Guided acoustic Brillouin scattering measurements in optical communication fibers

Fatih Yaman; Kohei Nakamura; Eduardo Mateo; Shinsuke Fujisawa; Hussam G. Batshon; Takanori Inoue; Yoshihisa Inada; Ting Wang

doi:10.1364/OE.426557

1. Introduction

Optical transmission capacity is limited by several impairments that arise from the optical fibers including loss, dispersion, nonlinearity, polarization-mode dispersion, etc. Recently it has been showed that guided acoustic Brillouin scattering (GAWBS) induces signal-to-noise SNR degradation that is not negligible [1], and also that it cannot be removed or mitigated with digital signal processing effectively [2]. It is important to be able to estimate the capacity of a link at the design stage accurately, therefore it is important to know how much SNR degradation is expected from GAWBS [3]. The scattered light, which becomes the GAWBS noise in the context of optical communication, generated in short lengths of fibers are quite small, and difficult to measure [3]. On the other hand, after long distances, it is difficult to separate it from other noise sources such as fiber nonlinearity contributing to variations in the estimations of the magnitude of the GAWBS noise [1,2]. It was shown that GAWBS depends on the fiber type used, in particular the effective area of the fiber [4,5]. Even though there are a few modern fiber types that are commonly used for optical transmission systems, there has not been reports tabulating the GAWBS in these fibers. For the fibers that were reported, the GAWBS noise was not categorized in detail, for instance the level of noise generated in the parallel and orthogonal polarizations with respect to the original signal [1,5,6]. Since the experimental results were not compared to the theoretical expectations [1], or they were compared with wrong models [5,6], it was not possible to confirm the veracity of the measurements.

In this paper we report the level of GAWBS noise generated in 4 types of commonly used fibers, SMF28-ULL, Vascade 2000, Vascade 3000, Z+130ULL in both polarizations. A novel homodyne measurement technique is presented that can measure the GAWBS noise spectrum in separate polarizations from just a single span and not limited by instrument dynamic range. We also confirm that GAWBS noise scales linearly with transmission distance by comparing single span measurement with loop transmission up to trans-Pacific distance. Furthermore, the inverse dependence of GAWBS noise on the fiber effective area is confirmed [1,4,5]. We report excellent fitting of the measurements with the theory only after making necessary corrections to the model which was originally outlined in [7]. In particular, it is found that the total GAWBS noise and its spectral shape can be calculated quite accurately from fiber parameters. The relationship between the power level of the GAWBS noise in parallel and orthogonal polarizations with respect to the optical carrier is established through theory and verified by measurements for the first time. Through measurements, the correlation between the magnitude of the radial component of the acoustic displacement vector and the broadening of the GAWBS peaks is confirmed. Impact of polarization diffusion due to random fiber birefringence on the depolarized GAWBS noise is determined and a method for reducing GAWBS noise is proposed. Dependence of GAWBS noise on the location of the core within the cladding is calculated to estimate GAWBS noise in multicore fibers. We note that even though for optical communications, being able to distinguish the polarization nature of the GAWBS noise does not make a direct impact, being able to fit the polarization properties of GAWBS to the theoretical expectation improves the confidence in the measurement accuracy.

2. Theory

GAWBS is generated by transverse acoustic modes in the fiber. These acoustic modes are ever present as they are due to the ambient temperature [7–9]. These modes do not have appreciable longitudinal components, which means they only generate forward scattering. The theoretical treatment follows that of [7,8]. The mode field distribution of the acoustic modes are calculated assuming a cylindrical glass with infinite length, and a free surface, i.e., the polymer coating is ignored. From the acoustic mode distribution, the induced strain is obtained in the form of a tensor. At this point our approach deviates from that of [7]. The strain is related to the impermeability tensor through the photoelastic tensor of fused silica. Both phase and birefringence modulation caused by the acoustic modes are calculated using coupled-mode which leads to the correct fitting of the experimental results. Note that instead of coupled-mode theory it is also possible to use other techniques such as perturbation or variational approaches which leads to the similar conclusions as ours [10]. A second notable deviation from the Shelby’s derivation [7] is in regards to the dependence of GAWBS noise on fiber length for long fibers. Here it will be argued that the linear dependence on the fiber length originates from the fact that optical mode interacts with many uncorrelated acoustic modes with a finite coherence length. Finally the acoustic mode groups that significantly contribute to the GAWBS noise in fibers with a concentric core are identified and the characteristics of the GAWBS noise generated by each mode group is analyzed.

The acoustic modes giving rise to GAWBS can be described by the displacement vector field components [7–9]

(1)$$ \begin{aligned} U_r\left(r,\phi \right) &= C_{nm}\left\{ A_1 \left\lbrack J_{n+1}\left(\rho\right) +J_{n-1}\left(\rho\right) \right\rbrack + \alpha A_2 \left\lbrack J_{n+1}\left(\alpha \rho\right) - J_{n-1}\left(\alpha \rho\right) \right\rbrack \right\} \Theta_d \left( n \phi \right) , \\ U_{\phi}\left(r,\phi \right) &= C_{nm}\left\{ A_1 \left\lbrack J_{n+1}\left(\rho\right) - J_{n-1}\left(\rho\right) \right\rbrack + \alpha A_2 \left\lbrack J_{n+1}\left(\alpha \rho\right) + J_{n-1}\left(\alpha \rho\right) \right\rbrack \right\} \Psi_d \left( n \phi \right) , \end{aligned}$$

where $\overrightarrow {U} = \left \lbrack U_r, U_{\phi },0 \right \rbrack \sin \left ( \Omega _{nm}t + \zeta \right )$ is the displacement vector field in cylindrical coordinates defined by radial and angular coordinates $r$ and $\phi$ respectively, $C_{nm}$ is the amplitude of the acoustic mode vibrating with the frequency $f_{nm} = \Omega _{nm}/2\pi$, $\rho =\Omega _{nm} r/V_s$ is the normalized radial coordinate, $V_s$ is the speed of the shear sound waves, $\alpha = V_s/V_d$ is the ratio of the speed of shear and longitudinal acoustic waves. The vibration frequencies $f_{nm}$ can be obtained from the solutions of the characteristic equation obtained by requiring traction-free boundary conditions at the glass cladding surface $r = a$. The characteristic equation can be expressed in the form

(2)$$\begin{vmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{vmatrix} =0 ,$$

where $B_{ij}$ are the components of the matrix

(3)$$B = \begin{bmatrix} \left(n^2-1- \frac{y^2}{2} \right) J_n \left( \alpha y \right) & \left(n\left(n^2-1\right)- \frac{y^2}{2} \right) J_n \left( y \right) - \left(n^2-1\right) y J_{n+1}\left( y \right) \\ \left(n-1\right) J_n \left( \alpha y \right) - \alpha y J_{n+1} \left( \alpha y \right) & \left(n\left(n^2-1\right)- \frac{y^2}{2} \right) J_n \left( y \right)+ y J_{n+1}\left( y \right) \end{bmatrix} ,$$

with

(4)$$y = \frac{\Omega a}{V_s} .$$

Equation (3) accepts multiple solutions $\Omega _{nm}$ for mode groups $n$, with $n,m \geq 0$ integers. In Eq. (1), $A_1 = n B_{11}$, $A_2 = B_{12}$. The acoustic waves are doubly degenerate in their angular dependence and without loss of generality, they can be expressed as follows

(5)$$ \begin{aligned} \Theta_d \left(n \phi \right) = \left\{ \begin{array}{c} \cos\left(n \phi \right), \quad d = 0 \\ \sin\left(n \phi \right), \quad d = 1 \\ \end{array} \right. , \qquad \Psi_d \left(n \phi \right) = \left\{ \begin{array}{c} \sin\left(n \phi \right), \quad d = 0 \\ -\cos\left(n \phi \right), \quad d = 1 \\ \end{array} \right. \end{aligned} .$$

In Eq. (5), the upper or lower solution or any combination of them can be chosen in general.

