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Axicon is known to be one of the most convenient optical means for the generation of Bessel beams36-38. However, achieving high-angle conical Bessel beams is very crucial for material processing applications22. The conventional methods use image-relayed beams with a telescopic system for demagnification. However, this approach takes up working space, is sensitive to alignment imperfections, and remains always limited by the entrance pupil diameter and damage threshold of the last optical element28,31. In this section, we numerically investigate an alternative simple approach aiming at the generation of good-quality infrared Bessel-like beams. The approach involves a controlled perturbation of the spherical wavefront produced by an ideal focusing lens. It is based on combining a lens with a modest angle axicon so that a linear radial-dependence phase contribution is added to the quadratic phase associated with the lens. The ray diagram of such optical configuration is schematically depicted in Fig. 1a. Ray-tracing shows that the combination of an axicon and a lens as a doublet (i.e. axicon-lens doublet) has the ability to efficiently produce high-angle pseudo-Bessel beams with extended-depth-of-focus. In the case of a modest angle axicon (small perturbation), we expect the typical converging angle to be imposed by the focusing power of the lens and the obtained profile can be seen as an aberrated focus. We desire a Bessel-like elongated focus resulting from the angular components not all collapsing at the same distance, and an axial intensity distribution exhibiting a slowly varying leading edge before a sharp fall on its trailing edge. Moreover, ray-tracing of Fig. 1a directly showcases the relative focal shift in the pre-focal region in comparison to single-lens focusing. This corresponds to a situation where the converging angle of the beam components even exceeded the maximum half-angle that would be derived from the NA value of the focusing lens.
Fig. 1 a Schematic ray diagram for a Gaussian beam focused by an axicon-lens doublet resulting in a pseudo-Bessel beam. b Cross-sectional view of intensity distributions across the optical axis for beams calculated with different phase perturbation conditions. The beams are focused with axicon-lens doublets including a lens of 400 mm focal length. The parameters $ \alpha $ and $ c_{s} $ are the base angle of the axicon and spherical aberration strength, respectively.
To confirm the general characteristics of the pseudo-Bessel beams originating from the axicon-lens doublets, we perform propagation calculations. We simulate the propagation of beams with the consideration of an initial phase described by:
$$ \begin{array}{*{20}{l}} \Phi(r) = k \Bigg[\dfrac{r^2}{2f} + c_{s} \dfrac{r^4}{8f^3} + \alpha (n-1) r \Bigg] \end{array} $$ (1) where $ k $ is the free-space propagation constant, r is the space coordinate of the studied problem exhibiting a cylindrical symmetry, and $ \alpha $ is the base angle of the axicon with refractive index $ n $. The first term in Eq. 1 represents the quadratic phase for an ideal focusing lens with focal length $ f $. The second and third terms correspond to added phase perturbations to describe the studied situations. The second term is added for encountering potential spherical aberrations in practical situations. The latter is introduced by accounting for the second term of a Tailor series expansion of the spherical wavefront curvature (i.e. w(r) = [r2/2R + r4/8R3 +...]; R being the radius of the spherical wavefront) with strength arbitrarily defined by the parameter $ c_{s} $. The third term describes the effect of the addition of a thin axicon.
The simulations are performed based on a numerical model using the Fresnel transfer function method as described in references39,40. For valid calculations under the paraxial approximation, we consider here an input collimated Gaussian beam of beam-waist $ \omega_{0}=2.4 $ mm and wavelength $ \lambda $ = 1550 nm illuminating the axicon-lens doublet (at $ z $ = 0) including a lens with focal length $ f=400 $ mm. For varying levels of phase perturbations corresponding to different axicon-lens combinations, we numerically calculate the normalized longitudinal intensity maps ($ x $–$ z $ view) of the resulting beams in air along the propagation direction ($ z $), as shown in Fig. 1b. For a specific value of $ \alpha $, we calculate the beam intensity distributions by introducing different strengths of spherical aberration using the parameter $ c_{s} $. The values of $ c_{s} $ are arbitrarily set so that the beam focus is significantly affected, as it is commonly observed for bulk interaction in Si due to the high refractive index (~3.5) of the material causing strong refraction at the air-material interface. In this way, we can test the robustness of the produced beams to the spherical aberration level that will depend on the processing depth.
