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In the study of a diffraction field of a light wave passing through a symmetrical paraxial optical system, the Collins formula and its inverse are convenient for calculation. The algorithm study of the Collins formula demonstrates that both a single fast Fourier transform algorithm and a double fast Fourier transform algorithm can be used in diffraction calculation. But, whichever algorithm is adopted, only by meeting some specific conditions can the amplitude and phase distributions of a diffracted wave be calculated correctly. Based on the Nyquist sampling theorem, the indispensable conditions to calculate a diffraction field accurately are presented. OCIS codes: 050.1960, 70.2590, 00.4430.

We evaluate the eigenfields of an unstable Bessel--Gauss resonator (UBGR) by use of the transfer-matrix method in which the transverse profiles and their corresponding losses of the UBGR are considered as the eigenvectors and eigenvalues of a transfer matrix so that the dominant mode fields and their losses of the UBGR can be readily extracted in terms of the matrix eigenvalue algorithm. Moreover, based on the eigenfields across two mirrors that resulted from the transfer-matrix method, we simulate the field distributions in the cavity and the propagation of output beams by means of the angular spectrum method. The computation results show that the UBGR easily produces a fundamental Bessel--Gauss mode of good quality, and the output beams retain the original Bessel--Gauss distribution during propagation. OCIS codes: 140.3300, 140.3410, 140.4780.

A Gaussian-reflectivity mirror resonator is proposed to achieve high-quality laser beams. To analyze the laser fields in a Gaussian-reflectivity mirror resonator, the diffraction integral equations of a Gaussianreflectivity mirror resonator are converted to the finite-sum matrix equations. Consequently, according to the Fox-Li laser self-reproducing principle, we describe the mode fields and their losses in the proposed resonator as eigenvectors and eigenvalues of a transfer matrix. The conclusion can be drawn from the numerical results that, if a Gaussian-reflectivity mirror is adopted for a piano-concave resonator, a fundamental mode can easily be obtained from a transverse-flow C[O.sub.2] laser and high-quality laser beams can be expected. OCIS codes: 140.3300, 140.3410, 140.4130.