Using Zernike Circle polynomials, a study has been made of the mode structure of the diffraction limited open resonator. Firstly, the mode structure of open resonators, formed by mirrors with arbitrary cross-sections but with rotational symmetry, have been formulated under the scalar approximations. The arbitrary mirror surfaces were defined by reference to con-focal surfaces, the differences between the two surfaces expressed in a distortion function. The mode field distributions on the mirror surfaces have been expressed as a series of Zernike Circle polynomials for the resonators whose mode structures depend on the diffraction effects of the finite mirror apertures. The modes of the resonator were then solutions of a highly convergent matrix equation which yielded the coefficients of the series, diffraction losses and resonant condition for each mode. The resonators with large mirror apertures have been similarly formulated by replacing the Zernike Circle polynomials with Gaussian Laguerre functions. Resonator configurations, with equivalent mode structures, have been defined by relationships expressed in terms of the distortion function. Secondly, the mode structure of the complete range of stable, diffraction limited, quadratic resonators have been analysed using the Zernike Circle polynomial expansion. The results have been compared with those of other workers, who employed different techniques, to test the applicability of the expansion to the general resonator. The proposed method and those of other workers have been compared and the advantages of the Zernike Circle polynomial expansion, both in the simplicity of application and in the rate of convergence for the dominant and higher order modes, has been demonstrated. Thirdly, the mode structure of the quadratic cavities modified by phase plates, or dielectric layers partially covering the mirror surfaces, has been analysed. Phase plates of varying thickness transparency and area have been analysed for their effect on the mode structures of the resonators. It has been shown that the presence of the phase plates improve some of the properties of the quadratic resonators, and the possible practical applications of this simple modification have been outlined. Fourthly, a X-band microwave experiment has been performed to confirm the effect of quarter wave plates on the mode structure of the confocal resonator.