Parametric amplifiers based on coupled-core fibers and coupled integrated waveguides are promising devices for delivering broadband and flat gain spectrum with low noise figure in vital technological applications such as optical communications. To address potential practical issues that may arise in the experimental demonstration, in the present work, we derive analytical expressions for the impact of relative input pump phase and power fluctuations between two waveguides on pump waves stability. We, furthermore, evaluate, through numerical simulations, how the pump power oscillations between the two waveguides affect the amplifier spectrum, identifying optimal regimes of operation for coupled-core fibers and also for coupled integrated waveguides. Our findings hint at the resilience of these parametric amplifiers' architectures and at their possible practical implementation in realistic labs and real-world scenarios.
Examples of the pump power difference evolution without losses (using parameters typical of coupled-core fibers) and with losses (using parameters typical of silicon nitride waveguides) are shown in Fig. 2. To study the worst-case scenario, where pump and phase fluctuations contribute to a maximum power difference between the two waveguides, the signs of Pd and are kept the same in the following analysis. For each case, we considered the impact of various initial pump power differences— , the impact of different initial phase differences— (corresponding to an optical path difference of , and 10 nm at wavelength λ = 1550 nm in silica), and the combined impact of both initial power and phase differences. For the lossless case [Figs. 2(a)–2(c)], the analytical theory is in excellent agreement with numerical simulation results of the coupled NLSEs up to when most of the power is contained in a single waveguide ( ). The larger the input perturbations, the earlier (along the propagation) becomes substantial. It is worth mentioning that the simulation results show the power exchange between the two waveguides upon propagation: the power in one waveguide flows into the other one and then goes back at the critical point , where all the power is contained in a single waveguide. As the stability analysis is based on the small perturbations assumption, this phenomenon is not captured by the analytical calculations. In the lossy case [Figs. 2(d)–2(f)], we observe that performs damped oscillations, and agreement between simulations and theory seems quite robust for the parameters considered.
To see how the amplifier gain spectrum is affected by the relative input pump phase and power fluctuations, we performed numerical simulations of Eqs. (1) for parameters corresponding to coupled-core fibers and integrated lossy coupled-waveguide amplifiers, considering terms up to the second order in the waveguide and coupling dispersion. We have simulated amplifiers with characteristics analogous to those used for the perturbation analysis presented in Fig. 2, resulting in the key quantity determining the amplifier gain, namely, the nonlinear phase shift accumulated per waveguide , to be up to 4.5 for dual-core fibers and up to 5.4 for integrated waveguides. These values can be considered realistic for parametric amplifiers used in optical communications, where the accumulated nonlinear phase shift ranges approximately between 3 and 5.6,8,13
We calculated the phase-insensitive (PI) gain spectrum of supermodes choosing as the initial condition the signals—with frequency larger than the pump ( )—to be fully located in the supermode as , and considering idlers—with frequency smaller than the pump ( )—to be vanishing, . Here, are the sideband waves (subscripts s, i and 1, 2 denote signal and idler and waveguides 1 and 2, respectively). The gain spectrum of the supermodes is, hence, defined as and for positively and negatively detuned angular frequency from the pump, respectively. We first studied how the relative pump power or phase difference alone can impact the amplifier gain spectrum in a similar way: we compared the parametric gain spectrum obtained with (Pd = 600 mW, ) and (Pd = 0, mRad). From Eq. (8), we know that would be almost equal in both cases for the chosen value of the total pump power Pp = 6 W. The results are summarized in Fig. 3 for coupled-core fibers and also for coupled integrated lossy waveguide parameters.
