## Spectrally flat and broadband double-pumped fiber optical parametric amplifiers

Optics Express, Vol. 15, Issue 9, pp. 5288-5309 (2007)

http://dx.doi.org/10.1364/OE.15.005288

Acrobat PDF (2177 KB)

### Abstract

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

© 2007 Optical Society of America

## 1. Introduction

2. Y.B. Lu, P.L. Chu, A. Alphones, and P. Shum, “A 105-nm ultrawide-band
gain-flattened amplifier combining C-and L-band dual-core EDFAs in a
parallel configuration,” IEEE Photon.
Technol. Lett. **16**, 1640–1642
(2004). [CrossRef]

4. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric
amplifiers and their applications,” IEEE
J. Sel. Top. Quantum Electron. **8**, 506–520
(2002). [CrossRef]

5. S. Radic and C.J. McKinstrie, “Optical amplification and signal
processing in highly nonlinear optical fiber,”
IEICE Trans. Electron. **E88-C**, 859–869
(2005). [CrossRef]

6. M Yu, C.J. McKinstrie, and GP Agrawal, “Modulation instabilities in
dispersion flattened fibers,” Phys. Rev.
E , **52**,
1072–1080
(1995). [CrossRef]

_{0}) and lengths [10–13

10. K. Inoue, “Arrangement of fiber pieces for a
wide wavelength conversion range by fiber four-wave
mixing,” Opt. Lett. **19**, 1189–1191
(1994). [CrossRef] [PubMed]

13. L. Provino, A. Mussot, E. Lantz, T. Sylvestre, and H. Maillote, “Broad-band and flat parametric
amplifiers with a multi-section dispersion-tailored nonlinear fiber
arrangement,” J. Opt. Soc. Am. B. **20**, 1532–1539
(2003). [CrossRef]

14. M. Yu, C.J. McKinstrie, and G.P. Agrawal, “Instability due to cross-phase
modulation in the normal dispersion regime,”
Phys. Rev. E **52**, 1072–1080
(1993). [CrossRef]

_{4}) [15

15. M.E. Marhic, Y. Park, F.S. Yang, and L.G. Kazovsky, “Broadband fiber optical parametric
amplifiers and wavelength converters with low-ripple Chebyshev gain
spectra,” Opt. Lett. **21**, 1354–1356
(1996). [CrossRef] [PubMed]

_{4}and by increasing the fiber nonlinear coefficient (λ) and the pump power (

*P*

_{1}+

*P*

_{2}) (or alternatively by reducing the fiber length) [15

15. M.E. Marhic, Y. Park, F.S. Yang, and L.G. Kazovsky, “Broadband fiber optical parametric
amplifiers and wavelength converters with low-ripple Chebyshev gain
spectra,” Opt. Lett. **21**, 1354–1356
(1996). [CrossRef] [PubMed]

17. C.J. McKinstrie, S. Radic, and A.R. Chraplyvy, “Parametric amplifiers driven by two
pump waves,” IEEE J. Sel. Top. Quantum.
Electron. **8**, 538–547
(2002). [CrossRef]

_{4}∼ 10

^{-6}ps

^{4}/km and γ = 15 W

^{-1}/km [18

18. M.Y. Gao, C. Jiang, W. Hu, and J. Wang, “Two-pump fiber optical parametric
amplifiers with three sections fiber allocation,”
Opt. Laser Technol. **38**, 186–191
(2006). [CrossRef]

_{4}, a gain ripple of 3 dB over 33.8 nm bandwidth has been demonstrated [19

19. S. Radic, C.J. McKinstrie, R.M. Jopson, J.C. Centanni, Q. Lin, and G.P. Agrawal, “Record performance of parametric
amplifier constructed with highly nonlinear
fibre,” Electron. Lett. **39**, 838–839
(2003). [CrossRef]

_{4}[21

21. J.M. Chavez Boggio, J.D. Marconi, and H.L. Fragnito, “Double-pumped fiber optical
parametric amplifier with flat gain over 47-nm bandwidth using a
conventional dispersion-shifted fiber,”
IEEE Photon. Technol. Lett. **17**, 1842–1844
(2005). [CrossRef]

_{0}on gain flatness. In sections 6 and 7 we present our experimental results. By using a well designed highly nonlinear fiber having a variation of λ

_{0}of ∼0.1 nm we demonstrate high gain and flat spectrum (25 ± 1.5 dB) over 115 nm. Finally, in section 8 we draw our conclusions.

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

_{1}+ ω

_{2}= ω

_{s}+ ω

_{i}; where ω

_{1}, ω

_{2}, ω

_{s}, and ω

_{i}are the pumps, signal and idler frequencies, respectively. The propagation constant mismatch of this FWM process is given by

_{c}= (ω

_{1}+ ω

_{2})/2, Δ

_{s}= ω

_{s}- ω

_{c}, Δω

_{p}= ω

_{1}- ω

_{c}, and β

_{2c}= β

_{2}(ω

_{c}) and ω

_{4c}= ω

_{4}(ω

_{c}) are the second and fourth order dispersion coefficients evaluated at ω

_{c}, respectively. The pumps provide a nonlinear contribution to the phase of the waves, so that the total propagation constant mismatch is κ = Δβ + γ(

*P*

_{1}+

*P*

_{2}), where

*P*

_{1}and

*P*

_{2}are the pump powers, and γ is the fiber nonlinear coefficient. The scope of this paper is restricted to fibers with conventional dispersion profiles, having only one λ

_{0}and quartic dispersion relation in the spectral region of interest (i.e., we neglect fifth and higher order dispersion terms, then β

_{4c}= β

_{4}is frequency independent). If the fiber loss can be neglected, the parametric gain,

*G*, is given by [17

17. C.J. McKinstrie, S. Radic, and A.R. Chraplyvy, “Parametric amplifiers driven by two
pump waves,” IEEE J. Sel. Top. Quantum.
Electron. **8**, 538–547
(2002). [CrossRef]

*x*

_{0}= γ

*P*

_{0}

*L*,

*L*is the fiber length, and

*P*

_{0}= 2√

*P*

_{1}

*P*

_{2}.

