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This paper presents an improved excimer laser micromachining method for fabricating arrayed microstructures with a predesigned surface profile. The proposed method is developed from a conventional biaxial laser dragging method, but numerical analysis and optimal pattern design on the contour mask are introduced so that the machined surface profiles can be well controlled and matched to the designed profiles. To demonstrate the capability of this new approach, an array of aspheric microlenses that have analog surfaces for minimizing the focal spot sizes of the lenses is designed and fabricated. An array of 10×10 microlenses with an aperture size of 100 μm and a designed aspheric profile are obtained experimentally. The machined surface profiles are closely matched to their designed ones, with a profile deviation of less than 1 μm. Furthermore, the machined surfaces are smooth, with an average surface roughness of around 2 nm. Optical measurements on these machined aspheric microlenses show minimized focal spot sizes approaching their optical diffraction limits.
©2012 Optical Society of America
1. Introduction
Excimer laser micromachining techniques have been developed for fabricating a variety of microstructures based on mask projection and mask/sample movement methods [1,2]. Commonly used excimer lasers are ArF (193 nm), KrF (248 nm), and XeCl (308 nm) pulsed lasers with a wavelength in the UV or Deep UV region [3]. The mechanism of laser material removal is through laser material interaction, which may involve thermal ablation and/or photo ablation of the materials. Polymer materials are particularly suitable for excimer laser micromachining because their covalence bonding energy is relatively low, and hence the photo-ablation mechanism can dominate the material removal [4,5]. A smooth machined surface can be easily obtained because less thermal effect is involved when the laser source can directly break the covalence bonding between polymer molecules.
For excimer laser machining of three-dimensional (3D) microstructures, several approaches have been adopted and developed. For example, using a gray-scale mask [6] can directly modulate the spatial distribution of projected laser energy intensity, and hence the machining depth variation, to produce 3D microstructures. However, the fabrication of gray-scale masks is complicated and costly. A contour mask, which is a binary photo mask with a specially designed pattern for laser light to pass through, is typically applied along with a programmed scanning path so that the overall or overlapped laser energy projected on the sample surface has a predesigned spatial distribution [7]. If the scanning paths are just straight lines and the contour mask contains repeated or periodic patterns, it is possible to create arrayed 3D microstructures in a very easy and straightforward way known as the excimer laser dragging method [8]. Figure 1 , as an example, shows the simulated results of a rectangular array of convex 3D microstructures obtained by a contour mask and a simple biaxial (x-y) laser line scanning method. This laser dragging approach can also achieve hexagonally arrayed microstructures by laser scanning along three axes that are separated from each other by 120°. Although this laser dragging method is easy and convenient, the surface profiles of machined microstructures are subject to certain limitations. For example, conventional laser dragging methods have difficulty in producing microstructures with axial symmetry, and hence have not been very successful for preparing microlens arrays that have good optical performance.
Fig. 1 Excimer laser machining of arrayed microstructures using a contour mask and x-y biaxial laser dragging method.
Microlens arrays are important optical elements that play a vital role in advanced micro-optical devices and systems used in optical data storage, digital display, and optical communication. Typical methods used for obtaining microlens arrays are photoresist thermal reflow [9], a photo thermal method [10], photo-polymer etching [11], a micro-jet method [12], and a micro-molding or hot embossing method [13]. Although the above-mentioned methods are widely used, a common problem shared by all these methods is that the microlens surface profile is not accurately controllable. Excimer laser micromachining integrated with a planetary contour scanning method [14–16] has been applied for machining single spherical and aspheric microlenses with very accurate surface profile control. This method has also been applied for fabricating a microlens array [17], but the efficiency is low because it is not a batch type of machining, and the filling factor of arrayed microlenses is also limited.
