# YAG Laser Rod 3D Corrective Process Optimization through Tool Influent Function Shape Inspection
## 1 Introduction  
Yttrium aluminum garnet Y3 Al5O12 (YAG) is an optically isotropic crystal of cubic structure characterized by a high Mohs hardness of 8.5 and a relatively high thermal conductivity of approximately 13 W/mK. The crystals are grown by the Czochralski method by drawing an oriented crystal nucleus from a melt placed in an iridium or molybdenum crucible. Neodymium ion-doped YAG is the most widespread active medium in the field of solid-state laser application, where it is used in the form of rods with precisely machined (polished) faces with high demands on both minimum surface micro-roughness and shape error, which can be further minimized by 3D subaperture shape correction. However, in the process of 3D shape correction of optical surfaces by computer numerical control (CNC) technologies, where the subaperture tool moves along precisely defined paths, specific structures are generated, often falling into the category of so-called mid-spatial frequencies (MSFs) [1], which can significantly limit the optical properties of the resulting optical element. The causes of MSF generation are numerous [2] and include the geometry and material properties of the polishing tool. The resulting character of the surface is then determined by their interaction. In addition, the precision of 3D shape correction controlled by dwell time algorithms [3, 4] is significantly affected by the possibility of achieving the most accurate tool influence function (TIF) shape modeling using a suitable mathematical approach. A rotationally symmetric Gaussian function [5] is often used, which, in many cases, may not be accurate enough and thus may introduce a substantial systematic error into the process of 3 Dcorrection. The polishing tool geometry, its construction material, and the process settings used all have a significant effect on both the final surface quality and the final shape accuracy; therefore, their optimization must be given considerable attention. Besides non-contact shape correction methods such as ion beam figuring (IBF) [6] or fluid jet polishing (FJP) [7], the dominant method for dwell time shape correction consists of processes based on contact tool utilization, where the active tool part consists of a flexible elastic membrane (FEM) inflated by compressed air or of a viscoelastic head (VEH) [8]. The contact between the tool and the polished (corrected) surface is then realized by a suitably selected polishing pad. Material removal can generally be described using the traditional Preston relation (1):  
\[\Delta\hbar(xy)=k\times p(xy)\times v(xy)\times\Delta t(xy) \tag{1}\]  
where k is the process parameter, p is the pressure on the polished surface, v is the relative velocity of the tool-surface movement, and \(\Delta t\) is the dwell time. Based on Equation (1), in the case of a static tool action (without movement in the x and y axes), at a selected pressure, tool speed, and at a given dwell time, material removal can be simulated relatively easily (see Figure 1) because pressure is either constant (for the FEM tool) or given by the tool deformation caused by interaction with the workpiece surface [9]. However, the above-presented simulation models assume a relatively simple pressure distribution due to the perfectly elastic behavior of the tool material, which can be described by [PERSON]'s law (2):  
\[\delta=E\times\varepsilon \tag{2}\]  
where \(\delta\) is the stress in the material (\(\delta\) determines the polishing pressure), \(\varepsilon\) is the relative deformation, and E is the elastic modulus. Nevertheless, this assumption about tool construction materials based on rubbers or other organic polymers does not correspond with reality [10]; for a description of their behavior (under static stress), it is necessary to use, at minimum, more complex models such as [PERSON]'s model (3):  
\[\delta=E\times\varepsilon+\eta\times(d\varepsilon/dt) \tag{3}\]  
where \(\eta\) is the viscosity of a considered material and \(d\varepsilon/dt\) is the relative deformation change with time.