LDOS computation in 3d cartesian vs Cylindrical coordinates

[raksharuia]

Hello,

I’ve been running some tests comparing LDOS calculations in Meep between 3D Cartesian and cylindrical coordinates, and I’m seeing a behavior I don’t fully understand.

Setup

I’m simulating a point dipole at the origin in a homogeneous bulk medium with:

• identical bulk material
• identical source
  * 3d cartesian: origin
  * cylindrical: origin, on-axis dipole [src_pt: r=1.5/resolution], m = -1 in cylindrical)
• identical resolution
• same source spectrum
• I add an LDOS instance to my simulation run and extract the ldos_data.
• I vary the grid resolution and plotted the LDOS spectra for each case (image attached).
• Cell
  • 3D Cartesian: cell size (a, a, a)
  • Cylindrical: cell size (a/2, 0, a) with dimensions=mp.CYLINDRICAL

Observation

• In 3D Cartesian, the LDOS spectrum stabilizes and converges as I increase resolution.
• In cylindrical, the LDOS spectrum keeps shifting with increasing resolution and does not look like it converges to a curve.

Output LDOS vs wavelength for 3d cartesian case :

Output LDOS vs wavelength for cylindrical case :

Since both setups represent the same physical system (homogeneous medium + equivalent dipole), I expected them to behave similarly. I understand that in cylindrical coordinates, the source is not a true point dipole, but effectively a ring source. So I’m wondering -

• Could this “ring vs point dipole” representation be responsible for the resolution-dependent normalization?
• Is this the expected behaviour in cylindrical coordinates in MEEP?

Note:

• I have not applied any additional normalization here.
• I understand that for computing efficiencies or purcell enhancement factor, this won't affect the results as they are ratios of computation from structure vs. homogeneous.

Any insights or pointers would be really helpful, and I thank you in advance for your time.

[stevengj]

Yes, a dipole source in cylindrical coordinates is effectively a ring source with an amplitude that includes a 2πr factor if I recall correctly, so as you shift it towards the origin it effectively gets scaled by Δr so it ends up going to zero with resolution.

But I keep forgetting how it was set up, maybe @oskooi remembers better since he did a bunch of experiments with this.

[raksharuia]

Thank you for the explanation.

[oskooi]

Yes, a dipole source in cylindrical coordinates is effectively a ring source with an amplitude that includes a 2πr factor if I recall correctly, so as you shift it towards the origin it effectively gets scaled by Δr so it ends up going to zero with resolution.

I think that is correct as well: a point source placed at $r = 1.5 \Delta r$ in cylindrical coordinates will be scaled by $2\pi r$ which means its amplitude will converge to zero as $\Delta r \rightarrow 0$.

[raksharuia]

This helps a lot. Thank you :-)