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Abstract
Bessel beams are advantageous in high aspect-ratio microhole drilling because of their immunity to diffraction. However, conventional methods of generating Bessel beams result in poor adjustability of the nondiffraction length. In this study, we theoretically describe and experimentally demonstrate the generation of Bessel-like beams (BLBs) with an adjustable nondiffraction length by using a phase-only spatial light modulator. In this method, nondiffraction lengths varying from 10 to 35 mm can be achieved by changing the designed phase profile (curvature). High-quality, high aspect ratio (560:1) and length-adjustable microholes can be drilled by spatially shaping a femtosecond laser beam.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, extensive research has been carried out regarding microhole drilling due to its broad applications in diesel engines [1], aerospace [2], optical fiber sensors [3], biomedical devices [4], and many other areas [5]. Currently, numerous methods exist for fabricating microholes, such as mechanical drilling [6], electrochemical discharge machining [7], electrochemical micro drilling [8], focused electron beam machining [9], focused ion beam machining [10], and laser drilling [11]. Compared with conventional fabrication methods, femtosecond laser beams offer unique advantages, such as high-precision and high-quality material processing [12]. Thus, femtosecond laser beams allow the drilling of deep microholes with minimized recast/microcracks and heat-affected zones [13]. However, drilling microholes by using a traditional femtosecond Gaussian beam requires multiple pulses. The hole length is limited, and bending is observed during microhole drilling [14].
Bessel beams are typical nondiffraction beams and have inherent advantages, such as a long focal length [15], for the drilling of microholes with a high aspect ratio. Furthermore, Bessel beams have a higher processing efficiency than traditional femtosecond Gaussian beams [16]. However, the parameters and configurations of Bessel beams are fixed, which is not suitable in areas such as laser trapping [17], imaging [18], and phase matching. Therefore, Bessel-like beams (BLBs) were developed. By using BLBs, microholes with a tunable length and microring structures [19,20] can be obtained. BLBs are generated by using special structures such as fiber microaxicons fabricated using a modified chemical etching method for laser focusing [21,22], polymeric round-tip microstructures fabricated with four-beam interference lithography [23], and pyramid-shaped microtips fabricated with a combination of optical lithography and chemical etching [24]. However, due to the complex manufacturing process and uncontrollable morphology of these structures, adjusting the length of the nondiffraction region is difficult. BLBs can also be generated using a combination of optical components or a nonlinear medium. Belyi et al. proposed an optical design that comprised two axicons and a spherical lens to generate a BLB with a z–dependent cone angle [25]. Summers et al. used a shallow angle axicon and a spherical lens to generate BLBs with a tunable spot size and an effective focal length [26]. Cheng et al. used cross-phase modulation to generate BLBs with different sizes and intensities in a hot atomic rubidium vapor cell by tuning the sample temperature and pump power [27]. These methods require multiple optical lenses and cannot be used to obtain an adjustable nondiffraction scheduled length. Čižmár T et al. and Ouadghiri-Idrissi I et al. respectively use different design methods to achieve the adjustment of the intensity distribution of the Bessel beam propagation direction under fixed length [28,29], Igor A. Litvin et al. achieved the generation of long-distance Bessel beams with axial shape invariance [30]. However, the above light field is not suitable for length-adjustable high aspect ratio microhole drilling.
In this paper, we propose a novel method for generating a BLB by a phase-only spatial light modulator (SLM). By theoretically designing phase distributions with different curvatures (R), the nondiffraction length can achieve target length. To understand the variations in the nondiffraction length, we performed numerical simulations of the on-axis intensity distribution with different curvatures. Furthermore, we drilled samples of polymethyl methacrylate (PMMA) to prove the shaping effect. The length of the microholes varied from 248 μm to 904 μm. The proposed approach is a simple and flexible method for the fabrication of high-quality, high aspect ratio and length-tunable microholes.
2. Experimental generation of BLBs
To generate BLBs with adjustable nondiffraction lengths, we shaped the phase of the field in a well-controlled manner. For this purpose, we used a computer-controlled diffractive optical element and a phase-only SLM (Holoeye PLUTO-NIR-2) to generate shaping beam. The size of the liquid crystal screen is 15.36 mm × 8.64 mm. The detailed experimental setup and hologram design are described in the following text.
