# Synthesis of Super-Oscillatory Point-Spread Functions with Taylor-Like Tapered Sidelobes for Advanced Optical Super-Resolution Imaging

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Foundation

## 3. Theoretical Approach to Taylor-Like Super-Directive Patterns

#### 3.1. $\overline{n}$ Taylor-Like Super-Directive Patterns

#### 3.2. One-Parameter Taylor-Like Super-Directive Patterns

## 4. Reduction of the Maximum Excitation Weighting Ratio

## 5. Imaging Theory and Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Polynomials $P\left(q\right)={\prod}_{n=1}^{N-1}(q-\mathit{Re}\left[{z}_{n}\right])$ where $\mathit{Re}$ denotes the real part. (

**a**) Comparison of Chebyshev (blue line) and $\overline{n}$ Taylor-like polynomials where the red dashed line denotes a 0dB preset first sidelobe level and green dotted line denotes a −20 dB preset first sidelobe level. (

**b**) Comparison of Chebyshev (blue line) and one-parameter Taylor-like polynomials where the red dashed line denotes when $B=0$ and the green dotted line denotes a −20 dB preset first sidelobe level.

**Figure 2.**Comparison of the $\overline{n}$ (blue line) and the one-parameter (green dotted line) Taylor-like polynomials. The illustration of the pattern ratio (dB) defined by ${\left|P\left(q\right)\right|}^{2}\left(dB\right)-{\left|P\left(1\right)\right|}^{2}\left(dB\right)={\left|P\left(q\right)\right|}^{2}\left(dB\right)$.

**Figure 3.**(

**a**) Comparison of Taylor-like patterns produced by the Schelkunoff’s polynomial method (blue line) and the Taylor-pattern function (green dotted line) where $N=21$, $\overline{n}=3$ and $cos\pi A=20$ dB. (

**b**) Taylor-like patterns where $\overline{n}$ is 7 (red dashed line), 9 (green dotted line) and 10 (blue line) respectively, $cos\pi A=20$ dB and $N=21$. (

**c**) Taylor-like patterns where N is 13 (green dotted line) and 29 (blue line) respectively with the same $\lambda /2$ element spacing, $cos\pi A=20$ dB and $\overline{n}=3$. (

**d**) Taylor-like patterns where the first sidelobe level is −10 dB (green dotted line), −20 dB (blue line) and −30 dB (red dashed line) respectively, $N=21$ and $\overline{n}=3$.

**Figure 4.**(

**a**) Simulated 21-element Taylor-like super-directive pattern where $\overline{n}=3$ (blue line), Chebyshev super-directive pattern (green dotted line) and the reference uniform pattern (red dashed line) with the same element spacing, $d=0.29\lambda $. (

**b**) The zeros distribution of the Taylor-like super-directive pattern on the complex unit circle.

**Figure 5.**Comparison of Taylor-like super-directive patterns with different combinations of $cosh\pi A$ and pattern ratios, where the blue line denotes a −10 dB first sidelobe level when the pattern ratio is 4 dB and $cosh\pi A=6$ dB, the red dashed line denotes a −20 dB first sidelobe level when the pattern ratio is 12 dB and $cosh\pi A=8$ dB, the green dotted line denotes a −30 dB first sidelobe level when the pattern ratio is 15 dB and $cosh\pi A=15$ dB.

**Figure 6.**(

**a**) Comparison of one-parameter Taylor-like patterns (green dotted line for $N=21$ and red dashed line for $N=29$) and $\overline{n}$ Taylor-like patterns (blue line for $N=21$ and purple dot-dash line for $N=29$) produced by Schelkunoff’s polynomial method with the same $\lambda /2$ element spacing. (

**b**) One-parameter 21-element Taylor-like patterns where the first sidelobe level is −13.2 dB (blue line), −30 dB (red dashed line) and −50 dB (green dotted line) respectively.

