Waves and modes

David Miller’s research has led to a new way of looking at optics, one based on the concept of “modes” – essentially the “best” choices of functions for describing a system – but in a way quite different from the common “resonator” or “propagating” modes. This approach leads to new fundamental insights, practical results and even physical laws. As discussed more generally in Fundamentals of optics, the new technologies of nanophotonics and complex optical circuits, and the expanding applications of optics in communicating, sensing and information processing need just such a new approach and its results.

Optics can be viewed in many different ways. The earliest picture, and one still useful to this day, is that of rays. When combined with the ideas of reflection and refraction, we can understand elementary optical systems, like lenses and mirrors. Imaging optics adds useful formulas and concepts for designing and using lenses. Fourier optics takes us to the next level, adding wave ideas of diffraction that give limits and new behaviors in relatively conventional optical systems.

Of course, light is an example of electromagnetic radiation. So, if necessary, we can use full electromagnetic simulation to model behaviors down to sub-wavelength scales. But for conceptual understanding, such as numbers of channels or basic limits to devices, and for efficient design of these new devices, we need an approach that is both conceptually fundamental so we can draw reliable conclusions, including limits and new physical laws, yet economical so we can analyze and design efficiently.

The approach that gives us these benefits is what we could call “modal optics”. Modes are essentially the best sets of functions for describing our systems. Often we will practically only require some relatively finite number of these functions to describe our system, giving much greater economy and efficiency than a brute-force electromagnetic modeling based on the field at every one of a dense set of points.

Now, we are used to “modes” in optics, both in resonator modes, for example in lasers, and in propagating modes in fibers. But the modes we need for our purposes here are neither of those. We need quite a different approach. This different approach is quite straightforward, with elegant, powerful, and complete mathematics and deep physical meaning, but until recently, it was largely unknown in optics and in waves generally.

In this approach, rather than trying to find “beams” or “resonator” modes, we find instead the optimum input and output functions for describing the optics. This approach therefore finds two sets of functions or modes, which are coupled together in pairs, one input function and its corresponding output function. This approach opens a wide range of new practical and fundamental results, and connects to some particularly powerful mathematics, mathematics that has essentially not been exploited in optics before. This mathematics includes singular value decomposition (SVD) and key results from functional analysis, especially Hilbert-Schmidt norms. See this major tutorial and review [1], and the page SVD modal optics for a deeper discussion of this approach and its physical insights.

[1] 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