(a)
(a)
(b)
(b)
ε=1
?
ε=1
ε=2.25
N periods
(air)
J
?
ε=2.25
(semi-infinite)
(semi-infinite)
(metal)
(a)
Figure 2:
(a) A pentagonal air cavity surrounded
xy plane). (b) The same, with a
Figure 1:
rectional reector with respect to incident light
by metal (2d,
from air (ε
J at the indicated location (with J
point current
Schematic of a multilayer mirror
we suppose that our structure is an omnidi-
= 1)
, but not for
ε = 2.25
(for
which there is some reection and some trans-
pointing in the indicated direction).
mitted beam(s)).
N
18.325 Mid-term Exam
(b)
ε = 2.25, then
ε = 1, and want
We have
periods of the mirror, then
to know whether (and, qualitatively, how much)
light is transmitted through to the
Problem 1
ε = 1.
Suppose that we have a 2d metallic electro-
label them with 1/2, 2, etcetera accord-
magnetic cavity in the shape of a regular pen-
ing to whether their length is half, twice,
tagon (ve equal sides and angles) as shown in
etcetera of the original current amplitude.)
Fig. 1(a).
(c) Show how you can place a
gle
and character table for this system.
one
for
this
group
ple
2 × 2
matrices;
ends
up
coordinate
e.g.
sponds to
that
irreducible
a
cos θ
sin θ
being
the
single
sim-
by θ
rotation
.
J
dierent
to excite elds with a
two -dimensional irreducible represen-
tation.
rotation/reection
− sin θ
cos θ
Show how you can place a
single point current
representation
point cur-
one-dimensional irreducible representa-
tion.
Hint:
single
J to excite elds transforming as a sin-
rent
(a) Give the space group, conjugacy classes,
corre-
Problem 2
Also note
√
5−1
=
≈ 0.30902
cos( 2π
5 ) √
4
1+ 5
= − 4 ≈ −0.80902.
(a) We have a certain multilayer lm consisting
and
of two repeating layers of
cos( 4π
5 )
ε1 and ε2 .
Suppose
that this structure, repeated semi-innitely,
does
J as
J(x, y) is given
with the vector J
(b) Suppose that we have a point current
form an omnidirectional reector (all
shown in Fig. 1(b). (That is,
polarizations and angles) at a frequency
by a Dirac delta function
from an ambient medium
lying in the
xy
not
plane.) Decompose this into
ε = 1,
but
form an omnidirectional reector at
ε = 2.25,
ω
does
ω
partner functions of your irreducible repre-
from an ambient medium
sentations from above, using the projection
cated schematically in Fig. 2(a). Now, sup-
operators:
pose that we have semi-innite
lowed by
P̂ (α) =
dα X (α) ∗
χ (g) Ôg
|G| g
N
as indi-
ε = 2.25
fol-
bilayers of our periodic structure
followed by semi-innite
ε = 1,
as depicted
in Fig. 2(b). If a frequency-ω planewave is
incident on the layers from the
(The result should be a sketch, expressing
ε = 2.25
medium, how does the reected light and
ε = 1)
the original sketch as a sum of several pen-
transmitted (into the
tagon sketches with arrows at appropriate
on
places in the pentagons. Don't try too hard
tative description is sucient, but must be
to get the
clearly justied.)
length
of the arrows to scale; just
1
N,
light depend
angle, and polarization?
(A quali-
1.4
x
p-polarized
Frequency ωa/2πc
1.2
s-polarized
?
Λ
1
y
0.8
0.6
0.4
B
U
L
0.2
0
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
(semi-infinite)
Wave vector kya/2π
Figure 4: Reection(?)
Figure 3: Projected band diagram for a multi-
from a multilayer lm
layer lm vs. surface-parallel
whose surface is periodically corrugated with pe-
larization on the left, and
ky . TE ( p) poTM ( s) polarization
riod
on the right.This structure is an omnidirectional
from above (air) with some angle and polariza-
reector from the frequencies
spect to air (ε
= 1).
L
to
U
Λ
along the
y
direction.
Light is incident
tion.
with re-
Light line of air is shown in
red.
Problem 3
(b) More specically, suppose that the periodic
(that is, Hermitian under the usual inner product
Suppose that we have a
∗
n xn yn for
with eigenvectors:
hx|yi =
structure has the projected band diagram
shown in Fig. 3. Now, suppose we periodically
corrugate
the surface layer as shown

