Generalised dispersive phase and its effect on four wave mixing in fibers
R. Deepa 1, R. Vijaya
*
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Abstract
A generalised method of calculating the efficiency of four wave mixing products in an optical fiber is presented in this paper. Apart
from indicating the insufficiency of the existing theoretical model, this work brings out the importance of calculating the dispersive phase
shift more precisely in combination with the nonlinear phase shift to predict the efficiencies of sideband generation. The experimental
results for the generation of the first and second order sidebands are analysed. Contrary to expectations, higher powers and longer
lengths of fiber do not result in larger number of sidebands. An attempt is made to understand this aspect as well as the experimentally
observed fluctuations in the four wave mixing products.
PACS: 42.81.i; 42.65.k
Keywords: Four wave mixing in fibers; Phase matching; Conversion efficiency
1. Introduction
Nonlinear optical phenomena become prominently
observable in high-power fiber optic communication systems. On the one hand, this leads to deterioration of the
signal quality due to cross-talk effects. On the other hand,
these nonlinearities make fibers an excellent medium for
easy generation of new frequencies, which finds many
applications including parametric oscillation and multi
wavelength all-fiber sources for wavelength division multiplexed (WDM) communication systems.
One of the prominent nonlinear effects in fibers is four
wave mixing (FWM). FWM is the process by which, electromagnetic fields of different frequencies, propagating
simultaneously in a nonlinear medium interact through
the third order nonlinear optical susceptibility of the medium, resulting in the generation of new frequencies. FWM
in single mode fibers was first reported by Hill et al. [1].
Since it can significantly influence the multi channel transmission systems, this phenomenon has been studied extensively in different types of fibers [2–8]. FWM leads to
signal deterioration and cross talk in dense wavelength
division multiplexed (DWDM) systems [9–11], and many
methods are being used to reduce the deleterious effect
of FWM in communication [12–14]. It is interesting to
note that FWM in fibers has also been used favourably
in many applications. It is a widely used tool to estimate
simultaneously, the nonlinear refractive index, dispersion,
and dispersion slope of the fiber medium [15,16]. It is also
used for phase sensitive amplification [17], wavelength
conversion [18–20] and optical signal reshaping [21]. The
phenomenon also plays an important role in parametric
oscillation [9], generation of frequency comb and supercontinuum in suitable fibers [22]. Supercontinuum (SC)
generation is a process where different nonlinear mechanisms such as self phase modulation, cross phase modulation, stimulated Raman scattering and FWM interact
together to create a large spectral enrichment. If the generated frequencies are well separated, the process is also
207
referred to as frequency comb generation. The optical
power and the characteristics of the fiber medium will
decide the extent and quality of the spectral broadening.
SC and frequency comb generation have important applications in terms of multi wavelength source design in telecommunication and broad band spectral characterisation
of fiber optic components.
For FWM to be efficient, the phase matching condition
has to be satisfied [2,9]. It is often indicated that, in the
case of fibers, the FWM efficiency at any wavelength is
degraded due to the chromatic dispersion of the fiber
at that wavelength [7]. Hence, the use of dispersion
shifted fiber (DSF) and choice of the signal wavelengths
close to or at the zero dispersion wavelength are suggested as the best options to achieve phase matching
and hence higher FWM efficiencies. A number of
research papers have discussed this aspect [1,2,4]. However, at high signal powers, the non-linearity in the fiber
also contributes to the phase mismatch through self- and
cross-phase modulation. Therefore, it is the total phase
mismatch due to the dispersion and the nonlinearity of
the fiber which decides the FWM efficiency in optical
fibers [23,24].
In the presence of pump waves, the efficiency of generation of the Stokes (longer wavelength) and anti Stokes
(shorter wavelength) components (sidebands) is different,
since the dispersion at those wavelengths is not the same
in a fiber. Also, the contribution of the dispersive phase
to the total phase mismatch and hence, the efficiency is different in the normal and anomalous dispersion regions.
There are a multitude of research papers, some of them
quite recent, which analyze the experimental results ignoring the appropriate phase matching factor. The absence of
(or an incorrect choice of) the nonlinear phase shift leads to
a skewed spectral output in a super continuum experiment.
Complex designs of dispersion decreasing fiber have been
proposed by other workers to overcome this problem [25]
in supercontinuum generation.
In this paper, we bring out the insufficiency of the existing theoretical model to predict the power of the newly
generated FWM signals in the most general case. The
apparently straight forward method of calculation of
phase shift at the Stokes and anti Stokes wavelength as
given in [2] results in erroneous estimate of efficiencies in
light of our experiments. The efficiency of the newly generated waves in the FWM process depends on the pump
power, wavelength separation between waves undergoing
mixing, chromatic dispersion and length of the fiber.
