臺灣博碩士論文加值系統

隨著科技發展突飛猛進，資訊的傳輸與需求量更是不可同日而語，但傳統的通訊系統早已跟不上如此龐大的傳輸量，矽光子技術被視為目前的解決方案。非線性光學材料的飛秒等級響應時間可用於處理和操作數據信號是一大可以利用的優勢。為了實現非線性調變，需要較高的功率，倘若以矽作為元件的材料，雙光子吸收的特性會降低性能，而五氧化二鉭(Ta2O5)具有高折射率(Refractive index)、高非線性係數(n2)、寬能隙(Large bandgap)的優點，同時也得以和矽基板在製作過程中加以整合，適合作為非線性全光調變器的材料，因此本論文以五氧化二鉭(Ta2O5)為材料，並利用非線性多模態干涉波導的特性來實現非線性全光調變切換器的製作。首先以分步Fourier法考慮的多模態波導中非線性傳播方程式分析其特性，並在模擬結果中觀察到非線性切換的效果。設計的尺寸為寬8um高0.1um長0.4557(31L)及寬8um高0.1um長0.7497(51L)的多模態波導，以波長1064nm，脈衝寬度100fs，重複率為80MHz的脈衝雷射量測。比較切換功率的能力，51L長波導的模擬及量測觀察到的結果，皆比31L長的波導來的早發生。若在模擬上增加高階模態的傳輸損耗，可以得到出較為吻合實驗結果的分析。

The rapid advancement in information science and technology, such as big data and deep learning opens even higher demand for data processing and transmission, which is beyond the capability of currently mature technology based on electronic circuits. Although integrated optics and silicon photonics serves one of the solution to such a demand, the electronic response in data modulation and processing still set the limit to the speed. Femtosecond response time are easily achieved in nonlinear optical materials, for example the Kerr effect can be used for all optical processing in ultrafast regime. Even though silicon itself is one of the good nonlinear material, its two photon absorption (TPA) become an obstacle for implementing efficient nonlinear optical processing device. Ta2O5, on the other hand, is a wide band gap material providing high transmission and low loss in communication bandwidth. In the meanwhile, it has large Kerr nonlinear coefficient that is comparable to silicon but does not suffer from TPA. In this dissertation, we implement and demonstrate non-linear all-optical optical switcher using nonlinear multiple interference effect in Ta2O5 waveguide of sub-wavelength thickness. We design the device based on multiple-mode nonlinear Schrödinger equations and split-step Fourier method for simulation. We designed and fabricated multi-mode waveguides of width 8um, height 0.1um, length 0.4557cm(31L), and length 0.7497cm(51L) and characterize them by a femtosecond oscillator emitting 100fs mode locked pulses at 80MHz repetition frequency and the wavelength of 1064 nm. It is observed that the transmission through the nonlinear MMI waveguide drops as the peak intensity of the incident laser increases. The transition in the power depending transmission is consistent with the simulation when the modal losses are justified unequally.

中文審定書 i
英文審定書 ii
致謝 iii
摘要 iv
Abstract v
目錄 vi
圖次 ix
表次 xv
1 第一章 緒論 1
1.1 前言 1
1.2 矽光子技術 2
1.3 全光訊號調變 3
1.4 非線性效應 5
1.4.1 四波混頻(four-wave mixing, FWM) 5
1.4.2 三波混頻(Three Wave Mixing) 7
1.4.3 SPM和XPM： 10
1.5 光學邏輯元件 11
1.5.1 640-Gb / s的同步邏輯閘： 12
1.5.2 High-Base加法和減法： 13
1.6 文獻回顧以及研究動機 14
1.7 光波導材料 22
參考文獻 25
2 第二章 原理 31
2.1 Maxwell’s equations 31
2.2 脈衝傳播方程式 35
2.2.1 非線性脈衝傳播 35
2.2.2 高階非線性效應 42
2.3 多模態干涉 50
2.3.1 多模態波導 51
參考文獻 56
3 第三章 模擬 61
3.1 數值方法 61
3.1.1 Sellmeier equation 61
3.1.2 Eigenvalue equation 63
3.1.3 Split-Step Fourier Method 64
參考文獻 73
4 第四章 製程 76
4.1 基板 76
4.2 清洗 77
4.3 鍍膜 77
4.4 微影 80
4.5 蝕刻 82
4.6 沉積 84
4.7 切割 85
4.8 研磨 86
5 第五章 量測 88
5.1 量測系統 88
5.2 長直波導 89
5.3 多模態干涉波導 91
5.4 討論 93
6 第六章 結論 96

第一章
[1] https://zh.wikipedia.org/wiki/同轴电缆
[2] http://www.mondaq.com/australia/x/290662/Copyright/The+data+explosion+from+analogue+to+digital
[3] 林天送, “積體電路的發明”, 科學發展,447,72-74 (2010)
[4] https://en.wikipedia.org/wiki/Integrated_circuit#/media/File:EPROM_Microchip_SuperMacro.jpg
[5] W. N. Ye and Yule Xiong, “Review of silicon photonics: history and recent advances”,J. Mod. Opt. 60, 1299-1320 (2013).
