Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser

Abstract

We present a detailed analytical self-consistent theory based on wave kinetic equations that describes generation spectrum and output power of a Raman fiber laser (RFL). It is shown both theoretically and experimentally that the quasi-degenerate four-wave mixing (FWM) between different longitudinal modes is the main broadening mechanism in the one-stage RFL at high powers. The shape and power dependence of the intracavity Stokes wave spectrum are in excellent quantitative agreement with predictions of the theory. FWM-induced stochasticity of the amplitude and the phase of each of the ˜106 longitudinal modes generated in the RFL cavity is an example of a light-wave turbulence in a fiber.

Full text

Four-wave-mixing-induced turbulent spectral
broadening in a long Raman fiber laser
Sergey A. Babin, Dmitriy V. Churkin,
*
Arsen E. Ismagulov, Sergey I. Kablukov, and Evgeny V. Podivilov
Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences,
Novosibirsk, 630090 Russia
*
Corresponding author: dimkins@iae.nsk.su
Received October 27, 2006; revised February 26, 2007; accepted March 2, 2007;
posted March 15, 2007 (Doc. ID 76531); published July 19, 2007
We present a detailed analytical self-consistent theory based on wave kinetic equations that describes genera-
tion spectrum and output power of a Raman fiber laser (RFL). It is shown both theoretically and experimen-
tally that the quasi-degenerate four-wave mixing (FWM) between different longitudinal modes is the main
broadening mechanism in the one-stage RFL at high powers. The shape and power dependence of the intrac-
avity Stokes wave spectrum are in excellent quantitative agreement with predictions of the theory. FWM-
induced stochasticity of the amplitude and the phase of each of the ⬃10
6
longitudinal modes generated in the
RFL cavity is an example of a light-wave turbulence in a fiber. © 2007 Optical Society of America
OCIS codes: 140.3550, 030.7060, 190.4380
.
1. INTRODUCTION
Raman fiber lasers (RFLs) are attractive light sources
providing almost any wavelength in the near-infrared re-
gion [1]. They are widely used in WDM telecommunica-
tion systems as multiwavelength signal and pump
sources for distributed Raman amplifiers [2,3]. Raman fi-
ber lasers can also be applied in long-distance remote
sensing [4], supercontinuum generation [5], pulse com-
pression [6], and optical coherence tomography [7]. Re-
cently, high-power yellow output at 589 nm has been gen-
erated by frequency doubling of RFL radiation [8,9], thus
considerably extending the range of RFL applications.
The RFL spectral performance is of great importance for
the majority of applications, and especially for those that
require frequency doubling. At the same time, it is well
known that the RFL output spectrum is strongly broad-
ened; however, a spectral broadening mechanism is not
yet well understood.
A RFL usually has a long cavity providing at the same
time high pump and Stokes wave intensities in the fiber
core. That is why nonlinear effects could have a signifi-
cant impact on the laser performance. Moreover, in ultra-
long RFLs, the nonlinear effects can be comparable with
linear ones, making possible, for example, a quasi-lossless
transmission [10]. Though the stimulated Raman scatter-
ing itself is a basic nonlinear process important for RFL
operation, it cannot explain the formation and broadening
of the generated RFL spectrum. On the other hand, dif-
ferent nonlinear processes, such as stimulated Brillouin
scattering (SBS) or four-wave mixing (FWM), can also
lead to the spectral broadening.
It has recently been shown that SBS generation thresh-
old is not reached in the medium-power RFL because of
low spectral density [11]. Therefore the SBS effect does
not change the RFL spectral performance. On the con-
trary, several indications of the FWM influence on the
spectrum of RFLs [12–15] and of erbium-doped fiber la-
sers [16] have been reported. It has also been shown that
the coupling among longitudinal modes of the RFL cavity
must be taken into account [17], as generation in spectral
wings is observed under the threshold calculated within
the model of independent longitudinal modes generation.
Moreover, several authors [12,13] have promised to de-
velop and publish models that take into account FWM in
RFL. However, their work has not yet been finished, ow-
ing to the complexity of the involved processes. Only one
paper [18] reports on an attempt to explain theoretically
the RFL generation spectrum through the FWM interac-
tion between Stokes wave longitudinal modes. But this
model is semiempirical and is based on the arbitrarily
postulated relation among phases of different modes.
Since there has been no adequate theoretical descrip-
tion of RFL spectrum broadening until now, to the best of
our knowledge, we recently performed a special theoreti-
cal and experimental investigation. Our preliminary re-
sults have been published in [19]. It has been shown that
an approach based on the wave-turbulence formalism ad-
equately describes FWM interaction between multiple
longitudinal modes generated in an RFL cavity. In the
present paper, we extend this approach and give a de-
tailed description of the developed self-consistent theory
as well as a more complete comparison with experimental
results. Furthermore, the role of the nondegenerate FWM
between pump and Stokes wave modes in RFL spectrum
broadening at low powers has been analyzed. The possible
influence of FWM on pump wave spectrum broadening is
also discussed.
2. EXPERIMENT
We have studied spectral broadening of a Stokes wave
generated in a long cavity on the example of a one-stage
RFL based on a phosphosilicate fiber [1] (Fig. 1). The
phosphosilicate fiber has a distinct isolated P
2
O
5
-related
Babin et al. Vol. 24, No. 8/ August 2007/J. Opt. Soc. Am. B 1729
0740-3224/07/081729-10/$15.00 © 2007 Optical Society of America

Raman gain peak with a large Stokes shift that is free
from complications induced by overlapping of different
Raman gain peaks in germanosilicate fibers. The pump
radiation was generated in a 16 m long low-Q cavity of
the ytterbium-doped fiber laser (YDFL) at the wavelength
of 1.06
␮
m. In the high-Q RFL cavity having length L
=370 m and formed by two fiber Bragg gratings (FBGs)
with peak reflectivities R
1
⯝R
2
⬎99%, the pump wave is
converted to the first Stokes wave 共1.234
␮
m兲 due to the
stimulated Raman scattering process. The spectral profile
of effective FBG losses
␦
=−ln共R
1
R
2
兲 determined mainly
by transmission is shown in Fig. 2. The total intracavity
Stokes wave power increases almost linearly as the input
pump power P
0
increases (Fig. 3); remaining comparable
with the input pump power within the range up to 3 W.
