## Abstract

We study theoretically and experimentally spectrally flat and broadband
double-pumped fiber-optical parametric amplifiers (2P-FOPAs). Closed formulas
are derived for the gain ripple in 2P-FOPAs as a function of the pump wavelength
separation and power, and the fiber non-linearity and fourth order dispersion
coefficients. The impact of longitudinal random variations of the zero
dispersion wavelength (λ_{0}) on the gain flatness is
investigated. Our theoretical findings are substantiated with experiments using
conventional dispersion shifted fibers and highly nonlinear fibers (HNLFs). By
using a HNLF having a low variation of λ_{0} we demonstrate
high gain and flat spectrum (25 ± 1.5 dB) over 115 nm.

©2007 Optical Society of America

## 1. Introduction

High capacity dense wavelength division multiplexed (DWDM) systems require broadband optical amplifiers with low ripple gain spectrum. Raman amplifiers in a multi-wavelength pump configuration and Erbium doped fiber amplifiers (EDFAs) providing flat gain over ∼100 nm have been demonstrated and are commercially available [1,2]. However, it has been predicted that in the near future, the required bandwidth would be of several hundreds of nanometers [3]. Furthermore, in future optical networks additional functionalities (for example, wavelength conversion for all-optical networking) instead of only amplification will be required. Therefore, there is an increased interest in devices with multifunctional capabilities that could operate over very broad bands with flat spectral response.

A fiber optical parametric amplifier (FOPA) has the potential of providing high gain
over a very broad bandwidth and also offering other functionalities such as
wavelength conversion, optical reshaping, wavelength exchange, phase conjugation,
etc [4,5]. The single-pumped FOPA (1P-FOPA) can exhibit gain over
large bandwidths but with rather poor uniformity [6–9]. This problem can be solved by concatenating
fibers with different zero dispersion wavelengths (λ_{0}) and
lengths [10–13], where numerical simulations predict gain
bandwidths exceeding 200 nm with a ripple of 0.2 dB [13]. Double-pumped FOPAs (2P-FOPAs) can offer flat gain spectra
in a single fiber [14–17]. Theoretically, it has been shown that a very
flat gain spectrum can be obtained in fibers with positive fourth order dispersion
coefficient (β_{4}) [15]. The flatness in a given bandwidth is improved by reducing
the absolute value of β_{4} and by increasing the fiber nonlinear
coefficient (λ) and the pump power (*P*
_{1}
+ *P*
_{2}) (or alternatively by reducing the fiber
length) [15,17]. For example, uniform (ripple < 1 dB) gain over
more than 300 nm is predicted in 2P-FOPAs employing fibers with
β_{4} ∼ 10^{-6} ps^{4}/km and
γ = 15 W^{-1}/km [18]. Experimental results, however, have proved that
theoretical predictions are too optimistic in general. Using a highly nonlinear
dispersion shifted fiber (HNLDSF) with positive β_{4}, a gain
ripple of 3 dB over 33.8 nm bandwidth has been demonstrated [19] (and 41.5 nm for ASE amplification [20]). Recently 47 nm of flat gain was demonstrated using
conventional DSFs and 71 nm using a HNLF having negative β_{4} [21,22].

Several factors conspire against obtaining flat-broadband FOPAs in practice. One is
the lack of highly nonlinear fibers with the desired dispersion coefficients (for
example, the lowest β_{4} reported is
∼3×10^{-5} [23], while the highest γ is 30 W^{-1}/km [24]). Another factor is the fact that a real fiber exhibits
random variations of the zero-dispersion wavelength along its length [16, 25–27]. Calculations reported in [26] predicted that the bandwidth of flat operation of 2P-FOPAs
would be limited to less than 100 nm due to unavoidable fluctuations of
λ_{0}. Still another factor is the effect of polarization
mode dispersion (PMD) [28–31], which tends to produce distortions in the
gain spectrum when the PMD parameter of the fiber and the pump separation are large [29].

In this paper we study theoretically and experimentally spectrally flat and broadband
2P-FOPAs. The main purpose of the theoretical part of this paper is to obtain
analytical expressions of the gain ripple for the various types of 2P-FOPA gain
spectra, which we classify by their number of extrema in sections 2, 3, and 4. In
section 5 we analyze the impact of longitudinal variations of
λ_{0} on gain flatness. In sections 6 and 7 we present our
experimental results. By using a well designed highly nonlinear fiber having a
variation of λ_{0} of ∼0.1 nm we demonstrate high gain
and flat spectrum (25 ± 1.5 dB) over 115 nm. Finally, in section 8 we
draw our conclusions.

## 2. The extrema of the 2P-FOPA gain spectrum and calculation of the gain ripple

The FWM process responsible for parametric gain in a 2P-FOPA satisfies
ω_{1} + ω_{2} = ω_{s} + ω_{i}; where ω_{1}, ω_{2}, ω_{s}, and ω_{i} are the pumps, signal and idler frequencies,
respectively. The propagation constant mismatch of this FWM process is given by

where ω_{c} = (ω_{1} +
ω_{2})/2, Δ_{s} = ω_{s} - ω_{c}, Δω_{p} =
ω_{1} - ω_{c}, and
β_{2c} = β_{2}(ω_{c}) and
ω_{4c} =
ω_{4}(ω_{c}) are the second and fourth
order dispersion coefficients evaluated at ω_{c}, respectively.
The pumps provide a nonlinear contribution to the phase of the waves, so that the
total propagation constant mismatch is κ = Δβ
+ γ(*P*
_{1} +
*P*
_{2}), where *P*
_{1} and
*P*
_{2} are the pump powers, and γ is the
fiber nonlinear coefficient. The scope of this paper is restricted to fibers with
conventional dispersion profiles, having only one λ_{0} and
quartic dispersion relation in the spectral region of interest (i.e., we neglect
fifth and higher order dispersion terms, then β_{4c} =
β_{4} is frequency independent). If the fiber loss can be
neglected, the parametric gain, *G*, is given by [17]

where *x*
_{0} =
γ*P*
_{0}
*L*, *L*
is the fiber length, and *P*
_{0} =
2√*P*
_{1}
*P*
_{2}.

