Surface Figure Research Articles

Glass ceramic ZERODUR®: Even closer to zero thermal expansion: a review, part 2

Observational astronomy has sought better telescopes with higher resolution from its beginning. This needs ever-larger mirrors with stable, high-precision surfaces. The extremely low-expansion glass ceramic ZERODUR® has enabled such mirrors for more than 50 years with significant improvements in size and quality since then. We provide a survey of the progress achieved in the last 15 years. Equally important as the thermal expansion coefficient CTE is its homogeneity. The CTE variation in 4-m mirror blanks lies below 5 ppb / K in radial and axial directions on large and short scales. Improved measurement capabilities allow reduced bias, which in the past made variations look greater than they were. Isotropy and uniformity of ZERODUR are outstanding. A method for lifetime calculation increases reliability considerably with respect to mechanical loads. The production and metrology capability and capacity are greatly extended. Surface figure and texture of large blanks allow starting directly with polishing. Filigree structures with up to 90% weight reduction are well-suited for space mirrors. The progress with low thermal expansion and its measurement, the insensitivity of ZERODUR against ionizing radiation in space, and outstanding application examples are presented in the first part of our review.

Glass ceramic ZERODUR®: Even closer to zero thermal expansion: a review, part 1

Observational astronomy has striven for better telescopes with higher resolutions from its start. This needs ever-larger mirrors with stable high precision surfaces. The extremely low expansion glass ceramic ZERODUR® enables such mirrors with more than 50 years of significant improvements in size and quality since its development. We provide a survey of the progress achieved in the last 15 years. The narrowest coefficient of thermal expansion (CTE) tolerance is now ±7 ppb / K. It is possible to adapt the material for lowest expansion to temperature-time courses given for special environments. At cryo temperatures, expansion is low and adaptable. Improved measurement capabilities allow for absolute CTE uncertainty of 3 ppb / K and reproducibility of 1 ppb / K (2σ). The influence of ionizing radiation on the surface figure integrity is subject to new investigations. Improved measurement capabilities increase the reliability of structure designs. Some outstanding examples are given for applications of ZERODUR in astronomy and in the very important high technology industry. The progress of thermal expansion homogeneity, the mechanical strength of ZERODUR, production capabilities in melting, precision machining, light-weighting, and dimensional metrology is presented in the second part of the paper.

Design method of wide field-of-view imaging systems using Gaussian radial basis functions freeform surfaces.

High optical performance systems with wide field-of-view (FOV) have important applications in remote sensing. The radial basis functions, which have a prominent local characteristic in surface description, have attracted much attention in recent years. In this paper, an effective design method for the wide FOV imaging system using Gaussian radial basis function freeform surfaces is proposed. The FOV of the optical system is extended from a relatively small value to a larger one, and the Gaussian radial basis function surfaces are extended stepwise based on certain criteria. A high image quality and small distortion off-axis freeform three-mirror system with a wide FOV (${{60}}^\circ \times {0.6}^\circ$) is designed as an example. Tolerance analysis considering both surface figure error and assembly error is performed. The design results demonstrate the effectiveness of the proposed method.

Active alignment of complex perturbed pupil-offset off-axis telescopes using the extension of nodal aberration theory

This paper presents an optical alignment strategy for complex perturbed pupil-offset off-axis reflective telescopes, based on the extension of nodal aberration theory (NAT). First, the direct expansion of the wave aberration function in the vector form for perturbed off-axis systems is given, which is especially convenient for the expansion of the corresponding higher-order terms. The inherent vector relationships between the contributions generated by the aberrations of the on-axis parent systems through pupil transformation are disclosed in detail, which is helpful to understand the aberration behavior of off-axis systems. Then, according to the inherent vector relationships, an analytical alignment model based on NAT for complex cases of perturbed off-axis telescopes is established. It can quantitatively separate the effects of misalignments and surface figure errors from the total aberration fields. The alignment model is solved by using particle swarm optimization algorithm. Then, an optical alignment example of the off-axis three-mirror anastigmatic telescope with misalignments and complex surface figure errors based on the proposed method is demonstrated. After correction, the perturbed telescope can be nearly restored to the nominal states. Finally, Monte Carlo simulations are carried out to show the effectiveness and accuracy of the proposed method.

