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A"],"published-print":{"date-parts":[[2026,7,1]]},"abstract":"<jats:p>We review the algebraic definition of the efficiency of a polarization modulation scheme, which is commonly adopted for solar and stellar spectro-polarimetry applications, and generalize it to allow distinct states of the modulation cycle to have arbitrary throughput and different photon-noise statistics for each state. Such a generalization becomes necessary to model and optimize the polarimetric efficiency of instruments implementing spatial polarization modulation schemes, where different optical paths are assigned to different polarization analysis states, which may be characterized by different throughput values. The proposed algebraic extension is also applicable to the concept of the efficiency of a polarization calibration scheme, which can then be used to create a merit function for the optimization of calibration sequences that take into account the specific characteristics of the polarimetric instrument and of its calibration optics.<\/jats:p>","DOI":"10.1364\/josaa.598937","type":"journal-article","created":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T15:00:10Z","timestamp":1779116410000},"page":"1027","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal polarization modulation and calibration schemes"],"prefix":"10.1364","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6990-513X","authenticated-orcid":true,"given":"R.","family":"Casini","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03773p874","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3215-7155","authenticated-orcid":true,"given":"D. 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