J. Eur. Opt. Society-Rapid Publ. 22, 51( 2026) 501
Figure 1. Schemes of applied geometries for defined inducing of SSD in fused silica samples:( A) Vickers indentation [ 20 ] and( B) Rockwell-Diamond [ 21 ] for scratches.
the coordinates P 0 =[ x 0, y 0, z 0 ]. By calculating the surface normal for each point N 0 =[ n x, 0, n y, 0, n z, 0 ] the evolving surface at a time step Dt is obtained by:
P i ¼ P 0 þ N i�1 rt i ¼ 1... k ð1Þ
where r is the etching rate in lm / s and Dt is an appropriately chosen small time step. Hence, the total etching depth at positions with surface normal [ 0, 0, �1 ] is d ¼ i r t: After each iteration step i the obtained surface coordinates P i are interpolated to the grid( x 0, y 0) yielding a local etch depth z i, followed by a Gaussian filter of width to slightly smooth the surface. Although the smoothing filter has no strict physical meaning, such smoothing acts as a simplified approach to a microscopic isotropic etching effect mimicking the Huygens-principle of elementary wavelets as described in [ 12 ].
The calculation scheme has been implemented in a selfwritten Matlab Ò script( The MathWorks, Inc.). As an example, the projection of normals of a 2D V-groove is shown in Figure 2A. By adjusting the iteration number i at a given etching rate r and time step Dt a prospective isotropic removal and topography, respectively, can be simulated. Figure 2B shows the development of the V-groove with progressive etching. In order to be able to map both, small and large removal depths, with a comparable accuracy, the calculation was performed in several defined steps of about 1 lm.
After the calculation of an isotropically etched surface z calc( x, y) based on the initial topography z 0, the result is compared by the experimentally obtained surface z exp after etching to the same depth. As the areal measurements of two consecutive PJE steps do not perfectly overlap due to slight lateral mismatch in the microscope, the lateral alignment was corrected by a Matlab Ò image recognition routine prior to performing the calculation. All pixels of z exp that do not match the original z 0 were set to zero and can therefore influence the subsequent calculation. Since this effect only occurs in the peripheral area, which is approximately zero for the leveled surface measurement, its influence on the subsequent calculation can be neglected. The flowchart in Figure 3 shows the procedure of data processing.
The comparison of z calc and z exp is done by taking the ratio given in equation( 2) where IF can be designated as isotropic factor.
IFðx; y
Þ ¼ zexp ðx; yÞ z calc ðx; yÞ ð2Þ
Thus, the experimental result of a PJE step is related to the calculated result, assuming isotropic etching with the same etching removal d. A value IF 1.0 indicates loci of isotropic etching in the experiment, while values IF > 1indicate loci of anisotropic etching. The greater the deviation from 1, the stronger the anisotropic removal. Provided that morphological distortions in the surface are anisotropically etched by the plasma jet, IF maps provide information on the position and the extent of SSD. To evaluate the anisotropic removal as a function of etch depth, the value IF mean- calculated as the average across the entire topography – and its standard deviation are used.
In the following section the application of the procedure to different surfaces is shown.
In order to gain a more detailed understanding of the characteristics of the SSD( laterally and depth-dependent), the SSD-affected areas can be represented as 3D-plots, where the measured topography data added by the removal depth( z exp + d) are plotted for loci where IF exceeds a value of 1, i. e. for anisotropically etched regions.
4 Results
4.1 Vickers imprints
Localized indentations generated by a Vickers indenter exhibit a widespread damage shape, with the lateral expansion being approximately twice as large as the measurable depth, as shown exemplarily for the initial measurement of indentation of Sample # 1 in Figure 4.
Since the increasing etching depth leads to further expansion of the indentation [ 14 ], the void volume VV was analyzed after each process step. Equation( 3) specifies the mathematical description where Z is the height value after removing the tilt of the areal measurement with respect to the area that is not affected by the indentation.
VV ¼ X ZX ð; Y ÞdXdY ð3Þ
Z0
To quantify the SSD depth the roughness parameter Sv, defined as maximum pit height of a topographic