Design of biaxial flexural mount for a rectangular mirror with large aspect ratio of a space telescope

In this paper, the biaxial flexural mount is presented according to the requirements of the large aspect ratio mirror of an off-axis three-mirror anastigmatic (TMA) space-borne telescope, and the process of the optimal design of the flexural mount structure is discussed in detail. From the perspective of mechanics-based design of space-borne mirror assembly structure and integrated optical-mechanical analysis, the optimal axial mounting position, the optimal mounting angle and the optimal structural critical dimensions of the flexure hinge were determined to improve the surface shape accuracy. Finally, the mechanical vibration test further proved the rationality of the structure design and the accuracy of the simulation analysis. The flexural mount proposed in this paper is can be applied to other reflective optics with high surface figure requirements. The optimization method proposed in this paper can also be applied to the optimization of other forms of flexural mount.

Removal of Mo/Si multilayer coatings on fused silica substrates by wet chemical etching

This work examined a novel etching procedure for the complete removal of Mo/Si multilayer coatings, as a means of reusing valuable mirror substrates employed in extreme ultraviolet imaging. A multilayer coating deposited on a fused silica substrate was etched with an alkaline solution containing potassium ferricyanide and sodium hydroxide, and the entire coating was removed within 60 min. The root mean square roughness and power spectrum density characteristics were subsequently assessed and the resulting values were equivalent to those before the deposition of the coating. These results demonstrate that such coatings can be removed while maintaining the original surface figure.

Hybrid IPSO-IAGA-BPNN algorithm-based rapid multi-objective optimization of a fully parameterized spaceborne primary mirror

The surface figure precision, weight, and dynamic performance of spaceborne primary mirrors depend on mirror structure parameters, which are usually optimized to improve the overall performance. To realize rapid multi-objective design optimization of a primary mirror with multiple apertures, a fully parameterized primary mirror structure is established. A surrogate model based on a hybrid of improved particle swarm optimization (IPSO), adaptive genetic algorithm (IAGA), and optimized back propagation neural network (IPSO-IAGA-BPNN) is developed to replace optomechanical simulation with its high computational cost. In this model, a self-adaptive inertia weight and a modified genetic operator are introduced into the particle swarm optimization (PSO) and adaptive genetic algorithm (AGA), respectively. The connection parameters of BPNN are optimized by the IPSO-IAGA algorithm for global searching capability. Further, the proposed IPSO-IAGA-BPNN, based on a rapid multi-objective optimization framework for a fully parameterized primary mirror structure, is established. Moreover, in addition to the proposed IPSO-IAGA-BPNN model, the Kriging, RSM, BPNN, GA-BPNN, PSO-BPNN, and PSO-GA-BPNN models are also analyzed as contrast models. The comparison results indicate that the predicted value obtained by IPSO-IAGA-BPNN is superior to the six other surrogate models since its mean absolute percentage error is less than 3% and its R2 is more than 0.99. Finally, we present a Pareto-optimal primary mirror design and implement it through three optimization methods. The verification results show that the proposed method predicts mirror structural performance more accurately than existing surrogate-based methods, and promotes significantly superior computational efficiency compared to the conventional integration-based method.

Calculation of Effect Ratio of 21 Geometric Errors and Detection of Surface Figure Error

Traditionally, a machine tool is regarded as a rigid-multi body system, and it is studied by integrated geometric error modeling. Considering a three axis-machine, this study introduced the ratio and effects of the geometric errors on the surface figure error. First, based on synchronous iteration location, a surface matching model was employed to assess the coordinate measurement result. Subsequently, considering all the geometric error functions having the same parameters, the effect ratio and the surface figure error of a single geometric error were obtained. Concurrently, based on the effect ratio and surface figure error results, the main geometric errors were obtained, such as EZX, EZY, EAX, and COY. Moreover, the proposed method and the main errors were experimentally verified. By compensating the main errors, the sphere accuracy was improved by 53.4% and the flat accuracy by 70.0%. Additionally, the method proposed in this paper could be utilized to detect the surface figure error.

