brillouin diagram
I encountered the attached dispersion diagram when reading a paper "Periodic Analysis of a 2-D negative refractive index transmission line structure" IEEE Transactions on Antena and Prop. Vol. 51 No. 10 Oct. 2003. It is said to be a brillouin diagram. Could anyone explain to me what the parameters such as G, M X stands for and how can we plot such kind of diagram.
Any good tutorials on this topics?
Thanks in advance.
The Brillouin diagram, more commonly referred to as the Dispersion or K-Omega diagram. This can be calculated by solving for the eigenvalues of the unit-cell structure. Many algorithms have been implemented for obtaining this. But mature commercial tool such as Microwave Studio, HFSS and BandSolve are available, saving us time and hazzle for programming. In metamaterial, we are dealing with periodic structures. Periodic structure displays a certain lattice. In your case, it is a square-lattice. The small diagram on your graph tells us that. It is the so-called Reduced Brillouin Zone, while the triangle is the reduced Brillouin triangle. This basic diagram gives us vital information. The square lattice has period or lattice constant of 'a'.
The calculation of the dispersion diagram involves three major steps (see your diagram along the x-axis). The x-axis in actual fact are of two phase values, namely ThetaX and ThetaY. I will be using degrees. (Note: You must come across this if you have ever simulate a unit-cell structure in any full-wave EM simulator. One had to assign periodic boundaries to all sides, or two-pairs of Master and Slave boundaries. The simulator will then ask for the Phase Shift of each of the two pairs of Periodic Boundary. Normally the values are left as 0 degree.)
1. G-X
Vary ThetaX from 0 degree to 180 degree.
Keep ThetaY fix at 0 degree.
2. X-M
Vary Theta Y from 0 degree to 180 degree.
Keep ThetaX fix at 180 degree.
3. M-G
Vary simultaneously both ThetaX and ThetaY from 180 degree back to 0 degree.
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When I say VARY, I tend to vary by 20 degree increments. Therefore, I end up solving the 28 data points or eigenvalues for each mode of interest. The more points you have the more refine the curve would be.
X-axis point 1: ThetaX = 0, ThetaY = 0 --- G->X starts, M->G ends
X-axis point 2: ThetaX = 20, ThetaY = 0
X-axis point 3: ThetaX = 40, ThetaY = 0
X-axis point 4: ThetaX = 60, ThetaY = 0
X-axis point 5: ThetaX = 80, ThetaY = 0
X-axis point 6: ThetaX = 100, ThetaY = 0
X-axis point 7: ThetaX = 120, ThetaY = 0
X-axis point 8: ThetaX = 140, ThetaY = 0
X-axis point 9: ThetaX = 160, ThetaY = 0
X-axis point 10: ThetaX = 180, ThetaY = 0 --- G->X ends, X->M starts
X-axis point 11: ThetaX = 180, ThetaY = 20
X-axis point 12: ThetaX = 180, ThetaY = 40
X-axis point 13: ThetaX = 180, ThetaY = 60
X-axis point 14: ThetaX = 180, ThetaY = 80
X-axis point 15: ThetaX = 180, ThetaY = 100
X-axis point 16: ThetaX = 180, ThetaY = 120
X-axis point 17: ThetaX = 180, ThetaY = 140
X-axis point 18: ThetaX = 180, ThetaY = 160
X-axis point 19: ThetaX = 180, ThetaY = 180 --- X->M ends, M->G starts
X-axis point 20: ThetaX = 160, ThetaY = 160
X-axis point 21: ThetaX = 140, ThetaY = 140
X-axis point 22: ThetaX = 120, ThetaY = 120
X-axis point 23: ThetaX = 100, ThetaY = 100
X-axis point 24: ThetaX = 80, ThetaY = 80
X-axis point 25: ThetaX = 60, ThetaY = 60
X-axis point 26: ThetaX = 40, ThetaY = 40
X-axis point 27: ThetaX = 20, ThetaY = 20
X-axis point 28: ThetaX = 0, ThetaY = 0 --- M->G ends, G->X starts.
Therefore, ThetaX and ThetaY are defined as variable in most commercial EM Solver. Such analysis is called Parametric Analysis.
Sometime, this can take a long time to run. However, in most situation, the "X-axis point 10" governed the width of the complete band gap and hence solving only the first 10 points are sufficient. But of course you are not interested in the band gap width, instead for LHM one is interested in the negative slope on along the curve of mode 1.
I hope this help! I cannot suggest an ultimate background reading material for you simply because I have not come across one. However you could try the following:
1. Introduction to Solid State Physics - Kittel et. al.
2. Wave Propagation in Period Structures - Brillouin (out of print text)
3. This forum!
P.S. I hope other users can explain this perhaps in another perspective and efficient way.
Dear sassyboy:
Thanks a lot for your patient and detailed reply. You are really an expert on this topic, at least to me, and I guess you probably have some experience in dealing with LHM, am I right? If so, hope we can have more discussions in the future as I just started the stuty of LHM.
BTW: do you have any tutorials/examples to show how this diagram can be obtained in HFSS? I will try to get the reference you recommended and pick up the background.
Thanks again and wish you a nice day!
Best regards,
A good introductory textbook on Bloch modes and Brillouin zones is
"Photonic crystals" by J.D. Joannopoulos, R.D. Meade, J.N. Winn
see h**p://www.amazon.com/exec/obidos/tg/detail/-/0691037442/102-0408423-1181746
Added after 5 minutes:
Some references you can find on this website:
h**p://ab-initio.mit.edu
you can also take a look on MIT Photonic-Bands software (h**p://ab-initio.mit.edu/mpb/), I think it can be used for LHM as well
h**p://www.emtalk.com/tut_2.htm
This website have 2 example about the dispersion diagram.
the commercial tool is HFSS
How do you define the source for a periodic structure. The plane waves are out of phase with neighboring unit cells when the k-vector is not zero. How do you get past this problem?
hi you have explained calculation of dispersion very nice but if there s triangular lattice which unit cell is hexagonal, then w can we plot the dispersion fr this type of lattice? Can you explain this?
thx very much
you can also check this page out. it was helpful for me.
http://en.wikipedia.org/wiki/Brillouin_zone
Eigenmode solvers (like the ones in HFSS & CST) calculates Floquet modes as the solution of fields in a periodic structure. These solutions, known as eigenvalues w(k) or simply the resonant frequency, are periodic functions of wavevector "k" in 1D, 2D or 3D. This result helps restrict the number of wavevectors that need to be analysed to completely characterize the periodic structure. The restricted region of wavevectors is known as the first Brillouin zone. All solutions obtained for wavevectors outside this zone will be equal to a solution of a wavevector within the zone. The irreducible zone removes redundancy in periodicity and structure symmetry.
For a 2D structure, this zone is a triangle with corners labelled Gamma, X, M and these corners correspond to regions of high symmetry.