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The simulation of nonlinear self-focusing phenomenon using ray-tracing method can macroscopically provide an intuitive picture of the propagation of light in a self-focusing material, without adopting paraxial approximation or self-similar hypothesis. In this paper, propagation of light is sampled by discrete slices along a certain direction. Thus nonlinear propagation is turned into the combination of optical modulation of the refractive index on separate slices and linear propagation between each two adjacent slices. On each slice, after calculating the flux, we use a novel algorithm to suppress the quantized errors. For the linear propagating process, Adams method is adopted to solve the ray equations, which solve the problem that the widely used Runge-Kutta method cannot be used in simulation of light in nonlinear materials. The simulation results reveal that there are several foci along the propagating axis and the location of the first focus becomes closer to the incident plane as the power of light goes up. Furthermore, because the program traces real rays, it is possible to reach the non-paraxial region and reveal the phenomenon of ring-structure flux distributions caused by self-focusing. This is significant for the safety of high-power laser systems. Some commercial optical design and simulation software are also based on ray-tracing methods. Thus the systems including both nonlinear and linear materials are possible to simulate, which can guide people to set up the corresponding experimental systems.
[1] Askar’yan G A 1962 Soviet. Phys. JETP 15 1088
[2] Chiao R, Garmire E, Townes C H 1964 Phys. Rev. Lett. 13 479
[3] Kelley P L 1965 Phys. Rev. Lett. 15 1005
[4] Shen Y R (Translated by Gu S J) 1987 The Principles of Nonlinear Optics (1st Ed.) (Beijing: Science Press) p327 (in Chinese) [沈元壤著 (顾世杰译) 1987 非线性光学原理 (上册) (北京: 科学出版社) 第327页]
[5] Hercher M 1964 J. Opt. Soc. Am. 54 563
[6] Braun A, Korn G, Liu X, Du D, Squier J, Mourou G 1995 Opt. Lett. 20 73
[7] Liu W, Kosareva O, Golubtsov I S, Iwasaki A, Becker A, Kandidov V P, Chin S L 2003 Appl. Phys. B 76 215
[8] Agrawal G P (Translated by Jia D F, Yu Z H) 2002 Nonlinear Fiber Optics, Third Edition and Applications of Nonlinear Fiber Optics (1st Ed.) (Beijing: Publishing House of Electronics Industry) p33 (in Chinese) [阿戈沃著 (贾东方, 余震虹译) 2002 非线性光纤光学原理及应用 (第一版) (北京: 电子工业出版社) 第33页]
[9] Hardin R H, Tappen E D 1975 SIAM Rev. Chronicle. 15 423
[10] Sinkin O V, Holzlöner R, Zweck J, Menyuk C R 2003 J. Lightwave Technol. 21 61
[11] Godoy-Rubio R, Romero-García S, Ortega-Moñux A, Wangümert-Pérez J G 2011 J. Opt. Soc. Am. B 28 2142
[12] Feit M, Fleck J 1988 J. Opt. Soc. Am. B 5 633
[13] Chi S, Guo Q 1995 Opt. Lett. 20 1598
[14] Li Z D, Guo Q, Lin W G 1999 Chin. J. Lasers 26 711 (in Chinese) [李忠东, 郭旗, 林为干 1999 中国激光 26 711]
[15] Wen S H 2001 Ph. D. Dissertation (Shanghai: Graduate University of Chinese Academy of Sciences) (in Chinese) [文双春 2001 博士学位论文 (上海: 中国科学院研究生院)]
[16] Wen S C, Fan D Y 2000 Acta Phys. Sin. 49 460 (in Chinese) [文双春, 范滇元 2000 物理学报 49 460]
[17] Wen S C, Fan D Y 2000 Acta Phys. Sin. 49 1282 (in Chinese) [文双春, 范滇元 2000 物理学报 49 1282]
[18] Censor D 1977 Phys. Rev. A 16 1673
[19] Kasparian J, Solle J, Richard M, Wolf J P 2004 Appl. Phys. B 79 947
