© The Author(s), published by EDP Sciences, 2025

1 Introduction

Optical freeform surfaces are becoming increasingly important as they serve a pivotal role in the miniaturization of optical assemblies and the enhancement of optical imaging quality. The implementation of rotationally symmetrical and cylindrical optical functional surfaces would necessitate the positioning of a considerable number of lenses in a series, which would inevitably result in an increased prevalence of aberrations and a notable loss of intensity.

Freeform optical elements optimized for a specific purpose are applied to generate the requisite intensity distribution at the desired receiver position. Consequently, a system with the same performance can be achieved with a significantly reduced number of optical elements, or even higher performance can be attained. Thus, freeform surfaces are capable of addressing the diverse requirements of optical assemblies and developments in photonic technologies [1–3].

However, manufacturing processes must also be adapted to meet these demands. The conventional production chain for optical elements, which relies on mechanically abrasive processes using classic tools such as cup or shell tools, must be supplemented or replaced by newer technologies. The use of various processes with sub-aperture tools and improvements in system technology for ultra-precision machining is beneficial in this regard.

The shape generation of refractive, optical freeform elements is usually determined by the application of computerized numerical controlled (CNC) grinding processes using multi-axes machines and tools with punctual engagement. The grinding process, which employs bonded grain with a geometrically undefined cutting edge, is a crucial technique in optical production, as it can be used to realize the geometric shaping of the component as well as to produce polishable surfaces with a specific surface structure, thanks to its high precision [4]. Due to the brittle-hard properties of inorganic non-metallic materials, a large number of lateral, radial and axial microcracks form from a certain load and chip thickness, which spread and lead to the formation of small material particles [5–7]. This is the grinding regime of brittle fracture, which always leads to the formation of a certain roughness structure and thus to a matt surface on glasses, which exhibits both surface cracks and crack depth damage under the component surface, the so-called subsurface damage (SSD). These must be completely removed in subsequent ultra-fine processing steps to produce an optical surface quality [8, 9]. Material damage can be successively reduced through multi-stage grinding processes with reduction of the abrasive grain size (pre-grinding, medium grinding and fine grinding), whereby the removal rates also decrease with each finer sub-process [8]. A second grinding regime, ductile grinding, is of increasing interest in the field of ultra-fine machining. In addition to the introduction and propagation of microcracks, plastic deformation processes described as ductile behavior can also be detected when looking at the chip formation zone on a microscopic basis [5, 6]. The ductile grinding regime was first described by Bifano et al., and then substantiated mathematically and experimentally in a paper published in 1991. The critical cutting depth or chip thickness, below which predominantly ductile removal takes place instead of brittle fracture, is decisive for achieving this regime. This is based on the energy consideration of the material removal, whereby it has been shown that at low removal depths the plastic flow of the material is energetically advantageous compared to fracture formation [10]. For glass, this results in low mathematical values for the critical cutting depth of well below 100 nm for silicate glasses in some cases [8, 10]. In addition to achieving such low cutting depths, which is generally only possible with ultra-precision machines, fine grain sizes, high concentrations, high cutting speeds and low feed rates can also be used to obtain or support the ductile regime [6, 11–13]. Ductile grinding can thus prevent the formation of microcracks in the interaction zone between the component and the tool, enabling the production of workpieces with high surface qualities and damage-free surface zones [5, 6, 9, 12, 13].

In this paper, an approach is described using special resin bond tools with fine diamond grain sizes, where the properties of the resin matrix can be described as highly porous and slightly elastic. This elasticity reduces the penetration depth of grains into the glass material, while the pores improve the supply of cooling lubricant and chip removal. The process is further called ultra-fine grinding, since it offers the potential to create partially reflective and transparent surfaces with roughness Rq < 10 nm and waviness Wt < 1 μm observed for planar fused silica samples before [14–17].

Given the considerable variability of local curvatures, polishing of freeform optics requires the use of very small sub aperture tools to ensure the retention of the shape after grinding. In this context, the application of mechanical-abrasive sub-aperture polishing is limited due to permanently changing tool engagement conditions. However, efforts have been made to overcome this challenge through the integration of process simulation [18, 19] and the development of adaptive tools [20]. Alternatively, beam-based tools may offer a viable solution. These non-contact techniques are not susceptible to tool wear during the machining process. A considerable number of research groups around the globe have been engaged in the development of laser-based polishing techniques. The CO₂ laser has been established as a method for laser polishing of glass surfaces [21–23]. A fast-scanning laser beam heats the surface of the sample up to the material-specific softening temperature, which leads to smoothing due to elevated viscosity and minimized surface tension [24]. However, thermal stress often results in structures in the mid spatial frequency (MSF) range not being smoothed or new MSF errors occurring due to the fast scanning process. As a result, an additional fine correction step is required for the optical surface [25]. As a beam-based tool, the atmospheric plasma jet (PJ) is also capable of polishing optical freeform surfaces [26]. The preservation of surface figure and the ability to recover from surface damage have already been proven [27, 28]. Plasma jet polishing (PJP) is a thermal process that results in localized heating of the surface near the softening point. Convective heating by the PJ causes the surface of the glass to melt, resulting in a reduction in viscosity and a thermally induced redistribution of material on the surface. Furthermore, the minimization of surface tension leads to a reduction in roughness. While the micro- and nanoscopic plasma-surface interaction is not yet fully understood, the reduction in surface roughness can be compared to the principle of polishing with a laser beam [21, 25]. In PJP, the plasma-based polishing tool, with its highly localized effect and small contact area, operates at a scanning velocity of approximately 1 mm/s along a meandering tool path. This setup enables a uniform and gradual heating and cooling of the material through tangentially flowing plasma gas, preventing steep temperature gradients. Thus, the generation of MSF can be significantly reduced.

The polishing effect in PJP can be mathematically described by a two-dimensional Gaussian function [29]. Once, the polishing function is known, the polishing result for a given initial surface can be estimated.

