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 (λ0) 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 λ0 we demonstrate high gain and flat spectrum (25 ± 1.5 dB) over 115 nm.
©2007 Optical Society of America
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 . 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 . 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) . 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 . 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  (and 41.5 nm for ASE amplification ). 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 , while the highest γ is 30 W-1/km ). 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  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 .
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 = ωs + ωi; where ω1, ω2, ωs, and ωi are the pumps, signal and idler frequencies, respectively. The propagation constant mismatch of this FWM process is given by
where ωc = (ω1 + ω2)/2, Δs = ωs - ωc, Δωp = ω1 - ωc, and β2c = β2(ωc) and ω4c = ω4(ωc) are the second and fourth order dispersion coefficients evaluated at ωc, 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 β4c = β4 is frequency independent). If the fiber loss can be neglected, the parametric gain, G, is given by 
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(ωs) that are obtained from the zeros of the derivative of G with respect to Δωs
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/∂Δωs = Δωs(2β2c + β4Δω2 s/3) = 0 and κ = 0. The zeros of ∂Δβ/∂Δωs are located at Δωs = 0 and at Δωs = ±√-6β2c/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 , are local maxima if κ.
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 Δωs = 0 always exists, while the existence of the other extrema will depend on the particular values of the FOPA parameters β2c, β4, γP 0, and Δωp. (For example, it is easy to show that a necessary condition for the existence of the extrema at Δωs = ±√-6β2c/β4 is that β2c 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  κ as a function of Δωs being a fourth order polynomial, has minimum ripple in a given region (∣Δωs∣ < Δωt) if it is proportional to the Chebyshev polynomial T 4 = 1 – 8(Δωs/Δωt)2 + 8(Δωs/Δωt)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 (∣Δωs∣ > ∣Δωp∣).The Chebyshev bandwidth, Δωt, is then larger than Δωp, 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 - ωs and ω = ω1 - ω2 + ωi, drastically perturb the 2P-FOPA, generally reducing the gain . 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 Δωp, say Δωs < bΔωp (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 λc 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 β3c = β3(ωc) = 0.065 ps3/km, β4 = -8.5×10-5 ps4/km, λ2 = 1595 nm, and λ0 = 1545 nm. The shortest set of λc values result in spectra with 7 extrema (showed in Fig. 1a), then increasing λc yields spectra with 5 extrema (Fig. 1b). A further increase in λc results in spectra with 3 extrema, but with very low gain for these FOPA parameters. (Spectra with only one extremum at Δωs = 0, as can be straightforwardly derived from Eq. (4), only exist in fibers with β4 > 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 β2c for each case. This is obtained by equalizing the gain at Δωs = 0, which is a minimum (maximum) in spectra with seven (five) extrema, with the gain at a frequency Δωs = bΔωp, where 0 < b ≤ 1. From Eq. 2 we note that this corresponds to equalize κ2 at these wavelengths, i.e. κ (Δωs = 0) = ±κ (Δωs = bΔωp). This procedure yields two possible values of β2c:
By substituting each value of β2c 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 β2c in 6(b) we have four roots of κ = 0 (spectra with seven extrema). Thus, with those values of β2c it is possible to calculate κ (and the gain) at the extrema. For example, with the β2c in Eq. 6(a) we can calculate the gain at Δωs = 0 (which is the maximum, G max) and also at Δωs = ±√-6β2c/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 β2c in Eq. 6(b) we can calculate G min at Δωs = 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 ∼ e2x/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 Δωp are varied. This spectrum occurs when the three local minima have the same gain, i.e. when κ2 is the same when evaluated at Δωs = 0 or at Δωs = ±√-6β2c/β4. The condition κ(Δωs = 0) = -κ(Δωs = ±√-6β2c/β4) results in the Chebyshev spectrum characterized by
Figure 2(a) shows the gain spectra obtained with this value of β2c 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 Δωt = (-12β2c/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 β2c from Eq. (9) the phase mismatch at minimum gain is κmin/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 κmin/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
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 β2c 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.
