Optical Archimedes screw with acceleration of both trajectories and orbital angular momentum

Consider a superposition of two beams which differ in the magnitude of the axial component of their linear momentum and the magnitude of their OAM. Their interference results in both an axial and an angular standing wave patterns, combining to create a helical pattern. Using self-accelerating beams instead of beams propagating in a straight line keeps locally the principle of superposition of two modes with different linear and orbital angular momenta, while allowing their directions to change in space along a desired arbitrary trajectory. This was the idea behind the self-accelerating Archimedes Screw [30]. It was generated by a superposition of two Bessel beams with OAM values of 1 and 3 with different linear momentum along an arbitrary path. Both beams were designed separately according to an algorithm developed by Zhao et al [12] which created a curved vortex beam with a constant value of OAM. A generalization of the self-accelerating Archimedes screw allows engineering the OAM values along propagation in a curve to create a changing amount of strands in the interference pattern. Two Bessel-like beams are designed to propagate along an arbitrary trajectory, where one of them has a fixed OAM and the other one, an OAM that changes with propagation. To create such beams the input plane is divided into expanding circles, each one of them is mapped to a different axial value. The interference of rays emanating from these circles in the input plane leads to a Bessel-like pattern that propagates along the predesigned path. Further explanation for creating an appropriate phase pattern at the input plane as well as a code to implement it are given in [30] , which we adopted here. In order to vary the OAM with propagation each circle mapped to a different axial location is multiplied by the desired OAM function: $e^{il\phi}$, where l can be any arbitrary number, integer or fractional. In this manner it is possible to tailor the OAM value at each axial location. We designed two optical Archimedes screws with varying OAM values, accelerating along a specified trajectory: $x = 5.33\times10^{-5}\times \sin(59.53z)/\cosh(29.77z)$ (The units are given in meters.) The phase pattern at the input plane needed to realize the beams is shown in the left column of figure 1. The OAM as a function of propagation distance is set to be changing linearly for the first beam (first row) and step-wise for the second beam (second row). The middle column shows the OAM value as a function of propagation distance. A calculation using Fresnel field propagation shows iso-intensity surfaces of the beam in the right column.

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The experimental setup consists of a 532 nm CW laser (Laser Quantum Ventus 532 Solo) which reflects off a phase-only spatial light modulator (SLM) (Holoeye Pluto SLM) following expansion and collimation. After the SLM the beam is demagnified with a collimating telescope and imaged using a scientific CMOS camera (Ophir Spiricon SP620U Beam Profiling Camera) on a translation stage set on the optical axis of the beam. The two phase profiles for creating the two beams (shown in figure 1) were multiplied with a blazed phase grating, directing a background free replica of the beam to the first diffraction order. The overall phase pattern, was loaded to the SLM which was set in a 4f configuration between itself and the imaging area. The measured intensities and extracted phases are presented for three axial locations for both the beam with the linearly varying OAM (figure 2) and for the beam with step-wise varying OAM (figure 3). The two top rows of each figure show the simulation results obtained using Fresnel field propagation, where the first row is the intensity and the second row is the extracted phase. The two bottom rows of each figure show the experimental results, where again the first row is the intensity and the second row is the extracted phase. The center of the white crosshair at the intensity plots indicates the position of the optical axis (at which passes the center of a beam propagating along a straight line). It can be seen that the experimental results are in good agreement with simulation results. In order to extract the phase from the experimental measurements we created three additional phase masks to realize the following fields: $1+E_{in}$, $e^{i\pi/2}+E_{in}$ and $e^{i\pi}+E_{in}$, where E_in is the input field needed to generate the beams displayed in figure 1. For each of these fields, three intensity profiles were measured, added and subtracted to extract the phase, where $E = |E|e^{i\Psi}$ is the field E_in after propagation to each of the imaging planes. The phase extraction procedure for each of the propagation distances is as follows:

$$\begin{equation} I_0 = |1+E|^2 = 1 + |E|^2 + 2|E|\cos(\Psi). \end{equation} \tag{ 1 }$$

$$\begin{equation} I_{\pi/2} = |e^{i\pi/2}+E|^2 = 1 + |E|^2 + 2|E|\sin(\Psi). \end{equation} \tag{ 2 }$$

$$\begin{equation} I_\pi = |e^{i\pi}+E|^2 = 1 + |E|^2 - 2|E|\cos(\Psi). \end{equation} \tag{ 3 }$$

$$\begin{equation} I_1 = (I_0-I_\pi)/4 \end{equation} \tag{ 4 }$$

$$\begin{equation} I_2 = (I_{\pi/2}-(I_0+I_\pi)/2)/2 \end{equation} \tag{ 5 }$$

$$\begin{equation} \Psi = \arctan{(I_2/I_1).} \end{equation} \tag{ 6 }$$

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The OAM correlation representation is given with a correlation function $\Gamma_\mathrm {OAM}$ which is calculated for each OAM value using the following expression:

$$\begin{equation} \Gamma_\mathrm {OAM} = \frac{\left|\iint_{s} \psi(x, y) \cdot \psi_\mathrm {OAM}^{*}(x, y) d x d y\right|^{2}}{\iint_{s}\left|\psi(x, y)\right|^{2} d x y d y\iint_{s}\left|\psi_\mathrm {OAM}(x, y)\right|^{2} d x d y}. \end{equation} \tag{ 7 }$$

Here $\psi(x, y)$ is the transverse complex amplitude distribution of the analyzed beam. The (x,y) coordinates change through the propagation of the beam, and they are transverse to the z axis as was defined above, $\psi_\mathrm {OAM}$ is the transverse complex amplitude distribution of a beam with the same trajectory propagated to the same distance as $\psi(x, y)$, with a fixed OAM value specified by the subscript OAM. The range of OAM values was 0–10 with 0.1 increments. s is the area where the intensity of $\psi_\mathrm {OAM}$ is in the range between the maximum intensity and half of it. The integration is performed over s which is unique for each $\psi_\mathrm {OAM}$. The OAM correlation representation is a measure we use to estimate the local OAM at every propagation distance. It is not a decomposition, as the correlation values are not summed to 1. The results of the OAM correlation representation are shown for the beam with linearly varying OAM in figure 4 and for the beam with step-wise varying OAM in figure 5. In both figures the correlation calculated from the simulation obtained using Fresnel field propagation is shown with blue dots and the correlation calculated from the experimental data is shown with orange dots. The black dotted lines in these figures indicate the designated theoretical values of the OAM. It can be seen that the results derived from experimental data are in good agreement with simulation results and theoretical values.

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