{'fraction_non_alphanumeric': 0.024079320113314446, 'fraction_numerical': 0.0056657223796034, 'mean_word_length': 5.3125, '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}}
Realization of first-order optical systems using thin lenses of positive focal length.

An explicit decomposition of the most general first-order optical system characterized by an Sp(4,R) matrix is obtained in terms of free propagation, thin convex lenses, and thin cylindrical lenses of positive focal length. The Euler decomposition of an Sp(4,R) matrix is used in arriving at the decomposition. It is shown that not more than four convex lenses and 14 cylindrical lenses of positive focal length are required to realize the same. Explicit decompositions for the case of differential free and inverse free propagation, differential magnifiers, and differential fractional Fourier transform are provided.