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

*G*(ω

_{s}) that are obtained from the zeros of the derivative of

*G*with respect to Δω

_{s}

*x*is real and in this case we have that

*f*(

*x*) (≥

*f*(0) = 1/3) is monotonic crescent. The extrema are then given by ∂Δβ/∂Δω

_{s}= Δω

_{s}(2β

_{2c}+ β

_{4}Δω

^{2}

_{s}/3) = 0 and κ = 0. The zeros of ∂Δβ/∂Δω

_{s}are located at Δω

_{s}= 0 and at Δω

_{s}= ±√-6β

_{2c}/β

_{4}, while the zeros of κ are at

*G*

*G*= 1 + sinh

^{2}

_{x0}. The zeros of

_{s}= 0 always exists, while the existence of the other extrema will depend on the particular values of the FOPA parameters β

_{2c}, β

_{4}, γ

*P*

_{0}, and Δω

_{p}. (For example, it is easy to show that a necessary condition for the existence of the extrema at Δω

_{s}= ±√-6β

_{2c}/β

_{4}is that β

_{2c}and β

_{4}have opposite signs).

15. M.E. Marhic, Y. Park, F.S. Yang, and L.G. Kazovsky, “Broadband fiber optical parametric
amplifiers and wavelength converters with low-ripple Chebyshev gain
spectra,” Opt. Lett. **21**, 1354–1356
(1996). [CrossRef] [PubMed]

_{s}being a fourth order polynomial, has minimum ripple in a given region (∣Δω

_{s}∣ < Δω

_{t}) if it is proportional to the Chebyshev polynomial

*T*

_{4}= 1 – 8(Δω

_{s}/Δω

_{t})

^{2}+ 8(Δω

_{s}/Δω

_{t})

^{4}. This approach is very useful for fibers with β

_{4}> 0 and is further analyzed in section 3.1. If β

_{4}< 0, it follows from Eq. 4 that the two outermost roots of κ = 0 always exist and are located outside the pumps (∣Δω

_{s}∣ > ∣Δω

_{p}∣).The Chebyshev bandwidth, Δω

_{t}, is then larger than Δω

_{p}, i.e. includes always the pump frequencies. In practice, however, as shown in the experimental part, the region around the pumps cannot be used in general for parametric amplification, since other ‘spurious’ nonlinear effects are very strong in those regions. Around the pumps, the combined actions of processes satisfying ω = 2ω

_{1}- ω

_{s}and ω = ω

_{1}- ω

_{2}+ ω

_{i}, drastically perturb the 2P-FOPA, generally reducing the gain [17

17. C.J. McKinstrie, S. Radic, and A.R. Chraplyvy, “Parametric amplifiers driven by two
pump waves,” IEEE J. Sel. Top. Quantum.
Electron. **8**, 538–547
(2002). [CrossRef]

_{p}, say Δω

_{s}<

*b*Δω

_{p}(0 <

*b*< 1). Minimizing the gain ripple in this reduced region cannot be treated with the fourth order Chebyshev polynomial approach. This is considered next.

### 2.2 Gain ripple in 2P-FOPAs

_{4}< 0. Figure 1 shows a set of gain spectra obtained by tuning λ

_{c}from 1544.56 to 1544.87 nm in steps of 0.039 nm. We considered a FOPA with

*L*= 420 m, γ = 15 (W-km)

^{-1},

*P*

_{1}=

*P*

_{2}= 0.25 W, λ

_{2}- λ

_{1}= 100 nm, third order dispersion β

_{3c}= β

_{3}(ω

_{c}) = 0.065 ps

^{3}/km, β

_{4}= -8.5×10

^{-5}ps

^{4}/km, λ

_{2}= 1595 nm, and λ

_{0}= 1545 nm. The shortest set of λ

_{c}values result in spectra with 7 extrema (showed in Fig. 1a), then increasing λ

_{c}yields spectra with 5 extrema (Fig. 1b). A further increase in λ

_{c}results in spectra with 3 extrema, but with very low gain for these FOPA parameters. (Spectra with only one extremum at Δω

_{s}= 0, as can be straightforwardly derived from Eq. (4), only exist in fibers with β

_{4}> 0.)

_{2c}for each case. This is obtained by equalizing the gain at Δω

_{s}= 0, which is a minimum (maximum) in spectra with seven (five) extrema, with the gain at a frequency Δω

_{s}=

*b*Δω

_{p}, where 0 <

*b*≤ 1. From Eq. 2 we note that this corresponds to equalize κ

^{2}at these wavelengths, i.e. κ (Δω

_{s}= 0) = ±κ (Δω

_{s}=

*b*Δω

_{p}). This procedure yields two possible values of β

_{2c}:

_{2c}in Eq. (4) we note that with the value in Eq. 6(a) we can have only two roots of κ = 0 (i.e. spectra with five extrema), while with the value of β

_{2c}in 6(b) we have four roots of κ = 0 (spectra with seven extrema). Thus, with those values of β

_{2c}it is possible to calculate κ (and the gain) at the extrema. For example, with the β

_{2c}in Eq. 6(a) we can calculate the gain at Δω

_{s}= 0 (which is the maximum,

*G*

_{max}) and also at Δω

_{s}= ±√-6β

_{2c}/β

_{4}(which is the minimum,

*G*

_{min}). From these values we obtain the gain ripple: Δ

*G*=

*G*

_{max}-

*G*

_{min}. In the same way with β

_{2c}in Eq. 6(b) we can calculate

*G*

_{min}at Δω

_{s}= 0, and knowing that

*G*

_{max}= 1 + sinh

^{2}

*x*

_{0}in the spectra with 7 extrema, we can then find Δ

*G*.

*G*

_{max}and

*G*

_{min}, to take the limiting case sinh

^{2}x ∼ e

^{2x}/4, with error < 3 % for

*x*> 2. Using this approximation,

*G*in decibel units is

*G*for the most representative types of gain spectra.

