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2009. 04.
Hanjo Lim
School of Electrical & Computer Engineering
Lecture 3. Symmetries & Solid State Electromagnetism
2
Photonic crystals ; the structures having symmetric natures
important symm.; translat., rotat., mirror, inversion, time-reversal
symmetry of a system => general statements on system’s behavior,
classify normal modes. Example : a two-dimensional metal cavity with an inversion symm.
If is found, is an eigenmode of freq.
even mode
odd mode
proof) Inversion symm. means that metal pattern;
If is nondegenerate, must be the same mode with
with the eigenvalue of the inversion operation.
Likewise
)(rH
);,(
);,(),()(
1
11
rH
rHrHbemayrHThen
)(rH
.,.,1 ei
)(rH
),( rH
).(rH
),()( 1rHrH
)()( rmrm
),()()(),(),,()( rHrmrmrHrHrmif
)()( rHrH
1)()().()( 2 rHrHrHrH
3
Formal(mathematical) treatment of inversion symmetry
operator that inverts vector,
operator that inverts vector field and its argument
property of inversion symmetry ; for any operation
def) commutator operator of two operators and
Symm. system under inversion means with and any
;
;
IO
ILet
)()()()()() rfrfsystemaforrfrIfIrfOdef I
I
f
r
II OO
1
II
IIII
OO
OOfromorOO
1
1 ,1
A BAABBA ,
0,
IO .
IO
B
y
x
IO
IO
IO
IO
4
Let then means that
is an eigenfunction operator.
∴ If is a harmonic mode, is also a mode with freq
If there is no degeneracy, there can only be one mode per frequency
General aspect ; With two commuting operators, simultaneous
eigenfunctions of both operators can be constructed.
Ex) eigenvalue of classify eigenmode parity
What if there is degeneracy ? If then?
,)(
1
r 0)()(, HOHOHO III
),( 1rH
HOI
.1
HOieHOc
Hc
OHOHO IIIII
,),()()( 2
2
2
2
1, andHHOI
,)(,1; HHOO II
),,(),( 11112 HHrH
5
Continuous Translational Symmetry
def) A system with transl. symm.; unchanged by a translation operation through a displacement Let translational operator is for each
If a system is invariant under operation,
Let our system is translationally invariant, or
Then,
proof:
∴ The modes of can be classified by how they behave under
.d
dT ,d
dT )()()( rfdrfrfTd
).()()( rdrrTd
.)()(.,.,0, rfrfTeiitselftotransformsTTTT ddddd
)()()(, rHTrHTrHT ddd
)()()()(
1)(2
2
drrdrHr
rHc
Td
)()(
1)(2
2
drHdr
drHc
0)()( 2
2
2
2
drH
cdrH
c
.dT
6
def) a system with continuous translat. symm. ; invariant under all
ex) If is invariant under all the in the z-direction,
for any uniform plane wave propagation in z-direction
is the eigenfunction. Then
∴ Eigenvalue of operator ∴ We can classify the waves by
Note changes the phase only by
ex) If a system is invariant in all three directions (ex: free space),
eigenmodes; with any constant vector
Note) The eigenfunctions can be classified by their particular values for wave vector
Implication of on the plane wave : transversality condition
sTd '
)()( dzHzH
)(z
ikzo eHH
.)()( ikzikddzikikz
d eeeeT
.d
.; ikdd eT
ddzikikz Tee .)(
)exp()( rkiHrH ok
.oH
)exp( rki
),,( directionk
0 H
sTd '
/2k
./2 dkd
7
In conclusion, the plane waves are the solutions of master eq. with 3-dimensional continuous translational symmetry.
