Transfer matrix method for fundamental LP₀₁ core mode coupling in a tilted FBG sensor

Fibre Bragg gratings (FBGs) are obtained through a permanent and periodic refractive index modulation in the core of the single-mode optical fibre. For many years, they have been employed in telecommunication industry as a passive device for wavelength division multiplexing and dispersion compensation components, or in laser apparatus for laser fibre stabilization, Erbium amplifier gain flattening device and amplifier pump reflectors. In aerospace structures, FBGs are used as sensors for structural health monitoring of composite materials as they are able to perform measurements of several parameters inside the material in an elegant and low intrusiveness way. Based on the Bragg and optical fibre structure many kind of customizations can be applied on FBG sensors during the manufacturing process. Each of them gives to the FBG sensor different proprieties and sensing abilities. In this work, we address the numerical simulation of the reflected spectrum by a special FBG sensor called a tilted FBG (TFBG), in which the core refractive index modulation is performed in way to obtain a tilted Bragg super-structure. By considering the classic Coupled-Mode theory for weakly-guided waveguides, we solved the mode propagation equations with the Transfer Matrix method (TMM) obtaining the TFBG reflectivity for different tilt angles.