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 (λ₀) 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 (β₄) [15]. The flatness in a given bandwidth is improved by reducing the absolute value of β₄ 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 β₄ ∼ 10^-6 ps⁴/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 β₄, 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 β₄ [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 β₄ 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 λ₀. 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 λ₀ 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 λ₀ 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 ω₁ + ω₂ = ω_s + ω_i; where ω₁, ω₂, 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 = (ω₁ + ω₂)/2, Δ_s = ω_s - ω_c, Δω_p = ω₁ - ω_c, and β_2c = β₂(ω_c) and ω_4c = ω₄(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 λ₀ and quartic dispersion relation in the spectral region of interest (i.e., we neglect fifth and higher order dispersion terms, then β_4c = β₄ is frequency independent). If the fiber loss can be neglected, the parametric gain, G, is given by [17]

where x ₀ = γP 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 + β₄Δω² _s/3) = 0 and κ = 0. The zeros of ∂Δβ/∂Δω_s are located at Δω_s = 0 and at Δω_s = ±√-6β_2c/β₄ , 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 + sinh² _x0. The zeros of $\frac{\partial \mathrm{\Delta \beta }}{\partial \Delta {\omega }_s}$, are local maxima if κ$\frac{{\partial }^2\mathrm{\Delta \beta }}{\partial \Delta {\omega }_s^2}=\kappa \left\{{\beta }_{2c}+\frac{{\beta }_4}{2}\Delta {\omega }_s^2\right\}>0$.

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, β₄, γP 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/β₄ is that β_2c and β₄ 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 β₄ > 0 and is further analyzed in section 3.1. If β₄ < 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 = 2ω₁ - ω_s and ω = ω₁ - ω₂ + 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 < bΔω_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 β₄ < 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, λ₂ - λ₁ = 100 nm, third order dispersion β_3c = β₃(ω_c) = 0.065 ps³/km, β₄ = -8.5×10^-5 ps⁴/km, λ₂ = 1595 nm, and λ₀ = 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 β₄ > 0.)

 

Fig. 1. Gain spectra calculated using Eq. (2). By tuning λ_c from shorter to longer wavelengths we have the spectra in (a) Black (λ_c = 1544.56 nm), red (λ_c = 1544.6 nm), green (λ_c = 1544.64 nm) and blue (λ_c = 1544.68 nm). (b) Black (λ_c = 1544.72 nm), red (λ_c = 1544.75 nm), green (λ_c = 1544.79 nm), blue (λ_c = 1544.83 nm), and magenta (λ_c = 1544.87 nm).

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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 = bΔω_p, where 0 < b ≤ 1. From Eq. 2 we note that this corresponds to equalize κ² at these wavelengths, i.e. κ (Δω_s = 0) = ±κ (Δω_s = bΔω_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/β₄ (which is the minimum, G _min). From these values we obtain the gain ripple: ΔG = 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 + sinh² x ₀ in the spectra with 7 extrema, we can then find ΔG.

It is convenient, in order to have tractable expressions of G _max and G _min, to take the limiting case sinh²x ∼ e^2x/4, with error < 3 % for x > 2. Using this approximation, G in decibel units is

In sections 3 and 4 we analyze ΔG 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 β₄, 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 κ² is the same when evaluated at Δω_s = 0 or at Δω_s = ±√-6β_2c/β₄. The condition κ(Δω_s = 0) = -κ(Δω_s = ±√-6β_2c/β₄) results in the Chebyshev spectrum characterized by

Figure 2(a) shows the gain spectra obtained with this value of β_2c for fibers with β₄ > 0 (blue line) and β₄ < 0 (black line) for a 2P-FOPA with the same parameters used in Fig. 1 except that now β₄ = ± 8 × 10^-5 ps⁴/km. These parameters result in x ₀ = 3.15 and ξ = ±1.07. The fiber with β₄ > 0 exhibits a ripple of 0.045 dB over a region, given by Δω_t = (-12β_2c/β₄)^1/2, which we call the Chebyshev bandwidth and is indicated by dotted blue lines. The fiber with β₄ < 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/2γP ₀ = (√2∣u∣ - √0.5sgn(ξ)+∣ξ∣)². The sign function of ∣, sgn(ξ), is negative (positive) in fibers with β₄ < 0 (> 0). We then substitute this value of κ_min/2γP ₀ in Eq. 8 to obtain G _min. Since the maximum gain (at κ = 0) is given in dB by G _max ≅ 8.7x ₀ - 6 , the gain ripple is

 

Fig. 2. (a) Gain spectrum with Chebyshev shape for positive (blue line) and negative (black line) β₄. (b) ΔG _dB as a function of ξ for β₄ > 0 and (c) ΔG _dB as a function of ξ for β₄ < 0. Solid red lines: ΔG as calculated with Eq. (10) for G _max = 21.35 dB and G _max = 48 dB. Dotted black lines: fittings with the expressions quoted in table I.