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

### 3.1 The fourth order polynomial Chebyshev gain spectrum

_{4}, γ(

*P*

_{1}+

*P*

_{2}), and Δω

_{p}are varied. This spectrum occurs when the three local minima have the same gain, i.e. when κ

^{2}is the same when evaluated at Δω

_{s}= 0 or at Δω

_{s}= ±√-6β

_{2c}/β

_{4}. The condition κ(Δω

_{s}= 0) = -κ(Δω

_{s}= ±√-6β

_{2c}/β

_{4}) results in the Chebyshev spectrum characterized by

_{2c}for fibers with β

_{4}> 0 (blue line) and β

_{4}< 0 (black line) for a 2P-FOPA with the same parameters used in Fig. 1 except that now β

_{4}= ± 8 × 10

^{-5}ps

^{4}/km. These parameters result in

*x*

_{0}= 3.15 and ξ = ±1.07. The fiber with β

_{4}> 0 exhibits a ripple of 0.045 dB over a region, given by Δω

_{t}= (-12β

_{2c}/β

_{4})

^{1/2}, which we call the Chebyshev bandwidth and is indicated by dotted blue lines. The fiber with β

_{4}< 0 gives a much larger ripple of 3.6 dB.

_{2c}from Eq. (9) the phase mismatch at minimum gain is κ

_{min}/2γ

*P*

_{0}= (√2∣

*u*∣ - √0.5sgn(ξ)+∣ξ∣)

^{2}. The sign function of ∣,

*sgn*(ξ), is negative (positive) in fibers with β

_{4}< 0 (> 0). We then substitute this value of κ

_{min}/2γ

*P*

_{0}in Eq. 8 to obtain

*G*

_{min}. Since the maximum gain (at κ = 0) is given in dB by

*G*

_{max}≅ 8.7

*x*

_{0}- 6 , the gain ripple is

_{2c}and β

_{4}have opposite signs. From Eq. (9) it is straightforward to see that this occurs only when ∣ξ∣ ≥ ½. Therefore, Eq. (10) is not valid for ∣ξ∣ < ½.

*G*calculated in fibers with positive and negative β

_{4}values, respectively and for two representative values of the parametric gain:

*G*

_{max}= 21.35 dB (

*x*

_{0}= 3.15) and

*G*

_{max}= 48 dB (

*x*

_{0}= 6.3). In general, decreasing ξ flattens the 2P-FOPA and the Chebyshev spectrum can offer very low ripple when β

_{4}> 0. As a specific example, we consider a FOPA characterized by: γ = 30 (W-km)

^{-1},

*P*

_{1}+

*P*

_{2}= 0.6 W, β

_{4}= 1×10

^{-5}ps

^{4}/km, pump separation of ∼ 33 THz (250 nm centered at 1535 nm). These values results in ξ = 2.67, i.e. Δ

*G*∼ 0.8 dB. This small ripple corresponds to a flat gain spectrum over nearly 250 nm. In Fig. 2(b) we have also plotted the Chebyshev bandwidth normalized to the pump separation as a function of ξ. In this case a bandwidth larger than 0.85 is obtained if ξ > 1.5.

_{4}< 0, the smallest ripple (2.4 dB for

*G*

_{max}= 21.35 dB and 6 dB for

*G*

_{max}= 48 dB) is obtained when ∣ξ∣ = 1. For ½ < ∣ξ∣ < 1, the ripple increases.

*G*with simple expressions of the type Δ

*G*

_{dB}=

*a*× ξ

^{p}(or Δ

*G*

_{db}=

*a*

_{0}+

*a*× ξ

^{p}), where

*a*

_{0},

*a*, and

*p*are constants. Examples of these power law fits are represented by dotted lines in Figs. 2(b) and 2(c). For the case β

_{4}< 0 the fit was for ∣ξ∣ > 1. Table I quotes the respective values of

*a*

_{0},

*a*, and

*p*. These simple expressions can be used as a rule of thumb to estimate the amount of increase (or decrease) in Δ

*G*by increasing (or decreasing) ξ. For example, when

*G*

_{max}= 48 dB and β

_{4}> 0, increasing ξ by a factor of 2 (for instance by increasing β

_{4}by a factor of 2), should lead to a factor of 2

^{2.9}∼ 8 increase in Δ

*G*.

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

_{4}< 0 had a rather poor flatness, but the gain ripple can be minimized for the other spectral shapes discussed in Fig. 1 (a). Figure 3(a) shows the gain spectrum obtained with the same parameters as in Fig. 2(a) when the region of minimization is

*b*= Δω

_{s}/Δω

_{p}= 0.85.

_{2c}in Eq. 6(b) as a function of the region of ripple minimization,

*b*:

*G*

_{max}≅ 8.7

*x*

_{0}- 6 , while the minimum gain is calculated by combining Eqs. (11) and (8). In Figure 3(b) we plot the gain ripple for the case

*b*= 0.85 and for two values of

*x*

_{0}= 3.15 and

*x*

_{0}= 6.3. Comparing these results to those shown in Fig. 2, for values of ∣ξ∣ > 1.5 (where the gain ripple is high) the two results are very similar for both

*x*

_{0}= 3.15 and 6.3. For values ∣ξ∣ < 1.5 the spectrum analyzed in this subsection exhibits a smaller ripple. This means that it is possible to reduce the ripple by slightly reducing the bandwidth of amplification. (Note in Eq. 11 that the κ is reduced as long as we reduce

*b*.)

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

_{4}< 0 (The case of fibers with β

_{4}> 0 is discussed in Appendix B). Figure 4 shows two typical gain spectra with identical FOPA parameters (

*x*

_{0}, β

_{4}, and Δω

_{p}) as in Figs. 2(a) and 3(a). The spectra were obtained for two different ways of minimizing the gain ripple: the solid line corresponds to equalizing the gain at Δω

_{s}= 0 with that at Δω

_{s}= Δω

_{p}, while the dashed line is obtained by equalizing the gain at Δω

_{s}= 0 with that at Δω

_{s}= 0.85Δω

_{p}. This equalization leads to the value of β

_{2c}in Eq. 6(a) from which it is possible to calculate the phase mismatch at Δω

_{s}= 0 (maximum) and at Δω

_{s}= ±√-6β

_{2c}/β

_{4}(minima):

*G*

_{max},

*G*

_{min}, and then Δ

*G*as a function of

*b*. Figures 5(a) and 5(b) show Δ

*G*for

*b*= 1 and 0.85, respectively. As in the case of spectra with 7 extrema, it is apparent that low ripple is obtained for small values of ∣ξ∣. Also, the gain ripple is slightly smaller for a smaller value of

*b*. However, this slight improvement in flatness is obtained by reducing the overall gain as can be observed in Fig. 4. It is interesting comparing Figs. 5(b) and 3(b) (i.e. when the ripple is minimized in the region Δω

_{s}= 0.85Δω

_{p}): if ∣ξ∣ < 0.5 the case of spectra with 5 extrema produces a flatter spectrum if compared with the seven extrema case; on the other hand, if ∣ξ∣ > 0.5 similar values of Δ

*G*for both cases are obtained when

*x*

_{0}= 3.15; finally, when

*x*

_{0}= 6.3 the spectrum with 7 extrema exhibit a flatter gain.

