By: Ellen Schelew, Applications Engineer at Lumerical Inc.
Introduction:
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. [1]. In this approach, the Green’s function is calculated based on the quasinormal 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. [2] 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.
Simulation:
The gold nanorod in [2] 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 quasinormal fields. After the simulation is run, a script file is used to extract the quasinormal modes from the “QNM” analysis group, and is calculated to reproduce the results in Ref. [2].
Results:

Purcell factor of xpolarized emitter located at = (60.5 nm,0,0), as calculated directly from the power emitted by the source located (blue) and calculated from the quasinormal mode method (green). For more examples for similar structures see Fig. 2 of Ref [1].

Purcell factor of xpolarized emitter located at = (60 nm,0,0). Compare to Fig. 2( c ) of Ref. [2].

, and for an xpolarized emitter at = (60 nm,0,0) and a zpolarized emitter at = (45 nm,0,23 nm). Compare to Fig. 5(b) of Ref. [2].
References:
[1] M. K. Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities”, Phys. Rev. B 97, 115302 (2018)
[2] 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).
Please let us know if you wish to learn more about this example.