 
1
1
 0   0
 , 
 0   0
−1
1
in Fig. 4, parallel to the interface, with a
period
Λ
(uniform in the
z
P
direction, out of
plane).
4×4 Hermitian matrix O
4×1
vectors
xy
dent (from air) in the
 
0
0
  1   1
,  , 
  1   −1
0
0
picted in Fig. 4, for what
Λ
or
Λ's
or



O by the small

0 0 0 0
 0 δ 0 0 

∆O = 
 0 0 0 0 
0 0 0 0

erty maintained? (If you need to refer
ω

∆O:
is the omni-directional reection propto a particular numerical value
|yi),
1, 2, −1, −1.
(a) Now, suppose that we change
plane as de-
and
 
and corresponding eigenvalues:
(i) If we consider only planewaves inci-
|xi
ky
in Fig. 3, you can simply label the corresponding point on Fig. 3 rather than
where
trying to work out what the number
in the eigenvalues (to rst order in
is.)
the approximate new eigenvectors (to zero-
δ 1.
th order in
(ii) What about if we consider planewaves
Give the approximate change
δ ),
and
δ ).
which omnidirectional reection is pre-
new largest and
larger or smaller than
the corresponding exact eigenvalues? Why?
(Don't try to compute the exact eigenvalues,
served for
which would require you to nd the roots of
that are not in the
xy
(b) Are
plane (i.e. they
Is there any (nite,
all
your
smallest
k component)?
nonzero) Λ for
have some out-of-plane
three-dimensional angles
approximate
eigenvalues
a quartic equation.)
and polarizations? Why or why not?
2
(semi-infinite)
0.8
Frequency ωa/2πc
0.7
0.6
0.5
0.4
1/2 layer
TE modes
0.3
ε=1
M
0.2
0.1
X
Γ
TM modes
1/2 layer
0
Γ
X
Γ
M
Figure 5: TE (red) and TM (blue) band diagram
(semi-infinite)
of a square lattice of dielectric cylinders (inset) in
air, around the edges of the irreducible Brillouin
zone (inset).
Figure 6:
Two semi-innite multilayer mirror
separated by some distance of air
(ε = 1,
less
Problem 4
than both of the mirror materials). Each lm is
Consider the square lattice of dielectric rods,
for an isolated semi-innite structure would sup-
whose band diagram and irreducible Brillouin
port a single TM surface state in the rst gap.
terminated with half a high-index layer, which
zone are shown in Fig. 5.
Now, suppose that
we change the shape of the rods from cylindrical to
pentagonal
band diagram as a function of the surface-
(with the pentagons oriented
parallel
point-upward as in Fig. 1, keeping the area of
ky
(up to the rst gap,
i.e.
the rst
two continuum regions, is enough).
the rods the same (you can assume that this is
(That
is, the structure looks like the top half of
therefore a small perturbation).
Fig. .)
(a) What is the new irreducible Brillouin zone?
(b) Now, suppose that we have
two
such semi-
innite structures as shown in Fig. 1,
(b) Will any degeneracies remain in the band
diagram? If so, which?
that they would, by themselves, have surface states.
(c) Consider a solution in the perturbed (pentagonal) structure with
ment between
Γ, X ).
Γ
and
X
k
rated by
on the line seg-
(exclusive, i.e.
k 6=
k
between
Γ
and
one another, and sketch how this changes
as they get closer to one another.
What
M?
(a) Suppose we have a semi-innite 1d periodic
ε1
and
ε2
bounded by
ε=1
on
one side, we look at its TM modes only, and
we terminate it (with half of the high-index
layer) so that it has a single surface state
in the rst gap.
Clearly
indicate what happens to the surface states.
Problem 5
structure of
Sketch what the
they are some large but nite distance from
electromagnetic energy propagating (or directions, depending on the band)?
In between them is air (sepa-
ε = 1 < ε1 , ε2 ).
TM projected band diagram looks like when
In what direction is the solution's
about for
both
terminated with half a high-index layer so
Sketch the TM projected
3