical simulations of Eqs. (3) and (4). Our
modeling unambiguously demonstrates that indeed
the fast solitons do adiabatically transform to the
slow cutoff solitons. Figure 1(a) shows the evolution
of the spatial spectra of the soliton in a lossless fiber.
The spectrum preserves its shape, but the soliton
wavenumber is shifting toward k=0, corresponding
to zero group velocity. The velocity of the soliton as a
function of time is shown in Fig. 1(b) for three values
of W. One can see that the time required to slow and
stop the soliton increases as W decreases, in full
agreement with Eq. (11).
Losses in the proximity of the cutoff are expected to
be much higher than the typical ?10 dB/km losses in
the middle of the bandgap of the currently existing
photonic crystal fibers. ?=10−7in our model approxi-
mately corresponds to 300 dB/km, which is the num-
ber we are taking to illustrate the effect of losses on
the soliton dynamics. Soliton velocity versus time for
different values of ? is plotted in Fig. 2(a). One can
see that the finite losses prevent the soliton from
reaching the zero group velocity and that larger
losses imply smaller time delays achievable in our
schemes. Figure 2(b) shows how the soliton velocity
varies with the propagation distance in the fiber. The
zero loss line unambiguously confirms the existence
of the maximal penetration depth given by Eq. (12).
In summary, we have demonstrated analytically
and numerically how fast solitons can be adiabati-
cally slowed down by intrapulse Raman scattering in
fibers with a frequency cutoff. We also found that an
unavoidable presence of linear losses prevents the
soliton slowdown to zero speeds and introduces a fi-
nite theoretically achievable minimum of soliton ve-
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spectra of the soliton as a function of t. (b) Numerically
(dashed curves) and analytically (circles) computed soliton
velocity as a function of time for three different energies.
The parameters of the system are ?=0, ?=0.77, ?2=1/2,
and ?=0.1. Dimensionless energy W=0.1 corresponds to
?2 nJ and to input pulse duration ?0.5 ps. vs=1 corre-
sponds to the velocity ?0.15c and t=106to time ?1.2
(Color online) (a) Numerically computed spatial
Fig. 2. (Color online) (a) Soliton velocity vsversus time for
different values of loss parameter ? for W0=0.17. The other
parameters are the same as for Fig. 1. vs=1 corresponds to
physical velocity ?0.15c and t=106to time ?1.2?10−8s.
(b) Soliton velocity vsversus the soliton coordinate xs. xs
=105corresponds to the distance ?6 cm. The solid and
dashed curves in (a) and (b) are obtained from numerical
modeling of Eqs. (3) and (4), and the points are calculated
from Eq. (11). The agreement between the two is perfect.
OPTICS LETTERS / Vol. 31, No. 21 / November 1, 2006