#### 2.1 The extrema of the 2P-FOPA gain spectrum

As a first step to analyze the gain flatness of 2P-FOPAs, we calculate the
extrema of *G*(ω_{s}) that are obtained from
the zeros of the derivative of *G* with respect to
Δω_{s}

The gain is exponential when *x* is real and in this case we have
that *f*(*x*) (≥*f*(0) =
1/3) is monotonic crescent. The extrema are then given by ∂Δβ/∂Δω_{s} = Δω_{s}(2β_{2c} +
β_{4}Δω^{2}
_{s}/3) = 0 and κ = 0. The zeros of ∂Δβ/∂Δω_{s} are located at Δω_{s} = 0 and at Δω_{s} =
±√-6β_{2c}/β_{4}
, while the zeros of κ are at

In principle, there could be four roots of κ = 0. To know if the
extrema are maxima or minima (absolute or local) we calculate the second
derivative of *G*

From Eq. 5 we can see that the zeros of κ are all
absolute maxima. These are points of perfect phase matching where we have
*G* = 1 + sinh^{2}
_{x0}. The zeros of $\frac{\partial \mathrm{\Delta \beta}}{\partial \Delta {\omega}_{s}}$, are local maxima if κ$\frac{{\partial}^{2}\mathrm{\Delta \beta}}{\partial \Delta {\omega}_{s}^{2}}\phantom{\rule{.2em}{0ex}}=\phantom{\rule{.2em}{0ex}}\kappa \left\{{\beta}_{2c}+\frac{{\beta}_{4}}{2}\Delta {\omega}_{s}^{2}\right\}>0$.

Thus there can be spectra having 7 extrema (four maxima and three minima), 5
(three maxima and two minima), 3 (two maxima and one minimum), or 1 (one
maximum). The extremum at Δω_{s} = 0 always exists,
while the existence of the other extrema will depend on the particular values of
the FOPA parameters β_{2c}, β_{4},
γ*P*
_{0}, and
Δω_{p}. (For example, it is easy to show that
a necessary condition for the existence of the extrema at Δω_{s} =
±√-6β_{2c}/β_{4}
is that β_{2c} and β_{4} have opposite
signs).

It would be useful for FOPA design to have expressions of the gain ripple for
these kinds of spectra. As noticed in [15] κ as a function of Δω_{s} being a fourth order polynomial, has minimum ripple in a given region
(∣Δω_{s}∣ <
Δω_{t}) if it is proportional to the Chebyshev
polynomial *T*
_{4} = 1 –
8(Δω_{s}/Δω_{t})^{2}
+
8(Δω_{s}/Δω_{t})^{4}.
This approach is very useful for fibers with β_{4}
> 0 and is further analyzed in section 3.1. If
β_{4} < 0, it follows from Eq. 4 that the two outermost roots of κ = 0 always
exist and are located outside the pumps (∣Δω_{s}∣ >
∣Δω_{p}∣).The Chebyshev
bandwidth, Δω_{t}, is then larger than
Δω_{p}, i.e. includes always the pump
frequencies. In practice, however, as shown in the experimental part, the region
around the pumps cannot be used in general for parametric amplification, since
other ‘spurious’ nonlinear effects are very strong in
those regions. Around the pumps, the combined actions of processes satisfying
ω = 2ω_{1} - ω_{s} and ω = ω_{1} - ω_{2}
+ ω_{i}, drastically perturb the 2P-FOPA, generally reducing the gain [17]. Furthermore, as shown in appendix A, these are regions
of strong crosstalk when the 2P-FOPA is used for DWDM applications. In order to
avoid these ‘spurious’ effects one has to limit the
operation of the 2P-FOPA to a spectral region smaller than
Δω_{p}, say
Δω_{s} <
*b*Δω_{p} (0 <
*b* < 1). Minimizing the gain ripple in this
reduced region cannot be treated with the fourth order Chebyshev polynomial
approach. This is considered next.

#### 2.2 Gain ripple in 2P-FOPAs

The procedure for the calculation of the gain ripple in the various kinds of
spectra is introduced in this subsection with an example of a fiber with
β_{4} < 0. Figure 1 shows a set of gain spectra obtained by tuning
λ_{c} from 1544.56 to 1544.87 nm in steps of 0.039 nm. We
considered a FOPA with *L* = 420 m, γ = 15
(W-km)^{-1}, *P*
_{1} =
*P*
_{2} = 0.25 W, λ_{2} -
λ_{1} = 100 nm, third order dispersion
β_{3c} =
β_{3}(ω_{c}) = 0.065 ps^{3}/km,
β_{4} = -8.5×10^{-5}
ps^{4}/km, λ_{2} = 1595 nm, and
λ_{0} = 1545 nm. The shortest set of λ_{c}
values result in spectra with 7 extrema (showed in Fig. 1a), then increasing λ_{c} yields
spectra with 5 extrema (Fig. 1b). A further increase in λ_{c}
results in spectra with 3 extrema, but with very low gain for these FOPA
parameters. (Spectra with only one extremum at
Δω_{s} = 0, as can be straightforwardly derived
from Eq. (4), only exist in fibers with β_{4}
> 0.)

The spectral region near the pumps should be avoided due to cross-talk (see Appendix A). Note that the spectra depicted in green in Fig. 1(a) exhibit low ripple over a bandwidth, which not includes the region near the pumps. In comparison, over comparable bandwidths, the spectra in black and blue exhibit poorer gain flatness. In the case of Fig. 1(b), the spectra in blue and magenta exhibit also regions of lower ripple (but with a decreased gain if compared with the 7 extrema case) if compared with spectrum in blue.

To calculate the gain ripple of the low ripple regions observed in Fig. 1, we need to know the value of
β_{2c} for each case. This is obtained by equalizing the gain
at Δω_{s} = 0, which is a minimum (maximum) in
spectra with seven (five) extrema, with the gain at a frequency
Δω_{s} =
*b*Δω_{p}, where 0 <
*b* ≤ 1. From Eq. 2 we note that this corresponds to equalize
κ^{2} at these wavelengths, i.e. κ
(Δω_{s} = 0) = ±κ
(Δω_{s} =
*b*Δω_{p}). This procedure
yields two possible values of β_{2c}:

where

By substituting each value of β_{2c} in Eq. (4) we note that with the value in Eq. 6(a) we can have only two roots of κ = 0 (i.e.
spectra with five extrema), while with the value of β_{2c} in
6(b) we have four roots of κ = 0 (spectra with seven extrema). Thus,
with those values of β_{2c} it is possible to calculate
κ (and the gain) at the extrema. For example, with the
β_{2c} in Eq. 6(a) we can calculate the gain at
Δω_{s} = 0 (which is the maximum,
*G*
_{max}) and also at Δω_{s} =
±√-6β_{2c}/β_{4}
(which is the minimum, *G*
_{min}). From these values we
obtain the gain ripple: Δ*G* =
*G*
_{max} - *G*
_{min}. In the
same way with β_{2c} in Eq. 6(b) we can calculate *G*
_{min} at
Δω_{s} = 0, and knowing that
*G*
_{max} = 1 +
sinh^{2}
*x*
_{0} in the spectra with 7 extrema,
we can then find Δ*G*.

