耦合模理论

耦合模理论及其在微波和光纤技术中的应用（研究生课程用）钱景仁中国科学技术大学二零零五年目录绪言 (Preface) (1)第一章耦合模的一般理论§1.1 耦合模方程 (6)§1.2 强耦合与弱耦合 (11)§1.3 周期性耦合 (18)§1.4 耦合模与简正模 (29)§1.5 缓变参数情况下本地简正模广义理论 (33)§1.6 理想模、本地简正模和超本地简正模 (37)§1.7 耦合器应用举例 (42)§1.8 临界界面附近和稳相点附近的耦合模方程 (46)第二章闭合波导中的耦合模问题§2.1 介质填充波导 (51)§2.2 缓变表面阻抗和阻抗微扰 (59)§2.3 弯曲波导 (64)第三章光纤中的耦合模问题§3.1 光纤中的简正模式 (68)§3.2 耦合模理论的推广 (80)§3.3 非理想光纤的耦合模方程 (81)§3.4 用闭合波导理论来研究开波导 (86)第四章 螺旋光纤及弯曲光纤§4.1 螺旋光纤的耦合模分析 (89)§4.2 单模传输条件下的螺旋光纤 (93)§4.3 弯曲光纤 (98)第五章耦合功率方程§5.1多模波导和多模光纤的传输特性 (104)§5.2 多模波导中的耦合功率方程 (105)§5.3 多模光纤传输中的耦合功率方程 (107)中文参考文献 (109)英文参考文献 (110)PrefaceWhat is the coupled-mode theory? Is it a common theory in physics?Waves and vibration phenomena are popular in physics as we know such as mechanical vibrations, acoustic waves, light waves, microwaves and radio waves. Furthermore, connection or coupling among systems is also a general rule in universe. Everything presupposes the existence of some other thing. Cause-effect relations and action-reaction relations are generally existed among systems in the universe.It is obvious that there aren’t any ideal waves which exist independently and do not change their amplitudes and directions. A real wave or vibration is always connected with a source or other waves. Now, it is necessary to describe how these waves or vibrations (oscillations) couple to each other, and how their amplitudes change with the time or the distance. To illustrate the principle of the coupling between waves or vibrations (oscillations), let’s take pendulums as an example.Fig. aA pendulum can vibrate, that is to say it swings from side to side. We can give it a push and then it will vibrate at a fixed speed or at a certain frequency. If two pendulums with same frequency are hung on a string and one of them is set swinging as shown in Fig. a, it will swing less and less until it stops altogether, while the other pendulum will swing higher and higher until it reaches a maximum. Then the process will be reversed until the first pendulum reaches a maximum and the second comes torest once more. This cycle repeats itself again and again. It would repeat infinitely ifthere were no losses in the system.This is a typical experiment performed in most early physics courses. I had done it when I was in middle school.1Fig. b Frequencies are the same. Fig. c Frequencies are different.If these two pendulums have different frequencies, then transfer of energy between them will not be complete, and the first pendulum will not stop in the process. We can plot a graph to express the process as shown in Fig.b and Fig.c. The abscissa represents the time, and the ordinate A represents the amplitude of each pendulum. If the initial conditions at t =0 are as follows:()()1201,00A A ==,We can see the variations of the amplitudes of the two coupled pendulums in Fig.b and Fig.c, respectively, when their frequencies are the same and different. The time spacing between two adjacent maxima (or minima) is the period of the process, which is determined by the coupling between the two pendulums. The stronger the coupling is, the shorter the period is. The coupling between the two pendulums is caused by the fact that the pendulums are connected to a same string, and any vibration of one of the pendulums will have an effect on the other through the string.It has been recognized that coupled transmission lines, coupled electrical circuits, coupled optical fibers and coupled waveguides are analogous to coupled pendulums. The variations of the amplitudes of waves are the same as shown in the figures, but now the abscissa represents distance instant of time.Sometimes the coupling is not between the same kind of waves or oscillations, for example, in a traveling wave tube, a space-charge wave and an electromagnetic wavecouple to each other. In a crystal, an electrical vibration will cause a mechanical (or acoustic) vibration and vice versa.There should be some general rules or there is a generalized theory to describe these coupling problems. It is the so called coupled-mode theory. Here, mode means one of the models of wave forms.In the theory, all the coupled-mode or coupled-vibration problems are formulated by a set of coupled-mode equations, which are simultaneous differential equations of first order with variable or constant coefficients. In case of two modes, they can be written as follows:()()()()()()11122221j j j j dA z A z cA z dz dA z A z cA z dz ββ=−+=−+Where i β and c are functions of z in general case.When n modes or waves should be considered in a coupling problem, n differential equations will be used instead of two.A common method in electromagnetic theory is the modal approach in which the normal modes of the system (those fields which propagate unchanged