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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 β₄ 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 ΔG calculated in fibers with positive and negative β₄ 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 β₄ > 0. As a specific example, we consider a FOPA characterized by: γ = 30 (W-km)^-1, P ₁ + P ₂ = 0.6 W, β₄ = 1×10^-5 ps⁴/km, pump separation of ∼ 33 THz (250 nm centered at 1535 nm). These values results in ξ = 2.67, i.e. ΔG ∼ 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 β₄ < 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.

Table I. Expression for fitting ΔG.

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Even though Eq. (10) expresses the gain ripple as a complicated function of ξ, it is possible to approximate ΔG with simple expressions of the type ΔG _dB = a × ξ^p (or ΔG _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 β₄ < 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 ΔG by increasing (or decreasing) ξ. For example, when G _max = 48 dB and β₄ > 0, increasing ξ by a factor of 2 (for instance by increasing β₄ by a factor of 2), should lead to a factor of 2^2.9 ∼ 8 increase in ΔG.

3.2 Gain spectrum with seven extrema and arbitrary shape

The gain spectrum with Chebyshev shape in fibers with β₄ < 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.

 

Fig. 3. (a) Gain spectrum when the ripple is minimized in a region Δω_s = 0.85Δω_p. (b) In red lines: Calculated ΔG _dB as a function of ξ for b = 0.85 for two values of x ₀. Black dotted lines: Power law fits to ΔG. (For x ₀ = 3.15 we have ΔG = 0.9 + 0.9∣ξ∣^1.53, while for x ₀ = 6.3 we have ΔG = 2.2 + 2.3∣ξ∣^1.5.)

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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 ≅ 8.7x 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 β₄ < 0 (The case of fibers with β₄ > 0 is discussed in Appendix B). Figure 4 shows two typical gain spectra with identical FOPA parameters (x ₀, β₄, 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/β₄ (minima):

 

Fig. 4. Gain spectra having 5 extrema. The gain ripple was minimized in the region Δω_s = Δω_p (solid line) and Δω_s = 0.85Δω_p (dashed line).

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Equations 12(a) and 12(b) were used to calculate G _max, G _min, and then ΔG as a function of b. Figures 5(a) and 5(b) show ΔG 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 ΔG 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.

 

Fig. 5. ΔG _dB as a function of ξ when the ripple is minimized in the region (a) Δω_s = Δω_p. (b) Δω_s = 0.85Δω_p. The dotted lines in Figure (a) show the power law fits to ΔG. For x ₀ = 3.15 we have ΔG = 3.2 ∣ξ∣^1.15, while for x ₀ = 6.3 we have ΔG = 8.7 ∣ξ∣^1.2. The value of G _max is 19 dB for the case b = 1, while for the case b = 0.85 depends on ξ and we have plotted in Fig. (b).

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

For ∣ξ∣ ≪ 1 Eq. (13) reduces to ΔG_dB ≈ (-1.25x 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 ΔG _dB ≫ -0.05β₄Δω⁴ _p L.

5. Influence of variations of λ₀ 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 λ₀ that influence the efficiency of parametric amplifiers. In order to study the effects of variations of λ₀, we numerically solved the signal propagation Eqs. given in Ref [26] by dividing the fiber in 5000 segments of length Δz. In each segment of fiber we defined a variation of the zero dispersion wavelength as Δ₀(z_k) = 〈λ₀〉 + δ_λ0(z_k), where k = 1, 2,.., 5000, and the random variation δλ₀(z_k) was generated using [33]

where Δz = z_k - z _k-1, L _c is a parameter related to the correlation length of the random process, and r_k is a computer generated random number with normal distribution (zero mean and unit variance). By using this definition, δλ₀(z_k) is a Gaussian stochastic process with expected values of 〈δ〉₀〉 = 0, correlation length L _corr = L_c(1 - e ^-L/Lc), and standard deviation σ_λ0.