In the present investigations ultra-fine grinding is combined with PJP in order to achieve a time- and cost-efficient production of freeform optics made of glass. The aim of the investigations is the application of a new process chain, schematically given in Figure 1, which combines freeform generation by coarse grinding, fine grinding and ultra-fine grinding with a subsequent PJP step to obtain a fully functional optical surface. Due to the combination of the 3-stage grinding process and the thermal induced polishing the size of the workpiece is mainly limited by the dimensions of the machines and the CNC axis strokes. In addition, there are minor curvature limitations, which has to be considered in the machining set-up. Obviously, the smaller the grinding tool, the finer the curvature radii that can be produced. Furthermore, for PJP, the PJ and the attached temperature measurement system must be able to tilt parallel to the local surface normal. The ability to manufacture various freeform surface designs by preserving the ground geometry is one of the main advantages of PJP, together with the absence of tool wear.

Figure 1 Schematic representation of a new process chain for freeform fabrication with a combination of grinding processes, including ultra-fine grinding, and plasma jet polishing.

Aspects of the individual grinding steps as well as the PJP process are discussed with regard to process convergence. By selecting the optimal parameters, a successful manufacturing of an Alvarez-type freeform optic will be demonstrated.

2 Experimental

2.1 Grinding

The process chain that was investigated consists of three grinding steps and the final PJP process. Grinding was carried out on a 5-axis CNC machine (Ultrasonic 20 linear, DMG MORI). The three main grinding steps include pre-grinding for fast material removal, fine grinding for shape generation and finally ultra-fine grinding for a first surface smoothing. For the pre-grinding process coarser (D126 or D64) cylindrical diamond grinding tools are applied in combination with a 3-axis tool path movement, following the freeform contour. The material removal is achieved level by level, whereas layers of a certain cutting depth (in z-direction) are removed from the workpiece with the tool moving in the x-y-plane. This is done until the rough surface geometry is achieved, which at this point is characterized by a step-like appearance.

Afterwards the fine grinding process with ball shaped diamond tools is applied to remove these steps and further approximate the component surface to the required geometry. This manufacturing step is characterized by certain (kinematic) characteristics shown in Figure 2, such as the necessary 5-axis simultaneous path movements and point-like tool contact with the workpiece.

Figure 2 Visualization of kinematic properties for freeform grinding.

For this, D30 diamond grain, which can be described as a medium to fine diamond grain size, is used in a metal bond. The fine grinding step serves for shape correction. Also, mid-spatial frequency errors should be removed during this process.

For ultra-fine grinding however, special diamond tools with a resin bond and fine grain size (D16) from company Günter Effgen GmbH are applied to reduce surface roughness. The bond material has a high porosity and slight elasticity, that can reduce the penetration depth of the diamond grains into the workpiece material and thus enable a kind of ductile grinding regime. This results in already partially reflective surfaces. The main purpose of this process is to achieve a further reduction in roughness and depth damage to the freeform surface while the material removal rate has a minor significance. This should lead to a smooth surface sufficient for the polishing step afterwards.

To investigate and improve the freeform grinding processes several parameters were specifically varied and their influence on periodic surface errors, form deviation and surface roughness were examined. These parameters include machine and program settings like path tolerance in CAM, sample positioning in the machine, as well as kinematic settings and grinding parameters.

2.2 Plasma jet polishing

The principle of atmospheric PJP in general has been explained in detail in several publications during the last decades [30–32]. The PJ source is based on a coaxial conductor system equipped with a gas supply. The PJ is excited at the gas outlet nozzle by microwave power (2.45 GHz). The power provided by a solid-state microwave generator (CP250, Trionplas Technologies GmbH) is coupled into the source via a coaxial cable. The microwave propagates as a continuous wave (CW) along the inner electrode, through which gases are also passed. A high-frequency electric field generated by the microwave power at the tip of the nozzle ignites the gas flow and forms a PJ discharge. As a non-mechanical tool, the PJ is wear-free as long as MW-power and gases are supplied. Since only noble gases (He and Ar) are used, there is no chemical attack on the substrate and material is locally redistributed rather than removed.

A stable polishing process requires a uniform surface temperature. Hence, a closed-loop power control was established using a PID controller in a data acquisition system (RedLab 2408-2AO, Meilhaus Electronic GmbH) to maintain the maximum surface temperature at the contact point of PJ and the surface at a predefined setpoint by adjusting the microwave power accordingly.

The measurements of the local surface temperature were performed with an infrared (IR) camera (Pyroview 380L, DIAS Infrared Systems), which was attached to the PJ source body. A synchronous movement of PJ source and IR camera ensures that the measurement spot always coincides with the PJ-surface contact area. The principle of operation is shown in Figure 3(A).

Figure 3 Closed-loop power control: (A) scheme of the working principle and (B) comparison of the measured temperature curve with PID control (blue) and without PID control (red), respectively.

The PJ source was moved by a 3-axis CNC-controlled motion system along an x-y-meandering toolpath. The z-direction was varied according to the surface contour along the toolpath maintaining a constant working distance (i.e. the distance between PJ nozzle and the sample surface). Figure 3(B) shows an example of temperature measurement during PJP. While the power was maintained at a constant value, resulting in the red curve, which indicates a significant temperature fluctuation, PID control was employed, resulting in the blue curve. At the end of each meander line, which corresponds to the edge of the workpiece, the power was set to a fixed lower value to prevent the temperature from exceeding the setpoint temperature.

In order to reduce internal residual stress caused by local surface heating, all samples underwent annealing following PJP in a chamber furnace (Carbolite GPC1200, Carbolite Gero GmbH & Co. KG). An individualized temperature cycle was programmed in accordance with the thermal properties of the utilized sample materials.

