Analytical calibration of slope response of Zernike modes in a Shack–Hartmann wavefront sensor based on matrix product

The Shack-Hartmann wavefront sensor (SH-WFS) is widely used as a slope-based wavefront sensing device. The modal method is favored for wavefront reconstruction from SH-WFS output because of its excellent performance. In this case, the calibration of modal (commonly Zernike modes) slope is required in advance. Traditional numerical or symbolic integral-based methods are not satisfactory because of their low accuracy or efficiency, particularly when an extremely large number of microlenses are involved. In this Letter, a novel method based on matrix product is proposed in which two key matrix operators are utilized. The first, namely the derivative matrix operator, is used to obtain the derivative of the Zernike modes; the second, that is, the transformation matrix operator, is then used to map the Zernike derivative defined in the original, whole circular pupil into modes defined in a scaled, translated circle pupil enveloping a specific microlens. With these two operators, the evaluation of slope response of Zernike modes could be unified into a matrix-product framework, which contributes its high efficiency. Numerical simulations show the superior advantages of the proposed method in accuracy and efficiency over traditional ones.