Abstract
In this work we study the scattering and transfer matrices for electric fields defined with respect to an angular spectrum of plane waves. For these matrices, we derive the constraints that are enforced by conservation of energy, reciprocity and time reversal symmetry. Notably, we examine the general case of vector fields in three dimensions and allow for evanescent field components. Moreover, we consider fields described by both continuous and discrete angular spectra, the latter being more relevant to practical applications, such as optical scattering experiments. We compare our results to better-known constraints, such as the unitarity of the scattering matrix for far-field modes, and show that previous results follow from our framework as special cases. Finally, we demonstrate our results numerically with a simple example of wave propagation at a planar glass-air interface, including the effects of total internal reflection. Our formalism makes minimal assumptions about the nature of the scattering medium and is thus applicable to a wide range of scattering problems.
Highlights
Scattering and transfer matrices are important mathematical tools that have been applied in a variety of fields, including the transport of electrons in wires [1,2], telecommunications [3], acoustics [4,5], and photonic crystals [6,7]
In this paper we have derived the general set of constraints for the scattering and transfer matrices imposed by conservation of energy, reciprocity, and time reversal symmetry
Our formalism considers the general case of vectorial light and allows for fields containing evanescent components
Summary
Scattering and transfer matrices are important mathematical tools that have been applied in a variety of fields, including the transport of electrons in wires [1,2], telecommunications [3], acoustics [4,5], and photonic crystals [6,7]. The corresponding scattering matrix constraints due to reciprocity for vector evanescent waves has been considered separately [49] Matrix constraints such as those mentioned place limits on the set of all physically possible scattering and transfer matrices, and can serve as useful guides in determining whether a given experimental or simulated matrix satisfies the corresponding physical law. While previous works have explored matrix constraints pertinent to a continuous decomposition of an electric field containing an infinite set of modes, such as in a continuous angular spectrum decomposition [51,52], the corresponding constraints satisfied by the scattering and transfer matrices defined with respect to a finite set of modes are important, for experiments and simulations in which only a finite description is physically possible.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have