abcd matrix for reflection and refraction of laser beam at tilted concave and convex elliptic paraboloid interfaces and studying laser beam reflection from a tilted concave parabola of revolution

studying gaussian beam is a method to investigate laser beam propagation and abcd matrix is a fast and simple method to simulate gaussian beam propagation in different mediums. of the abcd matrices studied so far, reflection and refraction matrices at various surfaces have attracted a lot of researches. however in previous work the incident beam and the principle axis of surface are in parallel. as an extension to those investigations, a general scheme that the incident beam is oblique is discussed here and the full analysis of the reflection and refraction of a gaussian beam at the surface of a tilted concave/convex elliptic paraboloid surface is addressed. based on the optical phase matching, analytic mathematical equations are derived for the spot size and the wavefront radius of a beam. expressions are converted into the abcd matrices, which are more convenient and practical to use. finally, a practical case is analyzed by applying the obtained formulas. this analysis is important since paraboloid surfaces in optics or terahertz waves are used as mirrors or lenses.