波导模式基本理论及其应用
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摘要
随着光纤通信领域中波分复用技术的飞速发展，对各种高性能的分离光学元件和集成光学元件的市场需求，使得集成光学和光波导理论成为了该技术领域的热点研究课题。在这一领域中，存在着理论分析和数值计算两大类解决问题的方法。当人们试图通过理论分析的方法解决问题得不到理想结果的情况下，就采用数值模拟的方法来获得问题的求解。数值方法的基础是理论分析，而且数值分析只能给出特定条件下的结果，其结果具有评价性能参考意义，不能从根本上提供解决或者优化问题的方法。因此理论分析方法尤为重要。但是当前的理论分析方法并不完善，存在一些问题。本文主要针对目前光波导模式理论中不太被涉及到的模式完备性理论做了论述，并在解决实际问题方面做了一些探究。
本文首先从泛函分析的方法出发，研究了矢量偏微分算子本征函数系的性质，得到了矢量偏微分算子正交完备归一化本征函数的存在证明，研究了Maxwell方程组的情况，证明了电磁场在均匀和非均匀介质条件下的正交本征函数系的完备性质。接着推导了阶跃折射率平面介质波导和圆光纤的正交模式——包括束缚模式和辐射模式。由于从一个空间到另外一个空间的光束传播伴随着界面上各个模式能量之间的耦合，作为应用，本文介绍了完备性在三个情况下的应用：平面波导出射光束的衍射性质、波导的横向和纵向耦合以及标量光栅的测试。其中平面波导的出射光束衍射性质是光从波导模式到自由空间谐波耦合，波导的横向耦合和纵向耦合是从一种波导的模式耦合到另外一种波导模式的耦合，标量光栅可看作从一个谐波空间到另外一个谐波空间的耦合。
最后作为验证的实验，本文完成了波导—光纤耦合的第一部分准备工作：V型槽的制作。本实验为波导—光纤耦合提供了对准的条件，为进一步实验做好了准备。
As the development of wavelength division multiplexer and demultiplexer in optical communication system, the requirement for high performance of resolution elements and integrated circuits make integrated optics and waveguide theory the hot spot for research. In this area, theoretical analysis and numerical simulation are two basic methods. After the fail of theoretical analysis, people turn to numerical simulation. But numerical simulation is on the basis of theoretical analysis and can give the result of specific conditions. But the results it gives can only look as the standard for judgment, and cannot give us the essential method to solve problems or optimize current result. So the theoretical method is more important. But there still exist kinds of imperfectness. This thesis tries to modify the perfect set of waveguide mode theory and gives several examples of its application.
Firstly, from the functional analysis, the eigenfunction series of vectorial partial differential operator is studied. The property of complete space is gotten. And this result is extended to the eigenfunction of Maxwell's equations. Then the mode series of step index planar dielectric waveguide and circular optical fiber are studied, including propagation modes and radiation modes. As application, there are three examples: the emergent wave from planar waveguide to free space, the transverse and longitudinal coupling of waveguide and the measurement of scalar gratings. The emergent wave from planar waveguide to free space means the modes of waveguide coupling to the modes of free space, the transverse and longitudinal coupling of waveguide means the modes of two waveguide coupling between each other, and scalar gratings means the modes coupling between one kind of free space to another. At the end of this thesis, a experiment-etch V-grooves on silicon is presented. This is prepared for the fixation of fiber to waveguide, the further experiment to study the coupling between optical fiber and waveguide.
本文首先从泛函分析的方法出发，研究了矢量偏微分算子本征函数系的性质，得到了矢量偏微分算子正交完备归一化本征函数的存在证明，研究了Maxwell方程组的情况，证明了电磁场在均匀和非均匀介质条件下的正交本征函数系的完备性质。接着推导了阶跃折射率平面介质波导和圆光纤的正交模式——包括束缚模式和辐射模式。由于从一个空间到另外一个空间的光束传播伴随着界面上各个模式能量之间的耦合，作为应用，本文介绍了完备性在三个情况下的应用：平面波导出射光束的衍射性质、波导的横向和纵向耦合以及标量光栅的测试。其中平面波导的出射光束衍射性质是光从波导模式到自由空间谐波耦合，波导的横向耦合和纵向耦合是从一种波导的模式耦合到另外一种波导模式的耦合，标量光栅可看作从一个谐波空间到另外一个谐波空间的耦合。
最后作为验证的实验，本文完成了波导—光纤耦合的第一部分准备工作：V型槽的制作。本实验为波导—光纤耦合提供了对准的条件，为进一步实验做好了准备。
As the development of wavelength division multiplexer and demultiplexer in optical communication system, the requirement for high performance of resolution elements and integrated circuits make integrated optics and waveguide theory the hot spot for research. In this area, theoretical analysis and numerical simulation are two basic methods. After the fail of theoretical analysis, people turn to numerical simulation. But numerical simulation is on the basis of theoretical analysis and can give the result of specific conditions. But the results it gives can only look as the standard for judgment, and cannot give us the essential method to solve problems or optimize current result. So the theoretical method is more important. But there still exist kinds of imperfectness. This thesis tries to modify the perfect set of waveguide mode theory and gives several examples of its application.
