# Multilevel Spiral Axicon for High-Order Bessel–Gauss Beams Generation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{max}, the lateral energy is not available anymore for the self-reconstruction of the central lobes, so the wavefront of the beam is distorted. Bessel beams are nondiffractive optical beams associated with a Bessel function of order, m, in paraxial approximation [13].

_{max}= w/[(n − 1) δ], where w represents the waist of the incident beam, n the refractive index of the axicon, δ = arctan(q/R), and q and R are the cone’s height and radius, respectively. Axicons are used in various applications such as imaging, optical tweezers or optical spanners, beam shaping, material processing, and optical and quantum communication in free space or in fiber [13].

## 2. Numerical Methods

_{0}= 121 μm, and, in the case of m = 1, is x

_{1}= 681 μm for both z = 50 cm and z = 75 cm, which demonstrates the propagation invariance property of the Bessel beams. One can observe from the cross-section intensity distribution presented in Figure 3 how the Bessel function profile is kept along the propagation axis.

## 3. Results

#### 3.1. Design for the Photolithographic Masks for Axicons

_{max}, for which the Bessel–Gauss beam is diffraction-free (Figure 4c).

_{i}, corresponding to each of the m = 5 photolithographic masks is given by the equation:

_{max}, on which the Bessel–Gauss beam is diffraction-free depends on R, q, n, and δ, according to the equation:

_{1}= 696 nm, for the second photolithographic mask—h

_{2}= 348 nm, for the third photolithographic mask—h

_{3}= 174 nm, for the fourth photolithographic mask—h

_{4}= 87 nm, and for the fifth photolithographic mask—h

_{5}= 44 nm.

#### 3.2. Technological Flow for the Fabrication of the Multilevel Axicons

_{3}[41,42] or KOH solution [43,44], or a laser writing axicon written in the polymer [44], EBL [45]. These fused silica etchants present good results in terms of high optical quality and low roughness. The use of KOH solution presents the advantage of safety in comparison to the acid HF solution. The disadvantage of KOH solution for wet etching consists of the expense for masking—the use of Cr-Au and high temperatures for high etching rates. We investigate a low-cost solution for wet etching with the major advantage of a short fabrication time, which is convenient for rapid fabrication, taking into consideration all five wet etching processes needed for 32-level axicon fabrication. This solution also confers precise control of the etch depth. The multilevel axicons working in transmission are fabricated by wet etching fused silica in hydrofluoric acid and ammonium fluoride, HF:H

_{4}FN (1:6), T = 22–23 °C, and an etching rate of 100 nm/min.

_{4}FN (1:6), at T = 22–23 °C, for 7 min, at the etching rate of 100 nm/min. The photoresist was removed in acetone, and the wafer was cleaned in Piranha solution (H

_{2}SO

_{4}+H

_{2}O

_{2}: 660 mL + 220 mL). After that, the surface of the wafer was prepared for the photolithographic process corresponding to the second mask by conducting a thermal treatment at 140 °C for 2 h in plasma. The process of the exposure of the mask and development for M2 was performed in the same conditions as in the case of M1. The wet etching had the same parameters for the solution, with an etching time of 3 min and 35 s. The pattern of mask M3 was transferred on the fused silica wafer in the same manner as mask M2 was, with an etching time of 1 min and 45 s. During the fourth photolithographic process, corresponding to mask M4, there were some challenges regarding the adhesion of the photoresist to the fused silica substrate. We found a solution to overcome this issue by using HMDS (hexamethyldisilazane-controlled) as an adhesion promoter. Before the photolithographic process, the M4 wafer was in a vacuum for 30 min. After that, the promoter HMDS was introduced in the vacuum for 1 min. The photolithographic process for mask M4 was performed based on the same parameters as in the case of mask M3, followed by a wet etching process in HF:H

_{4}FN (1:6) solution for 50 s. The photolithographic process in the case of mask M5 was similar to the one performed for M4, and the wet process in HF:H

_{4}FN (1:6) solution was conducted for 50 s.

#### 3.3. Structural Characterization

#### 3.4. Functional Characterization

## 4. Divergence-Free Behavior

_{0}(k

_{r}x)|

^{2}, where the transverse momentum, k

_{r}, is determined by the geometrical parameters of the fabricated axicon. The results displayed in Figure 16 show that, within experimental errors, the maxima of the measured intensities are in the same positions and are in good agreement with the zeroes and maxima of the ideal Bessel beam.

_{1}(k

_{r}x)|

^{2}.

## 5. Conclusions

_{4}FN (1:6), T = 22–23 °C, and an etching rate of 100 nm/min. We succeeded in fabricating zero-order and spiral axicons with 16 and 32 levels, characterized by high mode conversion efficiency η = 98.72% and η = 99.68%, respectively, and good transmission for visible light (λ=633 nm wavelength). Although in this work the apertures of the axicons are in the range of millimeters, the maturity of the photolithographic techniques allows the realization of optical elements, specifically axicons, with diameters comparable with the wafer size. This will give the possibility to generate nondiffractive Bessel–Gauss beams with long propagation distances.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Zeroth-order axicon. (

**a**) Numerical axicon of zero order with discrete profile (32 levels), (

**b**) intensity of a Bessel–Gauss beam of zero order generated after the diffraction of a Gaussian beam on the axicon for z = 75 cm.