The effect of clamping stress on the measured surface figure of optical glass BK7

This work probes the effect of clamping stress induced by pad block’s support surface (PBSS) shape error in measured surface figure of optical glass BK7. Under given PBSS topography measured by three coordinate machine, the clamping stress loaded on the workpiece and its effect on workpiece’s surface deformation was simulated by finite element method. Furthermore, the surface figure residual error (SFRE) in simulation was in good agreement with experimental results despite the measurement repeatability error.

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.

Optical performance of aluminum mirror for cryogenic applications

Aluminum mirrors are most commonly used for cryogenic optic systems. A parameter simulation has been conducted to design the aluminum mirror. We have obtained an optimized mirror structure after optimization applied in the mirror design stage. A finite element model analysis was conducted to predict the gravity sag, mounting-induced error and cryo-deformation of the mirror. Then we have fabricated a 160 mm diameter aluminum mirror based on the optimization results, tested it at room and cryogenic temperature. An error budget developed based on the experimental results shows that the fastening preloads effect dominates the surface figure of the aluminum mirror. The total deformation of the mirror surface at 100 K meets the optical requirement, which indicates that the optimized aluminum mirror is stable enough to retain good surface figure at cryogenic temperature and the aluminum mirror is promising for its cryogenic applications.

Dwell Time Algorithm Based on Bounded Constrained Least Squares Under Dynamic Performance Constraints of Machine Tool in Deterministic Optical Finishing

The dwell time algorithm is one of the most important techniques within the deterministic optical surfacing technologies. The existing dwell time algorithms are generally based on non-negative least squares (NNLS) without considering the dynamic performance constraints of machine tools. This is a circumstance that leads to poor convergence accuracy. In this paper, a dwell time algorithm, based on bounded constrained least-squares (BCLS) under dynamic performance constraints of the machine tool, has been developed. The upper and lower constraints of the dwell time model could be derived through the acceleration and deceleration mechanism of the CNC (Computer Numerical Control) machine tools. A two-metric projection Newton iteration algorithm was used to solve the large-scale dwell time model, which greatly improved the computation efficiency. The results of the experiments and simulations showed that the proposed algorithm will give a very high convergence accuracy for optical finishing with machine tools with different dynamic performances. When the machine acceleration was set to a value as low as 0.1 g, the accuracies of the surface figures PV (Peak-to-Valley) and RMS (Root Mean Square) till improved by 40.8% and 55.2%, respectively, when using the BCLS algorithm. The influences of different dynamic performances of the machine tools on the dwell time solutions have also been investigated, which will provide a good guidance in the design of deterministic polishing machine tools.

Supporting structure design for primary and tertiary mirror of off-axis TMA system

Supporting structure for primary and tertiary mirror is one of the most challenging technical points during development of off-axis three-mirror anastigmatism (TMA) biological imaging system. In order to minish the shape error of the mirror and meet the stability of the supporting structure which was working, the mirror supporting structure was designed and optimized with finite element method. First, based on requirement of optical design, materials and supporting mode for the mirror and its supporting structure were decided. Then, a weight reduction structure of the mirror cell was designed which had three-point backside and six-point lateral support structure. The flexible hinge with circular and cantilever beam was made according to the optimized mathematical model. The influence of the parameters on the precision of mirror surface were analyzed under the load conditions of gravity and uniform temperature rise of 4 ℃. Then, the static analysis and thermal analysis for the mirror subassembly were performed. The Finite Element Analysis (FEA) indicated that the surface error (RMS) was 1.529 nm at self-weight. RMS value was 2.426 nm when the temperature rised 4 ℃. Finally, the surface figure test and the wavefront aberrations test were carried out by using Zygo interferometer. The test RMS value of the surface figure is 0.025 λ and the wavefront aberrations of the optical system is 0.102 λ (RMS). The results show that this supporting system could meet the technical indicator requirements of biological imaging system (RMS value of the shape error ≤λ/40, RMS value of the wavefront aberrations≤λ/10).

