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 [1012].

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 [1618] 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 [1921]. 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]:

$$\begin{aligned} \frac{\partial \hat{A}_{m}(z,\omega_k)}{\partial z} & =i\sum_{n}\tilde{R}_{mn}^{(1)}(\omega_k)\hat{A}_{n}(z,\omega_k)+\frac{i\hbar \omega_0}{{(2\pi)}^2}\sum_{nuv}\iint d\omega_1d\omega_2\tilde{R}_{mnuv}^{(3)}(\omega_2-\omega_1)\hat{A}_{n}^{{\dagger}}(z,\omega_1)\\ & \times \hat{A}_{u}(z,\omega_2)\hat{A}_{v}(z,\omega_k+\omega_1-\omega_2)+\frac{i\sqrt{\hbar \omega_0}}{2\pi}\sum_{n}\int d\omega_1\hat{M}_{mn}(z,\omega_k-\omega_1)\hat{A}_n(z,\omega_1). \end{aligned}$$
Here, $\omega _0$ is the carrier frequency of the slowly varying envelope while $\omega _k$ ($k =p,s,i$) stands for the frequency of pump ($p$), signal ($s$) and idler ($i$). Other subscripts $m,n,u$ and $v$ represent the polarization of the optical field, which could be either $x$- or $y$-direction. $\hat {A}_{mk}(z,\omega _k)$ is the normalized field operator for a monochromatic wave in the frequency-domain which satisfies the commutation relation $[\hat {A}_{m}(z,\omega ),{\hat {A}}^{\dagger }_{n}(z',\omega ')]=2\pi \delta _{mn}\delta (z-z')\delta (\omega -\omega ')$.

 figure: Fig. 1.

Fig. 1. Correlated photon-pair generation in birefringent fiber via a type-I degenerate FWM process. Two x-polarized pump photons (slow axis) generate photon pairs with orthogonal polarization in the y-direction (fast axis). Together with correlated photon pairs generated from the FWM process, the interaction between pump photons and molecular-vibration phonons generates Stokes and anti-Stokes Raman photons with polarization both in x- and y-direction near the pump wavelength, which adds considerable noise in FWM photon-pair collection.

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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]:

$$\begin{aligned} \tilde{R}_{mnuv}^{(3)}(\Omega) & =\frac{\gamma}{3}(1-f_R)(\delta_{mn}\delta_{uv}+\delta_{mu}\delta_{nv}+\delta_{mv}\delta_{nu})+\gamma f_R\tilde{R}_a(\Omega)\delta_{mn}\delta_{uv}\\ & +\frac{\gamma}{2}f_R\tilde{R}_b(\Omega)(\delta_{mu}\delta_{nv}+\delta_{mv}\delta_{nu}), \end{aligned}$$
where $\gamma$ is considered to be frequency-independent to simplify the calculation [24]. $\Omega$ represents the frequency detuning between two designated frequencies. $f_R\approx 0.18$ is the fractional contribution of the photon or molecular vibration to the Kerr coefficient. $\tilde {R}_a(\Omega )$ and $\tilde {R}_b(\Omega )$ is the Fourier transform for the isotropic and anisotropic Raman response function in the time domain, respectively. The corresponding Raman gain spectrum is defined as $g_{a(b)}(\Omega )=2\gamma f_R\text {Im}[\tilde {R}_{a(b)}(\Omega )]$. The combination $g_a(\Omega )+g_b(\Omega )$ and $g_b(\Omega )/2$ represent the Raman gain with the polarized direction parallel and orthogonal to the pump, respectively, which have been experimentally measured in the fused silica fiber [25]. $\hat {M}_{mn}$ is the noise operator describing the system-environment coupling of the photonic system and phonon environment, which is proportional to the convolution of Raman response function and Langevin noise operator in the time domain. The commutation relation is described by $\left [\hat {M}_{mn}(z,\Omega ),{\hat {M}}^{\dagger }_{uv}(z',\Omega ')\right ]=2\pi \delta (z-z')\delta (\Omega -\Omega ')\left [g_a(\Omega )\delta _{mn}\delta _{uv}+g_b(\Omega )/2(\delta _{mu}\delta _{nv}+\delta _{mv}\delta _{nu})\right ]$ [26].

