Mid-spatial Frequency Errors Research Articles

The Cause of Ribbon Fluctuation in Magnetorheological Finishing and Its Influence on Surface Mid-Spatial Frequency Error.

In the high-power laser system, the mid-spatial frequency error of the surface of the high-power laser component will affect the normal operation of the high-power laser system. In order to improve the mid-spatial frequency error of the high-power laser component after magnetorheological finishing, the causes and influencing factors of the ribbon fluctuation in magnetorheological finishing are studied, and the influence of different ribbon fluctuation on the mid-spatial frequency error of the surface is studied. Firstly, the influence of different ribbon fluctuations on the mid-spatial frequency error of the machined surface is simulated by a computer. Secondly, the magnetic field in the circumferential direction of the polishing wheel, the fluctuation amount and frequency of the magnetorheological polishing ribbon are measured, and then the causes of the fluctuation of the magnetorheological polishing ribbon are analyzed. Moreover, through the principle of a single variable, the influence of process parameters on the fluctuation of magnetorheological polishing ribbon is explored. Finally, the fused silica component is scanned uniformly under the process parameters of magnetorheological polishing ribbon fluctuation of 40 μm, 80 μm, 150 μm, and 200 μm. The experimental results show that the greater the ribbon fluctuation, the greater the surface mid-spatial frequency error of the component, and the ribbon fluctuation is approximately linear with the RMS of the PSD2 in the mid-spatial frequency band on the surface of the component. Therefore, the fluctuation of the ribbon can be controlled by controlling the magnetorheological processing parameters, and the mid-spatial frequency band error on the surface of the high-power laser component can be significantly reduced by optimizing process parameters after magnetorheological finishing.

Sparse bi-step raster path for suppressing the mid-spatial-frequency error by fluid jet polishing.

The periodic ripple errors (mid-spatial-frequency (MSF) error) produced by computer-controlled sub-aperture polishing severely limit the improvement of high-performance optical systems. At the same time, the fluid jet polishing (FJP) method is non-destructive and non-contact, but it is still hard to widely use it due to the defect of small spot-size and low efficiency. In this paper, we found that FJP has a significant advantage in removing the residual periodic ripples in sub-aperture polishing. The mathematical model developed by complex spectrum optimization verifies the existence of the sparse "bi-step raster path" (BSRP), which can achieve efficient periodic ripple error removal by suppressing the first two-order peaks of the error spectrum. In the experiments, it was observed that the MSF error has been significantly reduced after BSRP processing while the surface form and surface roughness have not been deteriorated, which demonstrates the validity of the proposed method. The proposal of the BSRP provides a new approach for the application of FJP and the suppression of the MSF error.

Two characterization methods of ripple errors for the large square aperture.

The specification and characterization of mid-spatial-frequency (MSF) ripples for the large-square-aperture optical elements, typically used in high-power laser systems, have received considerable critical attention. It is necessary to resort to a simple and robust way to characterize error surfaces for facilitating prediction of performance degradation and guiding the fabrication and tolerance settings. In this paper, we characterize residual periodic surface undulations called ripple errors for the large square aperture generated from modern subapertures and deterministic optical fabrication techniques through two methods, taking a step from qualitative judgment to quantitative analysis. The cross artifact reduction technology, instead of traditional windowed preprocessing, is introduced into power spectral density to suppress spectrum leakage while retaining the information about the part. An efficient algorithm to generate Legendre moments for two-dimensional Legendre polynomials is proposed to quantify ripple errors. This work contributes to understanding the optical degradation caused by MSF errors and associating the design and performance index with surface parametric description.

Reduction of mid-spatial frequency errors on aspheric and freeform optics by circular-random path polishing.

Pseudo-random paths are a useful tool to reduce mid-spatial frequency errors created in the processing of optical surfaces by sub-aperture polishing tools. Several types of patterns have been proposed, including hexagonal, square and circular, but prior literature has largely focused on flat and gently curved surfaces. Here, an extension of the circular-random path to strongly curved aspheric and freeform surfaces is proposed. The main feature of the algorithm is to cover the entire surface to be polished with a uniformly distributed tool path. Aspheric condenser lenses are then polished with a regular raster and circular-random path. Analysis of the optical performance shows that the random path can reduce the amplitude of mid-spatial frequency errors and relative intensity of satellite images. These features are particularly desirable in short wavelength applications, such as mirrors for EUV and X-ray.

High-efficiency smooth pseudo-random path planning for restraining the path ripple of robotic polishing.

