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Abstract
Parametric amplifiers relying on the nonlinear four-wave mixing process are known for their signature symmetric gain spectrum, where signal and idler sidebands are generated on both sides of a powerful pump wave frequency. In this article we show analytically and numerically that parametric amplification in two identically coupled nonlinear waveguides can be designed in such a way that signals and idlers are naturally separated into two different supermodes, hence providing idler-free amplification for the supermode carrying signals. This phenomenon is based on the coupled-core fibers analogue of intermodal four wave-mixing occurring in a multimode fiber. The control parameter is the pump power asymmetry between the two waveguides, which leverages the frequency dependency of the coupling strength. Our findings pave the way for a novel class of parametric amplifiers and wavelength converters, based on coupled waveguides and dual-core fibers.
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Optical parametric amplifiers are promising devices for signal amplification and wavelength conversion [1,2]. The physical principle underpinning their operation is four-wave mixing mediated by Kerr nonlinearity in presence of a powerful pump wave [3]. They have been demonstrated in silica fibers [2] and in integrated waveguides too [4–6]. The characteristic single-pump parametric amplifier gain spectrum exhibits two main sidebands symmetrically located with respect to the pump frequency. In the phase-insensitive (PI) mode, signal waves to be amplified are injected in the amplifier at frequencies different (blue or red detuned) from the pump one. Idler waves are then spontaneously generated on the other side of the pump frequency due to the nature of the four-wave mixing process. In the phase-sensitive (PS) operational regime [7,8], instead, idlers, with an appropriate phase relation with respect to pump and signals, have to be injected at the amplifier input to provide adequate interference with the signals in order to achieve larger gain. In both scenarios, in the context of optical communications, idlers have to be filtered out after the amplification to prevent detrimental nonlinear interaction during information transmission. Idler amplification is dictated by the signature symmetric nature of the parametric gain spectrum which is governed at the leading order by the second-order dispersion coefficient. Asymmetric gain spectrum in parametric amplifiers has been studied and observed as a consequence of either third-order dispersion [9–11], gain saturation [12], Raman gain [13,14] or asymmetric losses for signal and idler waves [15–17]. However, these asymmetries are not pronounced enough to reduce to a negligible value the idlers gain over a large bandwidth. Multimode fibers offer an appealing alternative way to provide amplification to signals while confining idlers in a separate optical mode. This occurs through the so-called intermodal four-wave mixing process [18–22]. In this work, we demonstrate analytically and numerically a parametric amplifier consisting of a dual-core silica fiber, showing that an operational regime exists where signals carried by one supermode of the system experience substantial gain, while idler waves amplification is negligible in the same supermode. The physical principle underpinning this process is the analogue of intermodal four-wave mixing for a multicore fiber system.
Parametric amplification in coupled nonlinear waveguides, suggested in pioneering contributions several decades ago [23,24], can provide broadband flat gain and low noise figure in both PS and PI operational mode [25–27]. It also enables compensation of pump attenuation induced phase-matching degradation via spatially dependent coupling [28]; as well as dispersion compensation through coupling dispersion engineering [29].
In this article, we show instead that coupled-core fibers parametric amplifiers operated in the regime when the two cores are pumped with different power, thanks to the frequency dependency of the coupling, can provide frequency asymmetric gain for the two supermodes of the system. This results in negligible idlers amplification in the supermode carrying the signals, relying on a purely conservative nonlinear dynamics. In Fig. 1 the amplifier concept is illustrated pictorially. Intermodal dispersion has been observed experimentally [30] and also studied in the context of soliton formation and parametric amplification [31,32] in arrays of coupled waveguides. Furthermore, modulation instability in coupled nonlinear Schrödinger equations (NLSEs) has been studied analytically [24,33] including the role of frequency-dependent coupling and in particular of the first order term in its Taylor expansion around the pump frequency [34,35] (see also [36–39] for various generalized cases). However, the remarkable frequency asymmetric gain spectrum for different supermodes which we report here has never been described so far to the best of our knowledge.
Fig. 1. Amplifier concept. Schematic of the difference between the standard parametric amplifier, which amplifies signals and generates idlers in the same optical mode, and the asymmetrically pumped dual-core fiber parametric amplifier, for which the signals encoded into one supermode are amplified and no idlers are generated in the same supermode.
