Study of Zernike Polynomials Properties for Oblique Elliptical Aperture at an Angle (? / 4) with X-Axis

In this research, some of the optical properties have been studied for oblique elliptical aperture at an angle ( / 4) with x-axis, by using Zernike polynomials. Zernike polynomials for circular aperture and Gram Schmidt orthogonalization method were adopted to find Zernike polynomials for the new aperture. And in this case the equations used are very complex, therefore, new coordinates m and n were used, that they were oblique at (/ 4) to both x and y axes respectively. The relationship between Zernike polynomials for the new aperture with first and third order aberrations was derived. And it found that aberrations of high orders were balanced with aberrations of lower orders, for example, third order coma aberration were balanced with first order tilt error, while the third order spherical and astigmatism aberrations were balanced with focus aberration of first order. The standard deviation is also found in this research for balanced and unbalanced aberrations for any value of aspect ratio.