It is convenient, in order to have tractable expressions of
*G*
_{max} and *G*
_{min}, to
take the limiting case sinh^{2}x ∼ e^{2x}/4, with
error < 3 % for *x* > 2. Using this
approximation, *G* in decibel units is

In sections 3 and 4 we analyze Δ*G* for the most
representative types of gain spectra.

## 3. Gain ripple in 2P-FOPA spectra with seven extrema

#### 3.1 The fourth order polynomial Chebyshev gain spectrum

In this subsection we calculate the gain ripple of the Chebyshev spectrum as the
parameters β_{4}, γ(*P*
_{1}
+ *P*
_{2}), and Δω_{p} are varied. This spectrum occurs when the three local minima have the
same gain, i.e. when κ^{2} is the same when evaluated at Δω_{s} = 0 or at Δω_{s} =
±√-6β_{2c}/β_{4}.
The condition κ(Δω_{s} = 0) = -κ(Δω_{s} =
±√-6β_{2c}/β_{4})
results in the Chebyshev spectrum characterized by

Figure 2(a) shows the gain spectra obtained with this
value of β_{2c} for fibers with β_{4}
> 0 (blue line) and β_{4} < 0 (black
line) for a 2P-FOPA with the same parameters used in Fig. 1 except that now β_{4} =
± 8 × 10^{-5} ps^{4}/km. These parameters
result in *x*
_{0} = 3.15 and ξ =
±1.07. The fiber with β_{4} > 0
exhibits a ripple of 0.045 dB over a region, given by
Δω_{t} =
(-12β_{2c}/β_{4})^{1/2}, which
we call the Chebyshev bandwidth and is indicated by dotted blue lines. The fiber
with β_{4} < 0 gives a much larger ripple of 3.6
dB.

With the value of β_{2c} from Eq. (9) the phase mismatch at minimum gain is
κ_{min}/2γ*P*
_{0} =
(√2∣*u*∣ -
√0.5sgn(ξ)+∣ξ∣)^{2}.
The sign function of ∣, *sgn*(ξ), is
negative (positive) in fibers with β_{4} < 0
(> 0). We then substitute this value of
κ_{min}/2γ*P*
_{0} in
Eq. 8 to obtain *G*
_{min}. Since the
maximum gain (at κ = 0) is given in dB by
*G*
_{max} ≅ 8.7*x*
_{0}
- 6 , the gain ripple is

Equation (10) expresses the gain ripple as a function of the
parameter ξ. Before discussing the results from Eq. (10), it is important to mention the range of FOPA
parameters for which the preceded analysis is consistent and meaningful. The
existence of four roots in κ occurs only if β_{2c}
and β_{4} have opposite signs. From Eq. (9) it is straightforward to see that this occurs only when
∣ξ∣ ≥ ½. Therefore, Eq. (10) is not valid for ∣ξ∣
< ½.

Figures 2(b) and 2(c) show Δ*G* calculated in
fibers with positive and negative β_{4} values, respectively
and for two representative values of the parametric gain:
*G*
_{max} = 21.35 dB (*x*
_{0}
= 3.15) and *G*
_{max} = 48 dB
(*x*
_{0} = 6.3). In general, decreasing ξ
flattens the 2P-FOPA and the Chebyshev spectrum can offer very low ripple when
β_{4} > 0. As a specific example, we consider
a FOPA characterized by: γ = 30 (W-km)^{-1},
*P*
_{1} + *P*
_{2} =
0.6 W, β_{4} = 1×10^{-5}
ps^{4}/km, pump separation of ∼ 33 THz (250 nm centered at
1535 nm). These values results in ξ = 2.67, i.e.
Δ*G* ∼ 0.8 dB. This small ripple
corresponds to a flat gain spectrum over nearly 250 nm. In Fig. 2(b) we have also plotted the Chebyshev bandwidth
normalized to the pump separation as a function of ξ. In this case a
bandwidth larger than 0.85 is obtained if ξ > 1.5.

For fibers having β_{4} < 0, the smallest ripple
(2.4 dB for *G*
_{max} = 21.35 dB and 6 dB for
*G*
_{max} = 48 dB) is obtained when
∣ξ∣ = 1. For ½ <
∣ξ∣ < 1, the ripple increases.

Even though Eq. (10) expresses the gain ripple as a complicated function of
ξ, it is possible to approximate Δ*G* with
simple expressions of the type Δ*G*
_{dB} =
*a* × ξ^{p} (or Δ*G*
_{db} =
*a*
_{0} + *a* × ξ^{p}), where *a*
_{0}, *a*, and
*p* are constants. Examples of these power law fits are
represented by dotted lines in Figs. 2(b) and 2(c). For the case β_{4} <
0 the fit was for ∣ξ∣ > 1. Table I quotes the respective values of
*a*
_{0}, *a*, and *p*.
These simple expressions can be used as a rule of thumb to estimate the amount
of increase (or decrease) in Δ*G* by increasing (or
decreasing) ξ. For example, when *G*
_{max} =
48 dB and β_{4} > 0, increasing ξ by a
factor of 2 (for instance by increasing β_{4} by a factor of
2), should lead to a factor of 2^{2.9} ∼ 8 increase in
Δ*G*.

#### 3.2 Gain spectrum with seven extrema and arbitrary shape

The gain spectrum with Chebyshev shape in fibers with β_{4}
< 0 had a rather poor flatness, but the gain ripple can be minimized
for the other spectral shapes discussed in Fig. 1 (a). Figure 3(a) shows the gain spectrum obtained with the
same parameters as in Fig. 2(a) when the region of minimization is
*b* =
Δω_{s}/Δω_{p} =
0.85.

The phase mismatch at minimum gain can be calculated using the value of
β_{2c} in Eq. 6(b) as a function of the region of ripple minimization,
*b*:

To calculate the gain ripple we note that the maximum gain is
*G*
_{max} ≅
8.7*x*
_{0} - 6 , while the minimum gain is calculated by
combining Eqs. (11) and (8). In Figure 3(b) we plot the gain ripple for the case
*b* = 0.85 and for two values of
*x*
_{0} = 3.15 and *x*
_{0} = 6.3.
Comparing these results to those shown in Fig. 2, for values of
∣ξ∣ > 1.5 (where the gain ripple is
high) the two results are very similar for both *x*
_{0} =
3.15 and 6.3. For values ∣ξ∣ < 1.5
the spectrum analyzed in this subsection exhibits a smaller ripple. This means
that it is possible to reduce the ripple by slightly reducing the bandwidth of
amplification. (Note in Eq. 11 that the κ is reduced as long as we reduce
*b*.)

