Rotatum of light

Abstract

Vortices are ubiquitous in nature and can be observed in fluids, condensed matter, and even in the formation of galaxies. Light, too, can evolve like a vortex. Optical vortex beams are exploited in light-matter interaction, free space communications, and imaging. Here, we introduce optical rotatum, a behavior of light in which an optical vortex beam experiences a quadratic chirp in its orbital angular momentum along the optical path. We show that such an adiabatic deformation of topology is associated with the accumulation of a Gouy phase factor, which, in turn, perturbs the propagation constant (spatial frequency) of the beam. The spatial structure of optical rotatum follows a logarithmic spiral-a signature that is commonly seen in the pattern formation of seashells and galaxies. Our work expands the previous literature on structured light, offers new modalities for light-matter interaction, communications, and sensing, and hints at analogous effects in condensed matter physics and Bose-Einstein condensates.

Fig. 1.. Rotatum of light.

(A) A phase mask converts a plane wave into a vortex beam whose OAM can grow (or decay) following a quadratic dependence along the direction of propagation. (B) The OAM, signified by ℓ, can follow any arbitrary z-dependent profile, which can be linear, quadratic, or cubic. The linear and quadratic evolution of OAM gives rise to the spatial self-torque (top) and rotatum of light (bottom), respectively. (C) The mechanism relies on introducing an azimuthal gradient in the spatial frequency of the beam, dk_z/dϕ (left). This can be realized by creating a spatial frequency comb in the k_z domain. Each comb tooth is weighted by a complex coefficient, ${\tilde{A}}_n\left(\mathrm{\varphi }\right)$ (right). (D) Amplitude and phase profiles of the coefficients ${\tilde{A}}_n\left(\mathrm{\varphi }\right)$, designed in this case to produce a vortex beam with linearly evolving OAM. Each coefficient, ${\tilde{A}}_n\left(\mathrm{\varphi }\right)$, is associated with a different spatial frequency, k_z,n. a.u., arbitrary units. (E) Upon propagation, different components of the beam, associated with different k-vectors, and weighted by ${\tilde{A}}_n\left(\mathrm{\varphi }\right)$, will interfere forming an envelope with unity amplitude and z-dependent phase profile, which adiabatically deforms its helical twist along the optical path via spatial beating.

Fig. 2.. Generation of a vortex beam with spatial self-torque.

(A) Experimental setup: An expanded and collimated laser beam is incident on a reflective phase-only SLM with the desired hologram. The reflected beam is filtered and imaged onto a CCD using a 4-f lens system. The CCD is mounted on a translation stage to capture the transverse profile of the beam at different z planes. (B) Simulated 2D profiles of the intensity and phase of a vortex beam in which the OAM increases linearly along the optical path. The red and yellow markers denote phase singularities with positive and negative handedness. Scale bar is 100 μm. (C) Measured 2D intensity and reconstructed phase profiles of the beam in (B). (D) The initial beam profile (at z = 1 cm) exhibits a horizontal line of darkness and bifurcation. The inset depicts the underlying chain of phase singularities associated with this dislocation line. (E) Measured and simulated evolution of OAM (effective charge, ℓ_eff) as a function of propagation distance.

Fig. 3.. Optical vortex beams with nonmonotonically varying topological charge.

(A) Measured transverse profiles (intensity and phase) of a vortex beam whose OAM locally grows then decays along the optical path. The beam acquires a helical phase, increasing its OAM adiabatically from ℓ = 0 to ℓ = 2, in a linear manner over a range of 20 cm. Afterward, the local OAM decreases continuously from ℓ = 2 to ℓ = 0 as the beam propagates for longer distance. Scale bar, 100 μm. (B) Measured and simulated spatial evolution of OAM (effective charge, ℓ_eff) in comparison with the target design for the optical vortex beam in (A). (C) Measured 2D intensity and phase profiles of an OAM beam whose vorticity decays and then grows with propagation. The beam starts with a local charge of ℓ = 2 and then slowly unwinds its helicity to ℓ = 0 before it acquires the same helical phase (ℓ = 2) again, albeit with different size. (D) Measured and simulated evolution of the corresponding charge (ℓ_eff) as a function of z. Vertical 1D cuts of the transverse profiles in (A) and (C), marked by the dashed circles, are plotted in (E to G), respectively. These cuts suggest that the beam’s size is perturbed (i.e., it experiences a k-shift) even when the topological charge is the same. The change in size depends on whether ℓ_eff increases or decreases with propagation.

Fig. 4.. Optical vortex beams with quadratic evolution of topological charge: Optical rotatum.

(A) Simulated transverse profiles (intensity and phase) of a vortex beam whose OAM locally grows in a quadratic manner along the optical path. The beam acquires a helical phase, increasing its OAM continuously from ℓ = 0 to ℓ = 5 over a range of 22.5 cm. The red and yellow markers denote phase singularities of opposite handedness (positive and negative helicity, respectively). Scale bar, 100 μm. (B) Measured 2D intensity and phase profiles at the z planes in (A). (C) Intensity profile of the vortex beam at z = 1.5 cm. The inset depicts a close up exhibiting a line of phase singularities feeding the beam’s center. (D) Comparison between the target, simulated, and measured local charge (ℓ_eff) showing its quadratic dependence on z.