1. Introduction

SBS is a nonlinear effect caused by interaction between optical and acoustic waves (for an in-depth description see, e.g., [1]). For many fiber-optic applications it has a negative impact. For example, SBS has been shown to restrict performance of soliton-based long-haul transmission systems [2] and analog CATV systems [3]. In addition, SBS is an undesirable effect in the optical power delivery, high-power fiber lasers, and unrepeatered transmission where high input optical powers are required. Once the input power reaches some critical value, the amount of backscattered power increases quickly with the input power. The onset of SBS is determined by that critical input power which is called the SBS threshold (SBST) [4]. An increase in the SBST power which is highly desirable for many applications can be accomplished in several ways. For example, one can broaden the spectrum of the input signal using phase modulation to reduce its overlap with the narrow Brillouin gain spectrum [5] or impose a variation of fiber parameters such as stress [6], temperature [7], the core radius [8], or the Brillouin shift frequency [9, 10] along the fiber length. A more practical solution, however, is to control the Brillouin gain by the fiber design because fibers with different index profiles have different Brillouin gain spectra and therefore different SBS thresholds [11, 12, 13, 14].

In this paper we describe an approach to estimate the SBST power of a single-mode optical fiber from its index profile. To study the acousto-optic interaction we use the approach of Refs. [1, 15]. We present a simple reasoning that allows us to deduce a single parameter, the acousto-optic effective area, that quantifies the SBST depending on the radial distribution of the core dopant concentration. We confirm our theoretical results by measurement of the SBST in several GeO₂-doped silica fibers. The paper is organized as follows. In Section 2 we derive the expression for the acousto-optic effective area which unlike the commonly used optical effective area determines the strength of the SBS process in optical fibers. We then calculate profiles of acoustic modes that are needed to evaluate the acousto-optic effective area (Section 3) and describe the calculation of SBST when input optical wave interacts with several acoustic modes (Section 4).

2. Key design parameter for fibers with enhanced SBS threshold

Similarly to bulk media [1], the equation for the acousto-optic interaction in an optical fiber can be written as [15]

where ρ is the material density fluctuation around its mean value ρ ₀, Γ=η ₁₁/ρ ₀ is the damping factor, $v_l^2$ (r)=Y(r)/ρ(r) is the squared longitudinal sound velocity that depends on the transverse radial coordinate r due to silica doping with GeO₂, Y(r) is the Young’s modulus, γ=n ⁴ ε ₀ p ₁₂ is the electrostriction constant, n is the glass refraction index, η ₁₁ and p ₁₂ are the respective components of the viscosity and electrostriction tensors, and ε ₀ is the vacuum permittivity. Unlike Ref. [15], our approach takes into account the radial variation of mechanical properties of glass and therefore addresses acoustic guiding by the core region. The electric field E in the right-hand side of (1) is represented as a superposition of forward and backward propagating electro-magnetic waves

where f(r) is the dimensionless fundamental optical mode profile, A_j (z,t), j=1, 2, are the slowly varying envelopes of the optical field, u _j are unit polarization vectors of the forward- and backward-propagating waves, ω_j and β _j are, respectively, frequencies and propagation constants of optical waves, and “c.c.” stands for complex-conjugate.

A rigorous treatment of the acousto-optic interaction requires the solution of the equation for the material displacement (see, e.g., Appendix I in Ref. [15]). But since the acoustic wave is predominantly longitudinal [14, 16], we neglect the transversal component of the acoustic wave for the SBST estimation and adopt the scalar approximation [1, 15] that takes the form of eqn. (1) where

Ω=ω ₁-ω ₂ is the acoustic frequency and q=β ₁+β ₂≈2ω ₁ n _eff/c≈4πn/λ with n _eff, λ, and c being the effective refractive index of the optical mode, the input signal wavelength, and the speed of light, respectively.

We look for the solution of (1) in the form

where M is the number of acoustic modes ξ_m (r,θ) which are solutions of the unperturbed (with zero right-hand side and Γ=0) equation (1) and thus satisfy the eigenvalue equation

where ${\nabla }_{\perp }^2$=∂ ²/∂ r ²+(1/r)∂/∂ r+(1/r ²)∂²/∂ θ ² is the transverse Laplacian operator in cylindrical coordinates. In what follows, we consider acoustic modes without axial variation (∂/∂ θ=0) since only those modes interact efficiently with the axially symmetric optical mode f(r). Upon substitution of (3), (4), and (5) in (1) we find after multiplying both sides of the resulting equation with ξ _m (r) and integrating over the transverse plane

where we have denoted integrated quantities by angled brackets 〈ζ(r)〉=2π${\int }_0^{\infty }$ ζ(r)rdr and used the fact that for relevant modal solutions of (5) 〈ξ_m (r)ξ_l (r)〉/〈${\xi }_m^2$ (r)〉≈10^-8 (m≠l) for all studied fiber index profiles. Since each acoustic mode interacts independently with the optical field, from (4) and (6) the full material density variation is