2.3 Sample surface geometry

A so-called Alvarez freeform lens [33] was chosen and designed as the experimental geometry. This is an adaptive optical component consisting of two identical workpieces for varying the focus. These two components can enable a variable focal length setting depending on their lateral displacement relative to each other, shown in Figure 4(B). In general, this type of geometry is described by the following mathematical formula:$$Z=A\frac{\mathrm{ }{\left(x\pm {\delta }_{x_0}\right)}^3}{3}+\left(x\pm {\delta }_{x_0}\right)y^2+E\left(x\pm {\delta }_{x_0}\right)+G$$(1)

Figure 4 Alvarez project geometry: (A) Constructed geometry with ø 25.4 mm in CAD program and (B) simulation of the focus variation.

For the specific geometry under consideration, a focal length f′ = 150 mm, a value δ_x0 = 3 mm (a shift of −3 mm in relation to each other results in the effect of a flat plane) and the other parameters A = 0.0012 and E = −0.1454 were defined accordingly, G was set to zero. By inserting the parameters into the general equation, the freeform surface is described by the following polynomial equation.$$Z=0.0004 x^3+0.0012 x{y}^2+0.1454 x$$(2)

A circular aperture with a diameter of 25.4 mm (1 inch) with a notch for a defined component alignment was selected as the outer geometry, as shown in Figure 4(A).

Due to the geometric complexity of the freeform surface, a digital process chain for freeform processing had to be developed, which has this mathematical definition of the surface shape to be created as a starting point. The surface was constructed using the optical design software Zemax and exported as a CAD file. This was followed by the data processing of the volumetric body in a CAD program in order to subsequently implement the machining programming (grinding parameters and tool path generation) using CAD/ CAM software (PTC Creo). In this step, the determination of the component coordinate system is also an important aspect in order to define the orientation of the geometry in the working area of the grinding machine. Since the samples are produced from a cylindrical workpiece, it is advisable to place the origin of the coordinate system centered in the x-y direction and on the top surface of the component in the z direction, in order to calibrate the sample position to the machine coordinate system using a tactile measuring probe (see Fig. 4(A)). After generation of the CAM data file and application of a post-processor, it was then possible to obtain a file in G-code format that is compatible to the data language of the grinding machine.

3 Material and methods

The investigations of the manufacturing chain combining grinding and PJP were carried out on fused silica (FS). Due to the well-defined composition, it is the material of choice for adjusting the parameters of processes. In addition to the fact that FS only consists of SiO₂, its low thermal expansion and low sensitivity to breakage in the presence of thermal gradients during thermal processes makes it beneficial for the PJP process.

In order to assess the form errors with respect to the surface design, a telecentric white light interferometer (TopMap Pro.Surf, Polytec GmbH) offering a measurement field of 23 × 17 mm² (stitchable to larger areas) was applied for areal measurement of the ground and ultra-fine ground surfaces. Areal form measurements before and after PJP were performed with the optical non-contact profilometer (CT350S, Cyber Technologies GmbH). The resulting figure error was calculated by matching with the nominal Alvarez design.

Birefringence induced by internal stress was determined using the StrainScope S3/180 (ilis GmbH). The measured optical path difference (OPD) was normalized to thickness, also to enable the comparison of different samples. Furthermore, the indication of the normalized OPD follows the conventions in the production of precision optics. Investigations on potential bulk material alterations were carried out using Fourier transform infrared (FT-IR) spectroscopy, where the spectra were recorded with a MIR spectrometer (TENSOR II, Bruker Corporation) equipped with a Golden Gate ATR unit and a DTGS detector. The spectra were processed using the OPUS software (version 8.1). X-ray diffraction (XRD) was applied for phase investigations. Here, the 2θ/ω scan was measured with a diffractometer (ULTIMA IV, Rigaku Holdings Corporation) in out-of-plane geometry with parallel beam optics. A copper anode (Kα λ = 0.15406 nm) was used as x-ray source. The diffractometer was equipped with a scintillation detector. The diffractograms were analyzed using the software package PDXL (Rigaku Holdings Corporation).

Microscopic white light interferometers (WLI, NPFLEX, Bruker Corporation and TopMap Micro.View, Polytec GmbH) were used to determine surface waviness and roughness. By using different magnifications (5×, 50×), topographical features on different lateral scales were measured. The 5× objective covers a spatial frequency range of 2.4 · 10⁻³ μm⁻¹ to 7.8 · 10⁻² μm⁻¹, while the 50× objective is applicable in a range of 2.4 · 10⁻² μm⁻¹ to 8.4 · 10⁻¹ μm⁻¹. Figure 5 gives an example of a mean isotropic power spectral density function (PSD) of a lapped and PJP surface (adapted from [29]). The spatial frequency ranges that can be measured using the two objectives have a slight overlap. Waviness and roughness Sq values (designated as Sq_W and Sq_R, respectively) were calculated from the areal measurements after removal of a 2nd polynomial surface, to subtract the curvature due to the sample surface geometry.

Figure 5 Roughness measurement given in a mean isotropic PSD: Spatial frequency ranges and calculated values of SqW (green area) and SqR (yellow area) having a slight overlap. (adapted from [29]).

4 Results

The following section presents the primary findings of the investigations into freeform machining of FS. It is divided into three parts: the experimental grinding tests, the transfer point to PJP and the optimization of PJP.

4.1 Grinding investigations on fused silica

The grinding of freeform geometries is characterized by a number of special kinematic features. For example, the use of spherical grinding tools results in near point-shaped tool engagement or a very small contact surface with the component. A defined tool inclination is required for machining outside the center of the spherical tool (where cutting speed = 0 m/s), while a constant angular contact with the surface normal is maintained by simultaneous axis movements. Along the surface, there is usually a descending or meandering path movement in the x/y-direction. This can result in unwanted periodic or mid-spatial frequency form errors. Theoretically, the surface structure is formed by superimposing the individual tracks with the ball shaped grinding tool, as illustrated in Figure 6(A), which leads to a wavy surface depending on the path distance PD and infeed depth a_p. This should be prevented as far as possible by optimizing the kinematic parameter settings, or minimizing the amplitude and wavelength of the structure, respectively, in order to provide sufficient conditions for the subsequent PJP process.

