《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Waves, modes, communications, and optics - a tutoral aop-11-3-679

684 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial frequency.This is a sense in which a“mode”isa“way”or“manner'”of oscillation Musical instruments offer many other examples of such modes,as in standing waves in a pipe,or resonances in the vibrations of plates or hollow bodies.Such a mode will have a specific frequency of oscillation,and the amplitude of the vibration will take a specific physical form-it can be a function of position along the string or pipe or on the surface of some plate or body. The underlying mathematical idea of modes is associated with eigenfunctions or eigenvectors in linear physical systems;in oscillating systems or resonators,the func- tion that gives the amplitude of oscillation at each position is the eigenfunction,and the frequency (or often the square of the frequency)is the eigenvalue.Indeed,we can state a useful,general definition of a mode [8-10]: A mode is an eigenfunction of an eigenproblem describing a physical system. (1) Conventional resonator and waveguide modes are each the eigenfunctions of a single eigenproblem.The fixed "shape"of this oscillation amplitude inside the resonator is often thought of as the "mode"or eigenfunction in this sense.Waveguide modes use the same mathematics,but the concept here is that the transverse shape of the mode does not change as it propagates.An analogous informal definition of a propagating mode is that everything that propagates is propagating with the same wave vector,which also implies that the(transverse)shape does not change as it propagates.That transverse shape is the eigenfunction.Though such waveguide modes may well be modes of a specific frequency that we have chosen,the eigenvalue is typically a propagation con- stant or wavevector magnitude (or,again,often the square of this quantity). Before going any further,to support these ideas of modes,we need good notations; they should be general enough to handle everything we need,but they should suppress unnecessary detail.Wherever possible,we use a Dirac"bra-ket"notation,which op- erates at just such a useful level of abstraction.We introduce this notation progres- sively (see also [9)).In this notation a function can be represented by a"ket"or"ket vector,"written as lus)or l),for example.Linear operators,such as Green's func- tions or scattering operators,are represented by a letter,and here we will mostly use "sans serif'capital letters such as G and D.Most simply,we can think of kets as column vectors of numbers and the linear operators as matrices.Dirac notation imple- ments a convenient version of linear algebra equivalent to matrix-vector operations with complex numbers,and indeed such a matrix-vector view can be the simplest way to think about Dirac notation. 1.3.Modes as Pairs of Functions To handle communications and complex optical devices,we need to go beyond just resonator or waveguide modes;fortunately,though,we can use much of the same mathematics.The key mathematical difference between resonator and waveguide modes on the one hand and our new modes on the other is that communications modes and mode-converter basis sets each result from solving a singular-value decomposition (SVD)problem,which corresponds to solving two eigenproblems. The physical reason for having two such eigenproblems is because we are defining optimum mappings between two different spaces. For example,in communications,we may have sources or transmitters in one"source" volume and resulting waves communicated into another "receiving"volume

frequency. This is a sense in which a “mode” is a “way” or “manner” of oscillation. Musical instruments offer many other examples of such modes, as in standing waves in a pipe, or resonances in the vibrations of plates or hollow bodies. Such a mode will have a specific frequency of oscillation, and the amplitude of the vibration will take a specific physical form—it can be a function of position along the string or pipe or on the surface of some plate or body. The underlying mathematical idea of modes is associated with eigenfunctions or eigenvectors in linear physical systems; in oscillating systems or resonators, the func￾tion that gives the amplitude of oscillation at each position is the eigenfunction, and the frequency (or often the square of the frequency) is the eigenvalue. Indeed, we can state a useful, general definition of a mode [8–10]: A mode is an eigenfunction of an eigenproblem describing a physical system: (1) Conventional resonator and waveguide modes are each the eigenfunctions of a single eigenproblem. The fixed “shape” of this oscillation amplitude inside the resonator is often thought of as the “mode” or eigenfunction in this sense. Waveguide modes use the same mathematics, but the concept here is that the transverse shape of the mode does not change as it propagates. An analogous informal definition of a propagating mode is that everything that propagates is propagating with the same wave vector, which also implies that the (transverse) shape does not change as it propagates. That transverse shape is the eigenfunction. Though such waveguide modes may well be modes of a specific frequency that we have chosen, the eigenvalue is typically a propagation con￾stant or wavevector magnitude (or, again, often the square of this quantity). Before going any further, to support these ideas of modes, we need good notations; they should be general enough to handle everything we need, but they should suppress unnecessary detail. Wherever possible, we use a Dirac “bra-ket” notation, which op￾erates at just such a useful level of abstraction. We introduce this notation progres￾sively (see also [9]). In this notation a function can be represented by a “ket” or “ket vector,” written as jψSi or jϕRi, for example. Linear operators, such as Green’s func￾tions or scattering operators, are represented by a letter, and here we will mostly use “sans serif” capital letters such as G and D. Most simply, we can think of kets as column vectors of numbers and the linear operators as matrices. Dirac notation imple￾ments a convenient version of linear algebra equivalent to matrix-vector operations with complex numbers, and indeed such a matrix-vector view can be the simplest way to think about Dirac notation. 