In this work,we have studied the propagation dynamics of a secant soliton travelling through six types of Fibonacci and parabolic-Fibonacci-ordered multilayers. Although the propagation of solitons in different mediums such as graded-index inhomogeneous media is considered, their propagation in the fractal geometries has not yet been addressed. In this way, we utilise a fractional nonlinear Schrödinger equation formalism. Then, a numericalsplit-step Fourier method is employed to solve the resulting equation. Besides, we used a wave expansion method to calculate the band structure of the system by solving the stationary Schrödinger equation at different wavenumbers,because the refraction index that governs the beam divergence can be described using the band structure curvature. Our motivation is to engineer the band structure curvature that is an important parameter in obtaining the desired waveguiding properties.