Proof;
∴ Master eq. becomes
fGGfGfrkiHH o
)(0)exp(
)exp()(
))(()exp()()exp( 00
rkikieakakaki
eeez
ay
ax
arkieHHrkiH
rkizzyyxx
zikyikxikzyxrki
zyx
kHrkikiH oo
0)exp(
)exp()( rkiHrH ok
)(/)()(/1 2 rHcrHr
0000 ))(exp()()( HkirkiHeHerkieH rkirki
)()/1()()(/1 0HkierHr rki
)()()()( 000 HkieHkieHkie rkirkirki
)()()( 0 baccabcbaHkikie rki
rkirki eHkkkHHkke 0
200 )()(
)exp()/()exp()( 02
0
2
rkiHcrkiHkrH
8
If
If
Holds.
∴ Spectifying ; propagation direction & => how mode behaves
ex) An infinite plane of glass with
Invariant under the continuous translation
operation in the x- or y-direction.
The eigenmodes should have the form with the in-plane
wave vector and that would not be determined by
the symmetry consideration only.
But a constraint on exists from the transversality condition
.)/()/2()/2(),(1 2222 ccfkfcspacefree
/2/,/,/,1 0 fkcnncvnindexrefractivewith g
.)exp()/()exp(// 02
0 rkiHcrkiHandck
k
);()( zr
y
z
x
)()( zherH ki
k
yyxx akakk
)(zh
.0 H
)(zh
9
∴ We can classify the modes by their values of and the band number
by if there are many modes for a given
Assume a glass plane of width and a mode with the x-polarized field
Then master eq. becomes
0)()(0)( zhkizhezh ki
k
.k
,., ein ),( nk
H
.01)()(,)()exp()(, azforzrandazyikrH xnynk y
xnyik
xnyik azecaze
zyy
)()/()(
)(1 2
00)(
)()()()()(
1
zzyx
aaa
azazeazez
n
zyx
xnxnyik
xnyik yy
y
nyikxn
yikyy a
dzzd
eazeaikz
yy )(
)()()(
1
y
nzny
yik adzzd
azikez
y )(
)()(
1
a
10
2
2 )(
0)(
0
)(
,)(00
))((*dz
zda
dzzd
zyx
aaa
adzzd
zikzyx
aaa
azik nx
n
zyx
yn
ny
zyx
xny
y
nzny
yiky
nzny
yik adzzd
azikez
adzzd
azikez
yy )(
)()(
1)()(
)(1
y
nzny
yikyiknx
yik adzzd
azikz
eezdz
zdae
zyyy
)()(
)(1
)(1)(
)(1
2
2
y
nznyyy
yikx
nyik adzzd
azikz
aikz
eadz
zde
zyy
)()(
)(1)(
)(1)(
)(1
2
2
)(
1)(
1)(/1*zdz
dazz
ay
ax
az zzyx
xnyikn
xxnyyik
xnyik azec
dzzd
zdzdaaz
z
kea
dz
zde
zyyy
)()/(
)()(
1)()(
)()(
1 22
2
2
0
11
∴ Master eq. becomes
Let If that is,
evanescent wave to the air, i.e.,
confined wave in the glass => discrete modes(bands)
If traveling wave
extending both in the glass and the air region.
The separation of continuous states and
discrete bands at light line.
)()/()()(
)()(
1)()(
1 22
2
2
zczz
k
dzzd
zdzd
dz
zdz nn
ynn
dzzd
zdzd n )(
)(1
)()(
)()(
1)(
12
22
zcz
k
dzzd
zdzd
r nyn
.)(
22
22
cz
k y
.0)(
1, 2
dzd
zdzd
glassy ),exp();( zz
:/ yck
.0)(
1,.,.,022
dzd
zdzdei glassy ),exp();( ziz
,/2/22
/2.,.,000
2glassy
y ff
ckei
12
If is large, i.e, the wave of short or propagation more in z-direction,
and the wave is well confined in the glass
Let then with
These modes decay ever more rapidly as increase, since Discrete translational symmetry
Photonic crystals actually have discrete translational symmetry(DTS).
