A novel index-guiding Silica glass-core hexagonal High-Birefringence Photonic Crystal Fiber (HB-PCF) is proposed, with five rings of standard cladding air circular holes arranged in four formations inspired by the Binary Morse-Thue fractal Sequence (BMTS). The form birefringence, confinement loss, chromatic dispersion, effective mode area, and effective normalized frequency are evaluated for the four PCFs operating within (1.8 - 2 mu m) eye-safe wavelength range. Modeling and analysis of the four PCF formations are performed deploying full-vector analysis in Finite Element Method (FEM) using COMSOL Multiphysics. Respecting fabrication and in light of commercial availability in designing the proposed PCF structures, a high birefringence of up to (6.549 x 10(-3) at 2 mu m) is achieved with dispersion free single-mode operation. Confinement loss as low as (3.2 x 10(-5) - 6.5 x 10(-4) dB/m for 1.8 - 2 mu m range) is achieved as well. Comparison against previously reported PCF structures reveals the desirably higher birefringence of our BMTS HB-PCF. The proposed PCFs are of vital use in various optical systems (e.g.: multi-wavelength fiber ring laser systems, and tunable lasers), catering for applications such as: optical sensing, LIDAR systems, material processing, optical signal processing, and optical communication.