Study of Epoxy Bonding

Bonding is used widely when the optical glasses are connected with metal structures. It plays a very important role in the design of space telescope. Improper bonding may induce great surface figure error for optics in telescope. Milbond and EC2216 are the most popular epoxies that are used in optics. Bubbles usually exist in the epoxy when two components are mixed or translated to syringe. In this paper, the methods and experiments of adhesive mixture and injection in vacuum environment were explained to reduce the bonding surface figure of the mirror. The results show that adhesive mixture and injection in vacuum environment can dramatically decrease the bubble in the epoxy and greatly reduce the bonding surface figure error.

Analytical Modeling of Surface Potential and Figure of Merit Computation for Planar Junctionless pH Sensing BioFET

Herein this paper we propose a surface potential based analytical model for planar junctionless field effect transistor (JL-FET) for pH sensing. The electrolyte considered is phosphate buffer saline (PBS) solution which has been modeled as three layered stacked structure consisting of stern layer, ion-permeable membrane and bulk electrolyte. The proposed model has been deduced considering Poisson's equation in the channel region. Relative shift in threshold voltage (V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Th</sub> ) and maximum drain current (I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DS,max</sub> ) have been used as sensitivity metrics. The low concentrations of electrolyte (0.01), yielded higher V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Th</sub> sensitivity of 63 mV/pH and 59 mV/pH for bottom and liquid gate respectively as compared to higher molar concentrations of electrolyte. For 0.01 PBS the aggregate drain current shift has been found to be 52.8 μA/pH and is larger for liquid gate operation while as for bottom gate, shift of 18.9 μA/pH is observed. Further considering pH range of 1-14, we computed various figure of merits (FOMs) that include sensitivity, linearity and signal to noise ratio for the device. The FOMs were computed and analyzed for independent operation of liquid and bottom gate for three different molarities of PBS (1, 0.1, 0.01) each with pH range from 1 to 14. Signal to noise ratio of drain current is found maximum for low molar concentrations of electrolyte and also is highest at point of maximum transconductance. The results obtained from analytical model are in good coherence with the TCAD simulation model.

Surface preparation and analysis on fused silica glass substrate with deterministic grinding method

Optical quality of glass components are determined by its surface finish either at lapping stage or at final polished stage. The criticality of optical components is defined by its application. For commercial application its surface finish can be of the order of few microns, whereas for special purpose applications such as aerospace, navigation and laser based sensors its surface finish requires super smooth polish ranging from angstrom to sub angstrom level. In order to achieve such high surface figure, the optical components has to be pre-processed from lapping stage. Two parameters play a major role for obtaining better quality of optical components; they are Surface Roughness (Ra) and Sub Surface Damage (SSD). These parameters have to be monitored from the initial stage of lapping. In this work, Fused Silica glass substrate has been considered and lapped with different deterministic grinding methods, such as Diamond Pellet, Diamond Trizact Tile and Diamond Cup Tools. The abrasives used are pre determined diamond particles embedded in different matrix, such as metal bound, resin bound, ceramic bound and electroplated. The processed samples were characterized for its Average Roughness (Ra) through White Light Interferometry technique and its corresponding depth of SSD was calculated. Images of processed samples were captured through Field Emission Scanning Electron Microscope (FESEM). Process optimization and comparative study between various deterministic lapping processes has been established. Average Surface Roughness (Ra) has been reduced from 274 nm to 3.3 nm and its corresponding depth of SSD values were also reduced from 14.5 µm to 0.151 µm. Earlier, studies determine that SSD depends on grit size of diamond whose relation was given by 0.5d, irrespective of processing parameter or methodology adopted. In this study results have been obtained with one order down with respect to earlier achieved results.

Mechanisms influencing and prediction of tool influence function spots during hemispherical sub-aperture tool polishing on fused silica.

