{'fraction_non_alphanumeric': 0.01455026455026455, 'fraction_numerical': 0.0, 'mean_word_length': 5.256198347107438, '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}}
Engineering anisotropy to amplify a long-wavelength field without a limit

We present a simple design of circular or spherical shells capable of amplifying a long-wavelength or static field. This design makes use of only two isotropic materials and is optimally restricted to the prescribed geometric and materials constraint. Furthermore, it is shown that the amplification factor of the structure can be made arbitrarily large as the ratio of the radius of the inner sphere to that of the outer sphere decreases to zero. It is anticipated that the presented design will be useful for high-gain antennae for telecommunications, magnets generating strong, local uniform fields and thermoelectric devices harvesting thermal energy, among other applications.