Propagation of Bessel and Bessel–Gaussian beams through an unapertured or apertured misaligned paraxial optical systems

Propagation of Bessel and Bessel–Gaussian beams through an unapertured or apertured misaligned paraxial optical system is investigated. Analytical formulas for Bessel and Bessel–Gaussian beams propagating through an unapertured misaligned paraxial optical system are derived based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial optical system in the cylindrical coordinate system. By expanding the hard aperture function into a finite sum of complex Gaussian functions, some approximate analytical formulas are derived for Bessel and Bessel–Gaussian beams propagating through an apertured misaligned paraxial optical system. Some numerical examples are illustrated. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of Bessel and Bessel–Gaussian beams through an unapertured or apertured misaligned paraxial optical system.