Displacement caused by the acoustic vibrations induces strain in the fiber. The modulating strain in return causes phase and birefringence modulations which results in generation of optical side bands at the same frequency. The induced strain tensor in cylindrical coordinates can be obtained using the following relations:

(6)$$ S_{rr} = \frac{\partial U_r}{\partial r}, \quad S_{\phi \phi} = \frac{1}{r} \frac{\partial U_{\phi}}{\partial \phi} + \frac{U_r}{r}, \quad S_{r \phi} = \frac{1}{2} \left( \frac{1}{r} \frac{\partial U_r}{\partial \phi}+ \frac{\partial U_{\phi}}{\partial r} - \frac{U_{\phi}}{r} \right) , $$

and the strain components in the z–axis is zero. Inserting Eq. (1) into Eq. (6) we get:

(7)$$ \begin{aligned} S_{rr} &= C_{nm}\frac{\Omega_{nm}}{ V_s} s_{rr}(r) \Theta \left( n \phi \right) \sin\left( \Omega_{nm}t + \zeta \right), \\ S_{\phi\phi} & = C_{nm}\frac{\Omega_{nm}}{ V_s} s_{\phi\phi}(r) \Theta \left( n \phi \right) \sin\left( \Omega_{nm}t + \zeta \right), \\ S_{r \phi} & = C_{nm}\frac{\Omega_{nm}}{ V_s} s_{r \phi} (r) \Psi \left( n \phi \right) \sin\left( \Omega_{nm}t + \zeta \right) , \end{aligned}$$

where

(8)$$ \begin{aligned} s_{rr} &= \frac{1}{2}\left\{ A_1 \left\lbrack J_{n-2}\left(\rho\right) -J_{n+2}\left(\rho\right) \right\rbrack + \alpha^2 A_2 \left\lbrack 2 J_{n}\left(\alpha \rho\right) - J_{n-2}\left(\alpha \rho\right) - J_{n+2}\left(\alpha \rho\right) \right\rbrack \right\} ,\\ s_{\phi\phi} &=\frac{1}{\rho}\left\{ A_1 \left\lbrack (n+1)J_{n+1}\left(\rho\right) -(n-1)J_{n-1}\left(\rho\right) \right\rbrack \right. \\ & \left. + \alpha A_2 \left\lbrack (n+1) J_{n+1}\left(\alpha \rho\right) + (n-1)J_{n-1}\left(\alpha \rho\right) \right\rbrack \right\} , \\ s_{r \phi} &= \frac{1}{2\rho}\left\{ -A_1 \left\lbrack (n+1)J_{n+1}\left(\rho\right) +(n-1)J_{n-1}\left(\rho\right) + \frac{\rho}{2} \left\lbrack 2 J_{n}\left(\rho\right) - J_{n+2}\left(\rho\right) - J_{n- 2}\left(\rho\right) \right\rbrack \right\rbrack \right. \\ & + \left. \alpha A_2 \left\lbrack (n-1) J_{n-1}\left(\alpha \rho\right) - (n+1)J_{n+1}\left(\alpha \rho\right)+ \frac{\alpha\rho}{2} \left\lbrack J_{n-2}\left(\alpha\rho\right) - J_{n+ 2}\left(\alpha\rho\right) \right\rbrack \right\rbrack \right\} . \end{aligned}$$

It should be noted that the angular the strain tensor components $S_{rr}$ and $S_{\phi \phi }$ has the same angular dependence as the radial component of the displacement vector $U_r$ and the strain tensor component $S_{r\phi }$ has the same angular dependence as $U_{\phi }$.

So far Shelby’s derivation was followed to calculate the strain from the acoustic modes [7]. The next step is to estimate how the strain is converted to optical modulation which will be presented in more detail. As stated, the variations in the impermeability tensor $\eta _{ij}$ can be obtained from the strain tensor through photoelastic tensor in the reduced notation in Cartesian coordinates as follows [11]:

(9)$$\begin{bmatrix} \Delta\eta_{xx}\left(r,\phi \right) \\ \Delta\eta_{yy}\left(r,\phi \right) \\ \Delta\eta_{zz}\left(r,\phi \right) \\ \Delta\eta_{yz}\left(r,\phi \right) \\ \Delta\eta_{xz}\left(r,\phi \right) \\ \Delta\eta_{xy}\left(r,\phi \right) \end{bmatrix} = \begin{bmatrix} p_{11} & p_{12} & p_{12} & 0 & 0 & 0 \\ p_{12} & p_{11} & p_{12} & 0 & 0 & 0 \\ p_{12} & p_{12} & p_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & p_{11} - p_{12} & 0 & 0 \\ 0 & 0 & 0 & 0 & p_{11} - p_{12} & 0 \\ 0 & 0 & 0 & 0 & 0 & p_{11} - p_{12} \end{bmatrix} \begin{bmatrix} S_{xx}\left(r,\phi \right) \\ S_{yy}\left(r,\phi \right) \\ S_{zz}\left(r,\phi \right) \\ S_{yz}\left(r,\phi \right) \\ S_{xz}\left(r,\phi \right) \\ S_{xy}\left(r,\phi \right) \end{bmatrix}.$$

The photoelastic tensor in Eq. (9) takes a particularly simple form as fused silica is well approximated as an isotropic solid. Impermeability tensor is related to the dielectric permittivity as $\pmb {\eta } = \epsilon _0 \pmb {\epsilon }^{-1}$ where $\epsilon _0$ is the free space permittivity [12]. Assuming that the variations in dielectric permittivity due to GAWBS $\Delta \pmb {\epsilon }$ is a small perturbation on the undisturbed permittivity $\epsilon$ of the fiber which is assumed to be close to isotropic, i.e., $\pmb {\epsilon } \approx \epsilon + \Delta \pmb {\epsilon }$ and $\pmb {\eta } \approx 1/\epsilon + \Delta \pmb {\eta }$

(10)$$\Delta\eta_{ij}\left(r,\phi,t \right) \approx{-}\epsilon_0 \frac{\Delta \epsilon_{ij} \left( r,\phi,t \right) }{\epsilon^2} \implies \Delta \epsilon_{ij} \approx{-}\Delta\eta_{ij}\left(r,\phi,t \right) n_0^4 \epsilon_0$$

where $n_0 = \sqrt {\epsilon /\epsilon _0}$ is the refractive index of the unperturbed fiber. Aligning the x-axis with $\phi =0$, the relevant strain tensor components can be converted from the cylindrical coordinates to the Cartesian coordinates as follows:

(11)$$ \begin{aligned} S_{xx} &= \cos\left(\phi \right)^2 S_{rr} + \sin\left(\phi \right)^2 S_{\phi\phi} - \sin\left(2\phi \right) S_{r\phi} , \\ S_{yy} &= \sin\left(\phi \right)^2 S_{rr} + \cos\left(\phi \right)^2 S_{\phi\phi} + \sin\left(2\phi \right) S_{r\phi} , \\ S_{xy} &= \sin\left(2\phi \right) \left( S_{rr} - S_{\phi\phi} \right)/2 + \cos\left(2\phi \right)S_{r\phi}, \end{aligned}$$

Combining Eqs. (11) and (9) we obtain

(12)$$ \begin{aligned} \Delta\eta_{xx}(r,\phi,t) &= p_p\left( S_{rr} +S_{\phi\phi} \right) + p_m \left\lbrack \left( S_{rr} -S_{\phi\phi} \right)\cos\left(2\phi \right) - 2S_{r\phi}\sin\left(2\phi \right) \right\rbrack ,\\ \Delta\eta_{yy}(r,\phi,t) &= p_p\left( S_{rr} +S_{\phi\phi} \right) - p_m \left\lbrack \left( S_{rr} -S_{\phi\phi} \right)\cos\left(2\phi \right) - 2S_{r\phi}\sin\left(2\phi \right) \right\rbrack ,\\ \Delta\eta_{xy}(r,\phi,t) &= p_m \left\lbrack \left( S_{rr} -S_{\phi\phi} \right)\sin\left(2\phi \right) + 2S_{r\phi}\cos\left(2\phi \right) \right\rbrack , \end{aligned}$$

where $p_p = \left ( p_{11} + p_{12} \right )/2$, and $p_m = \left ( p_{11} - p_{12} \right )/2$.

Equation (12) describes how the impermeability tensor varies across the fiber cross section due to the acoustic vibrations. In general, such spatial variations in the refractive index distribution would scatter the signal traveling in the fiber into all the modes supported by the optical fiber and radiation modes. In the framework of coupled-mode theory [13–16], when the modulations are very small, as in the case of GAWBS, power lost to these scattering would be negligible, and not of interest to us. The non-negligible contribution would be the optical signal coupling into the modes supported by the fiber in the absence of acoustic vibrations. The transverse part of the electric field in the fiber can be expanded in terms of the transverse components of the electric fields of the modes of the undisturbed fiber as follows [13]

(13)$$\vec{E_t}\left(r,\phi,z,t \right) = \sum_{\nu} h_{\nu}\left( z \right) \vec{E}_{\nu,t}\left(r,\phi,t \right).$$

The subscript $t$ denotes the transverse component. The variation of the electric field along the fiber is contained in $h_{\nu }\left ( z \right )$ which is governed by the coupled mode equation given by

(14)$$\frac{\partial h_{\mu}\left(z\right)}{\partial z} ={-}\frac{\alpha}{2}h_{\mu}+ i k_{\mu}h_{\mu} + i \sum_{\nu}\kappa_{\mu\nu}\left( t \right) h_{\nu}\left(z\right), \quad \quad i = \sqrt{-1}$$

where $\alpha$ is the fiber attenuation coefficient, $k_{\mu }$ is the propagation constant of the mode $\mu$ in the absence of GAWBS. The coupling coefficients $\kappa _{\mu \nu }$ caused by GAWBS are given by [13]

(15)$$\kappa_{\mu\nu} ={-}i\frac{\omega}{4P} \int_{0}^{2\pi}\int_{0}^{\infty} \vec{E}^*_{\mu}\left(r,\phi,t \right) \Delta\pmb{\epsilon}\left( r,\phi,t \right) \vec{E}_{\nu}\left(r,\phi,t \right) r \,\mathrm{d}r \mathrm{d}\phi$$

where $\omega$ is the optical carrier frequency, P is used as the normalization factor for the amplitudes of the coupling modes defined as

(16)$$P \delta_{\mu\nu}=\frac{1}{4} \int_{0}^{2\pi}\int_{0}^{\infty}\hat{e}_z\cdot\left\lbrack \vec{E}_{\nu,t} \times \vec{H}^*_{\mu,t} +\vec{E}^*_{\nu,t} \times \vec{H}_{\mu,t} \right\rbrack r \,\mathrm{d}r \mathrm{d}\phi$$

and $\delta _{\mu \nu }$ is the Kronecker delta function. A minor detail to note is that, in Eq. (15) the optical mode field terms are no longer only transverse components. However, for the case of GAWBS this small detail is has no consequence as $\Delta \epsilon _{rz}$, $\Delta \epsilon _{\phi z}$, and $\Delta \epsilon _{zz}$ are assumed to be zero.