Following the perturbative approach for the derivation of the nonlinear pulse propagation in an optical fiber [17, 18] one can obtain the propagation equation for the optical envelopes A _1,2. Details of the derivation are given in the Appendix. Finally, using the normalization P_j =|A_j |² ε ₀ cn〈f ²〉/2 so that P_j has the dimension of the optical power in Watts we obtain the ordinary differential equation for the back-reflected (Stokes) power in the stationary regime

In (8), α is the fiber loss coefficient and

is the Brillouin gain coefficient with w=Γq ²/(2π) being the FWHM of the Brillouin gain spectrum. The latter has a Lorentzian shape

with the peak shifted from the signal frequency ν ₁ by ν_B =Ω _m /(2π). A typical value of ν_B for most germania-doped fibers is ~11 GHz. It varies only slightly (by ~0.5 GHz) for different acoustic modes and different fibers. Hence, without loss of accuracy we take equal Brillouin shifts ν_B for all acoustic modes. Additional scaling factors can be introduced in (9) to account for polarization effects [19] and the finite spectral line width of the input signal [12]. We do not use these factors here though because, as will be discussed in Section 4, we will calculate the SBST power relative to a measured value for some reference fiber. Finally,

is the acousto-optic effective area (in µm²) associated with mth acoustic mode. Unlike the optical effective area defined as [17, 18]

and conventionally used to describe SBS in optical fibers, the quantity given by (11) actually determines the total Brillouin gain. As we will see in the following section, the acousto-optic effective area approximately equals A _eff for fiber profiles which are close to step-index. This explains why approximating A ^ao with A _eff gives a good agreement with numerous experimental data when step-index, standard single-mode fiber is used. However, if the fiber profile differs from the step-index, counter-intuitive results (higher SBS threshold for fibers with smaller effective area) can be obtained [11].

Although it was mentioned before that the Brillouin gain coefficient relates to the overlap integral between acoustic and squared optical modal profiles [14, 15], no such quantity as $A_m^{\text{ao}}$ was used to predict the SBS threshold in optical fibers. Another advantage of the definition (11) is that it does not require normalization of the mode profiles. Indeed, the dimensionless functions f(r) and ξ_m (r) appear to the same power in the numerator and denominator of (11). As a final comment on the Brillouin gain coefficient we mention that the same expression for the peak gain (9) was obtained when the guiding nature of acoustic waves was not accounted for, i.e., using the plane wave approximation for acoustic waves [1, 17]. Hence, the peak value of the SBS gain is the same in bulk and waveguide geometries. However, an important difference in the total Brillouin gain coefficient in the equation for the optical power evolution (8) is due to the modal overlap factor determined by the acousto-optic effective area (11).

In what follows we show that the acousto-optic effective area is a good measure for the SBST power of an optical fiber and that only several modes with the smallest $A_m^{\text{ao}}$ (i.e., the largest Brillouin gain) efficiently contribute to the back-reflected power. Therefore only those modes can be taken into account when calculating the SBST.

3. Computation of acoustic mode profiles

Even though the optical mode profile might change only slightly from one single-mode fiber profile to another, acoustic modes and corresponding acousto-optic effective areas can vary significantly. The SBS threshold in turn will depend on the fiber index profile. To calculate the acousto-optic effective area we numerically solve (5) for different refractive index profiles defined as ${\Delta }_{\%}\left(r\right)=100\left[n\left(r\right)-n_{{\mathrm{SiO}}_2}\right]⁄n_{{\mathrm{SiO}}_2}$. We used the following relation for the longitudinal sound velocity (in m/s)

derived from measured data [20, 21].

Fig. 1 shows the index profile together with calculated optical and acoustic mode profiles for two single-mode optical fibers. We only show the acoustic modes with the smallest acoustooptic effective areas whose values are given in Table 1 where we summarize results for five single-mode germania-doped optical fibers. All studied fibers have approximately the same attenuation coefficient α=0.2 dB/km.

The last two columns of Table 1 give calculated and measured values of the SBST power for 20 km long fibers. As can be seen from Fig. 1, the first acoustic mode of fiber 2 is more localized that results in a weaker overlap with the optical mode and hence a larger value of $A_1^{\text{ao}}$ than in fiber 1. It leads to a smaller Brillouin gain coefficient and consequently higher SBST power of fiber 2 despite its smaller optical effective area compared to fiber 1 (see Table 1).