[6] Rong, H.; Jones, R.; Liu, A.; Cohen, O.; Hak, D.; Fang, A.; Paniccia, M. Nature (London, U.K.) 2005, 433, 725–728.
[7] Fang, A.W.; Park, H.; Cohen, O.; Jones, R.; Paniccia, M.J.; Bowers, J.E. Opt. Express 2006, 14, 9203–9210.
[8] Liu, A.; Jones, R.; Liao, L.; Samara-Rubio, D.; Rubin, D. Cohen, O.; Nicolaescu, R.; Paniccia, M. Nature (London, U.K.) 2004, 427, 615–618.
[9] Liu, J.F.; Michel, J.; Giziewicz, W.; Pan, D.; Wada, K.; Cannon, D.D.; Jongthammanurak, S.; Danielson, D.T.; Kimerling, L.C. Appl. Phys. Lett. 2005, 87, 103501.
[10] Andrew P. Knights and J. K. Doylend, “Silicon Photonics- Recent Advances in Device Development”
[11] S. J. Ben-Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightw. Technol., vol. 14, no. 6, pp. 955–966, Jun. 1996.
[12] A.E. Willner et. al., “All-Optical Signal Processing”, J. Lightwave Techn. 32, 660 (2014).
[13] A. Yariv and P. Yeh, Optical Waves in Crystals. vol. 5, New York, NY, USA: Wiley, 1984.
[14] G. Agrawal, Nonlinear Fiber Optics. New York, NY, USA: Academic, 2001.
[15] C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M. Fejer, “All-optical signal processing using χ2 nonlinearities in guided-wave devices,” J. Lightw. Technol., vol. 24, no. 7, pp. 2579–2592, Jul. 2006. Opt. Lett., vol. 25, pp. 25–27, 2000.
[16] S. Radic, “Parametric Signal Processing,” IEEE J. Sel. Topics Quantum Electron., vol. 18, no. 2, pp. 670–680, Mar./Apr. 2012.
[17] J. Leuthold, L. Moller, J. Jaques, S. Cabot, L. Zhang, P. Bernasconi, M. Cappuzzo, L. Gomez, E. Laskowski, E. Chen, A. Wong-Foy, and A. Griffin, “160 Gbit/s SOA all-optical wavelength converter and assessment of its regenerative properties,” IEEE Electron. Lett., vol. 40, no. 9, pp. 554–555, Apr. 2004.
[18] J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon., vol. 4, no. 8, pp. 535–544, Aug. 2010.
[19] M. A. Foster, A. C. Turner, R. Salem, M. Lipson and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Exp., vol. 15, no. 20, pp. 12949–12958, 2007
[20] M. Galili, J. Xu, H. C. Mulvad, L. K. Oxenløwe, A. T. Clausen, P. Jeppesen, B. Luther-Davies, S. Madden, A. Rode, D.-Y. Choi, M. Pelusi, F. Luan, and B. J. Eggleton, “Breakthrough switching speed with an all-optical chalcogenide glass chip: 640 Gbit/s demultiplexing,” Opt. Exp., vol. 17, pp. 2182–2187, 2009.
[21] K. Uchiyama, T. Morioka, M. Saruwatari, M. Asobe, and T. Ohara, “Error free all-optical demultiplexing using a chalcogenide glass fiber based nonlinear optical loop mirror,” IEEE Electron. Lett., vol. 32, no. 17, pp. 1601–1602, Aug. 1996.
[22] C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M. Fejer, “All-optical signal processing using χ2 nonlinearities in guided-wave devices,” J. Lightw. Technol., vol. 24, no. 7, pp. 2579–2592, Jul. 2006. Opt. Lett., vol. 25, pp. 25–27, 2000
[23] J. C. Knight and D. V. Skryabin, “Nonlinear waveguide optics and photonic crystal fibers,” Opt. Exp., vol. 15, pp. 15365–15376, 2007.
[24] S. Radic, “Parametric Signal Processing,” IEEE J. Sel. Topics Quantum Electron., vol. 18, no. 2, pp. 670–680, Mar./Apr. 2012.
