Describe wave aberration function using modified Zernike polynomials

Computer simulation was carried out, to describe the entrance wavefront into the circular and hexagonal aperture of telescope. The wave aberration function can be expressed as a weighted sum of Zernike polynomials. The next generation very large ground base optical telescope has segmented aperture with sub segments almost hexagonal shape, therefore, it requires to find modified Zernike polynomials analyzer for hexagonal shape by utilizing convention form of Zernike polynomials for circular shape. The modified Zernike polynomials have the same properties of the original one across hexagonal aperture, like orthogonality, normality, mathematically independent, making the variance of the sum of modes equal to the sum of the variances of each individual mode, the invers of transformation matrix is the transpose of it and this make the compensation process in Adaptive Optics system much easier and faster. Derivation of the modified Zernike polynomials was been derived totally according to Gram-Schmidt orthogonalization process. The number of aberration modes that was derived is only ten modes because of that orthogonalization process is complex and needed highly mathematical calculation.