While the value $ \alpha $ = 0° represents the normal Gaussian focus distributions (first column of Fig. 1b), $ \alpha $ = 0.2° and 0.4° represent the formation of pseudo-Bessel focus distributions. For these two cases, the simulations reproduce the essential features expected from the ray-tracing configuration in Fig. 1a, that is, an elongated depth-of-focus with a slow leading edge followed by a sharp drop of intensity. In particular, the simulations confirm that a modest change in the spherical phase (visualized in Fig. S1 of the Supplementary Information) can significantly elongate the beam focus longitudinally without affecting the lateral resolution. With the calculated beam profiles exhibiting lobes, the numerical study emphasizes the relevance of the analogy made with Bessel beams. Fig. 1b shows the effectiveness of axicon-lens doublets as focusing elements to produce Bessel-like beams and also reveals the relative robustness of the produced features despite some spherical aberration levels inevitably occurring for the considered practical cases.
Ultra-high-aspect-ratio structures through silicon using infrared laser pulses focused with axicon-lens doublets
- Light: Advanced Manufacturing 5, Article number: (2024)
- Received: 02 November 2023
- Revised: 18 March 2024
- Accepted: 04 April 2024 Published online: 10 July 2024
doi: https://doi.org/10.37188/lam.2024.022
Abstract: We describe how a direct combination of an axicon and a lens can represent a simple and efficient beam-shaping solution for laser material processing applications. We produce high-angle pseudo-Bessel micro-beams at 1550 nm, which would be difficult to produce by other methods. Combined with appropriate stretching of femtosecond pulses, we access optimized conditions inside semiconductors allowing us to develop high-aspect-ratio refractive-index writing methods. Using ultrafast microscopy techniques, we characterize the delivered local intensities and the triggered ionization dynamics inside silicon with 200-fs and 50-ps pulses. While similar plasma densities are produced in both cases, we show that repeated picosecond irradiation induces permanent modifications spontaneously growing shot-after-shot in the direction of the laser beam from front-surface damage to the back side of irradiated silicon wafers. The conditions for direct microexplosion and microchannel drilling similar to those today demonstrated for dielectrics still remain inaccessible. Nonetheless, this work evidences higher energy densities than those previously achieved in semiconductors and a novel percussion writing modality to create structures in silicon with aspect ratios exceeding ~700 without any motion of the beam. The estimated transient change of conductivity and measured ionization fronts at near luminal speed along the observed microplasma channels support the vision of vertical electrical connections optically controllable at GHz repetition rates. The permanent silicon modifications obtained by percussion writing are light-guiding structures according to a measured positive refractive index change exceeding 10−2. These findings open the door to unique monolithic solutions for electrical and optical through-silicon-vias which are key elements for vertical interconnections in 3D chip stacks.
Research Summary
Shaped beams for the fabrication of vertical interconnections in next-generation silicon chips
Through-Silicon Vias (TSVs) are essential elements for 3D integration of high-performance electronic and optoelectronic devices. Machining with so-called Bessel beams is a relevant approach for producing high-aspect-ratio TSVs in dielectrics but not in semiconductors. Researchers at LP3, a joint laboratory between CNRS and Aix-Marseille University (France), made a critical step to deliver a technology for semiconductors. Elaborating a concept based on a simple combination of conical lenses (called axicons) and microscope objectives, they produced infrared pseudo-Bessel microbeams with high angles inaccessible with standard methods. Using appropriate ultrashort pulses, they found a percussion writing modality leading to extremely high-aspect ratio structures through silicon. The characteristics of the elongated microplasmas and subsequent material modifications created inside chips support a new range of solutions for ultrafast vertical communication channels (optical and electrical) in silicon chips.
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