Adaptive wavefront interferometry for unknown free-form surfaces

Fast, intelligent and high-precision adaptive null interferometry for optical freeform surfaces by backpropagation

In the past 10 years, adaptive wavefront interferometry (AWI) has been employed for measuring freeform surface profiles. However, existing AWI techniques relying on stepwise and model-free stochastic optimizations have resulted in inefficient tests. To address these issues, deterministic adaptive wavefront interferometry (DAWI) is firstly introduced in this paper based on backpropagation (BP), which employs a loss function to simultaneously reconstruct and sparsify initial incomplete interferometric fringes until they are nulled. Each iteration of BP requires two phase shifts. Through simulations, we have verified that freeform wavefront error with a peak-to-valley (PV) of up to 168 λ can be fully compensated in tens of iterations using a 1024 × 1024 pixel area of a liquid-crystal spatial light modulator. In experiments, we accomplished a null test of a freeform surface with 80% missing interference fringes in 39 iterations, resulting in a surface profile error PV of 66.22 λ and measurement error better than λ/4. The DAWI has at least 20 times fewer iterations in fringe reconstruction than the 3-step AWI methods, and nearly an order of magnitude fewer iterations in the whole process, paving the way for significantly enhanced efficiency, generality and precision in freeform surface adaptive interferometry.

Fast recovery of sparse fringes in unknown freeform surface interferometry

In the adaptive freeform surface interferometer, the adaptive algorithms were equipped to find the required aberration compensation, making interferogram with dark areas (incomplete interferogram) sparse. However, traditional blind search-based algorithms are limited by convergence rate, time consumption, and convenience. As an alternative, we propose an intelligent approach composed of deep learning and ray tracing technology, which can recover sparse fringes from the incomplete interferogram without iterations. Simulations show that the proposed method has only a few seconds time cost with the failure rate less than 4‰. At the same time, the proposed method is easy to perform because it does not require the manual intervention of internal parameters before execution as in traditional algorithms. Finally, the feasibility of the proposed method was validated in the experiment. We believe that this approach is much more promising in the future.

Discontinuous phase unwrapping based on the minimization of Zernike gradient polynomial residual

In many certain optical metrology cases, the pupil is usually divided into multiple connected domains by secondary mirror spiders, thus producing segment piston errors and leaving a false phase unwrapping result. In this paper, a method based on minimization of Zernike gradient polynomial residual (MZGR) is proposed to estimate segment piston errors and correct erroneous phase unwrapping results. Simulations and experiments demonstrated that this method can obtain the segment piston errors precisely under complex aberration forms and varied obscurations, indicating reliable practicality. Comparison to the 4D commercial solution, the RMS (root-mean-square) of the residual decreased from 0.154 λ to 0.020 λ.

Advanced Optical Wavefront Technologies to Improve Patient Quality of Vision and Meet Clinical Requests

Adaptive optics (AO) is employed for the continuous measurement and correction of ocular aberrations. Human eye refractive errors (lower-order aberrations such as myopia and astigmatism) are corrected with contact lenses and excimer laser surgery. Under twilight vision conditions, when the pupil of the human eye dilates to 5-7 mm in diameter, higher-order aberrations affect the visual acuity. The combined use of wavefront (WF) technology and AO systems allows the pre-operative evaluation of refractive surgical procedures to compensate for the higher-order optical aberrations of the human eye, guiding the surgeon in choosing the procedure parameters. Here, we report a brief history of AO, starting from the description of the Shack-Hartmann method, which allowed the first in vivo measurement of the eye's wave aberration, the wavefront sensing technologies (WSTs), and their principles. Then, the limitations of the ocular wavefront ascribed to the IOL polymeric materials and design, as well as future perspectives on improving patient vision quality and meeting clinical requests, are described.

