本文提出由兩階段微流道所構成的光流體分束器，利用氯化鈣水溶液以及去離子水由對稱入口所注入流道內進行混合時產生的擴散帶構成具對稱性質的折射率梯度，除了在原微流道後方加上一第二階段微流道使分離角增加以外，也調整注入氯化鈣水溶液的入口高度為整體流道高度的一半，使得混合流體在微流道內除了產生側向梯度外，在深度方向也產生梯度，產生如鞍形的折射率分布，具擴散角的光束在通過如此的分布後，能夠同時產生分離光束以及在垂直方向聚焦的效果。本研究利用計算流體力學數值套裝軟體 ANSYS Fluent 以及自行開發的C++ 程式碼進行數值模擬，模擬光線在微流道中受到折射率分布影響的行進狀況。數值模擬所得到的結果可用以討論各項參數對於光流體分束器效能的影響。此外，我們利用光微影製程製作出所設計之微流道，以光線在充滿具液體所形成的折射率梯度的微流道中行進的實驗結果與模擬的結果互相比對，此比較顯示本程式碼為有效。數值模擬的結果顯示增加第二段流道可以使得原本通過第一段流道的分離光束具有更大的分離角，且在目標平面上向中央平面聚集的效果更好。改變氯化鈣水溶液的入口深度，可以有效地使光束在垂直方向上產生聚集的效果。增加主流道的長度可以使得光線受到折射率梯度影響的時間變長，有利於光線在目標平面上向中央平面聚集以及分為兩光束的效果。適當的主流道漸擴角及長度的搭配可以使得光束的分離角增加，而主流道漸擴角的增加使得光現在目標平面上向中央平面聚集的效果越好。增加包覆溶液與核心溶液的流率比值會使得分離角變小，但是適當的增加流率比值可使得光線在目標平面向中央平面聚集的效果增加。

In this work, we proposed a two-stage microchannel to form an optofluidic splitter which can generate large split angle and vertically converged beams based on the gradient of refractive index (GRIN) of the diffusion zone between two fluids. The calcium chloride solution and deionized water are employed as the cladding and core fluid, respectively, and injected symmetrically to form the beam splitter. The heights of the cladding fluid inlets are designed to be a half of the main-channel height to achieve a saddle-shaped concentration distribution to realize the convergence of light to the middle horizontal plane. Besides, the second part of the microchannel is used to increase the split angle. We use ANSYS Fluent and self-developed C++ codes to simulate the light propagation in the proposed microchannel. Then, the results obtained by the simulation can be used to investigate the effects of the parameters on the performance of the proposed optofluidic beam splitter. Also, we fabricate the microchannel by using photolithography process and the light propagation in the microchannel filled with the GRIN fluid is compared with that obtained by numerical simulation. The comparison shows that the simulation is valid. The results obtained by numerical simulation show that the addition of the second part of the microchannel results in a larger split angle compared to that of a one-stage microchannel. This also enhances the effect of convergence of light to the middle horizontal line on the objective plane. Also, the change of the depth of the inlets of the cladding fluid can make light be condensed vertically. Besides, the increase of the main channel length can increase the duration for light to be influenced by the GRIN which is favorable for light to converge to the middle horizontal plane and to split into two beams. With appropriate combination of the divergent angle of the main channel, the effect of splitting light is enhanced. The increase of the divergent angle of the main channel enhances the effect of the convergence of light to the middle horizontal line on the objective plane. The increase of the ratio of the flow rate of the cladding fluid to that of the core fluid leads to the decrease of the split angle, while appropriate increase of the flow rate ratio can enhance the effect of the convergence of light to the middle horizontal line on the objective plane.

[1] V. R. Horowitz, D. D. Awschalom and S. Pennathur, "Optofluidics: field or technique ?," Lab on a Chip, vol. 8, pp. 1856-1863, 2008.
[2] D. Erickson, D. Sinton and D. Psaltis, "Optofluidics for enrgy application," Nature Photonics, vol. 5, pp. 583-590, 2011.
[3] L. Lei, N. Wang, X. M. Zhang, Q. Tai, D. P. Tsai and H. L. W. Chan, "Optofluidic planar reactors for photocatalytic water treatment using solar energy," Biomicrofluidics, vol. 4, 043004 (12pages), 2010.
[4] X. Fan and I. M. White, "Optofluidic microsystems for chemical and biological analysis," Nature Photonics, vol. 5, pp. 591-597, 2011.
[5] F. B. Myers and L. P. Lee, "Innovations in optical microfluidic technologies for point-of-care diagnostics," Lab on a Chip, vol. 8, pp. 2015-2031, 2008.
