Few-Mode Fiber with Low Spontaneous Raman Scattering for Quantum Key Distribution and Classical Optical Communication Coexistence Systems

Abstract

In this paper, the theoretical model of spontaneous Raman scattering (SpRS) in few-mode fiber (FMF) is discussed. The influence of SpRS on quantum key distribution (QKD) in FMF is evaluated by combining wavelength division multiplexing (WDM) and space division multiplexing (SDM) techniques. On this basis, an improved ring-assisted FMF is designed and characterized; the transmission distance can be increased by up to 54.5% when choosing different multi-channels. The effects of forward and backward SpRS on QKD are also discussed.

1. Introduction

As a highly secure and reliable encryption method, QKD can generate secure keys between remote communication entities based on the inherent unpredictability of quantum physics [1]. It provides a theoretical basis for secure symmetric key encryption systems and has the potential to fundamentally alter the methods of protecting information exchange in the future. In the past few years, significant progress has been made in QKD research in terms of protocols and networks [2,3,4], including improvements to QKD protocols [5,6], noise analysis in quantum channels [7] and exploration of quantum relay networks [8,9]. The transmission distance and security key rate (SKR) of QKD have been greatly improved. In ultra-low-loss fiber, the transmission distance of QKD can reach up to 404 km [8], and the longest transmission distance of twin-field QKD is extended to 1002 km [10]. The highest SKR of high-speed QKD systems can reach 110 Mbps [11]. These advances have paved the way for the implementation of QKD networks. However, the high cost of deploying dedicated fibers is a major obstacle to the widespread use of QKD [12,13]. A promising solution to reduce deployment costs is to coexist QKD with classic optical communication. The coexistence system was first proposed in single-mode fiber (SMF) and has undergone a series of theoretical and experimental studies in WDM [14,15]. Nevertheless, the SpRS generated by classical signals is the biggest challenge of the scheme [16,17]. The wavelength range of Raman scattering exceeds 200 nm, making it susceptible to affecting quantum channels. It hinders the practical application of coexistence solutions in the most advanced technologies.

At the same time, in order to further improve the capacity, research on SDM has received increasing attention. Several types of optical fibers can achieve spatial multiplexing, such as multi-core fiber (MCF) [18], FMF, and few-mode multi-core fiber [19]. Among them, the modes supported by FMF can be used as parallel channels for independent signals. Compared with MCF, FMF has a simple manufacturing process and large effective mode field area, which is beneficial for reducing nonlinear noise and suitable for transmitting sensitive quantum signals [20,21]. Unfortunately, despite numerous experimental studies on FMF [22,23], there is limited theoretical research evaluating the impact of SpRS on QKD in FMF. Therefore, this paper explores the issue based on previous research. According to the conclusion, an improved ring-assisted FMF is designed, which significantly increases the signal transmission distance.

2. The SpRS When QKD Coexists with Classical Signals in FMF

In the coexistence system based on FMF, each mode only transmits classical or quantum signals, and the classical signals in each mode are at different wavelengths. This section will discuss the properties of the SpRS between modes in FMF and analyze its impact on the SKR of QKD systems.

2.1. Derivation of the SpRS in FMF

The proposed SpRS is based on the inter mode crosstalk effect. Firstly, the process of mode crosstalk is derived as shown in Equations (l) and (2), where P_i and P_j are the transmitted powers in each channel along the z-axis, α is the attenuation coefficient in the channels, and h_ij is the power coupling coefficient [24].

$${\displaystyle \frac{d{P}_j(z)}{dz}}=-\alpha z+h_{ij}(P_j-P_i)$$

(1)

$${\displaystyle \frac{d{P}_i(z)}{dz}}=-\alpha z+h_{ij}(P_i-P_j)$$

(2)

Equations (3) and (4) can be obtained by solving Equations (1) and (2) for the conditions that P_j(0) = p_c and P_i(0) = p_q, which represent the initial power of the classical and quantum channel, respectively.

$$P_j(z)={\displaystyle \frac{1}{2}}\left[(p_c+p_q)\cdot \exp (-\alpha z)-(p_c-p_q)\cdot \exp (-(\alpha +h_{ij})z)\right]$$

(3)

$$P_i(z)={\displaystyle \frac{1}{2}}\left[(p_c+p_q)\cdot \exp (-\alpha z)+(p_c-p_q)\cdot \exp (-(\alpha +h_{ij})z)\right]$$

(4)

Then, the derivation of SpRS is carried out, that is, the classical signal of the nth wavelength channel in jth mode generates SpRS in the mth quantum channel of ith mode. The SpRS is mainly composed of two parts. One is that the classical signals generate crosstalk in the channel of the quantum signal, which produces SpRS. The other part is that the classical signal generates SpRS in its own channel, and then the SpRS crosstalk in the channel of the quantum signal.