#### 4. Gain ripple in 2P-FOPA spectra with five extrema

In this section we study minimization of the gain ripple in spectra having five
extrema in fibers with β_{4} < 0 (The case of
fibers with β_{4} > 0 is discussed in Appendix B). Figure 4 shows two typical gain spectra with identical
FOPA parameters (*x*
_{0}, β_{4}, and
Δω_{p}) as in Figs. 2(a) and 3(a). The spectra were obtained for two different ways of
minimizing the gain ripple: the solid line corresponds to equalizing the gain at
Δω_{s} = 0 with that at
Δω_{s} = Δω_{p},
while the dashed line is obtained by equalizing the gain at
Δω_{s} = 0 with that at
Δω_{s} =
0.85Δω_{p}. This equalization leads to the value
of β_{2c} in Eq. 6(a) from which it is possible to calculate the phase
mismatch at Δω_{s} = 0 (maximum) and at
Δω_{s} =
±√-6β_{2c}/β_{4}
(minima):

Equations 12(a) and 12(b) were used to calculate
*G*
_{max}, *G*
_{min}, and then
Δ*G* as a function of *b*. Figures 5(a) and 5(b) show Δ*G* for
*b* = 1 and 0.85, respectively. As in the case of spectra with 7
extrema, it is apparent that low ripple is obtained for small values of
∣ξ∣. Also, the gain ripple is slightly smaller
for a smaller value of *b*. However, this slight improvement in
flatness is obtained by reducing the overall gain as can be observed in Fig. 4. It is interesting comparing Figs. 5(b) and 3(b) (i.e. when the ripple is minimized in the region
Δω_{s} =
0.85Δω_{p}): if
∣ξ∣ < 0.5 the case of spectra with 5
extrema produces a flatter spectrum if compared with the seven extrema case; on
the other hand, if ∣ξ∣ > 0.5 similar
values of Δ*G* for both cases are obtained when
*x*
_{0} = 3.15; finally, when
*x*
_{0} = 6.3 the spectrum with 7 extrema exhibit a
flatter gain.

Note in Fig. 5(a) that Δ*G* scales
nearly linearly with ξ. It can be shown that the exact expression of
Δ*G* for the optimised spectrum with 5 extrema and
*b* = 1 is

For ∣ξ∣ ≪ 1 Eq. (13) reduces to Δ*G _{dB}*
≈ (-1.25

*x*

_{0}+1.45)ξ. In very high gain amplifiers (

*x*

_{0}≫ 1) the ripple becomes independent of pump power: for example, if

*P*

_{1}=

*P*

_{2}, the limiting ripple is Δ

*G*

_{dB}≫ -0.05β

_{4}Δω

^{4}

_{p}

*L*.

#### 5. Influence of variations of λ_{0} and polarization mode
dispersion

In general, there may be small random fluctuations of core radius and refractive
index along the fiber, resulting in fluctuations of λ_{0}
that influence the efficiency of parametric amplifiers. In order to study the
effects of variations of λ_{0}, we numerically solved the
signal propagation Eqs. given in Ref [26] by dividing the fiber in 5000 segments of length
Δ*z*. In each segment of fiber we defined a
variation of the zero dispersion wavelength as
Δ_{0}(*z _{k}*) =
〈λ

_{0}〉 + δ

_{λ0}(

*z*), where

_{k}*k*= 1, 2,.., 5000, and the random variation δλ

_{0}(z

_{k}) was generated using [33]

where Δ*z* = *z _{k}* -

*z*

_{k-1},

*L*

_{c}is a parameter related to the correlation length of the random process, and

*r*is a computer generated random number with normal distribution (zero mean and unit variance). By using this definition, δλ

_{k}_{0}(

*z*) is a Gaussian stochastic process with expected values of 〈δ〉

_{k}_{0}〉 = 0, correlation length

*L*

_{corr}=

*L*(1 -

_{c}*e*

^{-L/Lc}), and standard deviation σ

_{λ0}.

The gain ripple was calculated as a function of the standard deviation of the
variation of λ_{0} as follows: Eq. (14) was first used to generate a set of 25 to 35 simulated
fibers for each value of σ_{λ0}. In order to
obtain the minimum ripple in each fiber, the gain spectrum was calculated for 60
pump locations by finely tuning of one of the pumps in a range of 1.2 nm and
then keeping the flattest gain spectrum. Note that a similar procedure is
employed in laboratory experiments to minimize the ripple. We then obtained the
Δ*G* for each fiber and calculated the average of
those 25-35 Δ*G* values.

#### 5.1 The influence of third order dispersion on the impact of
λ_{0} fluctuations

The 2P-FOPA parameters in our numerical simulations are
γ(*P*
_{1} +
*P*
_{2}) = 28 km^{-1}, *L* =
0.2 km, β_{4} = -2 ×
10^{-4}ps^{4}/km, and average zero dispersion wavelength
〈λ_{0}〉 = 1570 nm. The pumps are
located at λ_{1} ≅ 1520 nm and
λ_{2} ≅ 1621 nm, so the wavelength separation is
∼100 nm. We assumed *L _{c}* = 100 m, then the
correlation length is

*L*≅ 86.5 m. With this set of parameters ∣ ≅ -0.6 and we considered a gain spectrum of the type having 5 extrema. We did simulations for two values of the third order dispersion. Figure 6(a) shows the gain ripple as a function of σ

_{corr}_{λ0}for β

_{3}(ω

_{0}) = β

_{30}= 0.065 ps

^{3}/km (red squares) and β

_{30}= 0.0325 ps

^{3}/km (black squares). Several interesting features can be observed. For both values of β

_{30}, the ripple decreases as the variation of λ

_{0}increases reaching a minimum value before increasing strongly. This means that for this kind of spectrum, adequate amounts of variations of λ

_{0}tend to flatten the gain spectrum (the ripple was reduced from 4.3 dB to ∼1.6 dB).

