Science Research Info: The topological aberrations of twisted light

Topology describes the study of properties of spaces that are invariant under any continuous deformation.
Topological aberration: It contains a high-order optical vortex which experiences not only geometrical shifts, but an additional splitting of its high-order vortex into a constellation of unit-charge vortices.
Multiple optical vortices indicate the presence of more than one optical vortex in a light beam. Each optical vortex is a point or region where the intensity of light is zero, and the phase of the light waves spirals around this point, creating a "twisted" or helical wavefront. These beams are characterized by their helical wavefronts.

Goos-Hänchen (GH) Shift: The GH shift is a lateral displacement of a reflected light beam along the plane of incidence. Instead of reflecting exactly along the predicted path, the beam's central position shifts slightly parallel to the interface. GH shift arises from changes in the reflection coefficient of the interface, which vary with the incidence angle. These changes affect the beam's overall phase, leading to a shift.

ΔGH is the lateral shift (Goos-Hänchen shift), λ is the wavelength of the light. ϕr is the phase of the reflection coefficient, θi is the angle of incidence.
Imbert-Fedorov (IF) Shift : It is a transverse displacement of a reflected beam that occurs perpendicular to the plane of incidence.
In the context of vortex constellations, the coordinates of the vortices can be represented as complex numbers. The authors use Elementary Symmetric Polynomials(ESP) to summarize these coordinates and understand how they change under reflection.
vectors eI and eR contain the ESPs of the input and aberrated constellations, respectively.
Wirtinger Derivative: It helps how a complex function (or light beam) changes, especially when dealing with distortions or shifts in the beam's structure.

R’ and R” are first and second Wirtinger derivatives of R(χ) at χ = χ* = 0.
Aberrations usually change the elementary symmetric polynomials (ESP), which describe the positions of the vortices in a group (constellation). From these changes, we can directly figure out the angular Wirtinger derivatives related to the aberration.

Science Research Info