Singular beams have circulating energy components. When such beams are focused by low numerical aperture systems suffering from astigmatic aberration, these circulating energy components get modified. The phase gradient introduced by this type of aberration splits the higher charge vortices. The dependence of the charge, the aberration coefficient, and the size of the aperture on the nature of the splitting process are reported in this paper. The transverse components of the Poynting vector fields that can be derived from the phase gradient vector field distributions are further decomposed into solenoidal and irrotational components using the Helmholtz-Hodge decomposition method. The solenoidal components relate to the orbital angular momentum of the beams, and the irrotational components are useful in the transport of intensity equations for phase retrieval.
It is shown that the vector field decomposition method, namely, the Helmholtz Hodge decomposition, can also be applied to analyze scalar optical fields that are ubiquitously present in interference and diffraction optics. A phase gradient field that depicts the propagation and Poynting vector directions can hence be separated into solenoidal and irrotational components.
We report an interesting observation in the formation of Young's fringes from a two pinhole arrangement illuminated by waves from the neighborhood of a zero of an optical phase singularity. Spacing of the Young's fringes appears to defy the dependence of pin-hole separation. But for larger pinhole separation such an anomalous phenomenon is not discernible. The experiments show that the fringe spacing is governed by the stronger local phase gradient near the vortex core that also has a radial part. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have only azimuthal and longitudinal components. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient.
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