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

$$\begin{aligned} & \frac{\partial \hat{A}_{m}(z,\omega_s)}{\partial z}=i\beta_m(\omega_s)\hat{A}_{m}(z,\omega_s)+i\gamma P_{\text{peak}}\epsilon_m(\Omega_{sp})\hat{A}_{m}(z,\omega_s)\\ & + i\gamma P_{\text{peak}}\iiint d\omega_id\omega_{p1}d\omega_{p2}\eta_m(\Omega_{sp_1})G(\omega_{p1})G(\omega_{p2})e^{i[\Phi_x(\omega_{p1},z)+\Phi_x(\omega_{p2},z)]}\delta(\Delta \omega)\hat{A}_{m}(z,\omega_i)\\ & +i\sqrt{P_{\text{peak}}}\int d\omega_p\hat{M}_{mx}(z,\Omega_{sp})G(\omega_p)e^{i\Phi_x(\omega_p,z)}. \end{aligned}$$
Here, $\beta _m(\omega _k)=\tilde {R}_{mm}^{(1)}(\omega _k)$ ($k =p,s,i$) is the propagation constant of pump ($p$), signal ($s$) and idler ($i$). $\delta (\Delta \omega )$ is a Dirac-delta function showing the energy conservation with $\Delta \omega =\omega _s+\omega _i-\omega _{p1}-\omega _{p2}$. It comes from the FWM component of the second term on the right side of the Eq. (1), where we inversely use the selectivity of the delta function to transform the double integral in Eq. (1) to triple integral in Eq. (3) with a delta-function introduced. The parameter $\epsilon _m(\Omega )$ and $\eta _m(\Omega )$ are related to the Raman-induced phase shift and FWM efficiency decrement separately, which are defined as:
$$\begin{aligned} & \epsilon_x(\Omega_{sp})=\frac{1}{\gamma}\left[\tilde{R}_{xxxx}^{(3)}(\Omega_{sp})+ \tilde{R}_{xxxx}^{(3)}(0)\right]=2(1-f_R)+f_R\left[1+\tilde{R}_a(\Omega_{sp})+\tilde{R}_b(\Omega_{sp})\right]\\ & \epsilon_y(\Omega_{sp})=\frac{1}{\gamma}\left[\tilde{R}_{yxxy}^{(3)}(\Omega_{sp})+ \tilde{R}_{yyxx}^{(3)}(0)\right]=\frac{2(1-f_R)}{3}+f_R\left[\tilde{R}_a(0)+\frac{\tilde{R_b(\Omega_{sp})}}{2}\right]\\ & \eta_x(\Omega_{sp})=\frac{1}{\gamma}\tilde{R}_{xxxx}^{(3)}(\Omega_{sp})=1-f_R+f_R\left[\tilde{R}_a(\Omega_{sp})+\tilde{R}_b(\Omega_{sp})\right]\\ & \eta_y(\Omega_{sp})=\frac{1}{\gamma}\tilde{R}_{yxyx}^{(3)}(\Omega_{sp})=\frac{1-f_R}{3}+\frac{f_R\tilde{R}_b(\Omega_{sp})}{2}. \end{aligned}$$
In Eq. (3), the right side of this equation consists of the four terms. The first term represents the linear evolution of the launched fields, the second and third terms describe the third-order nonlinear processes including the cross-phase modulation and the FWM process, and the last term governs the spontaneous Raman scattering process.

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:

$$\begin{aligned} \frac{\partial \hat{B}_{m}(z,\omega_s)}{\partial z} & =\frac{i\gamma P_{\text{peak}}}{2\sqrt{\pi}\sigma} \int d\omega_i \eta_m(\Omega_{sp0})e^{-\frac{{(\omega_s+\omega_i-2\omega_{p0})}^2}{4\sigma^2}}\hat{B}_{m}^{{\dagger}}(z,\omega_i)e^{{-}i\Delta \beta_{m} z}\\ & +\frac{i\sqrt{P_{\text{peak}}}}{\sqrt{2\pi}\sigma} \int d\omega_p \hat{M}_{mx}(z,\Omega_{sp})e^{-\frac{{(\omega_p-\omega_{p0})}^2}{2\sigma^2}}e^{{-}i\Delta \beta'_{m}z}. \end{aligned}$$
Here, $\Delta \beta _{m} z=\Phi '_m(\omega _s,z)+\Phi '_m(\omega _i,z)-2\Phi _x(\omega _p,z)$ is the phase mismatch between pump and photon pairs while $\Delta \beta '_{m} z=\Phi '_m(\omega _s,z)-\Phi _x(\omega _p,z)$ is the phase difference between pump and signal photons. Regarding the two terms at the right side of Eq. (5) as $\hat {B}_{0m}(z,\omega _s)$ and $\Delta \hat {B}_{m}(z,\omega _s)$, we find that the two terms are independent of each other if we ignore the interaction between the weak FWM and Raman-scattered fields. Therefore, using the existing quantized Raman scattering model [26] and the Taylor expansion of operator $\hat {B}_{0m}(L,\omega _i)$ [27], we can obtain the final expression of the signal optical field at distance L by integrating over the spatial coordinate z:
$$\begin{aligned} \hat{A}_{m}(L,\omega_s) & =\int_0^L \left[\hat{A}_{0m}(z,\omega_s)+\Delta \hat{A}_{m}(z,\omega_s)\right] dz\\ & =\left[\hat{B}_{m}(0,\omega_s)+\frac{i\gamma \eta_mP_{\text{peak}}L}{2\sqrt{\pi}\sigma}\int d\omega_i e^{-\frac{{(\omega_s+\omega_i-2\omega_{p0})}^2}{4\sigma^2}}\text{sinc}\left(\frac{\Delta \beta_m L}{2}\right)\hat{B}_{m}^{{\dagger}}(0,\omega_i)e^{-\frac{i\Delta \beta_m L}{2}}\right.\\ & \left.+\frac{i\sqrt{P_{\text{peak}}}}{\sqrt{2\pi}\sigma}\int_0^Ldz\int d\omega_p \hat{M}_{mx}(z,\Omega_{sp})e^{{-}i\Delta \beta'_{m}z} e^{-\frac{{(\omega_p-\omega_{p0})}^2}{2\sigma^2}}\right]e^{i\Phi'_m(\omega_s,L)}. \end{aligned}$$
Here, $\hat {A}_{0m}(z,\omega _s)=\hat {B}_{0m}(z,\omega _s)e^{i\Phi '_m(\omega _{s},z)}$ represents the photons generated by the FWM process while $\Delta \hat {A}_{m}(z,\omega _s)=\Delta \hat {B}_{m}(z,\omega _s)e^{i\Phi '_m(\omega _{s},z)}$ represents the photons generated by Raman scattering.

