## Abstract

We investigate the impact of fiber birefringence and spontaneous Raman scattering on the properties of photon pairs that are generated by the spontaneous four-wave mixing process in birefringent fibers. Starting from the formulation of the theory of four-wave mixing, we show a theoretical model for a generated optical field with the consideration of the Raman scattering and a Gaussian-distributed pump. The theoretical model is then applied for deriving the closed expressions of the photon-pair spectral properties as a function of the fiber birefringence. Also, with the modeled Raman gain, we evaluate the reduction of the pair production rate due to the presence of the Raman effect as well as the contributions of the Raman-scattered photons over a broad wavelength range. The predictions are experimentally verified with a commercial polarization-maintaining fiber.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Correlated photon-pair sources are of great importance for quantum technologies such as fundamental tests of quantum mechanics [1], quantum-enhanced measurements [2] and quantum communication [3,4]. Among various media for generating correlated photon pairs such as nonlinear bulk crystal [5], quasi-phase-matched crystal or waveguide [6,7], quantum dot [8], cold atomic ensemble [9], optical fiber-based sources have gained popularity in recent years due to their feasibility and good performance on generating photon pairs with a spatial mode perfectly compatible with single-mode optical fibers, which makes them suitable for fiber-based or long-distance quantum communication [10–12].

Practical applications often require specific features of photon pairs such as central wavelength, photon bandwidth, pair generation rate, etc. Considerable efforts have been made to generate photons with tailored properties via dispersion engineering among different nonlinear media [13]. As for non-birefringent fibers, one common approach to satisfy the phase-matching condition is to locate the pump and photon-pairs wavelengths near the zero-dispersion wavelength (ZDW) [10]. Thus, the flexibility of wavelength selection depends on the capability of shifting the ZDW, which could be realized by engineering the transverse structure, changing the distribution of refractive index or doping the fiber material [14,15]. However, the generated photon pairs still suffer from strong photon background from Raman scattering at room temperature because of the small frequency detuning from the pump. Eliminating this Raman scattering requires cooling systems that add extra complexity to experimental setups.

Alternatively, the birefringence-assisted phase-matching process has been utilized due to its flexibility on wavelength selections and the ability for avoiding Raman contamination [12]. Many recent studies including theoretical calculations and experimental implementations [16–18] show the unique properties of the birefringent fiber-based source. However, the spectral engineering and Raman scattering influence of birefringent fiber-based sources has been a major bottleneck toward practical applications. We discuss two aspects that are essential to tackle this problem in the following sections. First, the influence of fiber birefringence on the concerned spectral properties of photon pairs needs to be evaluated. It has already been widely established that the fiber birefringence depends on fiber structural properties such as the thickness of the stress-applying part, the distance between the stress-applying area and fiber core, the diameter of air holes in microstructure fibers, and geometrical imperfections at cross section [19–21]. Clarifying the role of fiber birefringence on photon spectral properties can be useful for obtaining tailored photon properties by making appropriate fiber manufacturing. Second, it is necessary to evaluate the influence of Raman scattering in a birefringent fiber-based four-wave mixing (FWM) process. Even though a large frequency detuning between the pump and generated photon pairs ensures a weak Raman gain at phase-matched wavelength, there may still have considerable remaining Raman-scattered photons which influence the photon-pair generation rate and the purity of FWM photons.

In this paper, we address the above two problems. We develop the coupled-mode equations of the generated optical field by taking the consideration of the Gaussian spectral distribution of the pump and Raman scattering impact. Based on this theoretical expression, we exhibit the birefringence dependence of the essential features belonging to the birefringent fiber-based phase-matching curves. The spectral biphoton properties including photon bandwidth and frequency correlation are derived and simplified by intuitive expressions for pump wavelengths in a certain regime. Besides, the Raman scattering impact on photon generation rate and FWM photon purity are accurately evaluated based on the theoretical expression of generated photon fields. To deal with the wide separation between signal and idler wavelengths, we employed a modeled Raman gain spectrum with a broad range of 80THz in both Stokes and anti-Stokes regimes. The results show that the decrement of photon generation rate is frequency-dependent and unavoidable at arbitrary phase-matched wavelengths while the contamination of Raman-scattered photons can be accurately assessed and further suppressed by choosing appropriate pump power and filter bandwidth.

