CPM Seminar

Bounding Green Function Singular Values and Shannon Capacity

Sean Molesky

Polytechnique Montréal

As demands for communication and computation continue to grow, outstanding questions related to electromagnetic information transfer are becoming increasingly relevant. Namely, for given sender, mediator and receiver regions of space, to what extent is communication possible, and by what means can this potential be realized? The noisy coding theorem provides a partial answer so long as the communication system in question is fixed: information transfer (for any encoding) is limited by the Shannon capacity, which can in turn be calculated from the Green function of the system. However, if aspects of the communication system are variable, e.g. the characteristics of the mediator can be modified by changing its geometric shape, workable prescriptions to generalize Shannon's results are missing.

We illustrate two results towards this goal of a wider understanding of information transfer via electromagnetics. First, building off the recently developed QCQP performance bounds recipe, we show how the nth singular value of any Green function between fixed sender, mediator and receiver regions can be bounded over the freedom of arbitrary structuring for a predefined collection of (linear) materials. Second, we describe several optimization formulations that can be used to bound Shannon capacity in the setting of nanophotonics, and establish that the connection to the Green function mentioned above can be utilized to facilitate practical calculations of at least one of these formulations at scales approaching technological relevancy.

Our findings suggest a number of directions for future work beyond technical improvements. In particular, with some (likely non-trivial) adaptations they may offer insights into the improvement of imaging systems and the potential of meta-optics to improve information retrieval.