A matrix differential operator for passage from scalar to vector wave optics

Abstract

One possible way to generalize scalar to wave optics, thus including polarization in the treatment consistent with the Maxwell equations was shown by Mukunda, Simon and Sudarshan (MSS) for paraxial systems, based on a group theoretical analysis. Later, the MSS theory for the passage from the Helmholtz scalar wave optics to the Maxwell vector wave optics was derived by casting the basic formalism in a framework very similar to the Dirac electron theory. Here, we show that the elegant MSS substitution rule is equivalent to a second-order 6 × 6 matrix differential operator for passage from scalar to vector wave optics. We explicitly obtained this matrix differential operator in a series and exponential forms respectively. Expressions for the electromagnetic field components are derived without making any assumptions on the form of the transverse beam profile. It is observed that all types of beams have certain characteristic relations among their field components. The power of the formalism is demonstrated by considering five specific beams.

Keywords

• Matrix representation of Maxwell’s equations
• Helmholtz scalar wave optics
• Maxwell vector wave optics
• Mukunda–Simon–Sudarshan substitution rule
• Beam propagation
• Cross polarization