A theory for the structure of modes in a laser resonator with astigmatic elements has been developed in a form that is amenable to experimental check. Experi-mental results have been given to support the theory. Thle aualyss llows that It Is relatively easy to produce high-order Hermite-gaussian distributions from con- ventional gas-laser oscillators, whereas exceeding care may be required to produce high-order Laguerre-gaussian distributions. For applications in which the latter distributions may be important, a design formula for the intracavity optics has been given.