The model for our system consists of two coupled NLSEs ruling the propagation of the electric field amplitudes $A_{1,2}$ along two identical linearly coupled waveguides:
Here we defined $E_\pm = (e_{s\pm },e_{i\pm }^*)^T=(e_{s1}\pm e_{s2},e_{i1}^*\pm e_{i2}^*)^T$, so that the sidebands of supermodes $A_\pm =(A_1\pm A_2)/\sqrt 2$ can be expressed as
The submatrices $M_\pm$, which describe the evolution of $E_\pm$, and $M_c$, representing the coupling between the modes $E_\pm$ induced by pump power difference $P_d$, read respectively
We now focus on the input regime where signal waves have opposite phases in two coupled waveguides so that $e_{s1}(0)+e_{s2}(0)=0$ and study the output of the supermode $A_-$. Thus the gain spectrum for the sidebands can be defined as
According to our definition of the gain, $G(z)<1$, at the amplifier output for idlers implies that idlers have not reached a power equal to the input signals. Firstly we considered $C_2=0$ to better highlight the necessary conditions for idler-free parametric amplification. We have compared the analytical predictions with numerical simulations of the coupled NLSEs (Eqs. (1)) performed using a split-step Fourier algorithm. In the PI regime, considering $e_{i1}(0)=e_{i2}(0)=0$, the analytical gain reads
We show the dependence of the amplifier gain spectrum versus $C_1$ in Fig. 2 for anomalous (a) and normal (b) waveguide dispersion respectively, while an example of the idler-free operation for the $A_-$ supermode is shown in panels c) and d). In the latter case, the idler-free operational mode is limited to the high-frequency range only, and the signal gain is smaller for the parameters considered. For the particular parameters selected, the idler-free regime exhibits a remarkable $\approx$20 dB gain difference between signal and idler sidebands in the anomalous and in the normal dispersion regime too. As in the normal dispersion scenario, the asymmetric gain coexists with symmetric low-frequency sidebands — which would require additional filtering — to quantify the amount of useful signal power contained in the idler-free spectral lobe, for input signal waves in the interval $(0,\Omega _m]$, we have computed the following quantities: $I_+=\int _0^{+\Omega _m} S_-(\Omega ')d\Omega '\big /\int _{-\Omega _m}^{+\Omega _m} S_-(\Omega ')d\Omega '$ and $I_s=\int _{s1}^{s2} S_-(\Omega ')d\Omega '\big /\int _{-\Omega _m}^{+\Omega _m} S_-(\Omega ')d\Omega '$. $I_+$ is the ratio between the integral of the output spectral power $S_-(\Omega ')$ of the supermode $A_-$ computed for frequencies larger than the pump one and the total power contained in the full simulations window $[-\Omega _m=-2\pi \cdot 6,+\Omega _m=2\pi \cdot 6]$ rad/ps, excluding the pump. $I_s$ is the ratio between the integral of the supermode $A_-$ output spectral power computed in the interval around the asymmetric signal spectral lobe delimited by $s_1$ and $s_2$ and the total power contained in the full simulations window. For the anomalous dispersion case — (Fig. 2(c)) with $C_1=-7.5\text { ps}\cdot \text {km}^{-1}$ — we obtain $I_+=0.96$ and $I_s=0.79$ meaning that 96$\%$ of the power of $A_-$ supermode is contained in the signal spectrum and that 79$\%$ of the power is in the asymmetric spectral lobe in the interval $[s_1=2\pi \cdot 3.85, s_2=2\pi \cdot 5.61]$ rad/ps. Performances are less significant in the normal dispersion regime — (Fig. 2(d)) with $C_1=7.5\text { ps}\cdot \text {km}^{-1}$ — due to large power contained in the low-frequency symmetric sidebands. In that case, we obtain $I_+=0.58$ and $I_s=0.9$ meaning that 58$\%$ of the power of $A_-$ supermode is contained in the signal spectrum and that 9$\%$ of the power is in the asymmetric spectral lobe in the interval $[s_1=2\pi \cdot 4.10, s_2=2\pi \cdot 5.53]$ rad/ps. Similar results are obtained for the PS amplifier shown below.
