Xi Hou

Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method

Based on our previously reported annular subaperture reconstruction algorithm with Zernike annular polynomials and matrix method, we provide an experimental demonstration by testing a parabolic mirror with the complementary annular subaperture interferometric method. By comparing the results of annular subaperture method with that of the classical auto-collimation method, it is shown that the reconstruction results are in good agreement with the auto-collimation measurement. In addition, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients in our algorithm, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. It is believable that the reported method can be extended to test the surface shape of some large concave aspheric mirrors with acceptable accuracy.

Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces

We propose a more accurate and efficient reconstruction method used in testing large aspheric surfaces with annular subaperture interferometry. By the introduction of the Zernike annular polynomials that are orthogonal over the annular region, the method proposed here eliminates the coupling problem in the earlier reconstruction algorithm based on Zernike circle polynomials. Because of the complexity of recurrence definition of Zernike annular polynomials, a general symbol representation of that in a computing program is established. The program implementation for the method is provided in detail. The performance of the reconstruction algorithm is evaluated in some pertinent cases, such as different random noise levels, different subaperture configurations, and misalignments.

Method to test rotationally asymmetric surface deviation with high accuracy

We have proposed a new absolute method to test rotationally asymmetric surface deviation. Relying on the high accuracy of Zernike polynomial fitting with least-squares algorithm for the low-frequency component and preserving the high-frequency component with the averaging method, the new method can guarantee the high accuracy of the measurement result with fewer rotational measurements compared to the traditional multiangle averaging method. It realizes a balance between the accuracy and efficiency of the measurements. It has been verified by experiments; the root mean square (rms) of residual figure between the two methods is ∼0.6 nm. Meanwhile, the new method can suppress environmental noise introduced in measurement results well.

Absolute measurement of optical flats based on basic iterative methods

Generalized basic iterative methods for absolute measurement of optical flats are presented. They are based on iterative scheme and can be classified into block Jacobi Successive Over-relaxation (SOR) method and block SOR method. Both methods are effective for solving the three-flat problem with pixel-level spatial resolution, without the usage of a fitting procedure. Compared to the block Jacobi SOR method, the block SOR method with an inexpensive choosing of relaxation factor (such as ω = 1.5) converges much faster and saves more computational costs and memory space without reducing accuracy. It has been proved by both simulation results and experimental results. The proposed basic iterative methods are generalized; can correctly reconstruct absolute figures with pixel-level spatial resolution; are easy to understand and implement; and computationally efficient.

Absolute calibration of a spherical reference surface for a Fizeau interferometer with the shift-rotation method of iterative algorithm

Abstract. We present an absolute testing of a spherical reference surface of a Fizeau interferometer with the method of shift-rotation, which relies on the decomposition of the surface deviation into rotationally asymmetric and symmetric components. With a new procedure to measure the rotationally asymmetric component and an iterative algorithm to calculate the symmetric component, the method can calibrate the spherical reference surface with high accuracy. An experiment is presented to verify the validity of the absolute testing method. The reference surface deviation measured with the absolute testing method agrees well with that of random-ball-averaging testing method; the root mean square of the residual figure between them is ∼0.8  nm.

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.

Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering

This paper presents a novel method to extract the phase shift and phase distribution from two interferograms simultaneously. By employing Hilbert-Huang transform based prefiltering, the background intensities and modulation amplitudes of the two interferograms are suppressed and normalized respectively. With the addition and subtraction operation of the two prefiltered interferograms, two parametric equations are achieved which can be regarded as the complex harmonic motion of the Lissajous figure. The phase of the Lissajous figure can be directly demodulated by the ellipse fitting algorithm. Apart from the advantages of other well-known two-step phase demodulation algorithms, i.e., high accuracy and efficiency of the Gram-Schmidt orthonormalization (GS) method and the less stringent requirement concerning the fringe number in the extreme value of interference (EVI) method, proposed Lissajous figure and ellipse fitting (LEF) approach has another bonus related to its robustness to the fluctuations of the fringe patterns noise, background intensity and modulation amplitude. Simulations demonstrate the outstanding performance of the proposed method, and experiments further corroborate its effectiveness.

Optimized absolute testing method of shift-rotation

The absolute testing method of shift-rotation that combines the traditional N-position method and Zernike polynomial fitting has been commonly employed in surface metrology. It preserves the high spatial frequency of the surface deviation with the N-position method to obtain the rotationally asymmetric surface deviation, but it suffers the kNθ order angular terms errors missed by the N-position method when it calculates the rotationally symmetric surface deviation with the Zernike rotationally symmetric polynomial fitting method. An optimized absolute testing method of shift-rotation is presented in this paper. It considers the missing kNθ order errors when the equations of the rotationally symmetric surface deviation are solved. As a result, it is more accurate than the traditional method. Experimental absolute results of spherical surfaces are given.

Annular subaperture interferometric testing technique for large aspheric surfaces

A new method to test large aspheric surface by using Annular Sub-aperture Interferometric Testing (ASIT) is described in this paper. It solves the two main difficulties, which are manufacture difficulty and high cost of auxiliary elements (optics compensator, CGH, large reflector) used generally in testing the large aspheric mirror. The method reduces departure between the reference wavefront and tested wavefront to the measurement range of the interferometer by testing annular subaperture of aspheric surface, which makes reference wavefronts with different curvature radius match corresponding annular subaperture, then sews all sub-aperture datum together with a suitable algorithm to get the whole information of the surface. A stitching mathematics model is provided. The feasibility of the technique has been demonstrated by comparing test results from an autocollimation test and ASIT of an 152-mm diameter paraboloidal mirror. This method extends the dynamic range of interferometric measurements, by which the large, fast aspheric surface can be directly tested without additional optical elements.

Removal of surface figure deformation due to gravity in optical test

For optical surface measurement with higher accuracy, a simple method for removing surface figure deformation due to gravity was adopted and its principle research and error analysis were conducted. Based on optical flat supporting case, properties of gravitational deformation of the tested sample was analyzed by finite element method (FEM). In order to verify high accuracy of FEM analysis, a method which could remove the original surface deviation of the reference and the tested from the test results was adopted. According to the method, difference between theoretical and experimental results represented by Root-Mean-Square (RMS) value was only 0.404 nm. The result shown that the FEM analysis is accurate enough and the surface figure deformation due to gravity could be removed from test results efficiently. The method discussed here could benefit the high accuracy optical measurement.