Visualization of surface figure by the use of Zernike polynomials

Workflow for modeling of generalized mid-spatial frequency errors in optical systems

We propose a workflow for modeling generalized mid-spatial frequency (MSF) errors in optical imaging systems. This workflow enables the classification of MSF distributions, filtering of bandlimited signatures, propagation of MSF errors to the exit pupil, and performance predictions that differentiate performance impacts due to the MSF distributions. We demonstrate the workflow by modeling the performance impacts of MSF errors for both transmissive and reflective imaging systems with near-diffraction-limited performance.

Wavefront Aberration Sensor Based on a Multichannel Diffractive Optical Element

We propose a new type of a wavefront aberration sensor, that is, a Zernike matched multichannel diffractive optical filter, which performs consistent filtering of phase distributions corresponding to Zernike polynomials. The sensitivity of the new sensor is theoretically estimated. Based on the theory, we develop recommendations for its application. Test wavefronts formed using a spatial light modulator are experimentally investigated. The applicability of the new sensor for the fine-tuning of a laser collimator is assessed.

Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology

The accuracy of phase-shifting interferometers (PSI) is crippled by nonlinearity of the phase shifter and instability of the environment such as vibration and air turbulence. A general algorithm, utilizing Lissajous figures and ellipse fitting, of correcting the phase extraction error in the phase shifting interferometry is described in this paper. By plotting N against D, where N and D represent the numerator and denominator terms of the phase extraction function (i.e. an arctangent function) respectively, a Lissajous ellipse is created. Once the parameters of the ellipse are determined by ellipse fitting, one can transform the ellipse to a unit circle (ETC). Through this process the phase extraction error caused by random phase shift errors can be corrected successfully. Proposed method is non-iterated, adapts to all phase shifting algorithms (PSAs), and has high accuracy. Some factors that may affect the performance of proposed method are discussed in numerical simulations. Optical experiments are implemented to validate the effectiveness of proposed algorithm.

Single shot interferogram analysis for optical metrology

We report a novel constrained optimization method for single shot interferogram analysis. The unknown test wavefront is estimated as a minimum L2-norm squared solution whose phase is constrained to the space spanned by a finite number of Zernike polynomials. Using a single frame from standard phase shifting datasets, we demonstrate that our approach provides a phase map that matches with that generated using phase shifting algorithms to within λ/100  rms error. Our simulations and experimental results suggest the possibility of a simplified low-cost high quality optical metrology system for performing routine metrology tests involving smooth surface profiles.

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

Combined Twyman-Green and Mach-Zehnder interferometer for microlens testing

A new interferometer design for microlens testing is presented. The phase-shifting system combines the advantages of a Twyman-Green and a Mach-Zehnder interferometer and permits full characterization of the aberrations of microlenses as well as radius of curvature and focal length measurements. The Twyman-Green system is applied to surface testing in reflection (single reflection), whereas the Mach-Zehnder system is used for lens testing in transmission (single pass). Both measurements are performed without removal of the test part, allowing for combination of the results without confusion of the actual lens and without an azimuthal orientation error. The interferometer setup is explained, the test procedure is described, and experimental results are given.

Absolute interferometric test of aspheres by use of twin computer-generated holograms

A complete absolute interferometric test of axially symmetric aspheres is presented. The method is based on a specially designed computer-generated hologram (CGH) that reconstructs an aspherical wave as well as a spherical auxiliary wave. Since both phase functions have the same symmetry and their pattern is simultaneously encoded, we call this type of multiplex hologram a Twin-CGH. The spherical wave is used for calibration. The aberrations of the spherical auxiliary wave are measured absolutely with either a spherical mirror or an absolute test for Fresnel zone plates. Thus the two types of aberration inherent in the CGH can be identified and separated from each other. The errors of the spherical wave can be transferred to those of the aspherical wave. Two different methods thatuse Twin-CGHs for absolute testing of aspheric surfaces are described. Test procedures are explained, equations are derived, and experimental results are presented. A mutual comparison of the two results and a comparison with the established N-position rotation test are given.

Absolute measurement of planarity with Fritz's method: uncertainty evaluation

Fritz's method [Opt. Eng. 23, 379 (1984)] of using Zernike polynomials to assess the absolute planarity of test plates is revisited. A refinement is described that takes into account the data decorrelation that appears in experiments. An uncertainty balance is defined by propagation of error contributions through the steps of the method. The resultant measuring procedure is demonstrated on a data set from experiments, and a nanometer level of uncertainty is achieved.