Figure 6 Schematic illustration of surface formation: (A) theoretical surface formation (side view; feed direction perpendicular to the image plane) in ball tool grinding by combining descending grinding paths and (B) schematic comparison between an ideal spherical tool impression and a deformed (pressed wide) tool impression into the workpiece surface.

To investigate kinematic interactions and reduce such errors, as well as generate a homogenous surface quality with low roughness, several experiments were carried out on FS samples (HPFS 7980, Corning, Inc.). In addition, the shape deviations between the actual and target geometry should be as small as possible. For this purpose, a logical and yet novel investigation approach was pursued for grinding applications, which successively progresses from the tool impression in a single grinding path, comparable to the generation of a tool removal function during polishing, to the surface generation of several grinding paths and then to complex real free-form machining.

In the preliminary tests using flat components, it was first analyzed to what extent the shape of the grinding paths with spherical tools corresponds to the theory, i.e. whether the topography formed actually reflects an ideal spherical impression. For this, several single grinding paths were generated and measured regarding depth, shape and width, while varying parameters like ball tool diameter (between 5 and 30 mm in diameter), cutting depth and inclination angle. This was done for fine grinding with metal bond and for ultra-fine grinding with resin bond tools.

The generation of grinding tracks with a circular cross-section has been generally confirmed. However, the paths ground using resin-bonded tools were up to 50% wider than the geometrically expected value, compared to a maximum of 20% higher width for the metal-bonded tools. This indicates an expectedly greater tool deformation in the area of contact with the surface in the resin-bonded tool, which can be explained by the higher elasticity and lower strength of the bond. It means that the tools get slightly pressed wider by the contact forces acting during grinding, which can actually be beneficial since a wider tool shape reduces the height of the grinding structures for a constant PD. This effect is illustrated in Figure 6(B).

No significant influence regarding inclination angle, tool diameter or cutting depth on the relative width deviation to the geometrical value of the grooves was observed. However, regarding the generation of a constant grinding depth and efficient production best results were achieved with an inclination angle of α = 30° and cutting depth of a_p = 60 μm for metal bond fine grinding and a_p = 10 μm for resin bond ultra-fine grinding.

4.1.1 Fine grinding investigations on shape correction and mid-spatial form errors

In the experiments on the Alvarez geometry generation, the influence of kinematic settings was first analyzed. The influence of the fine grinding step (D30 metal bond) was found to be most influential on shape accuracy and introduction or removal of periodic mid-spatial defects. Because of this, the following investigations are carried out in the fine grinding regime. For this purpose, the direction of the feed movement in the x/y-direction was varied in relation to the inclination angle plane, as shown in Figure 7. On the one hand, feed direction and tool inclination angle α were oriented vertical to each other (Fig. 7(A)) and on the other hand, they were oriented parallel to each other (Fig. 7(B)). Grinding parameters regarding removal depth (a_p = 60 μm), feed rate (v_f = 300 mm/min) and cutting velocity (v_c = 7.9 m/s) were set at constant, experience-based values. A ball tool diameter of 10 mm was used.

Figure 7 Schematic visualization of Alvarez freeform grinding with a variation of grinding kinematics: (A) feed direction vertical to tool inclination angle α and (B) feed direction parallel to tool inclination angle α.

The resulting microtopography of the samples was measured using WLI. In order to make the medium and high-frequency surface defects recognizable, a form subtraction was applied to the measurement images using a third-order polynomial. The result shows that in contrast to the vertical orientation, which leaves behind clearly recognizable grinding structures in Figure 8(A) with a period that exactly reproduces the chosen path distance of 100 μm, the parallel feed-to-tilt kinematics is highly advantageous, as this produces a relatively homogeneous surface (Fig. 8(B)) which was shown in these investigations for the first time.

Figure 8 Kinematic comparison in freeform grinding experiments showing resulting surface topographies after fine grinding (D30), measured by WLI: (A) feed direction vertical to tool inclination angle and (B) feed direction parallel to tool inclination angle.

The proposed explanation for this is that the contact behavior of the vertical feed-to-inclination setting can be approximately compared with lateral surface grinding on cylindrical tools, which also generally results in a scored surface. Here, all engaged grains are at the same distance from the tool center at a fixed position and thus all have the same cutting speed. The grains all have a minimally different grain protrusion, and each grain re-engages in the same area of the sample surface after each full tool rotation. More protruding grains are thus formed directly in the machined sample surface, shown in Figure 9(A). With a parallel feed rate to the inclination, however, the engagement behavior is comparable to face grinding, which can enable more homogeneous grinding results. Each grain covers a circular path on the sample surface, which is superimposed with the linear feed movement. Depending on the position of the grain on the tool face, the circular path size and cutting speed of individual grains also differ. In this case, all the paths of the grains overlap to a certain extent, which results in a relatively statistically distributed surface structure (Fig. 9(B)). It can be stated that the parallel kinematics are therefore advantageous for avoiding this described form of medium-frequency, periodic defect structures representing the path distance and thus producing a more homogeneous surface.

Figure 9 Comparative schematic representation of topography generation during lateral and face grinding: (A) lateral surface grinding with exemplary protruding grains resulting in linear scores and (B) face grinding with exemplary protruding grains resulting in a more statistically distributed surface structure.

In order to obtain an overview of the entire component surface, a stitching measurement was carried out using WLI. Once again, a form subtraction was carried out using a fitted third-order polynomial, which allowed an error topography to be visualized, showing low- and medium-frequency deviations and structures. In addition to other minor surface deviations, two dominant types of grinding structures are shown in Figure 10. Medium-frequency ripple structures aligned vertically to the tool feed direction occurred on the entire surface. In addition, a characteristic “arch structure” was apparent.

Figure 10 Calculated error topography obtained by measurement with telecentric WLI showing unwanted surface structures.