In the first part, assuming the transmission distance is dz, the power of forward SpRS is as follows [12]:

$$dP{L}_{ICXT-FSRS}(z)={\eta }_{mn}{P}_{ICXT}(z)dz$$

(5)

where η_mn is the Raman scattering factor between the mth and nth wavelength channels, and P_ICXT(z) is the crosstalk power at dz, which can be obtained from Equation (4).

Due to channel loss, when the power of SpRS is transmitted to the output end of the fiber, the power can be obtained as shown in Equation (6). The equation can be integrated to obtain the power of the first part of the forward SpRS, as shown in Equation (7).

$$d{P}_{ICXT-FSRS}(z)=dP{L}_{ICXT-FSRS}(z)\exp [-{\alpha }_q(L-z)]$$

(6)

$$\begin{array}{cc}\hfill P_{ICXT-FSRS}& ={\displaystyle {\int }_0^{L}d{P}_{ICXT-FSRS}(z)}dz\hfill \\ & ={\displaystyle \frac{1}{2}}{\eta }_{mn}\cdot e^{-{\alpha }_{i}L}\cdot \left\{{\displaystyle \frac{p_c+p_q}{{\alpha }_i-{\alpha }_j}}[e^{({\alpha }_i-{\alpha }_j)L}-1]\right.\left.-{\displaystyle \frac{p_c-p_q}{{\alpha }_i-{\alpha }_j-2h_{ij}}}[e^{({\alpha }_i-{\alpha }_j-2h_{ij})L}-1]\right\}\hfill \end{array}$$

(7)

A similar method is used to derive the second part of forward SpRS. The power of Raman scattering generated at the dz distance in jth mode is

$$dP{L}_{FSRS-ICXT}(z)={\eta }_{mn}{P}_{CS}(z)dz$$

(8)

P_cs(z) is the power of the signal at dz, which can be obtained from Equation (3). The power crosstalks into the ith mode and transmits to the output port of the FMF, as shown in Equation (9).

$$d{P}_{FSRS-ICXT}(z)=dP{L}_{FSRS-ICXT}(z)\exp [-{\alpha }_q(L-z)]$$

(9)

The power equation can be derived as [25,26]:

$$\begin{array}{cc}\hfill P_{FSRS-ICXT}& ={\displaystyle {\int }_0^{L}d{P}_{FSRS-ICXT}(z)}dz\hfill \\ & ={\displaystyle \frac{1}{2}}{\eta }_{mn}\cdot e^{-{\alpha }_{i}L}\cdot \left\{{\displaystyle \frac{p_c+p_q}{{\alpha }_i-{\alpha }_j}}[e^{({\alpha }_i-{\alpha }_j)L}-1]\right.\left.-{\displaystyle \frac{p_c-p_q}{{\alpha }_i-{\alpha }_j-2h_{ij}}}[e^{({\alpha }_i-{\alpha }_j-2h_{ij})L}-1]\right\}\hfill \end{array}$$

(10)

From the above derivation, it can be concluded that the forward SpRS is

$$\begin{array}{cc}\hfill P_{FSRS}& =P_{ICXT-FSRS}+P_{FSRS-ICXT}\hfill \\ & ={\eta }_{mn}\cdot e^{-{\alpha }_{i}L}\cdot \left\{{\displaystyle \frac{p_c+p_q}{{\alpha }_i-{\alpha }_j}}[e^{({\alpha }_i-{\alpha }_j)L}-1]\right.\left.-{\displaystyle \frac{p_c-p_q}{{\alpha }_i-{\alpha }_j-2h_{ij}}}[e^{({\alpha }_i-{\alpha }_j-2h_{ij})L}-1]\right\}\hfill \end{array}$$

(11)

For the case of multiple classical channels, the forward SpRS of the mth channel in ith mode can be calculated as

$$P_{F\_SpRS}={\displaystyle \sum _{j=1}^J{\displaystyle \sum _{n=1}^N{P}_{FSRS}^{j,n}}}$$