ex) 1D PhC : DTS for 1D and CTS for 2D, 2D PhC : DTS for 2D and CTS for 1D, 3D PhC : DTS for 3D.
ex) Fig. 4: 2D PhC with primitive lattice vector for with integer and unit cell: xz slab with the width in the y-direction
yk
,)(
1 2
dzd
zdzd ).0( az
),()( glassz ,022
2
dzd
0)()0( a
.sincossin zAzBzA
.,./0)( 2
22
2
22
2
222
an
c
k
anana y
.2
222
c
k y yk
,l
alR
a
)()(,ˆ rRryaa
0
13
That is, photonic crystals are composed of repetition of unit cells.
Tanslational symmetry means that and with
Eigenmodes of simultaneous eigenfunctions of and
∴ Modes can be classified by specifying and values.
But not all the values of yield different eigenvalues. With
and have
the same eigenvalue for if
∴ Any with integer gives identical eigenvqlue of
and is a degenerate set. That is, the addition of an integral multiple of on leaves the state unchanged.
called as “primitive reciprocal lattice vector”.
0, ˆ xdT
ylaR ˆ
;
xk
xdT ˆ RT
.;)(,)(,)( )()(ˆ eigenvalueeeeeTeeeeT yiklaiklayikyik
R
xikdikdxikxikxd
yyyyxxxx
yk
yk
.)()(2
ykilaikykilaa
ilaikykilakiyki
Ryyyyyy eeeeeeeeT
)exp( yik y )exp( yki
)exp( laik y RT ./2 akk yy
0, RT
,2a
kk yy
amkk yy
2 mRT
ykei .,.
),( laik ye
yk
ab 2 yk
:ybb
y
z
x
14
Bloch theorem, Bloch ftunction and Brillouin zone
Any linear combination of degenerate eigenmodes for and
is an eigenfunction. Therefore, the general solution of a system having a
DTS(CTS) in the y-direction (x-direction) is
periodic ft. in y-dir.
Bloch theorem ; a wave propagating through the periodic material in the y-direction can always be expressed as or more generally if dielectrics periodic in 3D.
Note) Discrete periodicity in the y-direction gives that is simply
the product of a plane wave in the y-direction and a y-periodic ft.
and thus, Thus mode frequencies must also be periodic in
m
imbymk
m
yikxikymbkimk
xikkk ezceeezcearH
y
yxy
y
x
yx)()()( ,
)(,,
yk mbkk yy
directiononpolarizatiyinftperiodiczyuLet
yk.);,(
);,()()(),( ,
2)(
, zyuezceezczlayum
imbymk
laa
im
m
layimbmk yy
rki
kerur
)()(
),,( zyxH
),,,(),,( zyuezyxHy
y
kyik
.ftBloch
.yy kmbk HH
15
so that ∴ Knowl. about (1st BZ) is sufficient.
When the dielectric is periodic in 3D, the eigenmodes have the form of
with the inside the first BZ and a periodic ft.
satisfying for all lattice vector Photonic band structures
EM modes of a photonic crystal should have a Bloch form
and all the informations about such a mode is given by and
To solve for let’s start from from the master eq.
ak
a y
.R
,yk
)()( ruerHk
rki
k
k
)(ru
k
)()( ruRrukk
rki
kkerurH
)()(
yk ).(ruk
.,.,/)(2
eiHckHkk
),(ru
k
).(/)()()(
1 2rueckrue
r k
rki
k
rki
).()( yy kmbk
)()()()()()(
1)()(
1 rukierueruer
ruer k
rki
k
rki
k
rki
k
rki
)()()(
1)()()(
1 rukir
erukir
ekik
rki
k
rki
16
∴ Master eq. becomes or
with
∴ Solving this eigenvalue problem for the unit cell & for each value of
=> photonic band structure
Restricting an eigenvalue problem to a finite volume leads to a discrete spectrum of eigenvlues (ex: nearly free electronics in the 1st BZ).
