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V.V. Kotlyar

443001, Samara, Russia, Molodogvardeyskaya 151,

**DOI: **10.18287/2412-6179-CO-816

**Pages: **19-28.

**Full text of article:** Russian language.

**Abstract:**

Here we show theoretically that a superposition of two Bessel-Gaussian (BG) beams with different topological charges (TC) and different scaling factors (radial components of the wave vectors) has the TC equal to that of the BG beam with the larger scaling factor. If the scaling factors of the BG beams are equal, then TC of the whole superposition equals TC of the BG beam with the larger (in absolute value) weight coefficient in the superposition (i.e. with larger power). If the constituent BG beams are also same-power, TC of the superposition equals the average TC of the two BG beams. Therefore, if the sum of TCs of both beams is odd, TC of the superposition is a half-integer number. In practice, however, TC is calculated over a finite radius circle and, hence, the half-integer TC for the degenerated case cannot be obtained. Instead of the half-integer TC, the lower of the two integer TCs is obtained. Numerical simulation reveals that if the weight coefficients in the superposition are slightly different, TC of the superposition is not conserved on propagation. In the near field and in the Fresnel diffraction zone, TC is equal to the highest TC of the two BG beams, while in the far field it is equal to the lower TC. What is more, TC changes its value from high to low not instantly, but continuously at some propagation distance. In the intermediate zone TC is fractional.

**Keywords**:

topological charge, Bessel-Gaussian beam, Fresnel diffraction, far field.

**Citation**:

Kotlyar VV, Kovalev AA. Topological charge of a superposition of two Bessel-Gaussian beams. Computer Optics 2021; 45(1): 19-28. DOI: 10.18287/2412-6179-CO-816.

**Acknowledgements**:

This work was supported by the Russian Foundation for Basic Research under projects No. 18-29-20003 (Sections "Calculation of the topological charge of the sum of two BG-beams" and "Topological charge of a superposition of two Bessel-Gaussian beams with the same weight and scaling coefficients") and No. 20-57-56006 (Sections "Simulation" and "Simulation in the case of nearly equal weight coefficients"), as well as the Ministry of Science and Higher Education within the government project of the FSRC "Crystallography and Photonics" RAS (Sections "Introduction" and "Conclusion").

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