By means of a specially designed intracavity coupler
with extinction ratio 5:95 at 1.234
␮
m, intracavity Stokes
wave spectra were measured near the input FBG at dif-
ferent pump powers. Since the spectral shape is compli-
cated, we have performed more detailed analysis in com-
parison with the first experiments [19]. Near the
generation threshold, the spectrum is quite narrow and
consists of several peaks, Fig. 4(a), which can be easily at-
tributed to the minima of the effective FBG losses profile.
The total spectral width is ⬃0.2 nm, remaining almost
constant up to the Stokes wave power ⬃0.5 W; see open
squares in Fig. 5(a). At the same time, the width of each
peak is as small as 0.08 nm, and it grows with increasing
power. The width of the left peak is marked by solid
squares in the Fig. 5(a) in the low-power region. The
peaks merge together at Stokes wave powers ⬎0.5 W, so
the total spectrum profile can be well characterized by a
single width at high powers.
At high Stokes wave power, the intracavity Stokes
wave spectrum is strongly broadened and has a rather
smooth shape with exponential wings (Figs. 4(b)–4(d)).
Nevertheless, the ripples in the spectrum still exist corre-
sponding to the ripples in the FBG losses profile; see Fig.
2. Since the ripples in the spectrum change their shape
with increasing power, they strongly affect the depen-
dence of the spectrum width versus Stokes wave power. In
preliminary studies [19], the width values were measured
only at a few power values, which was not enough to iden-
tify the power dependence. Here much more detailed
spectral measurements have been performed. The ex-
tracted Stokes wave spectral width is plotted in Fig. 5(a)
as a function of power. The jumps in spectral width values
Fig. 1. One-stage RFL based on a phosphosilicate fiber.
Fig. 2. Effective losses of FBGs forming the RFL cavity: Mea-
surements (thin curve) and parabolic approximation (thick
curve).
Fig. 3. Measured (䊏) and calculated (curve) upon Eqs. (11) and
(12) total intracavity Stokes wave power.
Fig. 4. a, Intracavity Stokes wave spectrum near the generation threshold: P
0
=0.4 W. b–d, Measured (dots) and calculated (solid curve)
upon Eqs. (8) and (9) intracavity Stokes wave spectrum at different input pump power P
0
:b,P
0
=1 W; c, P
0
=2 W; and d, P
0
=3 W.
1730 J. Opt. Soc. Am. B /Vol. 24, No. 8/ August 2007 Babin et al.

are attributable to the presence of ripples. The spectral
power density at maximum increases very fast near the
generation threshold and is smoothly saturated well
above the threshold (Fig. 5(b)).
The value of the effective FBG transmission defined as
the ratio of the output Stokes wave power to the intrac-
avity Stokes wave power near the output FBG has been
also directly measured (Fig. 6). Owing to spectral broad-
ening, the effective transmission grows with power. Note,
that the pump wave, which is essentially multimode, is
also broadened during its propagation in a long Raman fi-
ber (Fig. 7).
3. WAVE-TURBULENCE APPROACH
To explain the spectral broadening in RFLs, we performed
a theoretical analysis based on wave-kinetic equations
[20] originally used to describe wave turbulence. As
shown below, it is possible to apply the wave-turbulence
approach since multiple waves (up to 10
6
longitudinal
modes) interact with each other via multiple quasi-
degenerate FWM processes in a long RFL cavity and since
the phases of the waves remain stochastic. It is confirmed
by an analysis of the experimentally measured Stokes
wave RF spectrum [21]. Intermode beating peaks in the
RF spectrum are diffused, so the width D of the intermode
beating peak becomes of the order of the Stokes wave lon-
gitudinal modes spacing ⌬. Such behavior makes it evi-
dent that there are strong mechanisms that dephase
Stokes wave components during a round trip. These facts
allow us to use the method of averaging over many
共⬃10
4
兲 longitudinal modes while deriving the wave-
kinetic equation, which describes the Stokes wave spec-
trum. Note that experimentally measured Stokes wave
spectra are also averaged as there are ⬃10
4
longitudinal
modes within the standard spectral resolution of
⬃0.01 nm. In addition, at high-power Stokes wave being
well above the generation threshold, we can neglect for
simplicity the nondegenerate FWM processes that couple
the Stokes wave and the pump wave, i.e., neglect pump
wave fluctuations.
The Stokes wave electromagnetic field E in the cavity
of length L can be represented as the sum of copropagat-
ing and counterpropagating waves running with the ve-
locity c,
E共z,t兲 =
1
冑
2
共E
+
共z,t兲e
ik共ct−z兲
+ E
−
共z,t兲e
ik共ct+z兲
兲 + c.c.,
where the amplitude is normalized by the average total
intracavity Stokes wave power I共z ,t兲=兩E
2
共z , t兲兩, and
I
±
共z , t兲= 兩E
±
共z , t兲兩
2
denotes the power of forward and back-
ward propagating wave along the z axis of the fiber, ␭
0
=2
␲
n
0
/k is the wavelength at the FBGs reflection maxi-
mum, and c is the light speed in the fiber. The amplitude
variations of the copropagated and counterpropagated
Stokes waves E
±
obey an equation that includes disper-
sion and nonlinear phase modulation owing to FWM be-
tween different longitudinal modes in the Stokes waves;
see, e.g., [22],
冉
1
c
d
dt
±
d
dz
冊
E
±
共z,t兲 =
冉
g
R
P共z兲 −
␣
2
冊
E
+
共z,t兲
−
i
2
␥
关I
±
共z,t兲 +2I
⫿
共z,t兲兴E
±
共z,t兲 + i
␤
d
2
E
±
共z,t兲
dt
2
, 共1兲
where P共z兲 is the mean (over the time) pump wave power,
g
R
is the Raman gain coefficient,
␣
is the Stokes wave op-
tical losses in the fiber,
␥
is the nonlinear Kerr coefficient
(amplitude of FWM, i.e., the cross-phase modulation coef-
ficient), and
␤
=共1/2兲dc
−1
/d
␻
is the dispersion coefficient.
