# Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}, Λ

_{2}and Λ

_{3}are the periods of the axicon, X and Y components of the linear gratings respectively and A is the amplitude. The randomness of A, Λ

_{1}, Λ

_{2}and Λ

_{3}was controlled by an independent uniform random variable $C\xb7U\left[0,1\right]$, where C is a constant used to set limits for different parameters (namely amplitude and periods).

_{s}. The complex amplitude entering the optical modulator is given as ${C}_{1}\sqrt{{I}_{s}}L\left(\frac{\overline{{r}_{s}}}{{z}_{s}}\right)Q\left(\frac{1}{{z}_{s}}\right)$, where $Q\left(a\right)=exp\left[j\frac{\pi a}{\lambda}{R}^{2}\right]$ is a quadratic phase function and $L\left(\frac{\overline{s}}{u}\right)=exp\left[\frac{j2\pi \left({s}_{x}x+{s}_{y}y\right)}{\lambda {z}_{s}}\right]$ is a linear phase function and C

_{1}is a complex constant. The complex amplitude after the optical modulator is given as ${C}_{2}\sqrt{{I}_{s}}Q\left(\frac{1}{{z}_{s}}\right)L\left(\frac{\overline{{r}_{s}}}{{z}_{s}}\right){\mathsf{\Psi}}_{OM}$, where C

_{2}is a complex constant. The PSF is observed at a distance of ${z}_{h}$ from the optical modulator which is given as

_{s}is varied in Equation (2), the central maximum is contributed by different annular regions of the axicon until it reaches a point when the phase of any region of axicon cannot satisfy the imaging condition. At this point, the boundary of the focal depth and the pattern starts to change into a ring pattern, becoming larger in diameter with distance henceforth. Within the focal depth, the Bessel beam exhibits a sharp central maximum and rings around it with decreasing intensity values of its radius. When the optical modulator consists of many axicons with different linear phases as shown in Equation (1), multiple Bessel fields J

_{0}with different spatial frequencies, angles, and relative strengths were generated at different locations in the sensor plane. Wherever there is an overlap between the Bessel fields, there is self-interference. The intensity distribution at the sensor plane can be simplified as

_{O}and I

_{PSF}, given as

_{PSF}is not a Delta-like function, an indirect imaging method needs to be applied. It has been identified recently that a non-linear reconstruction (NLR) method gives the highest SNR compared to matched filter, phase-only filter and Weiner filter [9,24]. The NLR is given as ${I}_{R}=\left|{\mathcal{F}}^{-1}\left\{{\left|\tilde{{I}_{\mathrm{PSF}}}\right|}^{o}\mathrm{exp}\lceil j\xb7\mathrm{arg}\left(\tilde{{I}_{\mathrm{PSF}}}\right)\rceil {\left|{\tilde{I}}_{O}\right|}^{r}\mathrm{exp}\left[-j\xb7\mathrm{arg}\left({\tilde{I}}_{O}\right)\right]\right\}\right|$, where o and r are varied until the lowest reconstruction noise is obtained, arg(∙) refers to the phase and $\tilde{B}$ is the Fourier transform of B. So, in the indirect imaging framework for a single Bessel beam, the imaging PSF is not a Bessel distribution but a non-linear autocorrelation of a Bessel distribution which is a Kronecker Delta-like function [42].

## 3. Simulation Studies

^{2}respectively, where NA is the numerical aperture D/z

_{s}and D is the diameter of the entrance pupil which is 5 mm and z

_{s}and z

_{h}are 50 cm in this study. To have a reliable comparison, the simulation results are also compared with direct imaging using a Fresnel Zone Plate (FZP) with z

_{s}and z

_{h}in 2f configuration (f = 25 cm) and an axicon with a minimum period. The amplitude and phase of the optical modulators FZP, diffractive axicon (σ = 1) and optical modulators for generating self-interfering Bessel beams for σ = 0.2, 0.1, 0.05, 0.025, 0.0125, and 0.00625 and their axial distributions from the optical modulator to the sensor are shown in Figure 2. The axial distribution was calculated at every plane by varying z

_{h}in Equation (2) from the plane of the optical modulator to the sensor plane and the 2D information was accumulated into a cube data. As seen from Figure 2, with an increase in the sparsity σ, the density of the Bessel beams increases. The individual Bessel beam can be identified by the green lines while the side lobes and the interference effects are observed in orange color.

## 4. Three-Dimensional Imaging

## 5. Discussion

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Amplitude, phase and axial intensity distribution for FZP, diffractive axicon (σ = 1), self-interfering Bessel beams σ = 0.2, 0.1, 0.05, 0.025, 0.0125, and 0.00625.

**Figure 3.**PSF and MTF obtained for direct and indirect imaging modes for FZP, diffractive axicon (σ = 1), self-interfering Bessel beams σ = 0.2, 0.1, 0.05, 0.025, 0.0125, and 0.00625. The MTFs are normalized to 1.

**Figure 4.**Plot of I

_{a}(0,0) with respect to Δz for FZP, axicon and optical modulator with σ = 0.2, 0.1, 0.05, 0.025, 0.0125, and 0.00625.

**Figure 5.**Images of PSFs and the imaging results of the two planes with two test objects for FZP and diffractive axicon.

**Figure 6.**Reconstruction results obtained for two test objects located at two planes for σ = 0.33, 0.2, 0.1, and 0.05.

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Anand, V.
Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles. *Micromachines* **2022**, *13*, 1347.
https://doi.org/10.3390/mi13081347

**AMA Style**

Anand V.
Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles. *Micromachines*. 2022; 13(8):1347.
https://doi.org/10.3390/mi13081347

**Chicago/Turabian Style**

Anand, Vijayakumar.
2022. "Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles" *Micromachines* 13, no. 8: 1347.
https://doi.org/10.3390/mi13081347