# Study of Superoscillating Functions Application to Overcome the Diffraction Limit with Suppressed Sidelobes

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## Abstract

**:**

## 1. Introduction

## 2. Superoscillating Signal Based on Superposition of Spatial Harmonics

## 3. Superoscillating Field Based on the Power of the Complex Function

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Features of the superoscillating field: comparative distributions for the functions of Equation (1)—bold line, Equation (2)—dashed line, and of Equation (3)—thin line.

**Figure 3.**The spatial spectrum of the field of Equation (1): (

**a**) initial and (

**b**) truncated following function of Equation (5).

**Figure 5.**Field amplitude of Equation (1) propagated at the distance (

**a**) z = 1.5λ, (

**b**) z = 7λ, (

**c**) z = 15λ, (

**d**) z = 100λ.

**Figure 6.**Complex field distribution of Equation (6) for a = 2 at different values of the power degree: (

**a**) N = 2, (

**b**) N = 5, (

**c**) N = 10, (

**d**) N = 100 (the amplitude is shown by the solid line, and the phase by the dashed line).

**Figure 7.**Amplitude of the field of Equation (6) at a distance z = λ for N = 20 and T = 5λ at different values of the coefficient: (

**a**) a = 1.5, (

**b**) a = 5, (

**c**) a = 10, (

**d**) a = 20.

**Figure 8.**The amplitude of the field of Equation (6) with a = 10 and N = 20 propagated at the distance z = λ for different sizes of the input field: (

**a**) T = 2.5, (

**b**) T = 10.

**Figure 9.**The amplitude of the field of Equation (6) with a = 10 and T = 5λ propagated at the distance z = λ for different values of the power degree: (

**a**) N = 50, (

**b**) N = 100, (

**c**) N = 200, (

**d**) N = 300.

**Figure 10.**The amplitude of the field of Equation (6) with a = 10, N = 100 and T = 5λ propagated at the different distances: (

**a**) z = 0.5λ, (

**b**) z = 0.8λ, (

**c**) z = 3λ, (

**d**) z = 4λ, (

**e**) z = 5λ, (

**f**) z = 6λ.

**Figure 11.**Generation of a compact light spot in the near-field zone by the fractional axicon $\tau \left(r\right)=\mathit{exp}\left[-i{\left(k{\alpha}_{0}r\right)}^{\gamma}\right]$ with (

**a**) γ = 0.5, ${\alpha}_{0}=125$ and (

**b**) γ = 0.25, ${\alpha}_{0}=7937607$: graphs show normalized intensity sections at distance of z = 0.01λ (solid line), z = 0.1λ (dashed line), and z = 2λ (dotted line).

**Figure 12.**Generation of a compact light based on approximation of the input rectangle signal by eigenfunctions of the finite propagation operator (red color for the signal and blue color for an approximation): input signals of (

**a**) 1λ and (

**c**) 0.1λ width and corresponding diffraction results at z = λ distance (

**b**,

**d**), accordingly.

**Table 1.**Comparing results for a tightly focused radially polarized beam with different types of pupil function apodization (S is the sidelobe level).

Apodization (Amplitude and Phase Distribution) | Distribution in the Focal Plane (Intensity and Graph of Cross-Section) |
---|---|

Without apodization | FWHM = 0.544λ, S = 0.16 |

Narrow ring | FWHM = 0.378λ, S = 0.162 |

Ring with phase jump | FWHM = 0.371λ, S = 0.351 |

Optimized function | FWHM = 0.359λ, S = 0.28 |

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**MDPI and ACS Style**

Khonina, S.N.; Ponomareva, E.D.; Butt, M.A.
Study of Superoscillating Functions Application to Overcome the Diffraction Limit with Suppressed Sidelobes. *Optics* **2021**, *2*, 155-168.
https://doi.org/10.3390/opt2030015

**AMA Style**

Khonina SN, Ponomareva ED, Butt MA.
Study of Superoscillating Functions Application to Overcome the Diffraction Limit with Suppressed Sidelobes. *Optics*. 2021; 2(3):155-168.
https://doi.org/10.3390/opt2030015

**Chicago/Turabian Style**

Khonina, Svetlana N., Ekaterina D. Ponomareva, and Muhammad A. Butt.
2021. "Study of Superoscillating Functions Application to Overcome the Diffraction Limit with Suppressed Sidelobes" *Optics* 2, no. 3: 155-168.
https://doi.org/10.3390/opt2030015