Test Wavefront Research Articles

Aspherical Surface Wavefront Testing Based on Multi-Directional Orthogonal Lateral Shearing Interferometry.

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.

Design and wavefront test method for telescope of space-based gravitational wave detection “Taiji” program

Design and wavefront test method for telescope of space-based gravitational wave detection “Taiji” program

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

Two photon sieves with different focal lengths were used to construct a radial shearing interferometer. A patch-based UNet approach was utilized to extract the central skeleton of interferogram, and then an iterative algorithm was applied for wavefront reconstruction. The transmitted wavefront of a window glass was measured with different detectors, and their results were consistent with the measurement of ZYGO interferometer. Compared to zone plate, photon sieves can suppress higher-order diffraction by reasonably designing the arrangement of small holes, thereby improving the signal-to-noise ratio of the interferogram. Moreover, photon sieves that is self-supporting and multifunctional by use of design and optimization has great application in wavefront sensing and interferometric imaging.

Absolute testing of rotationally symmetric surfaces with computer-generated holograms.

Extremely high accuracy is demanded for optics working at very short wavelength. Interferometric testing of optical aspheres or freeform surfaces requires null optics, typically computer-generated holograms (CGHs), to balance the wave aberrations. The measurement uncertainty is primarily limited by the accuracy of the test wavefront, which is predominantly influenced by the CGH and the interferometer optics. Absolute testing is fundamental to achieving accuracy much higher than that of the test wavefront through error separation. This paper presents a method for absolute testing of rotationally symmetric surfaces with CGH null optics. The basic assumption is that the off-axis hologram fabricated by raster scanning beam writing has negligible error of rotationally symmetric component due to pattern error of the CGH. Consequently, the wavefront error contributed by the CGH and the transmission flat can be completely separated from the absolute surface shape by combining the N-position method and the shift-rotation method. A theoretical model for absolute testing is proposed under the assumption. Experimental cross test is then presented to validate the method with sub-nanometer uncertainty. The assumption is further confirmed by characterizing the fabrication error of the hologram structures using a white light interferometer. Finally, the effect of noise, translation error, rotation error and eccentricity of rotation on the absolute testing is analyzed.

Error correction analysis of wavefront testing in quadriwave lateral shearing interferometry

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 1.3789×10−4λ 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.

Wavefront measurements using the Ronchi test for high-NA lithography projection lenses.

The Ronchi test is widely used for wavefront measurements in advanced lithography tools, and a physical optics explanation of the Ronchi test based on scalar diffraction theory can be found in numerous publications. However, for high-numerical aperture (high-NA) lithography projection lenses, the vector nature of light should be considered when performing wavefront measurements, especially the effect of polarization aberrations on the wavefront test results. In this paper, a vector model for describing shearing interferometry for high-NA lithography projection lenses is established. In addition to considering the vector nature of light, the vector model also calculates the Ronchigram on the screen of a detector at any distance from a diffraction grating, as opposed to the distance restriction for the Fraunhofer diffraction approximation used by the existing methods. Using the developed mathematical model of the Ronchi test, the Ronchigrams of high-NA lithography projection lenses under non-polarized illumination are simulated, and the effect of the distance between the diffraction grating and the detection screen on the wavefront measurement accuracy are discussed.

Microlenses Fabricated by Two‐Photon Laser Polymerization for Cell Imaging with Non‐Linear Excitation Microscopy

AbstractNon‐linear excitation microscopy offers several advantages for in‐vivo imaging compared to conventional confocal techniques. However, tissue penetration can still be an issue due to scattering and spherical aberrations induced on focused beams by the tissue. The use of low numerical aperture objectives to pass through the outer layers of the skin, together with high dioptric power microlenses implanted in‐vivo close to the observation volume, can be beneficial to the reduction of optical aberrations. Here, Fibroblast cell culture plano‐convex microlenses to be used for non‐linear imaging of biological tissue are developed and tested. The microlenses can be used as single lenses or multiplexed in an array. A thorough test of the lenses wavefront is reported together with the modulation transfer function and wavefront profile. Magnified fluorescence images can be retrieved through the microlenses coupled to commercial confocal and two‐photon excitation scanning microscopes. The signal‐to‐noise ratio of the images is not substantially affected by the use of the microlenses and the magnification can be adjusted by changing the relative position of the microlens array to the microscope objective and the immersion medium. These results are opening the way to the application of implanted micro‐optics for optical in‐vivo inspection of biological processes.