Under the weakly guiding approximation which holds for long-distance optical fibers, the LP modes approximate the electrical field very well satisfying

(17)$$\vec{H}_{\mu,t} = c n_0 \epsilon_0 \hat{e}_z \times \vec{E}_{\mu,t}$$

where $c$ is the speed of light in vacuum. Combining Eqs. (10), (16), and (17) the mode coupling equation Eq. (15) simplifies to

(18)$$\kappa_{\mu\nu} ={-}i\frac{k_0 n_0^3}{2} \frac{\int_{0}^{2\pi}\int_{0}^{\infty} \vec{E}^*_{\mu}\left(r,\phi,t \right) \Delta\pmb{\eta}\left( r,\phi,t \right) \vec{E}_{\nu}\left(r,\phi,t \right) r \,\mathrm{d}r \mathrm{d}\phi } {\int_{0}^{2\pi}\int_{0}^{\infty} \vec{E}^*_{\mu,t}\left(r,\phi,t \right) \cdot \vec{E}_{\nu,t}\left(r,\phi,t \right) r \,\mathrm{d}r \mathrm{d}\phi},$$

with $k_0 = \omega /c$.

So far the Eqs. (1)–(16) are quite general in terms of the optical mode profile. For instance they can be used to estimate the scattering efficiency for single-mode, multimode or multicore fibers. From here on, the analysis will be limited to the special case of single-mode fibers. As single-mode fibers support two polarization modes, the coupled mode Eq. (14) reduces to a pair of equations that can be written in the Jones vector notation as follows [17]:

(19)$$ \begin{aligned} \frac{\partial| h(z) \rangle}{\partial z} &= -\frac{\alpha}{2}|h(z)\rangle + i \bar{k} |h(z)\rangle + i\Delta k \sigma_1|h(z)\rangle + i\bar{\kappa}(t) |h(z)\rangle + i \overrightarrow{\Delta\kappa}(t) \cdot\overrightarrow{\sigma} |h(z)\rangle, \\ | h(z) \rangle &= \begin{bmatrix} h_{x}(z) \\ h_{y}(z) \end{bmatrix}, \quad \sigma_1= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \, \sigma_2= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \, \sigma_3= \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} , \end{aligned} $$

where $\bar {k} = (k_x + k_y)/2$ is the average propagation constant, $\Delta {k} = (k_x - k_y)/2$ is the fiber birefringence in the absence of GAWBS, $\bar {\kappa } = (\kappa _{xx} + \kappa _{yy})/2$ and $\overrightarrow {\Delta \kappa } = [(\kappa _{xx} - \kappa _{yy})/2, \kappa _{xy},0]$ are the isotropic and anisotropic part of the coupling coefficients induced by the GAWBS, $\sigma _1$, $\sigma _2$, $\sigma _3$ are the 2x2 Pauli matrices, and $\overrightarrow {\sigma }=[\sigma _1, \sigma _2 ,\sigma _3]$. The “$\cdot$" stands for the dot product, yielding $\overrightarrow {\Delta \kappa }(t) \cdot \overrightarrow {\sigma } = \Delta \kappa _1 \sigma _1 + \Delta \kappa _2 \sigma _2 + \Delta \kappa _3 \sigma _3$ [17]. The isotropic part of the GAWBS induces pure phase modulation whereas the anisotropic portion induces birefringence with a birefringence vector given by $\overrightarrow {\Delta \kappa }$ which will be referred to as GAWBS birefringence vector. In the absence of fiber birefringence, in Stokes space this rotation due to GAWBS can be expressed as:

(20)$$\frac{\partial \overrightarrow{h}}{\partial z} ={-} 2\overrightarrow{\Delta\kappa} \times \overrightarrow{h} ,$$

where $\overrightarrow {h} = \langle h | \overrightarrow {\sigma } | h\rangle$ is the Stokes vector representation of the Jones vector $|h \rangle$ [17] and “$\times$" stands for cross product. Equation (20) shows that GAWBS induces maximum rate of rotation of the optical field when the Stokes vector of the optical field $\overrightarrow {h}$ is orthogonal to $\overrightarrow {\Delta \kappa }$. It also shows that there is no rotation when these two vectors are either parallel or anti-parallel in Stokes space, corresponding to parallel and orthogonal in Jones Space. This can be easily shown by noting that the Jones vector $|h\rangle$ is the eigenvector of the birefringence matrix as $\overrightarrow {\Delta \kappa }.\overrightarrow {\sigma } |h\rangle = \pm |\Delta \kappa ||h\rangle$ [17]. Therefore, per Eq. (19), the birefringence term does not induce rotation but pure phase modulation. For any case in between these extremes, the birefringence terms will induce both phase modulation and polarization rotation depending on the angle between Stokes vectors $\overrightarrow {h}$ and $\overrightarrow {\Delta \kappa }$. It should be noted that the x- and the y-axis are defined with respect to the angle $\phi$ in Eq. (1) that determines the orientation of the angular features of the acoustic waves. For the sake of simplicity, in the rest of the paper, it will be assumed that the Jones vector $| h(z) \rangle$ is normalized such that its magnitude is equal to the optical power, i.e., $P = \langle h| h \rangle = \sqrt {\overrightarrow {h}\cdot \overrightarrow {h}}$.

Given a short piece of fiber with known birefringence and GAWBS parameters, Eq. (19) is well defined and it can be integrated. So far the the variation of the acoustic modes along the fiber was ignored. In fact, $U_z$ is explicitly set to zero, and it is also implicitly assumed that the acoustic mode oscillates coherently along the entire fiber length as $\sin \left ( \Omega _{nm}t + \zeta \right )$ with a fixed phase of $\zeta$. These are only approximations that are justified by the fact that the acoustic modes that contribute to GAWBS extend much longer in the fiber compared to the fiber diameter. In reality, it is expected that the GAWBS modes separated by long distances in the fiber would be oscillating with independent and random phases. Here this random variation along the length will be modeled as the fiber consisting of short lengths that are much longer than the fiber diameter but short enough that the acoustic mode would be completely coherent within. At the same time it will be assumed that the acoustic modes would be completely independent from the modes in other sections. In other words, $\zeta$ will be treated as a random variable uniformly distributed between $(-\pi ,\pi )$ that is constant within the mode coherence length $l_c$, but completely independent between different sections satisfying

(21)$$ \langle \zeta(z_1) \zeta(z_2) \rangle = \left\{ \begin{array}{c} \frac{\pi^2}{3}, \quad |z_1-z_2| \le l_c \\ 0, \quad |z_1-z_2| > l_c \\ \end{array} \right.$$

Incorporating random variations in the acoustic mode phase and the birefringence along the fiber, Eq. (19) can be put in the following stochastic form

(22)$$ \frac{\partial| h(z) \rangle}{\partial z} = \left( -\frac{\alpha}{2} + i \bar{k} + i\overrightarrow{b}(z)\cdot\overrightarrow{\sigma} \right)|h(z)\rangle + \left( i\bar{\kappa} + i \overrightarrow{\Delta\kappa} \cdot\overrightarrow{\sigma} \right)|h(z)\rangle \sin\left( \Omega_{nm}t + \zeta(z) \right) ,$$

where the vector $\overrightarrow {b}$ is the random fiber birefringence vector that randomly rotates the polarization. Since GAWBS is a very weak effect, the optical field propagates largely unaffected by the scattering therefore a perturbative formal solution of the following form can be sought

(23)$$|h(z)\rangle = e^{-\frac{\alpha z}{2}+i\bar{k}z}U(z)\left(|h_0\rangle + |\delta h(z)\rangle \right), \quad \frac{\partial U(z)}{\partial z} = i\overrightarrow{b}\cdot\overrightarrow{\sigma}U(z) ,$$

which amounts to adopting the reference frame of optical field where $|h_0\rangle$ is the undepleted optical carrier, $|\delta h\rangle$ is the scattered field generated by GAWBS, $U(z)$ is the unitary matrix describing the rotation of the field due to fiber birefringence. Inserting the formal solution in Eq. (23) into Eq. (22) and ignoring the smaller terms, the equation for the GAWBS field can be expressed as