[20] Dragoman D 1996 Appl. Opt. 35 4142
[21] Zhang Y C, Lü B D 2004 Opt. Lett. 29 2710
[22] Gao H H, Tian L, Zhang B, Barbastathis G 2010 Opt. Lett. 35 4148
[23] Peng Z T, Jing F, Liu L Q, Zhu Q H, Chen B, Zhang K, Liu H, Zhang Q Q, Cheng X F, Jiang D B, Liu H J, Peng H S 2003 Acta Phys. Sin. 52 87 (in Chinese) [彭志涛, 景峰, 刘兰琴, 朱启华, 陈波, 张昆, 刘华, 张清泉, 程晓峰, 蒋东镔, 刘红婕, 彭翰生 2003 物理学报 52 87]
[24] Qiao Y T 1991 Gradient Index Optics (1st Ed.) (Beijing: Science Press) p170 (in Chinese) [乔亚天 1991 梯度折射率光学 (第一版) (北京: 科学出版社) 第170页]
[25] Dyshko A L, Lugovoi V N, Prokhorov A M 1967 ZhETF Pis’ma 6 655
[26] Garmire E, Chiao R Y, Townes C H 1966 Phys. Rev. Lett. 16 347
[27] Wanger W G, Haus H A, Marburger J H 1968 Phys. Rev. 175 256
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[1] Askar’yan G A 1962 Soviet. Phys. JETP 15 1088
[2] Chiao R, Garmire E, Townes C H 1964 Phys. Rev. Lett. 13 479
[3] Kelley P L 1965 Phys. Rev. Lett. 15 1005
[4] Shen Y R (Translated by Gu S J) 1987 The Principles of Nonlinear Optics (1st Ed.) (Beijing: Science Press) p327 (in Chinese) [沈元壤著 (顾世杰译) 1987 非线性光学原理 (上册) (北京: 科学出版社) 第327页]
[5] Hercher M 1964 J. Opt. Soc. Am. 54 563
[6] Braun A, Korn G, Liu X, Du D, Squier J, Mourou G 1995 Opt. Lett. 20 73
[7] Liu W, Kosareva O, Golubtsov I S, Iwasaki A, Becker A, Kandidov V P, Chin S L 2003 Appl. Phys. B 76 215
[8] Agrawal G P (Translated by Jia D F, Yu Z H) 2002 Nonlinear Fiber Optics, Third Edition and Applications of Nonlinear Fiber Optics (1st Ed.) (Beijing: Publishing House of Electronics Industry) p33 (in Chinese) [阿戈沃著 (贾东方, 余震虹译) 2002 非线性光纤光学原理及应用 (第一版) (北京: 电子工业出版社) 第33页]
[9] Hardin R H, Tappen E D 1975 SIAM Rev. Chronicle. 15 423
[10] Sinkin O V, Holzlöner R, Zweck J, Menyuk C R 2003 J. Lightwave Technol. 21 61
[11] Godoy-Rubio R, Romero-García S, Ortega-Moñux A, Wangümert-Pérez J G 2011 J. Opt. Soc. Am. B 28 2142
[12] Feit M, Fleck J 1988 J. Opt. Soc. Am. B 5 633
[13] Chi S, Guo Q 1995 Opt. Lett. 20 1598
[14] Li Z D, Guo Q, Lin W G 1999 Chin. J. Lasers 26 711 (in Chinese) [李忠东, 郭旗, 林为干 1999 中国激光 26 711]
[15] Wen S H 2001 Ph. D. Dissertation (Shanghai: Graduate University of Chinese Academy of Sciences) (in Chinese) [文双春 2001 博士学位论文 (上海: 中国科学院研究生院)]
[16] Wen S C, Fan D Y 2000 Acta Phys. Sin. 49 460 (in Chinese) [文双春, 范滇元 2000 物理学报 49 460]
[17] Wen S C, Fan D Y 2000 Acta Phys. Sin. 49 1282 (in Chinese) [文双春, 范滇元 2000 物理学报 49 1282]
[18] Censor D 1977 Phys. Rev. A 16 1673
[19] Kasparian J, Solle J, Richard M, Wolf J P 2004 Appl. Phys. B 79 947
[20] Dragoman D 1996 Appl. Opt. 35 4142
[21] Zhang Y C, Lü B D 2004 Opt. Lett. 29 2710
[22] Gao H H, Tian L, Zhang B, Barbastathis G 2010 Opt. Lett. 35 4148
[23] Peng Z T, Jing F, Liu L Q, Zhu Q H, Chen B, Zhang K, Liu H, Zhang Q Q, Cheng X F, Jiang D B, Liu H J, Peng H S 2003 Acta Phys. Sin. 52 87 (in Chinese) [彭志涛, 景峰, 刘兰琴, 朱启华, 陈波, 张昆, 刘华, 张清泉, 程晓峰, 蒋东镔, 刘红婕, 彭翰生 2003 物理学报 52 87]
[24] Qiao Y T 1991 Gradient Index Optics (1st Ed.) (Beijing: Science Press) p170 (in Chinese) [乔亚天 1991 梯度折射率光学 (第一版) (北京: 科学出版社) 第170页]
[25] Dyshko A L, Lugovoi V N, Prokhorov A M 1967 ZhETF Pis’ma 6 655
[26] Garmire E, Chiao R Y, Townes C H 1966 Phys. Rev. Lett. 16 347
[27] Wanger W G, Haus H A, Marburger J H 1968 Phys. Rev. 175 256
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