*G*scales nearly linearly with ξ. It can be shown that the exact expression of Δ

*G*for the optimised spectrum with 5 extrema and

*b*= 1 is

*G*≈ (-1.25

_{dB}*x*

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

*x*

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

*P*

_{1}=

*P*

_{2}, the limiting ripple is Δ

*G*

_{dB}≫ -0.05β

_{4}Δω

^{4}

_{p}

*L*.

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

_{0}that influence the efficiency of parametric amplifiers. In order to study the effects of variations of λ

_{0}, we numerically solved the signal propagation Eqs. given in Ref [26

26. F. Yaman, Q. Lin, S. Radic, and G.P. Agrawal, “Impact of dispersion fluctuations on
dual-pump fiber-optic parametric amplifiers,”
IEEE Photon. Technol. Lett. **16**, 1292–1294
(2004). [CrossRef]

*z*. In each segment of fiber we defined a variation of the zero dispersion wavelength as Δ

_{0}(

*z*) = 〈λ

_{k}_{0}〉 + δ

_{λ0}(

*z*), where

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

_{0}(z

_{k}) was generated using [33

33. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and
polarization drift in installed fibers,”
J. Lightwave Technol. **18**, 941–951
(2000). [CrossRef]

_{0}as follows: Eq. (14) was first used to generate a set of 25 to 35 simulated fibers for each value of σ

_{λ0}. In order to obtain the minimum ripple in each fiber, the gain spectrum was calculated for 60 pump locations by finely tuning of one of the pumps in a range of 1.2 nm and then keeping the flattest gain spectrum. Note that a similar procedure is employed in laboratory experiments to minimize the ripple. We then obtained the Δ

*G*for each fiber and calculated the average of those 25-35 Δ

*G*values.

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

*P*

_{1}+

*P*

_{2}) = 28 km

^{-1},

*L*= 0.2 km, β

_{4}= -2 × 10

^{-4}ps

^{4}/km, and average zero dispersion wavelength 〈λ

_{0}〉 = 1570 nm. The pumps are located at λ

_{1}≅ 1520 nm and λ

_{2}≅ 1621 nm, so the wavelength separation is ∼100 nm. We assumed

*L*= 100 m, then the correlation length is

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

_{corr}_{λ0}for β

_{3}(ω

_{0}) = β

_{30}= 0.065 ps

^{3}/km (red squares) and β

_{30}= 0.0325 ps

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

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

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

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

_{0}depends on the value of β

_{30}: reducing β

_{3}by a factor of two allows σ

_{λ0}to increase by a factor of two in order to have the same impact on gain ripple. Figure 6(b) shows a typical example of the 25 realizations (25 simulated fibers) having σ

_{λ0}≅ 0.525 nm and β

_{30}≅ 0.065 ps

^{3}/km. For comparison, the black bold line represents the gain spectrum without variations of λ

_{0}, i.e. σ

_{λ0}= 0. Note that a gain reduction occurs at signal wavelengths at the center of the gain spectrum (Δω

_{s}= 0) and at the outer peaks (where κ = 0); no gain variation occurs for signal wavelengths at the pumps. Interestingly, at Δω

_{s}∼ ±√-6β

_{2c}/β

_{4}the gain increases slightly, resulting in a flatter spectrum. Note that since the pumps are optimized to obtain the flattest gain, their locations do not necessarily coincide with those that give the minimum ripple when σ

_{λ0}= 0.

_{λ0}we observed, as expected, a strong gain reduction at the center of the spectrum resulting in a useless FOPA [25–27

25. J.M. Chavez Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber
optical parametric amplifier in a 10Gb/s×64 channel dense
wavelength division multiplexing system,”
Opt. Commun. **218**, 303–310
(2003). [CrossRef]

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

*P*

_{1}+

*P*

_{2}) = 28 km

^{-1},

*L*= 0.2 km, β

_{30}= 0.065 ps

^{3}/km, β

_{4}= -2 × 10

^{-4}ps

^{4}/km, 〈λ

_{0}〉 = 1570 nm, and pumps separation ∼100 nm), but now we change

*L*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

_{corr}_{λ0}increases reaching a minimum value for σ

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

*L*= 86.5 m. Note that decreasing

_{corr}*L*by a factor of 10 allows σ

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

*L*is much smaller than that with β

_{corr}_{30}.

*P*

_{1}+

*P*

_{2}) = 28 km

^{-1},

*L*= 0.2 km, β

_{30}= 0.065 ps

^{3}/km, 〈λ

_{0}〉 = 1570 nm,

*L*= 86.5 m, β

_{corr}_{4}= -1.25 × 10

^{-5}ps

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

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

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

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

_{λ0}four times smaller.

*P*

_{1}+

*P*

_{2}) to 56 km

^{-1}and the 2P-FOPA parameters are now:

*L*= 0.1 km, β

_{30}= 0.065 ps

^{3}/km, 〈λ

_{0}〉 = 1570 nm,

*L*= 86.5 m, β

_{corr}_{4}= -2.5 × 10

^{-5}ps

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

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

*P*

_{1}+

*P*

_{2}) = 28 km

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

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

_{λ0}two times larger.