The gain ripple was calculated as a function of the standard deviation of the variation of λ₀ 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 ΔG for each fiber and calculated the average of those 25-35 ΔG values.

5.1 The influence of third order dispersion on the impact of λ₀ fluctuations

The 2P-FOPA parameters in our numerical simulations are γ(P ₁ + P ₂) = 28 km^-1, L = 0.2 km, β₄ = -2 × 10^-4ps⁴/km, and average zero dispersion wavelength 〈λ₀〉 = 1570 nm. The pumps are located at λ₁ ≅ 1520 nm and λ₂ ≅ 1621 nm, so the wavelength separation is ∼100 nm. We assumed L_c = 100 m, then the correlation length is L_corr ≅ 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_λ0 for β₃(ω₀) = β₃₀ = 0.065 ps³/km (red squares) and β₃₀ = 0.0325 ps³/km (black squares). Several interesting features can be observed. For both values of β₃₀, the ripple decreases as the variation of λ₀ increases reaching a minimum value before increasing strongly. This means that for this kind of spectrum, adequate amounts of variations of λ₀ 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 λ₀ depends on the value of β₃₀: reducing β₃ 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 β₃₀ ≅ 0.065 ps³/km. For comparison, the black bold line represents the gain spectrum without variations of λ₀, 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/β₄ 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].

 

Fig. 6. (a) Gain ripple as a function of σ_λ0 for β₃₀ = 0.065 ps³/km (red squares) and β₃₀ = 0.0325 ps³/km (black squares). The lines are guides for the eye. (b) Optimized output spectra obtained with 25 simulated fibers having σ_λ0 = 0.52 nm and β₃₀ = 0.065 ps³/km. The bold black line indicates the spectrum for the case σ_λ0 = 0.

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5.2 The influence of γ(P₁ + P₂), L_corr, and Δω_p on the impact of λ₀ 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, β₃₀ = 0.065 ps³/km, β₄ = -2 × 10^-4 ps⁴/km, 〈λ₀〉 = 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_corr = 86.5 m. Note that decreasing L_corr by a factor of 10 allows σ_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_corr is much smaller than that with β₃₀.

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, β₃₀ = 0.065 ps³/km, 〈λ₀〉 = 1570 nm, L_corr = 86.5 m, β₄ = -1.25 × 10^-5 ps⁴/km, and pumps separation ∼200 nm. The value of β₄ 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 λ₀ 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, β₃₀ = 0.065 ps³/km, 〈λ₀〉 = 1570 nm, L_corr = 86.5 m, β₄ = -2.5 × 10^-5ps⁴/km, and pumps separation ∼200 nm. The value of β⁴ 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 λ₀ on the gain ripple, the fiber should have a value of σ_λ0 two times larger.

 

Fig. 7. Gain ripple as a function of σ_λ0 for γ(P ₁ + P ₂) = 28 km^-1, L = 0.2 km, pumps separation ∼100 nm, and L_corr = 8.65 m (cyan triangles). For γ(P ₁ + P ₂) = 28 km^-1, L = 0.2 km, pumps separation ∼200 nm, and L_corr = 86.5 m (blue triangles). For γ(P ₁ + P ₂) = 28 km^-1, L = 0.1 km, pumps separation ∼200 nm, and L_corr = 86.5 m (green triangles). The lines are guides for the eye. In all cases β₃₀ = 0.065 ps³/km and β₄ is varied in order to keep ξ constant. For comparison the data with red squares in Fig. 6(a) is plotted.

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The numerical simulations in Figures 6 and 7 showed the influence of the various parameters on the impact of λ₀(z) in 2P-FOPA gain. Similar conclusions can be derived from taking the derivative of G with respect to ω₀. To have tractable expressions it is convenient to consider that in the region of high parametric gain, κ/ 2γP ₀ ≪ 1. In this limit) $\log \left(1-{\left(\frac{\kappa }{2\gamma P_0}\right)}^2\right)\sim 0$ and x ≈ x ₀ (1 - κ²/8γ² P ₀ ²). Then the parametric gain can be written as G _dB ∼ 8.7x - 6. The gain fluctuation, δG, due to a variation of δω₀ in ω₀ is then

The amount of gain variation is proportional to β₃₀, βω₀, and L and inversely proportional to γP ₀. δG 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 ω₀. 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(ω₁).s(ω₂)〉 = exp⌈-4D _p ² LΔω_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_d = 3/(2D_pΔω_p)², where L_d 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 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 λ₁, λ₂, 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.

