# SVD modal optics

Modes are very useful mathematically for working with waves. Modes are functions – essentially the best choices of functions – that enable us to reduce what could be a very complicated problem to a much simpler one. Instead of having to deal with the amplitudes of waves at every point in space in our calculations, we can instead just work with a much smaller number of amplitudes of these “mode” functions.

We are used to modes for resonators and for propagating in fibers. In these cases, the modes can essentially be the “beams” in space inside the resonators or fibers. But to understand the modes we need for communications or for describing optical components, we need to think of a different kind of picture. This picture is based on the mathematical idea of singular value decomposition (SVD).

In these applications, we need to think about how best to connect, through free-space or some optical system, from one “source” surface or volume to another “receiving” surface or volume. In such a view, the “modes” of the system are not the beams between the “source” and “receiving” spaces. Rather they are the best choices of pairs of functions, one in the “source” space and one in the “receiving” space.

Communications modes are pairs of functions – a “source” function in the source space, and a “receiving” function in the receiving space. They are found from the singular-value decomposition of the coupling operator, G, between the sources in the input space and the resulting waves in the receiving space. They are the best possible, orthogonal channels.

In communicating between source and receiver spaces, we can think of these pairs of functions as what we can call “communications modes”. These are the best choices of source and receiver functions that also give us all the orthogonal channels through the system – the independent “ways” in which we can communicate, without cross-talk or interference between the channels.

Mode-converter basis functions are pairs of functions, and “input” function in the input space, and an “output” function in the receiving space. The are found from the singular-value decomposition of the linear operator, D, that describes the output wave in the output space that the device creates as a result of input sources or waves in the input space.

In thinking about some linear optical device or system, we can similarly describe the device or system in terms of sets of input and output functions. For any linear system, there is actually an essentially unique set of orthogonal “input” functions that connect, one by one, to a corresponding essentially unique set of orthogonal “output” functions. In this case, we can think of these sets of functions as the “mode-converter basis sets”. Indeed, we can completely describe a linear optical component or system in terms of such sets of functions and the connection strengths between them.

Communications modes and mode-converter basis sets are essentially the same thing, with the different names just corresponding to how we are thinking of the problem: for communications, we want to know the orthogonal channels through the system; for linear optical devices, we want to know the orthogonal input and output functions that give us the most economical and complete way of describing the device.

In both cases, we find these sets of functions by formally performing the singular-value decomposition (SVD) of the coupling operator between the source and receiving spaces. SVD actually corresponds to solving two eigenvalue problems, one of which gives the set of source functions and the other of which gives us the set of receiving functions. These functions exist in pairs, one member of the pair being in the source space and the other being in the receiving space.

This SVD modal approach to optics was introduced by David Miller in 1998 [1], and developed and extended progressively since then in a series of articles [2][7]. These have been extensively cited in the optical and wireless literature. The recent paper “Waves, modes, communications, and optics: a tutorial,” [7] gives a complete tutorial introduction to this approach and its consequences.

This approach has many practical and fundamental consequences. These are discussed and reviewed in “Waves, modes, communications, and optics: a tutorial,” [7].

• It proves that there is a set of orthogonal channels through any linear optical system, including even complex scatterers.
• It allows us rigorously to count the number and strength of the channels through any linear system.
• It establishes a sum rule for the number and strength of connections between input and output spaces. The fact that the sets of mode functions are generally formally infinite can introduce a paradox suggesting that there are correspondingly infinite numbers of channels. This sum rule resolves those paradoxes, “cutting off” those infinities in any practical wave system.
• It gives a general and overarching view of diffraction and diffraction limits, one that applies not just to large plane parallel surfaces, as in much conventional optics, but more generally to surfaces and volumes of arbitrary shapes and sizes, including nanophotonic systems.
• The mode-converter basis sets lead to a new set of fundamental laws that apply only to them.
• The necessary best modes for such optical systems can be both implemented physically using programmable optical interferometer meshes, and self-configuring meshes can even automatically find these best modes.
• We can use this modal approach, with its clear counting of channels, to understand how complex an optical system needs to be for some chosen purpose.
• This modal approach lets us quantify limits to optical components.

[1] D. A. B. Miller, “Spatial Channels for Communicating with Waves Between Volumes,” Opt. Lett. 23, 1645‑1647 (1998). https://doi.org/10.1364/OL.23.001645

[2] D. A. B. Miller, “Communicating with Waves Between Volumes – Evaluating Orthogonal Spatial Channels and Limits on Coupling Strengths,” Feature Issue on Information Theory in Optoelectronic Systems, Appl. Opt. 39, 1681-1699 (2000). https://doi.org/10.1364/AO.39.001681

[3] R. Piestun and D. A. B. Miller, “Electromagnetic Degrees of Freedom of an Optical System,” J. Opt. Soc. Am. A 17, 892–902 (2000)

[4] D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985-23993 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23985https://doi.org/10.1364/OE.20.023985

[5] D. A. B. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238-251 (2013) http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-2-238 https://doi.org/10.1364/JOSAA.30.000238

[6] D. A. B. Miller, “Better choices than optical angular momentum multiplexing for communications,” PNAS 114, no. 46, E9755–E9756 (2017) https://doi.org/10.1073/pnas.1712762114

[7] D. A. B. Miller, “Waves, modes, communications, and optics: a tutorial,” Adv. Opt. Photon. 11, 679-825 (2019) https://doi.org/10.1364/AOP.11.000679