Generation of Stokes singularities using polarization lateral shear interferometer.

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.

Calibration of geometrical aberration in transmitted wavefront testing of refractive optics with deflectometry.

Deflectometry, with its noticeable advantages such as simple structure, large dynamic range, and high accuracy comparable to interferometry, has been one of the powerful metrological techniques for optical surfaces in recent years. In the "null" deflectometric transmitted wavefront testing of refractive optics, ray tracing of the test system model is required, in which both the miscalibration of system geometrical parameters and optical tolerances on tested optics could introduce significant geometrical aberrations in the testing results. In this paper, the geometrical aberration introduced by a system modeling error in the transmitted wavefront testing is discussed. Besides, a calibration method based on polynomial optimization of geometrical aberration is presented for the geometrical aberration calibration. Both simulation and experiment have been performed to validate the feasibility of the proposed calibration method. The proposed method can calibrate the optical tolerances on tested optics effectively, and it is feasible even with a large geometric error, providing a viable way to address the uncertainty in system modeling in transmitted wavefront testing of freeform refractive optics with large dynamic range.

Comparison of processing speed of typical wavefront reconstruction methods for lateral shearing interferometry.

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.

A Toolbox for Isophase-Curvature Guided Computation of Metrology Hologram.

We describe the algorithmic foundations of an open-source numerical toolbox, written in the Octave language, for the creation of computer-generated binary and multi-level holograms used in interferometric form error measurements of complex aspheric and free-form precision surfaces and wavefronts. In a typical measurement setup for this type of surface, a hologram is used to generate a test wavefront that has the design shape of the surface, which is then compared to a fabricated part using an imaging laser interferometer. The optical function of the hologram in the measurement is generally modeled with optical ray-tracing software and it can be encapsulated by a scalar optical phase function φ : R2 →R. The toolbox converts phase functions into equivalent binary holograms that generate the desired test wavefronts for an interferometric form error measurement. The algorithms in this toolbox take advantage of the relationship between the local properties of phase functions and the local geometry (curvature) of isophase lines. It forms the core of an effcient algorithm for the computation of optical holograms. Holograms are created in a format that can be processed by most laser-or e-beam lithography systems. While the toolbox is chiefy aimed at the creation of hologram layouts needed for measurements of precision surfaces and wavefronts, we show that the isophase-following algorithm is easily extended to phase functions with singularities and discontinuities. Such phase functions result in holograms with zone bifurcation and they can be used to generate helical wavefronts. Light beams with helical wavefronts have applications beyond surface and wavefront metrology. The toolbox also includes a family of functions for the effcient estimation and evaluation of Zernike polynomials, which are widely used in optical applications.

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.

Zernike monomials in wide field of view optical designs.

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.

Single-shot multi-planar wave-front measurement with multi-focal Fibonacci sieves

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.

Simultaneous reconstruction of phase and amplitude for wavefront measurements based on nonlinear optimization algorithms.

The non-perfect determined amplitude distribution in the pupil would affect the convergence speed and accuracy of phase retrieval method, which depends on the amplitude of fields to reconstruct the phase. In this paper, we propose two kinds of phase retrieval methods based on hybrid point-polynomial and point-by-point nonlinear optimization algorithms to reconstruct simultaneously the amplitude and phase of the wavefront. Intensity quantized errors are avoided by using modified first derivatives. For simple and general wavefront testing, the accuracy and robustness of proposed algorithms are verified both numerically and experimentally.

Relationship between shear ratio and reconstruction accuracy in lateral shearing interferometry

The shear ratio is an essential parameter of the lateral shearing interferometry system that affects dynamic range, signal-to-noise ratio, and accuracy. Some qualitative conclusions about the shear ratio’s influence can be found in existing reports, but no systematical analysis or quantified universal conclusion has been given. Using the sheared phase to reconstruct wavefront is the unique feature of the shearing interferometry, but it also affects the measurement accuracy significantly. The wavefront reconstruction methods can be divided into zonal and modal methods. We use the FFT reconstruction method and the least-square reconstruction method as the typical methods of modal and zonal methods to analyze the source of the reconstruction error and its relationship with the shear ratio. We conclude that the FFT reconstruction method and the least-square reconstruction method are equivalent to each other. The lost information during the shearing corresponds to some incalculable spectral values in the frequency domain and the pistons of different subgrids in the spatial domain. More spectral information will be lost if the shear ratio increased, which leads to more significant reconstruction errors. When the shear ratios are 1.6% and 6.3%, the corresponding relative reconstruction errors are about 1% and 1%, respectively. The error will be lower if the test wavefront can be represented by a limited amount of basis functions.

