18.325 Problem Set 6
a line defect formed by a missing row of rods in
a triangular lattice of rods.
Due Thursday, 17 November 2005.
(a) Change
Problem
1:
Brillouin
zones
ε
to 8.9 to match problem 1. Com-
pute and plot the TM projected band dia-
and
gram of this mode. By increasing the super-
band diagrams
cell size (which folds more and more bands
In class, we derived the irreducible Brillouin zone
in the continuum regions but leaves the de-
for a lattice of cylindrical dielectric rods in air
fect mode unchanged), identify the contin-
with lattice constant
a:
uum regions on your plot (the projection of
either a square lattice,
where the lattice vectors dier by
◦
90
the bands of the perfect crystal).
, or a tri-
angular lattice, where the lattice vectors dier
by
60◦
60◦
120◦ ,
(in class we used
(b) Sketch the Brillouin zone of the triangular
but you can get
lattice of rods, and show how it is projected
just by ipping the sign of one of the lattice
for the line defect. Careful: for the line de-
vectors).
(a) If
fect, we have to project this onto the
you
look
in
the
MPB
tutorial
but the edge of the new 1d Brillouin zone is
(http://ab-initio.mit.edu/mpb/doc/user-
not
tutorial.html), it shows you how to compute
these
calculations,
but
use
K.
Where is the edge of the projected
1d Brillouin zone?
the TM band diagrams in these two cases.
Repeat
Γ−K
direction (the nearest-neighbor direction),
rods
(c) Consider the TM bands for the triangu-
with radius
lar lattice that you computed in problem
than the
r = 0.2a and ε = 8.9 rather
ε = 12 in the tutorial, and nd
1.
the size of the rst TM gap for both the
the irreducible Brillouin zone for the line-
square and triangular lattices.
Project each one of these bands onto
(See also
defect waveguide, and superimpose its pro-
the sq-rods.ctl and tri-rods.ctl les in the
jected plot onto your the projected band di-
/mit/mpb/examples directory.)
agram you computed in (a). For each band
(b) Now,
consider
lattice
of
the
an
intermediate
same
circular
case:
rods
in
of the original band diagram, you should
a
have more than one projected band, corre-
air
sponding to all rotations/reections of the
where the primitive lattice vectors both
have length
75◦ .
original band (e.g. there are six equivalent
a but the angle between them is
rotations of the
Figure out what is the Brillouin zone in
this case, and the irreducible Brillouin zone,
direction, although
−k
axis and
can be omitted.)
and then run MPB to compute the size of
the rst TM gap.
Problem 3: Galerkin methods
Be careful about units (see the note on units
in the MPB tutorial):
Γ−K
some of these project onto the
in MPB, the
k
Consider the 1d Schrodinger equation for an
points
eigenvalue (energy)
are specied in the basis of the reciprocal lat-
tice vectors. If you work out the Brillouin zone
in Cartesian coordinates, you can convert to the
reciprocal basis by dividing by
E:
d2
− 2 + V (x) ψ(x) = Eψ(x),
dx
2π and calling the
ψ(0) = ψ(L) = 0,
V (x). Now suppose
(cartesian->reciprocal (vector3 x y z)) function
with boundary conditions
in MPB as described in the reference section of
for some smooth potential
the manual.
we want to express this problem on a computer
via a discrete grid of
∆ = L/N ,
Problem 2: Line-defect modes
N −1
points with spacing
using a basis of tent functions
centered at each grid point:
For this problem, you should make use of the le
/mit/mpb/examples/line-defect.ctl in the MPB
ψ(x) ≈ ψN (x) =
locker on Athena, which computes the bands of
N
−1
X
n=1
1
pn t(x − n∆)
t(x)
with some unknown coecients
t(x) =
pn ,
where
plot them (on a single plot).
The trans-
mission spectrum should be normalized by
1 − |x|/∆ |x| < ∆
.
0 otherwise
ψN (n∆) = pn (i.e.
grid points) and ψN
dividing by the transmission for nx=0 (no
holes). Relate the features of this transmission spectrum to the band diagram of prob-
pn are the values at
Thus,
the
the
is linearly interpolated
lem 1.
(b) Compute the TE transmission spectrum for
in between grid points. Note that the boundary
nx=10 layers, and relate it to the TE band
conditions are automatically satised.
diagram (which you can compute yourself
(a) Using the Galerkin approach, express the
with MPB, or you can look up from the
Schrodinger equation above as an
band diagram in the handout). Careful: you
(N − 1) matrix eigenvalue
EBp in terms of Hermitian
B.
(N − 1) ×
problem Ap =
matrices A and
need to change three places in the control
d2 /dx2
le. (What happens to the symmetry?)
(c) Try making the pulse frequency spectrum
operator just turned
very broad, e.g. df=2 (computing the ux
into a discrete-grid (nite-dierence) ver-
in a correspondingly wide frequency range),
sion of the 2nd derivative. (Hint: in evalu-
and plotting the TM transmission spectrum;
ating this integral in the Galerkin approach,
in the original range from (a), does the new
it is probably easier if you integrate by parts
transmission spectrum match what you had
to avoid taking the derivative of a discontin-
before?
(b) Show that the
uous function.)
(c) For
V (x) = 0,
(d) Predict analytically at what frequency
recall that we solved this
orders in the reected wave (i.e.
equation in class in the rst week: the so-
ψ(x) = sin(mπx/L), with eigenEk = (mπ/L)2 for m = 1, 2, 3, · · ·.
Compute your matrices A and B in this
case for L = 1, V (x) = 0, and N = 10,
0◦
values
reection). Now, modify the simulation
to use a TM continuous-wave (CW) source
and output
Ez
at the end, as in the Meep tu-
torial, and show that there is a qualitative
and plug them into Matlab to get the cor-
change in the reected eld pattern if you
responding 9 eigenvalues (with the eig(A,B)
put in a frequency just below
command), and give the fractional error in
quency just above
each eigenvalue compared to the analytical
ω0 ,
result. Which is most accurate, and why?
ω0 .
ω0 versus a fre-
If you look just below
then you will have to increase the pad
parameter in order to see an undisturbed
0◦
Problem 4: Meep!
reection pattern far from the crystal
why?
For this problem, you will use the Meep nitedierence time-domain code, which is installed
on the Athena Linux/Intel machines in the mpb
locker (add mpb); see also the Meep manual at
http://ab-initio.mit.edu/meep. As your starting
point, you should use the rod-transmission.ctl
example le, which is posted on the course web
page, which computes the transmission spec-
x
reected
waves at angles in addition to the normal
lutions are
trum of TM planewave source in the
ω0
you should start to see additional diracted
direction
through nx layers of the square-lattice rod crystal from problem 1.
(a) Compute the transmission spectrum as a
function of nx, for nx=1, 2, 3, 4, 5, 6, and
2