## Abstract

A computer simulation for the fabrication of aspheric hexagonal micro lens arrays by use of the laser dragging process was investigated. An excimer laser dragging process is capable of manufacturing a microstructure array with a curved surface. Generally, this process was used to produce aspheric micro lens arrays with an arrangement of rectangular coordinates. The first step in the proposed dragging process is to generate micro channels whose cross-section shape is similar to that of the mask. Then the dragging procedure is repeated twice, each time after rotating the work piece by 60 deg relative to the previous channels. This hexagonal micro lens array may be used in the application of image processing. Our results show that the hexagonal micro lens array has a better axial symmetry and higher fill factor than a rectangle one.

© 2005 Optical Society of America

## 1. Introduction

This paper examines the fabrication of aspheric hexagonal micro lens arrays by use of the dragging process with an excimer laser. The micro-optical components are increasingly important in micro-opto-electro-mechanical systems (MOEMS) that integrate and combine detection or generation functions with light. The refractive micro lens arrays are applied as collimators or beam expanders for optical communication, image collection, biomedical inspection, and so on. Meanwhile, integrated micro lens arrays offer interesting applications for various technologies such as liquid-crystal displays (LCDs) and personal digital accessories (PDAs). The major objective of using micro lens arrays is to enhance the brightness of illumination and simplify the construction of light guide modules. In a laptop display, a 25% increase in light output was reported while using micro lens technology [1]. Accordingly, the micro lens array is one of the key components in MOEMS.

To date, the processes that have been applied to micro lens array fabrication are (1) reflow, (2) photothermal, (3) volume swelling, (3) droplet, (4) gray scale or gray level mask, (5) dragging with am excimer laser [2–4]. The photo resist reflow or droplet process was developed in 1988 for spherical micro lens array fabrication [2]. Because of the surface tension effect, the photo resists changes into a hemispherical shape to minimize the surface energy. As a result, it is difficult to change the shape to aspherical using this kind of process. In addition, the distance between two neighboring micro lenses must be great enough to prevent the neighboring micro lenses from joining together. Therefore, the lens array fill factor is restricted to a lower value. That is, based on the surface tension action, the hemispherical form and lower fill factor are the limitations of the reflow and droplet process. Consequently, how to fabricate the precise aspheric micro lens array with an improved or new process is a challenge for the scientist.

In recent years, excimer lasers have been utilized as an industrial tool for micro fabrication. When radiation is delivered onto the surface of a polymer, the incident energy is absorbed into a thin layer (e.g., 0.1μm), and then the layer is decomposed, heated, and ablated [5]. Moreover, the work piece dragging process uses a fixed-position mask with the sample moved while the laser is firing, as shown in Fig. 1. Dragging is a powerful technique used to fabricate micro channels, groves, concave and convex cylindrical profiles, and many other surface relief features [6]. In Fig. 1, the concave semi-circular shape of the mask generates the concave grove. The work piece moves at a constant speed while the laser fires repetitively. The mask shape determines the channel cross-section while the channel surface smoothness is dominated by the mask accuracy; the sample material and the laser shot overlap [5]. Where the moving speed of the sample and the Hz number of the laser affect the overlapping density of laser shot.

In the application of machine vision and image processing, the hexagonal coordinate has the unique characteristic of clear definition for connection. Accordingly, a new fabrication process of aspheric hexagonal micro lens arrays by use of the dragging process with an excimer laser was investigated in this paper. In particular, the characteristics of axial symmetry and fill factor for these micro lenses were discussed.

## 2. Dragging process

An excimer laser with mask projection machining has been successfully used for the fabrication of 2.5D micro parts. Additionally, excimer laser machining is capable of manufacturing microstructure arrays with curved surfaces using various mask shapes. In the dragging mode, a complex mask pattern (e.g., convex semi-circular shape) is projected onto the surface of the sample, which is moved at a constant velocity to form a constant profile linear structure. Rotating the sample 90 deg and repeating the process crossing the patterns gives rise to a periodic and regular 3D structure with a rectangular arrangement, as shown in Fig. 2. Using the dragging process, excimer laser machining can rapidly make a micro pyramid array for a graded-index antireflection coating or a micro lens array with high fill factor.

