This paper solves the four coupled equations describing non-degenerate four-wave mixing, with the focus on amplifying a signal in a fiber optical parametric amplifier (FOPA). Based on the full analytic solution, a simple approximate solution describing the gain is developed. The advantage of this new approximation is that it includes the depletion of the pumps, which is lacking in the usual quasi-linearized approximation. With the proposed model it is thus simple to predict the gain of a FOPA, which we demonstrate with a highly nonlinear fiber to show that an undepleted FOPA can produce a flat gain spectrum with a bandwidth in the 100-nm range, centered on the zero-dispersion wavelength. When running the FOPA in depletion, this range can be slightly increased.
© 2011 Optical Society of America
Lately, fiber-optical parametric amplifiers (FOPAs) have attracted significant attention. There are several reasons for this, including their ability to provide phase-sensitive amplification [1–3], multicasting , regeneration [3, 5–7] and wavelength conversion [8, 9]. It is noted that especially when applying the FOPA as an amplifier or regenerator, most emphasis has been on single-pumped amplifiers. However research on phase-sensitive amplification is often based on dual-pumped amplifiers [3,10].
In wavelength conversion as well as in regeneration, amplifier saturation is of particular importance. Consequently, it is essential to understand and to be able to predict the depletion behavior of FOPA’s. Until now, the focus has been on single-pumped FOPAs. However, in this work we consider depletion in dual-pumped FOPAs.
Previously Kylemark et al.  derived an approximation for a single-pumped FOPA operated at the specific phase mismatch which allows for total conversion of power from the pump. Their approach is based on selecting a specific phase matching condition where the coupling of power is only directed from the pump to the signal. Other phase matching conditions result in periodic coupling of the power between pump and signal, and are not treated. In this work we adopt a similar approach, though for a dual-pumped FOPA. In addition, we extend the approach to any phase mismatch by neglecting the periodic behavior of the power flow. Depletion and the periodic behavior of the flow of power have previously been described analytically by the use of Jacobi elliptic functions [12–14]. In this work we derive a simpler analytical model of saturated dual-pumped FOPA. This model provides a useful tool to understand and design a two-pump FOPA operated in saturation, as for example when designing a wideband regenerator based on operating the amplifier in depletion. The application of the model presented in this work is discussed with a focus on optimizing the bandwidth of a two-pump parametric amplifier. By locating the pumps symmetrically around the zero-dispersion wavelength (ZDW), a bandwidth in the 100-nm range is achievable.
In general we consider an electric field consisting of four continuous waves (CWs), at frequencies ω1 through ω4. In non-denegerate four-wave mixing (FWM), the four distinct waves interact with each other under the condition ω1 + ω4 = ω2 + ω3. Two of these, numbers 2 and 3, are in this paper used as pumps, while 1 is a signal, also denoted s, that is to be amplified, and 4 is the idler, i, that arises when the signal is amplified. Under the assumption that the four waves are sufficiently separated in frequency, the nonlinear Schrödinger equation (NLS) can be reduced to four coupled differential equations for the complex field amplitudes :16, 17]. The most common method of solving Eq. (1) is by assuming that the pump powers remain much greater than the signal and idler power at all times and that the power loss of the pumps is negligible . This will quasi-linearize the differential equations and for pumps with identical power, Pp, result in 1]. Generally, gain exists when −6γPp < Δβ < 2γPp, but the greatest gain, G(z) = Ps(z)/Ps(0), is obtained with κ = 0, thus requiring that Δβ = −2γPp. This condition effectively means that the self- and cross-phase-modulation and dispersion induced wave-number shifts cancel each other. However as the pumps are depleted, the effective wave-number mismatch, κ, will change and thus decrease the coupling efficiency. The problem with this solution is that it does not account for pump depletion and nonlinear detuning.
By rewriting the complex amplitude A in Eq. (1) in terms of the real variables, power P and phase ϕ, as , the differential equations are instead written asEq. (4), this relative phase is important as it shows in which direction the power flows. If the relative phase is positive, the power is transferred from the pumps to the signal and idler, whereas if it is negative, the opposite happens. As power is transferred from the pumps to the signal and idler, nonlinear detuning occurs, which reduces the coupling. At some point the relative phase can be changed so much that the power flow is reversed and the power begins to couple back to the pumps.
Because the parametric process is symmetric, the signal and the idler have the same increase in power (photon flux), whereas the pumps have the same decrease, also written as dz(Ps – Pi) = 0 and dz(P2 – P3) = 0. Finally, the process is power conserving, so ΣαPα(z) = Ptotal. Because of these facts, all the powers are related to each other, and can thus be described by only one variable according to the Manley-Rowe relations [18, 19]. This variable, F, is chosen to be the power of the idler which is assumed zero at z = 0, i.e.Eq. (6), two governing differential equations can be set up for the power, F, and phase, θ. Specifically, Eq. (11) is due to FWM, while the second contains the linear and nonlinear phase-modulation terms. Because the Hamiltonian is independent of z, it is constant. It is therefore used to reduce the problem to one variable. Since F is initially zero, the value of the Hamiltonian is also zero. Thus, the phase, θ, can be determined as Eq. (8) a potential equation may be obtained for F, by squaring the equation and inserting the expression for θ, resulting in 20]. The solution is Eq. (2), the initial growth rate for Δβ = −2γPp is G(z) = 1 + sinh2 [2γPpz], whereas the growth rate for Δβ = γPs is . With Δβ = γPs, the roots of Eq. (9) are f1 = f2 = Pp and f3 = −4Ps/3. In this case, m = 1, reducing the elliptic function, sn, to the hyperbolic tangent function, tanh. The solution is thus rewritten as Eq. (14) are found by treating Ps as a small perturbation to the potential equation. The three non-zero roots are Eq. (3). Consequently, we may apply the relations
If the two pumps have different powers, called Pa and Pb, where Pa > Pb, then (Pp – F)2 is replaced by (Pa – F)(Pb – F) in Eq. (14), and 2Pp is replaced by Pa + Pb in Eq. (10). When these changes are made, the solution is the same as Eq. (16). Because of the different pump powers, the wave-number mismatch that results in full power transfer from the weaker pump is Δβ = γPs +γ(Pa – Pb), whereas the fastest growing gain condition is Δβ = −γ(Pa + Pb).
