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As silicon photonics transitions from research to commercial deployment, packaging solutions that efficiently couple light into highly compact and functional sub-micrometer silicon waveguides are imperative but remain challenging. The 220 nm silicon-on-insulator (SOI) platform, poised to enable large-scale integration, is the most widely adopted by foundries, resulting in established fabrication processes and extensive photonic component libraries. The development of a highly efficient, scalable, and broadband coupling scheme for this platform is therefore of paramount importance. Leveraging two-photon polymerization (TPP) and a deterministic free-form micro-optics design methodology based on the Fermat’s principle, this work demonstrates an ultra-efficient and broadband 3-D coupler interface between standard SMF-28 single-mode fibers and silicon waveguides on the 220 nm SOI platform. The coupler achieves a low coupling loss of 0.8 dB for the fundamental TE mode, along with 1 dB bandwidth exceeding 180 nm. The broadband operation enables diverse bandwidth-driven applications ranging from communications to spectroscopy. Furthermore, the 3-D free-form coupler also enables large tolerance to fiber misalignments and manufacturing variability, thereby relaxing packaging requirements toward cost reduction capitalizing on standard electronic packaging process flows. 

We develop a framework for sampling from discrete distributions $$\mu$$ on the hypercube $$\{\pm 1\}^n$$ by sampling from continuous distributions supported on $$\mathbb{R}^n$$ obtained by convolution with spherical Gaussians. We show that for well-studied families of discrete distributions $$\mu$$, convolving $$\mu$$ with Gaussians yields well-conditioned log-concave distributions, as long as the variance of the Gaussian is above an $O(1)$ threshold. We then reduce the task of sampling from $$\mu$$ to sampling from Gaussian-convolved distributions. Our reduction is based on a stochastic process widely studied under different names: backward diffusion in diffusion models, and stochastic localization. We discretize this process in a novel way that allows for high accuracy and parallelism. As our main application, we resolve open questions Anari, Hu, Saberi, and Schild raised on the parallel sampling of distributions that admit parallel counting. We show that determinantal point processes can be sampled via RNC algorithms, that is in time $$\log(n)^{O(1)}$$ using $$n^{O(1)}$$ processors. For a wider class of distributions, we show our framework yields Quasi-RNC sampling, i.e., $$\log(n)^{O(1)}$$ time using $$n^{O(\log n)}$$ processors. This wider class includes non-symmetric determinantal point processes and random Eulerian tours in digraphs, the latter nearly resolving another open question raised by prior work. Of potentially independent interest, we introduce and study a notion of smoothness for discrete distributions that we call transport stability, which we use to control the propagation of error in our framework. Additionally, we connect transport stability to constructions of optimally mixing local random walks and concentration inequalities.