Computer-controlled subaperture polishing technology is limited by its propensity to introduce midspatial frequency (MSF) error (ripple error), which significantly inhibits the performance improvement of optical systems. The pseudo-random polishing path is an important method for suppressing MSF error. However, a pseudo-random path that ensures both path smoothness and planning efficiency is difficult to generate. This paper proposes a novel, to the best of our knowledge, pseudo-random path planning method employing a reconstructive points algorithm that efficiently achieves full coverage of the workpiece under massive sampling points at once. Moreover, the generation time for millions of path points is reduced to less than 3 minutes. Additionally, a path modification method is proposed that achieves smooth processing on a machine tool with few additional path points; the vibration magnitude under the proposed smooth path can be reduced to 0.749 g (gravity acceleration), which is the same as that of a raster path. A precise speed management method is also proposed to ensure precise surface error corrections. Overall, the experimental results show that the peak valley of the form error can be converted to 0.115λ using the proposed algorithm without introducing a periodic MSF error.

Investigation on the smoothing of surface spatial frequency errors for borosilicate glass by heat-treatment

The mid-spatial frequency error and high-spatial frequency error on the optical glass component surface limit the development of high-power laser devices. By heat-treatment in wet atmosphere, the surface can be smoothed spontaneously under the action of capillary force, which is positive to improve the quality of optical components. In this paper, the finite element simulation model of the glass surface smoothing process is established based on the level-set method. The factors affecting the smoothing process, such as morphology wavelength and viscosity, are analyzed by simulation. A series of surface smoothing heat-treatment experiments for borosilicate glass were carried out to verify the effectiveness of improvement on surface quality through heat-treatment. Finally, the reasons for the discrepancy between theory and experiment are investigated through the simulation, and the effects of wavelength and the water content in the material are further investigated.

Smoothing process of conformal vibration polishing for mid-spatial frequency errors: characteristics research and guiding prediction.

By combining the conformal polishing method with short stroke vibration, a novel, to the best of our knowledge, conformal vibration polishing (CVP) method is proposed. The CVP method is expected to be an efficient means of optical processing by its high material removal rate and smoothing characteristics of mid-spatial frequency (MSF) errors. A quantitative time-domain smoothing model and a convergence factor (${\rm CF}_C$) are presented based on the research of smoothing characteristics. The motion mechanism, material removal ability, solution, and expansion of the smoothing model are demonstrated theoretically and experimentally. The experimental results exhibited good agreement with the theoretical predictions for the proposed method. The research provides a certain theoretical foundation for parameter selection and process optimization of the CVP method.

Research and application on the key techniques of conformal vibration polishing on optics

In computer controlled optical surfacing (CCOS), the small-sized polishing tools make the process exhibit a weaker material removal capability and exacerbate the deterioration of mid-spatial frequency (MSF) errors. Consequently, this paper presents a full-aperture conformal vibration polishing method incorporating short-stroke forced vibration. The realization mechanism, removal ability and machining effect of it were expounded separately, and verified by experiments. The experimental results show that conformal vibration polishing can achieve uniform smoothing of the MSF errors without destroying the original surface shape while efficiently removing the materials. The conformal vibration polishing proposed in this paper not only extends the existing processing field, but also provides a new approach for the suppression of MSF errors.

Freeform optics for imaging

In the last 10 years, freeform optics has enabled compact and high-performance imaging systems. This article begins with a brief history of freeform optics, focusing on imaging systems, including marketplace emergence. The development of this technology is motivated by the clear opportunity to enable science across a wide range of applications, spanning from extreme ultraviolet lithography to space optics. Next, we define freeform optics and discuss concurrent engineering that brings together design, fabrication, testing, and assembly into one process. We then lay out the foundations of the aberration theory for freeform optics and emerging design methodologies. We describe fabrication methods, emphasizing deterministic computer numerical control grinding, polishing, and diamond machining. Next, we consider mid-spatial frequency errors that inherently result from freeform fabrication techniques. We realize that metrologies of freeform optics are simultaneously sparse in their existence but diverse in their potential. Thus, we focus on metrology techniques demonstrated for the measurement of freeform optics. We conclude this review with an outlook on the future of freeform optics.

