Error Calibration Method of Infrared Shack-Hartmann Wavefront Sensor

Objective The Shack-Hartmann wavefront sensor is widely used in adaptive optics and optical testing. It typically consists of a microlens array and a detector array located at the focal plane of the microlenses. When an aberrated wavefront is incident, the positions of the spots on the focal plane exhibit deviations. The wavefront can be reconstructed by utilizing the displacements of the spots. For an infrared Shack-Hartmann wavefront sensor, a cold shield is positioned in front of the photosensitive surface of the mid-wave infrared detector array, and the distance between the cold shield and the photosensitive surface exceeds the focal length of the microlens array. Therefore, a relay imaging system is necessary to transfer the focal spot array onto the photosensitive surface. However, the aberrations of the relay imaging lens group can lead to errors in spot position detection, thereby introducing errors in wavefront sensing. In this study, a method was proposed to eliminate these errors, enabling the calibration of Shack-Hartmann sensors that incorporated such a relay imaging system. Methods Each sub-lens of the microlens array and the relay imaging lens group constitute an imaging subsystem. By tracing the chief rays of these optical subsystems, the error distribution of spot positions after relay imaging was analyzed. Simulation results reveal that the relay system exhibits varying magnification ratios for spot displacements at different positions, showing a periodic variation pattern with a period equal to one sub-aperture size. Based on the error distribution characteristics, a three-step calibration procedure was developed. First, multiple planar wavefronts with different tilt magnitudes, which were approximately uniformly distributed within the dynamic range, were incident to calibrate the nonuniform response error of each sub-aperture. By using the spot displacement of the central sub-aperture as a reference, polynomial fitting was applied to establish the functional relationship between spot displacement errors and measured displacements for each sub-aperture, enabling non-uniform response error correction. Next, a spherical wavefront was introduced to calibrate the nonlinear error. By leveraging the linear variation of wavefront slope with spatial position in spherical waves, the spot displacements of all sub-apertures were linearly fitted against their spatial coordinates. The fitting results served as the baseline to compute nonlinear errors for each sub-aperture. Polynomial fitting was then performed to characterize the relationship between nonlinear errors and spot displacements for nonlinear error correction. Finally, multiple spherical wavefronts were employed to calibrate the proportional coefficient between spot displacements and wavefront slopes. The proposed error calibration method was validated through both simulations and experiments. Results and Discussions A simulation system for an infrared Shack-Hartmann sensor is established to validate the error calibration method. After non-uniform error calibration, the root mean square (RMS) of the non-uniform errors across all sub-apertures is reduced to within 0.012 pixels. When a defocus wavefront with a peak-to-valley (PV) value of 10.0λ is applied, the RMS of the nonlinear errors decreases from 0.031 pixel to 0.022 pixel after nonlinear error calibration. In the simulation, a wavefront containing only tilt and defocus is measured, and the RMS of the error wavefront is reduced from 0.023λ to 0.009λ. An experimental system is developed to study the error calibration method for an infrared Shack-Hartmann sensor. After error calibration, the PV of the wavefront reconstruction error decreases from 0.952λ to 0.106λ, while the RMS error decreases from 0.149λ to 0.012λ. Both simulation and experimental results demonstrate that this method effectively mitigates wavefront sensing errors caused by the inherent aberrations in the relay system and other types of distortion induced by assembly errors. We also discuss the limitations of the error calibration method, which requires the spot displacement of the incident wavefront to remain within the calibration range. Conclusions We propose and validate an error correction method to address wavefront sensing errors in infrared Shack-Hartmann wavefront sensors caused by relay imaging systems. The method systematically corrects the nonuniform response errors and the nonlinear errors and calibrates the proportional coefficient between spot displacement and wavefront slope. Notably, the calibration process does not require introducing wavefronts with known slopes, making it both convenient and practical. The proposed method is evaluated through simulations and experiments. In simulations, after error correction, the RMS of spot displacement inconsistency errors is reduced to below 0.012 pixel, and the RMS of nonlinear errors decreases from 0.031 pixel to 0.022 pixel. In experiments, the RMS of wavefront sensing errors is reduced from 0.149λ to 0.012λ. Both simulation and experimental results demonstrate that the method effectively mitigates wavefront sensing errors induced by radial distortion from inherent relay system aberrations and other distortions caused by assembly misalignments.