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This work demonstrates the fabrication of a simple, low-cost microlens array (MLA) diffuser film with controllable haze distribution (diffusion effect) by a combination of “breath figures” (BFs) and micro-replica molding methods. Polystyrene (PS) molds obtained by BFs method contain concave, hexagonal packed air holes formed by the condensation of water vapor on cooling surfaces in a chamber in which relevant influence factors can be controlled. The sizes of the air holes in the BFs PS molds can be controlled by varying such factors as chamber temperature, chamber relative humidity, substrate temperature and others. The temperature distribution on the substrate affects the distribution of diameters of the air holes formed in a BFs PS mold. Convex PDMS (poly-dimethylsiloxane) MLAs were obtained by molding from the BFs PS molds. The focal lengths of MLAs were measured and compared with theoretical values. The diffusion effect of the diffuser films with MLAs of diameters 6 µm and 3 µm were compared. The results indicate that an MLA with a smaller diameter has a larger diffusion effect.
©2008 Optical Society of America
1. Introduction
Refractive micro-lens arrays (MLAs) have attracted enormous interest in the fields of optics, optoelectronics, bio-chemistry and display. Their applications include laser beam shaping systems [1], optical fiber coupling [2], multiple optical tweezers [3], charge-coupled-devices (CCD) [4], light diffusers [5,6], and many others. Various methods have been applied to fabricate MLAs. They include thermal reflow [7–9], gray-scale masking [10], holographic lithography [11], ink-jet printing [12], soft replica molding [13], electro-beam lithography [14], and others. However, most methods require precise control, a complex manufacturing process and a long fabrication time, and costly to a high-cost in industry and scientific research.
Recently, a simple, low-cost molding method was employed to fabricate MLAs [15]. The molds used in this method were fabricated by a simple self-assembly procedure called “breath figures (BFs)” [16]. In BFs, mircroporous polymer molds that contain hexagonally arranged pores were obtained by simply casting various polymer solutions under humid conditions. During the BFs process, water droplets condensed on cold surfaces of polymer solutions and formed arrays of air holes with diameters from 500 nm to 50 µ m [16]. MLAs were fabricated simply by molding from the microporous polymer BFs molds. The BFs molds contained hexagonal packed concave air-holes with a high filling factor, and their sizes were controlled by varying such influencing factors as the quantity of solutions, the temperature and the relative humidity [17–20]. Therefore, microlenses with different diameters were obtained simply by varying these factors in the fabrication of BFs molds.
MLAs can spread and homogenize non-uniform illumination. This capability can be exploited in such applications as light diffuser films in an LED backlight system of a liquid crystal display (LCD) and LED lighting. In these applications, the diffuser needs to generate a particular radiation pattern that eliminates non-uniformity of the intensity distribution from an LED [5]. Such a radiation pattern cannot be obtained using conventional opal-coated and sandblasted diffusers [5,21]. MLAs represent a good solution because a desired radiation pattern can be obtained by tailoring a MLA light diffuser by controlling the distribution of surface profiles of the microlenses in the array. However, a good fabrication technique that enables the surface profiles distribution of microlenses to be controlled is still lacking.
This work presents a simple approach for fabricating MLA diffuser films with controllable haze distribution using a BFs molding method. The BFs molds were first fabricated in a chamber in which relevant influence factors could be controlled, and air holes of various sizes were formed in the BFs molds by varying these factors. Controlling the temperature distribution on the substrate yielded the air holes with a desired distribution of diameters in a BF mold. The BF molds were transferred into convex PDMS (poly-dimethylsiloxane) MLAs. The surface profiles of MLAs were examined under a scanning electron microscope (SEM) and their focal lengths and the diffusion effect were determined by optical microscopy and an expanding white light source system, respectively. The light spreading intensity of the diffuser films obtained by this process are controlled by controlling the diameters of MLAs.
This work is organized as follows. Section 2 presents the fabrication of BFs molds and PDMS MLAs diffuser films. Section 3 presents the fabrication results of BFs molds and demonstrates how the influence factors affect the formation of air holes in BFs molds. Section 4 presents the measurements of focal lengths of PDMS MLAs. Section 5 presents the diffusion effect measurements of PDMS MLAs. Section 6 draws conclusions.
2. Fabrication of BFs molds and PDMS MLAs diffuser films
Figure 1 depicts the fabrication process of a poly-dimethylsiloxane (PDMS) MLA diffuser film. Figures 1(a) to 1(c) present the formation process of a BFs mold in a chamber with controllable influence factors, including chamber temperature (TC), chamber relative humidity (RH), substrate temperature (TS), quantity of solutions (Q) and concentration of polystyrene (PS) (C). To prepare BFs molds, PS was dissolved in carbon disulfide (CS2) and then dropped on a glass substrate (1×1 cm2). In Fig. 1(a), moisture initially condensed on the cold PS-CS2 solution surface by the evaporation of the CS2 solvent. Then in Fig. 1(b), the moisture was nucleated and arranged into a hexagonal arrangement; the moisture then evaporated. The PS polymer then became solid and a BFs concave mold, containing hexagonal arrangement of air holes, was formed as showed in Fig. 1(c). The whole formation process of the BFs mold took a few minutes. In Fig. 1(d), liquid PDMS was mixed with a 10:1 weight ratio of prepolymer and curing agent (Sylgard 184, Dow Corning) and then spin-cast onto the PS concave BFs mold. After it had been heating at 70 °C in a vacuum chamber for 30 minutes, a solid PDMS convex MLA film was peeled off from the BFs mold. Figure 1(e) presents the obtained PDMS MLA. Notably, the transparency of a transferred MLA is important to the high transmission of optical devices. Therefore, PDMS is preferred as a MLAs material because of its transparency, flexibility, thermal stability and homogeneous refractive index as well as the ease with which it fills designed mold shapes.
