# Generation, Topological Charge, and Orbital Angular Momentum of Off-Axis Double Vortex Beams

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generation of Off-Axis Double Vortex Beams

#### 2.1. Mathematical Description

_{0}, y

_{0}) = (0, 0)). In the Cartesian coordinate system, the linearly polarized on-axis vortex beam at the plane of z = 0 can be simply expressed as [2,17]

_{0}is an amplitude constant, m is the azimuthal index, and w is the waist radius of the fundamental Gaussian beam. When m is positive, the sign of y is positive, and vice versa. As shown in Figure 1a, the intensity distribution of the on-axis vortex beam maintains cylindrical symmetry.

_{1}along the x-axis and y

_{1}along the y-axis, as shown in Figure 1b, an off-axis single vortex beam will be obtained. It is noteworthy that the amplitude distribution of the off-axis single vortex beam presents a cylindrical symmetry breaking, and its phase singularity located at (x

_{1}, y

_{1}) deviates from the geometric center (0, 0) (see Figure 1e). The electric field of the linearly polarized off-axis single vortex beam can be described as [17]

_{1}and m

_{2}are the azimuthal indices of off-axis double vortexes with two phase singularities located at (x

_{1}, y

_{1}) and (x

_{2}, y

_{2}), respectively.

_{1}, and y

_{1}in the off-axis single vortex beam, and six adjustable parameters m

_{1}, m

_{2}, x

_{1}, y

_{1}, x

_{2}, and y

_{2}in the off-axis double vortex beam. Clearly, by adjusting the two pairs of independent variables (i.e., the azimuthal index m

_{j}, the positions located at two phase singularities (x

_{j}, y

_{j}) where j = 1, 2) of the off-axis double vortex beam, one will exploit its rich physical properties. Note that the presented off-axis double vortex beam has two main differences from the on-axis double vortex beam: (i) off-axis and on-axis double vortex beams have six and two adjustable parameters, respectively; (ii) on-axis double vortex beams cannot carry a pair of phase singularities with opposite TC signs, while off-axis double vortex beams can.

#### 2.2. Experimental Arrangement

#### 2.3. Results and Discussion

_{1}and m when w = 2.5 mm and x

_{1}= 0. Clearly, the theoretically simulated intensity patterns (see Figure 3a) of off-axis single vortex beams are in good agreement with the experimentally measured results (see Figure 3b). Compared with the axially symmetric on-axis vortex beam (i.e., y

_{1}= 0), the phase singularity of the off-axis single vortex beam is not at the geometric center of the beam, and the symmetry of its intensity distribution is broken. Moreover, with the increase of the off-axis distance |y

_{1}|, this symmetry breaking becomes more serious, resulting in the intensity distribution of the off-axis single vortex beams changing from a doughnut-shaped pattern to a crescent-shaped pattern. In addition, as the azimuthal index |m| increases, the phase singularity region of the off-axis vortex beam increases.

_{2}when w = 2.5 mm, x

_{1}= x

_{2}= 0, and m

_{1}= 1. When the parameters are taken as Δ = 0.5 w and x

_{1}= x

_{2}= 0, the intensity patterns of off-axis double vortex beams with different azimuthal indices m

_{1}and m

_{2}are shown in Figure 5. Obviously, for the off-axis double vortex beams shown in Figure 4 and Figure 5, their theoretically simulated intensity patterns are basically consistent with the experimentally measured results. The reason for the difference is that the centers of two off-axis phases loaded on the imperfect Gaussian beam in the experiments are not completely symmetrical about the beam’s geometric center. In addition, it is found that two phase singularities of the off-axis double vortex beam propagate stably in free space. As shown in Figure 4 and Figure 5, all off-axis double vortex beams have mirror symmetric intensity distribution. Especially when |m

_{1}| = |m

_{2}|, the intensity distribution of this off-axis double vortex beam maintains the two-fold rotational symmetry. Furthermore, with the increase of Δ value, the intensity distribution of this vortex beam changes from a doughnut-shaped pattern to a bowtie-shaped pattern. It is noteworthy that the intensity distribution of the off-axis single and double vortex beams (see Figure 3 and Figure 4) is only related to the position of phase singularities and the magnitude of azimuthal indices but is independent of the sign of these azimuthal indices. Therefore, the sign of the azimuthal index cannot be directly determined from the intensity distribution of the off-axis vortex beam. It is very necessary to develop other methods to identify the magnitude, sign, and distribution of the azimuthal index of the off-axis vortex beam.

