Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam

TLDR

An integral representation of the electromagnetic field in the focal region of a coherent beam from an aplanatic system was derived by Ignatowsky (1919) and Richards & Wolf (1959). The paper employs this representation to analyze the focal region’s structure in a typical case. The authors examine axial sections of a 16 λ × 10 λ cylindrical region near focus, presenting meridional diagrams and a diagram of the longitudinal component’s behavior. The study reveals strong longitudinal electric fields up to ~10^5 V/cm in the focal plane, with detailed energy‑density contours and meridional sections for a system with a 45° semi‑aperture.

Abstract

An integral representation for the electromagnetic field in the region of focus of a coherent light beam that emerges from an aplanatic optical system has been derived by Ignatowsky (1919) and by Richards and Wolf (1959). In the present paper this representation is used to analyze the structure of the focal region in a typical case. Contours of the time-averaged electric energy density in the focal plane, in one defocused plane and in two meridional sections of the focal region of a system with angular semi-aperture 45\ifmmode^\circ\else\textdegree\fi{} are presented. The meridional diagrams refer to axial sections through a cylindrical region around the axis near focus, of length $16\ensuremath{\lambda}$ and cross-sectional diameter $10\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the wavelength of the light. It is found that the field has a strong longitudinal component at certain points of the focal plane and that longitudinal electric field strengths of the order of ${10}^{5}$ V/cm could now be attained with focused laser beams. A diagram illustrating the complete behavior of the longitudinal component in the focal plane is also given.