**Figure 7.**(

**a**) Simulated 29-element one-parameter (blue line) and $\overline{n}$ (green dotted line) Taylor-like super-directive patterns. (

**b**) The zeros distribution of the one-parameter Taylor-like super-directive pattern on the complex unit circle.

**Figure 8.**The simulated 33-element Taylor-like super-directive patterns with reduced maximum weighting ratio. (

**a**) Zeros distribution of $\overline{n}$ Taylor-like super-directive pattern. (

**b**) Zeros distribution of one-parameter Taylor-like super-directive pattern. (

**c**) Patterns of $\overline{n}$ (blue line) and one-parameter (green dotted line) Taylor-like super-directive arrays.

**Figure 9.**(

**a**) Antenna array in the spatial domain and super-directivity in the angular domain. (

**b**) SO mask in the frequency domain and super-oscillation in the spatial domain (SO denotes super-oscillatory and PSF denotes point-spred function).

**Figure 13.**The experimental results of the Taylor-like super-oscillatory point-spread function corresponding to an one-parameter Taylor-like super-directive array. (

**a**) Two-dimensional Taylor-like super-oscillatory point-spread function. (

**b**) Experimental (blue line) and theoretical (red dashed line) One-dimensional Taylor-like super-oscillatory point-spread functions. Experimental (green dotted line) and theoretical (purple dot-dashed line) diffraction-limited point-spread functions (red line) at the $y=0$ plane. The full widths at half maximum (FWHM) for the experimental and theoretical Taylor-like super-oscillatory point-spread functions, and the experimental and theoretical diffraction-limited point-spread functions are around 29.1 $\mathsf{\mu}$m, 24.4 $\mathsf{\mu}$m, 40.5 $\mathsf{\mu}$m and 37.6 $\mathsf{\mu}$m, respectively. The ratio of the experimental Taylor-like FWHM to the experimental diffraction-limited FWHM is 0.72.

**Figure 14.**(

**a**) Simulation results of super-oscillatory Chebyshev (red dotted line), super-oscillatory Taylor-like (blue line) and diffraction-limited (green dashed line) point-spread functions. (Note: the SO Chebyshev has slightly tapered sidelobes because the accuracy of the coefficients is limited by the bit-depth (8-bit in this SLM), and imperfect linearity of the applied voltage and modulated phase; ) (

**b**) Intensity distributions along the x-axis in (

**c**) (red dotted line) and (

**d**) (blue line) compared with the diffraction-limited one (green dashed line). (Note: the equivalent Rayleigh criterion for a coherent optical system, which is the one used for this paper, is $0.82\lambda /NA\approx $ 60 $\mathsf{\mu}$m [11]). The experimental results of imaging two apertures (

**c**) with super-oscillatory Chebyshev point-spread function; (

**d**) with super-oscillatory Taylor-like point-spread function. (Note: to clarify the effect of the sidelobe structure on the resolution, the super-oscillatory point-spread functions are designed to have super-oscillatory sidelobes and the same beamwidth as the diffraction-limited one. Thus, the super-resolution would only result from the sidelobe structure.)

**Figure 15.**The experimental results of imaging a mask with the letter E (

**a**) with the diffraction-limited point-spread function; (

**b**) with a super-oscillatory Chebyshev point-spread function; (

**c**) with a super-oscillatory Taylor-like point-spread function. (

**d**) Intensity distribution along the red line shown in (

**c**) compared with the corresponding ones in (

**a**,

**b**). Taylor-like (blue line) has the lowest two dips between three peaks (indicating the three branches of the letter E) compared to the Chebyshev (red dotted line). The diffraction-limited (green dashed line) cannot resolve the three peaks.

**Figure 16.**Two transmissive objects are etched on a nickel substrate, including (

**a**) an object of two apertures, and (

**b**) an object of the letter E.

$\mathit{N}=21$, $\overline{\mathit{n}}=3$ and $\mathit{d}=0.29\mathit{\lambda}$ | ||||
---|---|---|---|---|

Types | Taylor | Chebyshev | Uniform | |

Parameters | ||||

First sidelobe level (dB) | −13.27 | −13.27 | −13.27 | |

Directivity (dBi) | 11.84 | 11.29 | 10.86 | |

Half power beamwidth (DEG) | 6.52 | 6.14 | 8.44 | |

Maximum weighting ratio | 1.86 × ${10}^{3}$ | 2.12 × ${10}^{3}$ | 1 |

**Table 2.**Comparison of $\overline{n}$ Taylor-like super-directive patterns with different combinations of $cosh\pi A$ and pattern ratios.