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:

$$\Delta \beta_y= \frac{n(\omega_s)\omega_s}{c}+\frac{n(\omega_i)\omega_i}{c}-2\frac{(n(\omega_p)+\Delta n)\omega_p}{c}-\frac{2}{3}\gamma P_{\text{peak}}.$$
Here $c$ is the speed of light, $n$ is the effective refractive index for the fast axis and $\Delta n$ is the fiber birefringence. As shown in Fig. 2(a), compared with non-birefringent fibers (grey line), the phase-matching curves of birefringent fibers have distinct features in the normal dispersion regime, which has been shown in the experiment [29]. First, these curves have an extra group of available solutions that are shown as the thick solid lines, which are continuous over a broad range and have smaller frequency detunings. Second, unlike non-birefringent fibers, each birefringent fiber has its maximum phase-matched wavelength, which means phase-matching can no longer be achieved if the pump wavelength exceeds such a value in the normal dispersion regime. Here, we give a further investigation on the birefringence dependence of the two features.

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]:

$$\Omega_{sp} \approx \sqrt{\frac{2\omega_{p0}}{\beta_2(\omega_{p0})c}}\sqrt{\Delta n},$$
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).

 figure: Fig. 2.

Fig. 2. (a) Phase-matching curves for birefringence fibers. All the fibers simulated here have the same basic structures. The core diameter is set to be 4.5$\mu$m while the ZDW is calculated to be 1.53$\mu$m. A picosecond laser is applied with a peak power of 70W. Using the Sellemier equations from [30], we compare the phase-matching curves for fibers with different values of birefringence. The arrows show the maximum phase-matched wavelength for fibers with a birefringence of $4.6\times 10^{-4}$. (b) Birefringence dependence of the phase-matching properties. The black line shows the influence of birefringence on frequency detuning with pump wavelength at 940nm while the red lines show the relation between birefringence and maximum phase-matched wavelength.

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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:

$$\left.\frac{\partial n(\omega)}{\partial {\omega}}\right|_{\omega=\omega_{s0}}\omega_{s0}+n(\omega_{s0})=\left.\frac{\partial n(\omega)}{\partial {\omega}}\right|_{\omega=\omega_{i0}}\omega_{i0}+n(\omega_{i0}).$$
Combining this equation with energy and momentum conservation, the unique pump wavelength can be determined for a designated fiber birefringence. In Fig. 2(b), the dashed line shows that the maximum phase-matched wavelength nearly has an inverse linear correlation with the square root of the birefringence.

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

$$S(\Delta \omega_s, \Delta \omega_i)= \exp{\left[-\frac{{(\Delta \omega_s+\Delta \omega_i)}^2}{4\sigma^2}\right]}\text{sinc}\left(\frac{\Delta \beta_m L}{2}\right).$$
The above equation can be further simplified by taking the first-order Taylor expansion for each propagation constant at their respective central wavelength to obtain a simplified phase mismatch $\Delta \beta _m \approx \tau _s\Delta \omega _s+\tau _i\Delta \omega _i$ [31]. Here we have $\tau _j=\beta _1(\omega _{j0})-\beta _1(\omega _{p0})-\Delta n/c$. Also the phase-mismatch function can be approximated by a Gaussian function $\text {sinc}(x)\approx e^{-\gamma _0x^2}$, $\gamma _0\approx 0.193$ [32]. Then the bandwidth of the one in a paired photon can be calculated by integrating the joint spectrum over the other photon’s frequency:
$$Bw_{s(i)}= \frac{4\sqrt{\ln2}}{|\tau_s-\tau_i|}\sqrt{\frac{1}{\gamma_0L^2}+\sigma^2\tau_{i(s)}^2}.$$
Equation (11) shows that the photon bandwidth depends on pump bandwidth $Bw_p$, fiber length $L$ and the dispersion properties of birefringent fibers. Especially, the photon-pair bandwidth becomes infinite when the group velocities of the signal photons equal to that of the idler ($\tau _s=\tau _i$), which is the same situation when the pump reaches its maximum phase-matched wavelength.

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

$$R= \frac {\sqrt{(1+\gamma_0\sigma^2L^2\tau_s^2)(1+\gamma_0\sigma^2L^2\tau_i^2)}}{\sqrt{\gamma_0}|\tau_s-\tau_i|\sigma L}.$$
In the above equation, the degree of frequency correlation is proportional to the values of $R$ with the minimum value $R=1$ representing a perfectly decorrelated biphoton spectrum. It is noted that for bipartite double-Gaussian wavefunctions, which is exactly the approximation we make when discussing the two-photon joint spectrum, R has the same value compared with the general used Schmidt mode K [13,35].

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

$$\tau_{s(i)}\approx (-)\beta_2(\omega_{p0})\Omega-\frac{\Delta n}{c}.$$
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 as
$$Bw_{s(i)}\approx 4\sqrt{\ln2}\sqrt{\frac{F(\omega_{p0})}{\Delta nL^2}+\sigma^2}\quad;\qquad \qquad R\approx\frac{1}{2}\big( \sqrt{\frac{F(\omega_{p0})}{\Delta n\sigma^2L^2}}+\sqrt{\frac{\Delta n\sigma^2L^2}{F(\omega_{p0})}}\big).$$
Here $F(\omega _{p0})=\frac {c}{2\gamma _0\omega _{p0}\beta _2(\omega _{p0})}$. As an example, we apply Eq. (14) to a commercial PMF pumped with a picosecond laser (details in Table 1). The influence of birefringence on generated photon-pair bandwidth and frequency correlation are shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. The influence of birefringence on photon bandwidth and frequency correlation. The approximated results are shown by the dotted lines while the accurate results are described by the solid lines.