A second interesting feature is that the impact of the variation of
λ_{0} depends on the value of β_{30}:
reducing β_{3} by a factor of two allows
σ_{λ0} to increase by a factor of two in order
to have the same impact on gain ripple. Figure 6(b) shows a typical example of the 25
realizations (25 simulated fibers) having σ_{λ0}
≅ 0.525 nm and β_{30} ≅ 0.065
ps^{3}/km. For comparison, the black bold line represents the gain
spectrum without variations of λ_{0}, i.e.
σ_{λ0} = 0. Note that a gain reduction occurs
at signal wavelengths at the center of the gain spectrum
(Δω_{s} = 0) and at the outer peaks (where
κ = 0); no gain variation occurs for signal wavelengths at the pumps.
Interestingly, at Δω_{s} ∼
±√-6β_{2c}/β_{4}
the gain increases slightly, resulting in a flatter spectrum. Note that since
the pumps are optimized to obtain the flattest gain, their locations do not
necessarily coincide with those that give the minimum ripple when
σ_{λ0} = 0.

For large values of σ_{λ0} we observed, as
expected, a strong gain reduction at the center of the spectrum resulting in a
useless FOPA [25–27].

#### 5.2 The influence of γ(P_{1} + P_{2}),
L_{corr}, and Δω_{p} on the impact of
λ_{0} fluctuations

We analyze now the influence of the correlation length by considering the same
2P-FOPA as in the previous subsection (i.e.
γ(*P*
_{1} +
*P*
_{2}) = 28 km^{-1}, *L* =
0.2 km, β_{30} = 0.065 ps^{3}/km,
β_{4} = -2 × 10^{-4} ps^{4}/km,
〈λ_{0}〉 = 1570 nm, and pumps
separation ∼100 nm), but now we change
*L _{corr}* to 8.65 m. Our results are plotted in Fig. 7 by the cyan triangles. Again, the gain ripple
exhibits the same behavior: decreases as σ

_{λ0}increases reaching a minimum value for σ

_{λ0}= 0.71 nm before increasing strongly. For comparison, we have plotted in red squares the case with

*L*= 86.5 m. Note that decreasing

_{corr}*L*by a factor of 10 allows σ

_{corr}_{λ0}to increase by a factor of 1.4 in order to have the same impact on gain ripple. This result indicates that the dependency on

*L*is much smaller than that with β

_{corr}_{30}.

Now we turn our attention to analyze the impact of pump separation. The 2P-FOPA
parameters are: γ(*P*
_{1} +
*P*
_{2}) = 28 km^{-1}, *L* =
0.2 km, β_{30} = 0.065 ps^{3}/km,
〈λ_{0}〉 = 1570 nm,
*L _{corr}* = 86.5 m, β

_{4}= -1.25 × 10

^{-5}ps

^{4}/km, and pumps separation ∼200 nm. The value of β

_{4}was reduced in order to keep constant ξ. The blue triangles in Fig. 7 show the results. Note that the minimum ripple is obtained for σ

_{λ0}= 0.13 nm. Comparing with the case of pumps separation of 100 nm (red squares), it is noted that an increase of the pump separation by a factor of two, in order to have the same impact of variations of λ

_{0}on the gain ripple, the fiber should have a value of σ

_{λ0}four times smaller.

Finally, we change γ(*P*
_{1} +
*P*
_{2}) to 56 km^{-1} and the 2P-FOPA
parameters are now: *L* = 0.1 km, β_{30} =
0.065 ps^{3}/km, 〈λ_{0}〉 = 1570
nm, *L _{corr}* = 86.5 m, β

_{4}= -2.5 × 10

^{-5}ps

^{4}/km, and pumps separation ∼200 nm. The value of β

^{4}was reduced in order to keep ξ constant. The green circles in Fig. 7 show the results. Comparing with the case of γ(

*P*

_{1}+

*P*

_{2}) = 28 km

^{-1}(blue triangles), it is noted that an increase of γ by a factor of two, in order to have the same impact of variations of λ

_{0}on the gain ripple, the fiber should have a value of σ

_{λ0}two times larger.

The numerical simulations in Figures 6 and 7 showed the influence of the various parameters on the
impact of λ_{0}(*z*) in 2P-FOPA gain. Similar
conclusions can be derived from taking the derivative of *G* with
respect to ω_{0}. To have tractable expressions it is
convenient to consider that in the region of high parametric gain, κ/
2γ*P*
_{0} ≪ 1. In this limit) $\mathrm{log}\left(1-{\left(\frac{\kappa}{2\gamma {P}_{0}}\right)}^{2}\right)\sim 0$ and *x* ≈
*x*
_{0} (1 -
κ^{2}/8γ^{2}
*P*
_{0}
^{2}).
Then the parametric gain can be written as *G*
_{dB}
∼ 8.7*x* - 6. The gain fluctuation,
δ*G*, due to a variation of
δω_{0} in ω_{0} is then

The amount of gain variation is proportional to β_{30},
βω_{0}, and *L* and inversely
proportional to γ*P*
_{0}.
δ*G* depends also on the signal wavelength
location: signal wavelengths close to the pumps suffer low gain variations,
while signal wavelengths far from both pumps suffer of larger gain variations.
This behavior is in agreement with results shown in Figs. 6 and 7. Eq. (15) also indicates that signal wavelengths where there is
phase matching are less affected by variations of ω_{0}. This
is in disagreement with the findings in Fig. 6(b).

#### 5.3 The impact of polarization mode dispersion (PMD)

PMD produces a misalignment of the states of polarization of the pumps and the
signals changing the FWM efficiency as these waves propagate along the fiber.
The alignment of pumps can be quantified by the internal product of their
polarization vectors **s**(ω). If they have parallel states
of polarization at the fiber input then at the fiber output their internal
product is given by
〈*s*(ω_{1}).*s*(ω_{2})〉
= exp⌈-4*D*
_{p}
^{2}
*L*Δω_{p}
^{2}/3⌉, where *D _{p}* is the
PMD coefficient, and

*L*the fiber length [33]. The depolarization effects can be related to a diffusion length defined as

*L*= 3/(2

_{d}*D*Δω

_{p}_{p})

^{2}, where

*L*indicates the distance for which the scalar product is reduced from 1 to 0.37 (i.e. by 4.3 dB) [29]. It was shown through numerical simulations that, as a rule of thumb, if

_{d}*L*>

_{d}*L*, then the PMD decreases the gain spectrum uniformly without distortions [29]. If

*L*<

_{d}*L*, the PMD reduces the gain and induces distortions in the shape of the gain. These distortions are more pronounced at the center of the gain spectrum because the polarization of signal and pumps exhibit more misalignment. The impact of PMD, for a FOPA for which

*L*>

_{d}*L*, can be easily taken into account simply by considering in Eq. (2) an effective interaction length over which the polarizations of pumps and signals, are aligned. In section 7 we show experiments for which

*L*<

_{d}*L*.

