Nonparaxial singular beams inside the focal region of a high numerical-aperture lens

Abstract

Key optical technologies, including lithography, data storage, optical tweezers, microscopy, and ultrafast laser materials processing, rely on strongly fo- cused light beams. Such beams are often used to exploit a vectorial nature of light and so detailed knowledge of the field structure inside a tight focus becomes in- creasingly important. So far, theoretical studies of intra-focal optical field compo- nents have been mainly concentrated on spatially homogeneous states of light po- larisation. In this work we present a new development in the calculations of local polarisation structure of tightly focused singular beams, including radially and azi- muthally polarised hollow beams. A rapidly increasing number of various practical applications in the last few years have attracted significant interest to studies related to three-dimensional structure of the field inside a focus of high numerical-aperture optical systems (1-5). Experimental intrafocal mapping of the squared electric field components is usually indirectly achieved with near field probes (6), point scatterers (7), fluorescent molecules and beads (8, 9), and the knife edge method (10). Recently, a new ap- proach has been developed (2) for visualising nanoscale structure of the electric field inside the focal volume of tightly focused singular beams, which is based on permanent imprinting of the field in transparent media. From the theoretical point of view, the first comprehensive studies of tightly focused linearly and circularly polarised Laguerre-Gaussian beams have been performed in Refs. (11-13). In Refs. (1, 14), the authors have attempted to analyse the focal spot of both radially and azimuthally polar- ised beams. The problem has been qualitatively solved in Ref. (14) for the case of non-apodised cylindrical vector beams, hence providing basic understanding of the focal structure of real beams. The general method allowing one to solve such problems is based on Debye vector integrals. A complexity of the problem for some types of vector beams, e.g., radially or azimuthally polarised beams, is related to finding an integrand  which would satisfy the conditions for the paraxial waves ( 2 1 ( 4 ) 0 z i          ) and simultaneously describe the vector field components. In the