(24)$$ \begin{aligned} \frac{\partial |h_0\rangle}{\partial z} &\approx 0, \\ \frac{\partial |\delta h(z)\rangle}{\partial z} &\approx i\left(\bar{\kappa} + \overrightarrow{\Delta\kappa}(z)\cdot\overrightarrow{\sigma}' \right) |h_0 \rangle \sin\left( \Omega_{nm}t + \zeta(z) \right), \end{aligned}$$

where $\overrightarrow {\Delta \kappa }'(z)\cdot \overrightarrow {\sigma } = U(z) \overrightarrow {\Delta \kappa }(z)\cdot \overrightarrow {\sigma } U^\dagger (z)$. As the matrix $U(z)$ is a random unitary matrix, $\overrightarrow {\Delta \kappa }'(z)$ is a randomly rotated version of $\overrightarrow {\Delta \kappa }$ at a distance z [17]. Since the equation is expressed in the rotating reference frame of the optical carrier due to the fiber birefringence the GAWBS term is rotating with the same statistics. This allows to describe this random rotation in terms of the polarization diffusion length $l_p$ defined as the distance the where the average of Stokes vector over an ensemble decays by a factor of natural logarithm, i.e., $\langle \overrightarrow {\Delta \kappa }'(z) \rangle = \langle \overrightarrow {\Delta \kappa }'(0) \rangle \exp (-z/l_p)$. However, for the sake of simplicity, the polarization rotation will be modeled similar to the case of acoustic mode phase variation where $\overrightarrow {\Delta \kappa }'(z)$ will be assumed to be constant over a section of length $l_p$ and completely random orientation at a different section, i.e., $\langle \overrightarrow {\Delta \kappa }'(z_1) \cdot \overrightarrow {\Delta \kappa }'(z_2) \rangle = \Delta \kappa ^2$ for $|z_1 -z_2| \le l_p$ and $\langle \overrightarrow {\Delta \kappa }'(z_1)\cdot \overrightarrow {\Delta \kappa }'(z_2) \rangle = 0$ for $|z_1 -z_2| > l_p$. For typical transmission fibers that are nominally non-birefringent, the polarization coherence length is on the order of 10–100 m [18].

After a distance $L$ that is much larger than both correlation lengths $l_c$, and $l_p$ the GAWBS field can be approximated as a discrete sum in the following form

(25)$$|\delta h(L)\rangle = \Delta z\sum_{q=1}^{N} i\left( \bar{\kappa} + \overrightarrow{\Delta\kappa}'_q\cdot \overrightarrow{\sigma} \right)|h_0\rangle \sin\left( \Omega_{nm}t + \zeta_q \right) ,$$

where $\Delta z = L/N$. The GAWBS noise power defined as $P_G(L) = \langle \langle \delta h(L) |\delta h(L) \rangle \rangle$ where the outer brackets stand for combined ensemble and time average can be expressed as

(26)$$\begin{aligned} P_G(L) &= \Delta z^2\sum_{q=1}^{N}\sum_{p=1}^{N} \left[ \bar{\kappa}^2 P_0 + \langle \overrightarrow{\Delta\kappa}'_q \cdot \overrightarrow{\Delta\kappa}'_p \rangle P_0 + \bar{\kappa} \langle \left(\overrightarrow{\Delta\kappa}'_q + \overrightarrow{\Delta\kappa}'_p \right) \cdot \overrightarrow{P_0} \rangle \right. \\ &\left. + i \langle \left( \overrightarrow{\Delta\kappa}'_q \times \overrightarrow{\Delta\kappa}'_p \right)\cdot \overrightarrow{P_0} \rangle \right] \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle , \end{aligned}$$

where $P_0 = \langle \langle h_0 | h_0 \rangle \rangle$, $\overrightarrow {P_0} = \langle h_0 |\overrightarrow {\sigma } |h_0 \rangle$ and it is assumed that the variation in the fiber birefringence and the acoustic mode phases are independent processes. In deriving Eq. (26) the identity $(\overrightarrow {a}\cdot \overrightarrow {\sigma })(\overrightarrow {b}\cdot \overrightarrow {\sigma }) = \overrightarrow {a}\cdot \overrightarrow {b}I + (\overrightarrow {a}\times \overrightarrow {b})\cdot \overrightarrow {\sigma }$ is used, where $I$ is 2x2 identity matrix [17]. The third and fourth terms in the summation in Eq. (26) would vanish as the average of the dot product between GAWBS birefringence vector and the Stokes vector of the optical field would average to zero. The first term is independent of rotation due to fiber birefringence and its accumulation would only depend on the correlation length of the acoustic mode phase variation, but the second term would depend on both the phase variation and also on the polarization rotations. In the two extreme cases where either $l_c \ll l_p$, or $l_p \ll l_c$ Eq. (26) simplifies to

(27)$$ P_G(L) = \left\{ \begin{array}{c} \frac{P_0L}{2} \left(\bar{\kappa}^2 + \Delta \kappa^2 \right)l_c , \quad l_c \ll l_p \\ \frac{P_0L}{2} \left(\bar{\kappa}^2 + \Delta \kappa^2 \frac{l_p}{l_c} \right)l_c , \quad l_c \gg l_p \\ \end{array} \right. .$$

Equation (27) shows that the GAWBS noise power grows linearly with fiber length $L$ as expected. This linear dependence was forced by Shelby by arguing dimensional consistency requirement [7]. Here we arrive at the same conclusion as a consequence of the short correlation length of the acoustic modes responsible for the GAWBS scattering compared to the lengths of interest, i.e. $l_c \ll L$. However, we now have the $l_c$ term entering in the expression that affects the magnitude of the GAWBS noise power. Within $l_c$, the noise power grows quadratically with distance, but the noise power from each section add linearly. Therefore, larger the $l_c$ larger the noise power for the same total length. However, it will be shown later that under the assumptions that GAWBS is generated by ambient temperature and that $l_c$ is equivalent to the average acoustic mode length, the dependence on $l_c$ will be eliminated. What is new is that when $l_p \ll l_c$, effective correlation length for the growth of the GAWBS noise due to the birefringence term reduces to $l_p$ instead of $l_c$. It will be shown based on measurements that for typical transmission fibers $l_c \ll l_p$ holds.

When measuring GAWBS, the polarization direction of the optical carrier makes the most natural reference frame. In this reference frame, the GAWBS noise can be separated into two parts; the part that has the same polarization as the carrier, and the part that is orthogonal. These two will be referred to as the parallel and orthogonal GAWBS noise given by $P_{G\parallel } = \langle |\langle \hat {e}_{\parallel }|\delta h \rangle |^2 \rangle$ and $P_{G\perp }= \langle |\langle \hat {e}_{\perp }|\delta h \rangle |^2 \rangle$, respectively, where $|\hat {e}_{\parallel } \rangle$ and $|\hat {e}_{\perp } \rangle$ are the unit Jones vectors that are parallel and perpendicular to the carrier, i.e., $|\hat {e}_{\parallel } \rangle = |\ h_0 \rangle / \sqrt {P_0}$ and $\langle \hat {e}_{\perp }| \hat {e}_{\parallel }\rangle =0$ with $\langle \hat {e}_{\perp }| \hat {e}_{\perp }\rangle =1$. An expression for the parallel and orthogonal GAWBS noise power after a distance $L$ can be obtained similar to Eq. (27) under the same assumptions as

(28)$$ \begin{aligned} P_{G\parallel}(L) &= \Delta z^2P_0\sum_{q=1}^{N}\sum_{p=1}^{N} \left\{ \left[\bar{\kappa}^2 + \langle \left( \overrightarrow{\Delta\kappa}'_q\cdot\hat{p} \right)\left( \overrightarrow{\Delta\kappa}'_p\cdot\hat{p} \right) \rangle + \bar{\kappa} \langle\left(\overrightarrow{\Delta\kappa}'_q +\overrightarrow{\Delta\kappa}'_p \right)\cdot\hat{p} \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\} , \\ P_{G\perp}(L) &= \Delta z^2P_0\sum_{q=1}^{N}\sum_{p=1}^{N} \left\{ \left[\langle \left( \overrightarrow{\Delta\kappa}'_q\cdot \overrightarrow{\Delta\kappa}'_p \right) \rangle - \langle \left( \overrightarrow{\Delta\kappa}'_q\cdot\hat{p} \right)\left( \overrightarrow{\Delta\kappa}'_p\cdot\hat{p} \right) \rangle + i\langle\left(\overrightarrow{\Delta\kappa}'_q \times \overrightarrow{\Delta\kappa}'_p \right)\cdot\hat{p} \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\}. \end{aligned}$$

In deriving Eq. (29) the identity $|\hat {e}_{\parallel } \rangle \langle \hat {e}_{\parallel } |$ $= (I + \hat {p}\cdot \overrightarrow {\sigma } )/2$ was used, where $\hat {p} = \overrightarrow {P_0}/P_0$ is the unit Stokes vector along the Stokes vector of the carrier $\overrightarrow {P_0}$, and the fact that orthogonal Jones vectors are anti-parallel in the Stokes space, i.e., $\hat {p}_{\perp } = -\hat {p}$, and $\hat {p}_{\perp } = \langle \hat {e}_{\perp } |\overrightarrow {\sigma } | \hat {e}_{\perp } \rangle$ [17].

