Wavefront Testing Research Articles (Page 1)

To overcome the limitations of phase sampling points in testing aspherical surface wavefronts using traditional interferometers, we propose a high-spatial-resolution method based on multi-directional orthogonal lateral shearing interferometry. In this study, we provide a detailed description of the methodology, which includes the theoretical foundations and experimental setup, along with the results from simulations and experiments. By establishing a relational model between the multi-directional differential wavefront and differential Zernike polynomials, we demonstrate high-spatial-resolution wavefront reconstruction using multi-directional orthogonal lateral shearing interferometry. Theoretical calculations and simulations of aspherical surface wavefront testing are followed by experimental verification on an aspherical surface with a known asphericity. Comparing the measurement results with those from the LuphoScan profilometer, we achieve a relative measurement error with an RMS precision better than λ/100.

The Taiji program aims to achieve space gravitational wave interferometry measurement through quasi-equilateral triangle satellite groups and explore major phenomena in the universe. We detail the design and wavefront test of the space gravitational wave detection telescope, which is one of the core payloads of the Taiji program. It features an all-glass-ceramics structure aiming for an optical path stability of ≤1 pm/Hz1/2. The telescope, with a 400-mm aperture and an off-axis reflective optical system, comprises two large glass-ceramics components, easing assembly and connectivity challenges. The weight of the telescope is 19 kg, which meets the ≤20 kg system requirement. Ground wavefront test methods were optimized using a “hanging-ear” support approach. Under 1 g gravity, the wavefront error of the system is 0.018λ [root mean square (RMS), λ=1064 nm], and considering processing, assembly, and in-orbit temperature conditions, the space environment system wavefront distortion RMS is ∼0.045λ (λ/22), meeting the RMS ≤λ/20 requirement. This design and testing method lay the groundwork for subsequent assembly and in-orbit application of the telescope, providing key technical support for the implementation of the Taiji program.

Transmitted wavefront testing of window glass with confocal photon-sieves radial-shearing interferometry based on deep learning

Quadriwave lateral shearing interferometry (QWLSI) is based on double birefringent crystals of a beam displacer (DBCs-BD), which can generate the lateral shearing interference wavefront of four beams of overlapped replicas in the DBCs-BD orthogonal directions. When the replica waves are overlapped incident to the analyzer and the direction of the transmission axis is set as 45° or 135°, the QWLSI's polarization interferogram can be obtained. This paper deduces the principle of QWLSI based on the DBCs-BD and presents the analysis of orthogonal error influence based on the DBCs-BD and the phase retrieval error of QWLSI when shear displacement by tilted incident on the DBCs-BD. In our investigation, we have established the correction range of the PBD's orthogonal angle error is within -0.5∘-0.5∘, the maximum error in PV is 0.0012λ, and the maximum error in RMS is 1.3789×10-4 λ in wavefront reconstruction. Moreover, when the testing light tilts to incident on the PBD in the range of -0.4∘-0.4∘, correction of the shear distance is used for wavefront reconstruction to achieve a high-precision wavefront testing result. Finally, the experiment shows that QWLSI based on the DBCs-BD exhibits feasibility and high precision.

Lateral shear interferometer, being a self-referenced interferometer, has proven to be an important tool in scalar optics. Here we employ a vectorial counterpart - polarization lateral shear interferometer, in which the two interfering beams apart from being derived from the test wavefront, are in orthogonal states of polarization. Therefore when the test wavefront has spatially varying phase gradient across the beam cross-section, the resulting shearogram produces polarization fringes instead of intensity fringes. Further, the shearogram becomes inhomogeneously polarized. This polarization lateral shear interferometer may have potential uses in metrology, but in this article we demonstrate the ability of the interferometer in the generation of all Stokes singularities in the single beam by launching a phase singular beam into it. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results and they are discussed in the article.

