Stimulated Brillouin scattering (SBS) is a nonlinear process caused by acoustic phonon scattering propagating in the backward direction. Acoustic vibration across mediums scatters the incident light of the pump wave causing an acoustic frequency shift resulting in Stokes and anti-Stokes waves. The process of transferring energy from the pump wave to the Stokes wave is known as the scattering phenomenon. The Stokes wave counter propagates in the opposite direction to the pump wave. The presence of acoustic waves propagating on the medium’s surface is known as the surface acoustic wave. SBS theory was first explained by Leon Brillouin in 1922, and since then related experimental works have been performed in the past decade. ¹ This paper describes a numerical model that analyzes the interactions of longitudinal, shear and hybrid acoustic modes; and optical modes in optical fibers. The numerical model was developed using COMSOL Multiphysics. Two case studies were used to demonstrate the model’s utility. Based on this numerical model, we report the influence of core radius, clad radius and effective refractive index on the Brillouin frequency shift and gain coefficient.

Prior studies related to SBS have focused on the acoustic frequency shift of different types of fibers. In theory, peak Brillouin gain for a standard silica fiber is approximated to be 5 × 10 ⁻¹¹ m/W. ² In addition, fibers around 2.6 × 10 ⁻¹² m/W have been presented by Nikles et al. ³ Optimized SBS acoustic frequency shift for tellurite photonic crystal fiber (PCF) was recorded at 8.43 GHz, which gives 9.48 × 10 ⁻¹¹ m/W of Brillouin gain. ⁴ Experimental results for SBS across a chalcogenide fiber was demonstrated by Song et al. at 6.08 × 10 ⁻⁹ m/W, showing a higher Brillouin gain compared to a standard silica fiber. ⁵ Woodward et al. reported an experimental study on SBS in small-core PCF, ⁶ which discussed the complexity of acoustic wave dynamics for different wavelengths of light, overlap between optical waves were minimum at 532 nm. Consequently, at 1550 nm, a higher overlap is achieved, which contributes to a higher Brillouin gain, and therefore a lower threshold power of 1160 mW. ⁶ More recently, Tchahame et al. demonstrated a multimode Brillouin spectrum across a long tapered PCF. ⁷ Beugnot et al. successfully conducted both the first numerical and experimental work on surface acoustic waves (SAW) of silica microfiber in 2014, with a Brillouin gain equivalent to 1.4 × 10 ⁻¹² m/W. ⁸ The idea is greatly appreciated as SAW has outstanding potential contribution in the optical sensor due to its sensitivity on various physical perturbations. However, the limitation is that threshold power of the tapered fiber is relatively high and so it is not feasible in optical application. Acoustic confinement in the fiber core is required to ensure high overlap with the optic wave. Cherif et al. studied the Brillouin spectrum of SBS characterization for small core tellurite PCF with variation in air-filling ratio. ⁴ In addition, previous results by Hu et al. showed a low threshold power of 52 mW on chalcogenide fibers whose acoustic waves were confined to the core. ⁹ However, for SBS across tapered silica fibers an undesired high threshold power was observed due to a reduced interaction of the surface acoustic wave with the optical wave. To counter this effect, gold and silver cladding materials have been proposed. The finding from Kim et al. shows novel shear Brillouin scattering detection using microscopic resolution. ¹⁰ This finding is fundamental as previous experimental set ups focused on longitudinal acoustic wave propagation.

Several experimental studies have been carried out by researchers to study the behavior of SBS in optical fibers. However, there are several limitations in experimental studies such as fabrication certainty, environmental influences and access to laboratory equipment. An accurate modeling tool, developed for the purpose of aiding experimental studies, would be beneficial to speed up research while preserving a good level of accuracy and confidence. Finite Element Method (FEM), a method that applies meshing technique to solve partial differential equations (PDE) with a certain boundary condition. ¹⁴ FEM received great attention when improvements in handling boundary conditions with the implication of penalty functions was discussed. ¹⁵ This gave FEM the preference in optical fiber modeling considering the boundary issue as the problem that other methods fail in. Ham et al. ¹⁶ then introduced the complete numerical solution, where FEM is used with spectral method that provides accuracy and consistency for 2-D and 3-D cases involving harmonic functions. These findings made FEM more suitable for optical fiber numerical modeling. Sriratanavaree et al. ¹⁷ used FEM in the study of optical and acoustic wave interaction in silicon slot waveguides. Subsequently, Monfared et al. ¹⁸ showed how FEM modeled for composite behavior of bond particle at fiber interface. Findings from Liu et al. ¹⁹ showed a numerical solution using FEM to enumerate the tension and bending in optical fiber accurately. A report on optical fiber modeling and simulation of effective refractive index for tapered fiber with finite element method was deliberated by Lee et al. ²⁰ Rahman et al. ^{11, 12} reported numerical modeling of SBS, considering the optical fundamental mode in the optical fiber using FEM.

