# Determining Vortex-Beam Superpositions by Shear Interferometry

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Modal Superpositions

- When $|{\ell}_{1}|<|{\ell}_{2}|$, the modal pattern is quite predictable and shows the following features:
- -
- The center of the pattern has an optical vortex of charge ${\ell}_{1}$. This is what is theoretically predicted. In practice, a multiply-charged point is very susceptible to perturbations, and so, the center of the pattern may consist of $|{\ell}_{1}|$ singly-charged vortices of sign ${\ell}_{1}/\left|{\ell}_{1}\right|$ in close proximity.
- -
- The center is surrounded by $|{\ell}_{1}-{\ell}_{2}|$ vortices arranged symmetrically [28] and located at a radial distance ${r}_{v}$ that satisfies:$$tan\beta =\frac{|{u}_{{\ell}_{1}}({r}_{v},{\varphi}_{v})|}{|{u}_{{\ell}_{2}}({r}_{v},{\varphi}_{v})|}.$$$${r}_{v,LG}=\frac{w}{\sqrt{2}}{\left(\right)}^{\frac{|{\ell}_{2}|!}{|{\ell}_{1}|!{tan}^{2}\beta}}\frac{1}{2\left(\right|{\ell}_{2}|-|{\ell}_{1}\left|\right)}$$$${\varphi}_{v}=\frac{\gamma +n\pi}{{\ell}_{2}-{\ell}_{1}},$$

For example, when ${\ell}_{1}=+1$ and ${\ell}_{2}=-2$, the composite mode for $\beta ={45}^{\circ}$ consists of a central vortex of charge $+1$ surrounded by three vortices of charge $-1$ located at a radius ${r}_{v}$. - When ${\ell}_{1}=-{\ell}_{2}$ and $\beta \ne {45}^{\circ}$, the pattern contains a central vortex of charge ${\ell}_{1}/\left|{\ell}_{1}\right|$. At $\beta ={45}^{\circ}$, there is no central vortex, and the composite mode has $2|{\ell}_{1}|$ radial lines (nodes) of ${180}^{\circ}$ shear phase, evenly separated. The relative weights of the modes produce subtle variations in intensity, which yields greater uncertainty in the determination. The method presented in this article is much more effective for the first case.

#### 2.2. Shear Interference Pattern

- The pattern consists of conjoined forks formed by the interference of the vortex beam with a displaced and tilted copy of it. If the shear interferometer is air spaced, the centers of the vortices are displaced by:$$s=2tsin\alpha ,$$
- The overall phase of the pattern is determined by the optical path-length difference and the reflection phases, which for our case is given by:$$\psi =\frac{4\pi tcos\alpha}{\lambda}+\pi ,$$
- The fringe density of the pattern is given by:$$\rho \simeq \frac{\theta}{\lambda},$$$$\theta =2\delta cos\alpha $$

## 3. Results

#### 3.1. Mode Comparison

#### 3.2. Determining the Topological Charge of the Component Beams

- We first examine the fork pattern in the center of the mode. From it, we extract the magnitude $|{\ell}_{1}|$ and sign ${\sigma}_{1}={\ell}_{1}|/|{\ell}_{1}|$ of the mode with smaller topological charge (recall that we assume $|{\ell}_{1}|<|{\ell}_{2}|$). No vortices means ${\ell}_{1}=0$. In the case of Figure 2b, we see the conjoined-fork pattern of a $+1$ vortex, revealing that ${\ell}_{1}=+1$. In the table in Figure 2a, we give the correspondence between the sign of the topological charge of the vortex and its forked signature in the shear pattern.
- We count the number of peripheral vortices N (in Figure 2b, we see that $N=3$). The type of conjoined forks specifies their sign. If the sign of peripheral vortices is the same as the one at the center, then:$${\ell}_{2}={\sigma}_{1}\left(\right)open="("\; close=")">N+|{\ell}_{1}|$$$${\ell}_{2}=-{\sigma}_{1}\left(\right)open="("\; close=")">N-|{\ell}_{1}|$$
- The angular orientation of the vortices reveals the relative phase between the modes per Equation (5). In our example, $\gamma \sim 0$.

#### 3.3. Determining the Relative Amplitude of the Component Beams

## 4. Discussion

## 5. Apparatus and Methods

#### 5.1. Shear Interferometer

#### 5.2. Shear-Pattern Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Beam modes (

**a**–

**c**) and corresponding shear interferograms (

**d**–

**f**) of vortex beams generated using phase modulation only (a,c), phase and amplitude modulation (b) and far-field (a,b) and near-field imaging (c). All modes have $\ell =+2$.

**Figure 2.**(

**a**) Table showing the conditions that lead to distinct shear patterns of optical vortices along with their signature in the pattern. $\delta >0$ corresponds to the second reflection deflected downward relative to the first reflection off the shear interferometer. (

**b**) Phase pattern of the shear interference of the superposition of modes with topological charges ${\ell}_{1}=+1$ and ${\ell}_{2}=-2$. We label the arrangement of vortices produced by the superposition. The measured radial distance of the vortices ${r}_{v}$ is taken as the distance between the center of the central pattern and the center of each of the peripheral vortices.

**Figure 3.**Images of equal-amplitude superpositions of modes with topological charges $(1,-2)$ in (

**a**), $(1,-4)$ in (

**b**), $(2,-4)$ in (

**c**) and $(-1,-2)$ in (

**d**). The images in the second row (

**e**–

**h**) are the shear interferograms of the superpositions above them.

**Figure 4.**Top row: shear interferograms of the superposition of modes with topological charges ${\ell}_{1}=+1$ and ${\ell}_{2}=-2$ for several values of $\beta $: ${35}^{\circ}$ in (

**a**), ${45}^{\circ}$ in (

**b**) and ${60}^{\circ}$ in (

**c**). Bottom row (

**d**–

**f**): reconstructions of the phase of the light field corresponding to the shear patterns above them. False color encodes phase.

**Figure 5.**Graph of the radial position of the peripheral vortices relative to the beam radius as a function of the parameter $\beta $ that determines the ratio of the amplitudes of the modes in Equation (2). The data shown correspond to the case $(+1,-2)$. The solid line corresponds to the ${r}_{v-LG}/\left(\sqrt{2}w\right)$ in Equation (4).

**Figure 6.**Apparatus used to make the measurements. Components include the spatial light modulator (SLM) lenses (L${}_{i}$), fiber collimators (C), single-mode fiber (SMF), beam splitter (BS), polarizer (P), neutral density filters (F) and digital camera (DC). The insert shows a photo of the shear interferometer. The diagram also shows the relevant parameters of the interferometer: the angle of incidence $\alpha $, the shear displacement s, the shear-plate separation t and second plate tilt $\delta $.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Khajavi, B.; Ureta, J.R.G.; Galvez, E.J.
Determining Vortex-Beam Superpositions by Shear Interferometry. *Photonics* **2018**, *5*, 16.
https://doi.org/10.3390/photonics5030016

**AMA Style**

Khajavi B, Ureta JRG, Galvez EJ.
Determining Vortex-Beam Superpositions by Shear Interferometry. *Photonics*. 2018; 5(3):16.
https://doi.org/10.3390/photonics5030016

**Chicago/Turabian Style**

Khajavi, Behzad, Junior R. Gonzales Ureta, and Enrique J. Galvez.
2018. "Determining Vortex-Beam Superpositions by Shear Interferometry" *Photonics* 5, no. 3: 16.
https://doi.org/10.3390/photonics5030016