$\mathit{d}=0.21\mathit{\lambda}$, $\overline{\mathit{n}}=3$ and $\mathit{N}=29$ | |||||
---|---|---|---|---|---|

Parameters | c | Directivity (dBi) | Directivity Ratio | Maximum Weighting Ratio | |

Pattern Ratio (dB) and$\mathit{cosh}\mathit{\pi}\mathit{A}$ | |||||

4 dB and 6 dB | 1.01 | 12.89 | 1.59 | $1.57\times {10}^{6}$ | |

12 dB and 8 dB | 1.02 | 12.68 | 1.52 | $1.53\times {10}^{6}$ | |

15 dB and 15 dB | 1.03 | 12.09 | 1.32 | $1.50\times {10}^{6}$ |

$\mathit{N}=29$ and $\mathit{d}=0.21\mathit{\lambda}$ | |||
---|---|---|---|

Types | One-Parameter | $\overline{\mathit{n}}$ | |

Parameters | |||

First sidelobe level (dB) | −20 | −20 | |

Directivity (dBi) | 12.69 | 12.84 | |

Half power beamwidth (DEG) | 5.92 | 5.54 | |

Maximum weighting ratio | 1.54 × ${10}^{6}$ | 1.56 × ${10}^{6}$ |

**Table 4.**Comparison of the two types of Taylor-like super-directive patterns with reduced maximum weighting ratio.

$\mathit{N}=33$ with 8 Extra Zeros Outside the Visible Region or $\mathit{N}=25$ without Extra Zeros | |||||
---|---|---|---|---|---|

Types | One-Parameter ($\mathit{N}=\mathbf{33}$) | $\overline{\mathit{n}}$($\mathit{N}=\mathbf{33}$) | One-Parameter ($\mathit{N}=\mathbf{25}$) | $\overline{\mathit{n}}$($\mathit{N}=\mathbf{25}$) | |

Parameters | |||||

d | 0.18$\lambda $ | 0.18$\lambda $ | 0.24$\lambda $ | 0.24$\lambda $ | |

Array size | 5.82$\lambda $ | 5.82$\lambda $ | 5.76$\lambda $ | 5.76$\lambda $ | |

First sidelobe level (dB) | −20 | −20 | −20 | −20 | |

Directivity (dBi) | 12.27 | 12.02 | 12.34 | 12.16 | |

Directivity ratio | 1.38 | 1.30 | 1.40 | 1.34 | |

Maximum weighting ratio | 478.15 | 464.00 | $6.05\times {10}^{4}$ | $5.80\times {10}^{4}$ |

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**MDPI and ACS Style**

Yang, H.; Eleftheriades, G.V. Synthesis of Super-Oscillatory Point-Spread Functions with Taylor-Like Tapered Sidelobes for Advanced Optical Super-Resolution Imaging. *Photonics* **2021**, *8*, 64.
https://doi.org/10.3390/photonics8030064

**AMA Style**

Yang H, Eleftheriades GV. Synthesis of Super-Oscillatory Point-Spread Functions with Taylor-Like Tapered Sidelobes for Advanced Optical Super-Resolution Imaging. *Photonics*. 2021; 8(3):64.
https://doi.org/10.3390/photonics8030064

**Chicago/Turabian Style**

Yang, Haitang, and George V. Eleftheriades. 2021. "Synthesis of Super-Oscillatory Point-Spread Functions with Taylor-Like Tapered Sidelobes for Advanced Optical Super-Resolution Imaging" *Photonics* 8, no. 3: 64.
https://doi.org/10.3390/photonics8030064