J. Eur. Opt. Society-Rapid Publ. 22, 3( 2026) 27
Figure 2.( a) and( b) provide the obtained u ff function values after the iterative evaluations as described in Section 2.( c) Pointwise function difference Du ff between maps shown in( a) and( b) which encodes the local difference between both surface solutions.( d) provides convergence curves using the quantity J I, according to the definition provided in [ 1 ].
we performed non-sequential ray-tracing analysis with the reconstructed three-dimensional freeform lens models via equation( 2). These results are presented in Figure 3 with, the original target distributions, the obtained mapping functions m and the irradiance information extracted from the non-sequential analysis, being shown. As described in Section 2, our numerical implementation is based on standard local polynomial basis functions. This results in C 0-only continuous solutions, with potential detrimental effects induced in the reconstructed three dimensional surfaces. To further evaluate this, we make use of an alternative surface representation based on Forbes Q freeform polynomials as briefly introduced in Section 2. We employ least-squares approximations defined directly on the original surface domains, by projecting the solutions into basis functions characterized by indices n and m, with all terms satisfying 2m + n T being included in the expansion. From the obtained projections, we calculate residue function maps DS z by subtracting the polynomial fits from the FEM solutions. These residue maps, which are shown in Figures 4a and 4b were obtained with a T value of 54, resulting in a total of 1592 basis terms. From the chosen T, we estimate the minimum radial feature period being captured by the polynomial fit, by considering the shortest axis radius, yielding an approximate value of r 200lm ¼ 7: 40lm. At this
T = 2
minimum radial period, a clear distinction in terms of the residue surface height can be observed.
For the uniform target case, a peak residue of 67 nm was obtained, with the large majority of the surface height structure being captured by the smooth polynomial fit. Differently, for the modulated-cosine target, the obtained fit residue contains relevant spatial features with height contributions in the order of 160 nm. In other words, the unresolved surface structure consist of spatial features bounded by an extension of approximately Dr with height amplitude contributions not larger than a value of k / 4.
Additionally, power spectral density( PSD) analysis was performed from the extracted DS z residue maps. These results are presented in Figure 4c as a comparison between cross-sectional views of the obtained PSDs. Both curves reveal that spatial features located at frequencies larger than 1 lm �1 are highly suppressed. Complementary to this, in Figure 4d we compare root-mean-square( RMS) estimates for DS z as functions of the expansion index T. Inthis case, clear RMS reduction trends can be observed, with values of 3.38 and 7.75 nm achieved at T = 106, forthe uniform and cosine-squared targets, respectively. Finally, the freeform lens corresponding to the cosine-squared target pattern was fabricated following the methodology described in Section 3. The obtained freeform lens was characterized using a white-light interferometer( WLI, Zygo, Nexview NX2, 10X, Dxy: 750 nm) for which the recorded intensity map is presented in Figure 5a.
As performed for the design surface case, we make use of Forbes Q freeform polynomials( T = 54) to fit the measured surface height profile. In( b), a portion of the obtained height fit residue map DS z with contributions bounded by a value of k is presented. From the full extracted DS z, a total residue RMS of 110 nm, in addition to a peak-to-valley value of 4.67 lm were estimated, which compare to values of 37 and 263 nm for the designed surface case at the same T value, respectively. These parameter differences can be