#### Event Title

Sum Rule for the Induced Local Density of Photonic States

#### Start Date

8-11-2017 1:30 PM

#### End Date

8-11-2017 5:30 PM

#### Description

We present a sum rule for the induced local density of photonic states (LDOS). This quantity describes the availability of photonic states at a certain position in the vicinity of a nanostructure, and therefore determines the decay rate of an emitter placed at that location. The LDOS is also closely related to the strength of the near-field produced by the nanostructure. The sum rule presented here establishes that the integral over the whole spectrum of the LDOS at a certain position must be equal to the static field produced by a dipole placed at that same position. We confirm the validity of the sum rule by performing rigorous numerical calculations for a variety of nanostructures including, nanospheres, nanodisks and films, made of different materials described using Drude-Lorentz dielectric functions, as well as for finite and extended graphene structures. Furthermore, we show that the derived sum rule can be used as a guide to select the most favorable nanostructure geometries to achieve strong values of LDOS. In addition, we extend the derived sum rule to the induced cross density of photonic states (CDOS), which determines the coupling between quantum emitters and it is intimately connected to the spatial coherence of light near the nanostructure. The results of this work serve to improve our understanding of the fundamental limits of the photonic response of nanostructures and therefore have important implications for the field nanophotonics.

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Sum Rule for the Induced Local Density of Photonic States

We present a sum rule for the induced local density of photonic states (LDOS). This quantity describes the availability of photonic states at a certain position in the vicinity of a nanostructure, and therefore determines the decay rate of an emitter placed at that location. The LDOS is also closely related to the strength of the near-field produced by the nanostructure. The sum rule presented here establishes that the integral over the whole spectrum of the LDOS at a certain position must be equal to the static field produced by a dipole placed at that same position. We confirm the validity of the sum rule by performing rigorous numerical calculations for a variety of nanostructures including, nanospheres, nanodisks and films, made of different materials described using Drude-Lorentz dielectric functions, as well as for finite and extended graphene structures. Furthermore, we show that the derived sum rule can be used as a guide to select the most favorable nanostructure geometries to achieve strong values of LDOS. In addition, we extend the derived sum rule to the induced cross density of photonic states (CDOS), which determines the coupling between quantum emitters and it is intimately connected to the spatial coherence of light near the nanostructure. The results of this work serve to improve our understanding of the fundamental limits of the photonic response of nanostructures and therefore have important implications for the field nanophotonics.