Spiral Caustics of Vortex Beams
Abstract
:1. Introduction
2. Electromagnetic Field on a Curved Surface
2.1. Coordinate Systems
2.2. An Incident Wave in the Coordinates
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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Optical Element Phase λ = 532 nm, f1 = f0 = 100 mm, r0 = 1 mm, r1 = 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 | ||
Rd = 1.5 mm | |Ex(u,v)|2 | modeling | ||
|Ez(u,v)|2 | experimental | |||
Rd = 0.7 mm | |Ex(u,v)|2 | modeling | ||
|Ez(u,v)|2 | experimental | |||
Calculation Parameters λ = 532 nm, f1 = f0 = 100 mm | Calculation Method | Intensity Distribution of the Electric Field Components in the Plane z = 100 mm | |
---|---|---|---|
|Ex(u,v)|2 | |Ez(u,v)|2 | ||
Input field: Rd = 1.5 mm Output field: r0 = 1 mm, r1 = 1.25 mm, x, y ∈ [−1.5 mm, 1.5 mm] | Direct integration (30)–(31) | ||
Asymptotic (39)–(40) | |||
Input field: Rd = 1.0 mm Output field: r0 = 2 mm, r1 = 2.5 mm, x, y ∈ [−3 mm, 3 mm] | Direct integration (30)–(31) | ||
Asymptotic (39)–(40) |
Parameters λ = 532 nm, f1 = f0 = 100 mm | Optical Element Phase | Intensity Distribution of the Electric Field Components in the Plane z = 100 mm | ||
---|---|---|---|---|
Modeling |Ex(u,v)|2 | Modeling |Ez(u,v)|2 | Experimental |E(u,v)|2 | ||
Rd = 1.5 mm, r0 = 1 mm, r1 = 1.25 mm, r2 = 1.5 mm | Sectorial | |||
Compositional | ||||
Rd = 1.5 mm, r0 = 2 mm, r1 = 2.5 mm, r2 = 3 mm | Sectorial | |||
Compositional | ||||
Element Phase λ = 532 nm, Rd = 120λ, f0 = 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 | |
f1 = 100λ, p = 0 | modeling | |||
experimental | ||||
f1 = 110λ, p = −10 | modeling | |||
experimental | ||||
f1 = 120λ, p = −20 | modeling | |||
experimental | ||||
f1 = 150λ, p = −50 | modeling | |||
experimental | ||||
Element Phase λ = 532 nm, Rd = 1.5 mm, r0 = 1 mm, r1 = 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 | |
f0 = 100 mm, f1 = 150 mm | modeling | ||
experimental | |||
f0 = 100 mm, f1 = 200 mm | modeling | ||
experimental | |||
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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
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 StyleSoifer, 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