_{0}(

*z*) in 2P-FOPA gain. Similar conclusions can be derived from taking the derivative of

*G*with respect to ω

_{0}. To have tractable expressions it is convenient to consider that in the region of high parametric gain, κ/ 2γ

*P*

_{0}≪ 1. In this limit)

*x*≈

*x*

_{0}(1 - κ

^{2}/8γ

^{2}

*P*

_{0}

^{2}). Then the parametric gain can be written as

*G*

_{dB}∼ 8.7

*x*- 6. The gain fluctuation, δ

*G*, due to a variation of δω

_{0}in ω

_{0}is then

_{30}, βω

_{0}, and

*L*and inversely proportional to γ

*P*

_{0}. δ

*G*depends also on the signal wavelength location: signal wavelengths close to the pumps suffer low gain variations, while signal wavelengths far from both pumps suffer of larger gain variations. This behavior is in agreement with results shown in Figs. 6 and 7. Eq. (15) also indicates that signal wavelengths where there is phase matching are less affected by variations of ω

_{0}. This is in disagreement with the findings in Fig. 6(b).

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

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

_{1}, λ

_{2}, and λ

_{S}as pumps and signal sources. In the case of fibers A and B, the pumps were amplified using C-band or L-band EDFAs. In order to obtain high power from the EDFAs, the pump lasers were amplitude modulated in the form of pulses with durations in the range 5-45-ns. We used an additional short length of fiber as relative delay (τ) between the pump pulses to compensate for differences in optical paths, so that, within the FOPA fiber, the two pulses overlapped in time within 5 % of the width. Optical filters (OF) were used to reject most of the ASE from the EDFAs. Polarization controllers (PCs) were used to align the states of polarization of pumps and the signal so as to maximize the parametric gain. The spectra were characterized using an optical spectrum analyzer (OSA) with 0.1 nm resolution, and the peak pump powers were measured using a photodiode and a fast oscilloscope. The fibers were selected after estimating the value of σ

_{λ0}with the method reported in [32]. We estimate the error in the gain measurements to be ±0.7 dB.

*L*= 35 m and pumped with ∼30 W pulses as indicated in Fig. 8 with the dotted lines. Using this approach we were able to obtain up to 4 W peak powers at these wavelengths - more than enough to pump the 2P-FOPA. To select this pump 1 we used a WDM coupler that filtered out wavelengths larger than 1515 nm. Figure 8(b) shows an example of a 2P-FOPA output spectrum measured in fiber C with

*L*= 150 m. Note that amplified noise around the pumps comes from the noise (that was unfiltered with the WDM) generated in the 1P-FOPA. ‘Spurious’ FWM tones that are 26 dB smaller than the signals can be also observed.

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

*P*

_{1}≅

*P*

_{2}∼1.8 W (pump separation of 55 nm) to ∼2.1 W (68.9 nm). The results are plotted with blue circles in Figs. 9 (a), (b), and (c), respectively. In each case the pumps locations were optimized to minimize the gain ripple. In this conventional DS fiber the spectral region for ripple minimization was 75–80 % of the region between the pumps. Note that the gain ripple increases as the pump separation increases from: Δ

*G*≅ 3.3 dB for 55 nm pump separation to 5 dB for 68.9 nm. The diffusion lengths for each pump separation are

*L*

_{d(a)}= 1.88 km,

*L*

_{d(b)}= 1.48 km, and

*L*

_{d(c)}= 1.2 km. In each case

*L*>

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

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

*L*

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

*L*

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

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

_{0}and PMD are sufficiently low.

*P*

_{1}≅

*P*

_{2}∼ 2.1 W, and the wavelength separation to 40 nm. The state of polarization of the signal was then varied in order to measure the maximum,

*G*

_{max(pol)}and minimum gain

*G*

_{min(pol)}. The PDG =

*G*

_{max(pol)}-

*G*

_{min(pol)}was measured in the spectral region between λ

_{0}and the pump at λ

_{2}(the region between λ

_{0}and λ

_{1}should be a replica of this due to symmetry). The PDG was around 2 dB that is low but not negligible. The same measurement was made for a pump separation of 69 nm, but we were unable to obtain a PDG smaller to 5 dB in the region between the pumps.

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

_{1}≅ 1528.6 nm and λ

_{2}= 1613.75 nm, while the pump powers were

*P*

_{1}∼ 1.9 W and

*P*

_{2}∼ 1.3 W. These values correspond to ξ ≅ -0.28. The pump wavelengths were optimized to minimize the ripple in a spectrum having 5 extrema, as shown in Fig. 10(a) with blue circles. Note that high and flat gain (

*G*≅ 35 ± 1.5 dB) was obtained over 71 nm bandwidth. There is an appreciable tilt in the gain spectrum due to the Raman gain produced by the pump at λ

_{1}(the measured Raman gain at λ

_{2}is ∼1.4 dB). Using the experimental parameters in Eq. (2) we obtained the gain spectrum for two values of λ

_{0}: 1570.1 nm (red line) and 1570.15 nm (black line). The effective interaction length was 248 and 243 meters, respectively.

_{s}= 0.83Δω

_{p}(83 % of the region between the pumps) are Δ

*G*

_{exp}= 2.3 dB and Δ

*G*

_{calc}= 0.82 dB, respectively. The agreement is reasonable within the experimental error and the low impact of variations in λ

_{0}and PMD (

*L*

_{d}= 0.44 km for this case). Fig. 10(b) shows typical output spectra for two signal locations: λ

_{s}= 1539 nm (red dotted line) and λ’

_{s}= 1581 nm (blue line).

_{1}= 1524.75 nm and λ

_{2}= 1617.75 nm to minimize the gain ripple over the largest bandwidth. The pump powers were λ

_{1}∼ 1.7 W and

*P*

_{2}∼ 1.1 W. Figure 11(a) shows that

*G*≅ 26 ± 1.5 dB over 84 nm. We then used the experimental parameters to calculate the gain spectrum using Eq. 2. Two values of λ

_{0}were used to fit the data: 1570.05 nm (red line) and 1570.1 nm (black line). The effective interaction lengths were 230 and 220 meters, respectively, which should take into account the effects of PMD and longitudinal variations of λ

_{0}. The agreement with experiments is quite good confirming that variations of λ

_{0}and PMD may decrease the gain, but without introducing distortions in the spectrum.

