Zhishan Gao

Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture

Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

Focal length measurement based on the wavefront difference method by a Fizeau interferometer.

A method for measuring the focal length of the lens by a Fizeau interferometer is proposed. Based on the Gaussian imaging equation and the longitudinal displacements of the object point and image point, a precise formula for focal length calculation is deduced. The longitudinal displacement of the object points is determined by the wavefront difference method with a subnanometer resolution. An experimental system for focal length measurements is set up to verify the principle. The sources of uncertainty in measurement are discussed. Both the positive and negative lens experimental results indicate that the measurement accuracy is less than 0.16% under normal experimental environment.

Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

Carrier squeezing interferometry with π/4 phase shift: phase extraction in the presence of multi-beam interference

Multi-beam interference exists in testing high-reflectivity surfaces with a Fizeau interferometer. In this paper, the multi-beam interference intensity was estimated as the sum of the first six order harmonics using the Fourier series expansion. Then, by adopting carrier squeezing interferometry with a π/4 phase shift, an algorithm was proposed to extract the phase from multi-beam interferograms. To ensure the separation of the lobes of phase-shift errors and the phase in the frequency domain, conditions of the necessary linear carrier in the proposed algorithm were derived. Simulation results indicated that the phase retrieving precision is better than PV 0.008λ and RMS 0.001λ, even when the reflection coefficient of the test surface is as high as 0.9 and the phase shift varies within π/4±π/20. Compared with the other algorithms, the proposed algorithm for multi-beam interference was validated by its good performance in the experiments, especially when the phase-shift error exists.

Applying slope constrained Qbfs aspheres for asphericity redistribution in the design of infrared transmission spheres.

An achromatic transmission sphere (TS) must be provided as an accessory for an infrared interferometer with a broadband wavelength channel. When designing an F/0.75 infrared TS using the one-asphere model of the two-lens-only package, the first surface was optimized as a strong asphere beyond our current testing ability, using either a slope-unconstrained Qbfs asphere equal to a conventional even asphere, or using a slope-constrained Qbfs asphere. We applied the asphere-splitting theory to redistribute the asphericity. Adopting only the slope-constrained Qbfs asphere, the strong asphere was successfully split into two weaker aspheres within our testing ability, getting a more manufacturable two-asphere TS solution. Besides, the TS solution with two weaker aspheres presented the same sensitivity to assembly errors compared to the solution with one strong asphere, as demonstrated by the analysis of the wavefront aberrations of the TS solutions. This phenomenon coincides well with the asphere-splitting theory, given that the total asphericity remains almost equivalent between the two solutions.

Generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surfaces with global optimized wavefront reconstruction algorithm

In this paper, a generalized shift-rotation absolute measurement method is proposed to measure the absolute surface shape of high-numerical-aperture spherical surfaces. Based on the wavefront difference method, the high order misalignment aberrations can be removed from the measurements. Our generalized shift-rotation absolute measurement process only needs one rotational measurement position and one translational measurement position. A wavefront reconstruction method based on the self-adaptive differential evolution algorithm is proposed to calculate the Zernike polynomials coefficient ai of the absolute surface shape Wtest(x,y), the rotation angle Δθ, the translation δx along the x axis, and the translation δy along the y axis. The translation error and rotation error in other absolute measurement methods are avoided using our generalized shift-rotation absolute measurement method. Experimental absolute results of the test surface and reference surface are given and the difference of reference surface shapes between two testings in experiments is 0.12 nm root mean square.

Large radius of curvature measurement based on virtual quadratic Newton rings phase-shifting moiré-fringes measurement method in a nonnull interferometer

We have proposed a virtual quadratic Newton rings phase-shifting moiré-fringes measurement method in a nonnull interferometer to measure the large radius of curvature for a spherical surface. In a quadratic polar coordinate system, linear carrier testing Newton rings interferogram and virtual Newton rings interferogram form the moiré fringes. It is possible to retrieve the wavefront difference data between the testing and standard spherical surface from the moiré fringes after low-pass filtering. Based on the wavefront difference data, we deduced a precise formula to calculate the radius of curvature in the quadratic polar coordinate system. We calculated the retrace error in the nonnull interferometer using the multi-configuration model of the nonnull interferometric system in ZEMAX. Our experimental results indicate that the measurement accuracy is better than 0.18% for a spherical mirror with a radius of curvature of 41,400 mm.

Alignment analyses of a galvanometer-based scanner in free-space Fourier domain optical coherence tomography.

Free-space Fourier domain optical coherence tomography is adopted for biomedical imaging with ultrahigh resolution, in which the setup consists of an interferometer and a spectrometer. Two-dimensional lateral sampling in the sample arm of the interferometer is achieved by using a galvanometer-based scanner. Optical path difference (OPD) drift in the full scan field of view is observed in the assembly process of the scanner. A galvo mirror mount offset with respect to the rotation axis is demonstrated as the derivation of this OPD drift by both geometric analyses and model building. Then, an iterative assembly process of the scanner is proposed with the OPD drift taken as the alignment criteria.