## 3. TCs of Off-Axis Double Vortex Beams

#### 3.1. Calculation of the TC

_{1}and m

_{2}. It should be emphasized that if the magnitudes of two azimuthal indices are equal but the signs are opposite, this off-axis double vortex beam carries vortices, although the total TC is equal to zero. Consequently, the TC is not enough to describe the phase singularity of the off-axis double vortex beam, and the spatial distribution of the TC also needs to be known. In addition, it is worth noting that the TC of the vortex beam is usually independent of the propagation of the beam at different positions [23], which is the basis for detecting the TC of the vortex beam using the approaches such as interferometry [41], intensity analysis of vortex beams [42], and diffractometry [27].

#### 3.2. Electric Field of the Off-Axis Vortex Beam Focused by a Tilted Convex Lens

_{0}in free space. According to the Huygens–Fresnel integral formula, the electric field at the propagation distance z after the lens can be expressed as [27]

_{l}(x,y) is required. Then, the integrations over x and y for an integer n ≥ 0 can be accomplished by the integral theorem

_{n}(i,β) is the Hermite polynomial of a complex argument. In this way, we obtain the analytical expression of the electric field at the propagation distance z after the titled lens as

_{l}(u,v) is an elliptical Gaussian beam modulated by Hermite polynomials. Therefore, at a certain position behind the lens, the intensity spots and their orientation give the magnitude and sign of the TC of the vortex beams, respectively [27]. Note that the obtained electric field E

_{I}(u,v) for the on-axis vortex beam is consistent with the one reported previously [27].

#### 3.3. Detection of the TC

_{0}= 50 cm, and z = 32 cm. The intensity patterns near the focal plane of the tilted convex lens shown in Figure 6, Figure 7 and Figure 8 are measured by the experimental setup shown in Figure 2b. It is obvious that the theoretical simulations are consistent with the experimental measurements.

_{1}and m

_{2}carried by the off-axis double vortex beam will have a significant impact on the intensity distribution focused by the tilted lens. If the signs of two TCs are the same, the number of dark inclined stripes is equal to the algebraic sum of their respective values. In contrast, if the signs of two TCs are opposite, the measured intensity distribution will be separated in space. For example, as shown in the first column of Figure 7, the left and right parts of the intensity distribution come from the contributions of TCs with negative and positive signs, respectively. Especially, in the first row and the second column of Figure 7, the number of dark stripes on the left and right sides of the intensity pattern is 1, which means that this vortex beam carries two vortices with TCs of −1 and +1, respectively. As shown in Figure 7, with the increase of the off-axis distance Δ, the intensity at both ends of the bright inclined stripes will be weakened, which will reduce the accuracy of the detected TC. The main reasons are analyzed as follows: when the off-axis distance Δ increases gradually, the intensity distribution of the off-axis double vortex beam changes from the ring type to the half-moon type, so the intensity of the bright stripes after the beam passes through the tilted lens will also be reduced, but the original phase singularity still exists. In short, we demonstrate theoretically and experimentally that the tilted lens method can detect not only the magnitudes and signs of two TCs of the off-axis double vortex beam but also the spatial distribution of the TCs. This is because when the signs of two TCs are opposite, the inclined stripes of the off-axis double vortex beam passing through the tilted lens are separated in space.

## 4. OAM of Off-Axis Double Vortex Beams

#### 4.1. OAM Density and Average OAM

^{*}(x, y) denotes the complex conjugate of the electric field E(x, y).

_{z}is defined as the OAM of paraxial beams J

_{z}normalized to power W [44]

_{z}= m [44] by substituting Equation (1) in Equation (23). If L

_{z}is multiplied by Planck’s constant, the OAM per photon carried by the conventional vortex beam can be obtained. However, for the off-axis single and double vortex beams, their average OAM value needs to be calculated numerically. Numerical simulation shows that the magnitude of the average OAM of the off-axis vortex beam decreases nonlinearly with the increase of off-axis distance. When the phase singularity is close to the beam center (i.e., x

_{j}→0 and y

_{j}→0), the average OAM is close to the TC. On the contrary, the average OAM tends to be zero when the off-axis distance is large (i.e., x

_{j}→∞ and y

_{j}→∞).

#### 4.2. Average OAM Measurement

^{2}is the intensity distribution at the focal plane of the cylindrical lens.