Firstly, from the functional analysis, the eigenfunction series of vectorial partial differential operator is studied. The property of complete space is gotten. And this result is extended to the eigenfunction of Maxwell's equations. Then the mode series of step index planar dielectric waveguide and circular optical fiber are studied, including propagation modes and radiation modes. As application, there are three examples: the emergent wave from planar waveguide to free space, the transverse and longitudinal coupling of waveguide and the measurement of scalar gratings. The emergent wave from planar waveguide to free space means the modes of waveguide coupling to the modes of free space, the transverse and longitudinal coupling of waveguide means the modes of two waveguide coupling between each other, and scalar gratings means the modes coupling between one kind of free space to another. At the end of this thesis, a experiment-etch V-grooves on silicon is presented. This is prepared for the fixation of fiber to waveguide, the further experiment to study the coupling between optical fiber and waveguide.
引文
[1]吴重庆 光波导理论[M]．北京：清华大学出版社，2001．3
[2]Allan W．Snyder，John D．Love，光波导理论[M]．北京：人民邮电出版社，1991．6
[3]廖先炳，跨世纪的集成光学[J]．世界产品与介绍，2002，2：9
[4]陈益新，光器件集成化是光网络发展的必由之路[J]．网络电信，2003，1：38～42
[5]马树坤，程万君，苑立波，光波导理论中的若干基本问题[J]．黑龙江大学自然科学学报，1995，12(4)：67～71
[6]Erwin Kreyszig，泛函分析引论及应用[M]，北京重庆出版社，1986．6
[7]宋文淼，矢量偏微分算子[M]，北京：科学出版社，1999．4
[8]F．W．拜仑，R．W．富勒，物理学中的数学方法[M]，北京：科学出版社，1984．12
[9]陈军，现代光学技术(电磁篇)[M]，杭州：浙江大学出版社，1996
[10]文舸一，电磁理论的新进展[M]，北京：国防工业出版社，1999
[11]蔡伯荣，集成光学[M]，成都：电子科技大学出版社，1990
[12]马养武，王静环，包成芳，鲍超，光电子学[M]，杭州：浙江大学出版社，1999
[13]D．Marcuse，介质光波导理论[M]，北京：人民邮电出版社，1982
[14]王子华，李明生，对称平面波导辐射模的自由空间模近似及应用[J]，应用科学学报，1999，17(2)：194～198
[15]丁浩，多层平面介质波导辐射模[J]，光学学报，1994，14(12)：1233～1239
[16]刘德森，高应俊，变折射率介质的物理基础[M]，北京：国防工业出版社，1991
[17]Walter K.Kahn, Shuwen Yang, Hamiltonian Analysis of Beams in an Optical Slab Guide[J], J.Opt.Soc.Am,1983,73(5):684～690
[18]Xuejin Wen, Bin Lin, Fluid Model of Waveguide Transverse coupling[C],Proceedings of the 8~(th) Optoelectronics and Communication Conference, Shanghai,2003
[19]Y. Hibino. An Array of Photonic Filtering Advantages: An Array of Photonic Filtering Advantages Arrayed-Waveguide-Grating Multi/Demultiplexers for Photonic
Networks[J]. IEEE Circuits and Devices Magazine, 2000, 16(6): 21～27
[20]雷红兵，欧海燕，杨沁清，胡雄伟，余金中，王启明．阵列波导光栅复用／解复用器光栅孔径对器件性能影响的数值分析[J]．半导体学报，2000，21(4)：359～364
[21]K. A. McGreer. Diffraction from Concave Gratings in Planar Waveguides[J].IEEE Photonics Technology Letters, 1995, 7(3): 324～326
[22]赵春华，朱祖华，陈抗生，杨易．凹面光栅型波长解复用器的标量波动分析[J]．光学学报，1998，18(3)：308～313
[23]J. D. Gaskill. Linear System, Fourier Transforms[M], and Optics. John Wily & Sons, Inc., New York, 1978:385～390
[24]M. Born, E. Wolf, Principles of Optics, 6th Edition, Pergamon Press[M], Oxford,1980:412～414
[25]W．T，Tsang主编，江剑平等译．半导体集成光电子学[M]．北京：电子工业出版社，清华大学出版社，1993：4～10
[26]A.E.Siegman. Lasers[M], California, University Science Books[M], Mill Valley, 1988:558～922
