# Computationally efficient Green's function calculation using quasi-normal modes

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 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. [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 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. [2].

Results:

1. 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 [1].

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

3. , 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. [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).

11 Likes

Thank you for the post. I wonder what is the origin of the oscillations at low frequencies in the first figure with direct FDTD results. How can this be dealt with? We find similar artefacts in our simulations sometimes.

Hi Raziman,

Great question. There are a number of different simulation settings that can cause oscillations to appear in the Purcell factor spectrum as numerical artefacts. For this particular case, the oscillations appear because the source pulse offset is just a little too small, causing a subtle truncation of the pulse. Here is the result for the first figure when the pulse offset is lengthened**.

**In order to change the pulse offset, the source settings must be defined in the time domain. There is currently an instability in the Purcell factor calculation when this feature is used, however we have already fixed it for the upcoming release that will be available later this month. The image above was produced using a software version that is fixed.

Other common reasons why ripples appear in the spectrum:

1. The simulation time is not sufficiently long to capture the full field decay. You can read more about this here: Simulation time and Frequency domain monitors.
2. Reflections off PML boundaries. A related post is: Reflections from PML.

I hope this helps you for your simulations!
Cheers,
Ellen

2 Likes

Hi there,

To calculate the quasi-normal modes, $\mathbf{f(r})$, we follow the procedure outlined in Section “B. Integration-free QNM calculation” of Ref [1] above by Dezfouli et al. In the first plot above, the “From quasi-normal mode calculation” Purcell factor (green) is calculated from the Green’s function $\mathbf{G(r}_a,\mathbf{r}_a;\omega) = A(\omega)\mathbf{f(r}_a)\mathbf{f(r}_a)$. The “Directly from FDTD results” Purcell factor (blue) is extracted from the dipole source (see “Purcell” result in “Results returned”).

Cheers,
Ellen

1 Like

Hello Ellen,

nice job. this procedure will indeed help a lot to calculate the emission rates of emitters placed everywhere around a complex structure. What about periodic structures as many emitters are simultaneously in the unit cell?

When will this “QNM” analysis group be integrated in FDTD solutions?

Best,
Renaud

1 Like

Hi Renaud,

The emission rate of a single emitter at an arbitrary location can be directly calculated from the Green’s function $\mathbf{G(r,r})$ extracted using this method. If you wish to consider multiple emitters simultaneously in the volume of interest, special care must be taken, as coupling between two or more emitters will affect their emission properties (see Ref [2] above). The Green’s function is again a powerful tool for calculating these effects.

This method is best suited for structures supporting confined optical modes, or confined modes with weak coupling to loss channels (see Ref [1]). Optical cavities introduced as defects in periodic structures are suitable, like planar photonic crystal microcavities. I don’t believe this method applies to emitters in purely periodic structures that extend to infinity in one or more dimensions.

Cheers,
Ellen

1 Like

I am trying to find G(ra,rb;w) from the electric fields. I see oscillations in the G(ra,r;w). Can you please make this example available. I could not find a link

Hi @tboddeti,

I replied to your request with a private message.

Cheers,
Ellen

1 Like

Could you please send this example to me? I want to learn this method. Thanks a lot.

Dear @liujingfeng,

Thanks for your interest in this example!

I replied to your request with a private message.

Cheers,
Ellen

Hello,

I am also interested in this method. Could you please send this example to me as well?

Thank you for this topic

Hi @eschelew,
I am interested in learing this method of calculating the Purcell factor. Can you please send me this example file?

Regards

Hi @nke,

Please see my direct message to you.

Cheers,
Ellen

Hi @eschelew,
I am interested in learning this method Please share it with me.

Hi @eschelew,
Would you please send this example to me?

Best

I am trying to simulate a similar strauture and would like to learn more about the QNM analysis group. Could you share the file with me ?

Thank you.

I would also like to look at simulation. Could you please send me the example files.

Thanks,
Amit

I responded to you privately

I responded to you privately