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Tables Icon

Table 1. Simulation parameters

As 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:

$$N_{ms}=\sigma_t\left\langle{\hat{A}}^{{\dagger}}_{0m}(z,\tau)\hat{A}_{0m}(z,\tau)\right\rangle=\frac{\gamma^2{|\eta_m|}^2{P_{\text{peak}}}^2L\sigma_t}{\sqrt{2\pi\gamma_0}|\tau_s-\tau_i|}.$$
Here, $\sigma _t$ is the pulse duration of the pump. $\hat {A}_{0m}(z,\tau )=\frac {1}{2\pi }\int d\omega _s \hat {A}_{0m}(z,\omega _s)e^{-i\omega _s \tau }$ is the Fourier transform of the FWM part in Eq. (6). It is noted that the influence of spontaneous Raman scattering on third-order nonlinear response function is already included by the frequency-dependence parameter $\eta _m$. Therefore, the decrement of the photon-pair generation rate varies with the phase-matched wavelengths.

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].

 figure: Fig. 4.

Fig. 4. Raman influence in FWM photon-pair generation rate. In the figure, blue and red lines show the influence of Raman when the polarization of photon pairs are parallel or orthogonal to the pump, respectively while the dashed line is normalized photon-pair generation rate when $f_R=0$. Particularly, the phase mismatch $\text {Re}(\Delta \beta _m)$ is set to be zero at all frequency detunings in this simulation. Other parameters could be found in Table 1.

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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:

$$\begin{aligned} N_{mR}(\omega) & =\sigma_t\left\langle{\Delta \hat{A}}^{{\dagger}}_{m}(z,\tau)\Delta \hat{A}_{m}(z,\tau)\right\rangle\\ & =\frac{1}{2\pi}\int_{-\infty}^{+\infty} \int_{\omega-f_{bw}/2}^{\omega+f_{bw}/2} \frac{P_{\text{peak}}}{\sqrt{\pi}\sigma}e^{-\frac{{(\omega_p-\omega_{p0})}^2}{\sigma^2}}\frac{\left|g_{mR}(\Omega_{sp})\right|}{A_{\text{eff}}}N_0(\Omega_{sp})L\sigma_td\omega_pd\omega_s. \end{aligned}$$
Here $\Delta \hat {A}_{m}(z,\tau )=\frac {1}{2\pi }\int _{\omega -f_{bw}/2}^{\omega +f_{bw}/2}\Delta \hat {A}_{m}(z,\omega _s)e^{-i\omega _s \tau }d\omega _s$ is the Fourier transform of the Raman part in Eq. (6). The average Raman noise photons per unit length per pump power at the frequency in the above equation is $\langle \hat {M}^{\dagger }_{mx}(z,\Omega _{sp})\hat {M}_{mx}(z',\Omega ^{'}_{sp})\rangle =2\pi \delta (z-z')\delta (\Omega _{sp}-\Omega ^{'}_{sp})|g_{mR}(\Omega _{sp})|N_0(\Omega _{sp})/A_{\text {eff}}$. $g_{mR}$ is Raman gain coefficient in units of meter per Watt ($m/W$). $N_0(\Omega )$ stands for the average phonon number in the medium, which equals to $\left [e^{\frac {\hbar |\Omega |}{k_BT}}-1\right ]^{-1}$ in the anti-Stokes region ($\Omega >0$) while $\left [e^{\frac {\hbar |\Omega |}{k_BT}}-1\right ]^{-1}+1$ in the Stokes region ($\Omega <0$).

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.

 figure: Fig. 5.

Fig. 5. (a) Spectral density for FWM and Raman photons per pump pulse. A logarithmic coordinate is applied for the y-axis. Three lines in the center show the Raman intensity under different temperatures. The Raman intensity at signal wavelength is expected to be $10^{-8}$ at 300K. (b) Numbers of FWM and Raman photons with changing pump average powers. The filter bandwidth is set to be 1nm. (c) The numbers of FWM and Raman photons with changing filter bandwidths. Pump average power is set to be 20mW. Except for the changing parameters, the others could be found in Table 1.

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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.

 figure: Fig. 6.

Fig. 6. Experimental setup. In this figure, parts (a) and (b) stand for two measurement setups that are used for analyzing the spectral property and photon generation rate separately. The orange fiber shows the input for signal photons while the green for idler photons.

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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).

 figure: Fig. 7.

Fig. 7. (a) Phase-matched wavelength comparison: small circles stand for the experimental data while solid lines are part of the simulation curve in Fig. 2(a). (b) Photon-pair generation rate comparison: solid line comes from Eq. (15) while red spots are calculated from experimental counting rate by $N_{\textrm {Pair}} = N_{S}N_{I}/N_{C}$.

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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 $N_{\textrm {Pair}} = N_{S}N_{I}/N_{C}$.

 figure: Fig. 8.

Fig. 8. (a) Photon-pair counting rates with different average pump power. (b) Heralding efficiencies of signal and idler. Solid lines are the fitted models of experimental data and the shaded areas indicate the $95\%$ confidence intervals.