[25] A. Yariv and P. Yeh, Optical Waves in Crystals. vol. 5, New York, NY, USA: Wiley, 1984.
[26] G. Agrawal, Nonlinear Fiber Optics. New York, NY, USA: Academic, 2001.
[27] C. Langrock, S. Kumar, J. E. McGeehan, A. E. Willner, and M. M. Fejer, “All-optical signal processing using χ2 nonlinearities in guided-wave devices,” J. Lightw. Technol., vol. 24, no. 7, pp. 2579–2592, Jul. 2006. Opt. Lett., vol. 25, pp. 25–27, 2000.
[28] J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica micro-structured optical fibers with anomalous dispersion at 800nm,” Opt. Lett., vol. 25, pp. 25–27, 2000.
[29] A. Bogoni, L. Poti, R. Proietti, G. Meloni, F. Ponzini, and P. Ghelfi, “Regenerative and reconfigurable all-optical logic gates for ultra-fast applications,” IEEE Electron. Lett., vol. 41, no. 7, pp. 435–436, Mar. 2005.
[30] J.-Y. Kim, J.-M. Kang, T.-Y. Kim, and S.-K. Han, “All-optical multiple logic gates with XOR, NOR, OR, and NAND functions using parallel SOA-MZI structures: Theory and experiment,” J. Lightw. Technol., vol. 24, no. 12, pp. 3392–3399, 2006.
[31] A. Bogoni, X. Wu, I. Fazal, and A. E. Willner, “160 Gb/s time domain channel extraction/insertion and all optical logic operations exploiting a single PPLN waveguide,” J. Lightw. Technol., vol. 27, pp. 4221–4227, 2009.
[32] A. Bogoni, X. Wu, Z. Bakhtiari, S. Nuccio, and A. E. Willner, “640 Gbit/s photonic logic gates,” Opt. Lett., vol. 35, no. 23, pp. 3955–3957, 2010.
[33] T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P. T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett., vol. 15, no. 10, pp. 1422– 1424, Oct. 2003.
[34] J. Wang, S. R. Nuccio, J.-Y. Yang, X. Wu, A. Bogoni, and A. E. Willner, “High-speed addition/subtraction/complement/doubling of quaternary numbers using optical nonlinearities and DQPSK signals,” Opt. Lett., vol. 37, pp. 1139–1141, 2012.
[35] J. Wang, J.-Y. Yang, X. Wu, O. F. Yilmaz, S. R. Nuccio, and A. E. Willner, “40-Gbaud/s (120-Gbit/s) octal and 10-Gbaud/s (40- Gbit/s) hexadecimal simultaneous addition and subtraction using 8PSK/16PSK and highly nonlinear fiber,” in Proc. Opt. Fiber Commun. Conf., Mar. 2011, pp. 1–3.
[36] A. Bogoni, X. Wu, Z. Bakhtiari, S. Nuccio, and A. E. Willner, “640 Gbit/s photonic logic gates,” Opt. Lett., vol. 35, no. 23, pp. 3955–3957, 2010.
[37] J. Wang, J.-Y. Yang, X. Wu, O. F. Yilmaz, S. R. Nuccio, and A. E. Willner, “40-Gbaud/s (120-Gbit/s) octal and 10-Gbaud/s (40- Gbit/s) hexadecimal simultaneous addition and subtraction using 8PSK/16PSK and highly nonlinear fiber,” in Proc. Opt. Fiber Commun. Conf., Mar. 2011, pp. 1–3.
[38] F. Liu, et al., “Experimental sutdy of nonlinear switching characteristics of conventional 2 × 2 fused tapered couplers,” Chinese Opt. Lett. 3, 190 (2005).
[39] P.L. Chu et al., “Analytical solution to soliton switching in nonlinear twin-core fibers,” Opt. Lett. 18, 328 (1993).
[40] P.L. Chu et al., “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B. 10, 1379 (1993).
[41] Schmidt-Hattenberger, C., Udo Trutschel, and F. Lederer. “Nonlinear switching in multiple-core couplers.” Optics letters 16.5 (1991): 294-296
[42] Leuthold, J., C. Koos, and W. Freude. “Nonlinear silicon photonics.” Nature Photonics 4.8 (2010).
[43] Del’Haye, P., et al. “Optical frequency comb generation from a monolithic microresonator.” Nature 450.7173 (2007): 1214-1217.
[44] Kippenberg, T. J., S. M. Spillane, and K. J. Vahala. “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity.” Physical Review Letters 93.8 (2004): 083904.
[45] Heebner, John E., et al. “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators.” Optics letters 29.7 (2004): 769-771.