The more general expressions for the FWM efficiency
derived here have been analyzed with the help of experimental data. In addition, we demonstrate that, one can
design an appropriate length of the fiber, for a given wavelength separation and power of the input pump waves, for
which the Stokes and anti Stokes signals grow at the same
efficiency. Some of these results are validated by our
experiments and could prove to be useful in multi-wavelength source design [26].
2. Theory
Let fi, fj, fk be the frequencies of the CW inputs to the
fiber, with propagation constants ki, kj, kk and let Ai, Aj,
Ak be their corresponding amplitudes. These input waves
interact in the fiber due to the third order nonlinear susceptibility, resulting in the emission of new frequencies at
fijk ¼ fi þ fj fk
ð1Þ
(subscripts i, j, k can select 1, 2, 3, j 5 k) with amplitude,
Aijk and propagation constant, kijk. If two of the input frequencies are the same, it is referred to as the partially
degenerate FWM. If all the frequencies are equal, it is totally degenerate FWM [9]. Assuming that the pump waves
are not depleted due to the generation of the FWM products, the coupled differential equations for the propagating
amplitudes, including the contributions to phase mismatch
due to XPM and SPM can be written as [24]
dAi
1
¼ aAi þ 2icðjAi j2 þ 2jAj j2 þ 2jAk j2 ÞAi
2
dz
dAj
1
2
2
2
¼ aAj þ 2icð2jAi j þ jAj j þ 2jAk j ÞAj
2
dz
dAk
1
2
2
2
¼ aAk þ 2icð2jAi j þ 2jAj j þ jAk j ÞAk
2
dz
dAijk
1
2
2
2
¼ aAijk þ 2icðjAi j þ jAj j þ jAk j ÞAijk
2
dz
1
þ DicAi Aj Ak eiDkz
3
ð2Þ
where a is the fiber attenuation coefficient, c is the nonlinear coefficient given by
c¼
2pn2
kAeff
ð3Þ
with Aeff, the effective fiber core area, k, the vacuum wavelength and n2, the nonlinear index coefficient of the fiber.
Dk is the total phase mismatch at fijk, which has contributions from dispersion (Dkl) and nonlinearity (Dknl).
Dk ¼ Dk nl þ Dk l
ð4Þ
The dispersive phase mismatch at fijk is calculated as
Dk l ¼ k ijk ðk i þ k j k k Þ
ð5Þ
Expanding the propagation constants in a Taylor series
about xk(=2pfk) and retaining terms up to third order in
(x xk), the linear phase mismatch at a given wavelength
can be written in terms of the dispersion and dispersion
slope as [2]
2pk2k
k2k
dDc
Dfik Dfjk Dc þ ðDfik þ Dfjk Þ
Dk l ¼
ð6Þ
c
2c
dk
where Dc is the fiber dispersion,
(both evaluated at kk) and
Dfmn ¼ j fm fn j;
dDc
dk
is the dispersion slope
ðm; nÞ ¼ i; j; k:
The nonlinear contribution to the phase mismatch is
analytically derived in [24] as
208
1 eaLeff
Dk nl ¼ c P i þ P j P k
aLeff
ð7Þ
where Leff is the effective length of the fiber, defined in
terms of the total fiber length (L) and a as
1 eaL
ð8Þ
a
It is worth noting that, in Eq. (4), Dkl is positive in the
normal dispersion region and negative, in the anomalous
dispersion region. Since Dknl is always positive, it is easier
to phase-match (Dk = 0) in the anomalous dispersion
region than at the zero dispersion wavelength.
The FWM efficiency, g is given by
(
)
a2
4eaL sin2 ðDkL=2Þ
g¼ 2
1þ
ð9Þ
a þ Dk 2
ð1 eaL Þ2
Leff ¼
and power generated in the side bands is given by [2,24],
(
)
aL 2
g 2 2
aL ½1 e
ð10Þ
P ijk ¼ D c P i P j P k e
9
a2
where D is the degeneracy factor (D = 2,3,6 for totally
degenerate, partially degenerate and non degenerate case,
respectively).
Referring to Eq. (9), the case of absolute phase matching
(Dk = 0) gives the largest value of FWM efficiency (g = 1).
But the efficiency is an oscillatory function of Dk. Hence,
ignoring the nonlinear contribution Dknl or an approximate estimation of the linear combination, Dkl, can lead
to very erroneous results.