Fig. 1. PDMS MLAs formation process. (a) Moisture condensed on cold PS-CS2 surface. (b) Moisture was nucleated and arranged into a hexagonal array. (c) PS polymers formed a concave mold that contained a hexagonal arrangement of air holes. (d) PDMS was spin-cast onto PS concave mold. (e) After PS mold was peeled off, PDMS MLA was obtained.
Figure 2 shows SEM pictures of a BFs mold and PDMA MLA. The BFs mold was formed at TC=30 °C, RH=72 %, TS=26 °C and Q=71.5 µl. Figure 2(a) presents a top view of the BFs mold that comprised hexagonal closed-packed air holes. The diameters of the air holes are about 5.1 µm, and the distances between the nearest-neighbor air holes are about 8.1 µm. Figure 2(b) presents a top-view image of a PDMS MLA that was transferred from the BFs mold that is shown in Fig. 2(a); the diameters of the MLAs are about 6.6 µm; the inset presents the side view of the PDMS MLA. The filling factor of the MLAs is very high - about 100 % in most regions of the MLA. A high filling factor is an important requirement of an MLA diffuser film. An MLA with high filling factor effectively diffuses the incident light beam and yields no hot spot on the display.
3. Fabrication results of BFs molds
Figures 3(a) to 3(c) present the average air hole diameters of BFs molds obtained at different values of the influence factors in a chamber. The average air hole diameters were determined from optical microscopic images of BFs molds using a digital image analyze software (All data are averages over ~2,500 air holes over an area 1×1 cm2). In Fig. 3(a), the Q factor is varied from 60.5 µl to 91.3 µl, while TC, RH, TS and C, are fixed at 25 °C, 72 %, 25 °C and 1 wt.-%, respectively. As the Q value increases, the average air hole diameters also increase, because as the quantity of solution increases, the CS2 solvent evaporation time becomes longer and more water condenses on the surface of the solution, forming large air holes. Figure 3(b) plots the average air hole diameter as a function of the RH value with the other influence factors fixed at TC=TS=25 °C, C=1 wt-% and Q=71.5 µl. The average air hole diameters increase with RH value because more water is available to coalesce as larger water droplets. Figure 3(c) plots the change in the average air hole diameter with TS for three values of TC (20 °C, 25 °C and 30 °C), with the other influence factors fixed at RH=72 %, Q=71.5 µl and C=1 wt.-%. The average air hole diameter declines as TS increases. At higher TS, the solution evaporates rapidly, reducing the time taken by the moistures to coalesce. Therefore the average air hole diameter is smaller on a substrate with higher TS. A comparison of the results at the three TC values (20, 25, 30 °C) shows that the average air hole diameter increases with TC, because the water vapor is not fixed inside the chamber. The relative humidity RH also increases with the chamber temperature TC. Therefore, larger air holes are formed.
Fig. 3. Variation of average diameters of air holes in BFs molds with various influence factors: (a) Q, (b) RH and (c) TS at three values of TC - 20 °C, 25 °C and 30 °C.
Since the temperature of the substrate strongly affects the size of the air holes obtained, a non-uniform distribution of air hole diameters in a substrate can be obtained in the presence of a temperature gradient. Figure 4(a) shows the set-up for obtaining a non-uniform temperature distribution on a thin glass substrate (size ~1.5×1.1 cm2, thickness ~0.7 mm). The glass substrate was placed on the top of another glass plate (left side) and a copper plate (right side); a gap of about 10 mm was left between the glass and copper plates. Both glass and copper plates were heated by a heater. The difference between the thermal conductivities of the copper and glass plates produced a temperature gradient on the glass substrate. The temperatures of the glass and copper plates were 39 °C and 41 °C, respectively. After the PS-CS2 solution was dropped onto the thin glass substrate, a hexagonal arrangement of closed-packed air holes was formed on the glass substrate. Figure 4(b) shows the SEM pictures of the BFs mold at different positions on the substrate, obtained under the following conditions; RH=72 %, TC=25 °C, Q=75 µl and C=1 wt.-%. Figure 4(c) depicts the variation of the average air hole diameter in the horizontal direction of the substrate. The average air holes diameter is smallest (~1.6 µm) at position I (on top of the copper plate) and gradually increases from the right to the left to the center (E) of the substrate, from which it then gradually declines to position A. The average air holes diameter is greatest at position I, because TS at this point is lowest. This result indicates that the desired distribution of air hole diameters can be feasibly obtained by controlling the temperature distribution on the substrate.