(12)

Similarly, the backward SpRS can be obtained as

$$\begin{array}{cc}\hfill P_{BSRS}& =P_{ICXT-BSRS}+P_{BSRS-ICXT}\hfill \\ & ={\eta }_{mn}\cdot \left\{{\displaystyle \frac{p_c-p_q}{{\alpha }_i+{\alpha }_j+2h_{ij}}}[1-e^{-({\alpha }_i+{\alpha }_j+2h_{ij})L}]\right.\left.+{\displaystyle \frac{p_c+p_q}{{\alpha }_i+{\alpha }_j}}[1-e^{-({\alpha }_i+{\alpha }_j)L}]\right\}\hfill \end{array}$$

(13)

For the case of multiple classical channels, the backward SpRS of the mth channel in ith mode can be calculated as

$$P_{B\_SpRS}={\displaystyle \sum _{j=1}^J{\displaystyle \sum _{n=1}^N{P}_{BSRS}^{j,n}}}$$

(14)

2.2. Impact of the SpRS on QKD

The protocol used in the simulation is the BB84 protocol with decoy-state method. The SKR is lower-bounded by [27]

$$R=q\left\{-Q_{\mu }{f}_e{H}_2(E_{\mu })+Q_1[1-H_2(e_1)]\right\}$$

(15)

where H₂ is the binary Shannon entropy, q is set to 1/2 for the BB84 protocol, and f_e accounts for the efficiency of error correction, which is set to 1.15. μ is the average number of photons in one pulse. Q_μ and E_μ are the overall gain and the quantum bit error rate. Q₁ and e₁ are the gain and the error rate of a single-photon state. Q_μ and E_μ can be obtained by Y₀, which is the noise count including the dark count of the single photon detector and the SpRS noise. Considering the widely used dual-detector system, Y₀ can be expressed as Ref. [12],

$$Y_0=2p_{dark}+p_{SRS}$$

(16)

where p_dark is the dark count rate of each single photon detector. p_SRS is the noise photon caused by the SpRS, which can be calculated by [15].

$$p_{SRS}=\left\{\begin{array}{c}{\displaystyle \frac{P_{F\_SpRS}\times \Delta f\times \Delta t}{h\times f}}, forward\\ {\displaystyle \frac{P_{B\_SpRS}\times \Delta f\times \Delta t}{h\times f}}, backward\end{array}\right.$$

(17)

where h is the Planck’s constant, and f is the frequency of the quantum channel. Δf is the receiving bandwidth of the quantum channel, and Δt is the detector effective gating width.

3. Performance of QKD in FMFs

On the basis of the above derivation, the performance of QKD at multiple wavelengths in SMF is demonstrated. In simulation, the quantum channel is at 1550.12 nm, corresponding to ITU standard channel. The classical channels are 1539.77~1542.14 nm with a wavelength interval of 0.8 nm. η_mn between the quantum channel and classical channels can be obtained from Ref. [15]. The commercial SMF used is a step-index fiber, with the structure shown in Figure 1a and parameters listed in

Table 1. The radius and refractive index of each fiber.

	nco	ncl	acl (μm)	aco (μm)
SMF	1.452	1.444	62.5	3
FMF1	1.452	1.444	62.5	7
FMF2	1.459	1.444	62.5	6
RCF	1.459	1.444	62.5	aco1 = 4, aco2 = 7.5
6MF	1.456	1.444	62.5	8

. The power of classical signals is set to 0 dBm, and the power of the quantum signal is −80 dBm. According to Equation (15), the maximum transmission distance of the quantum signal is only 0.64 km due to the interference of SpRS, as shown in Figure 2. In order to increase the transmission distance, the coexistence system is applied in FMF.

As can be seen from Equations (11)–(14), the power of SpRS in FMF is related to the fiber length and the initial power of the signal, α and h_ij. Among them, the parameters determined by FMF itself are α and h_ij. The h_ij can be expressed as [28]

$$h_{ij}={\displaystyle \frac{2d_c{\left|{\kappa }_{ij}\right|}^2}{1+{[d_c({\beta }_i-{\beta }_j)]}^2}}$$

(18)

where β is the propagation constant. κ_ij is the mode coupling coefficient, which can be calculated as [29]

$${\kappa }_{ij}={\displaystyle \frac{k_0^2}{{\beta }_j}}{\displaystyle \iint (n^2-n_0^2)}{f}_i^{\ast }(x,y)f_j(x,y)dxdy$$

(19)

where (n² − n₀²) is the refractive index perturbation [30], and f(x,y) is the mode field distribution.