except in phase) are found. This involves solving the wave equation adapted to the particular geometry of the system, and matching solutions at the boundaries to give the normal modes or eigensolutions. Any field of the system can then be expanded in terms of the normal modes, with the expansion coefficients determined by certain boundary conditions e.g. initial conditions. This modal-expansion or eigenvector method is physically intuitive and straightforward in principle, but modal solutions of the wave equation can only be found for a limited number of ideal systems of relatively simple geometry, including slabs and circular cylinders.Coupled-mode theory attempts to preserve the concept of modes for non-ideal systems in which an exact modal solution is not possible but where the normal modes of a reference system of simple geometry are known. These modes, in general, form a complete set so that they can be used to expand the fields of the non-ideal system.Because they do not satisfy the boundary conditions of the non-ideal system, the modes coupled or exchange power as they propagate. To derive the coupled-mode equations, Maxwell’s equations are transformed to those which determine how the individual mode amplitudes vary as a function of the parameters of the system. There have been several methods of coupled-mode analysis to formulate the coupled-mode equations. In the early times, people used to start directly from Maxwell’s equations along with the boundary conditions to derive these equations. Later, many other methods were utilized, such as using reciprocity theorem, starting from a Green function or stimulating equations of waveguides, someone also used variation method and perturbation approach, all these are substantial agreement.The method of coupled-modes is most useful when the deviation of the non-ideal system from the known reference system is not too great e.g. small deviations in refractive index or small deformation of cross-section. Although the imperfections may be small they can still produce marked effects, such as total transfer of power from one mode to the other in a waveguide or one waveguide to another. Coupled-mode theory has also been used to treat a variety of problems, including the cross-sectional deformation of waveguides. In many of the problems where the power transfer between modes is small, solutions can also be obtained by other techniques. However, coupled-mode theory has particular application to systems in which a large fraction of modal power may be transferred to other modes, as in the case of neighbouring waveguides in which complete transfer of power between waveguides can take place. This is unique for coupled-mode theory.The primary idea of the coupled-mode theory was first introduced by Pierce in 1940’s, when he worked on microwave electronic devices. Later, this idea was extended its use to the waveguide transmission by Miller and then the theory was fully developed. Recently, the theory has been widely used to solve optical fiber transmission problems and fiber gratings. On the other hand, the coupled-mode theory supervises the practice and many new coupling principles have been discovered. According them, a variety of devices have been designed, such as mode transducers, broadband optical fiber couplers and etc.A lot of coupling problems involving optics, acoustics and microwaves have been being solved by scientists of many countries, including Chinese scientists. Prof. Huang Hong-Chia, vice-president of Shanghai University, has made important contributions to coupled-mode theory. Some of his papers are listed in the end of this book for reference.In this book, the first chapter begins with the coupled-mode equations and is followed by many treatments to solve these equations. In Chapter 2, many typical coupled-mode problems in closed waveguides are solved. Those all problems will lead to the coupled-mode equations and then the coupling coefficients are derived. Chapter 3 begins with a discussion of the normal modes in optical fibers. The remainder of the chapter deals with coupling between these normal modes in imperfect optical fibers. In Chapter 4 helical fibers and bending fibers are studied. In the fifth chapter the coupled power theory is introduced, it consists of Pierce’s theory and Marcuse’s theory which are used in waveguide and optical fiber transmission, respectively.On the whole, coupled-mode theory is a general theory. Mathematically, it bases on the expansion theorem of eigen-functions, the existence of expansion in terms of eigen-functions makes the theory to be carried out. The mathematic areas in the theory are differential equations and linear algebra.第一章 耦合模的一般理论在这一章中，将首先从一般概念出发，得到耦合模方程。