Fig. 4. (a). Set-up for generating temperature gradient on the substrate. (b). SEM pictures of BF mold formed at different positions on the substrate. (c). Average of air hole diameters obtained at different positions on substrate.
4. Measurements of focal lengths of PDMS MLAs
Figure 5(a) shows the setup for measuring focal lengths of MLAs. A collimated He-Ne laser beam with a wavelength of 632.8 nm and a diameter of 2 mm was used as the light source to illuminate MLAs from the back side of the glass substrate. A microscope system with an object lens (OL) (NA=0.85) and a CCD camera was used to capture images of MLAs. An MLA sample was scanned in the z-direction of the microscope to determine the focal plan of the MLA. Figure 5(b) shows the images of the MLA captured at different positions along the z-axis (from z=-1.6 µm to z=+5.6 µm). The zero point position of the z-axis (z=0) is defined as the focal point of the OL, which is located at the top surface of the substrate, and the negative and positive z positions represent the focal point of the OL below and above the top surface of the substrate, respectively. When z=0, the microscope system captures a nondiffracted image of the MLA with the largest magnification ratio. When the focal point of the OL is moved to the focal plan of the MLA, the diameter of the light in the CCD are the smallest and their intensity becomes the highest [22], as revealed by the image obtained at z=+3.2 µm. In this method, the distance (L) between the top surface of the substrate and focal plane of the MLA can thus be determined.
Fig. 5. (a). Experimental setup for measuring the focal lengths of MLAs. (b) Images captured by microscope system at different positions along the z-axis for a PDMS MLA with diameter of 3 µm.
The geometrical relationship between L and the effective focal length (EF) of a MLA (Fig. 6(a)) can be expressed as follows.
where D is the diameter of the MLA, and h is the height of the cut cap of the PDMS microlens. The SEM picture of the PDMS MLA shown in the inset of Fig. 2(b) yields h~0.15D. Experimental EF values for different diameters of PDMS MLAs were thus determined based on Eq. (1) and the L values that were measured by the microscope system.
Theoretically, the EF value of a PDMS MLA can be determined using the equation [7],
where n (n=1.45) is the refractive index of PDMS. As shown in Fig. 6(a), the back focal length (BF) of a MLA can be expressed as,
Equations (1) and (3) (or Eqs. (2) and (3)) yield experimental (or theoretical) BF values for PDMS MLAs of various diameters. Figure 6(b) plots the variation of both experimental and theoretical EF and BF values as a function of the diameter of PDMS MLAs. Both experimental EF and BF values agree well with the corresponding theoretical results.
Fig. 6. (a). Geometrical relationship among different parameters of a PDMS MLA. (b). Experimental and theoretical EF and BF values as functions of diameter of PDMS MLAs.
5. MLA’s diffusion effect measurements
Figure 7 shows the setup for measuring the diffusion and 3D haze distributions of a nearly point light source that passes through different PDMS MLAs and a glass substrate. Figure 7(a) shows the setup for measuring diffusion. A tungsten lamp was used as a light source and its output was collimated and expanded using a pair of lenses. An iris was then used to select a small part of the light beam to serve as an almost point light source. The light source output through the iris then propagated through an MLA and was imaged by a color CCD camera. Digital software was used to analyze the haze distribution of the light beam that was detected by the color CCD camera. Figure 7(b) shows the haze distribution of the near point light source after propagation through a plane glass substrate; no diffusion effect occurred. Figures 7(c) and 7(d) display the haze distributions of the near point light source that propagates through the MLAs with diameters of 6 µm and 3 µm, respectively. The haze distribution in Fig. 7(d) is more uniform than that in Fig. 7(c). Therefore, MLAs with a smaller diameter diffuse more strongly.
Fig. 7. (a). Setup for measuring optical diffusion. (b). Optical diffusion through plane glass substrate. (c). Optical diffusion through MLAs with diameter of 6 µm. (d). Optical diffusion through MLA with diameter 3 µm.
6. Conclusions
This work presents a simple, low-cost approach for fabricating PDMS MLAs using BFs molding. In this approach, the diameters of air holes in BFs molds with a high filling ratio can be controlled by varying the influence factors. Furthermore, the temperature distribution on the substrate affects the distribution of diameters of air holes in BFs molds. Convex PDMS MLAs were fabricated simply by molding from the BFs molds. The focal lengths and distribution of the diffused light by PDMS MLAs were measured using a microscope and light diffusion measurement system, respectively. The light diffusion measurements reveal that an MLA with a smaller diameter (shorter focal length) has a stronger diffusion effect. Carefully controlling temperature distribution on the substrate yields a desirable distribution of focal lengths of microlenses in the array. Tailoring PDMS MLAs is useful for such applications as diffuser films in LED lighting, LCD backlights and others.
Acknowledgments
The authors would like to thank the National Science Council of the Republic of China, Taiwan, (Contract Nos. NSC 95-2112-M194-014-MY3 and NSC 95-2120-M194-006) and AU Optronics Corporation for financially supporting this research.
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