According to Equations (18) and (19), the h_ij of several commercial FMFs is calculated, and the characteristics applied to coexistence systems are analyzed. The fibers used include two kinds of four-mode fiber (FMF₁ and FMF₂), RCF and six-mode fiber (6MF). The structures of the fibers are shown in Figure 1a,b, and the parameters are listed in Table 1. The modes supported by each fiber are shown in Figure 1c. Furthermore, the refractive index difference (Δn_eff) and h_ij between modes are listed in

Table 2. The Δn_eff of the FMFs.

Δneff (×10−3)
	LP01 and LP11	LP11 and LP21	LP21 and LP02	LP31 and LP21	LP02 and LP31	LP31 and LP12
FMF1	2.4	2.9	0.6	-	-	-
FMF2	3.5	4.5	1.2	-	-	-
RCF	0.7	2.1	-	3.1	-	-
6MF	2.1	2.8	0.9	-	2.4	1.6

and

Table 3. The h_ij of the FMFs.

	hij (km−1)
	LP01 and LP11	LP01 and LP21	LP01 and LP02	LP01 and LP31	LP01 and LP12
FMF1	3.72 × 10−4	5.40 × 10−5	3.55 × 10−5	-	-
FMF2	1.7962 × 10−4	2.7054 × 10−5	2.0182 × 10−5	-	-
RCF	1.41 × 10−1	9.14 × 10−3	-	1.74 × 10−3	-
6-MF	4.8449 × 10−4	7.1666 × 10−5	5.3517 × 10−5	1.9382 × 10−5	1.6384 × 10−5

, respectively. It is worth noting that, considering the practical application, the quantum channel is placed in LP₀₁ mode and set to 1550.12 nm, so only the h_ij between LP₀₁ and other modes is calculated in Table 3.

Figure 3 and Figure 4 show the relationship between SKR and transmission distance in a coexistence system based on FMF, where the quantum channel is LP₀₁ and the classical channels are other modes. WDM technology is used in each classical channel, which can support four wavelengths at the same time. The wavelength range is still 1539.77~1542.14 nm with an interval of 0.8 nm. η_mn can be obtained from Ref. [15]. Referring to the parameters provided on the official website, α is set to 0.2 dB/km (LP₀₁), 0.2 dB/km (LP₁₁), 0.21 dB/km (LP₂₁), 0.21 dB/km (LP₀₂), 0.215 dB/km (LP₃₁) and 0.215 dB/km (LP₁₂). Figure 3 shows the variation in SKR in each fiber when the classic channel is only in one mode. Figure 4 is the variation in SKR when the classical signals have multiple modes. Comparing the two sets of figures, it can be seen that the quantum channel will suffer less interference and have a longer transmission distance when there are fewer modes of classical signals. In addition, it is found that when the classical signal is in LP₁₁ mode, the transmission distance of the quantum signal will be greatly reduced, which is believed to be caused by the large mode coupling between LP₀₁ and LP₁₁.

According to previous research, mode coupling in FMFs is related to Δn_eff. As Δn_eff increases, Δβ also increases, resulting in a decrease in h_ij. Due to the fact that LP₀₁ and LP₁₁ are adjacent modes, the mode coupling is more pronounced. As reflected in Table 2 and Table 3, h_ij is at a low level when the Δn_eff is large. Therefore, in order to reduce the power of SpRS in coexistence systems and promote the performance of QKD, it is necessary to reduce h_ij, that is, increase the Δn_eff to a certain extent. Based on the above discussion, the optimization of FMF will be developed in the following section.

4. Performance of QKD in the Ring-Assisted FMF

The most intuitive way to improve Δn_eff is to increase the effective refractive index difference between core and cladding. However, for the given number of modes and parameters, a large Δn_eff means a small mode field area, making it difficult to suppress nonlinear effects. The trade-off between mode field area and Δn_eff in step-index fiber limits the overall performance [31]. In order to solve this contradiction, a ring-assisted FMF was designed [32,33]. The n_eff of modes is controlled by adding a refractive index ring in the fiber core. By carefully selecting parameters of the refractive index ring, the n_eff can be redistributed to optimize the effective refractive index difference between modes.