_{1}= 1524.6 nm and the pump power was decreased to have the same amount of gain (∼26 dB). The 3 dB bandwidth decreased to 78 nm, whereas the difference between gain for outer and inner signal wavelengths decreased to 4.5 dB. Figure 11(b) also shows fittings to the experimental data for two values of λ

_{0}: 1570.05 nm (black line) and 1570.1 nm (red line).

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

*C*(see Table II). This fiber had 2 km of length and was cut in several pieces, with lengths varying from 100 to 370 m and having estimated variations of λ

_{0}from ∼0.1 to ∼0.4 nm. Figure 12 shows gain spectra obtained with the fibers with the smallest variations of λ

_{0}. The pump at λ

_{1}was generated using a 1P-FOPA. Figure 12(a) shows the gain spectrum of a fiber with 150 meters pumped with

*P*

_{1}∼

*P*

_{2}∼ 2.1 W at λ

_{1}≅ 1495.9 nm and λ

_{2}≅ 1611.9 nm. We obtained high and flat gain,

*G*≅ 25 ± 2 dB, over ∼102 nm. We also show two spectra calculated using Eq. (2) with

*L*

_{int}= 119 m and λ

_{0}= 1552.73 (black) and λ

_{0}= 1552.78 nm (red). With our parameters we have ξ = -0.95 and, from Eq. 11, we expect a ripple of Δ

*G*= 2.4 dB.

_{1}could be tuned over a large region because it was generated with a 1P-FOPA; however, the L-band EDFA limited the tunability of pump at λ

_{2}and as a consequence the 2P-FOPA bandwidth. To further increase this bandwidth we cooled the fiber with liquid nitrogen and the λ0 was shifted to 1546.8 nm. Figure 12(b) shows the gain spectrum of a cooled fiber with

*L*= 100 m pumped with

*P*

_{1}∼

*P*

_{2}∼ 3.3 W. The pumps were at λ

_{1}≅ 1483.1 nm and λ

_{2}≅ 1613.6 nm and were again optimized in order to have the smallest gain ripple. Note that high and flat gain,

*G*≅ 25 ± 1.5 dB, over ∼115 nm. This is, to the best of our knowledge, a record performance in terms of amount of gain and flatness for an optical amplifier. Dotted lines show fittings to the experimental data using Eq. (2) with

*L*

_{int}= 76 m and using λ

_{0}= 1546.89 (red) and λ

_{0}= 1546.84 nm (black).

_{0}. This was further confirmed in our experiments: by tuning slightly one of the pump we could retrieve the different spectral shapes (with 7 and 5 extremes) as in Fig. 1. Also, we have measured gain spectra for pump separations larger than 120 nm with the other fibers. Even for a variation of λ

_{0}of σ

_{λ0}∼0.4 nm (fiber length of 370 m) we still observed gain spectra that were in good agreement with the theory.

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

_{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.

*P*

_{1}∼ 190 mW and

*P*

_{2}∼ 170 mW and the gain spectrum was measured for three pump wavelength separations of λ

_{2}- λ

_{1}= 18.3 nm, 24.8 nm, and 39.4 nm. The pumps were also tuned in order to minimize the gain ripple. The results are shown in Figs. 14(a), (b), and (c), respectively. Note that as λ

_{2}- λ

_{1}increases, the ripple (calculated in the region between the pumps) increases and the amount of gain decreases strongly:

*G*= 〈

*G*〉 ± ½Δ

*G*= 36.5 ± 1.3 dB for λ

_{2}- λ

_{1}= 18.3 nm, 31 ± 1.5 dB for λ

_{2}- λ

_{1}= 24.8 nm, and 14.5 ± 4.5 dB for λ

_{2}- λ

^{1}= 39.4 nm.

## 8. Conclusions

_{0}tend to flatten the gain spectrum. We show that this amount of variation depends on 1/β

_{30}, on γ(

*P*

^{1}+

*P*

_{2}), on

*L*

_{corr}, and on 1/Δω

^{2}

_{p}. We experimentally showed that by using well-designed highly non-linear fibers, 2P-FOPAs with flat spectral response over 115 nm can be obtained. Further improvement of fibers in terms of the value (and sign) of the fourth order dispersion and nonlinear coefficients would lead to 2P-FOPAs with flat operation over several hundreds of nanometers and without requiring pump powers larger than 0.5 W.

## Appendix A

*L*= 60 meters, λ

_{0}= 1543 nm, ∣λ

_{1}- λ

_{2}∣ ≅ 163.2 nm, γ(

*P*

_{1}+

*P*

_{2}) = 52.5 km

^{-1}, β

_{30}= 0.016 ps

^{3}/km, and β

_{4}= -5.2×10

^{5}ps

^{4}/km. With these values,

*x*

_{0}= 3.15 and ξ = -0.72. Figure A1 shows the spectra at the input and at the output of the 2P-FOPA: the WDM signals experience 20 dB of gain with a ripple of ± 1.4 dB over the 60 nm of bandwidth. The flattest gain spectrum is obtained when the pumps are located at λ

_{1}= 1462.37 nm and at λ

_{2}= 1625.55 nm. The red line in figure A1 shows the gain calculated using the Eq. 2 with identical parameters as in the NLSE. One important feature in Figure A1 is that spurious tones around the pumps and at the outer maxima are efficiently generated and are strong enough to produce considerable crosstalk [35

35. J.L. Blows and P. F. Hu, “Cross-talk-induced limitations of
two-pump optical fiber parametric amplifiers,”
J. Opt. Soc. Am. B **21**, 989–995
(2004). [CrossRef]

_{3}, Δω

_{p}, and the number of WDM channels. In our example, this forbidden band is ∼20 % of the bandwidth between the pumps.

## Appendix B

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

_{4}> 0 are differentiated from fibers with β

_{4}< 0 (which were analyzed in section 4), by the fact that the four roots of κ = 0 can be located in the region between the pumps. Thus, the maximum gain of this spectrum is now

*G*

_{max}= 8.7x

_{0}- 6. To minimize the gain ripple we need to maximize the gain at Δω

_{s}= ±√-6β

_{2}/β

_{4}(local minimum). This implies in maximizing the gain at Δω

_{s}= 0; i.e. setting κ(Δω

_{s}= 0) = 0, from which we obtain

_{2c}in Eq. (B1) with identical parameters as in Figs. 2–4 and for

*x*

_{0}= 3.15. With this value of β

_{2c}it is easy to calculate the gain at Δω

_{s}= ±√-6β

_{2c}/β4, and then Δ

*G*. The result is shown in Fig. B1(b).