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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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References

  • View by:

  1. A. Aspect, J. Dalibard, and G. Roger, “Experimental test of bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49(25), 1804–1807 (1982).
    [Crossref]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
    [Crossref]
  3. A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
    [Crossref]
  4. F. Deng, G. L. Long, and X. Liu, “Two-step quantum direct communication protocol using the einstein-podolsky-rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
    [Crossref]
  5. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
    [Crossref]
  6. S. Tanzilli, H. D. Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. D. Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
    [Crossref]
  7. A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength,” New J. Phys. 12(10), 103005 (2010).
    [Crossref]
  8. N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
    [Crossref]
  9. H. Yan, S. Zhang, J. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106(3), 033601 (2011).
    [Crossref]
  10. L. Wang, C. K. Hong, and S. R. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum Semiclassical Opt. 3(5), 346–352 (2001).
    [Crossref]
  11. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
    [Crossref]
  12. B. J. Smith, P. Mahou, O. Cohen, J. S. Lundeen, and I. A. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17(26), 23589–23602 (2009).
    [Crossref]
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2021 (1)

Y. S. Lee, M. Xie, R. Tannous, and T. Jennewein, “Sagnac-type entangled photon source using only conventional polarization optics,” Quantum Sci. Technol. 6(2), 025004 (2021).
[Crossref]

2020 (1)

M. Cordier, P. Delaye, F. Gérôme, F. Benabid, and I. Zaquine, “Raman-free fibered photon-pair source,” Sci. Rep. 10(1), 1650 (2020).
[Crossref]

2019 (1)

J. G. Koefoed and K. Rottwitt, “Complete evolution equation for the joint amplitude in photon-pair generation through spontaneous four-wave mixing,” Phys. Rev. A 100(6), 063813 (2019).
[Crossref]

2018 (1)

2017 (2)

J. G. Koefoed, J. B. Christensen, and K. Rottwitt, “Effects of noninstantaneous nonlinear processes on photon-pair generation by spontaneous four-wave mixing,” Phys. Rev. A 95(4), 043842 (2017).
[Crossref]

M. A. Finger, N. Y. Joly, P. S. J. Russell, and M. V. Chekhova, “Characterization and shaping of the time-frequency schmidt mode spectrum of bright twin beams generated in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. A 95(5), 053814 (2017).
[Crossref]

2016 (1)

H. Liu, W. Xiao, W. Cai, E. Liu, B. Feng, Z. Wang, T. Liang, S. Wang, and J. Liu, “Photonic quasi-crystal fiber with high birefringence,” Opt. Eng. 55(3), 036101 (2016).
[Crossref]

2015 (2)

R. Quan, M. Wang, F. Hou, Z. Tai, T. Liu, S. Zhang, and R. Dong, “Characterization of frequency entanglement under extended phase-matching conditions,” Appl. Phys. B 118(3), 431–437 (2015).
[Crossref]

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “Effects of self-and cross-phase modulation on photon purity for four-wave-mixing photon pair sources,” Phys. Rev. A 92(5), 053849 (2015).
[Crossref]

2014 (1)

A. Dot, E. Meyer-Scott, R. Ahmad, M. Rochette, and T. Jennewein, “Converting one photon into two via four-wave mixing in optical fibers,” Phys. Rev. A 90(4), 043808 (2014).
[Crossref]

2013 (1)

2011 (2)

H. Yan, S. Zhang, J. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106(3), 033601 (2011).
[Crossref]

C. Söller, O. Cohen, B. J. Smith, I. A. Walmsley, and C. Silberhorn, “High-performance single-photon generation with commercial-grade optical fiber,” Phys. Rev. A 83(3), 031806 (2011).
[Crossref]

2010 (1)

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength,” New J. Phys. 12(10), 103005 (2010).
[Crossref]

2009 (1)

2008 (1)

Y. M. Mikhailova, P. A. Volkov, and M. V.Fedorov, “Biphoton wave packets in parametric down-conversion: Spectral and temporal structure and degree of entanglement,” Phys. Rev. A 78(6), 062327 (2008).
[Crossref]

2007 (3)

R. Amezcua-Correa, N. G. R. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, “Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation,” Opt. Express 15(26), 17577–17586 (2007).
[Crossref]

M. Zghal and R. Cherif, “Impact of small geometrical imperfections on chromatic dispersion and birefringence in photonic crystal fibers,” Opt. Eng. 46(12), 128002 (2007).
[Crossref]

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of raman scattering and pump polarization,” Phys. Rev. A 75(2), 023803 (2007).
[Crossref]

2006 (5)

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. Phys. 8(5), 67 (2006).
[Crossref]

R. J. Kruhlak, G. K. Wong, J. S. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. 31(10), 1379–1381 (2006).
[Crossref]

A. U’Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. Grice, I. Walmsley, and M. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15(1), 146–161 (2006).

M. V. Fedorov, M. A. Efremov, P. A. Volkov, and J. H. Eberly, “Short-pulse or strong-field breakup processes: a route to study entangled wave packets,” J. Phys. B: At., Mol. Opt. Phys. 39(13), S467–S483 (2006).
[Crossref]

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

2005 (1)

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005).
[Crossref]

2004 (1)

M. V. Fedorov, M. A. Efremov, A. E. Kazakov, K. W. Chan, C. K. Law, and J. H. Eberly, “Packet narrowing and quantum entanglement in photoionization and photodissociation,” Phys. Rev. A 69(5), 052117 (2004).
[Crossref]

2003 (1)

F. Deng, G. L. Long, and X. Liu, “Two-step quantum direct communication protocol using the einstein-podolsky-rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

2002 (1)

2001 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
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S. Tanzilli, H. D. Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. D. Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
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L. Wang, C. K. Hong, and S. R. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum Semiclassical Opt. 3(5), 346–352 (2001).
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1998 (1)

C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by ir spectroscopy,” J. Non-Cryst. Solids 223(1-2), 158–163 (1998).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
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1994 (2)

1991 (1)

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
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1989 (1)

1986 (1)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4(8), 1071–1089 (1986).
[Crossref]

1982 (2)

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49(25), 1804–1807 (1982).
[Crossref]

G. E. Walrafen and P. N. Krishnan, “Model analysis of the raman spectrum from fused silica optical fibers,” Appl. Opt. 21(3), 359–360 (1982).
[Crossref]

1979 (1)

L. Cohen, C. Lin, and W. French, “Tailoring zero chromatic dispersion into the 1.5 − 1.6μm low-loss spectral region of single-mode fibres,” Electron. Lett. 15(12), 334–335 (1979).
[Crossref]

1977 (1)

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[Crossref]

1976 (1)

R. H. Stolen and G. E. Walrafen, “Water and its relation to broken bond defects in fused silica,” J. Chem. Phys. 64(6), 2623–2631 (1976).
[Crossref]

1975 (1)

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11(2), 964–967 (1975).
[Crossref]

Aggarwal, I.

