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J. Eur. Opt. Society-Rapid Publ. 22, 3( 2026) hybrid numerical scheme to obtain all gradient information required as part of the iterative routine. In this case, for all quantities in equation( 1) excepting m and u ff, we rely on automatic differentiation tools via an open source implementation of a differentiable ray-tracing framework [ 3 ]. On the other hand, for m and u ff we make use of the polynomial preserving recovery( PPR) technique [ 4 ] which allows obtaining least squares approximations to the gradients at the domain’ s degrees of freedom. Importantly, by making use of this approach, we obtain a sparse matrix operator which can be directly applied on each iteration. Once a set of solution functions m and u ff are obtained, these are used to reconstruct a three-dimensional freeform surface using S ¼ u 1 ðxÞI 0 ðxÞþ XN�1 ½ u i ðxÞI i�1 ðxÞŠþu ff I N ðxÞ: ð2Þ i¼2
Finally, as a means of comparison and to extract information on the smoothness associated with the obtained solutions, we have also chosen to use Forbes Q freeform polynomials [ 5 – 6 ], with these being commonly used to express freeform surfaces in terms of global orthonormal basis functions.
3 Fabrication details
The chosen freeform lens was fabricated by means of a commercially available 2PP system( Photonic Professional GT2 by Nanoscribe GmbH, Karlsruhe, Germany) in combination with a high numerical aperture objective( Plan-Apochromat 25 / 1.40 Oil DIC, Zeiss). This objective was chosen given the obtained freeform lens dimensions, which has an elliptical shape with a major axis half-diameter equal to 400 lm. At the same time, a slicing and hatching distance of 100 nm were chosen, in order to have close fidelity to the original surface design and minimize discretization artifacts. The lens was fabricated directly on top of an AlGaInP visible laser diode( LD) with a central emission wavelength at 650 nm. These LDs consist of an emission surface protected by a window glass with a thickness of 400 lm. In order to align the freeform lens center to the emission point, an additional alignment step was performed. For this, a 10 / 0.35 NA objective was used to find the lateral location of the LD’ s emission center point, which is found 700 lm below the LD’ s upper window interface. Consequently, two reference markers were fabricated, these being used in the subsequent freeform lens fabrication step to align the structure. In Figure 1, a schematic representation of the geometrical characteristics of an LD is provided, in addition to representative rays traced from the emission point location towards the freeform optical surface direction and indications of all involved terms directly contributing to equation( 1).
4 Results
In this section we present two freeform surface design examples obtained following the methodology introduced
Figure 1. Schematic representation for the laser diode system considered in this case. The emission region is modeled as a point emitter, positioned at the origin. 300 lm above this point, a window glass with a thickness of 400 lm is found. All of the interfaces u( x) and the associated intermediate ray-directions I( x) are directly included in the Hamilton’ s characteristic function H shown in equation( 1).
in Section 2. For this, two different characteristic target functions have been chosen. The first one consists of an uniform target irradiance distribution defined over a circular aperture, with a diameter of 5 cm, at a target distance of 4.5 cm from the origin. The second scenario consists of a modulated cosine-squared function, given by cosð 2pxt b Þ2,
where x t denotes the x cartesian coordinates at the target plane and b the desired fringe periodicity. Both distribution are defined over the same spatial extent. As in [ 1 ], we make use of a standard transformation to obtain intensity functions from both irradiance distributions. For the source, we use the model
I ðh x; h y Þ¼I 0 exp �2ððhx = axÞ2 þðh y = a yÞ 2 Þ
with h x and h y being perpendicular emission angles associated with a Gaussian-like far-field intensity profile, while a x and a y represent half angles of divergence along both x and y axes. These were computed from the full-width at half-maximum angles provided in the manufacturer’ s datasheet( 28 and 9, respectively) using the relationship
a ¼ h fwhm p ffiffiffiffiffiffiffi. Following, in Figure 2 we compare the Euclidean
2ln2
distance function solutions u ff obtained for both target functions. Additionally, we contrast the extracted convergence curves in both cases, using the same definition for J I as given in [ 1 ].
From a visual comparison between the functions shown in( a) and( b), a clear similarity can be observed. Interestingly, the overall global function shapes and values extend over the same range. To better contrast both u ff functions, on( c) we display the pointwise function difference Du ff which reveals the local distinctions between both functions, with Du ff not exceeding values beyond 1 lm. Consequently,
ð3Þ