This paper is organized as follows. At the beginning of section 2, we calculate the expression of photon-pair optical fields under the influence of Raman scattering and pump pulse broadening. Based on this expression, we show the role of fiber birefringence in the phase-matched process and joint spectrum formation in section 2.1.1 and section 2.1.2. In section 2.2, we present the influence of Raman scattering both on reducing photon-pair generation rate and on adding extra noise photons. Finally, in section 3, experimental results are presented to verify the theoretical models developed in the previous sections.

## 2. Theoretical analysis of FWM in birefringent fibers

The configuration of our theoretical investigation is depicted in Fig. 1. A strong pump light propagating through a birefringent fiber drives the spontaneous FWM process. In this study, we align the polarization of the pump to the slow-axis and generated photon pairs to the fast-axis, thereby the type-I phase-matching condition satisfied. In addition to the FWM process, we include the spontaneous Raman scattering as well as the self- and cross-phase modulation processes in our description. Therefore, the output optical fields are categorized into the remaining pump photons, FWM photon pairs and Raman-scattered photons, whose time evolution can be described by the coupled-mode equations [22]:

In Eq. (1), $\tilde {R}_{mn}^{(1)}$ and $\tilde {R}_{mnuv}^{(3)}$ are Fourier transforms of the linear and the third-order nonlinear response function, respectively. Considering a fused silica fiber which is treated as a homogeneous and isotropic dielectric medium, we use the nonlinear parameter $\gamma$ to describe the intensity of third-order nonlinear process. Generally, $\gamma$ is associated with $\tilde {R}_{mnuv}^{(3)}$ by [23]:

Equation (1) gives us a general description of the optical fields with arbitrary polarization and spectral distribution. In the following part, we focus on a special case with a set of assumptions to simplify the calculations. Suppose the input pump is an undepleted classical light with a gaussian spectral distribution $G(\omega _p)=\left (1/\sqrt {2\pi }\sigma \right )\exp {\left [-\frac {{(\omega _p-\omega _{p0})}^2}{2\sigma ^2}\right ]}$, we describe the pump optical field as $A_{px}(z,\omega _p)= 2\pi \sqrt {P_{\text {peak}}}G(\omega _p)e^{i\Phi _x(\omega _p,z)}$ ( Here $\Phi _x(\omega _p,z)=[\beta _x(\omega _p)+\gamma P_{\text {peak}}]z$). In this expression, $\omega _{p0}$ is the central wavelength of the pump. $P_{\text {peak}}$ is the peak power of a pulsed pump and $\sigma$ is related to the full width half maximum (FWHM) $Bw_p$ by $\sigma =\frac {Bw_p}{2\sqrt {2\ln 2}}$. Besides, we further ignore the nonlinear interactions between weak generated optical fields ( i.e., signal, idler and Raman-scattered photons) as well as single-photon level frequency conversion of the generated signal ( or idler) photons under a strong pump, then the signal optical field with polarization at the m-axis is

We can make an approximation on frequency detuning as $\Omega _{sp}\approx \Omega _{sp0}=\omega _s-\omega _{p0}$ when a narrowband pump and nearly invariant Raman gain near $\Omega _{sp}$ are employed. Combining with the substitution $\hat {A}_{m}(z,\omega _{s(i)})=\hat {B}_{m}(z,\omega _{s(i)})e^{i\Phi '_m(\omega _{s(i)},z)}$ ( here $\Phi '_m(\omega _{s(i)},z)=[\beta _m(\omega _{s(i)})+\gamma \epsilon _m(\Omega _{s(i)p0})P_{\text {peak}}]z$), the simplified expression of Eq. (3) is:

Starting with the same Heisenberg equation of motion for photonic fields in [22] and adopting the calculation technique for the Gaussian-distributed pump in [28], here we exhibit an expression for the generated FWM and Raman-scattered fields with a Gaussian-distributed pump. Various combinations of phase-matching frequencies with different possibilities and frequency-dependent influences of the Raman scattering effect are shown in the above calculation. In the following sections, we will use this specific expression of generated optical field to clarify the role of birefringence and Raman scattering in birefringent fibers.