The PS operational mode in two coupled waveguides parametric amplifiers can be implemented in various different scenarios [25,26]. Here, without loss of generality, we focus on the initial condition: $e_{s1}(0)=e_{i1}(0)=-e_{s2}(0)=-e_{i2}(0)$, and assume that the initial phase of signal wave in waveguide $1$ is $\phi _s+\phi _0$, then the gain reads:
The characterization of the PS amplifier is shown in Fig. 3. We have furthermore investigated the dependency of the gain spectrum on the pump power difference $P_d$ which is shown in Fig. 4 for both PI and PS operational mode, as playing with pump power is a practical way to control the amplifier gain experimentally. Results show that there is a trade-off between maximizing the gain and keeping negligible idler amplification. It is important to stress that the gain asymmetry for signal and idler waves between different supermodes reported in this work is not related to odd dispersion terms $\beta _\text {odd}$, which in our system results in a simple phase factor as it can be appreciated from Eq. (4). Furthermore, while the impact of $C_1$ on the MI spectrum in asymmetrically pumped dual-core fibers has been investigated in [34,35], in those works the frequency asymmetric gain spectrum for different supermodes was not observed, possibly because the focus was on the eigenvalue spectrum of the continuous wave stability problem (which is symmetric in frequency) and the parametric gain was not calculated by fully solving the signals and idlers equations. Both in the PI and PS regime, the amplifier asymmetric gain spectrum is due to the combined action of the unbalanced pump power distribution between the two waveguides and of the frequency dependency of the coupling. Notably, $P_d\ne 0$ causes supermodes mixing as it can be clearly seen from Eqs. (5) and (7) showing that the supermode coupling matrix $M_c$ is nonvanishing only if $P_d\ne 0$. In that case, even if one supermode only is injected inside the amplifier, the nonlinear dynamics couples energy into the other supermode too. If $C_1\ne 0$ too, while the four-wave mixing process keeps its symmetric spectral feature, then the signals remain located in one supermode while the idlers are located in the other supermode. In S2 of Supplement 1 we prove this point by showing indeed examples of the gain spectrum for the sidebands of supermode $A_+$ obtained for the same parameters of Fig. 2, which corresponds to initial conditions ($e_{s+}(0)=0$, $e_{s-}(0)\ne 0$), and similarly the gain for $A_-$ and $A_+$ with initial conditions ($e_{s+}(0)\neq 0$, $e_{s-}(0)=0$) too. We stress again that for normal dispersion the signal-idler separation between supermodes is not complete and it does not apply in the low-frequency range. This makes the anomalous dispersion regime the most attractive for a full separation of signals and idlers. An analytical expression for the gain difference in dB between the two supermodes as a function of the dominant eigenvector $\mathbf {v}_1=(v_1, v_2,v_3,v_4)^T$ of matrix $M$ has been derived (see Supplement 1, S7) and reads: $G^\text {diff}_\pm = 10\log \frac {|e_{s\pm }|^2}{|e_{i\pm }|^2} = 10\log \left |\frac {v_{2\mp 1}}{v_{3\mp 1}}\right |^2$. A further intuitive picture of the reason for the frequency asymmetric gain of the two supermodes can be provided by considering that in the asymmetric operational regime signal waves for the individual waveguides modes ($u_{s1,s2}$) exhibit a $\pi$ phase difference, while idlers ($u_{i1,i2}$) have no phase difference (see Supplement 1, S8). We can attribute the origin of this phase-shift to the interplay between the asymmetric pump power distribution between the two cores and $C_\text {odd}$ — the odd part of the frequency dependent coupling — which are the necessary ingredients for the intermodal four-wave mixing to occur. As supermodes “$\pm$" are defined by the sum/difference between individual waveguides modes respectively, i.e., $A_\pm =(A_1\pm A_2)/\sqrt 2$, this unequal phase difference translates, due to interference, into the fact that one supermode is idler-free and the other one is signal-free (at least in a certain frequency bandwidth). While the supermode carrying the signals can be used for idler-free amplification purpose, the supermode carrying the idlers, $A_+$ in our examples, can be used for wavelength conversion applications. The two supermodes can be easily extracted at the amplifier output by a simple combination of a phase shifter and a coupler, as shown in S1 of Suppelment 1, where a potential experimental setup schematic is illustrated. Presence of coupling second-order dispersion ($C_2\ne 0$) and changing magnitude of waveguide dispersion ($\beta _2$) do not affect qualitatively and significantly the amplifier performances and their impact is briefly summarised in Supplement 1, S4 and S5.