As expected the isotropic term only contributes to the parallel GAWBS and none to the orthogonal GAWBS, where as the anisotropic term contributes to both $P_{G\parallel }$ and $P_{G\perp }$. The new non-vanishing term that appears in Eq. (29) is $\langle \left ( \overrightarrow {\Delta \kappa }'_q\cdot \hat {p} \right )\left ( \overrightarrow {\Delta \kappa }'_p\cdot \hat {p} \right ) \rangle$, which can alternatively be expressed as $\Delta \kappa ^2\langle \cos \left ( \theta _q \right )\cos \left ( \theta _p \right )\rangle$ where $\theta$ is the angle between the GAWBS birefringence vector $\overrightarrow {\Delta \kappa }$ and $\hat {p}$. Using the case of $l_c \ll l_p$ as an example, and setting $\Delta z = l_c$, Eq. (29) can be expressed as

(29)$$ \begin{aligned} P_{G\parallel}(L) &= l_c^2 P_0\sum_{q=1}^{N}\sum_{p=1}^{N} \left\{ \left[\bar{\kappa}^2 +\Delta\kappa^2\langle\cos\left( \theta_q \right)\cos\left( \theta_p \right)\rangle + \bar{\kappa} \Delta\kappa \langle\cos\left( \theta_q \right) + \cos\left( \theta_p \right) \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\} , \\ P_{G\perp}(L) &= l_c^2 P_0\sum_{q=1}^{N}\sum_{p=1}^{N} \left\{ \left[\langle \left( \overrightarrow{\Delta\kappa}'_q\cdot \overrightarrow{\Delta\kappa}'_p \right) \rangle - \Delta\kappa^2\langle\cos\left( \theta_q \right)\cos\left( \theta_p \right)\rangle + i\langle\left(\overrightarrow{\Delta\kappa}'_q \times \overrightarrow{\Delta\kappa}'_p \right)\cdot\hat{p} \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\} . \end{aligned}$$

The assumption $l_c \ll l_p$ implies $\theta _p \approx \theta _q$, and $\overrightarrow {\Delta \kappa }'_q \approx \overrightarrow {\Delta \kappa }'_p$ therefore $\langle \cos \left ( \theta _q \right )\cos \left ( \theta _p \right )\rangle \approx \langle \cos \left ( \theta _p \right )^2\rangle$, $\langle \left ( \overrightarrow {\Delta \kappa }'_q\cdot \overrightarrow {\Delta \kappa }'_p \right ) \rangle \approx \Delta \kappa ^2$ and $\overrightarrow {\Delta \kappa }'_q \times \overrightarrow {\Delta \kappa }'_p \approx 0$. Furthermore $\langle \cos \left ( \theta _p \right )^2\rangle = 1/3$ and $\langle \cos \left ( \theta _p \right )\rangle = 0$ would hold as the GAWBS birefringence vector $\overrightarrow {\Delta \kappa }$ is expected to be distributed uniformly on the Poincare sphere due to random polarization rotation in the reference frame of the carrier Stokes vector. It should be noted that, to satisfy the condition of uniform distribution on the Poincare sphere, $\cos \left ( \theta _p \right )$ is uniformly distributed between -1 and 1, rather than $\theta _p$ uniformly distributed between 0 and $2\pi$, resulting in $\langle \cos \left ( \theta _p \right )^2\rangle = 1/3$ rather than $\langle \cos \left ( \theta _p \right )^2\rangle = 1/2$. Considering $\langle \sin \left ( \Omega _{nm}t + \zeta _q \right ) \sin \left ( \Omega _{nm}t + \zeta _p \right ) \rangle = \delta _{pq}/2$ Eq. (29) can be simplified to

(30)$$ \begin{aligned} P_{G\parallel}(L) &= \frac{P_0 L}{2}\left(\bar{\kappa}^2 + \frac{1}{3}\Delta \kappa^2 \right)l_c , \\ P_{G\perp}(L) &= \frac{P_0 L}{2}\left(\frac{2}{3}\Delta \kappa^2 \right)l_c, \end{aligned}$$

where $L = Nl_c$. For the other extreme case where $l_p \ll l_c$, Eq. (29) can be put in the form

(31)$$ \begin{aligned} P_{G\parallel}(L) &= l_c^2 P_0\sum_{u=1}^{N}\sum_{v=1}^{N}\bar{\kappa}^2 \langle \sin\left( \Omega_{nm}t + \zeta_u \right) \sin\left( \Omega_{nm}t+ \zeta_v \right) \rangle \\ &+ l_p^2 P_0\sum_{q=1}^{M}\sum_{p=1}^{M} \left\{ \left[\bar{\kappa}^2 +\Delta\kappa^2\langle\cos\left( \theta_q \right)\cos\left( \theta_p \right)\rangle + \bar{\kappa} \Delta\kappa \langle\cos\left( \theta_q \right) + \cos\left( \theta_p \right) \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\} , \\ P_{G\perp}(L) &= l_p^2 P_0\sum_{q=1}^{M}\sum_{p=1}^{M} \left\{ \left[\langle \left( \overrightarrow{\Delta\kappa}'_q\cdot \overrightarrow{\Delta\kappa}'_p \right) \rangle - \Delta\kappa^2\langle\cos\left( \theta_q \right)\cos\left( \theta_p \right)\rangle + i\langle\left(\overrightarrow{\Delta\kappa}'_q \times \overrightarrow{\Delta\kappa}'_p \right)\cdot\hat{p} \rangle \right] \right.\\ &\left. \langle \sin\left( \Omega_{nm}t + \zeta_q \right) \sin\left( \Omega_{nm}t + \zeta_p \right) \rangle \right\} , \end{aligned}$$

where $L = N l_c = M l_p$. In this case $l_p \ll l_c$ implies $\zeta _p \approx \zeta _q$ and hence $\langle \sin \left ( \Omega _{nm}t + \zeta _q \right ) \sin \left ( \Omega _{nm}t + \zeta _p \right ) \rangle =1/2$. Using $\langle \overrightarrow {\Delta \kappa }'_q\cdot \overrightarrow {\Delta \kappa }'_p \rangle = \Delta \kappa ^2\delta _{pq}$, and $\langle \cos \left (\theta _q \right ) \cos \left (\theta _p \right )\rangle = \delta _{pq}/3$ Eq. (31) reduces to

(32)$$ \begin{aligned} P_{G\parallel}(L) &= \frac{P_0 L}{2}\left(\bar{\kappa}^2l_c + \frac{1}{3}\Delta \kappa^2l_p \right) ,\\ P_{G\perp}(L) &= \frac{P_0 L}{2}\left(\frac{2}{3}\Delta \kappa^2 \right)l_p . \end{aligned}$$

Keeping in mind the assumption that this is only true when either $l_c \ll l_p$ or $l_c \gg l_p$, Eqs. (30) and (32) can be combined together as

(33)$$ \begin{aligned} P_{G\parallel}(L) &= \frac{P_0 L}{2}\left(\bar{\kappa}^2 + \frac{\min[l_c,l_p]}{3l_c}\Delta \kappa^2 \right)l_c , \\ P_{G\perp}(L) &= \frac{P_0 L}{2}\left(\frac{2\min[l_c,l_p]}{3l_c}\Delta \kappa^2 \right)l_c. \end{aligned}$$

Comparing Eq. (33) with Eq. (27) shows that $P_G = P_{G\parallel } + P_{G\perp }$ as expected.

Now that the GAWBS noise is obtained in terms of the GAWBS coupling coefficients and fiber length in Eqs. (27) and (33), it can be related to the acoustic strain. The GAWBS terms in Eq. (33), namely, $\bar {\kappa }$ and $\overrightarrow {\Delta \kappa }$ can be expressed in terms of overlap integrals between the GAWBS strain components and the square of optical mode field by combining Eqs. (7), (12), and (18) as

