Solution of complex characteristic equation of LPFG sensors coated with complex refractive index double-layer films

The theoretic mode of LPFG coated with the weak absorption film and the sensing film is first put forward. Based on the coupled-mode theory, the complex characteristic equation of this LPFG is set up. The question on how to resolve the complex characteristic equation is analyzed theoretically in detail. The complex roots of LPFG weak absorption film coated are coincident with the results of D. V. Ignacio. In addition, there are Inf and numerical overflow in the resolution of the complex characteristic equation of metal film coated LPFG. These problems are solved through approximate deformation and scaling implementation in Bessel function. The complex roots of metal film coated LPFG are coincident with the results of Z. J. Zhang. The study proves the correction of the theoretic model, and provides theoretical foundation for studying the coupling and transmission characteristics of this kind of LPFG.