C. Xiong, E. Magi, F. Luan, S. Dekker, J. Sanghera, L. Shaw, I. Aggarwal, and B. Eggleton, Raman response in chalcogenide as2s3 fiber, in 2009 14th OptoElectronics and Communications Conference, (2009), pp. 1–2.

Agrawal, G. P.

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of raman scattering and pump polarization,” Phys. Rev. A 75(2), 023803 (2007).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (4th edition) (Academic, 2007).

Ahmad, R.

A. Dot, E. Meyer-Scott, R. Ahmad, M. Rochette, and T. Jennewein, “Converting one photon into two via four-wave mixing in optical fibers,” Phys. Rev. A 90(4), 043808 (2014).
[Crossref]

Akopian, N.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Alibart, O.

A. Martin, A. Issautier, H. Herrmann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength,” New J. Phys. 12(10), 103005 (2010).
[Crossref]

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. Phys. 8(5), 67 (2006).
[Crossref]

Amezcua-Correa, R.

Aspect, A.

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49(25), 1804–1807 (1982).
[Crossref]

Avron, J.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Baldi, P.

S. Tanzilli, H. D. Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. D. Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[Crossref]

Banaszek, K.

A. U’Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. Grice, I. Walmsley, and M. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15(1), 146–161 (2006).

Bell, B.

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “Effects of self-and cross-phase modulation on photon purity for four-wave-mixing photon pair sources,” Phys. Rev. A 92(5), 053849 (2015).
[Crossref]

Benabid, F.

M. Cordier, P. Delaye, F. Gérôme, F. Benabid, and I. Zaquine, “Raman-free fibered photon-pair source,” Sci. Rep. 10(1), 1650 (2020).
[Crossref]

Berlatzky, Y.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Bourgoin, J.-P.

Broderick, N. G. R.

Cai, W.

H. Liu, W. Xiao, W. Cai, E. Liu, B. Feng, Z. Wang, T. Liang, S. Wang, and J. Liu, “Photonic quasi-crystal fiber with high birefringence,” Opt. Eng. 55(3), 036101 (2016).
[Crossref]

Cantrell, C. D.

Carlson, D. R.

Chan, K. W.

M. V. Fedorov, M. A. Efremov, A. E. Kazakov, K. W. Chan, C. K. Law, and J. H. Eberly, “Packet narrowing and quantum entanglement in photoionization and photodissociation,” Phys. Rev. A 69(5), 052117 (2004).
[Crossref]

Chekhova, M. V.

M. A. Finger, N. Y. Joly, P. S. J. Russell, and M. V. Chekhova, “Characterization and shaping of the time-frequency schmidt mode spectrum of bright twin beams generated in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. A 95(5), 053814 (2017).
[Crossref]

Chen, J.

H. Yan, S. Zhang, J. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106(3), 033601 (2011).
[Crossref]

Chen, J. S.

Cherif, R.

M. Zghal and R. Cherif, “Impact of small geometrical imperfections on chromatic dispersion and birefringence in photonic crystal fibers,” Opt. Eng. 46(12), 128002 (2007).
[Crossref]

Cherlow, J.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11(2), 964–967 (1975).
[Crossref]

Christensen, J. B.

J. G. Koefoed, J. B. Christensen, and K. Rottwitt, “Effects of noninstantaneous nonlinear processes on photon-pair generation by spontaneous four-wave mixing,” Phys. Rev. A 95(4), 043842 (2017).
[Crossref]

Cohen, L.

L. Cohen, C. Lin, and W. French, “Tailoring zero chromatic dispersion into the 1.5 − 1.6μm low-loss spectral region of single-mode fibres,” Electron. Lett. 15(12), 334–335 (1979).
[Crossref]

Cohen, O.

C. Söller, O. Cohen, B. J. Smith, I. A. Walmsley, and C. Silberhorn, “High-performance single-photon generation with commercial-grade optical fiber,” Phys. Rev. A 83(3), 031806 (2011).
[Crossref]

B. J. Smith, P. Mahou, O. Cohen, J. S. Lundeen, and I. A. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17(26), 23589–23602 (2009).
[Crossref]

Cordier, M.

M. Cordier, P. Delaye, F. Gérôme, F. Benabid, and I. Zaquine, “Raman-free fibered photon-pair source,” Sci. Rep. 10(1), 1650 (2020).
[Crossref]

Dalibard, J.

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49(25), 1804–1807 (1982).
[Crossref]

Dekker, S.

C. Xiong, E. Magi, F. Luan, S. Dekker, J. Sanghera, L. Shaw, I. Aggarwal, and B. Eggleton, Raman response in chalcogenide as2s3 fiber, in 2009 14th OptoElectronics and Communications Conference, (2009), pp. 1–2.

Delaye, P.