#### 2.1 Spectral features of FWM photon pairs

### 2.1.1 Central wavelength of FWM photon pairs

The central wavelength of photon pairs generated by the FWM process in birefringent fibers is determined by the condition of the energy conservation ($\Delta \omega =0$) and the momentum conservation ($\Delta \beta _m=0$). We consider optical fibers exhibiting single ZDW whose birefringence contribution in phase-shift is much greater than the phase-shift caused by the optical nonlinear process such as self-phase modulation, cross-phase modulation and Raman scattering. Then the phase-matching condition in this case becomes:

Using Eq. (7), we can calculate the approximate frequency detuning of the thick solid lines by making second-order Taylor expansion of each $\beta _j (j = p, s, i)$ at the pump frequency $\omega _{p0}$ in the region away from the maximum phase-matched wavelength. Then, the relations between fiber birefringence and frequency detuning can be expressed as [27]:

where $\beta _n(\omega _{p0})=\left .\frac {{\partial }^n \beta }{\partial {\omega }^n}\right |_{\omega =\omega _{p0}}$ ($n=1,2,3\cdots$) stands for the $n$-th order differential of the propagation constant. The corresponding linear relation is shown as the black line in Fig. 2(b).Another feature, the maximum phase-matched wavelength, also has a strong dependence on the fiber birefringence. This phenomenon happens when the group velocity of the signal photon equals that of the idler. More precisely, it can be calculated by solving the following equation:

In this part, we summarize two distinguishing features for the phase-matching curves of the birefringent fiber and exhibit their strong birefringence dependence. The result shows that the frequency detuning in the region away from maximum phase-matched wavelength has a linear correlation with the square root of fiber birefringence while the value of maximum phase-matched wavelength has an inverse linear correlation with it.

### 2.1.2 Photon-pair joint spectrum

The normalized two-photon amplitude joint spectrum can be directly defined through Eq. (6) [12], which is shown as

The frequency correlation of the paired photons is another important property aside from the photon bandwidth. Generally speaking, experiments that aim to build an entangled source based on a certain degree of freedom would prefer photon pairs with the same properties on all other degrees of freedom. Therefore, either high frequency correlation photon pairs [2] or decorrelated photon pairs [32] is pursued depending on different experimental requirements. The frequency correlation of two-photon joint spectrum could be calculated by the ratio between single-particle and coincidence distribution width [33,34], which is shown as

The impact of fiber birefringence on the photon-pair bandwidth and frequency correlation has already been implied by the birefringence-dependent parameter $\tau _{s(i)}$ in Eq. (11) and Eq. (12). However, we expect to find a more intuitive relation that directly shows their birefringence dependence. As previously discussed in section 2.1.1, we can obtain precise approximation by taking the second-order Taylor expansion of the propagation constants when the pump is away from the maximum phase-matched wavelength. Then the corresponding approximation of $\tau _{s(i)}$ is

Combining the above equation with the approximated frequency detuning in Eq. (8), we can obtain the expected intuitive relations. Particularly, when the fiber birefringence satisfies the relation $\Delta n \ll c\beta _2(\omega _{p0})\Omega$, which is appropriate for most commercial silica PMF, the expressions for photon-pair bandwidth and frequency correlation can be further simplified asAs seen in the figure, both of the two properties show an approximately inversed linear relation with the square root of birefringence. The comparison on the performance of approximate equation Eq. (14) with its corresponding accurate expressions Eq. (11) and Eq. (12) show a satisfactory agreement with each other.

In this part, we analyze the impact of the birefringence on photon-pair bandwidth and frequency correlation. Especially, we give approximate expressions that directly show their birefringence dependence when the pump is away from the maximum phase-matched wavelength. Our approximate expressions show that in the low-birefringence region, two spectral properties exhibit linear relations with $1/\sqrt {\Delta n}$, which is in good accordance with the accurate theoretical results. Similar to previous work which focused on changing photon-pair spectral distribution by engineering the dispersion properties in the photonic crystal fibers [36,37], here we determine the impact of fiber birefringence, showing that selecting appropriate birefringence can be a practical approach to adjust the photon-pair bandwidth and frequency correlation of a birefringent fiber-based photon-pair source.

#### 2.2 Photon-pair generation rate and spontaneous Raman scattering

The photon-pair generation rate directly shows the interaction strength between optical fields, which is one of the most significant parameters for evaluating the photon-pair source. According to Eq. (6), the number of FWM photons generated per pulse can be described as:

