# Spiral Caustics of Vortex Beams

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## Abstract

**:**

## 1. Introduction

## 2. Electromagnetic Field on a Curved Surface

#### 2.1. Coordinate Systems

#### 2.2. An Incident Wave in the Coordinates $\left(x,y,z\right)$

#### 2.3. Computation of the Field: General Case

## 3. Computing the Field in the Neighborhood of a Spiral Caustic Surface

#### 3.1. Asymptotic Relations for the Diffraction Integral in the Neighborhood of a Spiral Caustic

#### 3.2. An Analytical Solution for an Annular Caustic

## 4. Designing DOEs to Generate Spiral Caustics

## 5. Results of the Numerical Simulation and the Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**An optical setup for generating the laser beams under study: LASER—a solid-state laser (λ = 532 nm), L1, L2, L3, and L4—lenses with focal lengths of 25, 150, 500, and 250 mm, respectively, SLM—a spatial light modulator HOLOEYE PLUTO VIS, D—a circular pupil, and CAM—a video-camera.

**Figure 2.**x-components of the fields considered in Table 1: (

**a**) phase of the input field, (

**b**) amplitude and (

**c**) phase of the field in the focal plane, (

**d**) normalized moduli of the expansion coefficients $\left|{c}_{n}\right|$ for the input fields in the first line (red) and in the second line (blue).

**Figure 3.**Variations of the orbital angular momentum (OAM) for the 3D microspiral with p = −10 by means of an additional vortex phase m = 0 (top line), m = −5 (middle line), m = 10 (bottom line): (

**a**) phase of the input field, (

**b**) amplitude, and (

**c**) phase of the field in the focal plane, (

**d**) normalized moduli of the expansion coefficients $\left|{c}_{n}\right|$ for the input fields in the first (blue), second (green), and third (red) lines.

**Table 1.**Numerical modeling and experiment on focusing into a single-turn spiral curve for the linear x-polarization.

Optical Element Phase λ = 532 nm, f _{1} = f_{0} = 100 mm,r _{0} = 1 mm, r_{1} = 1.25 mm | Intensity Distribution, x, y ∈ [−1.5 mm, 1.5 mm] | |||
---|---|---|---|---|

z = 100 mm Modeling | |E(u,v)|^{2} | |||

z = 80 mm | z = 100 mm | z = 120 mm | ||

R_{d} = 1.5 mm | |E_{x}(u,v)|^{2} | modeling | ||

|E_{z}(u,v)|^{2} | experimental | |||

R_{d} = 0.7 mm | |E_{x}(u,v)|^{2} | modeling | ||

|E_{z}(u,v)|^{2} | experimental | |||

**Table 2.**Comparative modeling of focusing into a single-turn spiral curve: Equations (30)–(31) vs. Equations (39)–(40), for circular polarization.

Calculation Parameters λ = 532 nm, f _{1} = f_{0} = 100 mm | Calculation Method | Intensity Distribution of the Electric Field Components in the Plane z = 100 mm | |
---|---|---|---|

|E_{x}(u,v)|^{2} | |E_{z}(u,v)|^{2} | ||

Input field: R_{d} = 1.5 mmOutput field: r _{0} = 1 mm, r_{1} = 1.25 mm,x, y ∈ [−1.5 mm, 1.5 mm] | Direct integration (30)–(31) | ||

Asymptotic (39)–(40) | |||

Input field: R_{d} = 1.0 mmOutput field: r _{0} = 2 mm, r_{1} = 2.5 mm,x, y ∈ [−3 mm, 3 mm] | Direct integration (30)–(31) | ||

Asymptotic (39)–(40) |

Parameters λ = 532 nm, f _{1} = f_{0} = 100 mm | Optical Element Phase | Intensity Distribution of the Electric Field Components in the Plane z = 100 mm | ||
---|---|---|---|---|

Modeling |E _{x}(u,v)|^{2} | Modeling |E _{z}(u,v)|^{2} | Experimental |E(u,v)| ^{2} | ||

R_{d} = 1.5 mm,r _{0} = 1 mm,r _{1} = 1.25 mm,r _{2} = 1.5 mm | Sectorial | |||

Compositional | ||||

R_{d} = 1.5 mm,r _{0} = 2 mm,r _{1} = 2.5 mm,r _{2} = 3 mm | Sectorial | |||

Compositional | ||||

Element Phase λ = 532 nm, R _{d} = 120λ,f _{0} = 100λ | Distortion of the Total Intensity of the Components Electric Field in Different Planes (x, y = ∈ [−120λ, 120λ]) | |||
---|---|---|---|---|

z = 100λ | z = 110λ | z = 120λ | z = 130λ | |

f_{1} = 100λ, p = 0 | modeling | |||

experimental | ||||

f_{1} = 110λ, p = −10 | modeling | |||

experimental | ||||

f_{1} = 120λ, p = −20 | modeling | |||

experimental | ||||

f_{1} = 150λ, p = −50 | modeling | |||

experimental | ||||

Element Phase λ = 532 nm, R _{d} = 1.5 mm,r _{0} = 1 mm, r_{1} = 1.25 mm | Distortion of the Total Intensity of the Components Electric Field in Different Planes (x, y = ∈ [–1.5 mm, 1.5 mm]) | ||
---|---|---|---|

z = 100 mm | z = 150 mm | z = 200 mm | |

f_{0} = 100 mm, f_{1} = 150 mm | modeling | ||

experimental | |||

f_{0} = 100 mm, f_{1} = 200 mm | modeling | ||

experimental | |||

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**MDPI and ACS Style**

Soifer, V.A.; Kharitonov, S.I.; Khonina, S.N.; Strelkov, Y.S.; Porfirev, A.P.
Spiral Caustics of Vortex Beams. *Photonics* **2021**, *8*, 24.
https://doi.org/10.3390/photonics8010024

**AMA Style**

Soifer VA, Kharitonov SI, Khonina SN, Strelkov YS, Porfirev AP.
Spiral Caustics of Vortex Beams. *Photonics*. 2021; 8(1):24.
https://doi.org/10.3390/photonics8010024

**Chicago/Turabian Style**

Soifer, Viktor A., Sergey I. Kharitonov, Svetlana N. Khonina, Yurii S. Strelkov, and Alexey P. Porfirev.
2021. "Spiral Caustics of Vortex Beams" *Photonics* 8, no. 1: 24.
https://doi.org/10.3390/photonics8010024