[27]A.E.Siegman. New developments in laser resonators[A], Proc SPIE, Optical Resonators[C], 1990, 1224:2～20
[28]曹清，邓锡铭，郭弘，非傍轴光束的光束质量因子．Ⅰ．定义[J]，光学学报，1996，16(9)1217～1222
[29]曹清，邓锡铭，非傍轴光束的光束质量因子．Ⅱ．特性分析[J]，光学学报，1996，16(10)1345～1349
[30]M.A.Porras. Non-paraxial vectorial moment theory of light beam propagation[J], Opt. Commun, 1996, 127:79～95
[31]Guoquan Zhou, Daomu Zhao, Jinxin Xu, Shaomin Wang. Semiconductor laser with beam quality factor M2<1 [J], Opt Commun, 2001, 187:395～399
[32]李红军，吴凡，刘恩海．计量光栅信号质量评价系统[J]．光电工程，2002，29(4)：35～38
[2]Allan W．Snyder，John D．Love，光波导理论[M]．北京：人民邮电出版社，1991．6
[3]廖先炳，跨世纪的集成光学[J]．世界产品与介绍，2002，2：9
[4]陈益新，光器件集成化是光网络发展的必由之路[J]．网络电信，2003，1：38～42
[5]马树坤，程万君，苑立波，光波导理论中的若干基本问题[J]．黑龙江大学自然科学学报，1995，12(4)：67～71
[6]Erwin Kreyszig，泛函分析引论及应用[M]，北京重庆出版社，1986．6
[7]宋文淼，矢量偏微分算子[M]，北京：科学出版社，1999．4
[8]F．W．拜仑，R．W．富勒，物理学中的数学方法[M]，北京：科学出版社，1984．12
[9]陈军，现代光学技术(电磁篇)[M]，杭州：浙江大学出版社，1996
[10]文舸一，电磁理论的新进展[M]，北京：国防工业出版社，1999
[11]蔡伯荣，集成光学[M]，成都：电子科技大学出版社，1990
[12]马养武，王静环，包成芳，鲍超，光电子学[M]，杭州：浙江大学出版社，1999
[13]D．Marcuse，介质光波导理论[M]，北京：人民邮电出版社，1982
[14]王子华，李明生，对称平面波导辐射模的自由空间模近似及应用[J]，应用科学学报，1999，17(2)：194～198
[15]丁浩，多层平面介质波导辐射模[J]，光学学报，1994，14(12)：1233～1239
[16]刘德森，高应俊，变折射率介质的物理基础[M]，北京：国防工业出版社，1991
[17]Walter K.Kahn, Shuwen Yang, Hamiltonian Analysis of Beams in an Optical Slab Guide[J], J.Opt.Soc.Am,1983,73(5):684～690
[18]Xuejin Wen, Bin Lin, Fluid Model of Waveguide Transverse coupling[C],Proceedings of the 8~(th) Optoelectronics and Communication Conference, Shanghai,2003
[19]Y. Hibino. An Array of Photonic Filtering Advantages: An Array of Photonic Filtering Advantages Arrayed-Waveguide-Grating Multi/Demultiplexers for Photonic
Networks[J]. IEEE Circuits and Devices Magazine, 2000, 16(6): 21～27
[20]雷红兵，欧海燕，杨沁清，胡雄伟，余金中，王启明．阵列波导光栅复用／解复用器光栅孔径对器件性能影响的数值分析[J]．半导体学报，2000，21(4)：359～364
[21]K. A. McGreer. Diffraction from Concave Gratings in Planar Waveguides[J].IEEE Photonics Technology Letters, 1995, 7(3): 324～326
[22]赵春华，朱祖华，陈抗生，杨易．凹面光栅型波长解复用器的标量波动分析[J]．光学学报，1998，18(3)：308～313
[23]J. D. Gaskill. Linear System, Fourier Transforms[M], and Optics. John Wily & Sons, Inc., New York, 1978:385～390
[24]M. Born, E. Wolf, Principles of Optics, 6th Edition, Pergamon Press[M], Oxford,1980:412～414
[25]W．T，Tsang主编，江剑平等译．半导体集成光电子学[M]．北京：电子工业出版社，清华大学出版社，1993：4～10
[26]A.E.Siegman. Lasers[M], California, University Science Books[M], Mill Valley, 1988:558～922
[27]A.E.Siegman. New developments in laser resonators[A], Proc SPIE, Optical Resonators[C], 1990, 1224:2～20
[28]曹清，邓锡铭，郭弘，非傍轴光束的光束质量因子．Ⅰ．定义[J]，光学学报，1996，16(9)1217～1222
[29]曹清，邓锡铭，非傍轴光束的光束质量因子．Ⅱ．特性分析[J]，光学学报，1996，16(10)1345～1349
[30]M.A.Porras. Non-paraxial vectorial moment theory of light beam propagation[J], Opt. Commun, 1996, 127:79～95
[31]Guoquan Zhou, Daomu Zhao, Jinxin Xu, Shaomin Wang. Semiconductor laser with beam quality factor M2<1 [J], Opt Commun, 2001, 187:395～399
[32]李红军，吴凡，刘恩海．计量光栅信号质量评价系统[J]．光电工程，2002，29(4)：35～38