Apparently divergent field distribution from scuff-spectrum

[acmcclung]

I'm trying to use scuff-spectrum roughly following this guide. I get the same mode frequency as in the example, but when I plot the field, I get an apparently divergent distribution, which seems unphysical. Is this the correct behavior?

scuff-spectrum --geometry sphere.scuffgeo --omega0 1.1-0.63i --Rx 0.2 --Ry 0.2 --N 14 --L 5 --EPFile evalpts

sphere.scuffgeo Sphere.msh evalpts sphere.ModeFrequencies sphere_C0_w1.1+-0.63i_RX0.2_RY0.2_N15_Mode0.evalpts.ModeFields plot_fields.py

[HomerReid]

This seems correct to me---if \omega has a negative imaginary part, then so does the free-space wavevector k, so the Green's function describing the fields of localized sources scales like e^{ikr}/r, which blows up exponentially for large distances. It's complementary to what we do in Casimir physics, where we Wick-rotate the contours of frequency integrations so that omega develops a positive imaginary part, in which case the Green's function is exponentially decaying at large distances.

[acmcclung]

Ok, thank you for clarifying. I guess I expected the example to look for localized modes, which is ultimately what I am interested in. Also, the way quality factor is signed in havminv seems to expect resonant localized modes to have negative frequency, perhaps contributing to my confusion. In any case, I will set my contours to search Im(\omega) > 0.

[jfeist]

I don't think you'll find resonances with positive imaginary part (as this would correspond to exponential growth in time, not decay). The exponentially diverging field in space is actually "physical": If you think of a resonant mode as a localized field distribution decaying exponentially in time, the amplitude of emission also decays in time. Spatial distance from the structure is effectively mapped to the time of emission, so the parts further away in space stem from a time when the amplitude of emission was higher. When you calculate the resonant mode, you get the mode corresponding to this process having gone on for "infinite" time, and thus the field grows exponentially in space the further you get away from the system. This is a well-known problem for obtaining, e.g., normalized eigenmodes, with a lot of literature about how to treat it (and quite a bit of controversy). This publication (and references therein) is probably a good starting point if you want to read more about it: Modes and Mode Volumes of Leaky Optical Cavities and Plasmonic Nanoresonators

[HomerReid]

Thanks for chiming in @jfeist! That looks like a very useful and recent reference.

@acmcclung, just to reiterate what Johannes said, the frequency contours you were using before were fine as is---you don't want to look for eigenfrequencies in the upper half-plane. Also, this example is finding localized modes, and you should have no trouble plotting field profiles in and near the sphere---the subtlety involving exponential blowup at large distances doesn't preclude using the tool to study the frequencies and spatial profiles of cavity modes, although if you need to normalize the fields you may need to introduce some of the more complicated tools described in the paper Johannes cited.