#### 6. Experimental setup and experimental results: short length fibers

We built 2P-FOPAs using three different fibers, A, B, and C, whose parameters are
quoted in Table II. Fig. 8 shows the experimental setup. We used tunable
external cavity lasers at λ_{1}, λ_{2},
and λ_{S} as pumps and signal sources. In the case of fibers
A and B, the pumps were amplified using C-band or L-band EDFAs. In order to
obtain high power from the EDFAs, the pump lasers were amplitude modulated in
the form of pulses with durations in the range 5-45-ns. We used an additional
short length of fiber as relative delay (τ) between the pump pulses
to compensate for differences in optical paths, so that, within the FOPA fiber,
the two pulses overlapped in time within 5 % of the width. Optical filters (OF)
were used to reject most of the ASE from the EDFAs. Polarization controllers
(PCs) were used to align the states of polarization of pumps and the signal so
as to maximize the parametric gain. The spectra were characterized using an
optical spectrum analyzer (OSA) with 0.1 nm resolution, and the peak pump powers
were measured using a photodiode and a fast oscilloscope. The fibers were
selected after estimating the value of σ_{λ0} with
the method reported in [32]. We estimate the error in the gain measurements to be
±0.7 dB.

In the case of fiber C, pump 1 was obtained using a single pumped FOPA made with
a HNLF having *L* = 35 m and pumped with ∼30 W pulses
as indicated in Fig. 8 with the dotted lines. Using this approach we were
able to obtain up to 4 W peak powers at these wavelengths - more than enough to
pump the 2P-FOPA. To select this pump 1 we used a WDM coupler that filtered out
wavelengths larger than 1515 nm. Figure 8(b) shows an example of a 2P-FOPA output spectrum
measured in fiber C with *L* = 150 m. Note that amplified noise
around the pumps comes from the noise (that was unfiltered with the WDM)
generated in the 1P-FOPA. ‘Spurious’ FWM tones that are 26
dB smaller than the signals can be also observed.

#### 6.1 Conventional dispersion shifted fiber with L_{A} = 0.95 km

We measured the gain spectrum for three pump wavelength separations: 55, 62, and
68.9 nm. In order to keep the same gain in these three cases, the pump powers
needed to be increased from *P*
_{1} ≅
*P*
_{2} ∼1.8 W (pump separation of 55 nm)
to ∼2.1 W (68.9 nm). The results are plotted with blue circles in Figs. 9 (a), (b), and (c), respectively. In each case the
pumps locations were optimized to minimize the gain ripple. In this conventional
DS fiber the spectral region for ripple minimization was 75–80 % of
the region between the pumps. Note that the gain ripple increases as the pump
separation increases from: Δ*G* ≅ 3.3 dB
for 55 nm pump separation to 5 dB for 68.9 nm. The diffusion lengths for each
pump separation are *L*
_{d(a)} = 1.88
km, *L*
_{d(b)} = 1.48 km, and
*L*
_{d(c)} = 1.2 km. In each
case *L _{d}* >

*L*, so we expect that the effect of any PMD would be to decrease the gain as the pump wavelength separation increases, but without introducing noticeable distortion in the gain spectra. We did simulations using Eq. 2 to compare with the experimental data. To take into account the possible effect of variation of λ

_{0}and PMD, we considered an effective interaction length

*L*

_{int}that corresponds to the experimental gain for each pump separation. These lengths were:

*L*

_{int}= 0.78 km, 0.73 km, and 0.67 km, respectively. The results are plotted in Figure 9 using black and red lines, for λ

_{0}= 1568.25 and 1568.15 nm, respectively. There is a very reasonable agreement between experiments and Eq. (2), meaning that real fibers, that are less than perfect, can be modeled with simple analytical expressions if longitudinal variations of λ

_{0}and PMD are sufficiently low.

Table III shows the values of
Δ*G*
_{dB} obtained using the simple
expression derived by fitting Δ*G* (see caption in Fig. 3), together with the experimental values obtained
by measuring *G*
_{max} and
*G*
_{min} in a region ∼ 75–80 %
between the pumps.

Two additional measurements were made to further characterize the 2P-FOPA. In the
first, we verified that the measured gain was independent of which end of the
fiber was used to input the signal. In the second measurement, we analyzed
polarization dependent gain (PDG). The polarization states of pump1 and pump2
were adjusted to be perpendicular by minimizing the gain of ASE noise. The pump
powers were set to *P*
_{1} ≅
*P*
_{2} ∼ 2.1 W, and the wavelength
separation to 40 nm. The state of polarization of the signal was then varied in
order to measure the maximum, *G*
_{max(pol)} and minimum
gain *G*
_{min(pol)}. The PDG =
*G*
_{max(pol)} -
*G*
_{min(pol)} was measured in the spectral region
between λ_{0} and the pump at λ_{2} (the
region between λ_{0} and λ_{1} should be a
replica of this due to symmetry). The PDG was around 2 dB that is low but not
negligible. The same measurement was made for a pump separation of 69 nm, but we
were unable to obtain a PDG smaller to 5 dB in the region between the pumps.

#### 6.2 Highly nonlinear dispersion shifted fiber with L_{B} = 0.3 km

The pumps were first located at λ_{1} ≅ 1528.6 nm
and λ_{2} = 1613.75 nm, while the pump powers were
*P*
_{1} ∼ 1.9 W and
*P*
_{2} ∼ 1.3 W. These values correspond
to ξ ≅ -0.28. The pump wavelengths were optimized to
minimize the ripple in a spectrum having 5 extrema, as shown in Fig. 10(a) with blue circles. Note that high and flat
gain (*G* ≅ 35 ± 1.5 dB) was obtained over
71 nm bandwidth. There is an appreciable tilt in the gain spectrum due to the
Raman gain produced by the pump at λ_{1} (the measured Raman
gain at λ_{2} is ∼1.4 dB). Using the experimental
parameters in Eq. (2) we obtained the gain spectrum for two values of
λ_{0}: 1570.1 nm (red line) and 1570.15 nm (black line).
The effective interaction length was 248 and 243 meters, respectively.

The measured and calculated ripple in a region
Δω_{s} = 0.83Δω_{p}
(83 % of the region between the pumps) are
Δ*G*
_{exp} = 2.3 dB and
Δ*G*
_{calc} = 0.82 dB, respectively. The
agreement is reasonable within the experimental error and the low impact of
variations in λ_{0} and PMD (*L*
_{d} =
0.44 km for this case). Fig. 10(b) shows typical output spectra for two signal
locations: λ_{s} = 1539 nm (red dotted line) and
λ’_{s} = 1581 nm (blue line).