 

Fig. 8. (a) Experimental setup for measurements of gain in FOPAs. (b) 2P-FOPA output spectrum measured in fiber C with L = 150 m.

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Table II:. Parameters for the three fibers in the experiments.

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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: ΔG ≅ 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 λ₀ 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 λ₀ = 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 λ₀ and PMD are sufficiently low.

Table III shows the values of ΔG _dB obtained using the simple expression derived by fitting ΔG (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.

 

Fig. 9. 2P-FOPA gain spectrum for: (a) λ₁ = 1540.7 nm and λ₂ = 1595.65 nm; (b) λ₁ = 1537.3 nm and λ₂ = 1599.2 nm; (c) λ₁ = 1533.9 nm and λ₂ = 1602.8 nm. Blue circles: experimental points. Black and red lines: gain spectrum using Eq. 2 for λ₀ = 1568.15 nm and λ₀ = 1568.25 nm, respectively. The effective interaction lengths are (a) L _int = 0.78 km. (b) L _int = 0.73 km. (c) L _int = 0.67 km.

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Table III. Experimental and numerical ΔG for the three pump wavelength separations.

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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 λ₀ and the pump at λ₂ (the region between λ₀ and λ₁ 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 λ₁ ≅ 1528.6 nm and λ₂ = 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 λ₁ (the measured Raman gain at λ₂ is ∼1.4 dB). Using the experimental parameters in Eq. (2) we obtained the gain spectrum for two values of λ₀: 1570.1 nm (red line) and 1570.15 nm (black line). The effective interaction length was 248 and 243 meters, respectively.

 

Fig. 10. (a) 2P-FOPA gain spectrum: blue squares (measurement), black line (λ₀ = 1570.1 nm), red line (λ₀: 1570.15 nm). (b) Output spectra for two locations of λ_s = 1539 nm (red dotted line) and λ’_s = 1581 nm (blue line). The ellipse indicates unfiltered noise due to the 40 nm free spectral range of the Fabry-Perot filter.

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The measured and calculated ripple in a region Δω_s = 0.83Δω_p (83 % of the region between the pumps) are ΔG _exp = 2.3 dB and ΔG _calc = 0.82 dB, respectively. The agreement is reasonable within the experimental error and the low impact of variations in λ₀ 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).

 

Fig. 11. 2P-FOPA gain spectrum. (a) Blue squares (measurement), black line (λ₀ = 1570.1 nm), red line (λ₀: 1570.05 nm). (b) Black circles (measurement), black line (λ₀ = 1570.15 nm), red line (λ₀: 1570.1 nm).

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The pump separation was then increased to 93 nm in order to expand the bandwidth. In one experiment the pumps were located at λ₁ = 1524.75 nm and λ₂ = 1617.75 nm to minimize the gain ripple over the largest bandwidth. The pump powers were λ₁ ∼ 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 λ₀ 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 λ₀. The agreement with experiments is quite good confirming that variations of λ₀ 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 λ₁ = 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 λ₀: 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 λ₀ from ∼0.1 to ∼0.4 nm. Figure 12 shows gain spectra obtained with the fibers with the smallest variations of λ₀. The pump at λ₁ 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 λ₁ ≅ 1495.9 nm and λ₂ ≅ 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 λ₀ = 1552.73 (black) and λ₀ = 1552.78 nm (red). With our parameters we have ξ = -0.95 and, from Eq. 11, we expect a ripple of ΔG = 2.4 dB.

 

Fig. 12. 2P-FOPA gain spectrum measured with fiber C. (a) L = 150 m. (b) L = 100m.

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The pump at λ₁ 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 λ₂ 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 λ₁ ≅ 1483.1 nm and λ₂ ≅ 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 λ₀ = 1546.89 (red) and λ₀ = 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 λ₀. 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 λ₀ 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 λ₂ - λ₁ = 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 λ₂ - λ₁ increases, the ripple (calculated in the region between the pumps) increases and the amount of gain decreases strongly: G = 〈G〉 ± ½ΔG = 36.5 ± 1.3 dB for λ₂ - λ₁ = 18.3 nm, 31 ± 1.5 dB for λ₂ - λ₁ = 24.8 nm, and 14.5 ± 4.5 dB for λ₂ - λ¹ = 39.4 nm.