M. Cordier, P. Delaye, F. Gérôme, F. Benabid, and I. Zaquine, “Raman-free fibered photon-pair source,” Sci. Rep. 10(1), 1650 (2020).
[Crossref]

Deng, F.

F. Deng, G. L. Long, and X. Liu, “Two-step quantum direct communication protocol using the einstein-podolsky-rosen pair block,” Phys. Rev. A 68(4), 042317 (2003).
[Crossref]

Diddams, S. A.

Dong, R.

R. Quan, M. Wang, F. Hou, Z. Tai, T. Liu, S. Zhang, and R. Dong, “Characterization of frequency entanglement under extended phase-matching conditions,” Appl. Phys. B 118(3), 431–437 (2015).
[Crossref]

Dot, A.

A. Dot, E. Meyer-Scott, R. Ahmad, M. Rochette, and T. Jennewein, “Converting one photon into two via four-wave mixing in optical fibers,” Phys. Rev. A 90(4), 043808 (2014).
[Crossref]

Dougherty, D. J.

Du, S.

H. Yan, S. Zhang, J. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106(3), 033601 (2011).
[Crossref]

Eberly, J. H.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, and J. H. Eberly, “Short-pulse or strong-field breakup processes: a route to study entangled wave packets,” J. Phys. B: At., Mol. Opt. Phys. 39(13), S467–S483 (2006).
[Crossref]

M. V. Fedorov, M. A. Efremov, A. E. Kazakov, K. W. Chan, C. K. Law, and J. H. Eberly, “Packet narrowing and quantum entanglement in photoionization and photodissociation,” Phys. Rev. A 69(5), 052117 (2004).
[Crossref]

Efremov, M. A.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, and J. H. Eberly, “Short-pulse or strong-field breakup processes: a route to study entangled wave packets,” J. Phys. B: At., Mol. Opt. Phys. 39(13), S467–S483 (2006).
[Crossref]

M. V. Fedorov, M. A. Efremov, A. E. Kazakov, K. W. Chan, C. K. Law, and J. H. Eberly, “Packet narrowing and quantum entanglement in photoionization and photodissociation,” Phys. Rev. A 69(5), 052117 (2004).
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C. Xiong, E. Magi, F. Luan, S. Dekker, J. Sanghera, L. Shaw, I. Aggarwal, and B. Eggleton, Raman response in chalcogenide as2s3 fiber, in 2009 14th OptoElectronics and Communications Conference, (2009), pp. 1–2.

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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Figures (8)

Fig. 1.
Fig. 1. Correlated photon-pair generation in birefringent fiber via a type-I degenerate FWM process. Two x-polarized pump photons (slow axis) generate photon pairs with orthogonal polarization in the y-direction (fast axis). Together with correlated photon pairs generated from the FWM process, the interaction between pump photons and molecular-vibration phonons generates Stokes and anti-Stokes Raman photons with polarization both in x- and y-direction near the pump wavelength, which adds considerable noise in FWM photon-pair collection.
Fig. 2.
Fig. 2. (a) Phase-matching curves for birefringence fibers. All the fibers simulated here have the same basic structures. The core diameter is set to be 4.5$\mu$m while the ZDW is calculated to be 1.53$\mu$m. A picosecond laser is applied with a peak power of 70W. Using the Sellemier equations from [30], we compare the phase-matching curves for fibers with different values of birefringence. The arrows show the maximum phase-matched wavelength for fibers with a birefringence of $4.6\times 10^{-4}$. (b) Birefringence dependence of the phase-matching properties. The black line shows the influence of birefringence on frequency detuning with pump wavelength at 940nm while the red lines show the relation between birefringence and maximum phase-matched wavelength.
Fig. 3.
Fig. 3. The influence of birefringence on photon bandwidth and frequency correlation. The approximated results are shown by the dotted lines while the accurate results are described by the solid lines.
Fig. 4.
Fig. 4. Raman influence in FWM photon-pair generation rate. In the figure, blue and red lines show the influence of Raman when the polarization of photon pairs are parallel or orthogonal to the pump, respectively while the dashed line is normalized photon-pair generation rate when $f_R=0$. Particularly, the phase mismatch $\text {Re}(\Delta \beta _m)$ is set to be zero at all frequency detunings in this simulation. Other parameters could be found in Table 1.
Fig. 5.
Fig. 5. (a) Spectral density for FWM and Raman photons per pump pulse. A logarithmic coordinate is applied for the y-axis. Three lines in the center show the Raman intensity under different temperatures. The Raman intensity at signal wavelength is expected to be $10^{-8}$ at 300K. (b) Numbers of FWM and Raman photons with changing pump average powers. The filter bandwidth is set to be 1nm. (c) The numbers of FWM and Raman photons with changing filter bandwidths. Pump average power is set to be 20mW. Except for the changing parameters, the others could be found in Table 1.
Fig. 6.
Fig. 6. Experimental setup. In this figure, parts (a) and (b) stand for two measurement setups that are used for analyzing the spectral property and photon generation rate separately. The orange fiber shows the input for signal photons while the green for idler photons.
Fig. 7.
Fig. 7. (a) Phase-matched wavelength comparison: small circles stand for the experimental data while solid lines are part of the simulation curve in Fig. 2(a). (b) Photon-pair generation rate comparison: solid line comes from Eq. (15) while red spots are calculated from experimental counting rate by $N_{\textrm {Pair}} = N_{S}N_{I}/N_{C}$.
Fig. 8.
Fig. 8. (a) Photon-pair counting rates with different average pump power. (b) Heralding efficiencies of signal and idler. Solid lines are the fitted models of experimental data and the shaded areas indicate the $95\%$ confidence intervals.

