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I am trying to calculate a photonic crystal’s band structure for a very simply case of a 1-dimensional periodic medium, or rather I am trying to understand the calculations that I am looking at for this situation.
The end result of what I would like to do is get some function that I can graph and use to predict a photonic bandgap if one exists after specifying the needed parameters (lattice constant, index of refraction, …).
I am looking at the explanation / calculations here:
http://ab-initio.mit.edu/photons/tutorial/L1-bloch.pdf (~5 MB)
and here:
http://ab-initio.mit.edu/photons/tutorial/photonic-intro.pdf
But I don’t really understand what is going on or recognize the result when I see it.
I also am looking a derivation (paper copy, no link) which goes into much more detail.
In this derivation, a 1-dimensional PC’s band structure is analyzed by looking at the electric field in one of the layers in the crystal. Bloch’s theorem is applied and we get two eigenvalues (complex conjugates of each other) which look very similar to the one’s found in the above .pdf documents.
I guess what I am not understanding is how to turn these eigenvalues which are found into something more meaningful.
Also, I don’t really understand the significance of first Brillouin zone in all of this. I only have a vague idea of what it is and how to find it.
Does it really apply in the one-dimensional case? Would it not just be a line in 1-D as opposed to a square in a 2D square lattice, or a semi-round soccer ball looking thing in a FCC cubic lattice?
The end result of what I would like to do is get some function that I can graph and use to predict a photonic bandgap if one exists after specifying the needed parameters (lattice constant, index of refraction, …).
I am looking at the explanation / calculations here:
http://ab-initio.mit.edu/photons/tutorial/L1-bloch.pdf (~5 MB)
and here:
http://ab-initio.mit.edu/photons/tutorial/photonic-intro.pdf
But I don’t really understand what is going on or recognize the result when I see it.
I also am looking a derivation (paper copy, no link) which goes into much more detail.
In this derivation, a 1-dimensional PC’s band structure is analyzed by looking at the electric field in one of the layers in the crystal. Bloch’s theorem is applied and we get two eigenvalues (complex conjugates of each other) which look very similar to the one’s found in the above .pdf documents.
I guess what I am not understanding is how to turn these eigenvalues which are found into something more meaningful.
Also, I don’t really understand the significance of first Brillouin zone in all of this. I only have a vague idea of what it is and how to find it.
Does it really apply in the one-dimensional case? Would it not just be a line in 1-D as opposed to a square in a 2D square lattice, or a semi-round soccer ball looking thing in a FCC cubic lattice?