As shown in Figure 3 and Figure 4, the RCF has a large h_ij due to its structure characteristics, resulting in a short transmission distance. Six-mode fiber has difficulties in mode multiplexing and de-multiplexing in practical applications. In four-mode fiber, the transmission distance is longer, but there is also a large mode coupling between LP₀₁ and LP₁₁. Therefore, we used FMF₂ as the basic parameter for design, where a_co3 = 6 μm, a_cl = 62.5 μm, n_cl = 1.444 and n_co1 = 1.459. a_co1 and a_co2 are the radii of the refractive index rings, while n_co2 and n_co3 are the refractive indices of the two refractive index rings, as shown in Figure 5.

Considering the changes involving multiple parameters, the ergodic method was adopted for parameter design. Figure 6 shows the impact of variation in the refractive index ring radius and refractive index on Δn_eff. The radius ranges are 0.5~4.5 nm and 1.5~5 nm with an interval of 0.5 nm. The refractive index range is 1.445~1.454 with an interval of 0.001. According to the simulation results, the ring-assisted FMF parameters can be determined as a_co2 = 5 μm, a_co1 = 3.5 μm, n_co2 = 1.457, and n_co3 = 1.45. Correspondingly, the Δn_eff and h_ij of the ring-assisted FMF are listed in

Table 4. The Δn_eff and h_ij of the ring-assisted FMF.

	Δneff (×10−3)			hij (km−1)
	LP01 and LP11	LP11 and LP02	LP21 and LP02	LP01 and LP11	LP01 and LP02	LP01 and LP21
ring-assisted FMF	5.4	4.2	0.5	6.83 × 10−5	9.27 × 10−6	1.01 × 10−5

. Compared with the parameters in Table 2 and Table 3, the Δn_eff between LP₀₁ and LP₁₁ increases, while h_ij decreases significantly.

Then, the performance of QKD when using the ring-assisted FMF is calculated. Through WDM technology, each mode of the classical channel still contains four wavelengths. When the classic channel is only one mode, the variation between SKR and transmission distance is as shown in Figure 7a. The solid line represents the ring-assisted FMF, and the dotted line represents FMF₂. It can be seen that the transmission distance has been greatly improved, with the classical channel showing the greatest improvement at LP₁₁, reaching 58.54 km and showing an increase of 47.16%. When the classical channels are LP₂₁ and LP₀₂, the transmission distance of the quantum signal is increased by 23% and 10.9%, reaching 97.66 km and 100.42 km. Similarly, Figure 7b shows the variation in SKR with transmission distance when the classical channels have multiple modes. When it contains three modes (LP₁₁, LP₂₁, LP₀₂) or two modes (LP₂₁, LP₀₂), the transmission distance increases by 47% (up to 52.39 km) and 24.1% (up to 85.21 km), respectively.

In Figure 8, the number of wavelengths in each mode of classical channels increases to ten, ranging from 1539.77 to 1546.92 nm with an interval of 0.8 nm. There is a significant variation in transmission distance at different numbers of wavelengths. Figure 8a shows the comparison of ring-assisted FMF and FMF₂ performance in one mode. When the classic channels are LP₁₁, LP₂₁ and LP₀₂, the transmission distance increases by 54.5%, 37.15% and 17.43%, reaching distances of 38.21 km, 76.97 km and 80.56 km, respectively. Furthermore, when the classical channels include three modes (LP₁₁, LP₂₁, LP₀₂) or two modes (LP₂₁, LP₀₂), the performance comparisons are as shown in Figure 8b, where the transmission distances increase by 52.68% (up to 33.59 km) and 34.13% (up to 62.48 km).

The impact of forward and backward SpRS on SKR is also discussed in Figure 9 when using ring-assisted FMF. When there are four wavelengths in each mode, backward SpRS has more impairment of QKD, resulting in a shorter transmission distance. Increasing to ten wavelengths, forward SpRS causes greater damage. The number of wavelengths and channel modes can be selected according to actual applications.

5. Conclusions

In summary, this paper derives a simulation model of SpRS in FMF and discusses the impact of SpRS on QKD. The simulation results show that the influence of SpRS depends on the h_ij between modes. On this basis, a ring-assisted FMF is designed to effectively improve the transmission distance of signals in the coexistence system. Furthermore, the impact of forward and backward SpRS on SKR is also discussed. In practical applications, the number of wavelengths and channel modes can be selected according to the requirements.