本論文利用拉格朗日插值法推算出LED單元於直下式背光模組所需之照度分佈，進而獲得出相對應之光強度分佈做為後續二次光學透鏡設計之目標，使LED背光模組不論於方形或三角形排列下，皆可提供相當高的照度分佈分別為94%以及99%。
背光模組除了照度均勻度的要求外，也需考慮色度均勻度之表現，然而，在傳統LED光源的建立上，並未掌握真實光源之空間色彩特性。本論文提出以中場量測方法建立具有光強度與空間色彩特性之光色模型，其模擬與量測光源之光強度分佈與相關色溫分佈之相關係數分別為0.996與0.997之高相似程度；而光源加上雙凸透鏡於接受半角為1.523度、1.965度以及3.26度之條件下，其模擬與量測之光強度分佈之相關係數皆為0.991，相關色溫分佈之相關係數分別為0.996、0.995以及0.989。
最後，進而依據光源光色模型之特性與背光板理想照度與色度分佈做為設計目標，進行二次透鏡設計與最佳化之改善，有助於直下式背光模組開發，同時達到均勻之照度分佈與色度分佈，其中，在LED背光模組方形排列，寬高比為4之條件下，二次透鏡設計之最佳照度均勻度可達83.7%，色度均勻度為0.0039。

In our study, we use lagrange interpolate polynomial equation to calculate the LED unit illuminance distribution (LUID) in LED backlight module. The illuminance uniformity is 94% and 99% for square and triangle arrangement, respectively.
In addition to illuminance uniformity, color uniformity also is important requirement for backlight module design. However, the traditional LED light source modeling methods did not consider the color spatial characteristics. According to this issue, we propose a new method by measuring mid-field measurement to establish chromatic source model (CSM) which including the properties of intensity and color distribution for LED light source. According, the normalized cross correlation (NCC) between measurement and simulation of intensity distribution and color distribution are 0.996 and 0.997, respectively. In order to obtain the more accuracy CSM, we further compare the NCC of measurement and simulation through mid-field. In the conditions of 1.523-1.965 and 3.26 degree, the NCC of Intensity distribution all higher than 0.99,as well as the NCC of CCT distribution is 0.996, 0.995 and 0.989, respectively.
Furthermore, we design and optimize the secondary optical lens base on the property of CSM and the target including illuminance and color uniformity simultaneously. According the simulate process, the condition of square arrangement and DHR is equal to 4, the 83.7% of illuminance uniformity can be achieve, as well as the color uniformity ( ) can reduce to 0.0039 which below the requirement of 0.004.

[1] E. F. Schubert and J. K. Kim, "Solid-state light sources getting smart," Science 308, 1274-1278(2005).
[2] M. G. Craford, "LEDs for Solid State Lighting and other Emerging Applications: status, trends, and challenges," Proc. SPIE. 5941, 594101-1 - 594101-9(2005).
[3] I. Moreno, M. Avendano-Alejo and R. I. Tzonchev, "Designing light-emitting diode arrays for uniform near-field irradiance," Appl. Opt. 45, 2265-2272 (2006).
[4] Z. Qin, K. Wang, F. Chen, X. B. Luo and S. Liu "Analysis of condition for uniform lighting generated by array of light emitting diodes with large view angle," Opt. Express 18, 17460-17476 (2010).
[5] J. Zheng and K. Qian. "Designing single LED illumination distribution for direct-type backlight," Appl. Opt. 52, 7022-7027(2013).
[6] R. Hu, H. Zheng, C. Ji, S. Liu and X. B. Luo, "A method to design freeform lens for uniform illuminance in direct-lit LED backlight with high distance-height ratio," ICEPT-HDP, 1474-1478(2012).
[7] Z. Liu, S. Liu, K. Wang and X. Luo, "Analysis of factors affecting color distribution of white LEDs," ICEPT-HDP, 1-8(2008).
[8] C. C. Yang, C. L. Chang, K. C. Huang, and T. S. Liao, "The yellow ring measurement for the phosphor-converted white LED," ICOPEN 19, 182-187(2011).
[9] C. C. Yang, H. Y. Tsai, and K. C. Huang, "Yellow-ring measurement of white LED in various environments," OSJ 20, 232-235(2013).
[10] C. C. Yang, H. Y. Tsai, Y. J. Chen and K. C. Huang, "A study on the yellow ring distribution of MR-16 WLED triplet lens module," Proc. SPIE 8769, 876932-1-876932-7 (2013).
[11] K. Wang, D. Wu, F. Chen, Z. Y. Liu, X. B. Luo, and S. Liu, "Angular color uniformity enhancement of white light-emitting diodes integrated with freeform lenses," Opt. Lett. 35, 1860–1862(2010).
[12] C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, "Precise optical modeling for LED lighting verified by cross correlation in the midfield region," Opt. Lett. 31, 2193-2195 (2006).
[13] C. C. Sun, C. Y. Chen, H. Y. He, C. C. Chen, W. T. Chien, T. X. Lee and T. H. Yang, "Precise optical modeling for silicate-based white LEDs," Opt. Express 16, 20060–20066 (2008).
[14] C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee and S. M. Huang, "Optical modeling for LED mid-field region," Proc. SPIE 6342, 634217-1-634217-7 (2006).
[15] C. C. Sun, W.T. Chien, I. Moreno, C. C. Hsieh and Y. C. Lo, "Analysis of the far-field region of LEDs," Opt. Express 17, 13918-13927(2009).
[16] F. Kokubo and M. Yamaguchi, "Light Emitting Device and Lighting Device Having the Same," U.S. Patent No. 2009/0052192 (2009).
[17] S. Ohkawa, "Light Emitting Device, Surface Light Source Device, Display and Light Flux Control Member," U.S. Patent No. 7621657(2009).
[18] J. Schanda, Colorimetry : Understanding the CIE system, John Wiley & Sons, New York (2007).
[19] D. L. MacAdam, "Visual sensitivities to color differences in daylight," J. Opt. Soc. Am. 32, 247–274 (1942).
[20] N. Ohta and A. Robertson, Colorimetry : Fundamentals and Applications, John Wiley & Sons, New York (2006).
[21] C.S McCamy, "Correlated color temperature as an explicit function of chromaticity coordinates," Color Res. Appl. 17, 142-144(1992).
[22] I. Moreno, "Illumination uniformity assessment based on human vision," Opt. Lett. 35, 4030-4032(2010).
[23] Z. Qin, C. Ji, K. Wang and S. Liu, "Analysis of light emitting diode array lighting system based on human vision: normal and abnormal uniformity condition, " Opt. Express 20, 23927-23943(2012).
[24] S. Li, K. Wang, F. Chen, and S. Liu, "New freeform lenes for white LEDs with high color spatial uniformity," Opt. Express 22, 24418-24428(2012).
[25] ENERGY STAR, Program Requirements for solid state lighting luminaires, Eligibility Criteria–Version 1 (2008).
[26] T.W. Sederberg, Computer Aided Geometric Design, Brigham Young University, Utah(2012).
[27] X. Wan, J. Yi, Z. Zhang, W. Xiao and L. Liu, "Lagrange interpolation reprojection-revising reconstruction with incomplete data in optical computed tomography," Opt. Eng. 49, 087001-1-087001-7 (2010).
[28] S. B. Sadkhan and K. H. Ruma. "Evaluation of polynomial reconstruction problem using Lagrange interpolation method," ICCTA 1, 1399-1403(2006).
[29] W. Yiding, and W. Xiaojing, "A new method of progressive images transmission based on largrange interpolation polynomials over finite fields," APCIP 2, 88-91(2009).