Lateral shearing interferometry is widely applied in wavefront sensing, optical components testing, and defect inspection. The procedure of reconstructing the wavefront is the most specific difference between lateral shearing interferometry and other classical methods such as the Fizeau and Twyman interferometers. The speed and accuracy are two main features to evaluate the performance of one wavefront reconstruction method. In this work, optimized procedures for three typical wavefront reconstruction methods-the iterative FFT wavefront reconstruction method (FFT method), the partial differential least-squares method (LSQ method), and the difference Zernike polynomial fitting method (DZF method)-are designed. The calculation speeds of the three wavefront reconstruction methods are evaluated with different GPUs and CPUs. According to the test results, the DZF method is the fastest method both in the GPUs and CPUs. The shortest processing times of the DZF, FFT, and LSQ methods are 100, 449, and 494 ms, respectively, with the wavefront size of 1024×1024pixels. The calculation speeds of the FFT method and the LSQ method are similar in the CPUs, and the FFT method is faster in the GPUs. The relationship between the consumed time and the wavefront size is an exponential function in the CPUs and a power function in the GPUs. Generally speaking, GPUs' processing speeds are faster than CPUs'. But CPUs can be faster than GPUs when the test wavefront sizes are smaller than 64×64pixels. Besides, the differences between the consumed times of different CPUs are relatively smaller than those of the GPUs.

Zernike polynomials are universal in optical modeling and testing of wavefronts; however, their polynomial behavior can cause a misinterpretation of individual aberrations. Wavefront profiles described by Zernike polynomials contain multiple terms with different orders of pupil radius (ρ). Zernike polynomials are a sum of high and low orders of ρ to minimize the RMS wavefront error and to preserve orthogonality. Since the low-order polynomials are still contained in the net Zernike sum, there is redundancy in individual monomials. Monomial aberrations, also known as Seidel or primary aberrations, are useful in studying an optical design's complexity, alignment, and field behavior. Zernike polynomial aberrations reported by optical design software are not indicative of individual (monomial) aberrations in wide field of view designs since the low-order polynomials are contaminated by higher order terms. An aberration node is the field location where an individual (monomial) aberration is zero. In this paper, a matrix method is shown to calculate the individual monomial aberrations given the set of Zernike polynomials. Monomial aberrations plotted as a function of field angle (H) indicate the field order (Hn) and the location of true aberration nodes. Contrarily, Zernike polynomial versus field (ZvF) plots can indicate false aberration nodes, due to the polynomial mixing of high- and low-order terms. Accurate knowledge of the monomial aberration nodes, converted from Zernike polynomials, provides the link between a ray-trace model or lab wavefront measurement and nodal aberration theory (NAT). This method is applied to two different optical designs: (1) 120° circular FOV fish-eye lens and (2) 120∘×4∘ rectangular FOV, off-axis, freeform four-mirror design.

Wave-front measurement based on coherent diffraction imaging (CDI) is a promising method for measuring wave-front aberrations, which has wide applications ranging from optical testing to adaptive optics. This study proposes a single-shot multi-planar wave-front measurement with multi-focal Fibonacci sieves to reconstruct the wave-front distribution of small transmissive objects. A Fibonacci sieve was designed to simultaneously capture multi-planar diffraction patterns at a single recording plane; thus, a multi-planar CDI algorithm can be used to reconstruct the test wave-front by a set of extracted sub-graphs. Its feasibility was proved in the optical region experimentally. Since diffractive optical elements used in the experiment are amplitude-only elements, the proposed wave-front measurement method opens up the possibility of practical real-time and on-line wave-front measurement ranging from x rays to terahertz.

Wavefront reconstruction for multi-lateral shearing interferometry using difference Zernike polynomials fitting

In order to measure the instantaneous wavefront of large aperture optical elements, a method based on the structure of oblique incidence of reflective shearing point diffraction interferometer is proposed. A lateral displacement between the reference wavefront and the test wavefront is formed after passing this structure. The shear of two beams introduces linear spatial carrier frequency to the point diffraction interferogram. After receiving a good contrast interferogram, wavefront phase is retrieved by Fourier transform (FT) automatically to realize the dynamic measurement of instantaneous wavefront. The optical path is up to 20 m, so the air current is a significance factor to the result. Besides, because of the air current, the system itself can be seen as a instantaneous wavefront happening and measurement of large aperture optical elements. The results indicate that the root mean square value is in accord with that acquired by SID4 wavefront sensor (less than 1/50λ), so about the repeated accuracy. The method proposed can be applied in high resolution and accuracy measurement of instantaneous wavefront.