In this work we are trying to answer the following question: is it possible to apply a laser dragging method to create a microlens array with a lens surface profile very close to axially symmetrical so that the optical functionalities are preserved? The chance to achieve such a goal heavily relies on the optimal pattern design of the contour mask used in laser dragging. To demonstrate this, this work chooses an aspheric microlens array as a target that is designed to minimize the focal spot size of the lenses to approach its optical diffraction limit. In Section 2, a systematic way for deriving such an optimal mask contour design is given and modeled. Section 3 provides experimental details for achieving an aspheric microlens array using the proposed method. Optical performances of these arrayed aspheric microlenses are experimentally examined in Section 3. Section 4 gives some conclusions and future perspectives of this new type of modified excimer laser dragging machining method.
2. Analysis and modeling of laser dragging using an optimal contour mask
This work uses one contour mask in a biaxial (x-y) excimer laser dragging method to fabricate a squarely oriented array of microlenses that have a predesigned axially symmetrical surface profile. As shown in Fig. 1, the contour mask consists of a series of repeated opening patterns that defined the laser projected area on the sample surface. The demagnetizing of the imaging system of typical excimer laser workstations can be omitted at this stage. During an excimer laser dragging process, each laser pulse can remove a certain depth of material from the sample surface, and it is very close to a binary or digital machining process. Either the sample or the mask moves during the laser firing sequences, and the ultimate machining depth at a particular point is assumed to be proportional to the number of laser shots received at this point, which depends on the mask contour design and the interval distance between two adjacent laser pulses. After the first laser scanning and machining along one axis, a machined surface with a 2D surface profile is created. A second laser scanning along another axis perpendicular to and intersecting with the first scan will superimpose additional machining depth onto the original one and results in arrayed 3D microstructures.
Figure 2 shows the analysis of how the contour mask design will affect the final machined surface profile in an x-y biaxial laser dragging process. This work uses the same contour mask for both x and y laser dragging, and the pattern of window opening on the contour mask is defined by a function h(x). The machined depth D at point with the polar coordinate (r,θ) is
where f is a constant determined by the laser machining rate of the sample and the interval distance between the adjacent laser shots during the laser scanning. Since this work is interested in obtaining axially symmetrical microstructures, the contour mask pattern function h(x) is assumed to be an even function, that is,It should be mentioned that h(x) may contain a constant term that results in a constant value in D(r, θ) following Eq. (1). However, this constant value in D(r, θ) will not affect the final machined surface profile since it only adds a constant machined depth into the whole sample surface.
In general, the machined depth D(r, θ), and thus the machined surface profile, is θ-dependent and therefore not axially symmetrical. One exception is when the contour mask pattern h(x) is a parabolic function, for example,
where a and c are constants. Submitting Eq. (3) into Eq. (1) gives,which is a parabolic function in r, and one can have arrayed microstructures with axially symmetrical surface profiles.For an axially symmetrical surface profile other than a parabolic one, there is no exact contour mask pattern, and one can only approach the desired surface profile in an approximate way. For example, if a semispherical profile is desired, an intuitive choice on the contour mask pattern function h(x) is a semicircle [18],
where R is the radius of the semisphere. The simulated 3D machined surface profile following Eq. (1) is shown in Fig. 3(a) with all dimensions normalized to R and neglecting any constant term. Figure 3(a) shows several contour lines with a number of given profile depths, and the obtained surface profile is obviously not axially symmetrical. To quantitatively characterize how far away the machined depth deviates from axial symmetry, this work defines a function called average axially symmetrical error, AASE(r), aswhere is the mean value of the machining depth at radius r. Figure 3(b) shows the average axially symmetrical error when the semicircular contour mask pattern defined by Eq. (5) is used for the x-y laser dragging machining of arrayed semispherical microstructures. It shows that the machined surface profile will deviate from axial symmetry pronouncedly, especially when r is increasing. Figure 3(c) displays the desired axially symmetrical surface profile and several simulated machined surface profiles from 0° to 45°. The discrepancy between the desired surface profile and the machined ones obtained from the x-y laser dragging process is also shown in Fig. 3(c).Fig. 3 Simulated results using a semicircle mask to achieve a semispherical surface: (a) contour lines of machining depth; (b) average axial symmetrical error; (c) simulated surface profiles after laser dragging along several different directions in comparison with originally designed profile and the deviation in surface profiles.
