{'fraction_non_alphanumeric': 0.030303030303030304, 'fraction_numerical': 0.0025252525252525255, 'mean_word_length': 5.833333333333333, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Self-bending scalar and vector bottle sheets.

This work demonstrates the generation of auto-bending cylindrical/tubular Bessel-Gauss bottle beams in homogeneous two-dimensional (2D) space. The corresponding wave fields flow through a two-dimensional curved trajectory leaving a singularity hollow central region, exhibiting the characteristic of circumventing obstacles. Scalar and vector fields are derived based on the angular spectrum decomposition method, the Helmholtz equation, the Lorenz gauge condition, and Maxwell's equations. The profile and area of the 2D bottle beams, together with the location of the autofocusing spots, are controlled by the intrinsic parameters of the illuminating waves and polarizations of the vector potential forming the incident fields. The demonstrated auto-bending cylindrical bottle beam solutions may find potential applications in acoustical and optical cloaking, auto-bending beam tweezers, imaging around steep corners, therapeutic investigations with unconventional autofocusing beams, acoustical and light sheets (i.e., slice of beams in 2D), and other related particle manipulation, isolation, and sorting devices, to name a few examples.