Po-Chih Lin

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.

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 subaperture position correction algorithm

The subaperture stitching interferometry is a technique suitable for testing high numerical-aperture optics, large-diameter spherical lenses and aspheric optics. In the stitching process, each subaperture has to be placed at its correct position in a global coordinate, and the positioning precision would affect the accuracy of stitching result. However, the mechanical limitations in the alignment process as well as vibrations during the measurement would induce inevitable subaperture position uncertainties. In our previous study, a rotational scanning subaperture stitching interferometer has been constructed. This paper provides an iterative algorithm to correct the subaperture position without altering the interferometer configuration. Each subaperture is first placed at its geometric position estimated according to the F number of reference lens, the measurement zenithal angle and the number of pixels along the width of subaperture. By using the concept of differentiation, a shift compensator along the radial direction of the global coordinate is added into the stitching algorithm. The algorithm includes two kinds of compensators: one for the geometric null with four compensators of piston, two directional tilts and defocus, and the other for the position correction with the shift compensator. These compensators are computed iteratively to minimize the phase differences in the overlapped regions of subapertures in a least-squares sense. The simulation results demonstrate that the proposed method works to the position accuracy of 0.001 pixels for both the single-ring and multiple-ring configurations. Experimental verifications with the single-ring and multiple-ring data also show the effectiveness of the algorithm.

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

The Self-Calibration of High Overlapping Density Subaperture Stitching Interferometry

The high overlapping density stitching interferometry not only can measure the tested surface but also precisely calibrate the reference errors of the interferometer in a single phase stitching process. The simulation results prove the feasibility.

The fast measurement of a mild asphere by the vibration-modulated sub-aperture stitching interferometer

Sub-aperture testing methods are widely used in optical shops to test surface deformations of large diameter, high numerical aperture, or aspherical lens surfaces. We are proposing a novel 4 axis vibration modulated interferometer for subaperture testing. This interferometer takes advantage of the rotationally symmetric property of the optical lens and measures the lens surface against its symmetry axis rotationally. By adapting a synchronous random phase modulation measurement, interferometric data is acquired on the fly when the lens is being rotated. The vibration modulated interference phase is then calculated and stitched into a complete lens surface map by least squared fitting. This method has advantages over the prior methods in that it acquires the interferogram in a much shorter acquisition time, even with lower requirements on the optics and mechanical hardware. The stitch error is then significantly decreased by increasing both the lateral resolution of sub-aperture and the reduced position uncertainty of the stitched sub-aperture phase maps. A measurement on a mild asphere is demonstrated to prove the feasibility of the proposed interferometer.

A subaperture stitching algorithm for aspheric surfaces

A subaperture stitching algorithm was developed for testing aspheric surfaces. The full aperture was divided into one central circular region plus several partially-overlapping annuli. Each annulus was composed of partially-overlapping circular subapertures. The phase map in each subaperture was obtained through the phase-shifting interferometry and retrieved by an iterative tilt-immune phase-shifting algorithm and a Zernike-polynomial-based phase-unwrapping process. All subapertures in one annulus were stitched simultaneously in least-squares sense. By eliminating the relative piston and tilt between adjacent subapertures, the sum of squared errors in the overlapped regions was minimized. The phase stitching between annuli also utilized the least-squares method in the overlapped region. Simulation results on a test wavefront with 30-wave spherical aberrations demonstrated the effectiveness of the proposed algorithm. The rms phase residue after the phase-shifting, phase-unwrapping and phase-stitching processes was 0.006 waves, which met the precision requirement of common interferometers. This algorithm should be applicable to general surfaces in subaperture stitching interferometry.