Bessel Beam Design Methods

To simultaneously melt the materials on both sides of the interface and establish a high-strength micro-region bond, the laser focal point must be precisely focused on the sample, which imposes stringent demands on the processing accuracy of the welding system. Additionally, due to the large axial intensity gradient of the Gaussian beam after focusing, the focal field temperature is uneven, making it prone to forming micro- and nano-void defects in the laser-affected region, which in turn affects the welding quality of the sample.

Spatial light shaping technology can be used to generate zero-order Bessel beams to optimize the intensity distribution of the laser focal field. This approach reduces the axial intensity gradient and extends the focal length, thereby increasing the depth-to-width ratio of the thermal effect region formed by the laser. As a result, it reduces the focusing accuracy requirements of the laser welding system, improving both welding quality and efficiency

1. The Generation and Parameter Design of Non-Diffracting Bessel Beams

In 1987, Durnin first proposed the zero-order Bessel beam, which shows unique non-diffracting properties: its transverse light field intensity distribution remains unchanged during propagation, and the size of the central spot is always close to the diffraction limit. Additionally, Bessel beams also exhibit a self-healing property during propagation. When the central spot is obstructed, the surrounding light will converge towards the center to “repair” the central spot. The mathematical expression for the transverse light field distribution of a zero-order Bessel beam is:

In the expression:

• J0 ​represents the zero-order Bessel function.
• r and φ are the radial and angular coordinate elements, respectively.
• z is the propagation distance.
• Kr and Kz ​are the transverse and longitudinal wavevector elements, respectively.

The central main spot of a zero-order Bessel beam has a strong confinement capability, allowing for irradiation levels of the order of TW/cm² or higher, which can effectively excite nonlinear absorption in materials. More importantly, the non-diffracting propagation characteristic of zero-order Bessel beams provides a larger depth of focus and a smaller axial intensity gradient, thus creating a nearly uniform temperature field and suppressing the formation of welding defects.

The following figure shows a comparison of the focal length of Bessel beams and Gaussian beams under the same transverse confinement capability. Bessel beams possess a considerable depth of focus while maintaining a transverse micron-level focal spot diameter.

There are several methods of generating zero-order Bessel beams, and the following three main methods are common:

Annular Aperture Method: The annular aperture method, as the name suggests, involves using an annular slit to produce Bessel beams. This was also the first successful method for generating Bessel beams. The diagram below illustrates the annular aperture method for generating Bessel beams. A plane wave is incident perpendicularly onto the annular slit from the left and diffraction occurs.

Afterward, a positive lens performs a Fourier transform, resulting in the formation of a Bessel beam behind the lens. The non-diffracting propagation distance Zmax ​is related to the diameter d of the annular slit and the numerical aperture of the lens.

Although this method can generate zero-order Bessel beams, the energy conversion efficiency is extremely low, making it difficult to apply in laser processing fields.

Spatial Light Modulator method: The generation process of a zero-order Bessel beam is essentially a process of altering the phase distribution of the beam. Therefore, a zero-order Bessel beam can also be generated using a spatial light modulator. A spatial light modulator is a type of optoelectronic modulation device that controls the light field’s intensity and phase distribution through electrical signals. A zero-order Bessel beam can be generated by applying the conical lens phase, as shown in the figure below, to the working panel of the spatial light modulator.

Axicon method: An axicon is one of the most commonly used passive glass-based diffractive elements for generating Bessel beams. When a Gaussian beam is normally incident on and passes through an axicon, its phase distribution is modulated, transforming it into a zero-order Bessel beam without any energy loss, as shown in the figure below.

Due to the low cost, ease of use, and high laser damage threshold of glass axicons, as well as their exceptionally high energy utilization efficiency, axicons are the primary choice for generating ultrashort pulse Bessel beams in the field of laser processing. The figure below shows a schematic of the beam narrowing and transmission of a zero-order Bessel beam. By adjusting the magnification and orientation of the 4f imaging system, the non-diffractive propagation distance, the half-cone angle, and the tilt angle in the propagation direction of the Bessel beam can be easily controlled.

When a zero-order Bessel beam with a half-cone angle of Ɵ1 and a diffraction-free propagation distance of Zmax passes through a 4f system composed of a lens (L1) and an objective lens (L2), the geometric dimensions will be further compressed. The lateral magnification is approximately M=f1/f2=5, and the longitudinal magnification is approximately M2=25. Thus, the final imaging of the zero-order Bessel beam inside the sample can be represented by the geometric parameters:

Geometric parameters of the Bessel beam imaged inside a quartz glass sample under different cone angles and beam compression magnifications.

Focus field intensity distribution of a Bessel beam

• r and z: Radial and axial coordinate components, respectively.
• λ: Central wavelength of the laser.
• w: 1/e² radius of the incident Gaussian beam.
• P0: Peak power of the ultrashort pulse laser.
• β1: Half-cone angle of the Bessel beam after beam compression.
• k: Wave vector.
• J0: Zero-order Bessel function.

Intensity distribution of the zero-order Bessel beam inside quartz glass: On the left are the optical power density distribution along the propagation direction and the cross-sectional view, and on the right are the optical power density distribution along the axis and the cross-sectional view

2. Characteristics of the Femtosecond Pulse Bessel Beam in Fused Silica Glass

Figure (a) shows the micrographs of the interaction between femtosecond pulse Bessel beams and fused silica glass at different pulse energies. The laser pulse width is fixed at 220 fs, and the half-cone angle of the Bessel beam inside the sample is 12.4°. It can be observed that the laser-affected region exhibits a typical one-dimensional linear structure. When the laser pulse energy is less than 9.5 μJ, the refractive index of the material in the focal region increases, appearing as a black region in the micrograph.

When the laser pulse energy exceeds 9.5 μJ, the refractive index of the material in the focal region decreases, appearing as a white region in the micrograph, and the length of the white region increases with increasing pulse energy. By polishing the sample, we observed the morphological characteristics of the white region at a pulse energy of 15.4 μJ under a scanning electron microscope, as shown in Figure (b). It can be concluded that a nanopore with a diameter of approximately 200 nm is formed in the region with a reduced refractive index.

Through ion beam etching and in-situ scanning electron microscope observation systems, we further confirmed the presence of the nanopore (Figure c). Therefore, in order to minimize the generation of laser-induced defects, the single pulse energy should not exceed 9.5 μJ during laser welding.

3. Achieving High-Quality Micro-Welding Between Fused Silica Glasses using Bessel Ultrashort Pulse Laser.

Figure (a) shows a top-view micrograph of the sample’s welding surface. It can be seen that the laser weld line is uniform and smooth. Although there are still a few randomly distributed micropore defects in the welded area, overall, it is significantly better than the Gaussian laser weld line. Measurements show that the weld line width is approximately 18 μm, and the spacing between weld lines is 40 μm. Figure (b) shows a side-view micrograph of the sample’s weld line.

It can be seen that the gap between the samples completely disappears after laser processing, and the material near the interface has fused into a single entity after undergoing the thermal melting-cooling process. Measurements reveal that the depth of the laser-induced thermal melting region reaches up to 227 μm. This indicates that during laser welding with these parameters, the focal position’s axial depth can reach up to 227 μm, which is four times that of Gaussian laser welding under the same conditions.

4. Where to Buy Bessel Lenses?

Wavelength Opto-Electronic offers high-quality Bessel lenses that are used in laser processing applications. The tunability of the depth of focus of the output beam by adjusting the size of the input beam diameter is the most attractive feature of this Bessel beam optical system.