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

_{2c}and β

_{4}are positive. The condition to calculate Δ

*G*in this spectrum is: 1) maximizing the gain at Δω

_{s}= 0, then

*G*

_{max}= = 8.7x

_{0}- 6 and 2) calculate the gain at a frequency Δω

_{s}=

*b*Δω

_{p}. The value of the phase mismatch at Δω

_{s}=

*b*Δω

_{p}is κ

_{min}/2γ

*P*=

*b*

^{2}[0.5 + ξ(

*b*

^{2}- 1)]

^{2}. Replacing this in Eq. (7) leads to the calculation of

*G*

_{min}. Figure B2 shows the plot of Δ

*G*as a function of ξ. Note that spectra having only one extremum implies in ξ restraint to 0 < ξ < 0.5.

## Acknowledgments

## References and links

1. | Y. Emori, S. Matsushita, and S. Naminki, “1 THz-spaced multi-wavelength
pumping for broad-band Raman amplifiers,” in
Proc. European Conference on Optical Communications (ECOC) |

2. | Y.B. Lu, P.L. Chu, A. Alphones, and P. Shum, “A 105-nm ultrawide-band
gain-flattened amplifier combining C-and L-band dual-core EDFAs in a
parallel configuration,” IEEE Photon.
Technol. Lett. |

3. | E. Desurvire, “Optical communications in 2025,” in Proc. European Conference on Optical Communications (ECOC), September 2005, Glasgow, Scotland. |

4. | J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric
amplifiers and their applications,” IEEE
J. Sel. Top. Quantum Electron. |

5. | S. Radic and C.J. McKinstrie, “Optical amplification and signal
processing in highly nonlinear optical fiber,”
IEICE Trans. Electron. |

6. | M Yu, C.J. McKinstrie, and GP Agrawal, “Modulation instabilities in
dispersion flattened fibers,” Phys. Rev.
E , |

7. | M.E. Marhic, N. Kagi, T.-K. Chiang, and L.G. Kazovsky, “Broadband fiber optical parametric
amplifiers,” Opt. Lett. |

8. | C. Floridia, M.L. Sundheimer, L.S. Menezes, and A.S.L. Gomes, “Optimization of spectrally flat and
broadband single-pump fiber optic parametric
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9. | P. Dainese, G.S. Wiederhecker, A.A. Rieznik, H.L. Fragnito, and H.E. Hernandez-Figueroa “Designing fiber dispersion for broadband parametric amplifiers,” IEEE-SBMO, International Microwave and Optoelectronics Conference (IMOC) , 2005, pp. 1–3. |

10. | K. Inoue, “Arrangement of fiber pieces for a
wide wavelength conversion range by fiber four-wave
mixing,” Opt. Lett. |

11. | M.E. Marhic, F.S. Yang, M.C. Ho, and L.G. Kazovsky, “High-nonlinearity fiber optical
parametric amplifier with periodic dispersion
compensation,” J. Ligthwave Technol. |

12. | J. Hansryd and P. A. Andrekson, “Broad-band continuous-wave-pumped
fiber optical parametric amplifier with 49-dB gain and wavelength-conversion
efficiency,” |

13. | L. Provino, A. Mussot, E. Lantz, T. Sylvestre, and H. Maillote, “Broad-band and flat parametric
amplifiers with a multi-section dispersion-tailored nonlinear fiber
arrangement,” J. Opt. Soc. Am. B. |

14. | M. Yu, C.J. McKinstrie, and G.P. Agrawal, “Instability due to cross-phase
modulation in the normal dispersion regime,”
Phys. Rev. E |

15. | M.E. Marhic, Y. Park, F.S. Yang, and L.G. Kazovsky, “Broadband fiber optical parametric
amplifiers and wavelength converters with low-ripple Chebyshev gain
spectra,” Opt. Lett. |

16. | J.M. Chavez Boggio, S. Tenenbaum, and H.L. Fragnito, “Amplification of broadband noise
pumped by two lasers in optical fibers,”
J. Opt. Soc. Am. B |

17. | C.J. McKinstrie, S. Radic, and A.R. Chraplyvy, “Parametric amplifiers driven by two
pump waves,” IEEE J. Sel. Top. Quantum.
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18. | M.Y. Gao, C. Jiang, W. Hu, and J. Wang, “Two-pump fiber optical parametric
amplifiers with three sections fiber allocation,”
Opt. Laser Technol. |

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fibre,” Electron. Lett. |

20. | S. Radic, C.J. McKinstrie, R.M. Jopson, and J.C. Centanni, “Continuous wave parametric amplifier with 41.5 nm of flat gain,” in Proc. of Optical Fiber Communication Conference (OFC), 2004, Paper TuC4. |

21. | J.M. Chavez Boggio, J.D. Marconi, and H.L. Fragnito, “Double-pumped fiber optical
parametric amplifier with flat gain over 47-nm bandwidth using a
conventional dispersion-shifted fiber,”
IEEE Photon. Technol. Lett. |

22. | J.M. Chavez Boggio, J.D. Marconi, H.L. Fragnito, S.R. Bickham, and C. Mazzali, “Broadband and low ripple double-pumped fiber optical parametric amplifier and wavelength converters using HNLF,” in Proc. Optical Amplifiers and their Applications (OAA), June 2006, Whistler, Canada. |

23. | M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance,” in Proc. European Conference Optical Communication (ECOC), 2005, Glasgow, Scotland, PD paper Th 4.4.4. |

24. | T. Nakanishi, M. Hirano, T. Okuno, and M. Onishi, “Silica based highly nonlinear fiber with γ = 30 /W/km and its FWM-based conversion efficiency,” in Proc. Optical Fiber Communication Conference (OFC), 2006, Anaheim, paper OtuH7. |

25. | J.M. Chavez Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber
optical parametric amplifier in a 10Gb/s×64 channel dense
wavelength division multiplexing system,”
Opt. Commun. |