As mentioned above, excimer laser machining has been successfully applied in the fabrication of microstructures and micro optical systems [4–8]. However, manufacturing 3D continuous relief has been progressively and successfully developed for low-cost mass production in micro optics [9–10]. Tonshoff *et al*. [11] showed that a pre-processor calculates the numerical control (NC) data for laser trigger, three axial (*x*, *y*, and *z*) motions, and the NC mask motion control can generate the desired 3D contours. Therefore, the desired 3D microstructure could be manufactured using excimer laser machining with the appropriate mask design, precision motion, and laser control. In the following, we discuss how to modify the dragging process to manufacture the aspheric hexagonal micro lens array.

In the general dragging process, the work piece was moved in one direction, a channel with a cross section shape similar to the mask would be produced, as indicated in fig. 1. Repeating this procedure at the direction orthogonal to the first channel produces the microstructure array with rectangular coordinate, as shown in Fig. 2 and the left-hand side Fig. 3. In contrast, in the improved dragging process, machining the work piece first and then repeating the dragging operation after 60 deg rotation twice leads to the hexagonal microstructure array, as shown in the right-hand side of Fig. 3. The comparison between these two different dragging processes will be discussed using computer simulation in the following paragraph.

## 3. Simulation process

To investigate the geometrical property of the micro lens after the dragging process, the 3D geometrical shape of a single micro lens with varied dragging processes was examined by using computer simulation. In the computer simulation process, the volume removal execution is used as machining action in the dragging process. In other words, the volume minus process replaces the laser dragging action in the computer program. Figure 4 illustrates the simulated and machining results of the rectangular micro lens array with a convex mask shape. It is quite conceivable that the contour of the computer simulation matches the SEM picture for the real machining result.

For convenience of analysis, the geometrical shape of a single micro lens was discussed in the beginning. Therefore, the simulating contours of a single micro lens with different dragging process were discussed and compared, especially in axial symmetry. In Fig. 5, the overlap of the two machining processes (which are orthogonal to each other) leads to the rectangular result. Moreover, after the dragging process is completed three times, a contour graph of single micro lens with hexagonal coordinate is plotted in Fig. 6.

In the fabrication process of the micro lens array, the number of mask patterns in the laser projection region is more than one. That is, after the first dragging procedure, micro grooves with parallel arrangement are generated. Similarly, the three identical patterns (semi-circular shape) of the mask were used in the computer simulation. Accordingly, a mesh graph of a 3×3 micro lens array with rectangular coordinates was generated after dragging two times, as shown in Fig. 7. There are nine micro lenses in total that make up this rectangular array. At the same time, Fig. 8 illustrates the hexagonal micro lens array that was produced after the dragging process was completed three times with the same mask. There are seven micro lenses in this hexagonal array. For convenience of representation, the convex types of mesh graph were generated.

## 4. Results and discussion

Generally speaking, the axial symmetry of the single micro lens dominates the practical optical performance. In addition, the fill factor is an important characteristic of the micro lens array in real optical applications. To investigate the difference between these two dragging processes, the axial symmetry of the single micro lens and the fill factor of the micro lens array will be analyzed in this section.

#### 4.1 Axial symmetry

In the previous section, Fig. 5 shows the contour of the single micro lens for rectangular coordinates with the semi-circular mask shape in the dragging process. For a standard axial symmetrical lens, the contour consists of several concentric circles. But Figs. 5 and 9 indicate that the axial symmetry is not good, especially at the four corners of the single micro lens. It is evident that at the boundary region, the axial symmetry of the rectangular micro lens array is not satisfied.

For the purpose of clear analysis, the various directions of the cross sections in the concave lens surface were studied. In the left-hand side of Fig. 9, two cross sections are cut along the DD (oblique diagonal direction) line and AA (middle horizontal direction) line. For the calculation of fill factor in next paragraph, the inner part region of the largest circle would be defined as a possible effective area of a single micro lens. In contrast, the outer part of the largest circle would be viewed as a useless area for this micro lens. Consequently, these two cross section shapes in the region of the black circle are separately plotted in the right-hand side of Fig. 9. At the center or axial region of micro lens, the DD and AA curve are nearly overlapping. Accordingly, the axial symmetry is good at the middle part. But for the boundary part, what has to be noted is that the differences between these two cross sections are obvious, especially at the four corners of this contour, as shown in Fig. 5. This clearly means that the axial symmetry at the boundary of rectangle micro lens is bad.