Because the RFL has a high-Q cavity, we can represent
the Stokes wave as the sum of longitudinal cavity modes:
Fig. 5. a, Full width at 1/ cosh共1兲=0.648 level of the intracavity Stokes wave spectrum. b, Spectral power density at its maximum.
Experiment (䊐, 䊏) and calculation (curve) upon Eqs. (9) and (10).
Fig. 6. Measured (䊏) and calculated (solid curve) upon Eqs. (15)
effective transmission coefficient T
eff
.
Babin et al. Vol. 24, No. 8/ August 2007/ J. Opt. Soc. Am. B 1731

E
±
共z,t兲 =
1
冑
2
兺
n
E
n
共t兲exp共in⌬t ⫿ i
␬
zn兲exp共− i
␯
n
t兲, 共2兲
where ⌬=2
␲
/
␶
rt
is the frequency shift between adjacent
Stokes wave longitudinal modes,
␶
rt
=2L / c is the RFL cav-
ity round-trip time for the Stokes wave,
␯
n
=
␤
c共n⌬兲
2
+
␥
cI
is a small frequency shift that takes into account the dis-
persion and the mean nonlinear phase shift, and
␬
=
␲
/L.
The factor 1/
冑
2 is chosen in order to normalize the total
power in the nth longitudinal mode to I
n
=兩E
n
兩
2
=I
n
+
+I
n
−
.
Let us rewrite the generalized nonlinear Schrödinger
equation (1) in terms of amplitude E
n
of the nth longitu-
dinal modes defined in Eq. (2),
␶
rt
dE
n
dt
−
1
2
共g −
␦
n
兲E
n
共t兲 =−
i
2
␥
L
兺
l⫽0
E
n−l
共t兲
兺
m⫽0
E
n−m
共t兲
⫻E
n−m−l
*
共t兲exp共2i
␤
ml⌬
2
ct兲,
共3兲
where the effective FBG losses
␦
n
=−ln关R
1
共⍀
n
兲R
2
共⍀
n
兲兴 for
the nth longitudinal mode that is generated at the fre-
quency detuned by ⍀
n
=n⌬ from the FBG center fre-
quency have been taken into account. The integral Stokes
wave round-trip amplification g depends on the averaged
(over the fiber length) pump power P
¯
as
g =2g
R
P
¯
L −2
␣
L. 共4兲
The averaged pump power P
¯
is defined by the following
expression (see [23] for the details):
P
¯
= P
0
1 − exp
冉
−2
␣
p
L −2
␭
␭
p
g
R
LI
冊
冉
␣
p
+
␭
␭
p
g
R
I
冊
L
, 共5兲
where P
0
is the input (regarding the RFL) pump power,
␣
p
is the pump wave optical losses in the fiber, and ␭
p
and ␭
are the pump and Stokes wavelengths. Note that this ex-
pression is valid within the limits of assumption I共z兲
=const, which is well justified in our case of a highly re-
flective cavity for the Stokes wave [23].
The terms with l =0 and m=0 in Eq. (3) give the slip of
the carrier phase with respect to the envelope (the mean
nonlinear phase shift); thus they do not affect the Stokes
wave spectrum. An FWM process between copropagating
and counterpropagating waves leads to the terms oscillat-
ing at beat frequency 2l⌬ and should be omitted since
they are nonresonant at l⫽ 0.
The right-hand side (RHS) of Eq.(3) for the RFL cavity
length of 370 m contains ⬃10
12
different terms with ran-
dom amplitudes and phases in each term that lead to the
stochastic (turbulent) evolution of the amplitude and
phase of the longitudinal mode E
n
. It is obvious that such
an extremely complex nonlinear evolution can be de-
scribed neither analytically nor numerically. For an ad-
equate description of the Stokes wave spectrum, one
should use a statistical description of the spectral power
density instead of a dynamical description of the longitu-
dinal modes amplitudes E
n
, i.e., to derive and to solve the
wave-kinetic equation. Toward this end, we follow the
methods of Zakharov et al. [20], well developed and
widely applied for the description of the wave turbulence
of interacting acoustic waves, waves on fluid surfaces,
spin waves, waves in plasma, etc. After rather complex
calculations, which are partially presented in Appendix A,
we derive from Eq. (3) the simplified wave-kinetic equa-
tion for the averaged time-independent Stokes wave spec-
trum I共⍀兲,
关
␦
共⍀兲 +
␣
2L +
␦
NL
兴I共⍀兲 =2g
R
LP
¯
I共⍀兲 +
␦
NL
I
2
冕
I共⍀
1
兲I共⍀
2
兲
⫻I共⍀
1
+ ⍀
2
− ⍀兲d⍀
1
d⍀
2
, 共6兲
where
␦
NL
defines nonlinear FWM-induced losses [see Eq.
(A5)], which can be written in our experimental case
共4
␤
L/
␦
2
⯝4.5兲 in the following form:
␦
NL
=
冑
2
3
␥
IL
冑
1+共8
␤
L⍀
2
/3
␦
NL
兲
2
. 共7兲
Here I =兰I共⍀兲d⍀ is the total intracavity Stokes wave
power. The round-trip dispersion-induced phase differ-
ence between the longitudinal modes is substituted here
by the phase difference averaged over the spectrum,
␤
⍀
2
2L, where the mean-square spectral width ⍀
2
is de-
fined in Eq. (A8); see Appendix A.
Fig. 7. Pump wave spectrum, a, before and, b, after transmission trough RFL (input pump power P
0
=1.5 W).