Implementation of a null test for freeform optics using a high-definition spatial light modulator

We report the implementation of an interferometric null test using a high-definition spatial light modulator (SLM) as a reconfigurable alternative to a computer-generated hologram. We detail the alignment process chain, including novel techniques using the SLM to project alignment fiducials on the test part. To validate the alignment protocol, we measure a mild off-axis conic with the SLM-based system and cross-validate with conventional interferometry within 30 nm root-mean-square (RMS) surface figure. Finally, we report the null test of a 65 mm clear aperture concave freeform with 91 μm peak-valley sag departure from the base sphere. The measured surface figure of the freeform is within 40 nm RMS compared to the measurement with a commercial metrology instrument.

Adam SPGD algorithm in freeform surface in-process interferometry

The adaptive interferometer has been recently proposed to realize the metrology of unknown freeform surfaces with several restructured algorithms for feedback control. The adaptive moment estimation (Adam) stochastic parallel gradient descent (SPGD) algorithm is employed in this paper for fringes release. The proposed algorithm makes considerable progress in relieving conflict of the convergence rate, speed, and parameters intervention. Simulations and experiments show its 37% time saving and 99% convergence rate, with arbitrarily configured parameter increment, compared with the SPGD algorithm. It would have great potential in in-process tests in freeform surface fabrication or large-volume testing.

Phase response measurement of spatial light modulators based on a Shack-Hartmann wavefront sensor

It is known that the phase response of spatial light modulators (SLMs) measured by double-beam interferometers is sensitive to mechanical and environmental disturbances. This paper proposes a Shack-Hartmann wavefront sensor (SHWS) method to measure the phase response characteristics of the SLM. The results show that the phase modulation depth measured by the proposed method is 1.7581λ, and 1.7993λ by the Twyman-Green interferometer method. The difference in the phase modulation depth between the two methods is only 0.0412λ, and its relative error rate is 2.29%. It proves that the phase modulation accuracy obtained by the SHWS with lenslets of 73*73 used in this paper is equivalent to that of the Twyman-Green interferometer. Compared with the interferometer method, the SHWS method is simple, compact, and robust, has good real-time performance, and is relatively vibration insensitive. In the future, the SHWS method will play a more important role in the detection of the SLM's phase response.

Interferometric measurement of high-order aspheric surface parameter errors based on a virtual-real combination iterative algorithm

Aspheric surface parameters, including vertex radius of curvature, conic constant, and high-order aspheric coefficients, decide the optical properties of aspheric surfaces. The measurement of aspheric surface parameter errors (SPEs) is a substantial issue for the fabrication of aspheric surfaces. Interferometry is a mature high-accuracy method in aspheric surface figure error measurement, but challenges still exist in the measurement of SPEs for high-order aspheric surfaces or convex aspheric surfaces. We propose an interferometric measurement method for high-order aspheric SPEs based on a virtual-real combination iterative algorithm (VRCIA). We also propose a recommended measurement system including a partial compensation interferometer to obtain the partial compensated wavefront and a laser differential confocal system to obtain the best compensation distance for calculating SPEs through the VRCIA. A high-order convex aspheric surface is measured to demonstrate the feasibility of the method. The relative accuracy of vertex radius of curvature error, conic constant error and fourth-order aspheric coefficient error can reach 0.025%, 0.095% and 3.02%, respectively.

Simultaneous phase-shifting interferometer with a monitored spatial light modulator flexible reference mirror

A simultaneous phase-shifting interferometer with a monitored spatial light modulator (SLM) flexible reference mirror is proposed to balance the flexibility and accuracy of aspheric-surface in-process measurements. In this method, polarization simultaneous phase-shifting camera systems are applied to reduce the influence of environmental vibrations on the in-process measurements. An SLM reference mirror is employed to improve the flexibility of in-process measurements. A device is integrated to monitor the SLM surface in order to improve measurement accuracy caused by the spatial phase nonuniformity and modulation instability of the SLM. Thus, the SLM surface is monitored and the aspheric surface is measured simultaneously in only one interferometer, which presents the advantages of a compact structure and simple calibration. A flat acrylic mirror with an unknown surface figure error is measured by the proposed interferometer. Cross tests demonstrate the feasibility of this interferometer.

