Re: [MPB-discuss] 3D doubts (partly cleared up)
On Thu, 2006-11-09 at 11:15 -0500, Steven G. Johnson wrote: Your assumptions are incorrect. The 3d case *should* have additional bands compared to the 2d case. In two dimensions, you are only looking at light propagating in the plane (for kz = 0). When you include a finite cell size S in the z direction, then you are including out-of-plane wavevectors kz + 2*pi*n/S for all integers n, by the periodic boundary conditions. These out-of-plane wavevectors are what are giving you your additional states, which depend on S. This is not a numerical effect, it is a physical consequence of the question you are asking MPB to solve. You can think of this as the folding into the finite Brillouin zone of the vertical periodicity S (recall that MPB uses periodic boundary conditions). (In the 2d case where S=0 the Brillouin zone in the vertical direction is infinite and there is no folding.) Or, if you like, you can think of this in terms of higher-order standing-wave modes in the vertical direction. That is, the vertical mode profile need not be constant, it can be cos or sin of 2*pi*n/S for any n. See e.g. my paper in Phys. Rev B vol. 60, p. 5751 (1999) for further discussion of the consequences of slab thickness. Or some of my tutorial presentations at ab-initio.mit.edu/photons/tutorial Cordially, Steven G. Johnson ___ mpb-discuss mailing list mpb-discuss@ab-initio.mit.edu http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss Thanks for your most helpful answer. I've tried the case of a slab of silicon cylinders in n=1.5 medium (the previously attached ctl file), and I also end up with folded bands. So I guess MPB isn't most appropriate for treating slabs, or is there a way to discriminate the bands arising from the imposed periodicity? If not I think I'll allow myself to have a bit of fun with MEEP. I'm mostly interested in superprism effects and the like, so I don't care much about the bandgap. However the band profiles do matter. On the band diagrams in your paper, it looks like there's more impact in the higher frequency bands. Are they also a consequence of the periodicity assumptions of the PWEM or are they really due to the slab finite thickness? ___ mpb-discuss mailing list mpb-discuss@ab-initio.mit.edu http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss
Re: [MPB-discuss] 3D doubts (partly cleared up)
On Fri, 17 Nov 2006, Vincent Paeder wrote: I've tried the case of a slab of silicon cylinders in n=1.5 medium (the previously attached ctl file), and I also end up with folded bands. So I guess MPB isn't most appropriate for treating slabs, or is there a way to discriminate the bands arising from the imposed periodicity? If not I think I'll allow myself to have a bit of fun with MEEP. This has nothing to do with MPB vs. Meep, or any other computational package. It is a matter of physics (or mathematics). MPB is giving the correct answer to the physical question you are asking, and any other code will give the same answer to the same question. If you have a slab that is *infinitely uniform* in the z direction and want to distinguish between different wavevectors in the z direction, the correct thing to do is to use a 2d computational cell and specify the desired out-of-plane wavevector. I'm mostly interested in superprism effects and the like, so I don't care much about the bandgap. However the band profiles do matter. On the band diagrams in your paper, it looks like there's more impact in the higher frequency bands. Are they also a consequence of the periodicity assumptions of the PWEM or are they really due to the slab finite thickness? A slab of finite thickness is a completely different thing than what you are doing. If this is what you want, you are asking MPB the wrong question, which is why you are getting the wrong answer. If you want a slab of finite thickness, the correct thing to do in MPB (or any other program) is to use a supercell, where you have a slab of your desired thickness, plus some distance of air (or whatever your substrate is) above/below the slab. The guided modes are then the ones whose wavevectors lie below the light line of the substrate/superstrate. Because these guided modes are exponentially localized to the slab, whether you have periodic or some other boundary conditions in the vertical direction is exponentially irrelevant as you increase the vertical cell size (*without* increasing the slab thickness). See also the Phys. Rev. B paper I mentioned in my previous message. See also the hole-slab.ctl example file included with MPB. Steven ___ mpb-discuss mailing list mpb-discuss@ab-initio.mit.edu http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss
