In femtosecond laser machining, spatial beam shaping can be achieved with wavefront modulators. The wavefront modulator displays a pre-calculated phase mask that modulates the laser wavefront to generate a target intensity distribution in the processing plane. Due to the non-perfect optical response of wavefront modulators, the experimental distribution may significantly differ from the target, especially for continuous shapes. We propose an alternative phase mask calculation method that can be adapted to the phase modulator optical performance. From an adjustable number of Zernike polynomials according to this performance, a least square fitting algorithm numerically determines their coefficients to obtain the desired wavefront modulation. We illustrate the technique with an optically addressed liquid-crystal light valve to produce continuous intensity distributions matching a desired ablation profile, without the need of a wavefront sensor. The projection of the experimental laser distribution shows a 5% RMS error compared to the calculated one. Ablation of steel is achieved following user-defined micro-dimples and micro-grooves targets on mold surfaces. The profiles of the microgrooves and the injected polycarbonate closely match the target (RMS below 4%).
© 2016 Optical Society of America
Femtosecond laser utilisation for surface micromachining has been widely studied in past years [1–3]. The main advantage of ultrashort light pulses lies in their particular interaction with materials where very restricted thermal and collateral effects are observed compared to longer irradiations (nanosecond and beyond). Very accurate ablation profiles with a precise control of the removed layer thickness (down to a few tens of nm) can thus be achieved. Spatial beam shaping aims at controlling the intensity distribution in the processing plane by means of wavefront modulation . Dynamic wavefront modulators can be divided in two main families, the Spatial Light Modulators (SLM) based on an addressable liquid crystal layer and Deformable Mirrors (DM) based on a reflective surface.
Due to their high resolution, SLMs permit to deal with high spatial frequency phase masks involving sharp phase jumps. SLMs are highly suitable for multispot operation processes [5,6]. This has a major interest in reducing the processing time and cost of femtosecond machining . However, due to their imperfect optical response, these devices induce deformations of the experimental intensity distribution when compared to the desired one .
To compensate for these shaping discrepancies, research groups have proposed optical feedback loops based on a measurement of the experimental intensity distribution enabling homogenization of multispot distributions [9,10]. In the case of continuous intensity targets, solutions based on averaging the shaping discrepancies by beam scanning on the sample , phase mask change during irradiations  or spatial filtering in a Fourier plane  have been reported, though adding a constraint to the laser process. Nevertheless, the calculated phase mask (using Iterative Fourier Transform Algorithm, lens and grating , Multi Fresnel lenses ) cannot be perfectly reproduced by the SLMs. There is a need of a phase mask calculation strategy that takes into account the limitations of the SLMs optical response. These limitations, regardless of their physical origin, can be described with the help of Zernike polynomials  as they constitute an orthogonal basis of surface functions on the unit circle. Thus the limitations of any modulator’s optical response can be defined evaluating the ability of the modulator to correctly reproduce a set of Zernike polynomials [14, 15]. This characterization can also be conducted on any other orthogonal basis, e.g the spatial modes of a DM .
The DM can also be used to perform laser spatial beam shaping . Currently, the maximum resolution is attained by MEMS DM having a few thousands of actuators of 300μm pitch. If DM are able to generate continuous phase modulations, their restricted resolution limits their optical response and their capacity to reproduce a calculated phase mask. Since both DMs and SLMs cannot perfectly reproduce phase masks for beam shaping, there is a need to consider the wavefront modulator limitations in the phase mask computation.
We propose a calculation strategy of the modulation phase mask for a desired target intensity that takes into account the limited optical response of the phase modulator in terms of Zernike polynomials. We apply it to generate arbitrary continuous intensity shapes. It is an alternative to other phase mask calculation algorithms (such as Iterative Fourier Transform Algorithm, lens and grating, Multi Fresnel lenses...) that disregard the phase modulator ability to reproduce the phase mask. The advantage of our method is that the number of Zernike polynomials coefficients involved in the calculation can be adjusted to the performance of the wavefront modulator that is employed. Indeed, the optical performance of a wavefront modulator is often characterized by the number of well-reproduced Zernike polynomials [15, 16]. Taking this optical response into account is not easily achievable with the usual phase mask calculation algorithms. The proposed calculation determines the coefficients of a chosen number of Zernike polynomials, each of them having an effect on the laser intensity distribution after propagation . To illustrate the technique, an optically addressed liquid-crystal light valve is used as a programmable wavefront tailoring device . The phase mask is calculated using a least-square fitting algorithm on an adjustable number of Zernike polynomials coefficients , optimized to generate a user-defined intensity distribution in the far field. The latter is related to a desired groove profile on steel by the experimental ablation rate . We demonstrate the efficiency of the technique with numerical calculations of phase masks for target intensities and experimental intensity shaping along with characterizations of the laser-processed surfaces. We point out that no wavefront measurements was necessary as the phase modulation is numerically calculated and then displayed on the SLM for experimental evaluation of intensity distribution shaping. The comparison between experimental and numerically calculated laser distributions reveals a low RMS error (< 5%) on cross section. The technique is demonstrated in the case of surface processing, where laser-machined grooves and dimples on steel molds are evaluated using classic and interferometric microscopy. The groove depth closely matches the expected profile (RMS < 4%). As a further illustration, the surface of the corresponding injected polycarbonate is presented, showing the desired profile as well.
2. Experimental details
As depicted on Fig. 1, a femtosecond amplified laser system (Thales Bright) operating at a repetition rate of 5 kHz delivers 160 fs light pulses at 800 nm with a maximum average power of 2 W, a linear polarization and a spectral bandwitdh of 8 nm. The laser beam goes through an optically addressed SLM (OA-SLM). The phase modulation is achieved with a blue light beam impinging a photo-conductive layer stuck to the liquid crystal layer (parallel nematic). The nonpixelated OA-SLM has a spatial resolution close to 100 μm on a 1 cm diameter pupil. Its measured dynamic phase range, linked to the thickness of the liquid-crystal layer, is 6π or 3λ at 800 nm. The device can handle 1 W of the femtosecond laser power. We have experimentally verified that the OA-SLM does not affect the laser pulse duration with autocorrelation measurements. Further details can be found in . By displaying a grey level mask with a video projector imaged on the OA-SLM, the optical retardation due to the liquid crystal molecules orientation is modulated in two dimensions. The femtosecond beam wavefront thus is modulated and therefore spatially shapes the beam intensity distribution in the far field. The beam is enlarged before the OA-SLM and truncated by an iris of 1 cm diameter just before passing through the OA-SLM in order to approach an uniform illumination. Considering the 100 μm spatial resolution over the 1 cm active area and the phase stroke, we can securily say that this modulator permits to adress a very large number of Zernike polynomials, up to at least 30. After the OA-SLM, the beam is reduced in size passing through a telescope constituted of two converging lenses of 75 cm and 100 cm focal lengths. The telescope is positioned to optically conjugate the SLM plane with the object focal plane of the F-theta lens of a scanner (ScanLab). This lens focuses (focal length = 88 mm) the laser beam on a steel surface and allows for rapid beam scanning on the surface for groove micro processing. The beam profile is measured with a CCD sensor (Thorlabs DCU224M) at the focal plane of a long focal length lens and as well in the focal plane of the F-theta lens using a home made beam analyser. The steel (X105CrMo17) is a specific metal for injection molds.
First, the steel ablation rate (i.e the depth of ablated material per pulse with respect to the laser fluence ) was experimentally determined for fluences below 1 J/cm2. For fluences below 1 J/cm2, in the so-called gentle ablation region on steel (see ), the empirical relationship between the ablated depth per pulse and the laser fluence can be approximated with a logarithmic function . For the X105CrMo17 steel, the ablation rate relating the laser peak fluence F in J/cm2 to the depth D of the ablated layer per pulse in nm could be experimentally approximated by D = 40 × logF + 45. This permits to calculate the target fluence distribution from a desired ablation profile, see Fig. 2 (I) and 2 (II). Then, a numerical optimization algorithm (detailed hereafter) determines the Zernike polynomials coefficients permitting to obtain the intensity target, see Fig. 2 (III). The corresponding phase mask, see Fig. 2 (IV), is displayed on the OA-SLM. By going through the OA-SLM, the femtosecond beam is spatially modulated and reaches the surface to be processed with the desired intensity profile.
The optimization algorithm consists of a least square fitting method coupled to a Fresnel propagation code. The least square fitting algorithm iteratively optimizes a vector of Zernike polynomials coefficients Cj (radians) of the wavefront function in radians W(r, θ) = ∑jCjZj(r, θ) where r and θ are polar coordinates on the unity disk and Zj(r, θ) a Zernike polynomial following the Noll’s sequential indices convention in . At each iteration, the intensity is numerically calculated for the current Cj and the 2D root mean square (RMS) error εRMS between the calculated intensity I(i, j) and the target intensity It(i, j) is evaluated following:
The least square fitting algorithm iteratively minimizes εRMS by optimizing the vector of Cj. The starting vector is constituted of zeros for each Zernike coefficient. Each run of the least square fitting algorithm produces a new vector that is the starting vector for the subsequent run. We have performed 30 to 50 iterations for the results presented hereafter, lasting altogether a few minutes on a standard PC. We mention here that we have also used a simulated annealing approach with very similar phase mask results requiring however a few hundreds of iterations (thus lasting a few tens of minutes on the same PC). The stopping point of the algorithm was defined by the user, when the εRMS started to show little improvement (less than a few %).
The least square fitting method (see Fig. 2) determines the Cj by iteratively reducing εRMS to numerically determine the Cj yielding the desired beam shaping. The optimization is implemented from the third to the twentieth polynomials as the first (piston) and second (tilt) do not modify the shape of the diffracted beam. The piston only creates a constant phase shift equal for every part of the incident beam, and the tilt spatially shifts the diffracted beam.
We underline here that the least square fitting algorithm is not employed with a wavefront sensor for an experimental verification of the applied phase mask on the laser wavefront as for adaptive beam shaping. Instead, the least square fitting method is used for an optimization of the Cj, i.e the calculation of the phase mask. The corresponding calculated phase mask is applied to the optical valve that modulates the femtosecond laser wavefront. The experimental verification of the beam shaping success is conducted by recording the laser intensity distribution in the focal plane of a lens on a CCD sensor. Micro grooves were machined by scanning the shaped beam on the steel sample at a speed of 25mm/s with 500 passages.
3. Results and discussion
3.1. Phase mask calculations
The numerical results of optimization of Cj to generate a phase mask producing a top hat intensity distribution in the far field are shown in Fig. 3. The target includes the point spread function associated with the focusing lens by a convolution. The influence of number of optimized Cj on the beam shaping accuracy is numerically studied. As the top hat distribution has a circular symmetry, we have used only polynomials with a spherical symmetry, i.e defocus and spherical aberrations (C4, C11 and so on). When calculating the phase mask generating a top hat distribution with a minimum number of Zernike coefficients (see Fig. 3 first and second rows), we were not able to lower the 2D εRMS between the top hat target and the calculated intensity below 6%. When using just one Zernike coefficient (defocus - Noll’s index C4), it is clear that a top hat distribution is out of reach as adding an aberration of defocus simply enlarges the spot size in the geometrical focal plane of a lens, see Figs. 3(b) and 3(c). In that case, the algorithm finds a coefficient that enlarges the spot size to its best fit with the top hat distribution yielding a εRMS of 11%. When the optimization is conducted on the 3 first spherical Zernike polynomials coefficients, the εRMS drops to 6.4%, see Fig. 3(e) and 3(f). When 5 coefficients are involved, the performance increases with a reduced root mean square errors to 3.1%, see Fig. 3(h) and 3(i). The corresponding optimized phase masks are shown in Fig. 3(a), 3(d), and 3(g). Clearly, this numerical study shows that by using an increased number of optimized Cj, the beam shaping quality is superior in the case of the top hat target.
Additional results are depicted on Fig. 4 for the top hat 4(a) and the three-steps 4(b) intensity profiles. The number of optimized Cj was 5 and 10 respectively. The number of optimized Cj was determined after several trials of the optimization with various number of Cj just as for Fig. 3. Here also, the targets include the point spread function influence. The εRMS between the target and the intensity distribution obtained after phase mask optimization is below 5% in both cases. We point out here that the εRMS is calculated slightly differently for the three steps shape. This target comes from a desired ablation profile obtained by beam scanning on a surface. Thus, the calculated intensity levels are accumulated along the direction of beam scanning yielding a vector of accumulated intensity levels. The εRMS is calculated following Eq. (1) in only one dimension. The benefit of the method is that the number of optimized Cj can be tuned according to the SLM or DM’s ability to achieve high orders of Zernike polynomials. Naturally, for a SLM or a DM with a limited number of correctly reproduced Zernike polynomials, some intensity targets may remain out of reach. The Fig. 4(c) addresses this point for the three-steps target intensity and the top hat target. Fig. 4 (c) is a numerical calculation of the dependence of best attainable εRMS with the number of Cj. εRMS decreases with the number of optimized Cj for both targets. This result strengthens the intuitive expectation that the more Cj are optimized, the more precise the beam shaping is. Noteworthy, a higher number of optimized Cj is required to obtain a satisfactory three-steps profile (εRMS below 5% for 10 Cj and more), than for the top-hat profile (εRMS below 5% for 4 Cj and more). As a rule of thumb, one may infer that a more complex target (e.g with multiple fluence levels, as for the three-steps profile) requires a higher number of optimized Cj than a ’simple’ target. Though, a detailed study of this possible trend is outside the scope of this paper. In practice, the number of optimized Cj is a trade-off between a sufficiently low εRMS numerically and the number of Cj correctly reproduced by the SLM.
Among the limitations of the proposed technique, one should mention the reduction of the calculation performance when reducing the number of Zernike coefficients. Also, the computation cost may greatly increase for a large number of Cj and high resolution modulator requiring high sampling rates for the beam propagation calculations. Also, we mention here that this method requires that the phase modulator to be well described by its ability to reproduce Zernike polynomials [3,14] or other functions forming a basis (e.g spatial modes of a DM ) on which the same optimization can be conducted.
3.2. Experimental intensity profiles and surface processing
Experimental intensity distributions are compared to calculated distributions in Fig. 5. The Cj represented on Fig. 2 (III) and the phase mask represented on Fig. 2 (IV) are those that permitted to obtain the experimental shaped beam on Fig. 5 (b). The intensity distributions are close to the theoretical ones, although residual aberrations of the optical system worsen the comparison. The spatially shaped beam in a) was obtained by applying 3.7 of vertical trefoil (C9), −1.17 of 1st order spherical aberration (C11) and −0.61 of vertical astigmatism (C6). The spatially shaped beam in c) was obtained by applying 2.3 of astigmatism in the x direction ((C5), −3.7 of astigmatism in the y direction (C6), 0.3 of coma in the y direction (C7), −0.8 of coma in the x direction (C8) and 2.8 of 2nd order astigmatism in the x direction (C13). The RMS error between experimental (a) and calculated (b) right spatial is calculated as follows. We recall that the scope of this beam shaping is to generate a groove with a controlled section (three-steps shape). For that, the beam is translated during laser machining along the horizontal direction of Fig. 5. Therefore, for both calculated and experimental beam profiles, a sum of the intensity levels along the scanning direction is achieved, yielding an accumulated intensity profile, similarly to our previous work . The RMS error is then calculated between the two profiles and is below 3%. This is a sufficient precision for the surface machining reported hereafter. We underline here that the experimental verification does not require any wavefront sensor, nor adaptive optics loop, which eases the applicability of this method. Naturally, as for any wavefront modulation system, the experimental beam shaping quality can be improved by using an adaptive loop for corrections of the aberrations of the optical system (e.g as we have previously employed [19, 21]). This correction can also be done with the help of Zernike polynomials [14, 22]. However, this not the scope of this paper that presents a phase mask calculation based on Zernike polynomials as an alternative to more common techniques (Iterative Fourier Transform Algorithm, Lens an gratings...) and demonstrate its applicability to surface machining. Also, this would add complexity to the experimental set up by adding a wavefront sensor and running the adaptive loop. Lastly, the obtained beam shaping quality is more than sufficient for the surface machining application that is discussed hereafter.
To illustrate the potential of this technique in surface machining, some surface processing results (grooves and dimples) achieved using this beam shaping technique are presented in Fig. 6. With rather low peak fluences (below 1 J/cm2), the interaction takes place in the gentle ablation regime where the ablation shows very low collateral effects . It is well-known that a variety of detrimental effects take place during the light matter interaction e.g nonlinear propagation in the air before reaching the target , heat accumulation or particle shielding effect in the case of multiple pulses irradiation  with consequences on the energy coupling efficiency and on the post-irradiation surface topology. However, using the empirical fluence/ablation depth relationship presented above is sufficient to assess for the efficiency of the laser beam shaping technique presented here. Moreover, the fairly low repetition rate (5 kHz), peak fluence (1 J/cm2) and pulse energy puts the interaction on steel in a regime where any nonlinear propagation, heat accumulation or particle shielding effects can be neglected [23, 24].
Figure 6 shows an example of micro-groove (a) and dimples (b) machined on steel observed by optical microscopy. The inset shows the experimental beam profile. The horizontal profile of the micro-groove (a) compares well to the ablation target in Fig. 6(c). The goal here is to obtain a three-steps shaped groove. The micro-groove is obtained by translating the beam in the insert in Fig. 6 (a) along the white arrow at a speed of 25mm/s. The desired three-steps section appears clearly. The RMS error between these the target profile (red triangles) and the actual micro-groove profile (black line) is below 4%. We point out here that the beam was translated with respect to the sample surface with the scanner head while keeping small scanning angles to avoid beam distortions at high scanning angle. In practive the laser-induced micro grooves appeared uniform all along their length (2 cm). The spatially shaped beam was also implemented to generate dimples shown in Fig. 6 (b). The resulting three-steps cross sectional profile can not be easily obtained without spatial shaping considering the different intensity levels inside the distribution. These ablation profiles have a strong interest in machining injection molds to confer specific wettability properties on polymer parts. A detailed study of these surface effects will be the subject of a future report. We point out here that this technique is not limited to ultra-fast lasers as it relies on a modulation of the spatial wavefront. Indeed, lasers having a thinner spectral bandwidth are more adapted to spatial beam shaping . However, the possibility offered by ultrafast lasers to finely control the ablated layer with the number of pulses and the laser fluence level makes easier the illustration of the possibility to control the ablation profile according to a user-defined fluence distribution, which is a key element of this report.
We report on an efficient method to implement Zernike polynomials for ultrafast laser machining of surfaces, and experimentally demonstrate its efficiency with an optically addressed spatial light modulator. An wavefront modulation function resulting from the sum of different balanced Zernike polynomials is used to modulate the spatial phase of an incident femtosecond beam. The weight of each polynomial is determined by a least square algorithm to fit a user defined target intensity, matching an arbitrary ablation profile. This method allows for unique machining profiles very difficult to reproduce without spatial tailoring because of the different intensity levels inside the distribution. Thanks to the possibility to adjust the number of optimized Zernike coefficients, it can be implemented taking into account the optical performance of the phase modulator. Microtexturations with arbitrary profiles are achieved using this technique on steel injection molds for the fabrication of polymer objects.
We gratefully thank Kassem Saab and Razvan Stoian for fruitful discussions. The financial support of ANR Topoinjection (ANR-13-RMNP-0010) and Equipex Manutech-USD (ANR-10-EQPX-36-01) are gratefully acknowledged.
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