In the case of partially degenerate FWM, we have two
pump frequencies, say f1 and f2 (f1 > f2), to start with,
and they generate the first order FWM products, say at frequencies f3 and f4. The frequencies of the first order side
bands, as required by the energy conservation equations
are f3 = 2f2 f1 (Stokes wavelength k3) and f4 = 2f1 f2
(anti Stokes wavelength k4). Second order four wave mixing products are also formed due to further mixing of f1,
f2, f3 and f4. Let f5(=3f2 2f1) and f6(=3f1 2f2) be the
second order Stokes and anti Stokes frequencies corresponding to the wavelengths, k5 and k6, respectively. Mixing of more than one frequency combinations would result
in a second order product. For example, the prominent
contributions to f6 are from:
Fig. 1. Pictorial representation of the generation of I and II order stokes
and anti stokes waves due to FWM.
where the phase shift due to dispersion is exactly compensated by the phase shift due to the nonlinearity. It is often
implied that in case of phase matching due to nonlinearity,
the frequency of the sidebands depends on the power of the
input signals [9]. Instead it is worth noting that the frequency of the sidebands is always power-independent,
and is given by Eq. (1). In the subsequent sections, we
explain our experimental setup and the results.
3. Experimental details
The experimental setup used to observe FWM is shown
in Fig. 2. Continuous wave outputs from two narrow linewidth distributed feed back (DFB) lasers are combined
using a 3 dB coupler and the combination is amplified
through an erbium doped fiber amplifier (EDFA). The
amplified signal passed through a band pass filter (BPF)
of bandwidth 1 nm, is coupled to a 5 km long DSF and
the output is studied on an optical spectrum analyser.
The DSF used has a zero dispersion wavelength (kd) of
1544 nm (as given by the manufacturer) and a dispersion
slope of 0.072 ps/km nm2. The mode field diameter of the
fiber at the operating wavelength is 8.3 lm, and the attenuation coefficient is 0.2 dB/km. For calculations, the value
of n2 used is 2.4 · 1020 m2/W. The DFB lasers have center
wavelengths of 1550.12 nm and 1550.92 nm with a tunability of 2 nm each.
This setup appears to be similar to the continuous wave
self phase modulation (CW SPM) setup [27]. In CW SPM
method, two DFB laser diodes with parallel polarization
produce an optical beat signal which is propagated through
1. f1, f2, and f4 (non degenerate),
2. f1 and f4 (partially degenerate),
3. f1 and f3 (partially degenerate).
The positions of the first order and the second order
products are shown pictorially on the frequency axis in
Fig. 1.
The efficiency of generation of these frequencies can be
estimated using Eqs. (4), (6), (7), (9), and (10) by setting
their powers appropriately. Perfect phase matching at a
given power is possible only at those sideband frequencies
Fig. 2. Experimental setup to observe FWM.
209
4. Results and discussion
4.1. First order sidebands
The conversion efficiency of the first order sidebands
(Stokes and anti Stokes) at different wavelength separations between the pump waves were measured at pump
powers of 34 mW each. These are shown in Figs. 3 and 4
as discrete data. It has to be noted that the BPF was not
used for wavelength separations >1 nm. The output powers
and hence the conversion efficiencies calculated from Eq.
Conversion Efficiency (dB)
-10
-20
-30
-40
-50
0
0.5
1
1.5
2
2.5
3
Wavelength Separation (nm)
Fig. 3. FWM efficiency for the first order stokes component shown as a
function of wavelength separation between the pump waves. Experimental
data is shown as discrete points. Solid line gives the conversion efficiency
calculated using Eq. (6), considering only the linear phase mismatch.
Dashed line gives the conversion efficiencies calculated, considering linear
and nonlinear phase mismatch. Dotted line gives the efficiency calculated
using Eq. (11).
-10
-20
Conversion Efficiency (dB)
a test fiber and the non-linear phase shift induced by the
self phase modulation effect is measured while the dispersion is neglected [28]. The measurement result is hence
independent of the frequency separation of the laser
sources.
However in our experiment, we observed that the efficiency with which the sidebands grow is a function of frequency separation between the laser sources, in the
presence and absence of polarization controllers before
the 3 dB coupler, and hence we conclude that the phenomenon observed is due to FWM. It is important to note that
the DSF used in this experiment is not of polarization-preserving nature.
One of the pump wavelengths is kept fixed at
1549.12 nm and the other is varied from 1549.92 nm to
1551.92 nm
and the conversion efficiency is calculated as
10 log PP0i where Pi is the power measured in the Stokes
(anti Stokes) wavelength at ki (i = 3,5 for the first and second order Stokes and i = 4,6 for the first and second order
anti Stokes wavelength), respectively and P0 is the pump
power at the output at k2(k1).
-30
-40
-50
0
0.5
1
1.5
2
2.5
3
Wavelength Separation (nm)
Fig. 4. FWM efficiency for the first order anti Stokes component shown as
a function of wavelength separation between the pump waves. Experimental data is shown as discrete points. Solid line gives the efficiency
calculated using Eq. (6), considering only the linear phase shift. Dashed
line gives the conversion efficiencies, considering both the linear and
nonlinear phase mismatches. Dotted line gives the efficiency calculated
using Eq. (12).
