## Combined calculi for photon orbital and spin angular momenta

^{1} Zentrum für Astronomie der Universität Heidelberg, Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany
^{2} OAM Solutions, LLC, 9300 Stardust Trail, Flagstaff AZ 86004, USA

e-mail: nick.elias@oamsolutions.com

Received: 10 December 2013

Accepted: 12 May 2014

*Context. *Wavelength, photon spin angular momentum (PSAM), and photon orbital angular momentum (POAM), completely describe the state of a photon or an electric field (an ensemble of photons). Wavelength relates directly to energy and linear momentum, the corresponding kinetic quantities. PSAM and POAM, themselves kinetic quantities, are colloquially known as polarization and optical vortices, respectively. Astrophysical sources emit photons that carry this information.

*Aims. *PSAM characteristics of an electric field (intensity) are compactly described by the Jones (Stokes/Mueller) calculus. Similarly, I created calculi to represent POAM characteristics of electric fields and intensities in an astrophysical context. Adding wavelength dependence to all of these calculi is trivial. The next logical steps are to 1) form photon total angular momentum (PTAM = POAM + PSAM) calculi; 2) prove their validity using operators and expectation values; and 3) show that instrumental PSAM can affect measured POAM values for certain types of electric fields.

*Methods. *I derive the PTAM calculi of electric fields and intensities by combining the POAM and PSAM calculi. I show how these quantities propagate from celestial sphere to image plane. I also form the PTAM operator (the sum of the POAM and PSAM operators), with and without instrumental PSAM, and calculate the corresponding expectation values.

*Results. *Apart from the vector, matrix, dot product, and direct product symbols, the PTAM and POAM calculi appear superficially identical. I provide tables with all possible forms of PTAM calculi. I prove that PTAM expectation values are correct for instruments with and without instrumental PSAM. I also show that POAM measurements of “unfactored” PTAM electric fields passing through non-zero instrumental circular PSAM can be biased.

*Conclusions. *The combined PTAM calculi provide insight into mathematically modeling PTAM sources and calibrating POAM- and PSAM-induced measurement errors.

Key words: instrumentation: miscellaneous / methods: analytical / methods: miscellaneous / methods: observational / techniques: miscellaneous

*© ESO, 2014*