1.3. Modes as Pairs of Functions To handle communications and complex optical devices, we need to go beyond just resonator or waveguide modes; fortunately, though, we can use much of the same mathematics. The key mathematical difference between resonator and waveguide modes on the one hand and our new modes on the other is that communications modes and mode-converter basis sets each result from solving a singular-value decomposition (SVD) problem, which corresponds to solving two eigenproblems. The physical reason for having two such eigenproblems is because we are defining optimum mappings between two different spaces. For example, in communications, we may have sources or transmitters in one “source” volume and resulting waves communicated into another “receiving” volume 684 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 685 [Fig.1(a)].The solutions to our problem are then the set of optimum source functions in the source or input volume that couple,one by one,to the resulting optimal waves in the receiving or output volume;SVD solves for both of those sets of functions,and it is these two sets of functions that are the communications modes.So,a given com- munications mode is not one function but two. We can also view a communications mode as defining a communications "channel." A simple view of a "channel"is that,when we put an input in one end,the corre- sponding output comes of the other end,without "leaking"into any other such "channel,"as in the literal meaning of a channel as carrying a stream of water, separately from other such streams or channels.In the case of the communications modes,modulating the "source"function leads to an amplitude in the corre- sponding receiving wave;the "separateness"here is defined by some mathematical "orthogonality"of all the source functions and all the receiving functions,and we clarify this idea below.We will be able to have separate channels for information flow even if the actual waves are mixed in the space between the source and receiver.When we use the term "channels"we mean such independent "ways"for sending informa- tion from source to receiver.In this sense,a communications mode describes the physical carrier for such an information "channel." In practice we may only need to solve one of these two SVD eigenproblems,and we can then deduce the solutions to the other.But because we can view this through two eigenproblems,each of these sets of functions,one in the source space and one in the receiving space,therefore has all the useful mathematical properties of eigenfunctions, including this idea of"orthogonality";such properties have profound consequences for the physical interpretation and the mathematics that follows. 1.3a.Communications Modes Note immediately that,in this view, the communications mode is not the propagating wave (or what we will call the beam)between the source volume and receiver volume. Figure 1 (a)Source or input Receiving or output volume or space volume or space V g v) lx） Hs Hg (b)Source or input Receiving or output volume or space volume or space Device Vs 1)》 Hs Hg Conceptual view for (a)communications modes and (b)mode-converter basis sets.In both cases a source function lus)in a source or input volume Vs,or more generally in a mathematical (Hilbert)space Hs,results in a wave function )in a receiving or output volume VR,or more generally in a mathematical (Hilbert)space HR.In the commu- nications mode case(a)the coupling is through a Green's function operator Gsk as appropriate for the intervening medium between the spaces.In the mode-converter case (b),the coupling is through the action of a device (or scattering)operator D

[Fig. 1(a)]. The solutions to our problem are then the set of optimum source functions in the source or input volume that couple, one by one, to the resulting optimal waves in the receiving or output volume; SVD solves for both of those sets of functions, and it is these two sets of functions that are the communications modes. So, a given com￾munications mode is not one function but two. We can also view a communications mode as defining a communications “channel.” A simple view of a “channel” is that, when we put an input in one end, the corre￾sponding output comes of the other end, without “leaking” into any other such “channel,” as in the literal meaning of a channel as carrying a stream of water, separately from other such streams or channels. In the case of the communications modes, modulating the “source” function leads to an amplitude in the corre￾sponding receiving wave; the “separateness” here is defined by some mathematical “orthogonality” of all the source functions and all the receiving functions, and we clarify this idea below. We will be able to have separate channels for information flow even if the actual waves are mixed in the space between the source and receiver. When we use the term “channels” we mean such independent “ways” for sending informa￾tion from source to receiver. In this sense, a communications mode describes the physical carrier for such an information “channel.” In practice we may only need to solve one of these two SVD eigenproblems, and we can then deduce the solutions to the other. But because we can view this through two eigenproblems, each of these sets of functions, one in the source space and one in the receiving space, therefore has all the useful mathematical properties of eigenfunctions, including this idea of “orthogonality”; such properties have profound consequences for the physical interpretation and the mathematics that follows. 