(10) considering only the linear phase shift, (as given by
Eq. (6)), are shown as solid lines in the same figures. It is
found that, the experimental results do not fit to the calculated values in case of both the Stokes and the anti Stokes
components. To verify the effect of nonlinear phase shifts
in the generation of the FWM products, the conversion
efficiencies are recalculated, including the nonlinear phase
shifts as given by Eq. (7). These results are shown as dashed
lines in Figs. 3 and 4. The calculated values now match well
with the experimental values for anti Stokes component
(Fig. 4), while it is not so with the Stokes component
(Fig. 3). It is however evident that the nonlinear effects contribute significantly to the phase matching mechanism, at
the power levels considered. The Stokes and anti Stokes
components in our experiment being in the anomalous dispersion region, remain phase matched for larger wavelength separation, due to the contribution from nonlinear
phase shift. The discrepancy in the case of Stokes component requires a detailed analysis.
One possible reason that can be attributed to this is the
variation of kd along the length of the fiber [16,29]. Hence
the experiment was repeated by launching power into the
DSF from the opposite port and it was observed that the
variations in efficiencies between the two cases are within
the experimental errors. This ensures that the optical fiber
under consideration is characterised by good uniformity
and that the fluctuations of kd do not result in detectable
differences in the measurement. Moreover, if a kd is chosen
in the calculation so as to fit the experimental observation
for the Stokes line, then the data would not match with the
theoretical prediction for anti Stokes component. Hence, it
is concluded that in the test sample of fiber used, variation
in kd is not contributing to the observed discrepancy in the
first order spectrum.
210
Another possible reason is the inadequacy of the theoretical model given above. The phase mismatch at the generated frequency, fijk is calculated in Eq. (6) by expanding
the propagation constants of the mixing frequencies as a
Taylor series about xk. However, for larger wavelength separation between the pump waves, this approach may need
to be modified. We expanded the propagation constants
at frequencies f1, f2, and f4 as a Taylor series about the frequency f3 to obtain an expression for the linear phase mismatch (Dkl3) for the Stokes component. Similarly, the
propagation constants at f1, f2 and f3 are expanded about
the frequency f4 to obtain the linear phase mismatch
(Dkl4) for the anti Stokes component. The resulting expressions are given as follows:
4pk23
Dk l3 ¼
ðf1 f2 Þ2 Dc
c
6pk23
3
2 dDc
ðf1 f2 Þ 2k3 Dc þ k3
ð11Þ
c2
dk
c
evaluated at k3 and
with Dc and dD
dk
4pk24
Dk l4 ¼
ðf1 f2 Þ2 Dc
c
6pk24
3
2 dDc
þ
ðf1 f2 Þ 2k4 Dc þ k4
c2
dk
ð12Þ
c
evaluated at k4.
with Dc and dD
dk
It has to be noted that the Eq. (6) and the above equations match for the anti Stokes component while they are
different for the Stokes component. While deriving Eqs.
(11) and (12), terms only up to the third order were
included in the Taylor series expansion. It is verified that
the fourth order dispersion terms do not affect the efficiencies significantly for the wavelengths used in the experiment. The efficiencies calculated using Eqs. (11) and (12)
are shown in dotted lines in Figs. 3 and 4 and they are
now found to match with the experimental results reasonably well for both the Stokes and anti Stokes components.
Fig. 5 clearly shows equal efficiency for the Stokes and anti
Stokes components at a wavelength separation of 0.8 nm
between the pump wavelengths for a DSF length of
5 km. The pump powers were 34 mW each in this measurement and the pump wavelengths were 1550.12 nm and
1550.92 nm.
The linear phase mismatch at the Stokes and anti Stokes
components are different even in Eq. (6) since they are positioned differently with respect to the zero dispersion wavelength. In addition to this, Eqs. (11) and (12) indicate that
they have be calculated differently. This difference depends
on the position of the pump wavelengths with respect to
the zero dispersion wavelength as well as the wavelength
separation between the pump waves. This latter aspect is
well emphasised through the results presented in Figs. 3
and 4.
It is the total phase shift given by Eq. (4), which decides
the efficiency. The nonlinear phase shift, which depends on
Fig. 5. Spectrum at the output of DSF of lengths (a) 5 km and (b) 19 km.
the power, is the same for both the Stokes and anti Stokes
components. But, depending on the magnitude of the nonlinear phase shift, the total phase mismatch for the Stokes
component can be even smaller than the anti Stokes component since the linear and nonlinear phase shifts are subtractive in the anomalous dispersion region. This would
result in more efficient generation of the Stokes component
in the anomalous dispersion region under certain conditions. The presence of an appropriate phase shift due to
dispersion is better for efficient FWM than zero dispersion.
This also necessitates extreme caution in unambiguously
identifying the zero dispersion wavelength since its position
decides the normal and anomalous dispersion regions. The
simplistic model suggesting that the FWM efficiency at any
wavelength is degraded due to the chromatic dispersion at
that wavelength is incorrect. The power dependence of the
first order FWM products and its implications are discussed below.