To show the Raman influence on the photon generation rate at each frequency individually, we assume that the phase-matching condition can be satisfied under arbitrary frequencies, namely $\text {Re}(\Delta \beta _m)=0$. As seen in Fig. 4, we plot the variation of the photon-pair generation rate when the polarizations of paired photons are parallel (blue line) or orthogonal (red line) to the pump. The detailed experimental data for parallel and orthogonal Raman response functions and their corresponding Fourier transforms come from Ref [38]. For the photon pairs generated by birefringent fiber, the frequency detuning from the pump is possibly far beyond 40THz, but the decrease of the photon-pair generation rate with both polarization is unavoidable. Since there are no obvious peaks for both parallel and orthogonal Raman gain in the region beyond 40THz [16], the intensity of the Raman response can be assumed to be zero in this region. It is asymptotically shown that the corresponding FWM generation rate is reduced to $67\%$ compared with the situation when $f_R=0$, which is expected to remain even at cryogenic temperature. This phenomenon essentially comes from the contribution of the phonon or molecular vibration to the Kerr coefficient. The finite phononic response time corresponds with the frequency-dependent Raman susceptibility in the frequency domain, which explains the frequency dependence of the decrement on photon production rate and the distribution of the following-discussed Raman gain. Furthermore, at the anti-Stokes side, the decrement is similar to the Stokes side since the real and imaginary part of $\widetilde {R}_a$ and $\widetilde {R}_b$ is symmetric about the original point [24].

The spontaneous Raman scattering not only influences the photon-pair generation rate of the FWM process but also generates noise photons with a broad spectrum near the pump frequency. Different from generated photon pairs which have narrow and separate bandwidths, the Raman-scattered photons have a continuous spectrum because of the amorphous property of fused silica [25]. Therefore, the bandwidth of the bandpass filter needs to be defined before we calculate the number of Raman-scattered photons. Suppose the bandwidth of the filter is $f_{bw}$ in the frequency domain and the center wavelength is $\omega$, then the number of Raman scattering photons generated at frequency $\omega$ is:

In Fig. 5, we compare the intensity of FWM photon pairs and parallel Raman-scattered photons under different conditions. The parallel component of the broad Raman gain spectrum is obtained by combining the multiple-vibrational-mode model in Ref [39], the Gaussian model in Ref [40] and experimental data of a dry suprasil fiber in [41]. The orthogonal Raman-scattered photons are not included due to the absence of the experimental data to accurately build an orthogonal Raman model over 40THz. Since the parallel components are known to be dominant over the orthogonal component, the parallel Raman-scattered photons are the main contribution to the decrease of signal-to-noise ratio when collecting FWM photon pairs. Besides, the linear dependence of the frequency detuning using the above Raman gain model results in nonzero intensity at the low-frequency detuning regime. It is inaccurate compared with the cubic low-frequency dependence shown in the experiment that leads to the dip in the noise at the pump frequency [26]. However, since we focused on the Raman intensity at signal and idler’s frequencies that are far from the low-frequency detuning regime, the current model is enough for the estimation of Raman influence on the signal-to-noise ratio of the FWM photon-pairs.

As shown in Fig. 5(a), the photons generated from the birefringent fiber have large frequency detuning which helps to avoid the Raman-dominant region. However, considerable Raman photons still remain, especially at the Stokes side. Several approaches can be considered to increase the relative intensities between FWM photon pairs and Raman-scattered photons. As seen by the dashed line (green) and dot-dashed line (yellow), cooling the fiber is a feasible way to reduce Raman intensity by reducing the phonons in the excited state. However, the cooling reduces only anti-Stokes photons as the Stokes photons mainly come from the coupling to the ground-state phonons. Instead of cooling the fiber, changing pump power and selecting filter bandwidth are two practical ways to enhance the signal-to-noise ratio at room temperature. As shown in Fig. 5(b), the growth of pump power causes a quadratic increase in the number of FWM photon pairs and a linear increase in Raman-scattered photons. Properly selecting the power of the pump pulse improves the signal-to-noise ratio. For filter selection in Fig. 5(c), since Raman intensity is linearly increased with the filter bandwidth, filters with small bandwidth help to improves the signal-to-noise ratio as long as it covers the bandwidth of FWM photon pairs. Besides, since the polarization of FWM photon pairs is orthogonal to the pump and parallel Raman-scattered photons, using a polarizer to block the parallel noise photons is also beneficial for collecting pure FWM photon pairs.

Compared with previous work [42], we study the impact of Raman scattering in birefringent fibers, where both parallel- and orthogonal-polarized Raman effects are essential and a broad Raman gain spectrum is needed for the large frequency detuning of paired-photons. Our simulations on the reduction of photon generation rate both with parallel and orthogonal polarizations show that a 33$\%$ decrement is expected for birefringent fiber-based photon sources. The reduction of signal-to-noise ratio in birefringent fiber extends even to a frequency detuning at 73THz, however, it can be suppressed by increasing pump intensity and reducing filter bandwidth.

