We have generalized the ABCD propagation law, Q(2) = (AQ(1) + B)/(CQ(1) + D), for an optical system by introducing a generalized complex radius of curvature Q for a general optical beam. The real part of 1/Q is related to the mean radius of curvature of the wave front, while the imaginary part is related to the second moment of the amplitude of the beam.

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An axicon and a lens are combined to form an optical system producing a ring-shaped pattern. The purpose of this paper is to show that when a lens-axicon combination is illuminated by a Gaussian beam, the transverse distribution of the focal ring is also a Gaussian distribution. The typical width of this distribution was found to be, in the case of the lens-axicon combination, 1.65 times greater than the typical width of the Gaussian beam obtained by focusing the same beam using the lens alone. This focusing system is well suited for the drilling of good quality large diameter holes using a high power laser beam.

New optical combinations of axicons and axicons with spherical mirrors and lenses suitable for laser machining are presented. Linear and annular focusing, coaxially and radially to the laser beam, are possible. Most combinations allow continuous adjustment of exit beam parameters, focal line length, focal ring diameter, and magnification, by varying the relative position of one of the axicons. Potential new laser applications are also discussed in relation to these optical devices.

Optical resonators using graded-phase mirrors are analyzed with the help of the generalized ABCD propagation law for a real optical beam. This analysis gives the second-order moment gross characteristics of the eigenmode and indicates a design procedure. An example of a super-Gaussian output beam shows that this type of optical resonator might have large transverse-mode discrimination that could provide operation in a large fundamental-mode beamwidth.