The efficiencies calculated from Eq. (9) using the modified expressions (11) and (12) for Dkl are shown in Fig. 6. It
is clear from this figure that the Stokes and anti Stokes
components are generated with equal efficiency up to a
wavelength separation of 1.2 nm at a power level of
70 mW each, while they remain equal only up to 1 nm
for a lower power of 7 mW. At a wavelength separation
of 2 nm, the conversion efficiency of the Stokes component
again becomes equal to that of the anti Stokes component
for the lower power level. But at the same wavelength separation, 70 mW would give larger efficiency for the anti
Stokes component. On the contrary, at a wavelength separation of 2.15 nm, the results are reversed. A power of
7 mW gives a larger efficiency for the Stokes component
while 70 mW gives equal efficiencies. Hence it is essential
to choose these parameters carefully while attempting to
get equal powers in Stokes and anti Stokes components
in supercontinuum generation. These results would change
211
Dk l51
Conversion Efficiency (dB)
-10
-30
Dk l52
-50
’Stokes-7mW’
’AntiStokes-7mW’
’Stokes-70mW’
’AntiStokes-70mW’
-70
0
0.5
1
1.5
2
2.5
3
Wavelength separation (nm)
Fig. 6. Efficiency of generation of first order stokes and anti stokes
wavelength as a function of wavelength separation at two different power
levels, calculated for a fiber length of 5 km.
if the power or the center wavelength or the fiber length
were changed. It may be stressed that the calculations are
based on the undepleted pump approximation where the
decrease of the pump is assumed to be only due to the fiber
loss and also that the FWM efficiency has been calculated
on the basis of the output pump power.
The corrections we have introduced in the dispersive
phase shift of the first order Stokes and the anti Stokes
wavelengths, to generalise for all wavelength separations,
can be extended to the generation of the second order mixing products. The corrected formulae and the experimental
results for the second order Stokes and anti Stokes components are discussed in the following section.
4.2. Second order products
Second order FWM products can be formed due to the
mixing between more than one combination of frequencies.
The generation of the second order Stokes component and
the calculation of its efficiency is discussed below. The second order Stokes frequency, f5 is generated primarily due
to the mixing of:
1. f1, f2, and f3 (non degenerate),
2. f2 and f3 (partially degenerate),
3. f2 and f4 (partially degenerate).
In addition to the above possibilities, there could be contributions due to the mixing of higher order frequencies
also, which are not considered in this discussion. The individual linear phase mismatches due to each of the above
contributing term is calculated by expanding the propagation constants at each frequency as a Taylor series about f5.
Retaining up to the third order term, the analytical expressions for the dispersive phase mismatches (Dkl51, Dkl52 and
Dkl53, due to the different combination of frequencies) at
the generated wavelength, k5 can be written as follows:
Dk l53
4pk25
2
ðf1 f2 Þ Dc
¼
c
6pk25
3
2 dDc
ðf1 f2 Þ 2k5 Dc þ k5
c2
dk
2
32pk5
4
2
3 dDc
ðf
f
Þ
k
D
þ
k
þ
1
2
5 c
5
c3
dk
2
2pk5
ðf1 f2 Þ2 Dc
¼
c
2pk25
3
2 dDc
ðf1 f2 Þ 2k5 Dc þ k5
c2
dk
7pk25
dD
c
4
2
3
ðf
f
Þ
k
D
þ
k
þ
1
2
5 c
5
c3
dk
2
8pk5
2
ðf1 f2 Þ Dc
¼
c
16pk25
3
2 dDc
ðf1 f2 Þ 2k5 Dc þ k5
c2
dk
2
112pk5
4
2
3 dDc
ðf
f
Þ
k
D
þ
k
þ
1
2
c
5
5
c3
dk
ð13Þ
ð14Þ
ð15Þ
c
, evaluated at k5. It may be noted that all the
with Dc and dD
dk
three equations have been written only in terms of f1 and f2
using f3, f4 and f5 given in Section 2. In deriving the above
equations, the dispersion of the fiber is assumed to vary
2
linearly and hence, the effect due to ddkD2c is neglected. The
nonlinear phase mismatch and hence, the total phase mismatches Dk51, Dk52 and Dk53, at these generated frequencies
can be calculated using Eqs. 7 and 4 respectively. Thus the
efficiencies and powers (P51, P52 and P53) at f5 due to each
of the above contributions can be calculated using Eq. (10)
with appropriate degeneracy factors. Since the three contributing terms to the total power at f5 need not be in
phase, the resultant power is not the algebraic sum of the
individual powers [10]. This is a very important point to
be noted.