[6] M. Ebnali-Heidari, M. Mansouri, S. Mokhtarian and M. K. Moravvej-Farshi, "Design and numerical simulation of an optofluidic pressure sensor," Applied Optics, vol. 51, pp. 3387-3396, 2012.
[7] X. Tang, S. Liang and R. Li, "Design for controllable optofluidic beam splitter," Photonics and Nanostructures - Fundamentals and Applications, vol. 18, pp. 23-30, 2015.
[8] A. Constable, J. Kim, J. Mervis, F. Zarinetchi and M. Prentiss, "Demonstration of a fiber-optical light-force trap," Optics Letters, vol. 18, pp. 1867-1869, 1993.
[9] Y. C. Seow, A. Q. Liu, L. K. Chin, X. C. Li, H. J. Huang and T. H. Cheng, "Different curvatures of tunable liquid microlens via the control of laminar flow rate," Applied Physics Letters, vol. 93, 084101 (3pages), 2008.
[10] J. Shi, Z. Stratton, S.-C. S. Lin, H. Huang and T. J. Huang, "Tunable optofluidic microlens through active pressure control of an air-liquid interface," Microfluid and Nanofluid, vol. 9, pp. 313-318, 2010.
[11] X. Mao, S.-C. S. Lin, M. I. Lapsley, J. Shi, B. K. Juluri and T. J. Huang, "Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom," Lab on a Chip, vol. 9, pp. 2050-2058, 2009.
[12] N.-T. Nguyen, "Micro-optofluidic Lenses: A review," Biomicrofluidics, vol. 4, 031501 (15pages) , 2010.
[13] K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh and G. Manukyan, "Optofluidic lens with tunable focal length and asphericity," Scientific Reports, vol. 4, 6378 (4pages), 2014.
[14] L. Miccio, P. Memmolo, F. Merola, P. A. Netti and P. Ferraro, "Red blood cell as an adaptive optofluidic microlens," Nature Communications, vol. 6, 6502 (7pages), 2015.
[15] S. Xiong, A. Q. Liu, L. K. Chin and Y. Yang, "An optofluidic prism tuned by two laminar flows," Lab on a Chip, vol. 11, pp. 1864-1869, 2011.
[16] D. B. Wolfe, D. V. Vezenov, B. T. Mayers, G. M. Whitesides, R. S. Conroy and M. G. Prentiss, "Diffusion-controlled optical elements for optofluidics," Applied Physics Letters, vol. 87, 181105 (3pages), 2005.
[17] Y. Yang, A. Q. Liu, L. K. Chin, X. M. Zhang, D. P. Tsai and C. L. Lin, "Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation," Nature Communications, vol. 3, 651 (7pages), 2012.
[18] Y. Yang, L. K. Chin, J. M. Tsai, D. P. Tsai, N. I. Zheludev and A. Q. Liu, "Transformation optofluidics for large-angle light bending and tuning," Lab on a Chip, vol. 12, pp. 3785-3790, 2012.
[19] S. Q. Mawlud and N. Q. Muhamad, "Theoretical and experiment study of a numerical aperture for multimode PCS fiber optics using an imaging technique," Chinese Physical Letters, vol. 29, 114217 (4 pages), 2012.
[20] X. Mao, J. R. Waldeisen, B. K. Juluri and T. J. Huang, "Hydrodynamically tunable optofluidic cylindrical microlens," Lab on a Chip, vol. 7, pp. 1303-1308, 2007.
[21] P. A. Lyons and J. F. Riley, "Diffusion coefficients for aqueous solutions of calcium chloride and cesium chloride at ," Journal of the American Chemical Society, vol. 76, pp. 5216-5220, 1954.
[22] A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," Applied Optics, vol. 21, pp. 984-987, 1982.
[23] J. R. Dormand and P. J. Prince, "A family of embedded Runge-Kutta formulae," Journal of Computational and Applied Mathematics, vol. 6, pp. 19-26, 1980.
[24] M. F. Modest, Radiaitve heat transfer, ed. Academic press, New York, 2013.
[25] A. Nealen, "An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation," Computer Methods in Applied Mechanics and Engineering, vol. 130, 2004.
[26] T. Most and C. Bucher, "A Moving Least Squares weighting function for the element-free Galerkin Method which almost fulfills essential boundary conditions," Structural Engineering and Mechanics, vol. 21, pp. 315-332, 2005.
[27] J. L. Bentley, "Multidimensional binary search trees used for associative searching," Communications of the ACM, vol. 18, pp. 509-517, 1975.
[28] A. Sharma and A. K. Ghatak, "Ray tracing in gradient-index lenses: computation of ray-surface intersection," Applied Optics, vol. 25, pp. 3409-3412, 1986.