1.3a. Communications Modes Note immediately that, in this view, the communications mode is not the propagating wave (or what we will call the beam) between the source volume and receiver volume. Figure 1 Conceptual view for (a) communications modes and (b) mode-converter basis sets. In both cases a source function jψSi in a source or input volume VS, or more generally in a mathematical (Hilbert) space HS, results in a wave function jϕRi in a receiving or output volume VR, or more generally in a mathematical (Hilbert) space HR. In the commu￾nications mode case (a) the coupling is through a Green’s function operator GSR as appropriate for the intervening medium between the spaces. In the mode-converter case (b), the coupling is through the action of a device (or scattering) operator D. Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 685

686 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial Indeed,in general the beam will change shape as it propagates,and it is not itself the "eigenfunction"of the mathematical problem (though it is easily deduced from the ac- tual eigenfunctions in simple communication problems).In this SVD way of looking at communications,the jth communications mode is a pair of functions-s)in the source or input space,and in the receiving or output space.Explicitly,therefore, communications modes are pairs of functions-one in the source space and one in the receiving space. They are a set of communications mode pairs of functions-a pair ls)and o),a pair ls2)and lg2),and so on.To find these functions,we perform the SVD of the coupling operator Gsg between the volumes or spaces.For the communications prob- lems we consider first,this Gsk is effectively the free-space Green's function for our wave equation. 1.3b.Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device,we can consider how it scatters input waves to output waves [Fig.1(b)].By analyzing this also as an SVD problem,in this case of a device (or scattering)operator D,we can similarly deduce a set [11]of input source functions si))that couple one by one to a set of output wave functions{));these two sets of functions are the mode-converter basis sets. In this second case,we want to describe the device as one that converts from a specific input mode lus to the corresponding output mode l),and so on,for all such mode pairs;again,as in the case of communications modes,we think in terms of pairs of functions here,one in the source or input space,and one in the receiving or output space.We can consider these as mode-converter pairs-a pair si)and Ri),a pair lus2)and oR2),and so on,just as in the communications modes.In this way of look- ing at a linear optical device [6], any linear optical device can be viewed as a mode converter,converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space,giving the mode-converter pairs of functions. The device converts input mode lusi)to output mode R),input mode lus2)to output mode 2),and so on.In this case,though the mathematics is similar to the communications modes,this is more a way of describing the device,whereas the communications modes are a way of describing the communications channels from sources to receivers.For the device case,we may not have anything like a simple beam between the sources and receivers,but we do have these well-defined functions or "modes"inside the source space or volume and inside the receiving space or vol- ume.We could also view the mode-converter basis sets as describing the communi- cations modes“through”the device. In an actual physical problem for a device,there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12]without ever knowing exactly what the wave field is inside the device.Then we could know the mode-converter pairs as eigenfunctions without knowing the "beam";this point em- phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4.Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful

Indeed, in general the beam will change shape as it propagates, and it is not itself the “eigenfunction” of the mathematical problem (though it is easily deduced from the ac￾tual eigenfunctions in simple communication problems). In this SVD way of looking at communications, the jth communications mode is a pair of functions—jψSji in the source or input space, and jϕRji in the receiving or output space. Explicitly, therefore, communications modes are pairs of functions—one in the source space and one in the receiving space. They are a set of communications mode pairs of functions—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on. To find these functions, we perform the SVD of the coupling operator GSR between the volumes or spaces. For the communications prob￾lems we consider first, this GSR is effectively the free-space Green’s function for our wave equation. 