Diffraction of convergent spherical waves with all possible polarization states using the Luneburg integral method

Abstract

We present a complete electromagnetic study, which includes electric, magnetic, and Poynting vector fields of diffracted convergent spherical waves under all possible polarization states compatible with Maxwell's equations. Exit pupil boundary conditions for these polarizations were obtained by means of Hertz potentials. Using these boundary conditions, two-dimensional Luneburg diffraction integrals for the three components of electric and magnetic fields were formulated, and after some approximations, we showed that the complete electromagnetic description of the inhomogeneous polarization states of spherical waves is reduced to the knowledge of seven one-dimensional integrals. The consistency of the method was tested by comparison with other previously reported methods for linearly polarized (LP), TE, and TM polarizations, while the versatility of the method was showed with the study of nonstandard polarization states, for example, that resulting from the superposition of TE and TM dephased spherical waves, which shows a helicoidal behavior of the Poynting vector at the focalization region, or the inhomogeneous LP state that exhibits a ring structure for the Poynting vector at the focal plane.

Assessment About Luneberg Integrals and Application to Digital in-line Holography

“…From [10], Eqs (13) and (14) on p. 734-735 we have, the expression of the matrices M e and M m are M e (x, y, z) = exp (ikr)…”

“…So, here we recall the results obtained by adding some supplements and a few modifications. The definition of the vectorial wave in [10], Eqs (11) and 12on p. 734 are the following…”

“…The lower index z = δ indicates that the derivations in the matrices have to be taken at the the value z = δ with positive or negative value of δ. The authors of [10] have been developed these matrices in circular coordinates (ρ, θ) and taken a negative value of δ (focal lenght). In our case, the general case is considered for our theoretical developments.…”

Assessment about Luneberg integrals and application to digital in-line holography

“…The Gaussian model cannot be used in the context of wavelength or sub-wavelength particles [22] and the scalar model is not sufficient [4]. This is the reason why, based on the work of the authors of [10], we develop here a vector electromagnetic field model allowing to describe as precisely as possible the holographic patterns delivered by such small objects. In Ref.…”

“…In Ref. [10], a first order approximation of the Luneberg's kernel is proposed to obtain a solution to the propagation of the electromagnetic field vector. This is why in most publications authors use Luneberg, respectively Rayleigh-Sommerfeld, and then carry out a far field approximation.…”