## 3. Experiment

In this section, we demonstrate the correlated photon-pair generation via spontaneous FWM process using a commercial PMF to verify the results from the aforementioned theoretical calculations.

#### 3.1 Experimental setup

A schematic drawing of the experimental setup is depicted in Fig. 6. Linearly polarized pump light was generated from a Ti:Sapphire pulse laser (Mira900-P, Coherent) operating at the wavelength of 940nm with a time duration of 3ps. While propagating through 0.23m-long PMF (HB800G, Thorlabs) with the polarization aligned to the slow axis, the pump photons created correlated photon pairs with the polarization parallel to the fast-axis. A polarization beam splitter and notch filter were used to block the pump photons as well as the Raman-scattered photons parallel to the pump. The generated photon pairs then passed through two dichroic mirrors and bandpass filters, and were eventually coupled into single-mode fibers. Part (a) and (b) in Fig. 6 show the apparatus for measuring the spectral properties and intensity of the photons, respectively. The spectral properties of the signal photons (764nm) were characterized by a spectrometer/monochromator (Spectrapro 2750, Princeton Instruments) with a liquid-nitrogen-cooled silicon charge coupled device (Si-CCD). Due to the spectral coverage of Si-CCD, the idler photons (1221nm) were analyzed by combining a monochromator and an InGaAs avalanche photodiode (InGaAs-APD). For the same reason, in part (b), our signal photons were detected by a silicon avalanche photodiode (Si-APD) whereas the idler photons were detected by a superconducting nanowire single-photon detector (SNSPD). The single and coincidence detection rates were analyzed by a time-correlated single-photon detector (TCSPC) with a 1ns coincidence time window.

#### 3.2 Experimental result

Using a spectrometer with high sensitivity and low noise, we measured the central wavelengths of signal and idler photons under varing pump wavelengths. When the wavelength of the pump is tuned from 918nm to 942nm, we measured the corresponding signal and idler photons three times at each pump wavelength. The mean values and standard derivation are shown in Fig. 7(a). Using the simulation parameters in Table 1, the experimentally measured wavelengths are in satisfactory agreement with the phase-matched wavelengths theoretically modeled in the above section. The small difference possibly arises from the variation of birefringence when producing the fiber, which could be $\pm 0.2\times 10^{-4}$ between different fibers [12]. Note that the other group of solutions shown in Fig. 2(a) (thin solid line) has extremely large frequency detuning with pump wavelength far from the maximum phase-matching wavelength. The idler photons with wavelengths at the micron-scale are not in our common consideration for photon pair generation in the optical band. Also, the commercial PMF used in our experiment doesn’t support the propagation mode at this wavelength. Therefore, here we only exhibit the group of solutions with smaller frequency detuning in Fig. 2(a).

In Fig. 7(b), we show the experimental result by the red hollow dots under varying pump power. The black line is the theoretical assumption that already considered the 33$\%$ decrement from the Raman effect on FWM efficiency. The good agreement of pair generation rate between the simulation and the experimental result confirms that the influence from the Raman effect does exist with a frequency detuning far beyond the commonly-considered range from 0 to 40THz. To obtain the exact numbers of pair generation rates under different pump powers, the photon-counting rates of signal and idler photons were measured as a function of the average pump power, as shown in Fig. 8(a). The single ($N_{S}$ and $N_{I}$) and coincidence ($N_{C}$) counting rates were recorded for five times at each pump power. After substracted the detectors’ dark counting rates and the accidental coincidence rates from the raw counting rates, we present the quadratic scaling of photon-counting rates. The pair generation rates was thus calculated as $N_{\textrm {Pair}} = N_{S}N_{I}/N_{C}$.