Tables (1)

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Table 1. Simulation parameters

Equations (16)

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A ^ m ( z , ω k ) z = i n R ~ m n ( 1 ) ( ω k ) A ^ n ( z , ω k ) + i ω 0 ( 2 π ) 2 n u v d ω 1 d ω 2 R ~ m n u v ( 3 ) ( ω 2 ω 1 ) A ^ n ( z , ω 1 ) × A ^ u ( z , ω 2 ) A ^ v ( z , ω k + ω 1 ω 2 ) + i ω 0 2 π n d ω 1 M ^ m n ( z , ω k ω 1 ) A ^ n ( z , ω 1 ) .
R ~ m n u v ( 3 ) ( Ω ) = γ 3 ( 1 f R ) ( δ m n δ u v + δ m u δ n v + δ m v δ n u ) + γ f R R ~ a ( Ω ) δ m n δ u v + γ 2 f R R ~ b ( Ω ) ( δ m u δ n v + δ m v δ n u ) ,
A ^ m ( z , ω s ) z = i β m ( ω s ) A ^ m ( z , ω s ) + i γ P peak ϵ m ( Ω s p ) A ^ m ( z , ω s ) + i γ P peak d ω i d ω p 1 d ω p 2 η m ( Ω s p 1 ) G ( ω p 1 ) G ( ω p 2 ) e i [ Φ x ( ω p 1 , z ) + Φ x ( ω p 2 , z ) ] δ ( Δ ω ) A ^ m ( z , ω i ) + i P peak d ω p M ^ m x ( z , Ω s p ) G ( ω p ) e i Φ x ( ω p , z ) .
ϵ x ( Ω s p ) = 1 γ [ R ~ x x x x ( 3 ) ( Ω s p ) + R ~ x x x x ( 3 ) ( 0 ) ] = 2 ( 1 f R ) + f R [ 1 + R ~ a ( Ω s p ) + R ~ b ( Ω s p ) ] ϵ y ( Ω s p ) = 1 γ [ R ~ y x x y ( 3 ) ( Ω s p ) + R ~ y y x x ( 3 ) ( 0 ) ] = 2 ( 1 f R ) 3 + f R [ R ~ a ( 0 ) + R b ( Ω s p ) ~ 2 ] η x ( Ω s p ) = 1 γ R ~ x x x x ( 3 ) ( Ω s p ) = 1 f R + f R [ R ~ a ( Ω s p ) + R ~ b ( Ω s p ) ] η y ( Ω s p ) = 1 γ R ~ y x y x ( 3 ) ( Ω s p ) = 1 f R 3 + f R R ~ b ( Ω s p ) 2 .
B ^ m ( z , ω s ) z = i γ P peak 2 π σ d ω i η m ( Ω s p 0 ) e ( ω s + ω i 2 ω p 0 ) 2 4 σ 2 B ^ m ( z , ω i ) e i Δ β m z + i P peak 2 π σ d ω p M ^ m x ( z , Ω s p ) e ( ω p ω p 0 ) 2 2 σ 2 e i Δ β m z .
A ^ m ( L , ω s ) = 0 L [ A ^ 0 m ( z , ω s ) + Δ A ^ m ( z , ω s ) ] d z = [ B ^ m ( 0 , ω s ) + i γ η m P peak L 2 π σ d ω i e ( ω s + ω i 2 ω p 0 ) 2 4 σ 2 sinc ( Δ β m L 2 ) B ^ m ( 0 , ω i ) e i Δ β m L 2 + i P peak 2 π σ 0 L d z d ω p M ^ m x ( z , Ω s p ) e i Δ β m z e ( ω p ω p 0 ) 2 2 σ 2 ] e i Φ m ( ω s , L ) .
Δ β y = n ( ω s ) ω s c + n ( ω i ) ω i c 2 ( n ( ω p ) + Δ n ) ω p c 2 3 γ P peak .
Ω s p 2 ω p 0 β 2 ( ω p 0 ) c Δ n ,
n ( ω ) ω | ω = ω s 0 ω s 0 + n ( ω s 0 ) = n ( ω ) ω | ω = ω i 0 ω i 0 + n ( ω i 0 ) .
S ( Δ ω s , Δ ω i ) = exp [ ( Δ ω s + Δ ω i ) 2 4 σ 2 ] sinc ( Δ β m L 2 ) .
B w s ( i ) = 4 ln 2 | τ s τ i | 1 γ 0 L 2 + σ 2 τ i ( s ) 2 .
R = ( 1 + γ 0 σ 2 L 2 τ s 2 ) ( 1 + γ 0 σ 2 L 2 τ i 2 ) γ 0 | τ s τ i | σ L .
τ s ( i ) ( ) β 2 ( ω p 0 ) Ω Δ n c .
B w s ( i ) 4 ln 2 F ( ω p 0 ) Δ n L 2 + σ 2 ; R 1 2 ( F ( ω p 0 ) Δ n σ 2 L 2 + Δ n σ 2 L 2 F ( ω p 0 ) ) .
N m s = σ t A ^ 0 m ( z , τ ) A ^ 0 m ( z , τ ) = γ 2 | η m | 2 P peak 2 L σ t 2 π γ 0 | τ s τ i | .
N m R ( ω ) = σ t Δ A ^ m ( z , τ ) Δ A ^ m ( z , τ ) = 1 2 π + ω f b w / 2 ω + f b w / 2 P peak π σ e ( ω p ω p 0 ) 2 σ 2 | g m R ( Ω s p ) | A eff N 0 ( Ω s p ) L σ t d ω p d ω s .

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