[46] Yeom, Dong-Il, et al. “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires.” Optics letters 33.7 (2008): 660-662.
[47] Wang, Christine Y., et al. “Mid-infrared optical frequency combs at 2.5 μm based on crystalline microresonators.” Nature communication 4 (2013): 1345.
[48] Grudinin, Ivan S., Lukas Baumgartel, and Nan Yu. “Frequency comb from a microresonator with engineered spectrum.” Optics express 20.6 (2012): 6604-6609.
[49] Grudinin, Ivan S., Nan Yu, and Lute Maleki. “Generation of optical frequency combs with a CaF2 resonator.” Optics letters 34.7 (2009): 878-880.
第二章
[1] P. Diament, Wave Transmission and Fiber Optics (Macmillan, 1990), Chap. 3.
[2] Y. R. Shen, Principles of Nonlinear Optics (Wiley, 1984), Chap. 1.
[3] P. N. Butcher and D. N. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990), Chap. 2.
[4] R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Chap. 1.
[5] D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1991), Chap. 2.
[6] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983), Chaps. 12–15.
[7] J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, 2004), Chap. 3.
[8] D. Marcuse, J. Opt. Soc. Am. 68, 103 (1978).
[9] H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap. 10.
[10] P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Chap. 9.
[11] F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
[12] J. P. Gordon, Opt. Lett. 11, 662 (1986).
[13] Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[14] E. A. Golovchenko, E. M. Dianov, A. N. Pilipetskii, A. M. Prokhorov, and V. N. Serkin, Sov. Phys. JETP. Lett. 45, 91 (1987).
[15] R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[16] K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[17] P. V. Mamyshev and S. V. Chernikov, Opt. Lett. 15, 1076 (1990).
[18] S. V. Chernikov and P. V. Mamyshev, J. Opt. Soc. Am. B 8, 1633 (1991).
[19] P. V. Mamyshev and S. V. Chernikov, Sov. Lightwave Commun. 2, 97 (1992).
[20] R. H. Stolen and W. J. Tomlinson, J. Opt. Soc. Am. B 9, 565 (1992).
[21] S. Blair and K. Wagner, Opt. Quantum Electron. 30, 697 (1998).
[22] T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[23] N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, IEEE J. Quantum Electron. 37, 398 (2001).
[24] A. Gaeta, Phys. Rev. Lett. 84, 3582 (2000); Opt. Lett. 27, 924 (2002).
[25] G. Chang, T. B. Norris, and H. G. Winful, Opt. Lett. 28, 546 (2003).
[26] J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[27] R. W. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[28] N. Tang and R. L. Sutherland, J. Opt. Soc. Am. B 14, 3412 (1997).
[29] A. Martínez-Rios, Andrey N. Starodumov, Yu. O. Barmenkov, V. N. Filippov, and I. Torres-Gomez, J. Opt. Soc. Am. B 18, 794 (2001).
[30] Agrawal, Govind P. Nonlinear fiber optics. Academic press, 2007
[31] F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[32] N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
[33] D. Anderson and M. Lisak, Phys. Rev. A 27, 1393 (1983).
[34] G. Yang and Y. R. Shen, Opt. Lett. 9, 510 (1984).
[35] E. Bourkoff, W. Zhao, R. I. Joseph, and D. N. Christodoulides, Opt. Lett. 12, 272 (1987).
[36] A. K. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, J. Lightwave Technol. 17, 216 (1999).
[37] R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron., vol. 18, no. 7, pp. 1062–1072, Jul. 1982.
[38] K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. New York, NY, USA: Academic, 2006, ch. 7.
[39] M. Bachmann, M. K. Smit, L. B. Soldano, P. A. Besse, E. Gini, and H. Melchior, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsPnnP four-port Mach-Zehnder interferometer,” in Proc. Con$ Opt. Fiber Commun. (OFC), San JoSe, CA, 1993, pp. 32-33.
[40] J. E. Zucker, K. L. Jones, T. H. Chiu, and K. Brown-Goebeler, “Strained quantum wells for polarization-independent electroopic waveguide switches,” J. Lightwave Technol., vol. 10, no. 12, pp. 1926-1930, 1992.
[41] R. J. Den, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, and J. I. Song, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett., vol. 4, no. 11, pp. 1238-1240, 1992.
[42] M. J. N. van Stralen, R. van Roijen, E. C. M. Pennings, J. M. M. van der Heijden, T. van Dongen, and B. H. Verbeek, “Design and fabrication of integrated InGaAsP ring lasers with MMI-outcouplers,” in Proc. European Con$ Integrated Optics (ECIO), Neuchltel, Switzerland, Apr. 1993, pp. 2.24-2.25.