The SBST for fiber 1 in Table 1 is representative of commonly cited “standard single-mode fiber” (e.g., Corning^® SMF-28e^® Fiber and OFS/Furukawa Allwave^® Fiber) and serves as a baseline for SBST comparison consistent with common industry practice. Fiber 2 is Corning^® LEAF^® Fiber compliant with the International Telecommunication Union (ITU) G.655 standard. SBST for fiber 5 (Corning^® NexCor^™ fiber [22]) is indicative of capability realizable with application of concepts in Section 2 to a fiber with parameters otherwise consistent with “standard single-mode fiber” (ITU G.652 compliant) [23, 24]. Below we explain how the calculated values of the SBST were obtained.

 

Fig. 1. Normalized index profile, fundamental optical mode f(r), and calculated from (5), (13) acoustic modes ξ_m (r) with the three smallest $A_m^{\text{ao}}$ for fiber 1 (a) and fiber 2 (b).

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4. Calculation of the SBS threshold power

To calculate the SBST from the found $A_m^{\text{ao}}$ we have used the following approach. The SBST is reached when the total Stokes power P _S at the fiber input z=0 equals some predetermined fraction η of the input signal power P ₁(0) [4, 25, 26]. From the solution of (8) obtained with the undepleted signal approximation P ₁(z)≈P ₁(0)exp(-αz) this condition can be written as [27]

where T is the fiber temperature and κ is the Boltzmann constant. From the short fiber approximation (fiber length L<50 km, see Ref. [27] Section 2C)

Table 1. Three smallest $A_m^{\text{ao}}$ in µm², optical effective area A_eff [µm²] calculated from (12) and calculated and measured SBS threshold powers P_th in dBm for various GeO₂-doped fibers

View Table

Substitution of (15) into (14) with the subsequent integration by the steepest descent method (see, e.g., Ref. [28], pp. 477–481) results in the following transcendental equation for the SBST

Here

is a constant and the second index in the acousto-optic effective area denotes the fiber number k from Table 1, while the first index m denotes, as before, the number of the acoustic mode, r_mk =$A_{11}^{\text{ao}}$/$A_{\mathit{\text{mk}}}^{\text{ao}}$ (r _mk ≠0 assumed). The solution of (16) for x gives the SBST power as

Because accurate values of gB is difficult to obtain theoretically, we rely on relative calculations of the threshold. From the measured value of P _th for a reference fiber 1 we obtain gB from (16) with x=P _thgB /(${\alpha A}_{11}^{\text{ao}}$). Then we calculate the constant B from (17) and solve (16) for x for other fibers listed in Table 1. The corresponding threshold power is then obtained from (18). We therefore assume that the relative strength of the SBS interaction due to each acoustic mode is given by the ratio r_mk , i.e., the widths of the Brillouin gain spectral lines w near the threshold are approximately equal for the lowest acoustic modes and that they do not change significantly from fiber to fiber. For calculations we have taken w=12 MHz, ν_B =11 GHz, M=3, η=0.01 [25, 26], λ=1.55 µm, and T=300 K. The exact values of these parameters are not important, since the solution of (16) depends on B very weakly [4]. We have also accounted for the connector loss by adding 0.1 dB to calculated values of the SBST.

The obtained value of $P_{\text{th}}^{\text{calc}}$ are within 0.5 dB (~12%) from the measured values (Table 1). We attribute this difference to a slight variation of the widths of Brillouin gain spectra in different fibers and the approximate nature of eqn. (1) discussed above. In most cases the impact of the second and higher-order acoustic modes on the SBS threshold is weak and the increase in the SBST of the kth fiber compared to the reference fiber can be roughly evaluated as

This approximation works especially well when $A_{1k}^{\text{ao}}$≪$A_{2k}^{\text{ao}}$ but tends to underestimate the threshold difference for index profiles with comparable effective acousto-optic areas for the first and higher-order modes. Since no acoustic core modes exist for pure silica core and F-doped cladding fibers [14], one has to account for cladding modes that have maximum overlap with the optical mode to calculate relevant acousto-optic effective areas.

As a final remark, we emphasize that apart from the strength of the acousto-optic interaction, change in the index profile also affects other important fiber parameters such as dispersion, loss, bending sensitivity, etc. Thus, a smart fiber profile design requires manipulation with acoustic properties of the fiber without compromising its the key optical properties [22].

5. Conclusions

In conclusion, we have theoretically and experimentally studied SBS in optical fibers with different index profiles. We have introduced the acousto-optic effective area which can be used for prediction of the SBS threshold power in optical fiber with various index profiles. Large acousto-optic effective area is a prerequisite for enhanced SBS threshold of the optical fiber.