Hello Andrea,

I’m not 100% sure, so please correct me if I’m wrong.
By introducing the line defect you keep the translational symmetry along one 
axis, the direction of the line. E.g. if you remove one rod with its axis along 
the x-axis, k_x still remains conserved.
So I guess you should project the band diagram on that axis, having bands as 
omega(k_x). Of course I used k_x in terms of Cartesian coordinates, not the 
reciprocal ones. But you can do the transformation between both basis by MPB 
For a general example of the projected band diagram in 3D PhC I would recommend 
Chapter 6 in Photonic Crystals: Molding the Flow of 
Light<http://ab-initio.mit.edu/book/> by John D. Joannopoulos, Steven G. 
Johnson, Joshua N. Winn, and Robert D. Meade, but I guess you already had a 
look for that. A specific projected band diagram of a silicon woodpile can be 
found in the following:

Isabelle Staude, Christopher McGuinness, Andreas Frölich, Robert L. Byer, Eric 
Colby, and Martin Wegener, "Waveguides in three-dimensional photonic bandgap 
materials for particle-accelerator on a chip architectures," Opt. Express 20, 
5607-5612 (2012)

I hope I could help!

Best regards,

From: Andrea Locatelli<mailto:andrea.locate...@unibs.it>
Sent: Tuesday, September 19, 2017 00:31
To: mpb-discuss@ab-initio.mit.edu<mailto:mpb-discuss@ab-initio.mit.edu>
Subject: [MPB-discuss] Woodpile waveguide

Hello MPB users,
I'm going to simulate a waveguide consisting of a linear defect introduced into 
a woodpile photonic crystal.
I had many doubts about the definiton of the irreducible Brillouin zone, but in 
the end I found an example that Steven posted here for the case of the perfect 
crystal which works fine (even if I used as some kind of black box).
Now I must introduce the linear defect and other doubts come into my mind for 
what concerns the projected band structure, and in particular the correct 
definition of the direction of light propagation (and then the direction of 
projection of the band structure).
Are there any examples concerning this very tricky case? Do you know any 
references that clearly illustrate this issue? Indeed, it's very difficult to 
find in the literature a complete projected band structure of a woodpile 
waveguide (or better, up to now I haven't found it).



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