{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:B3DHQD2AEPG3GKNZMTFXBERPCI","short_pith_number":"pith:B3DHQD2A","schema_version":"1.0","canonical_sha256":"0ec6780f4023cdb329b964cb70922f1220d28ed7dd75e59ad565428f04bdbf2b","source":{"kind":"arxiv","id":"1108.5906","version":1},"attestation_state":"computed","paper":{"title":"Paraxial and nonparaxial polynomial beams and the analytic approach to propagation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.optics","authors_text":"Joerg B Goette, Mark R Dennis, Michael A Morgan, Miguel A Alonso, Robert P King","submitted_at":"2011-08-30T10:26:18Z","abstract_excerpt":"We construct solutions of the paraxial and Helmholtz equations which are polynomials in their spatial variables. These are derived explicitly using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable $z$ with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of"},"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":"1108.5906","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.optics","submitted_at":"2011-08-30T10:26:18Z","cross_cats_sorted":[],"title_canon_sha256":"ef7558a04a1b7b99e449d2797fadb22a8b74beb5be12a584205572a63a6dc591","abstract_canon_sha256":"5fd08022624ed94c5870845ba9a4081f0f1c36b570411a7f2615e509444dfbfa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:39.344592Z","signature_b64":"Hx71pyQhejv7UIp0OVjnOoMQME+DSHUpGL8PxKsJ1OBs6JeRx52SQxoeV0DrDaM/SHbeP/DGiDXHDlS2pXZ6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ec6780f4023cdb329b964cb70922f1220d28ed7dd75e59ad565428f04bdbf2b","last_reissued_at":"2026-05-18T03:54:39.343988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:39.343988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Paraxial and nonparaxial polynomial beams and the analytic approach to propagation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.optics","authors_text":"Joerg B Goette, Mark R Dennis, Michael A Morgan, Miguel A Alonso, Robert P King","submitted_at":"2011-08-30T10:26:18Z","abstract_excerpt":"We construct solutions of the paraxial and Helmholtz equations which are polynomials in their spatial variables. These are derived explicitly using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable $z$ with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5906","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.5906","created_at":"2026-05-18T03:54:39.344078+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.5906v1","created_at":"2026-05-18T03:54:39.344078+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5906","created_at":"2026-05-18T03:54:39.344078+00:00"},{"alias_kind":"pith_short_12","alias_value":"B3DHQD2AEPG3","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"B3DHQD2AEPG3GKNZ","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"B3DHQD2A","created_at":"2026-05-18T12:26:24.575870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI","json":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI.json","graph_json":"https://pith.science/api/pith-number/B3DHQD2AEPG3GKNZMTFXBERPCI/graph.json","events_json":"https://pith.science/api/pith-number/B3DHQD2AEPG3GKNZMTFXBERPCI/events.json","paper":"https://pith.science/paper/B3DHQD2A"},"agent_actions":{"view_html":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI","download_json":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI.json","view_paper":"https://pith.science/paper/B3DHQD2A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.5906&json=true","fetch_graph":"https://pith.science/api/pith-number/B3DHQD2AEPG3GKNZMTFXBERPCI/graph.json","fetch_events":"https://pith.science/api/pith-number/B3DHQD2AEPG3GKNZMTFXBERPCI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI/action/storage_attestation","attest_author":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI/action/author_attestation","sign_citation":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI/action/citation_signature","submit_replication":"https://pith.science/pith/B3DHQD2AEPG3GKNZMTFXBERPCI/action/replication_record"}},"created_at":"2026-05-18T03:54:39.344078+00:00","updated_at":"2026-05-18T03:54:39.344078+00:00"}