Novel magic angle-step state and mechanism for restraining the path ripple of magnetorheological finishing

Computer-controlled sub-aperture polishing technology is limited by its propensity to introduce mid-spatial-frequency (MSF) error (ripple error), which significantly inhibits the performance improvement of the optical system. Here, we propose a novel processing form of the magnetorheological finishing (MRF) tool and its generation mechanism. There are specific path angles and steps, with only tens of microns bandwidth (‘magic’ angle step), under which a surface without significant path ripple can be stably realized without affecting the convergence of other frequency errors. The magic angle step is precisely obtained via the scale-variant filtered spectrum analysis. Using the two-dimensional (2-D) Taylor approximation of the tool influence function (TIF), we verify that in the magic angle step state, owing to the shape asymmetry and the edge discontinuity of the TIF, the convolution of TIFs presents a complementary effect. It is first observed in experiments that the MSF error can be abnormally increased by decreasing the path step around the magic angle step state. Therefore, the magic angle step and corresponding analysis model have the potential of being extended to other polishing tools. Furthermore, they can significantly benefit the manufacturing efficiency of high-power laser systems, huge telescopes, and lithography systems, as well as improve existing polishing technologies.

A predictable smoothing evolution model for computer-controlled polishing

Quantitative prediction of the smoothing of mid-spatial frequency errors (MSFE) is urgently needed to realize process guidance for computer controlled optical surfacing (CCOS) rather than a qualitative analysis of the processing results. Consequently, a predictable time-dependent model combining process parameters and an error decreasing factor (EDF) were presented in this paper. The basic smoothing theory, solution method and modification of this model were expounded separately and verified by experiments. The experimental results show that the theoretical predicted curve agrees well with the actual smoothing effect. The smoothing evolution model provides certain theoretical support and guidance for the quantitative prediction and parameter selection of the smoothing of MSFE.

Mid-spatial frequency error generation mechanisms and prevention strategies for the grinding process

The research presented in this paper is focused on the link between manufacturing parameters and the resulting mid-spatial frequency error in the manufacturing process of precision optics. The goal is to understand the generation mechanisms of mid-spatial frequency errors and avoid their appearance in the manufacturing process. Also, a simulation which is able to predict the resulting mid-spatial frequency error from a manufacturing process is developed and verified.

Full aperture super-smooth polishing and surface topology evolution of the mandrels for X-ray optics used in eXTP mission

The X-ray focusing mirror used in enhanced X-ray Timing and Polarimetry(eXTP) mission are manufactured by the electroformed nickel replication process, and the machining accuracy of the mandrels are directly related to the focusing quality of the mirror. The optical performance of the focusing mirror is not only affected by the shape error, but also closely related to the mid and high-spatial frequency errors. In this research, the full-aperture polishing process of NiP die was investigated, and the evolution of surface waviness and the surface roughness under subsequent polishing processes were examined. The 7 × 4 h of full-aperture polishing experiment was conducted on the self-developed two-axis polishing machine with optical grade pitch as the polishing tool. The experimental results showed that the mid-spatial frequency error of the mandrel was effectively reduced and the root mean square value of the surface roughness was decreased to 0.266 nm, which meets the stringent requirements for the surface quality of mandrel in eXTP mission.

Development of a concave freeform surface measurement using transverse translation-diverse phase retrieval

Transverse translation-diverse phase retrieval (TTDPR), a ptychographic wavefront-sensing technique, is a viable method for freeform optical surface metrology due to its relatively simple hardware requirements, flexibility, and demonstrated accuracy in other fields. In TTDPR, a subaperture illumination pattern is scanned across an optic under test, and the reflected intensity is gathered on an array detector near focus. A nonlinear optimization algorithm is used to reconstruct the wavefront aberration at the test surface from which we can solve for surface error, using intensity patterns from multiple scan positions. TTDPR is an advantageous method for aspheric and freeform metrology because measurements can be performed without null optics. We report on the development of a concave freeform mirror measurement using this technique. Simulations were performed to test algorithmic performance as a function of various parameters, including detector signal-to-noise ratio and position uncertainty of the illumination, with <λ / 100 root-mean-square (rms) wavefront-sensing error achieved in most cases (λ = 632.8 nm). An experimental measurement is then demonstrated, and results of reconstructed surface form and midspatial frequency error are presented. Surface reconstructions from two disjoint datasets agree to 13-nm rms, or λ / 50 at λ = 632.8 nm.

Experimental investigation of mid-spatial frequency surface textures on fused silica after computer numerical control bonnet polishing

Automated bonnet polishing is achieved using computer numerical control (CNC) technology. However, owing to specific toolpaths, CNC bonnet polishing is generally accompanied by the mid-spatial frequency (MSF) of surface textures on polished surfaces that could produce surface ripples or waviness and degrade image quality. In this study, the MSF surface textures on fused silica are investigated by developing a CNC bonnet polishing technique using a cerium oxide-filled polyurethane pad (LP66)—a cellular polyurethane material designed to handle high flatness and surface finishing requirements for optical glass materials. To minimize the MSF, optimal combinations of various polishing parameters, including tool offset, head speed, track spacing, and surface feed rate, are studied. Experimental results demonstrate that the head speed and feed rate significantly affect the surface texture during bonnet polishing. Although the tool offset does not cause surface textures, the material removal rate is affected. A series of optimization experiments is conducted, consequently leading to the effective removal of irregular surface ripples and a reduction of MSF errors. By optimizing the polishing parameters, extremely accurate surface quality is achieved, along with a root mean square error of 1.6 nm. These results demonstrate the potential applications of LP66 in CNC bonnet polishing for highly accurate freeform optical components.

