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)

Preface

What 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. a

A 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 to

rest once more. This cycle repeats itself again and again. It would repeat infinitely if

there 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.

1

Fig. 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 wave