To show the validity of the proposed model, the evolution of the idler as predicted by Eq. (24) has been plotted in Fig. 1 against the full solution, described by Eq. (16), as well as against the quasi-linearized model, Eq. (2). From the figure it is seen that the new expression fits well until the power begins to couple back into the pump. In all cases the new approximate expression has a better correlation with the analytic solution than the conventional quasi-linear result in Eq. (2).
Figure 2 shows the gain for different input powers. The gain has been evaluated by using Eqs. (2), (21) and (24) and it also shows that the new approximation, Eq. (24), is a better approximation to Eq. (21) than the usual approximation, Eq. (2). The new approximation takes saturation into account, but since it does not include back coupling, it is only useful until back coupling becomes significant. Before the system is in depletion, the gain is symmetric around κ = 0 as predicted by Eq. (2), but when saturation sets in, the gain profile becomes asymmetric which the new approximation also takes into account. If a setup is constructed with two pumps placed symmetrically around the ZDW, then it is possible to obtain a flat gain spectrum over a range of wavelengths, with very high signal gain. The wave-number mismatch is estimated by Taylor expanding all the propagation constants around the ZDW. Because of the chosen pump symmetry, all βn = dnβ(ω)/dωn|ωZD, where n is odd, will cancel out, and since β2 is zero at the ZDW, the first term that has influence is β4. The mismatch is thus approximated as21], but to get a uniform gain over a wide range, the important factor will be to have β4 as close to zero as possible, which allows the mismatch to be maintained close to zero over a large bandwidth. This is demonstrated in Fig. 3, which shows gain profiles for a highly nonlinear fiber (HNLF), , with L = 250 m, λZD = 1560.5 nm, β4(λZD) = 8.82 × 10−55 s4/m and γ = 11.5 W−1km−1, calculated using Eq. (24). If Eq. (1) is solved numerically and a loss of 0.74 dB/km is included, the obtained spectrum is similar to that shown in Fig. 3, only the gain is lower by 1.5 to 2 dB. It is important for the pumps to be placed far from the ZDW, in this case a separation of roughly 60 nm was used, in order to avoid the creation of extra waves by cascaded FWM , which invalidate the approximation of only 4 distinct frequencies. Furthermore, simulations show that when the signal is placed relatively close to either of the pumps, then the coupling efficiency is greatly reduced because the signal undergoes degenerate FWM with the nearer pump. For this fiber setup, degenerate FWM begins to be significant at a separation of roughly 10 nm. This effect reduces the actual obtainable bandwidth.
In Fig. 3, it is also shown that even if the symmetry is slightly broken, the gain spectrum remains reasonably flat, and its width even increases slightly.
We have obtained the exact solution of the dual-pumped FOPA equations, as well as a simple, yet very useful, approximate solution which includes pump depletion and nonlinear detuning. We have shown with a concrete example, which shows that it is possible to obtain flat gain bandwidths in the 100-nm range, that the model provides a simple and fast way of modeling the gain of a FOPA.
We have also shown that although the gain increases more rapidly as a function of fiber length when Δβ = −2γPp, this wave-number mismatch does not result in complete power conversion, as is the case when Δβ = γPs.
EOARD is thanked for financial support under grant number 063094. Magnus Karlsson is acknowledged for fruitful discussions.
References and links
1. 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]
2. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University Press, 2007). [CrossRef]
3. R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010). [CrossRef]
4. C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009). [CrossRef]
5. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003). [CrossRef]
6. C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009). [CrossRef]
7. J. Kakande, F. Parmigiani, R. Slavík, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Saturation effect in degenerate phase sensitive fiber optic parametric amplifiers,” ECOC, paper Th.10.C.2 (2010).
8. M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron. 8, 527–537 (2002). [CrossRef]
9. J. M. Chávez Boggio, J. R. Windmiller, M. Knutzen, R. Jiang, C. Brès, N. Alic, B. Stossel, K. Rottwitt, and S. Radic, “730-nm optical parametric conversion from near- to short-wave infrared band,” Opt. Express 16, 5435–5443 (2008). [CrossRef] [PubMed]
10. K. Croussore and G. Li, “Phase regeneration of NRZ-DPSK signals based on symmetric-pump phase-sensitive amplification,” IEEE Photon. Technol. Lett. 19, 864–866 (2007). [CrossRef]
11. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. 24, 3471–3479 (2006). [CrossRef]
13. G. Cappelline and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1990). [CrossRef]
14. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993). [CrossRef]
15. G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed. (Elsevier, 2007).
16. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992). [CrossRef]
18. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956). [CrossRef]
19. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
20. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer, 1954).
21. M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009). [CrossRef]