After carrying out various iterative parameter and machine setting variations, a cause for the arch structure was identified and corrected. The arch structure is characterized by a positional dependency of the component in the machine room or on the mounting plate used for fixing, respectively. This surface deviation occurs if the component positioning is eccentric in relation to the center of the table axis. Possible causes include influences of axis stability and deviations in path planning with off-center sample fixation. If the sample is fixed centrical, the formation of such a structure in the grinding process can be prevented. In the areal measurement given in Figure 11(A) no arch-like structure can be recognized. As a result, central positioning was always used for the following freeform generation on the CNC machine employed.

Figure 11 Iterative reduction of unwanted grinding structures: (A) removed arch structure and (B) removed ripple structures.

The cause for the vertically aligned ripple structures, however, is to be found way earlier in the freeform generation chain, namely in CAM program settings. In the CAM parameters, the so-called tolerance setting determines the accuracy and number of coordinate points that map the grinding path. The setting accuracy of the grinding machine used is one micrometer. It was experimentally found and evaluated that if the tolerance is set to 1 μm, the ripple structures appear on the machined surface. A coarser tolerance leads to an increase of the ripple wavelength, while a tolerance setting of 0.1 μm, i.e. an order of magnitude finer than the machine accuracy, prevents the formation of such ripples, as shown in Figure 11(B) The finer resolution generates more reference points in the CNC program, which are then interpolated to apparently form a more continuous and smoother tool path. However, the finer resolution also increases the size of the machine data files significantly which can complicate data handling. Therefore, one aspect of Section 4.2 deals with the initial condition of the ground surface regarding tolerance setting that is required or suitable for the final PJP.

The iterative surface improvements are also apparent by observing the PSD function of Alvarez lenses with different optimization grades. Figure 12 compares the resulting surface deviations in dependency of their spatial frequency measured with the telecentric WLI for two fine ground Alvarez samples shown before. The sample seen in Figure 10, ground without process optimizations, is characterized by rather high deviations in the low spatial frequency region and distinct peaks in the mid- to high-spatial region, representing periodic errors in Figure 12(A) compared to a more flat and uniform PSD curve in Figure 12(B) for the sample shown in Figure 11(B) which was produced in an optimized process.

Figure 12 PSD comparison generated from telecentric WLI measurement of D30 fine ground Alvarez lenses: (A) Alvarez sample shown in Figure 10 before process optimizations and (B) sample shown in Figure 11(B) after process optimizations.

After carrying out the process optimizations described, it is furthermore possible to achieve sufficient shape accuracies of the freeform geometry between approx. 3–6 μm PV value with the grinding machine used, which is expressly not an ultra-precision machine.

4.1.2 Ultra-fine grinding for roughness reduction

After fine grinding the Alvarez samples are ultra-fine ground as described above, using a resin bond tool with fine grain size D16. The tool paths are not changed for this, only the removal depth (a_p = 10 μm), feed rate (v_f = 100 mm/min) and cutting velocity (v_c = 13.1 m/s) are adapted to the changed grinding regime. It is possible to achieve the Alvarez shape with a semi-transparent, homogeneous surface on fused silica after ultra-fine grinding, as shown in Figure 13.

Figure 13 Ultra-fine ground Alvarez freeform using optimized parameters as a reference for subsequent PJ polishing: (A) photography of a sample and (B) shape representation measured by WLI.

Manufactured Alvarez samples were again investigated regarding surface deviations of different spatial frequencies. Qualitative inspection of the sample surfaces with a telecentric WLI and depiction of the measured figure error topography images in Figure 14 show, that apparently, the ultra-fine grinding process does not significantly produce new form deviations and mid-spatial frequency errors in comparison to the fine ground state.

Figure 14 Topographical comparison (figure error topographies) for one Alvarez sample after (A) D30 fine grinding and the same sample (B) after D16 ultra-fine grinding measured by telecentric WLI.

This is confirmed by a quantitative evaluation of the surface topography using the PSD function given in Figure 15. It is apparent that the ultra-fine ground state produces a lower PSD curve along the whole spatial frequency range. This means that all surface deviations from waviness to mid-spatial errors and higher frequency errors get reduced homogeneously by application of the ultra-fine grinding process to the freeform surface.

Figure 15 PSD comparison generated from telecentric WLI measurement for the same Alvarez sample after D30 fine grinding and after D16 ultra-fine grinding.

The achievable roughness is around 100 nm Sq which is an order of magnitude higher than the possible roughness on planar samples described in the introduction. This can be explained by the mentioned differences regarding tool-sample interactions between planar grinding with big areal tool engagement using cup tools and punctual, sub-aperture tool engagement using ball tools. Still, the resulting surface quality after ultra-fine grinding is at a suitable level to apply the PJP process afterwards as shown in the next chapters. The ultra-fine grinding process is a relevant connection point in order to achieve a polishable surface starting from the smooth ground state with a D30 metal bond. The diagram in Figure 16 emphasizes the roughness reduction from fine grinding to ultra-fine grinding with D16 resin bond is approx. 86 %.

Figure 16 Roughness comparison of the fine and ultra-fine ground surface condition on Alvarez lenses of fused silica glass, measured by WLI with 20× objective [n = 4].

After the process optimizations have been carried out as described, freeforms are finally ground by varying selected factors, and several samples are produced using the grinding process optimizations found. The final step of the process chain, the PJP, is then examined and optimized on these samples, see Sections 4.2 and 4.3.

4.2 Linking of grinding and PJP process

The implementation of a process chain requires the establishment of appropriate transfer points. This ensures that the surface state generated in the pre-process is sufficiently precise for the subsequent process. In order to achieve high production efficiency, it is important to avoid the formation of grinding marks that are beyond the scope of possible smoothing. Conversely, it is imperative to refrain from “over-accuracy,” which entails the avoidance of grinding marks that would be undetectable after PJP. The initial estimations of the grinding marks to be avoided can be made using a filter function that describes the PJP [29]. Figure 17 shows an examples of topography measurements of ground surfaces that are not yet sufficiently prepared for PJP. The horizontal grinding marks in Figure 17(A) are clearly recognizable, but vertical marks are not concise. However, the PSD function along the x-direction in Figure 17(C) indicates a prominent peak at a spatial frequency of 2.5·10⁻³ μm⁻¹. The areal mapping after applying the filter function confirms that presence (see Fig. 17(B)). For completeness, the PSD function in y-direction, given in Figure 17(D), shows the prominent peak at about 5·10⁻³ μm⁻¹ which significantly decreases when applying PJP. The disappearance of the numerous peaks at spatial frequencies above 1·10⁻² μm⁻¹ is most obvious in Figure 17(D). These result from the engagement of grinding grains and can almost completely be smoothened by the PJP process.

