Zh. Zhejnov, J. Urumov. Modeling the Influence of Deformation on Bragg Fibers Losses

Key Words: Bragg fiber; bending; losses; Photonic Crystal Fiber; model of deformation.

Abstract. The article analyzes the losses in one type of Photonic Crystal Fiber – Bragg fiber. It is one-dimensional fiber, made as coaxial cylindrical layers. The fiber cladding is a dielectric mirror realized as a multilayer dielectric coating. The paper proposes a method based on geometric optics for the analysis of light propagation losses in a multilayer microstructured fiber with an air core, as a result of tunneling. A model of M-layer fiber with set refractive indices of the dielectric layers is proposed. The angles of incidence and reflection of the boundaries of each two layers of the fiber cladding according to Snelius’ law are calculated. The transverse reflection coefficients of each boundary between two layers for TM and TE polarization are calculated. The electric fields of the reflected and incident beams are calculated. The magnitudes of these vectors are recursively related to each boundary of the previous and next fiber layer. The input power of the light in the first fiber layer for TE and TM polarization is calculated. The losses from light reflection when it passing through all layers of the fiber as a function of the reflection coefficients are calculated. The normalized attenuation is calculated. The characteristic equation for the optical waveguide is decided. The angles of reflection in different modes are calculated. The distance between two consecutive reflections of the beam is calculated as a function of the reflection coefficients for the different polarizations and the reflection angles of the layers for TE and TM polarizations. They determine the reflection coefficient and the phase change when light passes through the whole fiber. The delays of the rays with TE and TM polarization are obtained. The expressions for the chromatic dispersion of the fiber for TE and TM polarization are then derived. A mathematical model of the deformation, presented as a change in the geometry of the fiber in this section as part of a circle is proposed. For this section of the deformed fiber by geometric transformations the connection between the angles of incidence and reflection in the straight and in the round section of the deformed fiber is calculated. The distance between two consecutive reflections from the fiber boundary surface of the rays trajectory is determined by the number of consecutive reflections in the core of the fiber. The reflection losses in TE and TM polarization, which are proportional to the number of beam reflections, are then determined. The phase delays for beams with TE and TM polarization for a fiber of a certain length are determined as the sum of the individual delays in the reflections of the beams. The proposed mathematical model and algorithm for calculating the attenuation of the fiber for different modes of TE and TM polarization, an example of M-layer PCF fiber with air core and alternating dielectric layers with two alternating coefficients and thicknesses is solved, so as to create at the average wavelength of light a phase shift of 900. A MATLAB program has been written. It simulates the attenuation and dispersion of a fiber with set optical parameters. After solving the characteristic equation for this fiber with the introduced parameters, the normalized attenuations for TE and TM polarization are calculated for fiber with and without deformation for different propagating modes. Calculations have been made for fibers with core radius of 20-200 μm for a light wavelength of 1559 nm. The attenuation and chromatic dispersion graphs for TE and TM polarization of several straight and deformed fibers with the same length and different number of layers were plotted. A conclusion about the influence of the core diameter, the bending radius and the number layers of the fiber cladding on the losses of the different propagated modes is made. A comparison of PCF losses and ordinary quartz single-mode fibers was made. In conclusion, the disadvantage of the proposed method is that only the meridional rays of the propagating light are analyzed and the dielectric losses in the fiber cladding are not taken into account. The advantage of the proposed method is the small computational complexity and the correct qualitative result of the PCF analysis with certain parameters. Guidelines for future development are proposed – analysis of the losses of the fiber, taking into account the non-meridional rays in the fiber. The possibilities for using PCF for information transmission and as dispersion compensators in telecommunications are pointed. The possibility of using the method for optimization of certain parameters of PCF of this kind is proposed.