(34)$$ \begin{aligned} \bar{\kappa}& = \kappa_0 p_p \iint_A\Theta_d\left(n\phi \right) \left( s_{rr} +s_{\phi\phi} \right) \bar{f} \left( r,\phi \right)^2 r\,\mathrm{d}r \mathrm{d}\phi , \\ \delta\kappa & = \kappa_0 p_m \iint_A \left[\Theta_d\left(n\phi \right)\cos\left(2\phi \right) \left( s_{rr} -s_{\phi\phi} \right)-2 \Psi_d\left(n\phi \right)\sin\left(2\phi \right) s_{r\phi} \right] \bar{f} \left( r, \phi \right)^2 r\,\mathrm{d}r \mathrm{d}\phi , \\ \kappa_{xy} & = \kappa_0 p_m \iint_A \left[\Theta_d\left(n\phi \right)\sin\left(2\phi \right) \left( s_{rr} -s_{\phi\phi} \right) + 2 \Psi_d\left(n\phi \right)\cos\left(2\phi \right) s_{r\phi} \right]\bar{f} \left( r, \phi \right)^2 r\,\mathrm{d}r \mathrm{d}\phi , \end{aligned}$$

where $\delta \kappa = (\kappa _{xx} - \kappa _{yy})/2$, $\kappa _0 = \frac {k_0 n_0^3}{2} \frac {C_{nm}\Omega _{nm}}{ V_s}$. It was assumed implicitly that the single mode core is circularly symmetric therefore both polarization modes have the same spatial distribution given by $f\left (r,\phi \right )$, and $\bar {f}\left (r,\phi \right )^2 \equiv f\left (r,\phi \right )^2/\iint _Af\left (r,\phi \right )^2r\,\mathrm {d}r \mathrm {d}\phi$ is the corresponding normalized distribution. The integration domain $A$ stands for the fiber cross-section. Since we assume circular symmetry for the core, it may seem that the $\phi$ dependence in $f\left (r,\phi \right )$ may be redundant. However, the coordinate frame that is adopted is centered at the center of the fiber. If the core is not concentric with the fiber, as in multi-core fibers, or due to residual core-clad concentricity error, $f\left (r,\phi \right )$ is not circularly symmetric in the chosen coordinate system. Therefore, Eq. (34) can be used to calculate GAWBS noise for cores that are not at the center of the fiber or to estimate the impact of core-clad concentricity error on GAWBS noise [19].

In the further special case of the core being concentric with the fiber, the mode field distribution becomes circularly symmetric and no longer has $\phi$ dependence. In this case the radial and angular integrals in Eq. (34) can be separated as

(35)$$ \begin{aligned} \bar{\kappa}& = \frac{\kappa_0 p_p}{2\pi} \int_{0}^{a}\left( s_{rr} +s_{\phi\phi} \right) \bar{f} \left( r\right)^2 r\,\mathrm{d}r \int_{0}^{2\pi}\Theta_d\left(n\phi \right) \mathrm{d}\phi , \\ \delta\kappa & = \frac{\kappa_0 p_m}{2\pi} \int_{0}^{a}\left( s_{rr} -s_{\phi\phi} \right)\bar{f} \left( r \right)^2 r\,\mathrm{d}r \int_{0}^{2\pi}\Theta_d\left(n\phi \right)\cos\left(2\phi \right)\phi \\ -&2\frac{\kappa_0 p_m}{2\pi}\int_{0}^{a}s_{r\phi} \bar{f} \left( r \right)^2 r\,\mathrm{d}r \int_{0}^{2\pi}\Psi_d\left(n\phi \right)\sin\left(2\phi \right) \mathrm{d}\phi , \\ \kappa_{xy} & = \frac{\kappa_0 p_m}{2\pi} \int_{0}^{a}\left( s_{rr} -s_{\phi\phi} \right)\bar{f} \left( r \right)^2 r\,\mathrm{d}r \int_{0}^{2\pi}\Theta_d\left(n\phi \right)\sin\left(2\phi \right)\phi \\ +&2\frac{\kappa_0 p_m}{2\pi}\int_{0}^{a}s_{r\phi} \bar{f} \left( r \right)^2 r\,\mathrm{d}r \int_{0}^{2\pi}\Psi_d\left(n\phi \right)\cos\left(2\phi \right) \mathrm{d}\phi , \end{aligned} $$

where $\bar {f}\left (r\right )^2 \equiv f \left (r\right )^2/\int _{0}^{\infty }f\left ( r\right )^2 r \,\mathrm {d}r$, and the angular integral is calculated and taken out. The angular integrals in Eq. (35) determine which acoustic modes can cause coupling between the two polarization components, and as such act as selection rules. The equations are written in a general form where the angular dependence can have both upper, $d=0$ and lover solutions $d=1$, as shown in Eq. (5). First thing to notice is that all angular integrals vanish unless $n=0$, or $n=2$. Moreover, when $n=0$, but for the lower solution in Eq. (5), i.e., $\Theta \left (0\right ) = 0$ and $\Psi \left (0\right ) = -1$, all the coupling coefficients still vanish. This choice of angular dependence corresponds to pure shear acoustic modes as $U_r=0$ in Eq. (1). On the other hand, for the case of the upper solution, i.e., $\Theta \left (0\right ) = 1$ and $\Psi \left (0\right ) = 0$ corresponds to pure radial dilation modes as $U_{\phi }=0$. These acoustic modes are denoted as R$_{0m}$ modes. For R$_{0m}$ modes the coupling coefficients do not vanish and carrying out the angular integrals in Eq. (35) they can be reduced to

(36)$$ \begin{aligned} d = 0: \quad &\bar{\kappa} = \kappa_0 p_p \int_{0}^{\infty}\left(s_{rr} +s_{\phi\phi} \right) \bar{f} \left( r\right)^2 r \,\mathrm{d}r, \quad &\delta\kappa = 0, \quad &\kappa_{xy} = 0 , \\ d = 1: \quad &\bar{\kappa} = 0, \quad &\delta\kappa = 0, \quad &\kappa_{xy} = 0 . \end{aligned} $$

In the case of $n=0$, optical field experiences phase modulation only since the phase shift is the same for both polarizations, therefore no birefringence i.e., $\delta \kappa =0$, and the cross coupling term $\kappa _{xy}=0$. Therefore, if a polarized field is incident, the GAWBS peaks created by $n=0$ mode group create only GAWBS peaks that are also polarized and in the same polarization as the incident optical field. As a result these GAWBS side bands created by the R$_{0m}$ modes are referred to as the polarized GAWBS peaks, or polarized GAWBS noise.

The remaining case where the coupling coefficients does not vanish is for $n=2$. These modes are mixed acoustic modes having both radial and shear (torsional) components, hence denoted as $TR_{2m}$ modes. For $n=2$, the upper and lower solutions in Eq. (5) are simply rotated versions of one another by $45^{\mathrm {o}}$. The upper solution, i.e. $d = 0$, has two isoclines at angles $\phi =0^{\mathrm {o}}, 90^{\mathrm {o}}$, whereas for the lower solution, $d=1$, the two isoclines are at angles $\phi =\pm 45^{\mathrm {o}}$. Again by carrying out the angular integrals in Eq. (35), the coupling coefficients for the case of $TR_{2m}$ modes can be expressed as

(37)$$ \begin{aligned} d = 0: \quad &\bar{\kappa} = 0, \quad &\delta\kappa = \frac{\kappa_0}{2}p_m\int_{0}^{\infty}\left(s_{rr} - s_{\phi\phi} - 2s_{r\phi}\right) \bar{f} \left( r\right)^2 r \,\mathrm{d}r, \quad &\kappa_{xy} = 0 , \\ d = 1: \quad &\bar{\kappa} = 0, \quad &\kappa_{xy} = \frac{\kappa_0}{2}p_m\int_{0}^{\infty}\left(s_{rr} - s_{\phi\phi} - 2s_{r\phi}\right) \bar{f} \left( r\right)^2 r \,\mathrm{d}r, \quad &\delta \kappa = 0 . \end{aligned} $$

Equation (37) shows that for the two degenerate cases $d=0$, and $d=1$, the contribution from $TR_{2m}$ modes are either through $\delta \kappa$ or $\kappa _{xy}$ depending on the relative orientations of the isoclines of the acoustic mode and the incident polarization of the optical field. However, since on average the total noise power is determined by $\bar {\kappa }^2$ and $\Delta \kappa ^2 = \delta \kappa ^2 + \kappa _{xy}^2$ according to Eq. (27), both degenerate cases contribute equally as expected. Since both degenerate modes are equally present their contributions should be summed to find the GAWBS noise contribution of $TR_{2m}$ modes. Therefore, for the case of $TR_{2m}$ modes $\Delta \kappa ^2$ simplifies to

(38)$$\Delta\kappa^2 = \frac{\kappa_0^2p_m^2}{2} \left[ \int_{0}^{\infty}\left(s_{rr} - s_{\phi\phi} - 2s_{r\phi}\right) \bar{f} \left( r\right)^2 r \,\mathrm{d}r\right]^2 .$$

According to Eq. (37), $TR_{2m}$ modes only contribute to the birefringence term $\Delta \kappa$ and none to the isotropic term $\bar {\kappa }$. Therefore, $TR_{2m}$ modes are solely responsible for the creation of optical noise in the orthogonal polarization with respect to the optical carrier. Since $TR_{2m}$ modes contribute to both parallel and orthogonal GAWBS, the total noise generated by these modes are referred to as the depolarized GAWBS [7] noise as opposed to the polarized noise generated by $R_{0m}$ modes. Previously, the total GAWBS noise was considered as the sum of parallel $P_{G\parallel }$ and orthogonal contributions $P_{G\perp }$ which is the most natural way to separate it from a measurement point of view. However, since for the special case of single-mode fiber with a circularly symmetric core at the center of the fiber there are only two mode groups that generates GAWBS noise, an alternative way to analyze it is to separate into parts based on their physical origin. Instead of calling them $R_{0m}$ and $TR_{2m}$ contributions, it was preferred to name them as the polarized GAWBS noise created by $R_{0m}$ modes denoted as$P_{GP}$ and depolarized GAWBS noise generated by $TR_{2m}$ modes denoted as $P_{GD}$ [7]. According to Eq. (33) these four parameters are related as

(39)$$ \begin{aligned} P_{G\parallel} &= P_{GP} + \frac{1}{3}P_{GD} \\ P_{G\perp} &= \frac{2}{3}P_{GD} \quad \quad \quad \implies \quad P_{GP} = P_{G\parallel} - \frac{1}{2}P_{G\perp} \\ & \qquad \qquad\qquad \qquad \qquad \qquad P_{GD} = \frac{3}{2}P_{G\perp} \end{aligned} .$$

Equation (39) shows that even if it may not be possible to measure the polarized and depolarized GAWBS directly, they can be obtained from the parallel and orthogonal measurements.