Figure 17 First approximation to determine the transfer point of the process chain after grinding: (A) areal WLI measurements (5×) of ground samples is (C) filtered using a Gaussian filter function describing the polishing effect. (B, D) PSD functions emphasize prominent peaks.

Further investigation of the influence of improved grinding parameters on PJP results was conducted by experiment with a number of samples. The variation of grinding parameters is outlined in Table 1.

4.2.1 Form measurement

As the objective is to maintain the surface form throughout the PJP process, an assessment was conducted after ultra-fine grinding and compared to the results after PJP, as illustrated in Figure 18 for the samples listed in Table 1. After fine grinding the PV ranged from 4 μm to 8.5 μm. The comparison to the form measurement taken directly after the PJP process and after the subsequent annealing step reveals an improvement in the PV, as presented in Figure 18(A). It is evident that the annealing process does not result in a significant change in PV values. A notable improvement can be detected in the case of both #S1 and #S2. To ensure a fair comparison, the root mean square (RMS) values, shown in Figure 18(B), were used to eliminate the impact of outliers in the measurement. There are minimal variations, but no significant alteration to the initial surface form. The discrepancies could be attributed to measurement errors, such as thermal imbalances in the measurement environment or deviations in the freeform fit routine.

Figure 18 Comparison of figure error of FS freeform Alvarez lenses after grinding und every subsequent step of PJP: (A) PV and (B) RMS.

Birefringence due to internal stress was investigated by OPD measurements. The measurements normalized to local sample thickness are shown in Figure 19 for sample #S2. It is evident that internal stress can already be detected in the ground sample depicted in Figure 19(A). Internal stress arises during the glass forming step, which is a thermal process. Defined cooling curves ensure that this is kept to a minimum. The grinding process can also generate further internal stresses due to the clamping of the sample. The birefringence was less than 5 nm/cm, which meets the requirements for precision optics.

Figure 19 Measurement of birefringence of FS freeform lenses: OPD for (A) ground state, (B) after PJP, and (C) after the final annealing step.

As shown in Figure 19(B) the PJP causes a significant increase in OPD. At x ≈ +10 mm there is a notable surge in the measured values. This marks the start position of the PJ polishing process, where the hot PJ rapidly heats the sample from room temperature to the set point of 2000 °C. As the process continues, the sample base temperature remains at an elevated level, resulting in a reduced increase in OPD. However, the process-related local heating and subsequent natural cooling after PJP, which does not follow a defined cooling curve, results in an increase in stress birefringence. The subsequent controlled annealing procedure reduces the stress birefringence through a programmed heating and cooling cycle that includes dwell times for relaxation. Afterwards, the samples again meet the requirements for precision optics, as demonstrated by the example of #S2 in Figure 19(C).

Further analysis of the material’s response to the thermal load from the PJ revealed no irregularities, provided that the samples after PJP were subjected to annealing. Figure 20 presents the results of XRD and FT-IR measurements on plasma-polished sample #S2, both without and with annealing, in comparison to a conventionally polished reference FS surface. As shown in Figure 20(A), there is no discernable difference in the XRD measurements with respect to the intensities, indicating that there is no significant change of the amorphous structure. The FTIR measurement shown in Figure 20(B) can detect a change at 3670 cm⁻¹ when the sample is not annealed after PJP. Repeating the measurement at different positions (m01, m02, m03) reveals minimal differences, but the general result was confirmed. It is likely that a change in the bulk material has occurred, as the peak indicates the stretching mode of hydroxyl (OH⁻) [34, 35]. After annealing, the overall spectrum is at the same level with the reference measurement, which highlights the importance of the additional thermal post-treatment.

Figure 20 Investigations of structural changes of FS bulk material: (A) XRD measurement (B) FT-IR measurement.

4.2.2 Waviness and roughness

It is essential to reduce the waviness and roughness significantly during PJP in order to obtain a functional optical surface. Therefore, the waviness and roughness of the freeform samples were measured before and after PJP at two defined positions (y = ±10 mm, labelled P (peak) and V (valley)), where the nominal surface prescription exhibits its global maximum and minimum, as shown in Figure 21(A).

Figure 21 Surface roughness measurement of FS freeform lenses: (A) visualization of the Alvarez geometry and indication of the positions for roughness measurements and comparison of (B) waviness and (C) roughness for ground and PJP state [n = 2].

Figure 21(B) illustrates the mean Sq values, accompanied by error bars that highlight the slight discrepancy in Sq values between the two measuring positions. This could be attributed to the local differences in sample thickness and curvature between this positions. The effective thickness as per the Alvarez design is 4 mm (at V) versus 6 mm (at P).

The results of Figure 21(B, C) demonstrate the effectiveness of the polishing process, which reduced the roughness by approx. 99.5% to values of Sq_R = (0.69 ± 0.14) nm in the high frequency range (see Fig. 5). Additionally, the waviness was reduced by more than 50% yielding values of Sq_W = (53.59 ± 20.45) nm. The best result was observed for sample #S2, which demonstrated a 75% improvement in waviness. For #S3, the best nominal waviness was achieved (Sq_W = 37.5 nm), though this came with an expense of the grinding process time. From the perspective of an efficient process chain, smoothing is not significantly better compared to sample #S2 ground in half of the time.