1.3b. Mode-Converter Basis Sets When we change from thinking just about waves in free space to trying to describe a linear optical device, we can consider how it scatters input waves to output waves [Fig. 1(b)]. By analyzing this also as an SVD problem, in this case of a device (or scattering) operator D, we can similarly deduce a set [11] of input source functions fjψSjig that couple one by one to a set of output wave functions fjϕRjig; these two sets of functions are the mode-converter basis sets. In this second case, we want to describe the device as one that converts from a specific input mode jψSji to the corresponding output mode jϕRji, and so on, for all such mode pairs; again, as in the case of communications modes, we think in terms of pairs of functions here, one in the source or input space, and one in the receiving or output space. We can consider these as mode-converter pairs—a pair jψS1i and jϕR1i, a pair jψS2i and jϕR2i, and so on, just as in the communications modes. In this way of look￾ing at a linear optical device [6], any linear optical device can be viewed as a mode converter, converting from specific sets of functions in the input space one by one to specific corresponding functions in the output space, giving the mode-converter pairs of functions. The device converts input mode jψS1i to output mode jϕR1i, input mode jψS2i to output mode jϕR2i, and so on. In this case, though the mathematics is similar to the communications modes, this is more a way of describing the device, whereas the communications modes are a way of describing the communications channels from sources to receivers. For the device case, we may not have anything like a simple beam between the sources and receivers, but we do have these well-defined functions or “modes” inside the source space or volume and inside the receiving space or vol￾ume. We could also view the mode-converter basis sets as describing the communi￾cations modes “through” the device. In an actual physical problem for a device, there are ways in principle in which we could deduce the mode-converter pairs of functions by experiment [7,12] without ever knowing exactly what the wave field is inside the device. Then we could know the mode-converter pairs as eigenfunctions without knowing the “beam”; this point em￾phasizes that it can be more useful and meaningful to use the pairs of functions in the source and receiving spaces as the modes of the system rather than attempting to use the beam through the whole system as the way to describe it. 1.4. Usefulness of This Approach There are several practical and fundamental reasons why these pairs of functions are useful. 686 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial

Tutorial Vol.11,No.3/September 2019/Advances in Optics and Photonics 687 1.4a.Using Communications Modes In communications,we continually want larger amounts of useful bandwidth.This need is strong for wireless radio-frequency transmission [13],for optical signals in fibers [14-17]or free space [17-20],and even for acoustic information transmission [21-23].Recent progress in novel optical ways to separate different [16]and even arbitrary modes [24-29],including automatic methods [24-29],gives additional mo- tivation to consider the use of different modes (or "spatial degrees of freedom")in communications. Increasingly,therefore,we need to understand the spatial degrees of freedom in such communications and the limits in their use;a natural way to describe and quantify those is in terms of communications modes.Specifically, we can understand how to count the number of useful available spatial channels. Essentially,this can also be viewed as a generalization of the ideas of diffraction lim- its,and we will develop these ideas below.A key novel result is that this SVD approach gives a sum rule that bounds the number and strength of those channels. As we solve the problem this way,we can also unambiguously establish just exactly what the best channels are;we do not need to presume any particular form of these modes to start with.So,specifically,we do not need to analyze in terms of plane-wave “modes,.”Hermite--Gaussian or Laguerre-.Gaussian beams,optical“orbital”angular momentum(OAM)[19,20,30-32]"modes,"prolate spheroidals [33],arrays of spots, or any other specific family of functions;specifically, the SVD solution will tell us the best answers for the transmitting and receiving functions-the communications modes-and those will in general be none of the standard mathematical families of functions or beams. 1.4b.Using Mode-Converter Basis Sets In analyzing linear optical devices or scatterers, if we establish the mode-converter basis sets by solving the SVD problem,we will have the most economical and complete description of a device or scatterer. Essentially,we establish the "best"functions to use here,starting with the most im- portant and progressing to those of decreasing importance.An incidental and univer- sal consequence of this approach is that we realize that there is a set of independent channels through any linear scatterer (which are the mode-converter basis sets),and that we can describe the device com- pletely using those.The implications of the mode-converter basis sets go beyond sim- ple mathematical economy: Mode-converter basis functions have basic physical meaning and implications, giving fundamental results that can be economically and uniquely expressed using them. They allow us,for example,to write new versions and extensions of Kirchhoff's radi- ation laws [7],including ones that apply specifically and only to the mode-converter pairs,and to derive a novel modal version of Einstein's"A&B"coefficient argument on spontaneous and stimulated emission(Subsection 11.2).Such results suggest that this mode-converter basis set approach is deeply meaningful as a way to describe optical

