Why does the imaginary part of my simulated electric field enhancement differ from the analytical result at short distances?

[wswrpd]

Hi Developer,

I am currently using MEEP to calculate the electric field enhancement for an electric dipole positioned above a surface with a permittivity of $\epsilon=9$. While the real part of the electric field enhancement aligns correctly with my analytical solution, the imaginary part exhibits incorrect behavior, particularly at short "intermolecular" distances. I have ruled out the use of cylindrical coordinates as the source of this discrepancy, as a Cartesian 3D simulation yielded similarly incorrect results.

I have attached the relevant figures and the MEEP source code for your reference and assistance in resolving this issue. Thank you for your time and help.

MEEP CODE for z-polarized dipole:

XZ_Surface_Cylind_Symmetry_Ez_Freq_Enhancement_V2.py

Figures:

[oskooi]

In your script, you are using an $E_z$ source positioned at $r = 0$ and $m = 0$ (the default value since m is not defined as a keyword argument in the initialization of the Simulation object constructor):

sources = [mp.Source(src=mp.ContinuousSource(frequency=fcen),
                     component=mp.Ez,
                     center=mp.Vector3(0, 0, 5),
                     amplitude = 1j) ]

sim = mp.Simulation(cell_size=cell,
                    resolution=resolution,
                    sources=sources,
                    boundary_layers=pml_layers,
                    dimensions=mp.CYLINDRICAL,
                    geometry=[]) # <-- NO GEOMETRY 

A simulation in cylindrical coordinates with m = 0 involves purely real fields by default. You will therefore need to pass force_complex_fields=True as a parameter to the Simulation object constructor.

For reference, an example of an $E_z$ dipole at $r = 0$ is described in Tutorial/Radiation Pattern of an Antenna in Cylindrical Coordinates which includes this script.

[wswrpd]

@oskooi , Thanks for your quick reply. I've modified my code according to your suggestion (the updated version is attached for your reference, I reduced the resolution to 5 for a quick check). However, the imaginary part of the results still exhibits incorrect behavior (indicated by the diamond symbols). Do you have any further insight into why this might be occurring?

XZ_Surface_Cylind_Symmetry_Ez_Freq_Enhancement_V3.py

[oskooi]

Have you verified the results for $D < ~50$ nm are converged? As a check, try repeatedly doubling the resolution parameter.

[wswrpd]

@oskooi I have done my best to increase the resolution, but the result still drops to 1 when the distance $D$ approaches $0\text{ nm}$.

XZ_Surface_Cylind_Symmetry_Ez_Freq_Enhancement_V4.py

I also ran a separate simulation using a BEM code, and that result was much better. I suspect there might be a crucial setting missing in the MEEP code—something beyond just setting force_complex_fields = true.

[stevengj]

A simulation in cylindrical coordinates with m = 0 involves purely real fields by default. You will therefore need to pass force_complex_fields=True as a parameter to the Simulation object constructor.

Not necessary. Even for real fields, the DFT fields should be the correct complex value.

[stevengj]

It looks like the higher-resolution calculation is closer to the analytical result, so maybe it is just converging slowly? I'm not sure what you mean by "high" resolution, but as @oskooi said I would try plotting for resolution = 10, 20, 40, 80, 160, … repeatedly doubling, to see if there is a trend in what it is converging towards.

Getting good discretization accuracy at the origin has been painful

[Guangming1998]

It looks like the higher-resolution calculation is closer to the analytical result, so maybe it is just converging slowly? I'm not sure what you mean by "high" resolution, but as @oskooi said I would try plotting for resolution = 10, 20, 40, 80, 160, … repeatedly doubling, to see if there is a trend in what it is converging towards.

Getting good discretization accuracy at the origin has been painful

Hi I am the colleague of wsw (original author), I have a question here: the real part has good convergence while imaginary part does not. My understanding is that the real and imaginary part should be equivalent during simulation. Why it gives such disparity about the resolution sensitivity?