Fig. 2. PI amplifier. Map of analytical parametric gain for $A_-$ versus frequency and $C_1$ in anomalous a) and normal b) dispersion regime, and comparison of gain spectrum versus frequency between theory (continuous line) and numerics (dots) for anomalous c) and normal d) dispersion regime with $C_1=\mp 7.5\text { ps}\cdot \text {km}^{-1}$ ($C_1=0$) shown in orange (gray). Other parameters used are $\beta _2=\mp 0.5\text { ps}^2\text {km}^{-1}$, $\gamma =10\text { W}^{-1}\text {km}^{-1}$, $P_p=5.4\text { W}$, $C_0=15\text { km}^{-1}$, $C_2=0$, and $z=0.1\text { km}$.
Fig. 3. PS amplifier. Map of analytical parametric gain for $A_-$ versus frequency and $C_1$ in anomalous a) and normal b) dispersion regime with $\phi _s=0$, and versus frequency and $\phi _s$ in anomalous c) and normal d) dispersion regime with $C_1=\mp 7.5\text { ps}\cdot \text {km}^{-1}$; gain spectrum versus frequency comparison between theory (continuous line) and numerics (dots) for anomalous e) and normal f) dispersion regime with $\phi _s=0$ and $C_1=\mp 7.5\text {ps}\cdot \text {km}^{-1}$ ($C_1=0$) shown in orange (gray). Remaining parameters are like in Fig. 2.
Fig. 4. $P_d$-dependency. Map of PI analytical parametric gain for $A_-$ versus frequency and $P_d$ in anomalous a) and normal b) dispersion regime; and map of PS parametric analytical gain versus frequency and $P_d$ in anomalous c) and normal d) dispersion regime with $\phi _s=0$. Other parameters used are $C_1=\mp 7.5\text { ps}\cdot \text {km}^{-1}$, $\beta _2=\mp 0.5\text { ps}^2\text {km}^{-1}$, $\gamma =10\ \text {W}^{-1}\text {km}^{-1}$, $C_0=15\ \text {km}^{-1}$, $C_2=0$, and $z=0.1\ \text {km}$. The variation of $P_d$ depicted corresponds to a change of $P_p$ in the interval $[3, 8.5]\text { W}$.
We have furthermore verified through numerical simulations that the undepleted pump approximation assumed in the analytical theory is excellent for the parameters used in this study (input signals power $\sim 10^{-12}$ W) as pump depletion is negligible, resulting in an about $10^{-9}\%$ of pump power reduction between output and input. The validity of this assumption can also be appreciated indirectly from the nice agreement between theory and simulations across the whole paper.
In conclusion, we have described parametric amplification enabled by intermodal four-wave mixing in a dual-core optical fiber. We have shown how the interplay between frequency-dependent coupling and pump power unbalance in two coupled nonlinear waveguides constitutes the leading factor to achieve signals and idler separation through a frequency asymmetric gain for the two system supermodes. We have shown that the latter is possible for realistic dual-core fiber parameters. Our rigorous analytical theory, in excellent agreement with numerical simulation results, will provide a guide for future design of experimental demonstrations of the concept presented, for optimising idler-free amplifier performances (including bandwidth, maximum gain, energy efficiency), and for studying possible impairments. Furthermore, our framework can be adapted to describe coupled waveguides made of different materials such as silicon and silicon nitride. The results discussed in this work will potentially enable a more efficient use of the bandwidth in optical parametric amplification and may find applications for wavelength conversion too.
Funding
Engineering and Physical Sciences Research Council (EP/W002868/1); Royal Academy of Engineering (Research Fellowship Scheme).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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