1.4a. Using Communications Modes In communications, we continually want larger amounts of useful bandwidth. This need is strong for wireless radio-frequency transmission [13], for optical signals in fibers [14–17] or free space [17–20], and even for acoustic information transmission [21–23]. Recent progress in novel optical ways to separate different [16] and even arbitrary modes [24–29], including automatic methods [24–29], gives additional mo￾tivation to consider the use of different modes (or “spatial degrees of freedom”) in communications. Increasingly, therefore, we need to understand the spatial degrees of freedom in such communications and the limits in their use; a natural way to describe and quantify those is in terms of communications modes. Specifically, we can understand how to count the number of useful available spatial channels. Essentially, this can also be viewed as a generalization of the ideas of diffraction lim￾its, and we will develop these ideas below. A key novel result is that this SVD approach gives a sum rule that bounds the number and strength of those channels. As we solve the problem this way, we can also unambiguously establish just exactly what the best channels are; we do not need to presume any particular form of these modes to start with. So, specifically, we do not need to analyze in terms of plane-wave “modes,” Hermite–Gaussian or Laguerre–Gaussian beams, optical “orbital” angular momentum (OAM) [19,20,30–32] “modes,” prolate spheroidals [33], arrays of spots, or any other specific family of functions; specifically, the SVD solution will tell us the best answers for the transmitting and receiving functions—the communications modes—and those will in general be none of the standard mathematical families of functions or beams. 1.4b. Using Mode-Converter Basis Sets In analyzing linear optical devices or scatterers, if we establish the mode-converter basis sets by solving the SVD problem, we will have the most economical and complete description of a device or scatterer. Essentially, we establish the “best” functions to use here, starting with the most im￾portant and progressing to those of decreasing importance. An incidental and univer￾sal consequence of this approach is that we realize that there is a set of independent channels through any linear scatterer (which are the mode-converter basis sets), and that we can describe the device com￾pletely using those. The implications of the mode-converter basis sets go beyond sim￾ple mathematical economy: Mode-converter basis functions have basic physical meaning and implications, giving fundamental results that can be economically and uniquely expressed using them. They allow us, for example, to write new versions and extensions of Kirchhoff’s radi￾ation laws [7], including ones that apply specifically and only to the mode-converter pairs, and to derive a novel modal version of Einstein’s “A&B” coefficient argument on spontaneous and stimulated emission (Subsection 11.2). Such results suggest that this mode-converter basis set approach is deeply meaningful as a way to describe optical Tutorial Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics 687

688 Vol.11,No.3/September 2019/Advances in Optics and Photonics Tutorial systems.These mode-converter basis functions can also be identified in principle for a given linear object through physical experiments [7],independent of the mathematics. 1.4c.Areas of Research and Application This approach to waves,though not yet very widely known,has a history that goes back some decades,and already has many applications.The earliest,and very successful, application of eigenfunction approaches in waves is for laser resonators [34-36]with some related work in imaging [33].Such applications are special cases of the present approach in which the"source"and"receiver"functions are essentially mathematically the same.Following the introduction of the full SVD approach [5,37,38],there has been a broadening range of applications in wireless communications [13,39-51],where space-division multiplexing is increasingly an important option,r.f.imaging [52,53], electromagnetic scattering [54-64],optical systems [65-81],acoustic wave communi- cations [23],finding strong channels though strong scatterers [82-98](which is related to earlier work on electron transport though disordered media [97]),multiple-mode op- tical fibers for communications [81,99]and imaging [100-105],and free-space com- munications [18].This approach can also resolve paradoxes and confusions in counting available communications channels generally,such as whether OAM leads to more channels(and we discuss this below).The growing availability of optical systems that can generate complex and controllable devices [12,24-29,106-119]also means this SVD approach is practically accessible for more applications because we can generate sets of sources and can separate sets of waves,and SVD is also a good way to describe and even design those devices themselves [25].We give an extended discussion of this history and the wider literature in Appendix A. Aspects of this field have developed somewhat independently,and different authors therefore refer to similar concepts with different terminology.Our "device operator" or Green's function coupling operator between spaces is similar to the channel matrix in wireless communications [13].and the communications modes there are referred to [l3]as“eigenmodes of the channel”or“eigenchannels.”n optics,the“optical ei- genmodes"of [76-80]are similar to our communications modes,or,for more com- plex optical systems,the mode-converter basis sets.In work in channels through strong scatterers [81-92,96-98],the coupling operator (the "device"or "scattering" operator in our notation)is often called a "transmission"matrix (see,e.g.,[97])(with our mode-converter basis sets or communications modes through a scatterer known in that work as“optical eigenchannels'”or“transmission eigenchannels'”[97])or,some times,a"transfer matrix"[81].For consistency in this paper,we will use our notation, but the link to this other independent work and terminology is important to clarify. 