Structure of a laser field of various polarizations in the focal region of an ideal focusing lens. Calculation by methods of scalar diffraction theory

Within the framework of the scalar diffraction theory, we elaborate the method of calculating the electromagnetic-field structure in the ideal-lens focal plane. Using the method of quaternions for describing the rotation in three-dimensional space and diffraction integrals, we derive analytic formulas for the intensity distribution in the fields in the vicinity of the lens focal plane. We generalize the Debye method for an arbitrary polarization of the incident laser beam, in particular, for the linear, radial, and azimuthal polarizations. Such an approach, in view of the analytic formulas obtained, allows us to decrease the processing speed by an order. We show that the algorithm elaborated allows us to calculate the field amplitude?Cphase structure in the case of multimode laser radiation with a random phase.