The remaining term left in determining the coupling coefficients in Eqs. (36) and (37) are $C_{nm}$. These terms can be determined following the approach of Shelby [7] which invokes the argument of equipartition theorem. The total energy taken by each acoustic mode should be equal to $k_{B} T$ where $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature. The energy of an acoustic mode covering the extent of the fiber cross section and a length $l_m$ is given by

(40)$$\mathcal{E}_{nm} = \int_0^{l_m}\int_0^{2\pi}\int_0^{a}\frac{1}{2}\rho_0\Omega_{nm}^2\left(U_{r\mbox{-}nm}^2+U_{\phi\mbox{-}nm}^2\right)r \,\mathrm{d}r\mathrm{d}\phi\mathrm{d}z = k_{B} T ,$$

where $\rho _0$ is the density of the glass. Inserting Eq. (1) into Eq. (40) $C_{0m}$ and $C_{2m}$ can be related to the fiber parameters as

(41)$$ \begin{aligned} C_{nm}^2 = \frac{2 k_B T}{\chi \pi l_m \rho_0 a^2 \Omega_{nm}^2 D_{nm}}, \quad \chi = \left\{ \begin{array}{c} 2, \, n=0 \\ 1, \, n \ne 0 \\ \end{array} \right. , \end{aligned}$$

with the term $D_{nm}$ is given by

(42)$$ \begin{aligned} D_{nm} &= A_1^2 \left\lbrack J_{n-1}\left(y\right)^2 + J_{n+1}\left( y \right)^2 - J_{n}\left(y \right) \left\lbrack J_{n+2}\left(y\right) + J_{n-2}\left(y\right) \right\rbrack \right\rbrack \\ &+ \alpha^2 A_2^2 \left\lbrack J_{n-1}\left(\alpha y\right)^2 + J_{n+1}\left(\alpha y\right)^2 - J_{n}\left(\alpha y\right) \left\lbrack J_{n+2}\left(\alpha y\right) + J_{n-2}\left(\alpha y\right) \right\rbrack \right\rbrack \\ & + A_1 A_2 \frac{2\alpha}{y\left(\alpha^2 -1 \right)} \left\lbrack \frac{2 n \left(\alpha^2 -1 \right)}{\alpha} J_{n}\left(y \right) J_{n}\left(\alpha y \right) + J_{n-2}\left(y \right) J_{n-1}\left(\alpha y \right) + J_{n+2}\left(y \right) J_{n+1}\left(\alpha y \right) \right. \\ & \left. -\alpha \left\lbrack J_{n-1}\left(y \right) J_{n-2}\left(\alpha y \right) + J_{n+1}\left(y \right) J_{n+2}\left(\alpha y \right) \right\rbrack \right\rbrack , \end{aligned}$$

where $y = \Omega _{nm}a/V_s$. Putting together Eqs. (33), (36), (41), and (42), polarized GAWBS power generated by R$_{0m}$ modes can be obtained in terms of fiber parameters as follows

(43)$$ \begin{aligned} P_{GP}\left(L,\Omega_{0m}\right) &= P_0 \frac{k_B T n_0^6 p_p^2 k_0^2 L}{4\pi \rho_0 a^2 V_d^2 } \left(\frac{l_c}{l_m}\right) \left\lbrack \frac{S\left(\Omega_{0m}\right)^2 }{ J_1 \left( \frac{\Omega_{0m}a}{V_d}\right) - J_0 \left( \frac{\Omega_{0m}a}{V_d} \right)J_2 \left( \frac{\Omega_{0m}a}{V_d} \right) } \right\rbrack, \\ S\left(\Omega_{0m}\right) &=\int_0^a J_0\left(\frac{\Omega_{0m}r}{V_d }\right) \bar{f} \left( r\right)^2 r \,\mathrm{d}r , \end{aligned}$$

Similarly for the GAWBS noise generated by the TR$_{2m}$ modes we get

(44)$$ \begin{aligned} P_{GD}\left(L,\Omega_{2m}\right) &= P_0 \frac{k_B T n_0^6 p_m^2 k_0^2 L}{2\pi \rho_0 a^2 V_s^2 } \left(\frac{\min[l_c,l_p]}{l_m}\right)\left\lbrack \frac{ S\left(\Omega_{2m}\right)^2 }{D_{2m} } \right\rbrack , \\ S\left(\Omega_{2m}\right) &=\int_0^a \left\lbrack A_1 J_0\left(\frac{\Omega_{2m}r}{V_s }\right) - \frac{V_s^2}{V_d^2} A_2 J_0\left(\frac{ \Omega_{2m}r}{V_d }\right) \right\rbrack \bar{f} \left( r\right)^2 r \,\mathrm{d}r , \end{aligned}$$

where we used the fact that the $s_{rr} + s_{\phi \phi }$, $s_{rr} - s_{\phi \phi } - 2s_{r\phi }$ and $D_{0m}$ simplifies to

(45)$$ \begin{aligned} s_{rr} + s_{\phi\phi} &= 2\alpha^2 A_2 J_{n}\left(\frac{\Omega_{nm}r}{V_d}\right), \\ s_{rr} - s_{\phi\phi} - 2 s_{r\phi} &= 2\left\lbrack A_1 J_{n-2}\left(\frac{\Omega_{nm}r}{V_s}\right) - \alpha^2 A_2 J_{n-2}\left(\frac{\Omega_{nm}r}{V_d}\right) \right\rbrack \\ D_{0m} &= 2\left(\alpha A_2\right)^2\left\lbrack J_1 \left( \frac{\Omega_{0m}a}{V_d} \right)- J_0 \left( \frac{\Omega_{0m}a}{V_d} \right)J_2 \left( \frac{\Omega_{0m}a}{V_d} \right) \right\rbrack . \end{aligned}$$

In deriving Eq. (44) the doubly degenerate nature of the TR$_{2m}$ modes is taken into account as described in Eq. (38).

The factor $S\left (\Omega _{nm}\right )$ in Eqs. (43) and (44) is the resulting overlap integral between the acoustic strain and the optical mode field squared in our derivation. In [7] several steps connecting the acoustic strain to the effective index modulation were skipped, and an unfortunate naming convention was used to describe the mode intensity profile as “The profile of the guided optical mode". This resulted in several papers following the derivation in [7] to conclude incorrectly that the overlap integral is between the acoustic mode profile and the optical mode-field profile, instead of the optical intensity profile [4–6,20]. Interestingly, others [10,15,21,22] that were not directly interested in GAWBS noise for long distance communication fibers, did not have this misunderstanding.

Equations (43) and (44) show explicit dependence of GAWBS noise power on the temperature which is expected as the acoustic modes causing GAWBS are driven by the ambient temperature. Therefore the measurements carried out at one temperature can be translated to an environment at a different temperature. A nontrivial example is that GAWBS noise for fibers in submarine cables at the bottom of the sea would have lower GAWBS noise compared to measured values at room temperature, e.g., 0.25 dB lower at 4 $^{\circ }$C compared to 20 $^{\circ }$C.

As mentioned earlier, the vibration frequencies of the acoustic modes can be found out by solving the characteristic equation given by Eq. (2). For the case of $n=2$, the two modes corresponding to the upper and lower solutions of the Eq. (5) are degenerate and have the same frequencies as the solution, however, this is not the case for the $n=0$ modes. The pure shear modes and the radial modes R$_{0m}$ have different solutions. For the case of $n=0$ the Eq. (2) factors into a product as follows

(46)$$|B| = \left\lbrack \frac{a^2 \Omega^2}{2V_s^2}J_{2}\left(\frac{a \Omega}{V_s} \right) \right\rbrack \left\lbrack \frac{a^2 \Omega^2}{2V_s^2}J_{0}\left(\frac{a \Omega}{V_d}\right) - \frac{a \Omega}{V_d}J_{1}\left(\frac{a \Omega}{V_d} \right) \right\rbrack = 0,$$

where the roots of the first parenthesis yield the frequencies of the pure shear acoustic modes, and the roots of the parenthesis on the right corresponds to the frequencies $\Omega _{0m}$ of the R$_{0m}$ modes supported by the fiber.