If the sag height of the machined surface is reduced, the axial symmetrical problem can usually be improved to some extent. For example, if we reconsider a spherical convex surface with a radius of 1.2 R and a sag height of only 0.5 R, the simulated results for using a circular window opening pattern in the x-y laser dragging method are shown in Fig. 4 , in which all dimensions are normalized with respect to R. The AASE(r) is improved, but the machined surface profile is still significantly different from an axially symmetrical surface.
Fig. 4 Simulated results using a semicircle mask to achieve a low-sag semispherical surface: (a) contour lines of machining depth; (b) average axial symmetrical error; (c) simulated surface profiles after laser dragging along several different directions in comparison with originally designed profile and the deviation in surface profiles.
To improve surface profile accuracy, it is necessary to re-examine the contour mask pattern design. To optimize the mask design, a polynomial function is first assigned to h(x) as
where ai, (i = 1~6) are constants to be determined so that the machined surface profile can be as close to the desired axially symmetrical profile as possible. Choosing a 6th-order polynomial function for h(x) is quite arbitrary. In fact, one can choose any other kind of function as long as it has the flexibility to match the desired machined surface through Eq. (1). To optimize the mask contour design, an object function Err is first decided aswhere Dd(r,θ) is the desired machined depth, while the D(r,θ) is the simulated machined depth calculated from Eq. (1) using the contour mask pattern defined by Eq. (7). It should be mentioned that the choice of the objective function is also quite arbitrary. Certain weightings can be assigned to different evaluating positions (ri,θj) if needed. In this work, no weightings are assigned, and the data points (ri,θj) are evenly sampled in the (r, θ) space. Once the objective function Err is decided, one can start a multivariable functional minimization algorithm on a1 to a6 to minimize the function value and hence determine the contour mask pattern design function h(x). In this work, the simplex method [19] is used in the minimization process.The optimal contour mask design approach is applied to the previous case as discussed above and in Fig. 5 . Using the 6th-order polynomial function, an optimal mask pattern function h(x) with the obtained coefficients of ai, (i = 1~6) is listed in the first line of Table 1 . Using this optimal mask pattern design, the results of the simulated machined surface profiles are displayed in Figs. 5(a) to 5(c). It shows that the surface profile accuracy is significantly improved. For example, the ASSE value drops by almost an order of magnitude. Also noticed in Fig. 5(c) is that, within the central part of the spherical surface with a normalized radius of less than 0.8, the surface profile accuracy is significantly improved. This can be very important for microlens application, since one can easily impose an iris to limit all input light to pass through the central area of the lens.
Fig. 5 Simulated results using an optimal mask design to achieve a low-sag semispherical surface: (a) contour lines of machining depth; (b) average axial symmetrical error; (c) simulated surface profiles after laser dragging along several different directions in comparison with originally designed profile and the deviation in surface profiles.
