Semicomputational calculation of Bragg shift in stratified materials

The fiber Bragg grating (FBG) may be viewed as a one dimensional photonic band-gap crystal by virtue of the periodic spatial perturbation imposed on the fiber core dielectric material. Similar to media supporting Bloch waves, the engraved weak index modulation, presenting a periodic “potential” to an incoming guided mode photon of the fiber, yields useful spectral properties that have been the basis for sensing applications and emerging quantum squeezing and solitons. The response of an FBG sensor to arbitrary external stimuli represents a multiphysics problem without a known analytical solution despite the growing use of FBGs in classical and quantum sensing and metrology. Here, we study this problem by first presenting a solid mechanics model for the thermal and elastic states of a stratified material. The model considers an embedded optical material domain that represents the Bragg grating, here in the form of an FBG. Using the output of this model, we then compute the optical modes and their temperature- and stress-induced behavior. The developed model is applicable to media of arbitrary shape and composition, including soft matter and materials with nonlinear elasticity and geometric nonlinearity. Finally, we employ the computed surface stress and temperature distributions along the grating to analytically calculate the Bragg shift, which is found to be in reasonable agreement with our experimental measurements.

Figure 1

A segment of the proposed computational domain, subdomains $\mathrm{\Omega }$, and boundaries $\partial \mathrm{\Omega }$ for modeling an arbitrarily embedded FBG in a stratified medium. The subscripts on $\mathrm{\Omega }$ and $\partial \mathrm{\Omega }$ are defined as $C$ (core), $CL$ (cladding), $J$ (jacket), $H$ (host), and $\infty$ (infinite domain). When solving the optical problem, subdomain ${\mathrm{\Omega }}_J$ becomes a perfectly matched layer (${\mathrm{\Omega }}_{PML}$). The general forms of the PDEs to be solved and the assumed boundary conditions are annotated, where the dependent variables $T, \mathit{u}$, and $\mathit{E}$, are temperature, displacement field, and electric field, respectively, while $\mathit{H}, {\mathit{\Phi }}_r,$ and $\mathit{v}$ are magnetic field, radiative heat flux, and the velocity vector of the subdomain translational motion, respectively. The terms $Q_i, i=1,2,...$ specify various sources including thermoelastic effect.

Figure 2

(a) Meshing of the subdomains for the computational analysis of the mechanical and thermal states of the stratified medium. The mesh shown hides specific outer subdomains for display of inner subdomains and boundary meshing patterns. For the subdomains with a free triangular mesh, a maximum element size of 70 $\mu$m and a minimum mesh element size of 12.6 $\mu$m were employed for subdomains ${\mathrm{\Omega }}_J, {\mathrm{\Omega }}_H$, and ${\mathrm{\Omega }}_{\infty }$. A denser mesh was employed for providing more resolute FEM analysis for ${\mathrm{\Omega }}_C$ and ${\mathrm{\Omega }}_{CL}$; here a free triangular mesh of 38.5 $\mu$m maximum element size and 2.8 $\mu$m minimum element size were used. All boundaries were meshed with a swept distributed pattern of spacing 1 $\mu$m. (b) Magnified core and cladding mesh.

Figure 3

Computed stress distribution in a $z=L=200\mu$m segment of an embedded FBG sensor for materials listed in Table 1. An arbitrary deformation is here simulated by the application of a boundary force of $f_y=-10$ GN ${\mathrm{m}}^{-2}$ on the $y>0$ segment of $\partial {\mathrm{\Omega }}_{H,\infty }$. To mechanically simulate the host bulk, external (infinite) subdomains ${\mathrm{\Omega }}_{\infty }$ with suitable scaling properties are considered. The outmost boundaries are constrained to suppress irrelevant (rigid) degrees of freedom with all other boundaries conditioned to deform freely.

Figure 4

Computed stress and strain as functions of material properties and temperature extracted from the 3D solutions along path $S$ defined on subdomains listed in Table 2. The principal stress component (a), (c) and strain component (b) exhibit significant variations as a function of core and cladding materials (a), (b) and as a function of temperature (c).

Figure 5

Computed total displacement visualized for the fiber core only. The displacement has been scaled by a factor of 20 to visualize the core curvature. The embedded fiber has a length of 200 $\mu$m. FBG simulator input data is extracted along a path $S$ on the core, shown here arbitrarily.

Figure 6

Computed Bragg spectral properties using ray tracing. Optical properties used are listed in Table 3.

Figure 7

Curvature-induced redistribution of the $z$ component of the first mode field $E_z$ in a SMF. Here, a radius of curvature of $\rho =200$ mm essentially yields the same core confinement as an undeformed fiber. Optical properties used are listed in Table 3, and $T=T_i=293$ K. The radius of curvature of $\rho =5.84$ mm was selected for its correspondence to the max deformation visualized in Fig. 5.

Figure 8

Meshing of the subdomains for the computational analysis of the optical states of the stratified medium. The mesh shown corresponds to ${\mathrm{\Omega }}_C$ and ${\mathrm{\Omega }}_{CL}$, indicated respectively by the inner and outer blue circles (a). The core and cladding are shown (a), with a mapped maximum mesh element size of 14 $\mu$m and minimum element size of 52 nm. A close-up of the free tetrahedral mesh that composes a small segment of the core center (b) is also visualized with a maximum mesh element size of 7 $\mu$m and a minimum element size of 14 nm.

Figure 9

The variation of fiber core principal stress as a function of cladding material and thickness (a) and temperature (b). An arbitrary deformation is here simulated by the application of a boundary force of $f_y=-10$ (GN ${\mathrm{m}}^{-2}$) (top), and elevated temperature with no applied force (bottom).

Figure 10

Refractive index components resulting from induced stress due to thermoelastic [columns (a) and (b)], and elastic [columns (c) and (d)] contributions.

Figure 11

Computed field changes of the single mode of an SMF. The unperturbed mode (a) exhibits significant elastic mode field changes (b) in response to $f$.

Figure 12

Temperature distribution in an embedded fiber segment of length 1000 $\mu$m. The core, cladding, jacket, and the embedding host subdomains are made of pure ${\mathrm{SiO}}_2$ assumed close to Ge-doped ${\mathrm{SiO}}_2$, pure ${\mathrm{SiO}}_2,$ acrylate, and Si, respectively. The plot displays the temperature on the $z$ axis at the center of the fiber core.

Figure 13

Birefringent optical mode changes (b) as a result of an asymmetrical geometry (a) and a uniform $\mathrm{\Delta }T=20$ K.

Figure 14

Validation of FBG-SimPlus temperature functionality with experimental measurement of the relative Bragg shift. The straight FBG, subjected to a variation in temperature, responds with a Bragg shift.