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Register a new userOptimal SMF packing in photonic lanterns: comparing theoretical topology to practical packing arrangements
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Photonic lanterns rely on a close packed arrangement of single mode fibers, which are tapered and fused into one multi-mode core. Topologically optimal circle packing arrangements have been well studied. Using this, we fabricate PLs with 19 and 37 SMFs showing tightly packed, ordered arrangements with packing densities of 95 % and 99 % of theoretically achievable values, with mean adjacent core separations of 1.03 and 1.08 fiber diameters, respectively. We demonstrate that topological circle packing data is a good predictor for optimal PL parameters.
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We present a new concept of an integrated optics component capable of measuring the complex amplitudes of the modes at the tip of a multimode waveguide. The device uses a photonic lantern to split the optical power carried by an $N$-modes waveguide among a collection of single-mode waveguides that excite a periodic array of at least $N^2$ single-mode evanescently-coupled waveguides. The power detected at each output of the array is a linear combination of the products of the modal amplitudes-a relation that can, under suitable conditions, be inverted allowing the derivation of the amplitudes and relative phases of the modal mixture at the input. The expected performance of the device is discussed and its application to the real-time measurement of modal instability in high power fiber lasers is proposed.
Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems. 1. We show that Vector Bin Packing has no polynomial time $Omega( log d)$ factor asymptotic approximation algorithm when $d$ is a large constant, assuming $textsf{P} eq textsf{NP}$. This matches the $ln d + O(1)$ factor approximation algorithms (Chekuri, Khanna SICOMP 2004, Bansal, Caprara, Sviridenko SICOMP 2009, Bansal, Eli{a}s, Khan SODA 2016) upto constants. 2. We show that Vector Scheduling has no polynomial time algorithm with an approximation ratio of $Omegaleft( (log d)^{1-epsilon}right)$ when $d$ is part of the input, assuming $textsf{NP} subseteq textsf{ZPTIME}left( n^{(log n)^{O(1)}}right)$. This almost matches the $Oleft( frac{log d}{log log d}right)$ factor algorithms(Harris, Srinivasan JACM 2019, Im, Kell, Kulkarni, Panigrahi SICOMP 2019). We also show that the problem is NP-hard to approximate within $(log log d)^{omega(1)}$. 3. We show that Vector Bin Covering is NP-hard to approximate within $Omegaleft( frac{log d}{log log d}right)$ when $d$ is part of the input, almost matching the $O(log d)$ factor algorithm (Alon et al., Algorithmica 1998). Previously, no hardness results that grow with $d$ were known for Vector Scheduling and Vector Bin Covering when $d$ is part of the input and for Vector Bin Packing when $d$ is a fixed constant.
We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 0.350389 of the squares area, improving the previous best value of {pi}/10 approx 0.31416; even in an offline setting, there is an upper bound of {pi}/(3 + 2 sqrt{2}) approx 0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm achieves the higher value 0.375898. As a byproduct, we give an online algorithm for packing circles into a 1times b rectangle with b geq 1. This algorithm is worst case-optimal for b geq 2.36.
We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $deltaleq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $varepsilon>0$ there are sets of disks of area $1/2+varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $delta=frac{8}{5pi}approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A leq frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $varepsilon>0$ there are sets of squares of total area $frac{8}{5}+varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square $left(frac{1}{2}right)$, circles in a square $left(frac{pi}{(3+2sqrt{2})}approx 0.539right)$ and circles in a circle $left(frac{1}{2}right)$ have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.
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