Table 1. Six Constant-obtained Coefficients for Optimal Contour Mask Design
3. Fabricating aspheric micro-lens array with focal spot size close to optical diffraction limit
In this section, the optimal mask design method introduced in Section 2 is applied for fabricating an array of microlenses. These microlenses have an aspheric surface profile for minimizing the focal spot size to approach the theoretical limit predicted by optical diffraction theory. The analog surface profile is characterized by a sag function as [20]
where S(r) is the sag height of the cross-section profile of the aspheric surface at radius r, Cv is the base curvature, k is a conical constant, and c2i*x2i is the high-order aspheric term. For aspheric surface profiles, the conical constant k is used for correcting 3rd-order spherical aberrations and other low-order aberrations, and c2i is used for higher-order aberrations. In this work, we will use the conical constant k and the 2nd-order c2 and 4th-order c4 teams for designing and optimizing the aspheric lens profile. The optimization process is done automatically by using a built-in module OPTMIZATION in the optical simulation software Zemax-EE (ZEMAX Development Co., San Diego, CA). Experimentally, a thin polycarbonate (PC) plate with a thickness of 0.12 mm will be used for excimer laser machining of the aspheric microlens array. The refraction index of PC material is 1.585 for PC and the Abbe number is 34 in the Zemax simulation. The aperture diameter of microlenses is chosen to be 100 μm and the base curvature Cv in Eq. (6) is 12.5 (1/mm), which will result in a focal length of around 139 μm. After numerical iteration and simulation by Zemax, the optimal sag profile of this aspheric microlens is shown in Fig. 6(a) with the obtained parameters: 0.286362 for k, 2.607568 for c2, and 61.130984 for c4. As shown in Fig. 6 (b), the point-spread function (PSF) given from ZEMAX shows that such a lens profile can minimize the optical focal spot size to its diffractive limit of about 1.23 μm.Fig. 6 Zemax simulation of an aspheric microlens with an analog surface profile for minimized focal spot size: (a) sag profile of the lens and light tracing, and (b) light intensity distribution at focal plane and focal spot size.
Once the desired or designed aspheric profile is determined, its corresponding mask shape hd(x) can be obtained by using the numerical optimization process discussed in Section 2. The coefficients for the 6th-order polynomial for the optimal mask pattern design are given in the second row of Table 1. Figure 7(a) shows that the obtained contour mask pattern of hd(x) by assuming the machining factor f in Eq. (4) is one. Figure 7(b) shows the whole pattern of the contour mask, which will be used in the actual excimer laser machining discussed next. The size of the contour mask is 1×1 cm2, and there is a demagnifying factor of 10X in the projection system of the excimer laser machining system in use. Simulated results on the machined surface profile using this optimal contour mask design and the biaxial x-y laser dragging method are displayed in Fig. 7(c), in which several cross-section profiles taken along different directions are compared with the designed surface profile. Their deviation is also shown in Fig. 7(c). It shows that both surface profile accuracy and the axial symmetry are well achieved within an error of less than 1 μm.
Fig. 7 (a) The optimal mask pattern design, h(x), for excimer laser micromachining of an aspheric microlens array; (b) the whole contour mask pattern design; (c) simulated surface profiles along different directions in comparison with the desired one and the discrepancy between machined surface profiles and desired one.
The excimer laser micromachining system used in this work consists of a COMPex Pro 210 (Lambda Physik, Germany) laser source with a wavelength of 248 nm (KrF). This pulsed laser has a rectangular laser beam shape with a non-uniform energy distribution and a typical beam divergence of about 1–5 mrad. A telescope lens system is used to rectify the problem of beam divergences. An optical homogenizer system can reshape the laser energy distribution into a flat-top one over a cross-section area of 1.5×1.5 cm2 in the laser beam. The laser beam with a uniformly distributed light intensity is used to illuminate the contour mask, and the mask pattern is projected onto a sample surface with 10X demagnification onto a sample using a high-resolution projected lens. The sample is placed in a 4-axis (x-y-x-θ) servo-controlled automatic stage. The sample movement is controlled by a personal computer and can be programmed and synchronized with a laser trigger or laser firing sequence.
The sample material used in this work is a 0.125-mm-thick PC sheet, which has good optical transparency and is known to have a smooth surface after excimer laser machining [21]. The machining rate (μm/pulse) of PC is first determined experimentally. At a laser fluence[Au: Do you mean ‘fluency’ here? Please clarify.] of around 250 mJ/cm2, the machining rate was determined to be 0.12 μm. Once the machining depth rate is decided, the relative laser firing interval is chosen to achieve the desired machining depth and surface profile. Typically, the laser machining rate is about 5 Hz, and the sample is moving at a speed of 25 μm /second. The overall machining time for fabricating the aspheric microlens array is about 10 min. The machining area is 1×1 mm2, which contains an array of 10×10 micro-lenses that have an aperture size of 100 μm. Figure 8 shows the SEM images of the machined microlens array on a PC substrate right after the excimer laser dragging along the x and y axes using the designed contour mask. The machined surface profiles were measured by a color noncontact 3D topography laser scanning system (VK 9700, Keyence Ltd., Osaka, Japan). Figures 9(a) , and 9(b) show the measured results displayed in a contour graph and 2D cross-section profile, respectively.
