18.325 Problem Set 3
d1
thickness
ε1 − ∆ε rea−d1 , repeated with
compute ∆ω as a func-
and that the
gion has thickness
Due Tuesday, 18 October 2005.
tion of
Problem 1: Perturbation theory
(ii)
(a) In class, we derived the 1st-order correction
the quarter-wave stack thicknesses?
in the eigenvalue for an ordinary Hermitian
Ô |ui = u |ui for a small per∆Ô. Now, do the same thing for
generalized eigenproblem  |ui = uB̂ |ui.
eigenproblem
Problem 2: Band gaps in MPB
turbation
a
Consider the 1d periodic structure consisting of
Â(0) |u(0) i = u(0) B̂ (0) |u(0) i
thicknesses
to an un-
both Â
and
and
∆B̂ .
B̂
le
when we change
∆Â
u(0) is
that is posted on the course
TM gap size (of the
vs.
(ii) Now, apply this solution to the gen-
d1
d1
for
d1
rst
gap, i.e lowest
ranging from
gives the largest gap?
0
to
a.
ω)
What
Compare to the
∇×∇×E =
ω
c2 εE for a small change ∆ε, and show
that the rst-order correction ∆ω is the
(b) Given the optimal parameters above, what
same as the one derived in class using
would be the physical thicknesses in order
eralized eigenproblem
quarter-wave thicknesses.
2
the
H
eigenproblem.
for the mid-gap vacuum wavelength to be
λ = 2πc/ω = 1.55µm?
L×L
metal box that we solved in class for the Hz
(b) Recall the problem of the modes in an
problem set 1 for the
Ez
cations.)
Now, suppose that we increase
small constant
∆ε
(c) Plot the 1d TM band diagram for this struc-
(TM) polarization.
Originally, this box was lled with air (ε
ture, with
=
ε by some
L
L
2 × 2
compute it for
ω)
special features does the quarter-wave band
diagram have?
(c) In class, we calculated the band gap
∆ω
Problem 3: Space group of k
that appeared in a uniform (1d) material
changed to
half
ε1 − ∆ε
(which I just
lines and the other bands as dashed). What
degenerate modes?
when
d1 = 0.12345
plots (plot the quarter-wave bands as solid
modes of
the TM polarization? What happens to the
ε(x) = ε1
given by the quarter wave
chose randomly), and superimpose the two
What is the rst-order
for the rst four (lowest
d1
thickness, showing the rst ve gaps. Also
in the lower-left
corner of the box.
(This is the wave-
length used for most optical telecommuni-
(TE) polarization, and which you solved in
∆ω
bandgap1d.ctl
(a) Using MPB, compute and plot the fractional
non-degenerate, for simplicity.
1).
and
web page.
by small amounts
You may assume that
d1
help you with this, I've created a sample input
perturbed system and nd the rst-
u(1)
ε1 = 12 and ε2 = 1, with
d2 = a − d1 , respectively. To
two alternating layers:
(i) That is, assume we have the solution
order correction
a, and
d1 .
At what d1 values does this gap at k =
0 disappear (∆ω = 0)? What is ∆ω for
period
a was
∆ε, by
of the unit cell
for some small
Consider the structure of problem 2. What are
using perturbation theory. In particular, we
the symmetry operations of its space group?
derived the gap between the rst two bands
How are these reduced (if at all) if we consider
k = π/a,
the edge of the
Θ̂k that is, what is the space group as a function of k ? In class, when we derived the band
gap, we took the modes at k = π/a to be of
(i) Now, you should compute the gap that
even/odd (cos/sin) form; why was this justied?
that appears at
Brillouin zone.
appears between band 2 and band 3 at
k = 0,
the
center
Give an example of a 1d periodic structure in
of the Brillouin zone,
which the modes at
to rst order. However, do it more generally: assume that the
ε1
k = π/a
are
not
even or
odd. Is the band diagram of this structure still
region has
symmetric (i.e.
1
ω(−k) = ω(k))?
Problem 4: Defect modes in MPB
your conjecture
(Don't forget that, like in problem set 2, as
In MPB, you will create a (TM polarized) defect
the mode becomes less localized you must
mode by increasing the dielectric constant of a
single layer by
∆ε,
increase the computational cell size.
pulling a state down into the
curately you know the location of the gap
the one from problem 2, with the quarter-wave
thickness
d1 = 1/(1 +
12).
edgecompute the gap edge using the same
To help you with
this, I've created a sample input le
resolution as you are using to compute the
defect1d.ctl
defect, and you may need to increase both
that is posted on the course web page.
no
(a) When there is
band diagram
resolutions.)
defect (∆ε), plot out the
ω(k)
for the
N =5
An-
other thing to be careful about is how ac-
gap. The periodic structure will be the same as
√
1 with numerical evidence.
(f ) The mode must decay exponentially far
supercell,
from the defect (multiplied by an
π
ei a x
sign
and show that it corresponds to the band
oscillation and the periodic Bloch envelope,
diagram of problem 2 folded as expected.
of course).
(b) Create a defect mode (a mode that lies in
eld computed
κ if the eld decays ∼ e−κx )
and plot this rate as a function of ω , for
the rst defect mode, as you increase ε2 as
above (vary ε2 so that ω goes from the top
ε of a single ε1 layer by ∆ε =
Ez eld pattern. Do the
by increasing a single ε2 layer.
increasing the
and plot the
same thing
Ez
decay rate (i.e.
the band gap of the periodic structure) by
1,
From the
by MPB, extract this asympotic exponential
of the gap to the bottom).
Which mode is even/odd around the mirror
plane of the defect? Why?
ε of a single ε2 layer,
ω as a function of ∆ε
(c) Gradually increase the
and plot the defect
as the frequency sweeps across the gap. At
what
∆ε
do you get two defect modes in
the gap? Plot the
Ez
of the second defect
mode. (Be careful to increase the size of the
supercell for modes near the edge of the gap,
which are only weakly localized.)
(d) Via rst-order perturbation theory, derive
an
exact
expression for the rate of change
dω
of the defect-mode frequency ω of a
d(∆ε)
single defect layer ε2 + ∆ε layer, as ∆ε
is changed.
Your expression should be in
terms of the eigeneld
E of the localized de∆ε. Verify your for-
fect state at the current
mula numerically by showing, at a couple of
∆ε, that it correctly preω vs. ∆ε curve above.
can export E to Matlab and
dierent values of
dicts the slope of your
(Note that you
compute the necessary integrals there or,
for the Scheme lovers among you, you can
compute the integral directly in MPB using
the compute-energy-in-objects or computeeld-integral function.)
(e) Is there a minimum
∆ε
(of an
ε2
layer) to
1 In
create a defect mode, or is a defect mode localized even for innitesimal
∆ε?
Support
mathematics, we never merely guess.
jecture, or perhaps we postulate an ansatz.
2
We con-