4.3 PJP process optimization

Accordingly, two additional samples (#S4 and #S5) were ground with the same parameters used for #S2 to further optimize the PJP process by varying PJ scan velocity and temperature set point. The nomenclature is provided in Table 2. Depending on PJ scan velocity polishing time is 24 min and 46 min, respectively.

4.3.1 Form measurement

All samples show no appreciable change in figure error, which proves the surface form preservation being an important property of the thermal polishing process. Characteristic values such as PV, RMS and power error are retained.

The OPD result in fulfilling the benchmark value of rms-OPD < 5 nm/cm is corresponding to the requirement for precision optics all after PJP including annealing.

4.3.2 Waviness and roughness

Figure 22 compares the averaged waviness and roughness measured at the two positions, designated as “P” and “V”, as shown in Figure 21(A), after grinding and after PJP, respectively. The error bars in Figure 22 indicate the range of deviations between the two measurement positions. Due to the geometry-related surface curvatures, comparable roughness measurements could only be carried out at the two positions exhibiting near-zero slope. Otherwise, significant measurement artifacts showing interference fringes are visible, which deteriorate the measurement and make it unusable. Despite all three samples are ground with the same parameters, there is a notable discrepancy in the levels of initial waviness and roughness. This is likely a result of the dressing condition and successive wear of the grinding tool, as the samples were ground in the order of their respective numbering.

Figure 22 Surface roughness measurement of FS freeform lenses: (A) waviness and (B) roughness of ground and PJP state [n = 2].

While the overall waviness values Sq_W of the ground samples are lower than the roughness values Sq_R, they cannot be reduced by the same extent during PJP. However, an improvement in the Sq_W range from 50 nm to 15 nm was achieved.

The mean isotropic PSD functions in Figure 23 allow a more precise look at the measurement of each sample regarding the process steps and the two distinct measurement position, which were shown in Figure 21(A). First, it is obvious that after grinding more peaks occur at position “P” (see Fig. 23(A)) compared to position “V”. Both PSD functions also reveal that peaks located at higher spatial frequencies can be reduced more efficiently. The peaks at spatial frequencies higher than 0.01 μm⁻¹ have largely disappeared. Hence, low frequency errors must be avoided in the grinding process since they cannot be corrected later in the thermal polishing step. It is obvious that PJP parameters T = 2050 °C and v = 0.5 mm/s are beneficial to achieve the best values here (see #S5). This is also evident from the overall lowest Sq_W values given in Figure 22(A).

Figure 23 Waviness of FS freeform lenses presented in mean isotropic PSD functions given for the (A) position P and (B) position V (according to Fig. 21(A)).

As presented in Figure 22(B), the roughness of the ground samples is Sq_R = (179.6 ± 23.0) nm. Applying PJP, the roughness of the Alvarez lenses can be reduced very effectively. The slower scanning velocity of the PJ (v = 0.5 mm/s, see Table 2) seems to enable a more uniform smoothing of the sample. The areal measurements given in Figure 24. clarify this finding for sample #S2 which was the worst of the optimization batch and #S5 which was the best overall. The damage, which probably resulted from micro-chipping caused by the abrasive grain, is no longer apparent after the PJP. However, the Sq_R values can be improved for all samples to less than 1 nm by applying PJP. This indicates the stronger impact of the PJP to higher spatial frequency errors.

Figure 24 (A, B, E, F) Roughness of FS freeform lenses given as areal measurement maps for ground surfaces; (C, D, G, H) corresponding measurements at the same positions after PJP (position P and V according to Fig. 21(A)).

It was shown that the PJP with an increased temperature of 2050 °C in combination with a reduced scan velocity of the PJ (v = 0.5 mm/s) is beneficial. Considering both roughness and waviness, the best PJP results have been obtained for sample #S5. The birefringence due to internal stresses was also in accordance with the requirements for precision optics.

For the best sample mentioned above, the experimental results are compared with the corresponding theoretical polishing, applying the convolution function to the ground initial surface. So far, the predictability of the PJP result has been demonstrated for planar samples [29]. A more precise look at the mean isotropic PSD functions is given in Figure 25. The PSD curve obtained after PJP fits very well with the graph resulting from the filtered ground state, in the spatial range, where the convolution function can be applied. The comparison of the initial graph (black) and the PSD after PJP (blue) demonstrates the efficacy of the smoothing process in the higher spatial frequency range. Additionally, the application of PJP to the freeform element has been shown to result in a notable reduction in waviness (see ranges in Fig. 5).

Figure 25 Predictability of PJP result applying a filter function on the initial ground state for sample #S5: Comparison of mean isotropic PSD of the initial ground (black), filtered ground (golden) and the experimental PJP (blue).

It can be stated that the predictability of the polishing result is also sufficient for the freeform lenses and can therefore be applied for future optimization of the process chain, especially with regard to the linking of grinding and PJP.

5 Conclusion

As part of the investigations described above, it was possible to develop and systematically investigate a new type of process chain for freeform production using a combination of machining processes (grinding) and a beam-based thermal process (PJP). Both of these technology types were examined in detail and optimized for the specific application of an Alvarez lens.

With regard to the grinding processes, it was possible to identify influences on surface deviations in the complex process of freeform production and thus optimize the process step by step. After several improvement steps, sufficient shape accuracies of the freeform geometry between approx. 3–6 μm PV value were achieved. Among other things, different kinematic approaches, the machine accuracy and CAM program solution have a major influence on the resulting surface deviations and offer a high potential for optimization. Ultra-fine grinding with resin-bonded tools can achieve low roughness and good surface qualities suitable for the final PJP process regarding workpieces out of fused silica. They can be polished in one step while retaining their original form. High-frequency defects in particular can be removed so that roughness of Sq_R < 0.5 nm can be achieved. Grinding marks having longer wavelengths can only be removed to a limited extent (Sq_W < 20 nm). Here, the grinding process should be further optimized according to the target specifications of the desired optics. The application of the convolution function to theoretically assess the outcome of a PJP process can be helpful to minimize the amount of sample material used. It was shown that freeform manufacturing of the chosen Alvarez design can be performed within 150 min processing time.