1732 J. Opt. Soc. Am. B / Vol. 24, No. 8/ August 2007 Babin et al.

4. ANALYTICAL SOLUTION FOR THE
STOKES WAVE SPECTRUM AND
COMPARISON WITH EXPERIMENT
Equation (6) has a simple physical meaning. The left-
hand side (LHS) gives the Stokes wave attenuation, and
the RHS gives the Stokes wave amplification. The last
term in the LHS is the nonlinear attenuation of the
Stokes wave longitudinal mode I共 ⍀兲 owing to the scatter-
ing of this mode on the mode with frequency ⍀
1
+⍀
2
−⍀ to
other longitudinal modes with frequencies ⍀
1
and ⍀
2
. The
last term in the RHS describes the scattering of the
modes with frequencies ⍀
1
and ⍀
2
to the longitudinal
modes with frequencies ⍀ and ⍀
1
+⍀
2
−⍀. The amplitude
of the mode I共⍀兲 is increasing in this quasi-degenerate
FWM process. So this term is a FWM-induced gain, and it
is the origin of the spectrum broadening. It is important
that FWM-induced losses are homogeneous, i.e., do not
depend on frequency, while FWM-induced gain is inhomo-
geneous; i.e., depends on frequency. Note that the integral
of the FWM-induced terms of Eq. (6) over ⍀ is zero, which
is equivalent to conservation of the total energy in FWM
processes, i.e., the total FWM-induced losses are equal to
the total FWM-induced gain.
The solution to the integral equations (6) and (7),as
well as the solution to their extended versions [Eqs.
(A3)–(A6); see Appendix A], can be found numerically, but
it is always of interest to solve them analytically. If the
effective FBG losses profile is close to the parabolic form
␦
共⍀兲=
␦
0
+
␦
2
⍀
2
(which corresponds to the widely applied
Gaussian-shaped FBG’s reflection spectrum) (see Fig. 2),
we find an analytical solution to Eq. (6) for the intracavity
Stokes wave spectrum,
I共⍀兲 =
2I
␲
⌫ cosh共2⍀/⌫兲
, 共8兲
where the spectral width ⌫ is
⌫ =
2
␲
冑
2
␦
NL
␦
2
共9兲
and the spectral power density at maximum is
I共0兲 =
2I
␲
⌫
. 共10兲
By calculating the mean-square spectral half-width
from Eq. (A8) in Appendix A and Eq. (8),
⍀
2
=
␦
NL
/2
␦
2
, and
substituting it into Eq. (7),wefind
␦
NL
=
冑
2
3
␥
IL
冑
1+共4
␤
L/3
␦
2
兲
2
. 共11兲
From Eqs. (9) and (11) it follows that the spectral width
⌫ increases as the square root of the Stokes wave power I.
The spectral power density at maximum I共0兲 increases
also as the square root of I; see Eqs. (10) and (11).
The obtained results should be compared with the pre-
liminary results published in [19]. It has been shown [see
Eq.5of[19]] that the spectral width is proportional to the
product of the intracavity Stokes wave power I and the
square root of the parameter
⑀
=
␶
/3
␶
rt
. Under the assump-
tion
⑀
=const, the spectral width should exhibit linear
growth with power. However, it was noted that the experi-
mentally measured value of
⑀
decreases with increasing
power. One of the most important results of the present
paper is the fact that the correlation time
␶
is almost com-
pletely defined by the nonlinear losses; see Eq. (A6) of Ap-
pendix A. Thus we can rewrite
⑀
=2/共3
␦
NL
兲 through the
nonlinear losses coefficient and then show that
⑀
is in-
versely proportional to the intracavity Stokes wave
power. As a result, we have discovered the square-root
law that governs the increase in the spectral width and in
the peak power as the Stokes wave power grows.
By substituting Eq. (8) for Eq. (6) and integrating over
⍀, one can obtain how the intracavity Stokes wave power
I depends on the Stokes wave power P
0
,
␦
0
+
␦
NL
共I兲
2
+2
␣
L =2g
R
P
0
1 − exp
冉
−2
␣
p
L −2
␭
␭
p
g
R
LI
冊
␣
p
+
␭
␭
p
g
R
I
,
共12兲
where Eq. (5) has been used. Equation (12) means that
the gain (RHS) and losses (LHS) are equal.
From Eqs. (11) and (12), one can plot the value of the
intracavity Stokes wave power I using only fiber and FBG
parameters:
␣
p
=2.5 dB/km (including lumped losses, i.e.,
losses on intracavity coupler, excess losses on FBGs, and
splice losses);
␣
=2.5 dB/km (including lumped losses); the
normal dispersion
␤
⬇13.3 nm
−2
km
−1
; the nonlinear Kerr
coefficient
␥
⬇3km
−1
W
−1
; the Raman gain coefficient g
R
=1.3 km
−1
W
−1
; the length L=370 m; and the fitted curva-
ture of the effective FBGs losses
␦
2
=4 nm
−2
(see Fig. 2).
There is very good agreement between the calculated and
measured intracavity Stokes wave power I (see Fig. 3).
Finally, using Eqs. (8), (9), and (11), one can easily cal-
culate the Stokes wave spectral profiles I共⍀兲 at different
intracavity Stokes wave powers I (Figs. 4(b)–4(d)), as well
as the spectral width and spectral power density values
(Fig. 5). There is excellent agreement between the ana-
lytical theory predictions and the experiment in all power
range except the threshold. Let us note that absolutely no
fitting was performed, and only real fiber and FBG pa-
rameters were used. The fluctuations in the experimen-
tally measured profiles correspond to the ripples in the
FBG losses profile. So we can conclude that our theory
gives the perfect analytical description for the RFL gen-
eration spectrum. Thus the Stokes wave spectrum is
broadened owing to the quasi-degenerate FWM between
longitudinal modes of the Stokes wave.