Sub-aperture tool polishing of precision optics requires a detailed understanding of the local material removal [tool influence function (TIF)] at the contact spot between the workpiece and tool to achieve high removal determinism and hence precision of the optic relative to the desired/design surface figure. In this study, the mechanisms influencing and the quantitative prediction of the removal rate and shape of TIF spots during polishing of fused silica glass with cerium oxide slurry using a rotating hemispherical pad-foam tool for a wide variety of process conditions (including tool properties, kinematics, and applied displacements) are investigated. The TIF volumetric removal rate can be estimated utilizing the average relative velocity and contact area using a simple analytical model. In addition, stability of the volumetric removal rate for fixed process conditions is shown to be greatly dependent on the pad preparation and amount of tool use (affecting both pad topography and slurry buildup), whose general behavior shows an increase in removal rate followed by stabilization with polishing time. The determination of the TIF removal shape is more complex. An extended version of the Preston removal model is developed to explain a comprehensive set of measured TIF removal shapes to within ∼22%. This model incorporates a number of phenomena impacting the TIF removal shape including: (a) temporal and spatial dependent relative velocity between the workpiece and tool; (b) an elastic mechanics based, as well as hydrodynamic, pressure distribution; (c) a spatially dependent friction coefficient possibly caused by both reduced slurry replenishment in low velocity regions and pad slurry islands (100 µm scale) and porosity (millimeter scale); and (d) a shear-based removal mechanism on the periphery of the contact spot.

Higher-order Hermite-Gauss modes as a robust flat beam in interferometric gravitational wave detectors

Higher-order Laguerre-Gauss (LG) modes have previously been investigated as a candidate for reducing test-mass thermal noise in ground-based gravitational-wave detectors like Advanced LIGO. It has been shown however that LG modes' fragility against mirror surface figure imperfections limits their compatibility with the current state-of-the-art test masses. In this paper we explore the alternative of using higher-order Hermite-Gauss (HG) modes for thermal noise reduction, and show that with the deliberate addition of astigmatism they are orders of magnitude more robust against mirror surface distortions than LG modes of equivalent order. We present simulations of Advanced LIGO-like arm cavities with realistic mirror figures which can support HG$_{33}$ modes with average arm losses and contrast defects in a Fabry-Perot Michelson interferometer configuration which are well below the typical measured values in Advanced LIGO. This demonstrates that the mirror surface flatness errors will not be a limiting factor for the use of these modes in future gravitational-wave detectors.

Φ1.05 m轻量化反射镜设计与制造

针对Ф1.05 m空间光学系统主镜的设计指标要求，提出了轻量化反射镜结构优化设计的新方法，并建立了反射镜结构自动化仿真分析与优化设计平台，基于此平台确定了性能优异的主镜结构设计方案。主镜重量小于50 kg，轻量化率已接近国外先进水平；主镜在三球铰支撑下的第一阶模态频率为361.2 Hz，自由状态下的一阶非零模态频率为501.9 Hz；在1 ℃均匀温度变化下，不去离焦和去除离焦之后的面形RMS分别为0.55 nm和0.10 nm；主镜在30g过载加速度作用下的最大应力为16.1 MPa，均满足设计要求。采用目前最先进的第三代大口径反射镜加工工艺，路线为超精密铣磨—小磨头数控研抛—离子束精修，实现主镜面形误差的确定性去除。为保证面形检测结果的天地一致性，发展了重力卸载技术和面形误差数据后处理技术，剔除重力和其他系统误差对检测的影响。主镜最终面形精度达到0.011λ RMS，获得了高精度的光学面形，也证明了方案的合理性。

Uniform Polishing Method of Spherical Lens Based on Material Removal Model of High-Speed Polishing Procedure

Although the high-speed polishing technology has been widely applied to obtain an ultra-smooth surface in the field of spherical optical manufacture, it is still mainly used in small-size or easily polished lenses. In the infrared optical system, large-size silicon lenses are often used to increase the luminous flux. As is known, the material is hard-polished, it is time-consuming to reduce the surface roughness by iterative polishing and it is difficult to avoid the form accuracy getting worse. To produce an ultra-smooth surface efficiently without destroying the figure, a scientific understanding of material removal in the high-speed polishing process is necessary, which would lead to the process being more deterministic. In this paper, a mathematical model of material removal is developed based on the classic Preston equation. The predicted results of the proposed model show good agreement with the experimental data. Further, a method to achieve uniform polishing can be addressed with a systematic analysis of the key factors affecting material removal and their contribution to spatial non-uniform removal. Finally, the experimental results indicate that the surface roughness of hard-polished spherical optics can be improved efficiently using the uniform polishing method without the surface figure being destroyed.