Testing the mid-spatial frequency error of a large aperture long-focal-length lens with CGH.

We present the use of a computer-generated hologram (CGH) to test the mid-spatial frequency error of a large aperture long-focal-length lens. In order to verify this test approach, a 450 mm × 450 mm reflective CGH is designed and fabricated for testing the 440 mm × 440 mm spatial filter lens. Both 0th and 1st order diffraction wavefront of CGH are measured, and the 0th order diffraction wavefront is used to calibrate the substrate error. The mid-spatial frequency wavefront error caused by the CGH fabrication errors are evaluated using the binary linear grating model and power spectral density (PSD) theory. Experimental results and error analysis show that the CGH test approach is also feasible for the measurement of mid-spatial frequency error, and the measurement accuracy of PSD1 can reach 0.8832 nm RMS.

Modeling and analysis of the mid-spatial- frequency error characteristics and generation mechanism in sub-aperture optical polishing.

In the field of ultra-precision manufacturing, the mid-spatial-frequency (MSF) error can severely affect the performance of the optical elements, but it is rather difficult to quantitatively predict the MSF error distribution. In this paper, the piecewise-path convolution (PPC) analysis is established to investigate the characteristic and the mechanism of the MSF error. The path type, tool influence function (TIF), feed rate, movement type, etc. are all considered mathematically in the analysis. This method can quantitatively predict the MSF error distribution. The coupling relationship among the path type, TIF and the MSF error are proved through the filtering theory. Besides, the analysis reveals the mathematical relationship between the tool movement type (orbital motion, radial runout) and the MSF error; the results show that the tool motion can also introduce non-negligible MSF error. Based on the research above, two selection formulae of path type, TIF and polishing parameters are provided for low MSF error polishing, which gives the theoretical guidance for the parameter selection in deterministic polishing. Practical experiments demonstrate the validity of the analysis results and conclusions.

Six-directional pseudorandom consecutive unicursal polishing path for suppressing mid-spatial frequency error and realizing consecutive uniform coverage.

Planning the polishing path is an important part of the polishing process. Reasonable selection of the polishing path can improve surface quality and reduce the mid-spatial frequency (MSF) error. In this paper, a six-directional pseudorandom consecutive unicursal polishing path and the corresponding generation method are proposed. The proposed path is especially useful for suppressing the MSF error of the surface and realizing consecutive uniform coverage. Compared with traditional polishing paths such as scanning, spiral, and fractal, the proposed path has the following characteristics: multi-directionality, high randomness, smoothness, and consecutiveness. In order to verify that the proposed path has the ability to suppress the MSF error, a group of contrast experiments were conducted and the measurement data was processed by the empirical mode decomposition (EMD) method and the fast Fourier transform (FFT) method. The results show that the fluctuation amplitude of the MSF error and the fluctuation area with MSF error are obviously reduced by using the proposed path. Moreover, the corresponding wavelength of the crests in the spectrum shows multi-values. The experimental results show that the proposed path has high randomness and the ability to suppress the MSF error.

Simple methods for estimating the performance and specification of optical components with anisotropic mid-spatial frequency surface errors.

Specification and tolerancing of surfaces with mid-spatial frequency (MSF) errors are challenging and require new tools to augment simple surface statistics to better represent the structured characteristics of these errors. A novel surface specification method is developed by considering the structured and anisotropic nature of MSF errors and their impact on the modulation transfer function (MTF). The result is an intuitive plot of bandlimited RMS error values in polar coordinates which contains the surface error anisotropy information and enables an easy to understand acceptance criterion. Methods, application examples, and the connection of this surface specification approach to the MTF are discussed.

Desensitization to mid-spatial-frequency surface errors.

Mid-spatial-frequency (MSF) surface errors are left after processing aspheres or freeforms by small bits. MSF errors sometimes have significant effects on the image quality of optical systems. The efforts on the analysis and tolerancing of the MSF errors have been paid. This paper focuses on the desensitization to the MSF errors. The design target for the desensitization to the MSF errors is defined. A possibility of the desensitization is shown for optical systems in which freeforms are used with spheres.