Compact adaptive interferometer for unknown freeform surfaces with large departure

The newest experimental validation report of the coverage for the rotationally non-symmetric departure of a freeform surface in adaptive interferometry is about 20 µm. A compact adaptive interferometer is introduced to test unknown freeform surfaces with larger departure. The cascaded DMs (woofer and tweeter) can effectively double the measurable rotationally non-symmetric departure, to ∼80 µm using current DM technology. With a constrained decoupling control algorithm, the woofer and tweeter can averagely share the aberrations without coupling. DM surface monitoring is addressed by a time-division-monitoring (TDM) technique, which avoids separate monitoring devices and configurations and thus makes a compact configuration. Measurements of two different surfaces are presented: a nearly flat freeform with ∼40 um departure, and an off-axis paraboloid with ∼50 um of asymmetric departure.

Surface Topography Measurement of Mirror-Finished Surfaces Using Fringe-Patterned Illumination

Mirror-finished surface products have a wide range of applications in different engineering industries, such as power generation, aerospace, semiconductors and optics. The surface topography of mirror-finished products is typically measured in a metrology laboratory, which is typically time consuming and cannot be integrated into the manufacturing process. To allow for in-situ product quality assurance and automatic tool change for manufacturing processes, a more accurate and responsive surface-measurement method is needed. For highly polished surfaces, a sub-micron surface fluctuation makes it possible to use light-scattering effects and image processing for surface texture analysis. A non-contact surface inspection system using a fringe-patterned illumination method is proposed in this paper. A predesigned pattern was projected onto the target surface, and its reflected image was captured by a camera. It was found that the surface parameters Sa and Sq, which are widely used to evaluate surface quality, are significantly correlated with luminous-intensity distribution. Another parameter, Str, which quantifies the uniformity of surface-texture directions due to polishing or grinding marks, was traditionally quantified after a complete-surface topographic measurement. In this research, a new approach is proposed to determine surface isotropy through a luminance-intensity distribution analysis. By rotating the test coupon, the variation of specular reflection showed correlation with the significance of surface-texture direction. The experimental results demonstrate that mirror-finished surfaces with a large deviation in luminance intensity across the pattern possess low Str values, which indicates low uniformity in surface texture.

Deep neural network based calibration for freeform surface misalignments in general interferometer

In interferometric testing for optical freeform surfaces, the calibration of surface misalignment aberrations is a tremendous challenge. A complex surface figure with a six-axis degree of freedom introduces complex non-linear misaligned aberration variations with slight misalignments. The traditional sensitive matrix method provides considerable precision in only the linear area, which is unacceptable. In this paper, a deep neural network (DNN)-based calibration method is introduced. The well-trained DNN can treat this non-linear relation and, thus, yield accurate misalignment estimation. Subsequently, the estimated misalignments are simulated in a model to predict all the misalignment aberrations by ray tracing. These aberrations are then removed by a simple wavefront data subtraction. The simulation and experimental results show the feasibility of the proposed DNN-based method.

High-resolution few-pattern method for 3D optical measurement

Accurately and quickly obtaining the three-dimensional (3D) shape information of objects has become increasingly important in various scientific fields. However, simultaneously achieving the high-resolution and high-speed 3D shape measurement of unknown objects remains challenging in practice. In this Letter, we propose a novel variant shifting-phase method for 3D optical measurement. Based on a digital fringe projection system, the method performs a point-wise high-resolution measurement of objects by projecting only four intensity-coded patterns. We can retrieve the wrapped phases and their corresponding fringe orders simultaneously from these four patterns, and do not require any pre-acquired information of the object. The experimental results successfully demonstrate the effectiveness of the easy-to-operate method.

