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"volumefrequency. 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