Besides pair generation rates, we also estimate the heralding efficiencies $H_{S(I)} = N_{C}/N_{S(I)}$ of signal and idler from the measured single and coincidence rates. In Fig. 8(b), it is observed that the heralding efficiencies vary as pump power changes. To explain this feature, we modeled the single and coincidence detection rates including the Raman scattered photons ($N_{\textrm {Raman},S(I)}$) which scales linearly with the incident pump power [43]. We employed the least-squares method to globally fit the model to the measured values of $\{N_{S}, N_{S}, N_{C}, H_{S}, H_{I}\}$ with a set of fitting parameters of $\{N_{\textrm {Pair}}, \eta _{S}, \eta _{I}, N_{\textrm {Raman},S}, N_{\textrm {Raman},I}\}$, where $\eta _{S}$ and $\eta _{I}$ are the total detection efficiencies of signal and idler photons, respectively. From the optimized fitting parameters of 140(3), 0.220(1), 0.187(2), 60(59), 368(23), we estimated the pair production rate and Raman-scattered photon rates to be $1.84(4) \times 10^{-6}$cps/mW$^2$ per pulse and $4.8(3) \times 10^{-6}$cps/mW per pulse, respectively. From the fit of the model, it is shown that spontaneous Raman scattered photons in the orthogonal direction still significantly contribute to the idler photon detections. At the low pump power level where Raman-scattered photons dominate the idler photon detection, the photon detections at the idler arm poorly herald the presence of the signal photons. As the pump power increases, the quadratic increment of the correlated photon pairs dominates over the linear dependence of Raman-scattered photons, and therefore the idler photon detection heralds the signal photons with higher probability. Also, because the filter bandwidth at the idler’s side is 25nm in our experiment, we are able to further suppress the Raman-scattered photons by using a narrower bandpass filter.

So far, we have shown the influence of birefringence and spontaneous Raman scattering in birefringent silica fiber. The experimental results on the phase-matched wavelength as well as the photon generation rate under the consideration of Raman scattering impact show a good agreement with our theoretical simulations. Our experiment shows that with a large frequency detuning far beyond 40THz in birefringent silica fiber, the Raman effect still has a 33$\%$ decrement in FWM efficiency and adds considerable Raman Stokes photons, which are rarely considered in a birefringent fiber-based photon-pair source. It can be noted that although the presented parameters are focused on the optical fiber media, the theoretical framework and the clarified relationships are applicable for other third-order nonlinear processes inside a homogeneous, isotropic and transparent material with their respective phase-matching conditions and Raman gain. For example, chalcogenide $As_2Se_3$ exhibits three orders of magnitude greater third-order nonlinearity than the fused silica. Its Raman gain has been studied in Refs. [44], which could directly be used to our model. Another good candidate is the silicon nitride waveguides [45] where one can engineer the birefringence by adjusting the geometric parameters, e.g., width and thickness. The material is known to be isotropic, and therefore our derived expressions could be used to generate spectrally pure photon pairs at the desired wavelengths.

## 4. Conclusion

In this paper, we investigated the features of nondegenerate correlated photons generated by birefringent fiber both theoretically and experimentally. Starting from coupled-mode equations, we combined the effect of finite pump bandwidth and Raman scattering to find a comprehensive expression for signal and idler optical field. Based on this expression, we analyzed the influence of fiber birefringence on the photon-pair central wavelength and joint spectrum. From our model, the frequency detuning and the maximum phase-matched wavelength in the normal dispersion regime showed the linear dependence on the square root of fiber birefringence. The formulas of photon-pair bandwidth and frequency correlation were also derived and analyzed to show a more detailed relation between fiber birefringence and the two-photon joint spectrum. Additionally, based on the expression of the optical field, we further analyzed the photon-pair generation rate and the influence of Raman scattering. We experimentally confirmed that the existence of Raman scattering reduced the photon-pair generation rate to approximately 67$\%$, which showed that the theoretical model with commonly considered frequency detuning 40THz works well in our case at 73THz. At the same time, the noise from Raman scattering at idler’s wavelength had considerable intensity even if we filter the Raman photons with parallel polarization to the pump. The intensity of the pump and the filter bandwidth should be carefully chosen to balance the production rate and signal-to-noise ratio.

The presented theory could be refined with the inclusion of the dual-pump configuration and more precise Raman model with the consideration of environmental humidity, dopings and broader distribution range. Recently, Koefoed et al. [46] reported the numerical evaluation of the coupled-mode equations using split-step Fourier methods. This allows including arbitrary pulse shapes, self- and cross-phase modulations, and higher harmonics of the pumps. Also, with the released approximations by the numerical solver, the studied relationship could be further verified under more realistic conditions. We believe that our study provides a set of guidelines to engineer the photon-pair source with birefringence, offers a general Raman scattering assessment model, and contributes to the development of fiber-based quantum light sources toward long-distance quantum communications.

## Funding

Industry Canada; Natural Sciences and Engineering Research Council of Canada (RGPIN-386329-2010); Ontario Research Foundation (098, RE08-051); Canada Foundation for Innovation (25403, 30833); National Natural Science Foundation of China (11974205); Tsinghua Initiative Scientific Research Program; Beijing Innovation Center for Future Chip; Key-area Research and Development Program of Guangdong Province (2018B030325002); National Key Research and Development Program of China (2017YFA0303700).

## 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.

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