Complex frequency (wavelength)

[myurkin]

In a discussion Helge Dobbertin mentioned that to simulate certain physical phenomena (like Casimir forces) it is required to solve classical electromagnetic problem for complex frequency (wavelength or wavenumber). In this respect I see the following questions:

• Will ADDA break down if we just redefine corresponding variables as complex? What additional changes are required and for what computed quantities? I guess, internal fields will be the easiest, while far-fields or radiation forces may require additional effort.
• How the calculation of Sommerfeld integrals (for surface mode) would be affected?
• Is it possible to define wavelength etc. always as complex without significant increase of computational time? (and without excessive use of if-s).

This issue may also be related to simulations for particles inside an absorbing medium - the latter is mentioned in #91.

[myurkin]

@Sunmosk

[Sunmosk]

For now, I'm going to make the WaveNum variable complex and change all the downstream variables and functions accordingly. Not sure it will solve this issue, but let's try :)

[myurkin]

Most of ADDA internal calculations depend only on wave number in the host medium, and relative refractive index (of particle to that of the host medium). Implementing absorbing host medium (#91) makes any complex wave number possible, which is exactly what we need to simulate complex frequency. So frequency w=a*w0 (complex a, real w0) in the host medium with refractive index mh (and particle refractive index m) is equivalent to frequency w0, host medium a*mh and particle refractive index a*m. So one can already simulate complex frequency with ADDA using -mhost command line option. There are, however, two issues:

1. Some quantities inside ADDA, such as wavelength inside the medium, are determined through Re(mh), which may be zero for purely imaginary frequency (which is relevant for calculation of Casimir forces). Then this wavelength is effectively infinite. These quantities are not critical for ADDA calculations, but are shown in the output for informational purpose. Thus, we should, at least, avoid infinities there.

2. Some simulation results do depend on the frequency (or on k0). For now, it seems that only EELS and CL probabilities (Simulation of EELS spectra #155) are such quantities (since the electron energy is another relevant parameter, which breaks standard scaling invariance). Thus, there is motivation to enable additional control of frequency in addition to mh. For instance, we can allow complex arguments to -lambda command line option).