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 × ξp (or ΔG db = a 0 + a × ξp), 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 22.9 ∼ 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 = Δωs/Δωp = 0.85.
The phase mismatch at minimum gain can be calculated using the value of β2c 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 Δωp) 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 Δωs = 0 with that at Δωs = Δωp, while the dashed line is obtained by equalizing the gain at Δωs = 0 with that at Δωs = 0.85Δωp. This equalization leads to the value of β2c in Eq. 6(a) from which it is possible to calculate the phase mismatch at Δωs = 0 (maximum) and at Δωs = ±√-6β2c/β4 (minima):
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 Δωs = 0.85Δωp): 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.
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 ΔGdB ≈ (-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  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(zk) = 〈λ0〉 + δλ0(zk), where k = 1, 2,.., 5000, and the random variation δλ0(zk) was generated using 
where Δz = zk - z k-1, L c is a parameter related to the correlation length of the random process, and rk is a computer generated random number with normal distribution (zero mean and unit variance). By using this definition, δλ0(zk) is a Gaussian stochastic process with expected values of 〈δ〉0〉 = 0, correlation length L corr = Lc(1 - e -L/Lc), and standard deviation σλ0.
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 σλ0. 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 Lc = 100 m, then the correlation length is Lcorr ≅ 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 σλ0 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 σλ0 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 σλ0 ≅ 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. σλ0 = 0. Note that a gain reduction occurs at signal wavelengths at the center of the gain spectrum (Δωs = 0) and at the outer peaks (where κ = 0); no gain variation occurs for signal wavelengths at the pumps. Interestingly, at Δωs ∼ ±√-6β2c/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 σλ0 = 0.
For large values of σλ0 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(P1 + P2), Lcorr, and Δωp 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 Lcorr 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 σλ0 increases reaching a minimum value for σλ0 = 0.71 nm before increasing strongly. For comparison, we have plotted in red squares the case with Lcorr = 86.5 m. Note that decreasing Lcorr by a factor of 10 allows σλ0 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 Lcorr 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, Lcorr = 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 σλ0 = 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 σλ0 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, Lcorr = 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 σλ0 two times larger.
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) 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 LΔωp 2/3⌉, where Dp is the PMD coefficient, and L the fiber length . The depolarization effects can be related to a diffusion length defined as Ld = 3/(2DpΔωp)2, where Ld indicates the distance for which the scalar product is reduced from 1 to 0.37 (i.e. by 4.3 dB) . It was shown through numerical simulations that, as a rule of thumb, if Ld > L, then the PMD decreases the gain spectrum uniformly without distortions . If Ld < 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 Ld > 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 Ld < 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 λS 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 σλ0 with the method reported in . 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 LA = 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 Ld > 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.
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 LB = 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.
The measured and calculated ripple in a region Δωs = 0.83Δωp (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: λs = 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 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 LC = 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.
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 σλ0 ∼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 (LD = 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 σλ0 ∼ 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.
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 Ld = 20.9 km, 11.4 km, and 4.5 km, respectively. These values of Ld 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.
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.
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 . 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, Δωp, and the number of WDM channels. In our example, this forbidden band is ∼20 % of the bandwidth between the pumps.
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 Δωs = ±√-6β2/β4 (local minimum). This implies in maximizing the gain at Δωs = 0; i.e. setting κ(Δωs = 0) = 0, from which we obtain
Figure B1(a) shows the gain spectrum obtained with β2c in Eq. (B1) with identical parameters as in Figs. 2–4 and for x 0 = 3.15. With this value of β2c it is easy to calculate the gain at Δωs = ±√-6β2c/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 β2c and β4 are positive. The condition to calculate ΔG in this spectrum is: 1) maximizing the gain at Δωs = 0, then G max = = 8.7x0 - 6 and 2) calculate the gain at a frequency Δωs = bΔωp. The value of the phase mismatch at Δωs = bΔωp is κmin/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.
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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