It is also important that the main RFL characteristic,
the output RFL power I
1,2
out
共⍀兲=−ln共R
1,2
兲I
⫿
共⍀兲, can be also
easily calculated within the developed theory. In the case
of identical Gaussian FBGs, −ln共R
1,2
兲⯝共
␦
0
+
␦
2
⍀
2
兲/2, the
running Stokes wave power is I
±
共⍀兲⯝I共⍀兲/2, which re-
sults in
I
1,2
out
共⍀兲 =
␦
0
+
␦
2
⍀
2
2
I
␲
⌫ cosh共2⍀/⌫兲
. 共13兲
Note that for an adequate description of the RFL out-
put power, the effective transmission coefficient T
eff
of
Babin et al. Vol. 24, No. 8/ August 2007/ J. Opt. Soc. Am. B 1733

each FBG has been introduced in a phenomenological way
earlier [17,23,24]. Now it is possible to find it theoreti-
cally. The effective transmission coefficient for each FBG
in the case of small losses is T
eff
⯝
␦
eff
/2, where the mean
effective FBG losses
␦
eff
can be derived from Eq. (13):
␦
eff
⬅
2共I
1
out
+ I
2
out
兲
I
=
冕
␦
共⍀兲I共⍀兲d⍀/I =
␦
0
+
␦
NL
共I兲
2
.
共14兲
Thus
T
eff
⯝
␦
NL
/4, 共15兲
where
␦
0
⬇0 (see Fig. 2). The effective transmission coef-
ficient grows linearly with increasing Stokes wave power
owing to the Stokes wave spectrum broadening induced
by the quasi-degenerate FWM processes (see Fig. 6, solid
line). However, there is some discrepancy at high powers
between the theoretical prediction and experimentally
measured values that can be attributed to the difference
between the mean intracavity Stokes wave power I and
the power I共z=0兲 measured near the input FBG; this dif-
ference increases with increasing transmission. One more
reason is the difference between R
1
共⍀兲 and R
2
共⍀兲 that
grows with detuning ⍀.
5. DISCUSSION
The obtained results allow us to reach agreement be-
tween apparently contradictory experimental and theo-
retical results on the Raman gain saturation and the gen-
eration spectrum. The first fact is related to the question
about the RFLs’ multiwavelength generation. It is well
known that the multiwavelength generation is easily
achievable in RFLs experimentally, see, for example,
[25,26]. Usually it is deemed that this possibility exists
due to the inhomogeneous nature of the Raman gain spec-
trum. Nevertheless, it has been recently experimentally
shown that Raman gain saturates homogeneously even at
high pump and Stokes wave powers [27]. On the other
hand, because of homogeneous nature of the Raman gain
one should expect single-frequency RFL generation in-
stead of the multiwavelength one. However, single-
frequency operation has not been observed in RFLs to this
date.
Second, it is well known that the Stokes wave spectrum
is fairly broad even near the generation threshold. At
high power, the Stokes wave spectrum is well described
by the phenomenological model of the independent gen-
eration of different longitudinal modes [24,28]. These
facts do not agree directly with the homogeneous nature
of the Raman gain saturation. Thus some inhomogeneous
mechanisms must exist. Their manifestations in RFLs
have been reported by several authors recently
[15,17,29,30]. In [15,29], it is supposed that the multi-
wavelength generation is possible owing to the intraline
FWM-induced inhomogeneous losses.
In the present paper, we have proved experimentally
and theoretically that inhomogeneous mechanisms are in-
deed based on FWM. Moreover, mixing between different
longitudinal modes of the Stokes wave (quasi-degenerate
FWM) induces both nonlinear losses and gain. However,
we have clearly shown that FWM-induced losses in our
case 共4
␤
L/
␦
2
⯝4.5兲 have a homogeneous nature in spite of
assumptions of some papers [15,29]. At the same time, the
FWM-induced gain is an inhomogeneous gain. So multi-
wavelength generation in RFL can be achieved owing to
the inhomogeneous gain, in particular.
The developed analytical theory has its applicability
limits. The quasi-degenerate FMW processes are efficient
at high powers. However, near the generation threshold
when the power of the Stokes wave is small, the contribu-
tion of quasi-degenerate FWM processes to the spectral
broadening can also be small. At the same time, the RFL
spectrum is broadened even near the generation thresh-
old; see Fig. 4(a). Thus, some other efficient spectral
broadening mechanisms should exist at low powers.
We suppose that at low powers the Stokes wave spec-
trum is broadened owing to nondegenerate FWM between
the Stokes wave and different longitudinal modes in the
pump wave generated in the multimode YDFL. In this
case, the spectral width of the Stokes wave should be pro-
portional to the spectral width of the effective FBG losses
profile near its minimum and pump power fluctuations
that are determined by its own longitudinal mode beat-
ing. This mechanism explains why the single-frequency
regime has not yet been achieved in the RFL. In the usual
YDFL-pumped RFL, there are nonzero fluctuations of the
pump wave power at the Stokes wave generation thresh-
old [21]. Therefore, the Stokes wave spectrum is broad-
ened owing the nondegenerate FWM with different longi-
tudinal modes of the pump wave, and single-frequency
generation is not possible. The low-power theory details
are presented in Appendix B.
The developed wave-turbulence approach for the de-
scription of the RFL spectrum can be useful for solving
other problems of fiber optics such as, for example, a mul-
tilongitudinal mode wave propagation in a fiber [31]. One
can qualitatively affirm that the multimode pump wave
propagating in a long RFL cavity is broadened by its own
quasi-degenerate FWM processes between different pump
wave longitudinal modes. The experimental results con-
firm this supposition; see Fig. 7. Based on the wave-
turbulence approach of the wave-kinetic equation that
takes into account the existence of a small but nonzero
phase correlation, it is possible to perform analytical
analysis of the wave spectral broadening during propaga-
tion in long fibers.