1.5.Approach of This Paper Because the ideas here go beyond conventional textbook discussions,and because we are combining concepts and techniques that cross several different fields,the approach of this article is quite tutorial.Most algebra steps are written explicitly,and many "toy" examples illustrate the key steps and points.I have tried to write the main text so that it is readable,and with a progressive flow of ideas.I introduce core mathematical ideas in the main text,but relegate most other derivations and mathematics to appendices. This article has been written to be accessible to readers with a good basic undergraduate knowledge of mathematics and some physical science,such as would be acquired in a subject such as electrical engineering or physics or a discipline such as optics(for specific presumed background,see [120)),but I explicitly introduce all other required advanced mathematics and electromagnetism.Wherever possible,I take a direct approach in der- ivations,working from fundamental results,such as Maxwell's equations or core math- ematical definitions and principles,without invoking intermediate results or methods

systems. These mode-converter basis functions can also be identified in principle for a given linear object through physical experiments [7], independent of the mathematics. 1.4c. Areas of Research and Application This approach to waves, though not yet very widely known, has a history that goes back some decades, and already has many applications. The earliest, and very successful, application of eigenfunction approaches in waves is for laser resonators [34–36] with some related work in imaging [33]. Such applications are special cases of the present approach in which the “source” and “receiver” functions are essentially mathematically the same. Following the introduction of the full SVD approach [5,37,38], there has been a broadening range of applications in wireless communications [13,39–51], where space-division multiplexing is increasingly an important option, r.f. imaging [52,53], electromagnetic scattering [54–64], optical systems [65–81], acoustic wave communi￾cations [23], finding strong channels though strong scatterers [82–98] (which is related to earlier work on electron transport though disordered media [97]), multiple-mode op￾tical fibers for communications [81,99] and imaging [100–105], and free-space com￾munications [18]. This approach can also resolve paradoxes and confusions in counting available communications channels generally, such as whether OAM leads to more channels (and we discuss this below). The growing availability of optical systems that can generate complex and controllable devices [12,24–29,106–119] also means this SVD approach is practically accessible for more applications because we can generate sets of sources and can separate sets of waves, and SVD is also a good way to describe and even design those devices themselves [25]. We give an extended discussion of this history and the wider literature in Appendix A. Aspects of this field have developed somewhat independently, and different authors therefore refer to similar concepts with different terminology. Our “device operator” or Green’s function coupling operator between spaces is similar to the channel matrix in wireless communications [13], and the communications modes there are referred to [13] as “eigenmodes of the channel” or “eigenchannels.” In optics, the “optical ei￾genmodes” of [76–80] are similar to our communications modes, or, for more com￾plex optical systems, the mode-converter basis sets. In work in channels through strong scatterers [81–92,96–98], the coupling operator (the “device” or “scattering” operator in our notation) is often called a “transmission” matrix (see, e.g., [97]) (with our mode-converter basis sets or communications modes through a scatterer known in that work as “optical eigenchannels” or “transmission eigenchannels” [97]) or, some￾times, a “transfer matrix” [81]. For consistency in this paper, we will use our notation, but the link to this other independent work and terminology is important to clarify. 1.5. Approach of This Paper Because the ideas here go beyond conventional textbook discussions, and because we are combining concepts and techniques that cross several different fields, the approach of this article is quite tutorial. Most algebra steps are written explicitly, and many “toy” examples illustrate the key steps and points. I have tried to write the main text so that it is readable, and with a progressive flow of ideas. I introduce core mathematical ideas in the main text, but relegate most other derivations and mathematics to appendices. This article has been written to be accessible to readers with a good basic undergraduate knowledge of mathematics and some physical science, such as would be acquired in a subject such as electrical engineering or physics or a discipline such as optics (for specific presumed background, see [120]), but I explicitly introduce all other required advanced mathematics and electromagnetism. Wherever possible, I take a direct approach in der￾ivations, working from fundamental results, such as Maxwell’s equations or core math￾ematical definitions and principles, without invoking intermediate results or methods. 688 Vol. 11, No. 3 / September 2019 / Advances in Optics and Photonics Tutorial