2.1 Experimental setup
Figure 1(a) illustrates the setup used for the light field measurement. The light source was a femtosecond laser system (Spectra-Physics) consisting of a mode-locked Ti: Sapphire oscillator and a regenerative amplifier, which produced 50-fs full width at half maximum (FWHM) linearly polarized laser pulses at a repetition rate of 1 kHz and a central wavelength of 800 nm. The shutter is used to control the on and off of the laser. The energy of the laser pulses was tuned with a neutral density (ND) filter. The function of the beam splitter is to split the beam into two parts, one of which is perpendicular to the SLM for beam shaping and the other part is blocked by a black damper. The phase-only SLM was used to shape the Gaussian beam into the beam we designed. The output light beam passed through a 4f system consisting of two identical plano-convex lenses (f = 150 mm). The 4f system, which translates the initial light field as it is behind the second plano-convex lens, were used to conveniently measure different locations of the shaping light field. A camera-based beam profiling system (Ophir Spiricon SP620U) was used to record the shaping field, which can realize the real-time viewing and measuring of a laser’s structure. By moving the system, we can record the shaping field at different distances from the SLM. For the microdrilling, we replace the green dotted frame in Fig. 1(a) with the optical components in Fig. 1(b). In additional, we limited the beam diameter to 7 mm and the measured energy is 60 μJ before it is incident on the SLM. The shaping beam was statically focused onto a polished PMMA bulk sample through a telescope system, which included a plano-convex lens (f = 150 mm) and a 20 × microobjective [NA = 0.45 and equivalent focal length = 9 mm], as displayed in Fig. 1(b).
Fig. 1 (a) Measurement setup. S: shutter, ND: neutral density filter, BS: beam splitter, and BD: black damper. (b) Fabrication setup. MO: micro-objective.
2.2 The hologram design and implementation
An axicon can be used to generate Bessel beams and is triangular in sectional shape. The optical delay increases with an increase in the thickness of the material. Therefore, the corresponding phase profile (black part in Fig. 2) is thin at the center and thick at the edges with the linear change. If the phase distribution no longer changes in a linear form but in a curved form, the BLBs can be generated. In this paper, we have studied the case where two phase distributions change in a curved form. Comparing with the non-diffraction length formed by the Bessel beam, we define the curvature radius R μm that lengthens the length of the shape as “+”, and the shorter one we define it as “-”, which is convenient for the distinction and use below. As with adding a convex (concave) lens to the two sides of an axicon, the original parallel light converges (diverges), which affects the length of the intersection of the two sides beams on the axis (red solid line in Fig. 2). Because the designed optical fields were azimuthally independent, we selected half of the longitudinal section of the phase distribution. We set the same phase difference between the different beams at r = 0 and r = 3.5 mm. The phase between the two points changes in the form of a curve. As shown in Figs. 3(a) and 3(b), we use MATLAB to design different curves with the radius of curvature of + 9 × 105 μm, + 1.6 × 106 μm, + 5 × 106 μm, −1 × 106 μm, −5 × 105 μm and −3 × 105 μm, respectively. The insets are magnified view of black dotted frame. The above six curvature radius is used for the light field measurement and microhole drilling. Compared with the phase profile of a Bessel beam with a uniform concentric ring, the ring spacing gradually increases from the inside to the outside for the longer BLBs. In the shorter BLBs, the ring spacing gradually decreases from the inside to the outside as shown in the last row of Fig. 2.
Fig. 3 Phase profiles for Bessel beams (2° cone angle) and BLBs, R: curvature radius. The insets are magnified view of black dotted frame.
We loaded a hologram on the SLM and measured the light field distribution at different locations by moving the laser beam profiler. Figure 4 illustrates the hologram and measured intensity profiles of the Bessel beam and BLBs at 5, 10, 15, 20, 25, and 30 cm from the SLM. The results indicated that the diameter of the main lobe for the Bessel beam was a little smaller than that for the longer BLB and larger than that for the shorter BLB. Furthermore, the lengths of the nondiffraction propagation regions for the two types of beams were different. The on-axis intensity for the Bessel beam was almost zero when the propagation distance was 25 cm. Under the same energy, the nondiffraction length of the longer BLB (R = + 9 × 105) was more than 30 cm, whereas the on-axis intensity of the shorter BLB (R = −3 × 105) was relatively weak at 10 cm.
Fig. 4 Phase and cross-section intensity profiles at 5, 10, 15, 20, 25, and 30 cm from the SLM for the three types of beams. (a) Bessel beam (2° cone angle), (b) BLBs: R = + 9 × 105, and R = −3 × 105. The scale bar represents 100 μm.
