  # 1D photonic band structure

Dear all,
i was wondering if someone could explain to me following issues:

1. The bandstructure analysis group (https://kb.lumerical.com/en/pic_passive_bragg_initial_design_with_fdtd.html) returns only two peaks. But the grating structure should have spectrum in between those two peaks also. i looked at the script written in bandstructure analysis group but i don’t understand how it suppresses the intermediate spectrums. And is it necessary that the 2 peaks always lie at band edge (to get the correct value of BW)?
2.Band structure of Bragg grating for the wide range of Bloch wavevector:a
Is the line mentioned in the figure light line? it does not look like higher order band.
2. is it correct to calculate the effective index at a particular frequency by the expression neff=ck/(fperiod) from this band diagram? or do i need to plot band diagram for a normal waveguide to calculate effective index?

thank you.

Dear @ktwayana

These are all great questions, and I will do my best to address them properly.

1. the bandstructure analysis group uses couple of time monitors to record the electric field in time, and then adds the Fourier transformed data from all of these monitors (with appropriator apodization). So, it does not suppress any spectrum (unless if it is outside the spectrum). So, the two peaks that you are seeing are the only peaks available. I examined this by right clicking the time monitors and plotting spectrum (for K=0.3): This is being said that I could not see the other spectrum (f~1.9THz).

I think that since the two spectrum get closer at K=0.5 (as you can see in your plot), we only need to study them at the band edge. The results at this point will give us the band gap.

1. It can’t be light line as there is no spectra expected beyond light line. I couldn’t see that spectrum in my simulations. Can you please explain how you got it? Here is the spectrum for K=0.45 (where that spectrum seems to have the highest peak): 1. I think you can use neff=cK/(fperiod), where K varies from 0 - 0.5. I put some numbers and I got n_eff~2.2 - 2.5.

Please let me know if you have further questions.

Thanks

Dear @bkhanaliloo,
Actually, my confusion is what you explained in your post.
Yes, the randomly distributed time monitors inside the simulation region collect the field over time. I think, if we are simulating Bragg grating structure the field decays quickly due to the destructive interference except at the band gap (stop band). At the stop band there should be constructive interference. So the time monitors should have field value for the band of spectrum (stop band) rather than only at two frequency points.
Again my question is why time monitors do not record the field value at the frequency in between?

i got the band diagram by running a parameter sweep for the range of bloch wavevector (K), as mentioned in https://www.lumerical.com/support/video/waveguide-bragg-gratings-res.html video after 24:40 min time.

i did the effective index calculation from dispersion (band) diagram of strip waveguide at 1.55 microns wavelength and i got neff 2.3419 which is very close to the effective index 2.344915 calculated from mode solver.
Thank you.

Dear @ktwayana

First, let me clarify what I have said in my previous comment. I think the line that you have mentioned is not the light line. If you plugin the numbers into n_subfa/K at any point of the line, it should be equal to speed of light ©. By looking at your plot at K=0.3, f~1.5e14, n_sub=1.44, and a=0.32e-6; we have:

1.44 * 1.5e14 * 0.32e-6 /0.3 = 230400000

I think this means that this can not be the light line. Again, I could not see this mode in my simulations. Maybe you can upload your simulation file for a review?

Now, lets come back to your questions:
In FDTD, we use Bloch boundaries and set the kx value:

This means that any time that light leaves the FDTD region from right (left) side, it will enter the simulation region from left (right) side with a phase corresponding to kx * periodicity. This means that only certain frequencies will satisfy the condition and the rest will be suppressed. This is also how we construct the band diagram. The plots that I provided earlier for specific K values match with band diagram and are what we expect to see.

I hope I could answer your question and clarify why you should not expect to see more than two peaks.

Thanks

Dear @bkhanaliloo,
thank you very much, that makes sense.
could you please show me mathematically the frequency (frequencies) at which phase condition matches for the specific value of kx and period for eg. in your previous post for k = 0.45 there was two phase match condition (peaks) at approx. 185 THz and 206 THz.

Simulation file for band diagram calculation by running parameter sweep is here.Grating1.fsp (2.5 MB)
if you plot in logarithmic scale you get the similar (not same because dimension is different) diagram that i posted early.
thank you.

A post was split to a new topic: Photonic band structure and how to separate TE and TM modes

Dear @ktwayana

The phase that light picks up at one period is Ka where K is the wave vector and is equal to n_eff * w /c and a is the periodicity. Using Bloch boundaries, when light wants to enter from right to left side, it picks up a phase equal to -kx * a where kx is what we have defined in software. At certain frequencies where these two phases cancel out, over many cycles, we have a constructive interference and light will be amplified. This will be simplified to neff=ck/(f*period) which you have already used this approach to calculate neff in your earlier post. In this cases, you will not be able to calculate n_eff directly (because geometry changes along x), but probably you can come up with other design to verify this.