The operation of many photonic devices is based on interactions between optical guided waves. We have discussed the electromagnetic analysis of the modes in individual planar and channel waveguides in Chapter 1. From that discussion, it is clear that solving Maxwell's equations rigorously for several coupled modes or waveguides is very difficult. Only approximate and numerical solutions are available. In this chapter, we will introduce several approximate electromagnetic techniques for analyzing the interactions of guided waves. These methods include the perturbation method and coupled mode analyses. Practical devices such as the grating filter, the directional coupler, the Y-branch coupler, the Mach–Zehnder modulator, and the multimode interference coupler will be discussed as specific examples. In addition, analysis of coupled waveguides as super modes of the total structure in the effective index approximation is presented. This analysis will allow us to view the interactions between coupled waveguides from another point of view.

In Chapter 1, we have shown that the guided-wave modes together with the radiation modes comprise a complete set of modes. In guided-wave devices, radiation modes are excited at any dielectric discontinuity. Rigorous modal analysis of propagation in a waveguide with varying cross-section in the direction of propagation should involve, in principle, all the modes. However, radiation modes usually fade away at some reasonable distance from the discontinuity. They are important only when radiation loss must be accounted for.

In order to create the electro-optical effects discussed in Chapter 3, a voltage is applied to the electrodes of the devices through electrical circuits to produce the electrical field. Most of the voltages that control the modulation, switching and signal processing functions are time varying, their frequency spectra range from MHz to tens of GHz. In analog applications, it is the frequency response of the device that is important whereas in digital applications, it is the time response of the device to a voltage (or current) pulse that is important. Pulse modulators are usually large signal devices. The time response of devices such as intensity modulation in a Mach–Zehnder or electro-absorption modulator is usually non-linear with respect to the magnitude of the applied voltage. Thus it is difficult to give a general discussion of the time response of electro-optical devices. However, pulses can be represented as a summation of their frequency components. Section 4.2.6 discusses the relation between frequency response and pulse propagation. Therefore, only the response of the devices to a time harmonic small voltage signal at different frequencies will be discussed in this book.

There are two major causes for frequency variation of the small signal response of electro-optical devices.

1. (1) The voltage across the device supplied by the electrical circuit is frequency dependent. For example, when the electrical source has a time harmonic variation, the fraction of the source voltage that appears across the device is frequency dependent. There is an electrical bandwidth of the voltage produced by the circuit driving the optoelectronic device.

2. […]

Fields in channel waveguides are confined to the vicinity of the core within a few μm in both the lateral and the depth directions. There are two main advantages of the lateral confinement of channel guided-wave modes:

1. (1) The RF electric field required to obtain an electro-optical effect such as electro-optic change of index or electro-absorption needs only to exist in a small region around the core. The required electric field in a small region can be achieved with just a moderate RF voltage applied to the electrodes. Furthermore, when the electrodes are fabricated parallel to the channel waveguides, the electro-optical change of index or electro-absorption produced by a propagating RF signal can be synchronized with the propagation of the guided wave in a traveling wave interaction as discussed in Chapter 4. Thus the electro-optical modulation at high frequencies may be carried out effectively.

2. (2) Most optoelectronic devices are eventually connected to single mode optical fibers. The optical field pattern of the channel waveguides can be designed such that it matches well with the field pattern of single mode optical fibers or tapered fibers, providing high efficiency transmission of optical power to and from the low loss single mode fibers.

Traditionally, guided-wave devices have been discussed in the literature according to the type of optical interactions they utilize, such as directional coupling or electro-absorption.