The effective power at f5 can be written as
pffiffiffiffiffiffiffiffiffiffiffiffiffi
P 5 ¼ P 51 þ P 52 þ P 53 þ 2 P 51 P 52 cos½ðDk 51 Dk 52 ÞL
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ 2 P 52 P 53 cos½ðDk 52 Dk 53 ÞL
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ 2 P 51 P 53 cos½ðDk 51 Dk 53 ÞL
ð16Þ
The contribution due to each term in the above expression is significant for larger pump powers, and hence, none
of the terms can be neglected for our experimental
conditions.
The above procedure may be repeated for the second
order product at the anti Stokes wavelength. The phase
mismatches due to the three contributions at the anti
Stokes wavelength take a similar form as that of Eqs.
(13)–(15), with all the terms positive. The total power at
the second order anti Stokes wavelength can be calculated
using Eq. (16), with the corresponding phase mismatches.
The variation of conversion efficiencies with wavelength
separation, for a pump power of 30 mW each, and a fiber
212
AntiStokes
Conversion Efficiency (dB)
First Order
Second Order
-20
Stokes
-20
-40
-40
-60
First Order
Second Order
0
0.5
1
1.5
2
2.5
3
-60
0
0.5
1
1.5
2
2.5
3
Wavelength Separation (nm)
Fig. 7. Variation of conversion efficiencies at the anti Stokes wavelength
with the wavelength separation between the pumps, at a pump power of
30 mW each and fiber length 5 km (k1 is fixed at 1549.12 nm and k2 is
varied). The corresponding variation for the Stokes wavelength is shown
in the inset.
length of 5 km for the first and second order Stokes and
anti Stokes wavelength are as shown in the Fig. 7.
Similar to the first order FWM products, the variation
in the conversion efficiencies with wavelength separation
of the second order Stokes and the anti Stokes wavelengths
are not identical. It is also clear from Fig. 7 that the second
order products are as efficient as the corresponding first
order products for smaller wavelength separations. At this
power, the efficiency of generation of the second order
Stokes and anti Stokes components are also the same, for
a wavelength separation 60.8 nm. This indicates that, the
generation of higher order FWM components with efficiencies as high as the first order components is theoretically
feasible. Thus, a flat supercontinuum generation is possible, starting with two CW frequencies by carefully design-
10
Conversion Efficiency (dB)
0
First Order
Second Order
ing the fiber length, power, the position of pump
wavelengths with respect to the zero dispersion wavelength
and their wavelength separation.
Fig. 8 shows the variation of efficiency of generation of
the first and second order Stokes wavelength with the
pump power, for a wavelength separation of 0.8 nm and
a fiber length 5 km. The corresponding results for anti
Stokes wavelength are exactly identical for these parameters. It is clear that for a wavelength separation of
0.8 nm, the second order FWM product even becomes
more efficient than the first order, for a power >15 dBm.
It has to be reiterated that the above theory is valid under
the assumption that the powers in the generated first and
second order sidebands are much less than the pump
waves. This undepleted pump power approximation is
not quite valid for larger conversion efficiencies, and hence
these results show only the expected trends. In the case of
higher conversion efficiencies, the exact powers in the sidebands can be estimated using the split step Fourier transform method.
The experimental results for the second order FWM
products are now analysed. High efficiencies predicted for
the second order products by the theory discussed above,
could not be observed experimentally. The theoretically
estimated values for the experimental conditions of Fig. 5
and the experimentally observed values are tabulated in
Table 1. This shows that, the results match reasonably well
for the first order FWM products for a 5 km long fiber,
while they are way off in the case of second order. It is also
seen from Table 1 that the estimated efficiencies for the
19 km fiber are higher than that of the 5 km fiber, in the
first and second order. However, the experimentally
observed conversion efficiencies of the first and second
order components for the 19 km fiber are much lower than
the theoretically predicted values. Considerable fluctuations are observed at longer lengths and at higher pump
powers, and this could indicate the reasons for the mismatch between the experimentally observed results and
the estimated values. These fluctuations can be attributed
to more than one factor, which are discussed in the next
section.
-10
-20
Table 1
Theoretical estimate and experimental results of conversion efficiencies of
the stokes and anti stokes components for different lengths of the fiber
-30
-40
Wavelength (nm]
Conversion efficiency (dB)
-50
Length = 5 km
Length = 19 km
-60
Observed
Estimated
Observed
Estimated
36.5
24.1
< 36.0
17.2
17.0
17.6
17.0
12.1
17.0
17.6
18.0
13.5
36.0
24.5
< 36.0
23.2
-70
-80
0
2
4
6
8
10
12
14
16
18
20
Pump Power (dBm)
Fig. 8. Variation of conversion efficiencies with pump power for the first
and second order Stokes wavelength, for a pump wavelength separation of
0.8 nm, in a fiber of length 5 km.