Flexible interferometric null testing for concave free-form surfaces using a hybrid refractive and diffractive variable null

Free-form surfaces have been applied in a wide range of modern optical systems. As a supporting technique for fabricating free-form surfaces, the interferometric null method for testing the surface figure error has very limited flexibility. In this Letter, we report a flexible interferometric null test method which can test free-form surfaces with a very broad departure varying range. In the presented flexible null method, a hybrid refractive and diffractive variable null (HRDVN) is utilized as the flexible null. The HRDVN has superb aberration types adaptability, amplitude adaptability, and moderate phase generating accuracy. A flexible interferometric null testing setup was established using the HRDVN. Its superb adaptive capacity and moderate test accuracy were successfully demonstrated by measuring a free-form surface with rotationally symmetric departure of 173.486λ (λ=632.8  nm) and non-rotationally symmetric departure of 23.786λ.

System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology

We report the simulation of an adaptive interferometric null test using a high-definition phase-only spatial light modulator (SLM) to measure form and mid spatial frequencies of a freeform mirror with a sag departure of 150 μm from its base sphere. A state-of-the-art commercial SLM is modeled as a reconfigurable phase computer generated hologram (CGH) that generates a nulling phase function with close to an order of magnitude higher amplitude than deformable mirrors. The theoretical uncertainty in form measurement arising from pixelation and phase quantization of the SLM is 50.62 nm RMS. The calibration requirements for hardware implementation are detailed.

Intelligence enhancement of the adaptive wavefront interferometer

The adaptive wavefront interferometer (AWI) we have reported recently is utilized to test in-process surfaces with severe surface figure error which is beyond dynamic range of conventional interferometers [S. Xue, S. Chen, Z. Fan, and D. Zhai, Opt. Express26, 21910 (2018).]. However, it shows low intelligence when Monte-Carlo simulation is conducted to apply AWI on various surface figure error. In some simulation cases, the unresolvable fringes keep still or cannot be turned into completely resolvable fringes. To troubleshoot this issue, we studied AWIs in a general framework of global optimization for the first time. Under this framework, we explained that three optimization issues contribute to the poor performance of AWI. On this basis, we proposed a machine vision and genetic algorithm combined method (MV-GA) to control AWI to realize efficient and robust tests of various surface figure error. Monte-Carlo simulation and experiment verify the robustness has been greatly enhanced.

General equations for the null-screen test for aspherical surfaces with deformation coefficients

A modified approach to calculate the null screen for testing fast convex/concave aspherical surfaces with deformation coefficients is presented. Introducing the aberration polynomial into the equation of the sagitta, the null screens can be generated in a simple way, in contrast to the conventional design method. This approach is easy to implement for the calculation of the null screen since the equations presented here are easy to program computationally. The validation of this approach was done by analyzing the fast aspherical surface of a convex condenser lens (f/0.18). We obtained a percentage error smaller than 1.3% in the recovery of the coefficients that describe the shape of the surface.

Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres

A null lens moving back and forth relative to a point source can generate variable spherical aberration for flexible test of aspheres. Different from the previous methods, variable spherical aberration null theory was developed by us to optimize the null lens. The optimized null was a plano-convex singlet containing a high order even asphere. Its attractive advantages are the simple structure and the broad range of testable surfaces. Most concave prolate conic and near conic surfaces with k∙R value varying between 0 and about 70000mm and with smaller relative aperture than that determined by each k∙R value can be tested. The testable asphericity range is between 0 and about 230λ. Relations among these testable surfaces were revealed as different groups of equidistant surfaces. To explicitly show the ability of the null, the measurable surfaces range map that contains all parameters defining a conic surface was offered. A practical near-null test system using this null was established. Alignment, near-null data processing, and error sources are analyzed in detail. To verify the broad testable surfaces range, three surfaces with widely varying amounts of asphericity were tested. Cross tests were provided to verify the test system accuracy.