26. | F. Yaman, Q. Lin, S. Radic, and G.P. Agrawal, “Impact of dispersion fluctuations on
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IEEE Photon. Technol. Lett. |

27. | X.M. Liu, W. Zhao, K.Q. Lu, T.Y. Zhang, Y.S. Wang, M. Ouyang, S.L. Zhu, G.F. Chen, and X. Hou “Optimization and comparison of
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28. | A. Legrand, C. Simonneau, D. Bayart, A. Mussot, E. Lantz, T. Sylvestre, and H. Maillotte, in Proc. Optical Amplifiers and their Applications (OAA), July 2003, Otaru, Japan. |

29. | F. Yaman, Q. Lin, and G.P. Agrawal, “Effects of polarization-mode
dispersion in dual-pump fiber-optic parametric
amplifiers,” IEEE Photon. Technol. Lett. |

30. | C.J. McKinstrie, H. Kogelnik, R.M. Jopson, S. Radic, and A.V. Kanaev, “Four-wave mixing in fibers with
random birefringence,” Opt. Express |

31. | M.E. Marhic, K.K.Y. Wong, and L.G. Kazovsky, “Parametric amplification in optical fibers with random birefringence,” in Proc. Optical Fiber Communication Conference (OFC), February 2004, Anaheim, paper TuC2. |

32. | J.M. Chavez Boggio, S. Tenenbaum, J.D. Marconi, and H.L. Fragnito, “A novel method for measuring longitudinal variations of the zero dispersion wavelength in optical fibers,” in Proc. European Conference on Optical Communication (ECOC), September 2006, Cannes, France, paper Th1.5.2. |

33. | M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and
polarization drift in installed fibers,”
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34. | M. Farahmand and M. de Sterke, “Parametric amplification in presence
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35. | J.L. Blows and P. F. Hu, “Cross-talk-induced limitations of
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J. Opt. Soc. Am. B |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.2330) Fiber optics and optical communications : Fiber optics communications

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 5, 2007

Revised Manuscript: April 5, 2007

Manuscript Accepted: April 10, 2007

Published: April 16, 2007

**Citation**

J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, "Spectrally flat and broadband double-pumped fiber optical parametric amplifiers," Opt. Express **15**, 5288-5309 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5288

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### References

- Y. Emori, S. Matsushita, and S. Naminki, "1 THz-spaced multi-wavelength pumping for broad-band Raman amplifiers," in Proc. European Conference on Optical Communications (ECOC) vol. 2, 2000, paper 4.4.2, pp. 73-74.
- Y.B. Lu, P.L. Chu, A. Alphones, P. Shum, "A 105-nm ultrawide-band gain-flattened amplifier combining C-and L-band dual-core EDFAs in a parallel configuration," IEEE Photon. Technol. Lett. 16, 1640-1642 (2004). [CrossRef]
- E. Desurvire, "Optical communications in 2025," in Proc. European Conference on Optical Communications (ECOC), September 2005, Glasgow, Scotland.
- J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002). [CrossRef]
- S. Radic and C.J. McKinstrie, "Optical amplification and signal processing in highly nonlinear optical fiber," IEICE Trans. Electron. E88-C, 859-869 (2005). [CrossRef]
- M Yu, C.J. McKinstrie, GP Agrawal, "Modulation instabilities in dispersion flattened fibers," Phys. Rev. E, 52, 1072-1080 (1995). [CrossRef]
- M.E. Marhic, N. Kagi, T.-K. Chiang, and L.G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573− 575 (1996). [CrossRef] [PubMed]
- C. Floridia, M.L. Sundheimer, L.S. Menezes, and A.S.L. Gomes, "Optimization of spectrally flat and broadband single-pump fiber optic parametric amplifiers," Opt. Commun. 223, 381-388, 2003. [CrossRef]
- P. Dainese, G.S. Wiederhecker, A.A. Rieznik, H.L. Fragnito, and H.E. Hernandez-Figueroa "Designing fiber dispersion for broadband parametric amplifiers," IEEE-SBMO, International Microwave and Optoelectronics Conference (IMOC), 2005, pp. 1-3.
- K. Inoue, "Arrangement of fiber pieces for a wide wavelength conversion range by fiber four-wave mixing," Opt. Lett. 19, 1189-1191 (1994). [CrossRef] [PubMed]
- M.E. Marhic, F.S. Yang, M.C. Ho, and L.G. Kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Ligthwave Technol. 17, 210-215 (1999). [CrossRef]
- J. Hansryd and P. A. Andrekson, "Broad-band continuous-wave-pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency," 13, 194-196 (2001).
- L. Provino, A. Mussot, E. Lantz, T. Sylvestre, and H. Maillote, "Broad-band and flat parametric amplifiers with a multi-section dispersion-tailored nonlinear fiber arrangement," J. Opt. Soc. Am. B. 20, 1532-1539 (2003). [CrossRef]
- M. Yu, C.J. McKinstrie, and G.P. Agrawal, "Instability due to cross-phase modulation in the normal dispersion regime," Phys. Rev. E 52, 1072-1080 (1993). [CrossRef]
- M.E. Marhic, Y. Park, F.S. Yang, and L.G. Kazovsky, "Broadband fiber optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra," Opt. Lett. 21, 1354-1356 (1996). [CrossRef] [PubMed]
- J.M. Chavez Boggio, S. Tenenbaum and H.L. Fragnito, "Amplification of broadband noise pumped by two lasers in optical fibers," J. Opt. Soc. Am. B 18, 1428-1435 (2001). [CrossRef]
- C.J. McKinstrie, S. Radic, and A.R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum. Electron. 8, 538-547 (2002). [CrossRef]
- M.Y. Gao, C. Jiang, W. Hu, and J. Wang, "Two-pump fiber optical parametric amplifiers with three sections fiber allocation," Opt. Laser Technol. 38, 186-191 (2006). [CrossRef]
- S. Radic, C.J. McKinstrie, R.M. Jopson, J.C. Centanni, Q. Lin, and G.P. Agrawal, "Record performance of parametric amplifier constructed with highly nonlinear fibre," Electron. Lett. 39, 838-839 (2003). [CrossRef]
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