Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 683 Waves,modes,communications, and optics:a tutorial DAVID A.B.MILLER 1.INTRODUCTION The idea of modes is common in the world of waves,especially in optics.Modes are very useful in simplifying many problems.But,there is much confusion about them. Are modes“resonances'”?Are they“beams"?Do they have to stay the same“shape"? Are they“communication channels'”?How do we“count'”modes?Are they properties of space or of objects such as scatterers?Just what is the definition of a mode?The purpose of this paper is to sort out the answers to questions like these,and to clarify and extend the idea of "modes."In particular,we want to use them for describing waves in communications and in describing sophisticated optical devices.Such ap- plications are increasingly important:communications may require mode-or space- division multiplexing to increase capacity,and we are able to fabricate progressively more complex optical devices with modern micro-and nano-fabrication. 1.1.Modes and Waves At their simplest,modes can be different shapes of waves.Some such modes arise naturally in waveguides and resonators;these modes are well understood and are taught in standard texts (see,e.g.,[1-4]).A key benefit of modes is that,when we choose the right ones,problems simplify;instead of describing waves directly as their values at each of a large number of points,we can just use the amplitudes of some relatively small number of modes.But when we want to use modes to under- stand communications with waves more generally,or when we want to describe some linear optical device or object economically using modes,we need to move beyond the ideas of just resonator or waveguide modes.Specifically,we can introduce the ideas of communications modes in communicating with waves [5]and mode-converter basis sets [6,7]in describing devices.These modes are not yet part of standard texts,nor is there even any broad and deep introduction to them.Further,many of their details and applications are not yet discussed in the literature. The reason for writing this paper is to provide exactly such an introduction.As well as sorting out the ideas of modes generally,we explain the physics of these additional forms of modes,which brings clearer answers to our opening questions above. We show how these ideas are supported by powerful and ultimately straightforward mathematics.We introduce novel,useful,and fundamental results that follow.This approach resolves many confusions.It reveals powerful concepts and methods,gen- eral limits,new physical laws,and some simple and even surprising results.It works over a broad range of waves,from acoustics,through classical microwave electromag- netism,to quantum-mechanical descriptions of light. 1.2.Idea of Modes One subtle point about modes is that it can be difficult to find a definition or even a clear statement of what they are.We should clarify this now. Modes are particularly common in describing oscillations of physical objects and sys- tems.Simple examples include a mass on a spring,or waves on a string,especially one with fixed ends.In these cases,an informal definition of an oscillating mode is that it is a way of oscillating in which everything that is oscillating is oscillating at the sameWaves, modes, communications, and optics: a tutorial DAVID A. B. MILLER 1. INTRODUCTION The idea of modes is common in the world of waves, especially in optics. Modes are very useful in simplifying many problems. But, there is much confusion about them. Are modes “resonances”? Are they “beams”? Do they have to stay the same “shape”? Are they “communication channels”? How do we “count” modes? Are they properties of space or of objects such as scatterers? Just what is the definition of a mode? The purpose of this paper is to sort out the answers to questions like these, and to clarify and extend the idea of “modes.” In particular, we want to use them for describing waves in communications and in describing sophisticated optical devices. Such ap￾plications are increasingly important: communications may require mode- or space￾division multiplexing to increase capacity, and we are able to fabricate progressively more complex optical devices with modern micro- and nano-fabrication. 1.1. Modes and Waves At their simplest, modes can be different shapes of waves. Some such modes arise naturally in waveguides and resonators; these modes are well understood and are taught in standard texts (see, e.g., [1–4]). A key benefit of modes is that, when we choose the right ones, problems simplify; instead of describing waves directly as their values at each of a large number of points, we can just use the amplitudes of some relatively small number of modes. But when we want to use modes to under￾stand communications with waves more generally, or when we want to describe some linear optical device or object economically using modes, we need to move beyond the ideas of just resonator or waveguide modes. Specifically, we can introduce the ideas of communications modes in communicating with waves [5] and mode-converter basis sets [6,7] in describing devices. These modes are not yet part of standard texts, nor is there even any broad and deep introduction to them. Further, many of their details and applications are not yet discussed in the literature. The reason for writing this paper is to provide exactly such an introduction. As well as sorting out the ideas of modes generally, we explain the physics of these additional forms of modes, which brings clearer answers to our opening questions above. We show how these ideas are supported by powerful and ultimately straightforward mathematics. We introduce novel, useful, and fundamental results that follow. This approach resolves many confusions. It reveals powerful concepts and methods, gen￾eral limits, new physical laws, and some simple and even surprising results. It works over a broad range of waves, from acoustics, through classical microwave electromag￾netism, to quantum-mechanical descriptions of light. 1.2. Idea of Modes One subtle point about modes is that it can be difficult to find a definition or even a clear statement of what they are. We should clarify this now. Modes are particularly common in describing oscillations of physical objects and sys￾tems. Simple examples include a mass on a spring, or waves on a string, especially one with fixed ends. In these cases, an informal definition of an oscillating mode is that it is a way of oscillating in which everything that is oscillating is oscillating at the same Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 683