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 used is 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, ω² denotes the eigenvalue where ω is the optical angular frequency of the wave and ε and µ are the permittivity and permeability, respectively (21).

(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=2πn_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 =2β_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 =2β 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:

                          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_ij  = c_ijklS_kl ; ?,?, ?? = ?, ?, ?                                                                  (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 ?, ? and ? 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).

       (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):

(5)

Where ρ₀ is fiber core density, p₁₂ is elasto-optic coefficient which contributes to the periodic light scattering and is FWHM of acoustic wave in SBS (23).

 

COMSOL PARAMETERS

Table 1: Tapered fiber parameters

 

Table 2: Acoustic Velocity Parameters for Acoustic Model Simulation (Sriratanavaree et al., 2014) (21)

 

 

 

 

 

 

 

 

 

 

 

 

METHOD IN COMSOL

Based on the parameters in table 1, SBS characterization using the fully vectorial approach was performed on various core diameters of the tapered region of the silica microfiber and was verified against earlier results by H.J Lee (23). The n_eff were calculated for all core diameters. Refractive index for the core and clad used were 1.4502 and 1.445 respectively considering the measured values in a typical single mode fiber (21). Following that, three cases of acoustic wave, shear, longitudinal and hybrid behavior were analyzed.

 

Optical Mode Solver

To obtain the n_eff results for the 2 case studies covered in this research, the optical mode equation was solved using the COMSOL RF module. In the RF module the electromagnetic waves, frequency domain physics engine was used. The geometric structures of the fibers were generated using the 2 dimensional space dimension. From there, the parameters as denoted in table 1 were entered into the solver. The density, refractive index of both the core and clad values are declared under the materials section. Thereafter, the mode analysis frequency is set to the desired wavelength of 1550nm and the perfect electrical conductor boundary condition was used. As with all FEM solvers meshing is required. For this case, the triangular mesh and the “finer” element size was used. The effective mode index or n_eff for the fundamental mode can thereafter be obtained after computing the solver.

 

Acoustic Mode Solver

As for the acoustic model, the acoustic waves of shear and longitudinal were evaluated independently to record the findings. Thereafter, the hybrid acoustic wave across the fiber in which both shear and longitudinal acoustic waves co-exist were examined. The hybrid mode model was the model used to determine the Brillouin Shift Frequency v_B as both these waves propagate together in real life conditions. Table 2 shows the velocity assigned for core clad region in each respective case. In the case of pure shear acoustic, longitudinal velocity for the core region were made equivalent to 5736 m/s, this is to prevent interruption of longitudinal acoustic wave. Similarly, for pure longitudinal acoustic, the shear acoustic of the core was made to 3625 m/s. For the hybrid acoustic wave, the velocities of both the longitudinal and shear waves of the clad are defined to be slightly higher than the core (21). This is to prevent interruption of acoustic wave between the two regions. The core-clad ratio was taken to be 1:3 so that the clad region is long enough to prevent wave reflection from the outer side of clad back to the inner core. Equation 3 being a partial differential equation is solved using the Mathematics physics engine. The weak form PDE interface was used to solve equation 3. Like the optical model, the geometric structures were setup and defined in the model and the values for the acoustic wave for both shear and longitudinal velocities as tabulated in table 2 were used. The Drichelet boundary condition was used for acoustic model simulation and the mesh setting for the acoustic model is similar with the optical model whereby the triangular mesh was used with the “finer” element size used. Computing the solver would return multiple results of eigenvalues. The eigenvalue returning the fundamental mode plots would be the one selected as the Brillouin Shift Frequency v_B result.