[43] R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, and J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett., vol. 64, no. 14, pp. 1753-1755, 1994.
[44] R. M. Knox and P. P. Toulios, “Integrated circuits for the millimiter through optical frequency range,” in Proc. Symp. Submillimiter Waves, J. Fox, Ed., New York, Mar./Apr. 1970, pp. 497-516.
[45] P. N. Robson and P. C. Kendall, Eds., Rib Waveguide Theory by the Spectral Index Method, Optoelectronic Series, Research Studies Press Ltd. New York: Wiley, 1990.
[46] C. M. Weinert and N. Agrawal, “Three-dimensional simulation of multimode interference devices,” in Proc. Integr. Phot. Res. (IPRC), San Francisco, Feb. 1994, pp. 287-289
[47] N. S. Kapany and J. J. Burke, Optical Waveguides. New York: Academic, 1972.
第三章
[1] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
[2] V. I. Karpman and E. M. Krushkal, Sov. Phys. JETP 28, 277 (1969).
[3] N. Yajima and A. Outi, Prog. Theor. Phys. 45, 1997 (1971).
[4] R. H. Hardin and F. D. Tappert, SIAM Rev. Chronicle 15, 423 (1973).
[5] R. A. Fisher and W. K. Bischel, Appl. Phys. Lett. 23, 661 (1973); J. Appl. Phys. 46, 4921 (1975).
[6] M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55, 213 (1976).
[7] I. S. Greig and J. L. Morris, J. Comput. Phys. 20, 60 (1976).
[8] B. Fornberg and G. B. Whitham, Philos. Trans. Roy. Soc. 289, 373 (1978).
[9] M. Delfour, M. Fortin, and G. Payre, J. Comput. Phys. 44, 277 (1981).
[10] T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[11] D. Pathria and J. L. Morris, J. Comput. Phys. 87, 108 (1990).
[12] L. R. Watkins and Y. R. Zhou, J. Lightwave Technol. 12, 1536 (1994).
[13] M. S. Ismail, Int. J. Comput. Math. 62, 101 (1996).
[14] K. V. Peddanarappagari and M. Brandt-Pearce, J. Lightwave Technol. 15, 2232 (1997); J. Lightwave Technol. 16, 2046 (1998).
[15] E. H. Twizell, A. G. Bratsos, and J. C. Newby, Math. Comput. Simul. 43, 67 (1997).
[16] W. P. Zeng, J. Comput. Math. 17, 133 (1999).
[17] I. Daq, Comput. Methods Appl. Mech. Eng. 174, 247 (1999).
[18] A. G. Shagalov, Int. J. Mod. Phys. C 10, 967 (1999).
[19] Q. S. Chang, E. H. Jia, and W. Sun, J. Comput. Phys. 148, 397 (1999).
[20] W. Z. Dai and R. Nassar, J. Comput. Math. 18, 123 (2000).
[21] S. R. K. Iyengar, G. Jayaraman, and V. Balasubramanian, Comput. Math. Appl. 40, 1375 (2000).
[22] Q. Sheng, A. Q. M. Khaliq, and E. A. Al-Said, J. Comput. Phys. 166, 400 (2001). [23] J. B. Chen, M. Z. Qin, and Y. F. Tang, Comput. Math. Appl. 43, 1095 (2002).
[24] J. I. Ramos, Appl. Math. Comput. 133, 1 (2002).
[25] X. M. Liu and B. Lee, IEEE Photon. Technol. Lett. 15, 1549 (2003).
[26] W. T. Ang and K. C. Ang, Numer. Methods Partial Diff. Eqs. 20, 843 (2004).
[27] M. Premaratne, IEEE Photon. Technol. Lett. 16, 1304 (2004).
[28] G. M. Muslu and H. A. Erbay, Math. Comput. Simul. 67, 581 (2005).
[29] O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, J. Lightwave Technol. 21, 61 (2003).
[30] T. Kremp and W. Freude, J. Lightwave Technol. 23, 1491 (2005).
[31] J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).
[32] G. H. Weiss and A. A. Maradudin, J. Math. Phys. 3, 771 (1962).
[33] M. Lax, J. H. Batteh, and G. P. Agrawal, J. Appl. Phys. 52, 109 (1981).
[34] W. Sellmeier, Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen, Annalen der Physik und Chemie 219, 272-282 (1871).
[35] C.-L. Wu et. al., "Four-wave-mixing in the loss low submicrometer Ta2O5 channel waveguide," Opt. Lett. 40, 4528 (2015).