Snapshot phase-shifting lateral shearing interferometer

We propose a novel, low-cost snapshot phase-shifting lateral shearing interferometer (SPLSI) with full-range continuously adjustable lateral shear ratio. A tunable shear plate, consisting of a polarizing beam splitter (PBS) plate and a reflective mirror, separates the orthogonally polarization states of the test wavefront into original and sheared wavefronts. The lateral shear ratio can be tuned by adjusting the spacing between PBS plate and the mirror. A pixelated polarization camera simultaneously captures four π/2 phase-shifted interferograms for phase-shifting interferometric measurement. In addition to tunable shear ratio and snapshot capture of phase-shifted interferograms, one more key feature is that SPLSI can measure the wavefront of low-coherence light. The feasibility of the proposed system is demonstrated experimentally. The proposed system provides a simple, compact, and fast way to achieve the instantaneous phase-shifting lateral shearing interferometric testing of wavefronts with minimal impact from environmental disturbance.

A new method to evaluate the effectiveness of impulse voltage for detecting insulation defects in GIS equipment

Lightning impulse (LI) test is an effective means to detect insulation defects in gas-insulated metal-enclosed switchgear (GIS) equipment. However, the LI test wavefront time Tf will exceed the standard value of 1.2 μs ± 30% for large test load capacitance. Whether extending the Tf will influence defect detection effectiveness should be a concern. In this study, a generating system of impulses with different wavefront parameters was established. The insulation characteristics of a coaxial cylinder structure with an SF6 gas gap with conductive protrusion in a bus under impulses with different wavefront parameters were studied. Experimental results show that the voltage–time curve of the gap shows a bathtub trend that has a flat part in the range of 1 μs to 5 μs. With a Tf increase, the 50% breakdown voltage increases. Whether extending the Tf will influence defect detection effectiveness is analyzed from the standpoint of the different meaning of the breakdown voltage in voltage-time curve and 50% breakdown voltage. The essence of detection effectiveness under impulse voltages with different wavefront parameters is a comparison of the probability of detecting the same insulation defect. Based on the discharge probability distribution, a new quantitative method to evaluate the insulation defect detection effectiveness of impulse voltages for GIS equipment is proposed. The method, based on the 90% discharge voltage value of an insulation defect under a standard LI, normalizes the discharge probability under different wavefront parameters.

Misalignment correction for free-form surface in non-null interferometric testing

Abstract Unlike rotational symmetric surfaces, free-form surfaces have sophisticated degrees of freedom, therefore, are more difficult to be aligned. In this work, a practical and accurate correction method is proposed for the misalignment removal of the free-form surface in a non-null interferometric testing system based on computer modeling. The misalignment aberrations, introduced by axial location error, rotation error and non-axial attitude error, are modeled and corrected step by step. The axial and non-axial positions of the free-form surface in the computer model are adjusted from the ideal position to the misaligned one, matching the actual positions in experiment for the correction. Since the modeled position are tuned to be consistent with the actual one, all the misalignment aberrations can be removed from the test wavefront with the reconstruction algorithm. Computer simulations are displayed to verify the accuracy of the proposed method. Experimental results after alignment show basic consistency with the results of Taylor Hobson Profilometer, in which the peak-to-valley (PV) value error of the profile line is better than 1/30 λ .

High resolution full field wavefront measurement of a misaligned miniature lens.

Compared with the industry standard MTF consequential testing result, the full field transmitted wavefront testing is more analytical for field aberration analysis. A novel wavefront measuring device specialized for the miniature lens testing application is developed to measure the full field aberration in a high resolution of 35x36 radial-azimuthal fields. The device adapts the high dynamic range Shack-Hartman wavefront sensor to minimize the alignment uncertainty induced from collimator under high field angle. The plane symmetrical aberration due to elements misalignment is identified and quantified throughout the measured field. The field constant coma and field linear astigmatism contributes most aberration errors to the edge of the field as expected. Through the field dependent aberration analysis. This device proves that the miniature lens image quality near the edge of the field is practically limited by the misalignment of the optical elements.