6. CONCLUSION
Thus we have developed an analytical theory that de-
scribes the Stokes wave spectrum formation and broaden-
ing in RFLs. This self-consistent theory takes into ac-
count FWM and complies with all experimental and
theoretical results regarding the RFL generation and the
Raman gain saturation nature. It has been shown that
the main spectral broadening mechanism is the quasi-
degenerate FWM between different Stokes wave longitu-
dinal modes. The spectral shape of the Stokes wave is de-
fined mainly by the effective FBG losses profile. The exact
analytical solution for the RFL intracavity spectra formed
by longitudinal modes interacting via the FWM process
results in a specific hyperbolic secant shape at high pow-
1734 J. Opt. Soc. Am. B / Vol. 24, No. 8/ August 2007 Babin et al.

ers. The theory explains generation of longitudinal modes
at the spectral wings that must be forbidden within the
model of independent mode generation. The spectral
width of the Stokes wave is proportional to the square
root of the intracavity Stokes wave power as well as the
maximum spectral power densities. The predictions of the
theory are in excellent quantitative agreement with the
experimental data.
The perfect theoretical description of the generated
Stokes wave spectrum became possible since the dynami-
cal description of ⬃10
6
amplitudes and phases of longitu-
dinal modes that change their values stochastically (tur-
bulent behavior) was discarded in favor of statistical
description of the spectrum averaged over a large number
共⬃10
4
兲 of longitudinal modes.
The turbulent behavior observed in RFL is a funda-
mental effect that may be important in other fields of fi-
ber optics and may have a significant impact on the per-
formance of practical fiber systems. We believe that
predicted simple square-root law for the spectral width
and peak power density growth should be valid for other
types of fiber lasers as well.
APPENDIX A: SPECTRAL BROADENING
WELL ABOVE THE THRESHOLD
Multiplying Eq. (3) by E
n
*
and taking the real part, one
can obtain the equation describing how the power of the
nth Stokes wave longitudinal mode changes,
␶
rt
dI
n
dt
− 共g −
␦
n
兲I
n
共t兲 =−Re
冋
i
␥
L
兺
l⫽0, m⫽0
E
n−l
共t兲E
n−m
共t兲
⫻E
n−m−l
*
共t兲E
n
*
共t兲exp共2i
␤
ml⌬
2
ct兲
册
.
共A1兲
If one assumes that phases of different longitudinal
modes are random (uncorrelated) then averaging of the
RHS of Eq. (A1) gives zero value as all terms include a
random phase difference. To take into account a phase
correlation induced by the FWM processes, we calculate
the correction to E
n
by integration of the RHS of Eq. (3),
␦
E
n
共t兲 =
− i
␥
L
2
冕
−⬁
t
兺
l
⬘
⫽0,m
⬘
⫽0
E
n−l
⬘
共t
⬘
兲E
n−m
⬘
共t
⬘
兲E
n−m
⬘
−l
⬘
*
共t
⬘
兲
⫻exp共2i
␤
m
⬘
l
⬘
⌬
2
ct
⬘
兲
dt
⬘
␶
rt
. 共A2兲
In the next step, we substitute this correction (with index
n changed to n −l) instead of E
n−l
in the RHS of Eq. (A1)
and then instead of E
n−m
, etc. As a result, we obtain the
sum of four terms, and after averaging them over ⬃10
4
spectral components near the frequency ⍀=n⌬, we derive
a wave-kinetic equation for the stationary Stokes wave
spectrum I共⍀兲=具I
n
典/⌬:
␶
rt
dI共⍀兲
dt
= 关g −
␦
共⍀兲兴I共⍀兲 + S
FWM
共⍀兲 =0, 共A3兲
where FWM-induced terms are
S
FWM
共⍀兲 =−
␦
NL
I共⍀兲
+ 共
␥
L兲
2
冕
I共⍀ − ⍀
1
兲I共⍀ − ⍀
2
兲I共⍀ − ⍀
1
− ⍀
2
兲
共3
␶
rt
/
␶
兲关1+共4
␶
L
␤
/3
␶
rt
兲
2
⍀
1
2
⍀
2
2
兴
d⍀
1
d⍀
2
,
共A4兲
and nonlinear FWM-induced losses
␦
NL
have the follow-
ing form:
␦
NL
= 共
␥
L兲
2
冕
关I共⍀ − ⍀
1
兲 + I共⍀ − ⍀
2
兲兴I共⍀ − ⍀
1
− ⍀
2
兲 − I共⍀ − ⍀
1
兲I共⍀ − ⍀
2
兲
共3
␶
rt
/
␶
兲关1+共4
␶
L
␤
/3
␶
rt
兲
2
⍀
1
2
⍀
2
2
兴
d⍀
1
d⍀
2
. 共A5兲
Here we assume an exponential decay of the correlation
function 具E
l
共t兲E
l
*
共t
⬘
兲典= I
l
e
−兩t− t
⬘
兩/
␶
with correlation time
␶
,
and the Gaussian statistics for field E
n
共t兲.
The exponential decay of the correlation function re-
sults in the Lorentzian shape of the peaks in the Stokes
wave intermode beating RF spectrum:
F共⍀兲 =
兺
n⫽0
DF
n
␲
关D
2
+ 共⍀ − n⌬兲
2
兴
,
where F
n
=兺
l
I
l
I
l+n
, and D =2/
␶
. So the correlation time
␶
=2/D can be found from the experimentally measured
HWHM of the peaks in the RF intermode beating spec-
trum of the Stokes wave.
The last term in the RHS of the Eq. (A4) describes an
increase of intensity in the mode with frequency ⍀ owing
to scattering of modes with frequencies ⍀ −⍀
1
and ⍀
−⍀
2
to modes with frequencies ⍀ and ⍀−⍀
1
−⍀
2
, i.e., this
term represents the FWM-induced nonlinear gain. The
physical meaning of
␦
NL
is the round-trip nonlinear at-
tenuation coefficient of the Stokes wave with frequency ⍀
induced by its scattering on other longitudinal modes
with frequencies ⍀− ⍀
1
−⍀
2
. It is exactly the nonlinear at-
tenuation that leads to exponential decay of the correla-
tion function. The nonlinear attenuation increases with
power, and completely determines the correlation time at
high Stokes power:
␶
rt
␶
⯝
␦
NL
/2, 共A6兲
This relation makes Eqs. (A4), (A3), and (A5) self-
consistent.
