By: Ellen Schelew, Applications Engineer at Lumerical Inc.
Green’s functions are a useful tool for extracting information about the emission and coupling characteristics of classical and quantum emitters in complex dielectric environments. The Green’s function, , describes the field at point “b” when a point dipole is located at point “a”. For a given point, is commonly calculated using an FDTD simulation where a point dipole is placed at point . When the Green’s function is desired for many points, multiple simulations are required. This becomes cumbersome for computationally intensive simulations, for example, high Q modes, dielectric structures with small or metallic features requiring fine meshing, etc.
This post gives a brief introduction to an alternative approach that requires only a single simulation to extract the general Green’s function, , which is based on the work of Dezfouli et al. in Ref. . In this approach, the Green’s function is calculated based on the quasi-normal mode field profile of an optical resonator, , such that . While the mode profile is easily extracted from FDTD simulations, it is generally nontrivial to properly normalize to calculate the appropriate Green’s function values. A computationally efficient method is applied here to complete the normalization.
In the following, results for the gold nanorod in Ref.  are reproduced, where the Green’s function is used to predict the total decay rate of a quantum emitter near the metal nanoparticle, , and the associated Purcell factor. It is also used to predict the incoherent and coherent coupling rates between two quantum emitters, and , respectively.
The gold nanorod in  studied here has a radius of 15 nm and length of 100 nm, as shown below. A “QNM” analysis group contains all the components necessary to calculate the resonant frequency, field decay rate and quasi-normal fields. After the simulation is run, a script file is used to extract the quasi-normal modes from the “QNM” analysis group, and is calculated to reproduce the results in Ref. .
Purcell factor of x-polarized emitter located at = (60.5 nm,0,0), as calculated directly from the power emitted by the source located (blue) and calculated from the quasi-normal mode method (green). For more examples for similar structures see Fig. 2 of Ref .
Purcell factor of x-polarized emitter located at = (60 nm,0,0). Compare to Fig. 2( c ) of Ref. .
, and for an x-polarized emitter at = (60 nm,0,0) and a z-polarized emitter at = (-45 nm,0,23 nm). Compare to Fig. 5(b) of Ref. .
 M. K. Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities”, Phys. Rev. B 97, 115302 (2018)
 R.-C. Ge and S. Hughes, “Quantum dynamics of two quantum dots coupled through localized plasmons: An intuitive and accurate quantum optics approach using quasinormal modes,” Phys. Rev. B 92, 205420 (2015).
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