1548.52 (Second order
anti stokes)
1549.32 (First order
anti stokes)
1551.72 (First order
stokes)
1552.52 (Second order
stokes)
213
4.3. Fluctuations in FWM efficiencies
It is observed in our experiments that, for longer lengths
of fiber and larger pump powers, the generated FWM
products start fluctuating in their power levels. The pump
powers are also found to fluctuate under these conditions.
FWM is a process which is highly sensitive to the polarisation states of the mixing frequencies, their phase matching and the individual pump powers. Changes in any of
these parameters from the expected values can lead to fluctuations in efficiencies. There are other factors which result
in the stochastic phase fluctuations like fiber medium inhomogeneities due to variations in refractive index, caused by
temperature, density fluctuations in the fiber and intrinsic
thermodynamic fluctuations [31]. For our experimental
conditions, we feel that these latter factors do not contribute significantly to the fluctuations.
The theory of FWM and Eq. (10) are developed under
the assumption that the pumps are linearly polarised along
the same direction and their polarisation states are maintained throughout the length of the fiber. Since the DSF
used in the experiment is not polarisation preserving, the
polarisation states of the pump and the generated frequencies evolve randomly as they propagate along the fiber.
This would lead to fluctuations in phase and hence in mixing efficiencies.
As the pump propagates through the fiber, the power in
the individual wavelength can fluctuate due to the process
of stimulated Brillouin scattering (SBS). It is well known
from the theory of Brillouin scattering that light from a
narrow line width source and longer lengths of fiber reduce
the threshold of stimulated Brillouin scattering [9]. Due to
this, when the incident power exceeds the threshold for
SBS, a part of it is reflected back from the fiber. To study
the influence of this process for our experimental conditions, power from a single DFB laser source was launched
into the DSF used for the previous experiments and the
transmitted power was measured. The experiment was
Transmitted Power (mW)
70
Length = 0.5 km
Length = 3.5 km
Length = 5.0 km
60
50
40
30
20
10
0
repeated for different lengths of the fiber. The variation
of the transmitted power with input power for different
fiber lengths is as shown in Fig. 9.
It is clear from Fig. 9 that, for short lengths of fiber, the
variation is linear, while for longer fiber lengths, the output
power saturates, indicating strong SBS. It has been
reported in the literature [32–35] that, pump power exceeding the SBS threshold leads to fluctuations in the scattered
as well as transmitted powers. Since the pump powers used
in generating the second order products are above the SBS
threshold, it fluctuates, resulting in fluctuations of the powers in the generated mixing products.
The degrading effect due to the phase and power fluctuations is expected to be more for the second order FWM
mixing products, since the powers in the second order
product is proportional to the fifth power of pump power
as given by Eq. (10). This is the possible reason for the discrepancy between the theoretically estimated and experimental values for the power of second order FWM
products. It is reported widely in the literature that SBS
threshold can be increased by phase modulating the pump
waves. High efficiencies for the second order products are
obtained by Song [30], but by modulating the pump signals
with data at 2.5 GB/s. But the validity of the theory discussed above (which is applicable to CW) for modulated
signals is not clear to the authors.
Thus, it is a combination of the two factors namely, non
polarisation preserving nature of the DSF used, and SBS
which lead to phase and hence power fluctuations. The
fluctuations start affecting the generation of higher order
products more severely. The fluctuations observed in first
order FWM products for the results discussed in Section
4.1 were within the margin of experimental errors.
It is conclusive that efficient generation of more number
of sidebands is possible by choosing a highly nonlinear
fiber (so that the fiber length required is small, and SBS
can be minimised), or by using DSF with appropriate
SBS suppression technique, with the right choice of powers, wavelength and length (and hence, the total phase
shift) of the fiber. The presence of an appropriate phase
shift due to dispersion is better for efficient FWM than zero
dispersion. The zero dispersion wavelength has to be
unambiguously identified since it decides the normal and
anomalous dispersion regions. The simplistic model suggesting that the FWM efficiency at any wavelength is
degraded due to the chromatic dispersion at that wavelength is incorrect. When a new frequency is generated
due to multiple frequency combinations such as in the second order, its power should be calculated with the inclusion of appropriate phase factors.
5. Conclusions
0
10
20
30
40
50
60
70
80
Input Power (mW)
Fig. 9. Variation of transmitted power with the launched power for
different lengths of DSF.
The phenomenon of FWM is studied in DSF experimentally. Generation of the first and second order products are
analysed independently. The importance of calculating the
phase shift correctly to estimate the conversion efficiency of
214
the FWM products is demonstrated. The frequently used
formulae for calculating the dispersive phase shift at the
Stokes and anti Stokes wavelength has to be modified for
larger wavelength separations. It is found that the nonlinear phase shift plays an important role in deciding the efficiency of generation of sidebands, even where the
dispersion effects are non-zero. Operating at the zero dispersion wavelength is not always the best way to achieve
high FWM efficiencies. By adjusting the power and hence
the nonlinear phase shifts, or by an appropriate choice of
wavelength separation between the pumps, one can generate sidebands in the anomalous dispersion region with
equal efficiencies. With the appropriate choice of power,
second order products can be generated as efficiently as
the first order products. However, limitations may arise
due to fluctuations in power levels as observed in our
experiments. The possible causes of fluctuations are analysed and solutions are suggested. This will find application
in flat super continuum generation.