It is relevant to note that the wave-kinetic equation
[Eqs. (A3)–(A6)] is valid when the nonlinearity is much
less than the dispersion:
Babin et al. Vol. 24, No. 8/ August 2007/ J. Opt. Soc. Am. B 1735

␥
IL  4
␤
L⍀
2
, 共A7兲
where
⍀
2
is a mean-square spectral half-width,
⍀
2
=
冕
⍀
2
I共⍀兲d⍀
I
. 共A8兲
Condition (A7) provides a phase mismatch between re-
mote spectral components (longitudinal modes), as a re-
sult, the FWM-induced phase synchronization is sup-
pressed owing to dispersion, and phases of different
longitudinal modes can be considered as weakly corre-
lated.
By evaluating the integrals in Eqs. (A3)–(A5) at the
limit of condition (A7), one can find from the wave-kinetic
equation the spectral width and the spectral power of the
Stokes wave if the spectrum is bell shaped. For simplicity,
let us examine the case of Gaussian-apodized FBGs in
which effective FBG losses have parabolic form:
␦
共⍀兲 =
␦
0
+
␦
2
⍀
2
.
The integral over the frequency of S
FWM
共⍀兲 is equal to
zero, which means that energy is conserved in FWM pro-
cesses. That is why from Eq. (A3) it follows that
␦
0
+
␦
2
⍀
2
= g =2g
R
LP
¯
−2
␣
L. 共A9兲
This equation has a simple physical meaning of equal-
ity between the Stokes wave power gain and losses in a
round trip. In particular, Eq. (A9) connects the spectral
half-width with the saturated gain g.
To obtain one more relation, let us evaluate the nonlin-
ear losses
␦
NL
in the case of a bell-shaped intracavity
Stokes wave spectrum. The main contribution to the inte-
gral [Eq. (A5)] over ⍀
1
accumulates near zero frequencies.
The residual integral over ⍀
2
gives a logarithm bounded
by the spectral width. As a result, we obtain a relation be-
tween the nonlinear losses and the spectral half-width:
3
2
␦
NL
⬇ B
共
␥
IL兲
2
4
␤
L⍀
2
ln
冉
8
␤
L⍀
2
3
␦
NL
冊

␥
IL  4
␤
L⍀
2
,
共A10兲
where B is a constant that accounts for the particular
spectral shape. For example, the Gaussian shape results
in B=0.7, and the hyperbolic secant shape gives B= 0.9.
In evaluation of nonlinear losses [Eq. (A10)], we have ne-
glected small terms ⬃
␦
NL
/共4
␤
L⍀
2
兲 1. Inclusion of the
frequency dependence of the nonlinear losses is beyond
the framework of this approximation.
It remains to obtain one more relation. For this pur-
pose, let us take ⍀=0 in Eq. (A3), which gives us g =
␦
0
−S
FWM
共0兲/I共0兲. After substituting Eq. (A6) for Eqs. (A4)
and (A5),wefind
S
FWM
共0兲
I共0兲
= 共
␥
L兲
2
冕
d⍀
1
d⍀
2
I共⍀
1
兲I共⍀
2
兲
冋
I共⍀
1
+ ⍀
2
兲
I共0兲
+1
册
− 关I共⍀
1
兲 + I共⍀
2
兲兴I共⍀
1
+ ⍀
2
兲
共3
␦
NL
/2兲关1+共8L
␤
/3
␦
NL
兲
2
⍀
1
2
⍀
2
2
兴
. 共A11兲
At ⍀
1
=0 or ⍀
2
=0, the integrand numerator takes the
zero value, therefore all spectral components give a con-
tribution to the integral. For a bell-shaped intracavity
Stokes wave spectrum, we obtain in the approximation of
condition (A7) with the result of Eq. (A10) the following
relation:
g −
␦
0
=
S
FWM
共0兲
I共0兲
=
3
2
A
␦
NL
冉
␥
IL
4
␤
L⍀
2
冊
2

␦
NL
, 共A12兲
where A is a constant that accounts for the spectral
shape, so A =0.3 for the Gaussian shape, and A=1.5 for
the hyperbolic secant shape. Equation (A12) connects to-
gether the saturated gain g, the nonlinear losses
␦
NL
, and
the spectral half-width.
Finally, from Eqs. (A9), (A12), and (A10), we find ex-
pressions for the spectral half-width,
⍀
2
=
␥
I
4
␤
冋
4
␤
L
␦
2
AB ln
冉
4
␤
LA
␦
2
B
冊
册
1/4
共A13兲
for the nonlinear attenuation
␦
NL
=
2
3
␥
IL
冋
B ln
冉
4
␤
LA
␦
2
B
冊
册
3/4
冉
4
␤
LA
␦
2
冊
−1/4
, 共A14兲
and for the Stokes wave power I,
2g
R
P
¯
共I兲L =2
␣
L +
␦
0
+
␥
IL
冋
AB ln
冉
4
␤
LA
␦
2
B
冊
册
1/4
冉
4
␤
L
␦
2
冊
−3/4
,
共A15兲
where the averaged pump power P
¯
共I兲 is defined by Eq. (5).
It is important that the spectral width and the spectral
power density depend equally according a square-root law
on the intracavity Stokes wave power I. Spectral broad-
ening leads to increase in the output losses, which grow
linearly with I [the last term in Eq. (A15)].
It should be noted that the inequality [condition (A7)]
for the RFL conditions is equivalent to the inequality
4
␤
L
␦
2
 1, 共A16兲
i.e., is valid for a long cavity, a large normal dispersion, or
wide FBGs forming the cavity. In the performed experi-
ment, the condition (A16) is fulfilled; 4
␤
L/
␦
2
⯝4.5. Never-
1736 J. Opt. Soc. Am. B / Vol. 24, No. 8/ August 2007 Babin et al.

theless, this parameter is not as large as for Eqs.
