Nathan Zechar what makes you think that superluminal phase velocity is non-physical?

There are many discussions on the topic. A classic is Sommerfeld's work (https://en.wikisource.org/wiki/Translation:An_Objection_Against_the_Theory_of_Relativity_and_its_Removal), but a quick google search reveals many others:

- https://www.mathpages.com/home/kmath210/kmath210.htm

- https://en.wikipedia.org/wiki/Faster-than-light

- https://physics.stackexchange.com/questions/6912/in-superluminal-phase-velocities-what-is-it-that-is-traveling-faster-than-light

By definition, vacuum has a relative permittivity of 1exactly. This is the normalised expression for the causality in the universe, which then gives the "speed of light". Setting a lesser permittivity simply means you are setting the speed of causality in your FDTD equations at faster than "our" speed of light.

If you do that, you will get "non-physical phenomena" because you have left the realm of physics.

Mathematics is not limited by real world physical constraints, and FDTD is just mathematics so can you set the constraints to anything you want. Then, the interesting, mathematically possible, but the physically impossible can be observed.

If you want realistic FDTD results, you need to set realistic physical constraints...

There is no problem. Wave fronts can travel faster than light without violating relativity or Maxwell's equations. This is not non-physical. They travel faster than light in all hollow metal waveguides. Look up metal waveguide propagation. The energy does not travel faster than light, and if you do a transient simulation you will see that the increase in amplitude doesn't travel faster than light, but the wave fronts travel through the amplitude envelope, which travels more slowly. There is some similarity with Moire fringes, patterns that can be made by the crossing of two combs. These patterns can travel with unlimited speed as the combs are moved slowly.

Malcolm White , since he is doing FDTD, Nathan is pushing the pulse through the media. I guess, his pulse is traveling faster than light, or he would not see the phase front at longer than possible distances. And that is weird. There is a mistake should be somewhere in the math, or in the code. But let's see for his clarification on this point.

When he says he sees wave fronts travelling through the medium faster than light he almost certainly means the fields or animated fields plotted at one frequency. Not many people look at the excitation used in the time domain. This will go no faster than light. I have never looked at that, using CST for 15 years, although I learn a lot seeing the successive pulses exiting at the ports, and they don't get there faster than light.

I realise that I was wrong, and if Nathan Zechar actually has er less than 1 the pulse (and all frequencies) will travel faster than light. Actual artificial dielectrics with er less than 1 "lose" part of the wave inside the fine detail of the material, so that the wave looks like it has moved 30 deg (1/12 wavelength in the macroscopic picture), for example, but actually has moved 390 deg (and took 390 deg worth of time). Your large-scale results may be realistic, but local detailed fields that actually happen in a metamaterial won't be there, and the time for a time domain signal to travel through the material will not be realistic (for a metamaterial). A similar thing happens with travelling wave tube (or magnetron or backward wave oscillator or gyrotron etc) slow wave structures, which were early 1D metamaterials. Interaction is calculated using the apparent wave traveling slowly, or backwards, not the actual wave that is travelling at lightspeed but in a convoluted path. Some apparent waves may travel faster than lightspeed (in the upper branch of the dispersion diagram).

Stuart Barth, I understand phase velocity can have values faster than light, however a single frequency, 1 cycle sinusoidal signal traveling at a velocity faster than light.... I'm operating under the assumption that information cannot travel faster than light.

Zoltan Vilagosh, Permittivity less than 1 isn't leaving the realm of physics so long as a dispersion relation is satisfied where group velocity is at or less than the speed of light.

Andrey Porokhnyuk, Yes, I see single frequency sinusoid, 1 cycle, travel faster than light for value of permittivity - 1> er > 0, which actually isn't too weird considering how FDTD is typically described and coded up. Do you use MATLAB or Python? I can find a quick 1D example for you. Just plug in a value for permittivity (1> er > 0) and you will observe a single frequency propagate faster than light. I don't think the way Maxwell's equations are used in FDTD code accounts for this.

Malcolm White - Yes this is correct. They should not be there. Once a system has reached a steady state over several single frequency periods, then the electric field everywhere should match real world results. However for the initial wavefront, this should not be the case.

c= sqrt (1/e0*μ0) , so changing e to a number less than one increases the speed of light to greater than that in a vacuum in our universe.

group velocity is not bounded by c, but causality is.

you can do anything with mathematics, e.g. a mathematician is never hungry,

0= pizza-pizza, eat the pizzas! :)

Malcolm White , "almost certainly means the fields or animated fields plotted at one frequency".

Sorry. FDTD does not work like that. TD is for time domain. You synthesize a pulse, and see how it propagates. You can not see anything above zero amplitude where the pulse has not propagated yet. Yes, you can take a Fourier transform for getting a spectral image, but form zero amplitude, it gives zero Fourier amplitudes.

In artificial media, faster than light phase results in infinitely small oscillations appear and grow in front of the pulse(negative), and pulling it forward, or the carrier(0

Nathan Zechar "FDTD is typically described and coded up. Do you use MATLAB or Python?"

Sorry, I do not solve numeric problems from the University wehre we were using Matlab. Butduring PhD I was playing a lot with metamaterials. And I am using other solvers applying to antennas and arrays. Speaking of FDTD, I like FreeEMS, how it is done and linked to Matlab/Octave

Andrey Porokhnyuk

"In artificial media, faster than light phase results in infinitely small oscillations appear and grow in front of the pulse(negative), and pulling it forward, or the carrier(0

well, you are right. I was captured by practical considerations, and did not imagine the purely synthetic case where you have a nondispersive material of eps=1/2. Malcolm is right, in that case it will propagate FTL. But it is impossible on practice since you have to make a bandgap structure that pulls the dispersion to cutoff, effectively slowing the wave, while increasing its phase velocity, so it always stays slower than light.

Thanks Andrey Porokhnyuk . It is an interesting question and I have learnt a lot answering it. I think the (as you rightly say, impossible) low er must increase the distance of influence for each time step in the code.