_{1}= 0) and off-axis double vortex beam (m

_{1}= 1, m

_{2}= 2, and x

_{1}= x

_{2}= 0), respectively. The numerical simulations by Equation (23) are also plotted in Figure 9. Apparently, the measured OAM values are in good agreement with the theoretical results. It is shown that the average OAM value decreases nonlinearly as the off-axis distance (y

_{1}or Δ) increases. This result can be understood as follows: the average OAM value of off-axis single Gaussian vortex beams is L

_{z}= mexp[−2(r

_{0}/w)

^{2}], which decreases nonlinearly as the off-axis distance r

_{0}increases [20,22]. As described in Equation (3), an off-axis double vortex beam is equivalent to the superposition of two off-axis single vortex beams. Obviously, OAM density (see Equation (22)) and average OAM have similar conclusions. When the phase singularity is close to the beam center, the average OAM value is approximately equal to the TC. When the off-axis distance Δ is large, the average OAM trends to zero, although the TC of the off-axis single and double vortex beams is independent of the off-axis distance of the phase singularities. The results show that changing the off-axis distance of the off-axis vortex beam can easily manipulate the average OAM of the vortex beam, thereby realizing the applications of particle manipulation, optical communication, etc.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**f**) The intensity and phase distributions of the on-axis vortex, off-axis single vortex, and off-axis double vortex beams. The parameters describing the vortex beams are marked in the figures.

**Figure 2.**Schematic experimental setup. The setup consists of three parts: (

**a**) the generation of the vortex beams, (

**b**) TC detection, and (

**c**) OAM measurement. (

**d**) Exemplary phase masks projected on the SLM for generating (I) on-axis vortex, (II) off-axis single vortex, and (III) off-axis double vortex beams.

**Figure 3.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis single vortex beams with different values of y

_{1}and m when w = 2.5 mm and x

_{1}= 0.

**Figure 4.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis double vortex beams with different values of Δ and m

_{2}when w = 2.5 mm, x

_{1}= x

_{2}= 0, and m

_{1}= 1.

**Figure 5.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis double vortex beams with different values of m

_{1}and m

_{2}when w = 2.5 mm, x

_{1}= x

_{2}= 0, and Δ = 0.5 w.

**Figure 6.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis single vortex beams with different values of y

_{1}and m focused by a tilted convex lens, when λ = 1064 nm, w = 2.5 mm, f = 30 cm, z

_{0}= 50 cm, z = 28 cm, θ = 21°, and x

_{1}= 0.

**Figure 7.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis double vortex beams with different values of Δ and m

_{2}focused by a tilted convex lens when λ = 1064 nm, w = 2.5 mm, f = 30 cm, z

_{0}= 50 cm, z = 28 cm, θ = 21°, x

_{1}= x

_{2}= 0, and m

_{1}= 1.

**Figure 8.**(

**a**) Theoretically simulated and (

**b**) experimentally measured intensity patterns of off-axis double vortex beams with different values of m

_{1}and m

_{2}focused by a tilted convex lens when λ = 1064 nm, w = 2.5 mm, f = 30 cm, z

_{0}= 50 cm, z = 28 cm, θ = 21°, x

_{1}= x

_{2}= 0, and Δ = 0.5 w.

**Figure 9.**The average OAM values of (

**a**) off-axis single vortex beams with different values of y

_{1}when m = 2 and x

_{1}= 0, and (

**b**) off-axis double vortex beams with different values of Δ when m

_{1}= 1, m

_{2}= 2, and x

_{1}= x

_{2}= 0. The scatters are the experimental data, while the solid lines are the numerical simulations by Equation (23). The inserts are exemplary intensity distributions at the focal plane of the cylindrical lens.

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**MDPI and ACS Style**

Guo, M.; Le, W.; Wang, C.; Rui, G.; Zhu, Z.; He, J.; Gu, B. Generation, Topological Charge, and Orbital Angular Momentum of Off-Axis Double Vortex Beams. *Photonics* **2023**, *10*, 368.
https://doi.org/10.3390/photonics10040368

**AMA Style**

Guo M, Le W, Wang C, Rui G, Zhu Z, He J, Gu B. Generation, Topological Charge, and Orbital Angular Momentum of Off-Axis Double Vortex Beams. *Photonics*. 2023; 10(4):368.
https://doi.org/10.3390/photonics10040368

**Chicago/Turabian Style**

Guo, Mingxian, Wei Le, Chao Wang, Guanghao Rui, Zhuqing Zhu, Jun He, and Bing Gu. 2023. "Generation, Topological Charge, and Orbital Angular Momentum of Off-Axis Double Vortex Beams" *Photonics* 10, no. 4: 368.
https://doi.org/10.3390/photonics10040368