Chemical Preparation Laboratory for IND Candidate Compounds

Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond

“…The Husimi function at the system boundary of closed systems was introduced in Refs. [9,10] by projection of the conventional Husimi function from full phase space [coordinates (r, Φ), momentum (k j sin χ j , k j cos χ j )] onto the reduced phase space at the boundary r = r c with coordinates φ = Φ and sin χ j [11]. The four different Husimi functions (corresponding to the incident and emerging wave at both sides of a dielectric interface) can be constructed by the same procedure when the appropriate boundary conditions are employed.…”

“…The conventional Husimi function for a given wave function Ψ(r, Φ) of the dielectric system is obtained as the overlap with a wave packet with minimal uncertainty in the variables (r, Φ) for real space and (k j sin χ j , k j cos χ j ) for momentum space. The projection onto the boundary can be formulated rigorously [10]: The wave function is expressed by means of advanced and retarded Green's functions, which in turn allow to distinguish between incident and emerging waves. Green's formula is then used to express the solution Ψ j of the Helmholtz equation in region j = 0, 1 as an integral over the boundary, involving both Ψ j and its normal (radial) derivative Ψ ′ j .…”