Fig. 9 Experimentally measured surface profiles of the machined aspheric microlens array using a confocal microscope: (a) contour graph and (b) cross-section profile.
To carefully examine the profile accuracy of the machined aspheric microlens array, Figs. 10(a) to 10(f) show several 2D cross-section profiles of one microlens along several different directions from 0° to 150° with respect to the x axis and in comparison with the original desired or designed axially symmetrical surface profile obtained from the Zemax simulation. To examine the profile differences in detail, the discrepancy between the desired and machined surface profiles as shown in Figs. 10(a) to 10(f) are calculated and displayed in Figs. 11(a) to 11(f), respectively. It is observed that the machined surface profile accuracy is very good. Also shown in Figs. 11(a) to 11(f) is that the profile is more accurate in the central area of the lens aperture. For example, within the range of a radius of 45 μm, the maximum profile error is mostly within ± 1 μm.
Fig. 10 Comparison between machined cross-section 2D profiles with their original design profile in different directions.
For optical components, the surface roughness is of great importance and hence needs to be characterized. The surface roughness of the aspheric machined microlens is characterized with an AFM system (SPA-400, Seiko Instruments Inc., Tokyo, Japan). The surface roughness is measured over 20×20 μm2 areas that are randomly picked. The average roughness (Ra) is obviously less than 5 nm as shown in Fig. 12 , which is quite smooth considering that the structures are directly machined from the laser beam without any modification or surface treatment.
Finally, the optical performance of the machined microlens array for minimizing the focal spot sizes are experimentally measured and examined. The measurement is carried out using a He-Ne laser with a wavelength of 632.8 μm, a 40X objective lens, and a CCD camera. The light intensity distribution of these microlenses at the focal plane is measured. The experimentally measured light intensity distribution and its focal spot size for the fabricated aspheric lens are shown in Fig. 13 . One can clearly see that a very small focal spot size is achieved with a spot size of 1.5 μm, which is slightly larger than the simulated one of 1.23 μm. It should be mentioned that in the focal spot size measurement the whole aperture area of microlens is used and therefore is slightly different from the simulated case shown in Fig. 6. It shows that we have successfully applied the x-y laser-dragging method to fabricate an array of aspheric microlenses that possess the optical characteristics and performance we designed.
Fig. 13 Experimentally measured light intensity distribution at the focal plane of an aspheric microlens for determining the focal spot size.
4. Conclusions
In this work we present a new excimer laser micromachining method for fabricating arrayed microstructures with a predesigned surface profile. It modifies conventional laser line scanning or laser dragging methods by including a numerical analysis process that can yield an optimal contour mask design to closely match the machined surface profiles to their theoretical counterparts. Based on this new approach, we have successfully fabricated an array of 10×10 microlenses that have an aspheric surface profile for minimizing the focal spot size close to the diffraction limit. Details on the numerical analysis, mask pattern design, excimer laser machining processes, machined surface profile measurements and characterization, and optical performance measurements are given. It shows that the new excimer laser dragging method is indeed capable of producing arrayed microstructures with accurate surface profiles in a simple and straightforward way. Larger machining areas can be easily obtained with longer line scanning and precise stitching between adjacent scanning lines. Once a prototype sample is fabricated, mass production or replication of the arrayed microstructures can be easily done using standard LIGA-like processes [22] that may include electroforming, injection molding, and hot embossing. Hence, the proposed method has great potential in many industrial applications such as micro-optics, electro-optics, displays, and lighting. Further applications are under study.
Acknowledgment
This work is supported by the National Science Council (NSC) of Taiwan through projects NSC 098-2811-M-006-012 and NSC 100-2120-M-006 −007 -CC1.
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