 

Fig. 14. 2P-FOPA gain spectra using fiber D for P ₁ ≅ 190 mW and P ₂ ≅ 170 mW (a) Δλ_pumps = 18.2 nm, (b) Δλ_pumps = 24.8 nm, and (c) Δλ_pumps = 39.4 nm. The lines are guides for the eye.

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If the gain is calculated using Eq. (2), we find that for the region between the pumps G = 51 ± 0.3 dB for λ₂ - λ₁ = 18.3 nm and G = 50 ± 3 dB for λ₂ - λ₁ = 39.4 nm. The disagreement between Eq. (2) and the experimental data is considerable and is likely related to both longitudinal variations of λ₀ 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_d indicate that PMD could uniformly decrease the gain spectrum for the case λ₂ - λ₁ = 18.3 nm and also introduce increased distortions as λ₂ - λ₁ increases to 24.8 nm and 39.4 nm. To assess the contribution that variations of λ₀ have on the observed gain reduction, we performed numerical simulations including the estimated variations of λ₀. For the case λ₂ - λ₁ = 18.3 nm we found now that drops to around 43–45 dB. This indicates that we can roughly 〈G〉 attribute to variations of λ₀ as being responsible for 5–7 dB gain reduction. Numerical simulations were also performed for the case λ₂ - λ₁ = 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 λ₀ tend to flatten the gain spectrum. We show that this amount of variation depends on 1/β₃₀, on γ(P ¹ + P ₂), on L _corr, and on 1/Δω² _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, λ₀ = 1543 nm, ∣λ₁ - λ₂∣ ≅ 163.2 nm, γ(P ₁ + P ₂) = 52.5 km^-1, β₃₀ = 0.016 ps³/km, and β₄ = -5.2×10⁵ ps⁴/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 λ₁ = 1462.37 nm and at λ₂ = 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 β₃, Δω_p, and the number of WDM channels. In our example, this forbidden band is ∼20 % of the bandwidth between the pumps.

 

Fig. A1. Solid lines: Numerical solution of the NLSE for the amplification of 80 signals by a 2P-FOPA. Red line: Gain spectrum obtained with Eq. (2) using identical parameters in the NLSE.

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Appendix B

B.1 Calculation of gain ripple in spectrum having 5 extrema and β₄ > 0

Spectra having 5 extrema in fibers with β₄ > 0 are differentiated from fibers with β₄ < 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 = 8.7x₀ - 6. To minimize the gain ripple we need to maximize the gain at Δω_s = ±√-6β₂/β₄ (local minimum). This implies in maximizing the gain at Δω_s = 0; i.e. setting κ(Δω_s = 0) = 0, from which we obtain

(B1)$${\beta }_{2c}=\Delta {\omega }_p^2{\beta }_4\frac{\left(1-2\xi \right)}{\left(24\xi \right)}.$$

 

Fig. B1. (a) Gain spectrum obtained with β_2c from Eq. B1. Inset: zoom of gain spectrum. (b) Gain ripple as a function of ξ for two values of x ₀. Continuous lines in red: analytical calculation. Dashed lines: power law fits to ΔG. We have ΔG = 0.54ξ^3.85 for x ₀ = 3.15 and ΔG = 0.26ξ^3.55 for x ₀ = 6.3.

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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 ΔG. The result is shown in Fig. B1(b).

B.2 Calculation of gain ripple in spectrum having 1 extremum and β₄ > 0

By looking at Eq. (4) it can be deduced that spectra having 1 extremum only occurs when both β_2c and β₄ are positive. The condition to calculate ΔG in this spectrum is: 1) maximizing the gain at Δω_s = 0, then G _max = = 8.7x₀ - 6 and 2) calculate the gain at a frequency Δω_s = bΔω_p. The value of the phase mismatch at Δω_s = bΔω_p is κ_min/2γP = 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 ΔG as a function of ξ. Note that spectra having only one extremum implies in ξ restraint to 0 < ξ < 0.5.

 

Fig. B2. (a) Spectrum with one extreme (β₄ > 0). (b) ΔG as a function of ξ for two values of x ₀. Continuous lines in red: analytical calculation. Dashed lines: power law fits to ΔG. We have ΔG = 3.1 - 2.8ξ^0.9 for x ₀ = 3.15 and ΔG = 1.25 - 1.1ξ^0.9 for x ₀ = 6.3.

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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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