**Figure 2.**Higher-order axicons with discrete profile (32 levels): (

**a**) numerical axicon of first order, m = 1, (

**b**) intensity distribution of a Bessel–Gauss beam of order m = 1 generated after the diffraction of a Gaussian beam on the spiral axicon for z = 75 cm, (

**c**) numerical axicon of order m = 4, (

**d**) intensity distribution of a Bessel–Gauss beam of order m = 4 generated after the diffraction of a Gaussian beam on the spiral axicon for z = 75 cm.

**Figure 3.**Intensity distribution in cross-section generated by: (

**a**) zero-order axicon at z = 50 cm (black line) and z = 75 cm (red line), (

**b**) spiral first-order axicon at z = 50 cm (black line) and z = 75 cm (red line).

**Figure 4.**(

**a**) Refractive axicon with continuous profile; (

**b**) DOE—diffractive (Fresnel) axicon; (

**c**) z

_{max}—propagation distance for which the Bessel–Gauss beam is diffraction-free.

**Figure 5.**Photolithographic masks for multilevel axicon of order zero (

**a**–

**e**) M1–M5 for 5 photolithographic processes with different ring widths: (

**a**) a1 = 160 μm, (

**b**) a2 = 80 μm, (

**c**) a3 = 40 μm, (

**d**) a4 = 20 μm, (

**e**) a5 = 10 μm.

**Figure 6.**Photolithographic masks for multilevel axicon of order 1 (

**a**–

**e**) M1–M5 for five photolithographic processes.

**Figure 7.**Flow chart for the fabrication of multilevel axicon with a number of

**#i**photolithographic masks.

**Figure 8.**Cross-section profile of a zero-order axicon with 16 levels, measured with mechanical profilometer.

**Figure 9.**Cross-section profile of a zero-order axicon with 32 levels, measured with mechanical profilometer.

**Figure 10.**White light interferometry tridimensional maps for (

**a**) zero-order axicon with 32 levels in fused silica, (

**b**) spiral axicon with 32 levels in fused silica, (

**c**) cross-section profile for zero-order axicon with 32 levels from (

**a**).

**Figure 11.**(

**a**) AFM cross-section image for 32-level spiral axicon m = 4 in fused silica, 44.7 nm step, (

**b**) roughness 2D profile—0.8 nm for 7 μm

^{2}area.

**Figure 12.**Optical setup for the generation of intensity and wavefront distribution for Bessel–Gauss beams.

**Figure 13.**Measured intensity distribution of the beam generated by (

**a**) a spiral axicon of order 1, and (

**b**) a spiral axicon of order 4.

**Figure 14.**Fork interference patterns for beams generated by higher-order axicons with (

**a**) m = 1, (

**b**) m = −1, (

**c**) m = −4.

**Figure 15.**The 2D plots of measured intensities of zero-order Bessel–Gauss beam generated with zero-order axicon recorded at (

**a**) z = 30 cm, (

**b**) z = 37 cm, (

**c**) z = 42 cm.

**Figure 16.**Intensity cross-sections of the zeroth Bessel-Gauss beams for z = 30 cm (red line), z = 37 cm (blue line), and z = 42 cm (green line). With the black line is represented the intensity of an ideal Bessel beam of order zero, |J

_{0}(k

_{r}x)|

^{2}.

**Figure 17.**2D plots of measured intensities of zero-order Bessel–Gauss beam generated with zero-order axicon recorded at (

**a**) z = 30 cm, (

**b**) z = 40 cm, (

**c**) z = 50 cm.

**Figure 18.**Intensity cross-sections of the first-order Bessel–Gauss beams for z = 30 cm (red line), z = 37 cm (blue line), and z = 42 cm (green line). The black line represents the intensity of an ideal Bessel beam of order zero, |J

_{1}(k

_{r}x)|

^{2}.

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**MDPI and ACS Style**

Tudor, R.; Bulzan, G.A.; Kusko, M.; Kusko, C.; Avramescu, V.; Vasilache, D.; Gavrila, R.
Multilevel Spiral Axicon for High-Order Bessel–Gauss Beams Generation. *Nanomaterials* **2023**, *13*, 579.
https://doi.org/10.3390/nano13030579

**AMA Style**

Tudor R, Bulzan GA, Kusko M, Kusko C, Avramescu V, Vasilache D, Gavrila R.
Multilevel Spiral Axicon for High-Order Bessel–Gauss Beams Generation. *Nanomaterials*. 2023; 13(3):579.
https://doi.org/10.3390/nano13030579

**Chicago/Turabian Style**

Tudor, Rebeca, George Andrei Bulzan, Mihai Kusko, Cristian Kusko, Viorel Avramescu, Dan Vasilache, and Raluca Gavrila.
2023. "Multilevel Spiral Axicon for High-Order Bessel–Gauss Beams Generation" *Nanomaterials* 13, no. 3: 579.
https://doi.org/10.3390/nano13030579