3. Theory
Any optical field U(x,y,z) can be decomposed into its plane-wave components through Fourier transformation. Each plane-wave component is defined by a wave-vector k = [kx,ky,kz] in the orthogonal system of coordinates [x,y,z]. We can express this decomposition as follows:
This plane-wave representation is known as the spatial spectrum. If we set a zero value at a selected position on the axis, the Fourier transform for azimuthally independent optical fields can be written in the form of a zero-order Hankel transform as follows:
where and J0 is the zero-order Bessel function. Considering the inverse transformation and positive transformation symmetry of the Hankel transform, the inverse transform of Eq. (2) is given as follows:As indicated in Eq. (3), any azimuthally independent optical field can be decomposed into ideal zero-order Bessel beam components. Each of these ideal Bessel beam components then propagates in free space as follows: , where is the axial wave-vector component (). By using the inverse transform of Eq. (3), the on-axis field is as follows:
The detailed simulation procedure is as follows. The expression for the Bessel beam intensity [31] after the Gaussian beam is incident on the axicon is given as follows:
where , , is the refractive index of the axicon, is the bottom angle of the axicon, is the radius of incident laser beam waist, and is the electric field intensity. Here, is designed as a variable of the radius. The specific design of the phase curve expression is given in Eq. (6).where , R is the curvature of the phase profile, and is a constant. Therefore, we can adjust R to obtain different , following which we can obtain different initial transverse optical fields . Substituting the fields into Eqs. (2) and (4), we obtain the final longitudinal optical field .4. Analysis and application
We theoretically simulated the on-axis intensity profiles for BLBs with different curvatures, as illustrated in the continuous curves of Fig. 5. Moreover, we recorded the on-axis peak intensities at different positions. The measurement interval was 1 cm. The results are represented by the red points in Fig. 5. The comparison indicates the consistency of the experimental and theoretical results. For the longer BLBs, the comparison between the theoretical and experimental axis intensity distribution is displayed in Figs. 5(a), 5(b), and 5(c), where the curvatures are + 9 × 105, + 1.5 × 106, and + 5 × 106, respectively. When the curvature of the longer BLB was relatively small [as shown in Fig. 5 (a)], the nondiffraction length was approximately 35 cm. This was longer than the nondiffraction length formed by the corresponding axicon (approximately 22 cm), which allowed for a large quasi nondiffraction propagation region along the propagation length of the longer BLB. Simultaneously, the measurement results exhibited a certain shift to the right. There may be two reasons for this shift. The first cause is that deviations may occur in the selection of the initial position when measuring the light field. The second cause is that we did not consider the effect of light diffraction on the simulation results. Moreover, a certain oscillation occurs in the rear section of the nondiffraction region. We speculate that the oscillation occurs because of the intersection of the beam at different radii [32]. Further research could eliminate the impact of the oscillation. For the shorter BLBs, the comparison between the theoretical and experimental axis intensity distribution is displayed in Figs. 5(d), 5(e), and 5(f), where the curvatures are −1 × 106, −5 × 105, and −3 × 105, respectively. When the curvature was small for the shorter BLB, the nondiffraction length was approximately 10 cm [as shown in Fig. 5 (f)], which was shorter than that formed by the corresponding axicon. Therefore, a rapid cutoff for the nondiffraction region was observed. The measurement result was observed to possess a certain shift that can be explained in terms of the aforementioned causes.
Fig. 5 Comparison between the simulation and experimental results for BLBs with different propagation distances.
To achieve a controllable nondiffraction length, we measured the non-diffraction length under several curvatures and compared them with the theoretical calculations as shown in Fig. 6. The curvature on the left graph is “+” and the one on the right is “-”. The theoretical calculation formula was . The experimental results agreed with the theoretical calculation. The phase distribution can be designed with different curvatures to obtain a continuously adjustable length of the nondiffraction region.
Fig. 6 The theoretical simulation and experimental results of the non-diffraction length versus curvature for the case where the curvature is “+” (left) and the curvature is “-” (right).
Considering the influence of the diameter of the incident light on the length of the nondiffractive region, we limited the diameter of the beam to 7 mm and drilled microholes in PMMA with a femtosecond laser. Femtosecond laser has an extremely high transient power, and the energy will undergo significant nonlinear absorption when interacting with the material. Therefore, for transparent dielectric like PMMA, multi-photon ionization is required to generate enough seed electrons in the early stages of laser-material interaction and then avalanche ionization generates a large amount of free electrons. When the critical electron density of the material is reached, the materials ablate. The results are displayed in Fig. 7. The lengths of the microholes varied from 248 μm [R = −3 × 105] to 904 μm [R = + 9 × 105, continuous part] with a small change in the radius [~1.6μm]. The experimental results verified that the microholes had adjustable lengths.
Fig. 7 Drilling results for the three types of beams [the longer BLBs: R = + 9 × 105, + 1.5 × 106, and + 5 × 106; the shorter BLBs: R = −1 × 106, −5 × 105, and −3 × 105; and Bessel beam. The length scale bar is 100 μm, the diameter scale bar is 2 μm.].
5. Conclusions
In this paper, we have demonstrated a novel method for fabricating high-quality and length-tunable microholes in PMMA by using femtosecond laser beam shaping. Numerical simulations demonstrated that the envelope of the on-axis intensity distribution matched perfectly with the light field profile. Two types of BLBs can be generated. The longer BLBs feature a large quasi nondiffraction propagation region along its propagation length, and the shorter BLBs have a rapid cutoff for the nondiffraction region. The depth of the microholes can be varied in the range of several hundred micrometers by adjusting the phase curvature and leaving the radius almost unchanged. Finally, the fabrication results have considerable potential for high aspect ratio microhole drilling.
Funding
National Key R&D Program of China (2018YFB1107200); National Natural Science Foundation of China (NSFC) (51675049); the Natural Science Foundation of Beijing Municipality (3172027); the Young Elite Scientists Sponsorship Program (2016QNRC001), and the Science; Technology Co-ordinating Innovation Project of Shaanxi Province of China (No.2016KTZDGY-02-02).
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