Accurate and Effective Model for Coexistence of Classical and Quantum Signals In Optical Fibers

TL;DR

This work addresses the challenge of transmitting quantum signals alongside classical data in optical fibers by developing a semi-analytical model based on coupled-field equations that capture key interference mechanisms: SpRS, SRS tilt, FWM (degenerate and non-degenerate), spatial crosstalk, and Rayleigh backscattering in both SMF and SDM contexts. The model derives differential equations for the accumulated interference power $P^{\mathrm{Int}}_{n,i}(z)$ and introduces efficient approximations for FWM through an averaged efficiency factor $\tilde{\\rho}^{(n)}_{ihkl}$, incorporating SRS tilt effects. Numerical results demonstrate when and where interference is minimized (e.g., at the high end of the band for both co- and counter-propagation) and show substantial SDM-based isolation (about 40 dB) in co-propagation, with FWM most problematic in short-reach, co-propagating links and mitigable by spectral spacing. The findings provide practical insights for quantum-channel placement and coexistence strategy, enabling fast evaluation of a wide range of parameter choices in next-generation SDM optical networks.

Abstract

The rising interest in quantum-level communication has resulted in proposals for coexistence schemes with classical signals within the same fiber optic channel, where the most recent proposals leverage novel fibers designed for space-division multiplexing (SDM) transmission. In all cases the large power difference between classical and quantum channels presents challenges for such schemes, as the classical signals generate interfering noise that corrupts the quantum signal. In this work, we discuss the main interference mechanisms in coexistence scenarios and provide a model to quantify their impact on the quantum signal quality. Analytical approximations in the model allow accurate and fast numerical solutions in the millisecond time-scale. The model accounts for out-of-band non-linear interference effects, namely spontaneous Raman scattering (SpRS) and four-wave-mixing (FWM) in both cases of single-mode and SDM fibers with weakly-coupled degenerate mode groups. Rayleigh and SpRS backscattering are considered in counter-propagating scenarios. Since broadband classical transmission is targeted, the model also accounts for the effect of stimulated Raman scattering (SRS)-induced power tilt. Use of the model in sample scenarios indicates that the interference noise power is minimized at the high end of the transmission band in both cases were the quantum is co- and counter-propagating with respect to the classical signals, with a preference of one or the other scheme depending on the link length and quantum signal center frequency. Our model reveals that FWM has negligible impact in counter-propagating schemes, but can be relevant in co-propagating schemes under certain scenarios. Nevertheless, the FWM interference can be mitigated by deallocating the classical signals adjacent to the quantum channel.

Figures (10)

• Figure 1: (a) Power evolution of the classical signals and the predominant interference effects under coexistence schemes. The fwm generated from the crosstalked signal is disregarded due to its negligible intensity. mcf experience crosstalk from adjacent cores, and mmf from distinct spatial mode groups. (b) Coexistence scenarios with corresponding signal directions and dominant interference effects on the quantum signal. In single-mode scenarios crosstalk is disregarded. (c) Reported skr in various qkd implementations with interference psd values. For DV-qkd, explicit interference power values were not provided. Instead, the values were inferred from reported noise photon counts or from the qber and transmission rates, assuming the detector bandwidth equals the qkd symbol rate, unit detector efficiency, and a 50% photon discard rate due to basis misalignment.
• Figure 2: Frequency- and mode-group-dependent attenuation profiles for single-mode and sdm scenarios. The attenuation values are obtained using the model from walker1986rapid.
• Figure 3: fwm interference noise psd at the center frequency of the allocated spectrum (${f=(f_{\mathrm{Max}}+f_{\mathrm{Min}})/2}$) versus propagation distance. The solid line represents the analytical expression \ref{['eq:fwmcomplete']}, while the dot-dashed lines correspond to the solutions using the approximated expression from \ref{['eq:chiavg']} and the upper and lower envelope boundaries from \ref{['eq:chilimits']}. The inset graph shows the fwm interference noise psd at the fiber end. Simulation parameters are provided in Table \ref{['tab:parameters']}, and attenuation values are shown in Fig. \ref{['fig:Attenuation']} for a single-mode scenario.
• Figure 4: Total fwm interference noise power in a 50 GHz channel at the (a) lower and (b) upper edges of the allocated spectrum versus distance in a single-mode scenario. The figure shows the actual fwm interference noise power, along with curves considering the proposed approximations for the fwm efficiency factor given in \ref{['eq:rhosrs']} and \ref{['eq:rhoappsrs']}. For reference, the fwm interference noise power neglecting srs is also presented. We consider only the strongest fwm contribution, arising from the three neighboring channels. A total launch power of 30 dBm is assumed, uniformly distributed across all channels. For the actual fwm curve, we utilize $10^{6}$ steps. The remaining simulation parameters are listed in Table \ref{['tab:parameters']}, and the attenuation profile is shown in Fig. \ref{['fig:Attenuation']}. To improve visual clarity, the curves are plotted for the second half of the fiber span. The inset illustrates the actual fwm interference and the approximation using \ref{['eq:rhoappsrs']} across the full span length.
• Figure 5: Profiles of the coefficients for (a) Raman gain efficiency and spectral Raman cross-section, and (b) frequency-dependent crosstalk. The Raman gain profile is adapted from stolen1989ramanlin2006raman. The Raman cross-section is derived from \ref{['eq:etagr']} for a signal at the center frequency of the allocated spectrum (${f=(f_{\mathrm{Max}}+f_{\mathrm{Min}})/2}$), and is normalized by the signal bandwidth $B_{s}$. The crosstalk is derived from hayashi2011characterization, for a linear slope of 1 dB/km across the allocated spectrum in wavelength scale, and crosstalk of $-60$ dB/km at the center wavelength.