The pump separation was then increased to 93 nm in order to expand the bandwidth.
In one experiment the pumps were located at λ_{1} = 1524.75
nm and λ_{2} = 1617.75 nm to minimize the gain ripple over
the largest bandwidth. The pump powers were λ_{1}
∼ 1.7 W and *P*
_{2} ∼ 1.1 W. Figure 11(a) shows that *G* ≅
26 ± 1.5 dB over 84 nm. We then used the experimental parameters to
calculate the gain spectrum using Eq. 2. Two values of λ_{0} were used to fit
the data: 1570.05 nm (red line) and 1570.1 nm (black line). The effective
interaction lengths were 230 and 220 meters, respectively, which should take
into account the effects of PMD and longitudinal variations of
λ_{0}. The agreement with experiments is quite good
confirming that variations of λ_{0} and PMD may decrease the
gain, but without introducing distortions in the spectrum.

Note further in Fig. 11(a) that a consequence of having flat gain is
reduction in the overall gain: the maximum gain (occurring at the outer signal
wavelengths) is 10 dB higher compared to the parametric gain in the region
between the pumps. Any attempt to increase the gain in the region between the
pumps leads to an increased gain ripple. This is shown clearly in Fig. 11(b) where pump1 was detuned to
λ_{1} = 1524.6 nm and the pump power was decreased to have
the same amount of gain (∼26 dB). The 3 dB bandwidth decreased to 78
nm, whereas the difference between gain for outer and inner signal wavelengths
decreased to 4.5 dB. Figure 11(b) also shows fittings to the experimental data
for two values of λ_{0}: 1570.05 nm (black line) and 1570.1
nm (red line).

#### 6.3 Highly nonlinear dispersion shifted fiber with L_{C} = 0.1 and
0.15 km

To investigate the 2P-FOPA gain flatness in the case where the pumps are
separated by more than 100 nm we used the fiber *C* (see Table II). This fiber had 2 km of length and was cut in
several pieces, with lengths varying from 100 to 370 m and having estimated
variations of λ_{0} from ∼0.1 to ∼0.4
nm. Figure 12 shows gain spectra obtained with the fibers
with the smallest variations of λ_{0}. The pump at
λ_{1} was generated using a 1P-FOPA. Figure 12(a) shows the gain spectrum of a fiber with 150
meters pumped with *P*
_{1} ∼
*P*
_{2} ∼ 2.1 W at
λ_{1} ≅ 1495.9 nm and λ_{2}
≅ 1611.9 nm. We obtained high and flat gain, *G*
≅ 25 ± 2 dB, over ∼102 nm. We also show two
spectra calculated using Eq. (2) with *L*
_{int} = 119 m and
λ_{0} = 1552.73 (black) and λ_{0} =
1552.78 nm (red). With our parameters we have ξ = -0.95 and, from Eq. 11, we expect a ripple of Δ*G* =
2.4 dB.

The pump at λ_{1} could be tuned over a large region because
it was generated with a 1P-FOPA; however, the L-band EDFA limited the tunability
of pump at λ_{2} and as a consequence the 2P-FOPA bandwidth.
To further increase this bandwidth we cooled the fiber with liquid nitrogen and
the λ0 was shifted to 1546.8 nm. Figure 12(b) shows the gain spectrum of a cooled fiber
with *L* = 100 m pumped with *P*
_{1}
∼ *P*
_{2} ∼ 3.3 W. The pumps were
at λ_{1} ≅ 1483.1 nm and λ_{2}
≅ 1613.6 nm and were again optimized in order to have the smallest
gain ripple. Note that high and flat gain, *G* ≅ 25
± 1.5 dB, over ∼115 nm. This is, to the best of our
knowledge, a record performance in terms of amount of gain and flatness for an
optical amplifier. Dotted lines show fittings to the experimental data using Eq. (2) with *L*
_{int} = 76 m and using
λ_{0} = 1546.89 (red) and λ_{0} =
1546.84 nm (black).

The good agreement between experiments and the simple analytical theory observed
in Fig. 12 indicates, even for pump separations larger than
120 nm, the good quality of the HNLF in terms of low PMD and small fluctuations
of λ_{0}. This was further confirmed in our experiments: by
tuning slightly one of the pump we could retrieve the different spectral shapes
(with 7 and 5 extremes) as in Fig. 1. Also, we have measured gain spectra for pump
separations larger than 120 nm with the other fibers. Even for a variation of
λ_{0} of σ_{λ0}
∼0.4 nm (fiber length of 370 m) we still observed gain spectra that
were in good agreement with the theory.

## 7. Gain spectrum in long length fibers (*L*_{D} = 13.8 km)

One motivation for using long fiber lengths is to use the 2P-FOPA as distributed
amplifier and distributed wavelength converter. The parameters of this fiber are the
same that fiber A, but now σ_{λ0} ∼ 0.25
nm. The experimental setup is similar to that shown in Fig. 8; however, instead of using the amplitude modulator we
used a phase modulator driven by three sinusoidal electrical signals (0.41, 1, and
2.4 GHz) in order to suppress the stimulated Brillouin scattering (SBS). We estimate
the error in the measurements with this setup to be around±0.5 dB.

The fiber was pumped with *P*
_{1} ∼ 190 mW and
*P*
_{2} ∼ 170 mW and the gain spectrum was
measured for three pump wavelength separations of λ_{2} -
λ_{1} = 18.3 nm, 24.8 nm, and 39.4 nm. The pumps were also
tuned in order to minimize the gain ripple. The results are shown in Figs. 14(a), (b), and (c), respectively. Note that as
λ_{2} - λ_{1} increases, the ripple
(calculated in the region between the pumps) increases and the amount of gain
decreases strongly: *G* = 〈*G*〉
± ½Δ*G* = 36.5 ± 1.3 dB
for λ_{2} - λ_{1} = 18.3 nm, 31 ±
1.5 dB for λ_{2} - λ_{1} = 24.8 nm, and 14.5
± 4.5 dB for λ_{2} - λ^{1} = 39.4
nm.