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

Phase-shifting Jamin double-shearing interferometer based on polarization beam splitting film

In this paper, we describe a zonal wavefront sensor in which the photodetector array can have a smaller number of rows. The test wavefront is incident on a two-dimensional array of diffraction gratings followed by a single focusing lens. The periodicity and the orientation of the grating rulings of each grating can be chosen such that the +1 order beam from the gratings forms an array of focal spots in the detector plane. We show that by using a square array of zones, it is possible to generate an array of +1 order focal spots having a smaller number of rows, thus reducing the height of the required detector array. The phase profile of the test wavefront can be estimated by measuring the displacements of the +1 order focal spots for the test wavefront relative to the +1 order focal spots for a plane reference wavefront. The narrower width of the photodetector array can offer several advantages, such as a faster frame rate of the wavefront sensor, a reduced amount of cross talk between the nearby detector zones, and a decrease in the maximum thermal noise. We also present experimental results of a proof-of-concept experimental arrangement using the proposed wavefront sensing scheme.

I describe and demonstrate a rapid and simple technique for calibrating spherical reference surfaces in Fizeau interferometry. A flat mirror at the focus of the test wavefront and a partial occlusion of the test beam enable a double-pass measurement of one half of the surface against the other, which gives an error map for half the aperture. Subsequent rotations of the beam stop and stitching together of several measurements yields the full-aperture calibration file. The method always produces point-symmetrical surface maps and is therefore not exact. However, most actual reference surfaces are approximately point symmetric and therefore the trade-off of accuracy for expediency is reasonable in practice.

To investigate analytically whether higher-order wavefront errors comprising combinations of trefoil along 30 degrees (trefoil30), vertical coma, and spherical aberration could provide cues to sphero-cylindrical refractive error development. A total of 25 test wavefronts, subdivided into five different types and five levels of higher-order root mean square errors (HO-RMS), were created for the study. One type contained spherical aberration only, producing HO-RMS levels between 0.1 and 0.5 microm. Four wavefront types contained coma, trefoil, and spherical aberration of various sign combinations also producing HO-RMS levels between 0.1 and 0.5 microm. From the 25 wavefronts, refractive power maps were created and 2025 different sphero-cylindrical combinations were added to each refractive power map. For each sphero-cylinder combination, the visual Strehl ratio based on the modulation transfer function (VSMTF) was calculated. Retinal images and refractive power histograms were calculated for the refractive power maps corresponding to the peak of the VSMTF. Spherical aberration affected the best focal plane thereby inducing spherical or defocus cues. The VSMTF produced by vertical coma and trefoil30, in combination with spherical aberration, could be improved with sphero-cylinders of various magnitudes and directions (i.e., with-the-rule, against-the rule, myopic astigmatism, or hyperopic astigmatism). Clinical significance of sphero-cylinders (i.e., >or=0.25 D) was reached at HO-RMS levels between 0.2 and 0.3 microm for a 5-mm pupil zone. In the context of compensatory blur driven eye growth, commonly occurring combinations of the three considered higher-order aberrations have the potential to produce cues to eye growth resulting in myopia and with-the-rule astigmatism.

Interferometry at grazing incidence allows one to perform macroscopic shape tests of rough surfaces with submicrometer precision. The highest measurement accuracy is achieved with a null test: the object wave front is adapted to the ideal surface under test through the use of diffractive optical elements. Deviations between the ideal and the real surface shape result in characteristic phase distributions at the interferometer output. Rod objects with rather arbitrary cross-section profiles are especially suitable for this type of measurement. The design of appropriate test wave fronts can be carried out by means of geometrical considerations. Expressions describing aberrations that are due to misalignments of the work piece can easily be derived. Misalignment aberrations and genuine surface deviations are separated by least-squares fitting.

A radial shearing interferometer with two equispaced circular gratings is described for obtaining a constant radial displacement of the test wavefront. The procedure for the fringe analysis is shown by applying the Fourier transform method to the circular fringe pattern for precisely obtaining the differentiation of the test wavefront. A novel method is also shown for holographically making the equispaced circular gratings by using a cylindrical mirror and a collimating lens. It is shown that the deflection and misalignment of the gratings will not appear in the interferogram by using the holographic production method.