Exploring a Gröbner basis approach to modeling wavefront aberrations using Zernike polynomials

Decades of research have explored the benefits and limitations of using Zernike polynomials to fit wavefront aberrations and there are a plethora of commercial software packages available for this purpose. Automation is particularly efficacious when combined with hardware that measures the aberrations (using a Shack-Hartmann sensor) and adaptive optics to correct them. To gain insight into using these polynomials, we have adopted a first principles approach to fitting the aberrations. This enables: (1) characterization and determination of the accuracy of the approach (e.g. least squares); (2) easy modification of the approach and (3) an understanding of the conditions that limit the application of these fitting polynomials (e.g. scarcity of data). As an alternative method of fitting the polynomials, we have implemented and explored using a Gröbner basis reduction. Instead of treating the data as a fitting problem, this approach uses a set of simultaneous polynomial equations. Our results indicate that this approach offers better performance as compared to least squares with fewer data points and improved accuracy. We have applied this approach to data acquired using a commercial adaptive optics system with a deformable mirror and a Shack-Hartmann wavefront sensor (Imagine Optic Inc.).