Acknowledgements
Consistent help in the experimental work from Amit
Kumar and the financial support from the Department of
Information Technology, Government of India, are gratefully acknowledged.
References
[1] K.O. Hill, D.C. Johnson, B.S. Kawasaki, R.I. MacDonald, J. Appl.
Phys. 49 (1978) 5098.
[2] Nori Shibata, Ralph P. Braun, Robert G. Waarts, IEEE J. Quant.
Electron. 3 (1987) 1205.
[3] K. Inoue, H. Toba, IEEE Photon. Technol. Lett. 4 (1992) 69.
[4] K. Inoue, J. Lightwave Technol. 10 (1992) 1553.
[5] S. Watanabe, T. Chikama, Electron. Lett. 30 (1994) 163.
[6] K. Inoue, H. Toba, J. Lightwave Technol. 13 (1995) 88.
[7] R.W. Tkach, A.R. Chraplyvy, F. Forghieri, A.H. Gnauck, R.M.
Derosier, J. Lightwave Technol. 13 (1995) 841.
[8] S. Gao, C. Yang, G. Jin, Opt. Commun. 206 (2002) 439.
[9] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, San
Diego, CA, 2001 (Chapter 10).
[10] T. Schneider, Nonlinear Optics in Telecommunications, Springer
Verlag, 2004.
[11] F.Di. Pasquale, P. Bayvel, J.E. Midwinter, Electron. Lett. 31 (1995)
998.
[12] I. Neokosmidis, T. Kamalakis, A. Chipouras, T. Sphicopoulos, IEEE
J. Lightwave Technol. 23 (2005) 1137.
[13] A. Bogoni, L. Poti, IEEE J. Quant. Electron. 10 (2004) 387.
[14] K. Nakajima, M. Ohashi, K. Shiraki, T. Horiguchi, K. Kurokawa, Y.
Miyajima, J. Lightwave Technol. 17 (1999) 1814.
[15] H. Chen, Opt. Commun. 220 (2003) 331.
[16] M. Eiselt, R.M. Jopson, R.H. Stolen, J. Lightwave Technol. 15 (1997)
135.
[17] C.J. McKinstrie, S. Radic, Opt. Express 12 (2004) 4973.
[18] F. Curti, F. Matera, G.M. Tosi Beleffi, Opt. Commun. 208 (2002)
85.
[19] B.C. Sarker, T. Yoshino, S.P. Majumder, Optik 113 (2003) 541.
[20] M. Islam, O. Boyraz, IEEE J. Sel. Top. Quant. Electron. 8 (2002) 527.
[21] E. Ciaramella, F. Curti, S. Trillo, IEEE Photon. Technol. Lett. 13
(2001) 142.
[22] T. Morioka, K. Mori, M. Saruwatari, Electron. Lett. 29 (1993) 862.
[23] T. Yamamoto, M. Nakazawa, IEEE Photon. Technol. Lett. 9 (1997)
327.
[24] S. Song, C.T. Allen, K.R. Demarest, R. Hui, J. Lightwave Technol.
17 (1998) 2285.
[25] K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, T. Morioka,
Electron. Lett. 33 (1997) 1806.
[26] D.D. Shah, Nimish Dixit, R. Vijaya, Opt. Fiber. Tech. 9 (2003) 149.
[27] B. Batagelj, IEEE ICTON (2002) 103.
[28] A. Boskovic, S.V. Chernikov, J.R. Taylor, L. Gruner Nielson, O.A.
Levring, Opt. Lett. 21 (1996) 1966.
[29] F. Yaman, Q. Lin, S. Radic, G.P. Agrawal, IEEE Photon. Technol.
Lett. 16 (2004) 1292.
[30] S. Song, Opt. Express 7 (2000) 166.
[31] B. Khubchandani, P.N. Guzdar, R. Roy, Phys. Rev. E 66 (2002)
066609-1.
[32] K. Inoue, T. Hasegawa, H. Toba, IEEE Photon. Technol. Lett. 7
(1995) 327.
[33] A.A. Fotiadi, R. Kiyan, O. Deparis, P. Megret, M. Blondel, Opt.
Lett. 27 (2002) 83.
[34] I. Bar-Joseph, A.A. Friesem, E. Litchman, R.G. Waarts, J. Opt. Soc.
Am. B 2 (1985) 1606.
[35] A.L. Gaeta, R.W. Boyd, Phys. Rev. A 44 (1991) 3205.