(A13)–(A15) to give us sufficient accuracy of the evalua-
tion. On the other hand, in this case, we can substitute
the averaged over-the-spectrum values of frequencies ⍀
1,2
2
instead of its real values in all denominators of the inte-
grands in the RHS of Eqs. (A4) and (A5). In that way, Eqs.
(A3)–(A6) can be simplified to Eqs. (6) and (7) on the sta-
tionary intracavity Stokes wave spectrum I共⍀兲. This al-
lows us to find the analytical solution [Eq. (8)].
APPENDIX B: SPECTRAL BROADENING AT
LOW POWERS
Near the Stokes wave generation threshold, the intracav-
ity Stokes wave power I is much smaller than the pump
wave power P. So we can neglect the Stokes wave influ-
ence on pump wave fluctuations
␦
P共z ,t兲. On the other
hand, we can also neglect the Stokes wave influence on
the Stokes wave phase modulation as compared with the
pump wave influence. Thus, we take into account nonde-
generate FWM involving the Stokes and pump waves but
neglect the quasi-degenerate FWM processes between dif-
ferent Stokes wave longitudinal modes.
One can obtain equations on the amplitudes E
n
共t兲 of
the longitudinal modes that are connected with each
other by means of the pump wave power fluctuations:
␶
rt
dE
n
共t兲
dt
=
g −
␦
n
2
E
n
共t兲 −2i
␥
L
兺
l⫽0
p
l
共t兲exp关il共
␬
c
1
− ⌬兲t兴 E
n−l
共t兲. 共B1兲
Here we have made an expansion of fluctuations of the
pump wave running with speed c
1
in the positive direc-
tion inside the RFL cavity:
␦
P共z ,t兲=兺
l⫽0
e
il
␬
共c
1
t−z兲
p
l
共t兲. The
expansion coefficients have the following form: p
l
共t兲
=p
−l
*
共t兲=兰
0
L
␦
P共z ,t兲e
−il
␬
共c
1
t−z兲
dz / L.
Note that the coefficients p
l
共t兲 in Eq. (B1) have the
meaning of the coupling coefficients of different Stokes
wave longitudinal modes. Therefore, the more compo-
nents the pump wave fluctuations expansion includes, the
more remote Stokes wave longitudinal modes are coupled
with each other. Thus the Stokes wave spectrum should
be broadened owing to the pump wave fluctuations.
Multiplying Eq. (B1) by E
n
*
and taking the real part,
one can obtain an equation on I
n
⬅兩E
n
兩
2
, i.e., an equation
on the Stokes wave spectrum component:
␶
rt
dI
n
dt
= 共g −
␦
n
兲I
n
−2
␥
L
兺
l⫽0
关ip
l
共t兲e
il共
␬
c
1
−⌬兲t
E
n−l
共t兲E
n
*
共t兲
+ c.c.兴. 共B2兲
Repeating the derivation given in Appendix A, we per-
form direct integration of Eq. (B1), which gives us the cor-
rection for the Stokes wave longitudinal mode amplitude:
␦
E
n
共t兲 =−2i
␥
L
冕
−⬁
t
兺
l
⬘
⫽0
p
l
共t
⬘
兲e
il
⬘
共
␬
c
1
−⌬兲t
⬘
E
n−l
⬘
共t
⬘
兲
dt
⬘
␶
rt
.
After that, we can substitute the integral representations
for the RHS of Eq. (B2), first instead of E
n−l
, and then in-
stead of E
n
*
, and after averaging over spectral compo-
nents, we obtain the equation on the Stokes wave spec-
trum I共⍀兲=具I
n
/⌬典 at frequency ⍀⯝n⌬,
␶
rt
dI共⍀兲
dt
− 关g −
␦
共⍀兲兴I共⍀兲 =2
␶
1
␶
rt
共2
␥
L兲
2
冕
−⬁
⬁
d⍀
⬘
⫻
F
p
共⍀
⬘
兲关I共⍀ − ⍀
⬘
兲 − I共⍀兲兴
1+关
␶
1
共c
1
/c −1兲⍀
⬘
兴
2
,
共B3兲
where
␶
1
=2/D
1
is the dephasing time that is inversely
proportional to the pump wave RF spectrum width D
1
near the Stokes wave generation threshold, F
p
共⍀兲 is the
pump wave RF spectrum profile.
Equation (B3) is a wave-kinetic equation near the
Stokes wave generation threshold. The first term in the
integrand of the RHS describes the nonlinear FWM-
induced gain: a Stokes wave with some frequency ⍀ −⍀
⬘
scatters on the pump wave to the Stokes wave with fre-
quency ⍀ and other pump wave. Such processes lead to
the Stokes wave spectral broadening. The second term de-
scribes the losses induced by the nondegenerate FWM
processes between the Stokes wave and different longitu-
dinal modes of the pump wave. These losses correspond to
the scattering of the Stokes wave longitudinal mode ⍀ on
the pump wave, so the intensity of the scattered Stokes
wave is reduced.
From Eq. (B3), it follows that if there are nonzero fluc-
tuations of the pump wave power, then the Stokes wave is
broadened even near the generation threshold. Such spec-
tral broadening owing to nondegenerate FWM processes
does not depend on the type of the gain saturation.
ACKNOWLEDGMENTS
The authors acknowledge financial support by the Inte-
gration program of the Siberian Branch of the Russian
Academy of Sciences, the governmental program of sup-
port for leading scientific schools and young scientists in
Russia, the programs of the Russian Academy of Science,
Civilian Research and Development Foundation grant
RUP1-1509-NO-05; and the Fiber Optic Research Center
(Moscow, Russia) for the supply of the phosphosilicate fi-
ber. We also thank V. V. Lebedev and A. M. Shalagin for
fruitful discussions.
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