An FPGA Architecture for Extracting Real-Time Zernike Coefficients from Measured Phase Gradients

Abstract

Zernike modes are commonly used in adaptive optics systems to represent optical wavefronts. However, real-time calculation of Zernike modes is time consuming due to two factors: the large factorial components in the radial polynomials used to define them and the large inverse matrix calculation needed for the linear fit. This paper presents an efficient parallel method for calculating Zernike coefficients from phase gradients produced by a Shack-Hartman sensor and its real-time implementation using an FPGA by pre-calculation and storage of subsections of the large inverse matrix. The architecture exploits symmetries within the Zernike modes to achieve a significant reduction in memory requirements and a speed-up of 2.9 when compared to published results utilising a 2D-FFT method for a grid size of 8 × 8. Analysis of processor element internal word length requirements show that 24-bit precision in precalculated values of the Zernike mode partial derivatives ensures less than 0.5% error per Zernike coefficient and an overall error of <1%. The design has been synthesized on a Xilinx Spartan-6 XC6SLX45 FPGA. The resource utilisation on this device is <3% of slice registers, <15% of slice LUTs, and approximately 48% of available DSP blocks independent of the Shack-Hartmann grid size. Block RAM usage is <16% for Shack-Hartmann grid sizes up to 32×32.

Adaptive output regulation of the adaptive optics system

“…In other words, a transformation matrix between the nodal coordinates of the DM and the Zernike modal coordinates of the distorted wavefront is required. The construction of the transformation matrix using the measured phase gradients is explained in Moser et al (2015). By ignoring the first mode, the phase ( w ( x , y , t )) and its gradients ( θ x ( x , y , t ), θ y ( x , y , t )) are expressed in terms of the Zernike modal coordinateswhere n z is the number of the Zernike polynomials, and the transformation matrix can be expressed in as followswhereThen, the DM model in equation (10) can be transformed into modal coordinates as followswhere y∈ℝnz, A=T†AT, ℬ=T†B, and C=CT.…”

“…In other words, a transformation matrix between the nodal coordinates of the DM and the Zernike modal coordinates of the distorted wavefront is required. The construction of the transformation matrix using the measured phase gradients is explained in Moser et al (2015). By ignoring the first mode, the phase (w (x, y, t)) and its gradients (θ x (x, y, t), θ y (x, y, t)) are expressed in terms of the Zernike modal coordinates wðx, y, tÞ ¼…”

Optimal actuator placement in adaptive optics systems

“…In order to express the output of the DM in terms of Zernike coordinates, a transformation matrix is required. In Moser et al (2015), the construction of such a transformation matrix using the measured phase gradients is explained. Once the transformation matrix is applied, the output matrix of the DM model (20) may be defined as C1∈ℝ2Nz×6n.…”