Is there a limit to the performance of linear optical components? Suppose we asked a specific question, such as how much glass would we need to make a device that would split 32 wavelengths in the telecommunications C-band near 1.5 Î¼m wavelength. Intuitively, we would probably agree that 1 Î¼m3 of glass would not be enough. We could, however, certainly achieve this goal with a roomful of optics and, in fact, we know we could purchase a commercial arrayed waveguide grating device, with a scale of centimeters, to make such a splitter. So our intuitive experience suggests that there is some limit on performance of such optical components, though historically we have not had a general limit we could use.
The need for a limit is not merely academic. With modern nanophotonic techniques, we can make a very broad variety of devices, some with little or no precedent. In part because of the high refractive index contrast available to us in photonic nanostructures, the design of such devices is often quite difficult, and we would at least like to know when to stop trying to improve the performance. Classes of devices of interest to us could include dispersive structures, slow light elements, holograms, or any kind of device that separates different kinds of input beams or pulses to different positions in space or time.
Recently, we have been able to devise quite a general approach to limits for the performance of linear optical components. This approach gives upper limits to performance that are quite independent of the details of the design, being dependent instead only on the overall geometry of the device and, for example, the largest dielectric constant variation anywhere in the structure at any wavelength. This overall limit has already been applied to calculate limits to dispersive devices and to slow light. Here we will introduce this limit, summarizing its derivation and the applications thus far.
This limit is based on the idea of counting possible orthogonal wave functions that can be generated when an optical component acts to âscatterâ an incident wave into a receiving volume. This idea in turn is based on some earlier work that is a generalization of diffraction theory to volumes, in which we can count the orthogonal âcommunications modesââthe best choices of sources in one volume and the resulting waves in another for communicating between the two.
In this chapter, after summarizing the background to the need for a new limit, especially in nanophotonics, we will then introduce the underlying mathematical methods, including a discussion of communications modes and their applications. We will give the proof of a new general theorem for strong andâor multiple scattering, a theorem that underlies our limit to optical components. Then we will summarize two applications of the new limit, namely, one to slow light devices and the other to dispersion of pulses, before indicating future directions and drawing conclusions.
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