We have then explored the impact on the parametric gain of different magnitudes of pump power and pump phase fluctuations and compared different parametric gain profiles obtained with the unperturbed scenario. The sidebands' gain for the lossless dual-core fiber amplifier is shown in Fig. 4 for supermode and in Fig. 5 for supermode . For the lossy integrated waveguide amplifier, the gain is shown in Fig. 6 for supermode and in Fig. 7 for supermode . Up to Pd = 40 mW and mRad, we observe that for propagation distance 100 m and 1 m, for coupled-core fibers and integrated waveguides, respectively, the main impact of amplitude and phase perturbations is to slightly reduce the maximum gain without substantially affecting the ideal amplifier performance. However, for larger propagation distances and/or for larger values of input power and phase differences (here, we have considered Pd = 200 mW and mRad to illustrate a more extreme case), substantial modifications of the gain spectrum and coupling of energy from the input supermode to supermode occur. This results, as well, in a frequency asymmetric gain spectrum for different supermodes due to the interplay between power asymmetry in the two waveguides and C1, the first-order term of the coupling Taylor expansion around the pump frequency. It can be indeed shown that the combination of coupling dispersion and asymmetric pump power among the two waveguides can lead to intermodal four-wave mixing, resulting in coupling between different supermodes and also frequency asymmetric parametric gain spectrum of individual supermodes.37 We note that significant changes in the gain spectrum start to occur when most of the pump power is contained in a single waveguide ( ) as shown by the evolution of plotted in each case considered.
In the antisymmetrically pumped (ξ = 1) parametric amplifier, the evolution of the pump power difference is still given in Eq. (9), but ρ keeps imaginary for any distance, so that the pump power difference oscillates around zero. This is illustrated in Fig. 8. In the antisymmetric scenario, the same initial conditions for signals and idlers as in the symmetric case have been considered, and a more stable parametric amplification is achieved—without the flat gain profile though—as depicted in Figs. 9 and 10, where we can appreciate that spectral gain degradation occurs for much larger values of the perturbations compared to the symmetric case. Analogously to the symmetric case, substantial changes in the gain spectrum for mode correspond to energy transfer to mode (not shown here). We have presented the impact of the input pump phase and power fluctuations in the PI operational regime. It is worth mentioning that in the phase-sensitive scenario (where idler waves are present at the input too), we find results that are analogous, and we do not report here for the sake of brevity. A similar scenario occurs as well for a PI amplifier where the input signals are located in supermode instead that in .
Achieving two almost identical optical paths for the waves injected into the coupled-nonlinear-waveguide amplifier may result in alignment issues that require special equipment for precise compensation. However, existing commercial phase-meters38 enable phase control up to the order of 6 μrad. This makes our parameters choice, regarding the gain spectrum degradation caused by large relative phase difference illustrated in the previous figures, a very pessimistic scenario. A very pessimistic scenario has been indeed considered for the cases of gain degradation caused by large values of input power differences too. An analysis based on current 50/50% couplers commercially available,39 which have a coupling ratio tolerance of ±5%, would predict relative differences of the order of ±50 mW for a 1 W pump in the worst-case scenario. However, it is very important to stress that since the resolution of typical power meters is of the order of sub μW, it is possible to correct this imbalance, as it is not a fundamental one, without sophisticated technology, for instance, by using variable optical attenuators in each output port of the 50/50% couplers. Hence, it is likely feasible to achieve the performance of Figs. 4, 6(a), and 6(b) with minimum engineering effort and a negligible impact on the noise figure.
In conclusion, we have provided the analytical theory of pump waves stability for an equally pumped dual-waveguide parametric amplifier under the influence of relative input power and phase fluctuations. Our analytical theory predicts the evolution of pump power difference along the waveguides, and it is in good agreement with numerical simulations for realistic parameters. We have, furthermore, shown numerically how pump waves instability caused by power and phase fluctuations translates into amplifier gain spectrum changes, which occur for the magnitude of the input perturbations much larger than what can be realistically controlled in experiments. The results presented in this work hint at the resilience of equally pumped coupled dual-waveguide parametric amplifiers against pump fluctuations and can help the design of photonic technologies relying on this amplification scheme.
A.M.P. acknowledges support from the Royal Academy of Engineering through the Research Fellowship Scheme.
Conflict of Interest
The authors have no conflicts to disclose.
Minji Shi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal). Vitor Ribeiro: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Auro M. Perego: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.