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On a periodic cell $[-1,1[^2$, the refractive index is defined by $n^2= 1+ g_1(x_1)+g_2(x_2)$ with $g_i(x_i)= r^2\\quad {\\rm for} \\quad 0\\leq x_i<1, \\hbox{and} g_i(x_i)= 0\\quad {\\rm for} \\quad -1\\leq x_i\\leq 0$ where $r^2$ is constant. We find that for $r^2>-1$ the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. However, the case whereby $r^2=-1$ is very unusual, as the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1102.0136","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.optics","submitted_at":"2011-02-01T12:00:34Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"c8dab4b836dc5e94cfd9f6ca9f62ef77ea729f68eee1f96d70387502c56d4ce3","abstract_canon_sha256":"941045ad16c28c01e7b483ba7dfa93a0102303398f68d8574f26f1ffe8697453"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:19.677759Z","signature_b64":"eoqVRdEGbsZRHZ0pWB3VufDLJ1qIsjqLvswQKR432/baHHFIb5UtXbJITjTG3ks9W4aMqLNTvW6+nZ9asw/BCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94a2865ad3a52d08d24df39e14f86dbdfde931fae24471650a6c08618c442d4e","last_reissued_at":"2026-05-18T02:03:19.677323Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:19.677323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a class of three-phase checkerboards with unusual effective properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"physics.optics","authors_text":"Evgeniya Nolde, Julius Kaplunov, Richard V. Craster, S\\'ebastien Guenneau","submitted_at":"2011-02-01T12:00:34Z","abstract_excerpt":"We examine the band spectrum, and associated Floquet-Bloch eigensolutions, arising in a class of three-phase periodic checkerboards. On a periodic cell $[-1,1[^2$, the refractive index is defined by $n^2= 1+ g_1(x_1)+g_2(x_2)$ with $g_i(x_i)= r^2\\quad {\\rm for} \\quad 0\\leq x_i<1, \\hbox{and} g_i(x_i)= 0\\quad {\\rm for} \\quad -1\\leq x_i\\leq 0$ where $r^2$ is constant. We find that for $r^2>-1$ the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. 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