Chao-Wen Liang

Iterative phase-shifting algorithm immune to random phase shifts and tilts

An iterative phase-shifting algorithm based on the least-squares principle is developed to overcome the random piston and tilt wavefront errors generated from the phase shifter. The algorithm iteratively calculates the phase distribution and the phase-shifting map to minimize the sum of squared errors in the interferograms. The performance of the algorithm is evaluated via computer simulations and validated by the Fizeau interferometer measurements. The results show that the proposed algorithm has a fast convergence rate and satisfactory phase-estimation accuracy, improving the measurement precision of the phase-shifting interferometers with significant phase-shifter errors.

Vibration modulated subaperture stitching interferometry

A novel subaperture stitching interferometry is developed to measure the surface deformation of the lens by utilizing the mechanical vibration induced from a motorized stage. The interferograms of different subapertures are acquired on the fly while the tested optics is rotating against its symmetrical axis. The measurement throughput and the subaperture positioning accuracy are improved simultaneously by adopting both the synchronous rotational scanning mechanism and the non-uniform phase shifting algorithm. The experimental measurement shows the stitched phase RMS error of 0.0037 waves proving the feasibility of the proposed phase acquisition method.

Measurement improvement by high overlapping density subaperture stitching interferometry

The vibration-modulated subaperture stitching interferometer acquires the interferogram on the fly dynamically. With its highly improved measurement throughput, we applied the device for high overlapping density subaperture stitching interferometry to acquire hundreds of overlapping subapertures in a single phase stitching measurement. The averaging effect of the high overlapping density stitching interferometer is discussed. In the experiment, the proposed high overlapping density stitching interferometer is also proved to reduce measurement uncertainty and improve measurement quality effectively.

Interferometer reference error suppression by the high-overlapping-density phase-stitching algorithm.

The subaperture stitching interferometer is a flexible testing device that measures either high-numerical-aperture or large aperture optics without the requirement of additional auxiliary optics. In the measurement, the interferometer reference optics error can contaminate the stitched phase of the complete tested optics and reduce measurement accuracy. We propose high-overlapping-density subaperture stitching interferometry (HOD-SSI) to reduce the impact of reference optics errors on the stitched phase. The tested optics surface deformation phase is determined by averaging the multiple subaperture measurements taken at different rotational angles. Simulation and experiment show that HOD-SSI can effectively reduce the stitched phase errors due to the static reference optics errors.

Wavefront measurement made by an off-the-shelf laser-scanning pico projector.

Focal plane testing methods such as the Shack-Hartmann wavefront sensor and phase-shifting deflectometry are valuable tools for optical testing. In this study, we propose a novel wavefront slope testing method that uses a scanning galvo laser, in which a single-mode Gaussian beam scans the pupils of the tested optics in the system. In addition, the ray aberration is reconstructed by the four-step phase-shifting measurement by modulating the angular domain. The measured wavefront is verified by a Fizeau interferometer in terms of Zernike polynomials.

Aberration compensation and position scanning of a subaperture stitching algorithm

An aspheric testing system based on subaperture stitching interferometry has been developed. A procedure involving subaperture aberration compensation and radial position scanning was established to resolve discrepancies in the overlapped regions. During the aspheric measuring process, the Fizeau-interferometer axis, the optical axis of the asphere, and the mechanical rotation axis have to be aligned. Due to the tolerance of alignment mechanisms, subaperture interferograms would be contaminated by various amounts of aberrations associated with the rotation angle. These aberrations introduce large inconsistencies between adjacent subapertures in the stitching algorithm. Zernike coefficients of the subapertures in one annulus were examined and each coefficient term was found to be a sinusoidal function of the rotation angle. To eliminate the influence of misalignments, each subaperture was compensated with appropriate amounts of coma and astigmatism to make the resulting Zernike coefficients converge to the mean values of the sinusoidal functions. In addition, the determination of the overlapped regions relies on the precise estimate of the distance between the center of each subaperture and the center of the aspheric optics. This distance was first provided by the encoder and then estimated by position scanning along the radial direction pixel-by-pixel in numerical computations. The means of the standard deviation in the overlapped regions in the simulation and the experimental measurement of an aspheric lens were 0.00004 and 0.06 waves, respectively. This demonstrates the reliability of the subaperture aberration compensation and position scanning process.

An Iterative Tilt-Immune Phase-Shifting Algorithm

An iterative least-squares-based phase-shifting algorithm was developed to cope with random pistons and tilts from the phase shifter. The allowable tilt amount was 0.5 waves over the pupil. The achieved rms accuracy was 0.0024 radians.

Line scanning deflectometry using a laser pico projector

In our previous publications, we had successfully made a deflectometry measurement by using a portable laser projector. In this research, we propose the beam weighting centroid method rather than previous the phase shifting method for quantification of the angular direction of the testing beam in the tested optics entrance pupil. By projecting the angular sequential lines on tested optics entrance pupil, the wavefront aberration is reconstructed from two orthogonal directions measurements, in a similar way to the line scanning deflectometry. The limited gray scale problem of laser projector during the phase shifting measurement is therefore eliminated. The reconstructed wavefront is proven to yield a more accurate result than the phase shifting methods at the cost of more image frames and acquisition time.

Solar concentrator with a toroidal relay module

III-V multijunction solar cells require solar concentrators with a high concentration ratio to reduce per watt cost and to increase solar energy transforming efficiency. This paper discusses a novel solar concentrator design that features a high concentration ratio, high transfer efficiency, thin profile design, and a high solar acceptance angle. The optical design of the concentrator utilizes a toroidal relay module, which includes both the off-axis relay lens and field lens design in a single concentric toroidal lens shape. The optical design concept of the concentrator is discussed and the simulation results are shown. The given exemplary design has an aspect ratio of 0.24, a high averaged optical concentration ratio 1230×, a maximum efficiency of 76.8%, and the solar acceptance angle of ±0.9°.

Characterization of field dependent aberrations in Fizeau interferometer using double Zernike polynomials

Fizeau interferometer is widely used to test the surface deformation of the optical lens surface profile. However, in some measurement circumstances the common path condition of the Fizeau configuration does not hold. For example, the subaperture scanning interferometry of asphere or the non-null aspherical element testing has dense fringe spacing. Systematic aberrations of non-null testing are introduced into the measurement wavefront with the high wavefront slope of the returning beam. We propose to use a two-dimension scanning device to drive a test ball to different fields of the Fizeau interferometer for the the interference phase at each field. By least square fitting the measurement, we can get the double Zernike polynomial coefficients representing the field dependent aberrations in the interferometer system. According to the coefficients, the off-axis aberrations in the interferometer can be identified