Acknowledgments

HM and TA thank T. Liebeskind for chemical sample preparation but also N. Schönherr and J. Griebel für FT-IR and XRD measurements. SH and JB thank M. Binder for PSD evaluation of ground sample topographies.

Funding

This work was supported by financial support by German Federal Ministry of Education and Research (BMBF) within the framework of the VIP+ program (No. 03VP08631) “Validation of a new process chain for the production of freeform optics by Plasma Jet smoothing of ultra-finished freeform surfaces”.

Conflicts of interest

The authors declare that they have no competing interests.

Data availability statement

All data generated or analyzed during this study are included in this published article.

Author contribution statement

HM conducted the PJP experiments and measurements, performed analysis of the results and handled the paper documentation. SH and CS conducted the grinding experiments and measurements, SH was also mainly responsible for the evaluation of the research and the writing of the paper for this part. SF contributed the design and construction of the Alvarez geometry. JB contributed significantly to the evaluation and interpretation of the results with his expertise in the field of manufacturing technologies, and corrected and expanded the manuscript. TA wrote MATLAB fitting routines, offered valuable advice for the interpretation of PJP data and took extensive care of correction of the manuscript to present the data more clearly. The authors have read and approved the final manuscript.

References

All Tables

All Figures

	Figure 1 Schematic representation of a new process chain for freeform fabrication with a combination of grinding processes, including ultra-fine grinding, and plasma jet polishing.
In the text

	Figure 2 Visualization of kinematic properties for freeform grinding.
In the text

	Figure 3 Closed-loop power control: (A) scheme of the working principle and (B) comparison of the measured temperature curve with PID control (blue) and without PID control (red), respectively.
In the text

	Figure 4 Alvarez project geometry: (A) Constructed geometry with ø 25.4 mm in CAD program and (B) simulation of the focus variation.
In the text

	Figure 5 Roughness measurement given in a mean isotropic PSD: Spatial frequency ranges and calculated values of SqW (green area) and SqR (yellow area) having a slight overlap. (adapted from [29]).
In the text

	Figure 6 Schematic illustration of surface formation: (A) theoretical surface formation (side view; feed direction perpendicular to the image plane) in ball tool grinding by combining descending grinding paths and (B) schematic comparison between an ideal spherical tool impression and a deformed (pressed wide) tool impression into the workpiece surface.
In the text

	Figure 7 Schematic visualization of Alvarez freeform grinding with a variation of grinding kinematics: (A) feed direction vertical to tool inclination angle α and (B) feed direction parallel to tool inclination angle α.
In the text

	Figure 8 Kinematic comparison in freeform grinding experiments showing resulting surface topographies after fine grinding (D30), measured by WLI: (A) feed direction vertical to tool inclination angle and (B) feed direction parallel to tool inclination angle.
In the text

	Figure 9 Comparative schematic representation of topography generation during lateral and face grinding: (A) lateral surface grinding with exemplary protruding grains resulting in linear scores and (B) face grinding with exemplary protruding grains resulting in a more statistically distributed surface structure.
In the text

	Figure 10 Calculated error topography obtained by measurement with telecentric WLI showing unwanted surface structures.
In the text

	Figure 11 Iterative reduction of unwanted grinding structures: (A) removed arch structure and (B) removed ripple structures.
In the text

	Figure 12 PSD comparison generated from telecentric WLI measurement of D30 fine ground Alvarez lenses: (A) Alvarez sample shown in Figure 10 before process optimizations and (B) sample shown in Figure 11(B) after process optimizations.
In the text

	Figure 13 Ultra-fine ground Alvarez freeform using optimized parameters as a reference for subsequent PJ polishing: (A) photography of a sample and (B) shape representation measured by WLI.
In the text

	Figure 14 Topographical comparison (figure error topographies) for one Alvarez sample after (A) D30 fine grinding and the same sample (B) after D16 ultra-fine grinding measured by telecentric WLI.
In the text

	Figure 15 PSD comparison generated from telecentric WLI measurement for the same Alvarez sample after D30 fine grinding and after D16 ultra-fine grinding.
In the text

	Figure 16 Roughness comparison of the fine and ultra-fine ground surface condition on Alvarez lenses of fused silica glass, measured by WLI with 20× objective [n = 4].
In the text

	Figure 17 First approximation to determine the transfer point of the process chain after grinding: (A) areal WLI measurements (5×) of ground samples is (C) filtered using a Gaussian filter function describing the polishing effect. (B, D) PSD functions emphasize prominent peaks.
In the text

	Figure 18 Comparison of figure error of FS freeform Alvarez lenses after grinding und every subsequent step of PJP: (A) PV and (B) RMS.
In the text

	Figure 19 Measurement of birefringence of FS freeform lenses: OPD for (A) ground state, (B) after PJP, and (C) after the final annealing step.
In the text

	Figure 20 Investigations of structural changes of FS bulk material: (A) XRD measurement (B) FT-IR measurement.
In the text

	Figure 21 Surface roughness measurement of FS freeform lenses: (A) visualization of the Alvarez geometry and indication of the positions for roughness measurements and comparison of (B) waviness and (C) roughness for ground and PJP state [n = 2].
In the text

	Figure 22 Surface roughness measurement of FS freeform lenses: (A) waviness and (B) roughness of ground and PJP state [n = 2].
In the text

	Figure 23 Waviness of FS freeform lenses presented in mean isotropic PSD functions given for the (A) position P and (B) position V (according to Fig. 21(A)).
In the text

	Figure 24 (A, B, E, F) Roughness of FS freeform lenses given as areal measurement maps for ground surfaces; (C, D, G, H) corresponding measurements at the same positions after PJP (position P and V according to Fig. 21(A)).
In the text

	Figure 25 Predictability of PJP result applying a filter function on the initial ground state for sample #S5: Comparison of mean isotropic PSD of the initial ground (black), filtered ground (golden) and the experimental PJP (blue).
In the text