Hyperbolic fractional-order Fourier transformations in scalar theory of diffraction

We define hyperbolic fractional-order Fourier transformations by replacing the circular trigonometric functions in the integral expressions of conventional fractional-order Fourier transformations with hyperbolic trigonometric functions. We establish the composition laws of these hyperbolic transformations. We then use hyperbolic fractional-order Fourier transforms to mathematically represent Fresnel diffraction phenomena that cannot be described by conventional fractional Fourier transforms, due to their geometric configurations. Additionally, we apply appropriate compositions of these transformations to coherent optical imaging.