Further investigations

[tsoos99dev]

Hi! First of all awesome video! You inspired me to make my own simulation, and investigate further. I'd be interested in your opinion about my approach, and I'd also like to ask a few questions about the video. This is not meant to be an issue, but I think discussions aren't enabled in the repo.

Model I'd like to find the simplest setup in which the slowdown of light can be presented. Just like your last simulation, I'd like to start with 1D. The medium will be made of equally spaced nodes. The nodes will effectively stay in place, but as the external electromagnetic wave passes through them a dipole moment will be induced on the node. The nodes' state will be an effective separation that is proportional to the electric field created by each node locally. The dynamics of this separation will be modeled as a mass-spring-damper system. Everything will be in the time domain. As I later found out this is what's called the Lorentz oscillator model.

Initial calculations Main assumptions about the properties of the system:

• IR pulse wavelength : 905nm

• IR pulse duration: 1ns

• Node / molecular spacing: 0.3nm

Based on this approximately 3000 nodes fit in a single cycle, and there's close to 3e5 cycles in a single pulse. This would definitely result in behaviour closer to what I'd expect from an ideal plane wave. Therefore I'll start with these longer pulses. Would you agree with these estimates?

Choosing the oscillator model parameters is quite tricky for me. From what I read about the resonant frequencies of water molecules or silicon dioxide, it looks like we're very far from that frequency here. I guess otherwise it wouldn't be transparent, but I'll just have to try out different natural frequencies and damping ratios in the model. What do you think about the right choice here?

I don't know nearly enough about the subject to know if this is the right approach physically, so I'd appreciate any kind of feedback. I'm going to base my simulation on the FDTD approach described here

[tsoos99dev]

After some more research, I think I'll be able to simplify the model further. The slowdown has to be demonstrable with a lossless dielectric. So essentially what you were explaining in the video with the electrons following the wave instantaneously. My current best explanation for the phenomenom in the lossless medium goes something like this:

1. The initial wave is fully canceled by the opposing waves created by the material after traveling a fairly short distance.
2. Simultaneously the waves created by the material travel backwards.
3. These backward traveling waves change the motion of previous atoms, creating the delayed wave.
4. How this intuitively leads to a decrease in speed is still a mystery to me.

My next goal is to implement this in a simulation. I have most of the basics ready here. I'll definitely implement more advanced material models later though.

@mithuna-y

[mithuna-y]

That's very cool you're doing your own simulation! These all seem like great ideas though I have to admit I don't understand it well. I wish I could comment on the physics but I don't know enough about this sort of simulation- it's over my head unfortunately. I would love to hear about your progress though so please keep me updated. I'd love if you could show us video from your simulation. Thank you for doing this, I really appreciate it!

[tsoos99dev]

Here's my first fairly interesting result using the Lorentz model: (click), and here's the supplementary visualisation for the model: (click). In the video I was trying to recreate the situation in which the refractive index is less than unity. The effect isn't super obvious, but it's clear that the maxima move faster than the source wave, which enters from the left. It's also visible that the wavelength increased compared to the source. The slightly gray patch represents the material and the source starts out in vacuum. The graphics isn't top notch, but it works. The values I'm using are not quite representative of your experiment. I was focusing on this effect specifically. The relevant resonant frequency is slightly higher in water than what I'm using so I could keep the unit of distance at 0.1nm. There's a lot of attenuation as predicted by the model in this particular situation, since I used a frequency right above resonance.

• The unit of distance in the final animation was increased to 1 nanometers, but the simulation was using 0.1nm.
• The unit of time is ~0.3 attoseconds which is pretty crazy
• The total simulated length in space is 30000 units or 3 microns.
• The total simulated length of time is 30000 units or 3 femtoseconds.
• The source wavelength is 72.4nm. According to the model this gives a refractive index around 0.75.
• Molecular spacing is 0.5nm, so there's 4 cells in between every node where there's only vacuum.

The wavefront is super interesting. The dispersion is definitely visible. I'm not certain the shape is physically accurate though. The simulation has its own perks, but there's something small traveling at the speed of light in vacuum there. I know that because it reaches the end of the window. It looks like even the transients are visible before assuming the steady state exponentially decaying waveform.

[tsoos99dev]

I'll study the model a little more, and create other videos too. Then I'll try to look at the situation in your experiment specifically. The IR pulse has a much longer wavelength than the equivalent wavelength at the resonant frequency of water. This is based on measurements made by others I found in papers, but this model could be a good way to describe what's happening. I'll also be able to look at the fields created by individual nodes. I might be able to add a few representative oscillating dipoles to the animations. Since your experiment was pretty far from resonance it should simplify the behaviour a lot.

[tsoos99dev]

In this simulation I reduced the absorption coefficient and the index of refraction further. In some sense the light takes time to penetrate into the material. I mean the wavefront is obviously traveling at a finite speed, but because of the transients it takes some time to reach the final shape even after the wavefront passes. The maximum field value keeps increasing with consecutive oscillations way after the wavefront passes. Here's a gaussian pulse. It nicely disperses, but the wavefront still travels at the same speed as before. The simulation time is a little longer.