本篇論文中，使用數值方法分析空間光孤子如何在Kerr非線性、非局域性介質，及在與光傳播方向垂直的橫截面中，具有週期性排列的光晶結構下之傳播。並且探討其穩定性和移動性。我們發現了，除在能帶結構中的全反射區裡，非局域效應會穩定光孤子的傳播外，在布拉格禁止帶當中所生成的光孤子，更能夠明顯的降低一維亮光孤子的不穩定性，且增加其移動的能力。在全反射區，慢慢增加非局域效應，光孤子就能夠脫離原本像是被週期性位能束縛的位井，跑到下一個位井當中。而調變非局域效應參數能加增光孤子的可移動性。在布拉格禁止帶當中，這樣的效應會表現的更加明顯。這樣的發現，也說明了光孤子能夠在晶格當中幾乎沒有輻射損耗的跨過晶格的位障。

In this thesis,we analyze the existence,stability and mobility of gap solitons in photonic crystals with diffusion mechanism of the nonlinearity numerically.For the bands of Bragg gap,solitons with nonlocal effects are more stabilized and become more movable due to the combinations of non-locality effect and the oscillation tails of the wave packets. We show that gap solitons can revive an elastic-like collision even in the photonic systems due to non-locality.

[1] J.S.Russell,in“14th meeting of the British Association Reports”,York(1844).
[2] D. J. Korteweg and F. de Vries, ”On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves.” Philosophical Magazine, 39, 422–443 (1895).
[3] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform,SIAM, Philadelphia (1981).
[4] Yuri S. Kivshar, Govind P. Agrawal. Optical solitons : from fibers to photonic crystals (Academic Press, London,2003).
[5] Govind P. Agrawal. Nonlinear fiber optics (San Diego : Academic Press, 2001).
[6] R. Y. Chiao, E. Garmire and C. H. Townes, Phys.Rev.Lett.,13,479
(1964).
[7] W.Krolikowski et al. , phys.Rev. E 64, 016612 (2001);
J.Wyller et al., Phys. Rev. E 66, 066615 (2002);
W.Krolikowski et al., J.Opt. B 6, S288 (2004);
M.Peccianti,C.Conti,and G.Assanto,Phys.Rev. E 68 ,R025602 (2003).
[8] S.K.Turitsyn, Theor. Math. Phys. (Engl. Transl.) 64,797 (1985);O.Bang et al., Phys.Rev E 66, 046619 (2002).
[9] D.Neshev et al., Opt.Lett. 26, 1185 (2001);
W.Krolikowski,O.Bang,and J.Wyller, Phys.Rev.E 70, 036617 (2004).
[10] A.I.Yakimenko,Y.A.Zaliznyak,and Y.Kivshar,Phys.Rev E 71,065603(R)(2005).;B.Brieds et al.,Opt.Express 13, 435 (2005).
[11] L.F.Mollenauve et al., Phys.Rev.Lett. 45 ,1095 (1980).
[12] C. Kittle, Introduction to Solid State Physics, John wiley (1996).
[13] J.A.C. Weideman and S.C.Reddy, A MATLAB differentiation matrix
suite, ACM Trans. Math. Software, to appear
[14] Lloyd N.Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathemetics.
[15] B. T. Polyak , Journal of Mathematical Sciences, Vol. 133, No. 4, 2006
[16] J.W.Cooley and J.W. Tukey, Math. Comput. 19, 297(1965)
[17] Zhiyong Xu, et al. Phys.Rev.Lett. 95,113901(2005)
[18] Yuri S.Kivshar and David K.Campbell , Phys.Rev.E 48 , 3077(1993)