If the gain is calculated using Eq. (2), we find that for the region between the pumps
*G* = 51 ± 0.3 dB for λ_{2} -
λ_{1} = 18.3 nm and *G* = 50 ± 3 dB
for λ_{2} - λ_{1} = 39.4 nm. The disagreement
between Eq. (2) and the experimental data is considerable and is likely
related to both longitudinal variations of λ_{0} and PMD. Since
this fiber has a long length, it is reasonable to suppose that PMD will produce a
considerable misalignment of pump and signal polarizations, thus reducing the gain.
The calculated diffusion lengths for these pump separations are
*L _{d}* = 20.9 km, 11.4 km, and 4.5 km, respectively.
These values of

*L*indicate that PMD could uniformly decrease the gain spectrum for the case λ

_{d}_{2}- λ

_{1}= 18.3 nm and also introduce increased distortions as λ

_{2}- λ

_{1}increases to 24.8 nm and 39.4 nm. To assess the contribution that variations of λ

_{0}have on the observed gain reduction, we performed numerical simulations including the estimated variations of λ

_{0}. For the case λ

_{2}- λ

_{1}= 18.3 nm we found now that drops to around 43–45 dB. This indicates that we can roughly 〈

*G*〉 attribute to variations of λ

_{0}as being responsible for 5–7 dB gain reduction. Numerical simulations were also performed for the case λ

_{2}- λ

_{1}= 24.8 nm and we found that 〈

*G*〉 ∼ 42–45 dB. Therefore, as the pump separation increases the main contribution for gain reduction is PMD.

## 8. Conclusions

We have studied numerically and experimentally broadband double-pumped fiber optical
parametrical amplifiers (2P-FOPAs) having flat spectral response. Expressions for
the gain ripple as a function of the FOPA parameters were deduced for the most
representative kinds of 2P-FOPA spectra, which classified by their number of extrema.
The impact that longitudinal variations of the zero dispersion wavelength has on the
gain spectrum was studied in detail through numerical simulations. We showed that
adequate amounts of variations of λ_{0} tend to flatten the gain
spectrum. We show that this amount of variation depends on
1/β_{30}, on γ(*P*
^{1}
+ *P*
_{2}), on *L*
_{corr},
and on 1/Δω^{2}
_{p}. We experimentally showed that by using well-designed highly non-linear
fibers, 2P-FOPAs with flat spectral response over 115 nm can be obtained. Further
improvement of fibers in terms of the value (and sign) of the fourth order
dispersion and nonlinear coefficients would lead to 2P-FOPAs with flat operation
over several hundreds of nanometers and without requiring pump powers larger than
0.5 W.

We stress that 2P-FOPA for real applications should use cw pumps and should also be implemented in order to have low PDG. Our measurements with co-polarized pulsed pumps, however, exemplify well the potentialities of this device in terms of bandwidth.

## Appendix A

We analyze the amplification of 80 WDM signal channels located from 1478 nm
to 1538 nm (with 100 GHz spacing and -30 dBm input power) by solving the
non-linear Schrödinger equation (in order to take into account
all FWM processes). The parameters are: fiber length *L* = 60
meters, λ_{0} = 1543 nm,
∣λ_{1} - λ_{2}∣
≅ 163.2 nm, γ(*P*
_{1}
+ *P*
_{2}) = 52.5 km^{-1},
β_{30} = 0.016 ps^{3}/km, and
β_{4} = -5.2×10^{5}
ps^{4}/km. With these values, *x*
_{0} = 3.15
and ξ = -0.72. Figure A1 shows the spectra at the input and at the
output of the 2P-FOPA: the WDM signals experience 20 dB of gain with a
ripple of ± 1.4 dB over the 60 nm of bandwidth. The flattest gain
spectrum is obtained when the pumps are located at λ_{1}
= 1462.37 nm and at λ_{2} = 1625.55 nm. The red line in figure A1 shows the gain calculated using the Eq. 2 with identical parameters as in the NLSE. One
important feature in Figure A1 is that spurious tones around the pumps and
at the outer maxima are efficiently generated and are strong enough to
produce considerable crosstalk [35]. In fact, in the NLSE simulation, the channels were
located from 1478 nm up to 1540 nm, because for wavelengths smaller than
1478 nm the tones due to pump-signal spurious FWM are only ∼15 dB
smaller than signal and would introduce prohibited distortion. In the NLSE
simulation we assumed pump and signal with parallel polarizations, which
would normally enhance the generation of spurious tones; however we also
used a low output signal power (∼ -10 dBm).

The extent of the region of high crosstalk will depend on
β_{3}, Δω_{p}, and the
number of WDM channels. In our example, this forbidden band is
∼20 % of the bandwidth between the pumps.

## Appendix B

## B.1 Calculation of gain ripple in spectrum having 5 extrema and
β_{4} > 0

Spectra having 5 extrema in fibers with β_{4} >
0 are differentiated from fibers with β_{4} <
0 (which were analyzed in section 4), by the fact that the four roots of
κ = 0 can be located in the region between the pumps. Thus, the
maximum gain of this spectrum is now *G*
_{max} =
8.7x_{0} - 6. To minimize the gain ripple we need to maximize
the gain at Δω_{s} =
±√-6β_{2}/β_{4}
(local minimum). This implies in maximizing the gain at
Δω_{s} = 0; i.e. setting
κ(Δω_{s} = 0) = 0, from which
we obtain

Figure B1(a) shows the gain spectrum obtained with
β_{2c} in Eq. (B1) with identical parameters as in Figs. 2–4 and for
*x*
_{0} = 3.15. With this value of
β_{2c} it is easy to calculate the gain at Δω_{s} = ±√-6β_{2c}/β4,
and then Δ*G*. The result is shown in Fig. B1(b).

## B.2 Calculation of gain ripple in spectrum having 1 extremum and
β_{4} > 0

By looking at Eq. (4) it can be deduced that spectra having 1 extremum
only occurs when both β_{2c} and β_{4}
are positive. The condition to calculate Δ*G* in
this spectrum is: 1) maximizing the gain at
Δω_{s} = 0, then
*G*
_{max} = = 8.7x_{0} - 6 and 2) calculate
the gain at a frequency Δω_{s} =
*b*Δω_{p}. The value of the
phase mismatch at Δω_{s} = *b*Δω_{p} is κ_{min}/2γ*P* =
*b*
^{2}[0.5 +
ξ(*b*
^{2} - 1)]^{2}. Replacing
this in Eq. (7) leads to the calculation of
*G*
_{min}. Figure B2 shows the plot of
Δ*G* as a function of ξ. Note that
spectra having only one extremum implies in ξ restraint to 0
< ξ < 0.5.

## Acknowledgments

We thank fruitful discussions with G.S. Wiederhecker. We thank Prof. Hypolito J. Kalinowski from CEFET-Paraná for making the Bragg gratings used in some of the experiments. We gratefully acknowledge J.B. Rosolem, A.A. Juriollo, C. Floridia, A. Paradisi, F. Simoes, and R. Arradi from CPqD Foundation for the loan of equipment used in this investigation. This work was financially supported by the Brazilian agencies Fapesp, Capes, and CNPq.

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