We present a numerical model to optimize Brillouin frequency shift and gain based on various core diameters of the tapered region of silica microfiber structures. To further understand the interactions, the COMSOL model should be capable of modelling solutions in the given structures of the optical fiber. Previous research was mainly done to enhance performance on spatial resolution and also sensing range, but there have not been many insights for gain. The ability to increase the Brillouin gain coefficient has opened new opportunities to control their interaction, and several new industrial and commercial applications. In this research work, we demonstrate the numerical model for a microfiber design that is expected to increase Brillouin gain coefficient. The fully vectorial method, developed in COMSOL, is used in this case to determine the contributions of various optical fibre parameters towards SBS, thus aiding the design of microfibers with enhanced SBS performance. The equation below is used for the analysis of light guidance where H is the full vectorial magnetic field, * represents the complex conjugate and transpose, ω ² denotes the eigenvalue where ω is the optical angular frequency of the wave and ε and μ are the permittivity and permeability, respectively. ²¹ ω 2 = ∫ ∇ × H ∗ . ∈ − 1 ∇ × H + ρ ∇ . H ∗ ∇ . H dxdy ∫ H ∗ μH dxdy (1)

Equation (1) solves for the propagation constant of optical modes in optical waveguides, which can also guide acoustic mode. The propagation constant of the optical wave, β, is defined as β _optic = 2 πn _eff/ λ. There are two basic types of acoustic waves, namely shear and longitudinal acoustic waves. Shear waves are associated with dominant material dispersion in the transverse directions, which is perpendicular to the direction of propagation, taken here as the z-axis. On the other hand, for a longitudinal wave, expansion and contraction of the wave is associated with particle movements along the z direction which is in parallel to the wave propagation. However, acoustic wave propagating through a waveguide can be a combination of shear and longitudinal acoustic waves. Brillouin frequency shift of the Stokes wave is given as f = 2n _eff V _ac / λ, where, V _ac is the acoustic velocity. The acoustic wave satisfies Hooke’s Law ¹² which relates to the stress (tensor) and strain (force) of the waveguiding materials. Electric field associated with a high power optical signal causes molecular movements due to electrostriction process. ¹³ Such a material movement can generate acoustic waves that leads to density variation along the waveguide. The time and space dependent density variation changes the refractive index profile and produces a moving optical grating. This grating can reflect incoming light when its wavelength matches the spatial period of the gratings generated by the acoustic wave. Above a threshold power, if phase matching conditions are satisfied, it can inhibit forward guidance of the incoming light. The backward scattered reflected wave is frequency shifted, which explains the occurrence of the Stokes Wave. The relationship between optic and acoustic propagation for phase matching condition can be given as: K _acoustic = 2 β _optic where K _acoustic is the acoustic propagation constant and this will be double of the β _optic , the optic propagation constant.

For the SBS characterization in the optical fiber, both its guided optical and acoustic modes can be obtained using the FEM. The n _eff for the optical mode in a fibre for a given radius is first calculated using H-field based FEM model. Eigenvector and eigenvalue of acoustic waves are also obtained and then the acoustic mode patterns are generated. At phase matched conditions, the acoustic wave propagation constant is double the value of the optical wave propagation constant: K _acoustic = 2 β optic.

In this research work, the fully vectorial approach was used to solve the optical wave equations for n _eff using the commercial COMSOL software. The optical parameters E(x,y) and n _eff , are obtained by solving the equation below: Δ t 2 + ( 2 π λ ) 2 ( n 2 − n e f f 2 ) E = 0 (2)

Where ∆t is transverse Laplacian operator in the ( x, y) direction, while n _eff is the effective refractive index of fundamental optical mode, ⁹ directly related to Brillouin frequency shift via the Bragg condition. ¹⁰ The acoustic wave, which consists of the stress and strain components, are governed by Hooke’s Law ¹² whereby solving the equation below would yield its displacement. T i j = c i j k l S k l ;     i , j , k l = x , y , z (3)

where T denotes the stress field and S represents the force field which is equivalent to partial differentiation of displacement. c _ijkl is the tensor relation of elastic stiffness where i, j, kl are equivalent to propagation in x, y and z direction respectively. ²² For an isotropic medium with uniform wave propagation, the elastic stiffness constant is given by the longitudinal and shear velocity that is dependent on the material properties of the optical fiber core and cladding. ¹²

For the purpose of calculating the Brillouin gain, the overlap factor from the fully vectorial approach was used. The overlap between optical wave and acoustic wave is given in equation (4) below. ²¹ Γij = ∫ H i 2 x y U j x y 2 dA ∫ H i x y 4 dA ∫ U j x y 2 dA (4)

where H _i ( x, y) is the fundamental mode in optical wave and U _j ( x, y) is the displacement vector of acoustic wave. Optic-acoustic wave overlap factor is influenced by the acoustic wave’s strain field and refractive index of the optical fiber. ¹¹ Both the optical and acoustic wave vectors have to be normalized to calculate the overlap factor.

The Brillouin gain coefficient is represented by the equation below: ²³ g B v = g p = 4 π 3 n eff 8 p 2 12 c λ p 3 ρ 0 v B Δ v B (5)

Where ρ ₀ is fiber core density, p ₁₂ is elasto-optic coefficient which contributes to the periodic light scattering and is FWHM of acoustic wave in SBS. ²³