On the other hand, Fig. 6 shows the contour of a single micro lens for hexagonal coordinates with the identical (semi-circular) shape of the mask in the dragging process. As in the previous step, two cross sections were cut along the BB (middle vertical direction), and the AA (middle horizontal direction) lines are plotted separately in the right-hand part of Fig. 10. In particular, the BB line replaced the original DD (oblique diagonal direction) line in the hexagonal array, because the two crossing micro groves were joined along the middle vertical direction. The result shows that these two cross sections are almost identical. Meanwhile, in Fig. 6, it is evident that the axial symmetry is good even at the four corners of this contour. For a qualitative comparison, this proves that the hexagonal coordinates of the micro lens array get the better axial symmetry than rectangle case.

To support a quantitative analysis of axial symmetry, the height difference between two cross sections for these two cases was discussed. Figures 11 and 12 depict the difference in value of two cross sections by computer simulation. Moreover, a strict definition of effective area was adopted in this paper. At one specified point in a micro lens, if the height difference is smaller than 1 unit, then this point is recognized as an effective point. For example, the effective radius of the rectangular coordinate is recognized as 34 units, and that of the hexagonal case is 43 units, as separately described in Figs. 11 and 12. Assume that the radius of the largest circle in a single micro lens is 1 (50 units), in Figs. 9 and 10. Accordingly, for the rectangular and hexagonal case the effective radii are 0.68 and 0.86, respectively. In summary, this new dragging method (hexagonal coordinate) generates the larger effective radius and valid area than the conventional process (rectangular coordinate).

Additionally, Table 1 presents three height-difference parameters for rectangular and hexagonal coordinates. The *Sum of difference* is the sum of the absolute height difference between these two cross sections for each of the 101 points, *D _{max}* denotes the maximum
absolute value of the height differences between these two cross sections for all 101 points, and

*Ratio of difference*is the value of

*D*/Sag. The results indicate that the hexagonal coordinate reduces these three height-difference parameters in this table.

_{max}#### 4.2 Fill factor

Figures 13 and 14 show contour graphs of micro lens arrays for rectangular and hexagonal coordinates, respectively. Comparing these two figures, we see the difference between the effects of dragging times and direction is evidently. In Fig. 13, there are nine micro lenses with rectangular coordinates in the whole contour. As described in Fig. 11, the effective radius of the rectangular case is 0.68. As a result, the fill factor equals 0.3632 (i.e., π*0.68*0.68/(2*2)) for the rectangular case. On the other hand, for the hexagonal case in Fig. 14, the whole hexagon (marked with bold black line) is defined the calculation area. According to the similar procedure, the effective radius of hexagonal case is 0.86. Therefore, the fill factor of hexagonal case equals to 0.5217 (i.e., 7* π*0.86*0.86/(6*6*sec(30°)-4*(1/2)*3*3*tan(30°))). Table 2 compares results for rectangular and hexagonal cases. Accordingly, the new dragging process is capable of making a micro lens array with hexagonal coordinates. In addition, adopting the strict definition for effective region, the hexagonal micro lens array has a higher fill factor than the rectangle coordinate.

These two dragging process have been compared in this section. In particular, the coordinates of the micro lens ware changed, because the crossing area of these micro grooves constructs the micro lenses. Meanwhile, the direction of dragging affects the coordinates of the micro lens. Thus, the number of micro grooves and directions of dragging dominate the amount and coordinates of the micro lenses. In view of optic performance, the new dragging process obviously improves both the axial symmetry and the fill factor, as shown in Tables 1 and 2. The various directions of dragging do reduce the deviation of axial symmetry. That is, the dragging processes with different directions average the form error of a single micro lens. But in geometrical mathematics, octagonal coordinates do not exist. Therefore, it cannot produce a micro lens array with octagonal coordinates using a rotating work piece with 45 deg.

## 5. Conclusion

A new dragging process for the fabrication of a hexagonal micro lens was examined. A computer simulation for the laser dragging process was performed. The new dragging process is the one performed after the first micro channel with a cross section shape similar to that of the mask is produced; repeating the dragging procedure after 60 deg rotation relative to the previous channel twice produces the hexagonal microstructure array. The computer simulation shows that the hexagonal micro lens array has a better axial symmetry and higher fill factor than rectangle micro lens array.

## Acknowledgment

The authors thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract NSC-91-2218-E-230-003.

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