Jörg B. Götte

Light beams with fractional orbital angular momentum and their vortex structure

Light emerging from a spiral phase plate with a non-integer phase step has a complicated vortex structure and is unstable on propagation. We generate light carrying fractional orbital angular momentum (OAM) not with a phase step but by a synthesis of Laguerre-Gaussian modes. By limiting the number of different Gouy phases in the superposition we produce a light beam which is well characterised in terms of its propagation. We believe that their structural stability makes these beams ideal for quantum information processes utilising fractional OAM states.

The analogy between optical beam shifts and quantum weak measurements

We describe how the notion of optical beam shifts (including the spatial and angular Goos–Hanchen shift and Imbert–Federov shift) can be understood as a classical analogue of a quantum measurement of the polarization state of a paraxial beam by its transverse amplitude distribution. Under this scheme, complex quantum weak values are interpreted as spatial and angular shifts of polarized scalar components of the reflected beam. This connection leads us to predict an extra spatial shift for beams with a radially-varying phase dependance.

Quantum formulation of fractional orbital angular momentum

The quantum theory of rotation angles [S.M. Barnett and D.T. Pegg, Phys. Rev. A 41 3427 (1990)] is generalized to non-integer values of the orbital angular momentum. This requires the introduction of an additional parameter, the orientation of a phase discontinuity associated with fractional values of the orbital angular momentum. We apply our formalism to the propagation of light modes with fractional orbital angular momentum in the paraxial and non-paraxial regime.

Topological aberration of optical vortex beams: determining dielectric interfaces by optical singularity shifts.

The splitting of a high-order optical vortex into a constellation of unit vortices, upon total reflection, is described and analyzed. The vortex constellation generalizes, in a local sense, the familiar longitudinal Goos-Hänchen and transverse Imbert-Federov shifts of the centroid of a reflected optical beam. The centroid shift is related to the centre of the constellation, whose geometry otherwise depends on higher-order terms in an expansion of the reflection matrix. We present an approximation of the field around the constellation of increasing order as an Appell sequence of complex polynomials whose roots are the vortices, and explain the results by an analogy with the theory of optical aberration.

Generalized shifts and weak values for polarization components of reflected light beams

The simple reflection of a light beam of finite transverse extent from a homogeneous interface gives rise to a surprisingly large number of subtle shifts and deflections which can be seen as diffractive corrections to the laws of geometrical optics (Goos–Hanchen shifts) and manifestations of optical spin–orbit coupling (Imbert–Fedorov shifts), related to the spin Hall effect of light. We develop a unified linear algebra approach to dielectric reflection which allows for a simple calculation of all these effects and lends itself to an interpretation of beam shifts as weak values in a classical analogue to a quantum weak measurement. We present a systematic study of the shifts for the whole beam and its polarization components, finding symmetries between input and output polarizations and predicting the existence of material independent shifts.

Achromatic vector vortex beams from a glass cone

The reflection of light is governed by the laws first described by Augustin-Jean Fresnel: on internal reflection, light acquires a phase shift, which depends on its polarization direction with respect to the plane of incidence. For a conical reflector, the cylindrical symmetry is echoed in an angular variation of this phase shift, allowing us to create light modes with phase and polarization singularities. Here we observe the phase and polarization profiles of light that is back reflected from a solid glass cone and, in the case of circular input light, discover that not only does the beam contain orbital angular momentum but can trivially be converted to a radially polarized beam. Importantly, the Fresnel coefficients are reasonably stable across the visible spectrum, which we demonstrate by measuring white light polarization profiles. This discovery provides a highly cost-effective technique for the generation of broadband orbital angular momentum and radially polarized beams.

Eigenpolarizations for giant transverse optical beam shifts.

We show how careful control of the incident polarization of a light beam close to the Brewster angle gives a giant transverse spatial shift on reflection. This resolves the long-standing puzzle of why such beam shifts transverse to the incident plane (Imbert-Fedorov shifts) tend to be an order of magnitude smaller than the related Goos-Hänchen shifts in the longitudinal direction, which are largest close to critical incidence. We demonstrate that with the proper initial polarization the transverse displacements can be equally large, which we confirm experimentally near Brewster incidence. In contrast to the established understanding, these polarizations are elliptical and angle dependent. We explain the magnitude of the Imbert-Fedorov shift by an analogous change of the symmetry properties for the reflection operators as compared to the Goos-Hänchen shift.

Beam shifts for pairs of plane waves

Following Hans Wolters treatment of the spatial Goos–Hanchen shift of a totally internally reflected light beam by the superposition of two plane waves, polarized perpendicular to the plane of incidence, we consider the reflection and refraction of several similar pairs of plane waves, with varying geometry and incident polarization. We consider explicitly the partial reflection analogue and the in-plane polarized analogue to Wolters example, as well as a pair of plane waves propagating slightly out of their mutual plane of incidence, revealing the transverse, Imbert–Fedorov shift. We find these simple cases have a complicated polarization structure, with a range of polarization singularities and complex orbital and spin current flows, generalizing Wolters discovery of an optical vortex and circulating energy flow at the heart of the net scalar interference pattern.

Are Fresnel filtering and the angular Goos–Hänchen shift the same?

The law of reflection and Snells law are among the tenets of geometrical optics. Corrections to these laws in wave optics are respectively known as the angular Goos–Hanchen shift and Fresnel filtering. In this paper we give a positive answer to the question of whether the two effects are common in nature and we study both effects in the more general context of optical beam shifts. We find that both effects are caused by the same principle, but have been defined differently. We identify and discuss the similarities and differences that arise from the different definitions.

Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable z with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of paraxially and nonparaxially propagating beams, especially in regard to the divergence of nonparaxial analogs to familiar paraxial beams.