In Eqs. (43) and (44), the GAWBS noise power is expressed in terms of fiber parameters that can be easily measured except for the length scales $l_m$ and $l_c$ which determine the average mode length, and the average mode coherence length, respectively. It is reasonable to assume that these two length scales are the same, as the acoustic modes would be coherent within the mode length and vice versa. Therefore, it will be assumed without further proof that $l_m =l_c$ and their ratio in these equations will be omitted. Furthermore, it will be shown that for the fibers measured for this paper, a good agreement is observed for the condition $l_c \ll l_p$. The remaining parameters can be divided into a group that depends on the waveguiding parameters of the fiber, e.g., core diameter, cut-off wavelength which is contained in the overlap integral term $\Gamma$, and those parameters that does not play a direct role on determining the optical mode profile, such as properties of the glass, cladding diameter, absolute temperature etc.. The overlap integral shows that a large overlap between strain from a particular acoustic mode and optical mode-squared creates a larger GAWBS noise power created at the frequency of that particular mode. It is worth noting that an important piece missing in the analysis is the impact of the polymer coating which plays a crucial role in mechanical reliability of optical fibers.

Equations (43) and (44) can be used to calculate the magnitude of GAWBS peaks generated by individual acoustic modes with vibration frequencies $\Omega _{nm}$. More often than not it is the relative power of the GAWBS noise with respect to the carrier power $P_0$ that matters, therefore when appropriate the results in the rest of the paper will be presented in terms of GAWBS scattering efficiency defined as $\gamma _{G}\left (\Omega _{nm} \right ) = P_{G}\left (\Omega _{nm} \right ) /P_0$. The same notation used to describe the polarization of the GAWBS noise power will be applied to the GAWBS efficiency, i.e., $\gamma _{Gx}\left (\Omega _{nm} \right ) = P_{Gx}\left (\Omega _{nm} \right ) /P_0$ where $x = \parallel , \perp , P, D$ depending on whether the GAWBS noise is parallel or orthogonal to the carrier, polarized or depolarized, respectively. In defining the spectral features of the GAWBS noise, the single side spectrum convention is adopted here as per Eq. (24) the spectrum is symmetric, and a time-domain average is implemented in deriving Eqs. (27) and (33).

For optical communications, what matters is the sum of GAWBS noise power with respect to the carrier from all the GAWBS modes, as this sum determines the signal to noise ratio due to GAWBS noise only. Connecting the notation of [3] to this paper, the total GAWBS efficiency is defined as

(47)$$\Gamma_{GAWBS} = \sum_{n,m=0}\gamma_{G}\left(L,\Omega_{nm}\right), \quad \Gamma_{GAWBS,x} = \sum_{n,m=0}\gamma_{Gx}\left(L,\Omega_{nm}\right),$$

where, again, $x = \parallel , \perp , P, D$ is the descriptor for the polarization.

Inspecting Eqs. (43) and (44), it may appear that there is a strong dependence on the fiber cladding radius $a$. Comparing the GAWBS peak powers for $a=62.5~\mu m$ and $a=40~\mu m$ as shown in Fig. 1(a), indeed the magnitude of GAWBS peaks generated by individual modes scales inversely with cladding radius $a$ however, the frequency spacing between GAWBS modes is reduced in proportion to $a$, resulting in the total GAWBS noise to remain the same. It is easy to see that the frequency spacing between modes reduces as cladding radius increases since the mode frequencies are roughly determined by the round-trip time of sound waves between the cladding air interface. The total GAWBS noise power is plotted in Fig. 1(b) with respect to the cladding radius, for polarized only ($\Gamma _{GAWBS,P}$), depolarized ($\Gamma _{GAWBS,D}$) only and their sum ($\Gamma _{GAWBS}$). The total GAWBS noise is virtually independent of fiber radius. Another point worth mentioning is that the total depolarized GAWBS noise power generated by TR$_{2m}$ modes is lower than the polarized GAWBS power generated by the R$_{0m}$ by only $\approx 2.5$ dB and should not be neglected as suggested in [4]. The parameters used to calculate the curves in Fig. 1 and the rest of the paper are listed in Table 1 unless specified otherwise. Moreover, the optical mode profile is assumed to be the fundamental mode of a weakly guiding circularly symmetric core given by [23]

(48)$$f\left( r \right) = \left\{ \begin{array}{@{}c@{}} \frac{J_0\left( \frac{p r}{w_c} \right)}{J_0\left( p \right)}, \, r \leq w_c \\ \frac{K_0\left( \frac{q r}{w_c} \right)}{K_0\left( q \right)}, \, r > w_c \\ \end{array} \right. \quad p = w_c k_0\sqrt{n_c^2 - n_0^2}, \quad q = w_c k_0\sqrt{n_0^2 - n_{cl}^2 } ,$$

where $w_c$ is the core radius, $n_c$ and $n_{cl}$ are the core and cladding refractive indices. Even though the mode profile is described in terms of the parameters $w_c$, $n_c$ and $n_{cl}$ in Eq. (48), it can also be determined by providing the effective area $A_{eff}$ and the cutoff wavelength $\lambda _c$ under the weakly guiding assumption. The latter parameters will be used in the rest of the paper as they are more likely to be quoted in fiber specifications, where the effective area is defined as [24]

(49)$$A_{eff} = 2\pi \frac{\left\lbrack \int_{0}^{\infty} f \left( r\right)^2 r \,\mathrm{d}r \right\rbrack^2}{\int_{0}^{\infty} f \left( r\right)^4 r \,\mathrm{d}r}.$$

Fig. 1. (a) GAWBS peaks generated by R$_{0m}$ modes (filled markers), and peaks generated in the perpendicular polarization by TR$_{2m}$ modes (empty markers), for fiber cladding radius $a = 40 \mu m$ (blue) and $a = 62.5 \mu m$ (yellow) (b) Polarized (blue), depolarized (yellow) and total (green) GAWBS noise power as a function of cladding radius $a$.

Parameter	Value	Description
a	62.5 $μ$ m	Fiber cladding radius
Aeff	80 $μ$ m $^{2}$	Effective area
$λ_{c}$	1.29 $μ$ m	Cut-off wavelength
n $_{0}$	1.445	Refractive index
p $_{11}$	0.116	Photo-elastic coeff.
p $_{12}$	0.255	Photo-elastic coeff.
$ρ_{0}$	2.2 g/cm $^{3}$	Glass density
V $_{d}$	5834 m/s	Longitudinal acoustic speed
V $_{s}$	3637 m/s	Shear acoustic speed
$λ$	1.55 $μ$ m	Optical wavelength
T	293 K	Absolute temperature

Abbreviation	Fiber type	Length (km)	Effective area ( $μ$ m $^{2}$ )
SMF28_80	SMF-28 ULL	80.3	80.8
ZPL_60	Z-PLUS Fiber 130	60.2	138
Ex3_60	Vascade EX3000	60.4	154.9
Ex2_84	Vascade EX2000	84.3	114.9
Ex2_40	Vascade EX2000	40.2	112.9
Ex2_10_1	Vascade EX2000	10	112.2
Ex2_10_2	Vascade EX2000	10.1	113.5
Ex2_10_3	Vascade EX2000	10	114.3
Ex2_10_4	Vascade EX2000	10.1	111.6

Fiber	Parallel	Orthogonal	Total	Total GAWBS
	GAWBS (dB)	GAWBS (dB)	GAWBS (dB)	@1000 km (dB)
SMF28_80	-40.6	-45.7	-39.4	-28.5
Ex2_40	-45.1	-50.0	-43.9	-29.9
ZPL_60	-44.2	-49.2	-43.0	-30.8
Ex3_60	-44.7	-49.5	-43.5	-31.3

Fiber	$Δ a$ ( $μ$ m)	$R^{2}$
SMF28_80	0.08	0.97
Ex2_40	0.09	0.92
ZPL_60	0.132	0.95
Ex3_60	0.173	0.99

	Optical Mode Parameters	Acoustic Parameters
Fiber	$λ_{c}$	$\frac{V_{d}}{a}$	$\frac{V_{s}}{a}$	$α$	$F_{G}$	$F_{G ⊥}$
	( $μ$ m)	(MHz)	(MHz)		(pm/kg)	(pm/kg)
SMF28_80	1.2	92.57	57.6	0.622	9.96	1.52
Ex2_40	1.4	93.24	58.08	0.622	9.96	1.6
ZPL_60	1.9	93.75	58.59	0.625	9.9	1.54
Ex3_60	1.45	93.84	58.53	0.624	9.92	1.6

p $_{11}$	p $_{12}$	$λ$ ( $μ$ m)	Medium	Source
0.126	0.26	0.633	Bulk silica	[36]
0.113	0.252	0.633	Silica core, B $_{2}$ O $_{3}$ cladding fiber	[31]
0.116	0.255	1.53	SMF28	[32]
0.115	0.266	1.55	Silica core, F cladding fiber	This work

Abstract

1. Introduction

2. Theory

3. Measurements

4. Comparison of theory and measurements

5. Conclusion

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (26)

Tables (6)

Equations (52)

Optics Express