∴ For each value of an infinite set of modes =>band index
has the only as a parameter in it. Thus is a continuously
varying ft. with for a given
k
)(kn
)(/)()(2
ruckrukkk
)()()(
)()(
1)(
ruRrusatisfyingruioneigenfunct
kir
ki
kkk
k
),(/)()()()(
1)(2
ruckrukir
kikk
k )(kn
,k
k
.n
.n
k
17
Rotational Symmetry and Irreducible BZ.
- Phonic crystal : usually have rotational, reflection, inversion symmetry.
ex) Assume a PhC with a 6-fold rotational symmetry.
Let the operator rotates vectors
by an angle about the
To rotate a vector field we need to transform so that and
1r
5r
3r
4r
2r
6r
)( 1rf
)( 2rf
)( 6rf
),ˆ( n f
.ˆ axisn
f
f
1r
6r
),(rf
ff
.1rr
)( 11 rf
)( 22 rf
1r
3r
2r
6r
)( 22 rf )( 11 rf
1r
2r
6r
rotationfieldafterrotationfieldbefore
18
def) vector field rotational operator
also satisfies the master eq. with the same eigenvalue as
Note) State is the Bloch state with
Proof; We need to prove
(sub proof) Without loss of generality (WLOG), let rotation
about the origin through the angle in the xy-plane.
Let displacement vector is then with the translation operator
)(/)()()(
0,2
knnknkn HOckHOHO
rotationthetoriantvaissystemtheifOthen
nkHO
nk
H
nkHO
).()(.,., )(
nk
Rki
nkRHOeHOTeikk
)()(: 1rfrfOO
cossin
sincos
cossin
sincos),0(
yx
yx
y
x
y
x
y
x
.)(.,.),()( 11111 RRRnkRnkRTTOOOTOeiHTOHOT
,
b
aR
),,0(
RT
19
corresponds to operator and to vector
Since is the Bloch state with and same eigenvalue as
it follows that
b
a
y
xM
1MLet
)cos()sin(
)sin()cos(
cossin
sincos111
Obyx
ayxO
y
xOTO
R
byx
ayx
cossin
sincos
cossin
sincos
11
M
b
aM
y
x
RkRkRk
baba
11
):.(.11 RrrTTofdefnoteQED
y
xTR
y
xRRR
)()()( )( 1
1 nk
Rki
nkRnkRHeOHTOHOT
)()( )()( 1
nk
Rki
nk
Rki HOeHOe
nkHO
).()( kk nn
k
1
b
a.R
,nk
H
20
In general, whenever a photonic crystal has a rotation, mirro-reflection,
or inversion symmetry (point group) have that symmetry as well.
Full symmetry of the point group => some regions of BZ have repeated
pattern => irreducible BZ( the smallest region not related by symmetry).
ex) -Real lattice has 4-fold symmetry and
reflection symmetries
-Field patterns in real space or
in the rest of the BZ is just the copies
of the irreducible BZ.
Mirror symmetry and Seperation of Modes
Mirror reflection symmetry => Separation of the eigenvalue equation for
into two separate equations ( to mirror plane) => Provides
immediate information about the mode symmetries (ex: Fig. 4).
)(kn
k //,,,
kkHE
)(kn
nkH
21
Mirror reflection in the changes to - and leaves and
Mirror reflection in the changes to - and leaves and
For a system to have mirror symmetry, it should be invariant under the simultaneous reflection of and
Def) mirror reflection operator
ex) Note 1)
with and thus
Note 2) with the reflected
wave vector and an arbitrary phase
Proof : Since
also satisfies the master eq. with the same eigenvalue as We thus need
to prove that
.z
.r
)()(; rMfMrfOO xxMM xx
f
xM planeyz x x
yM planexz y y x
y
.z
).()( rfrfOOxx MM
)()( rfrfO
xM
,12
kM
i
kMMxxx
HeHOO
0,r )(rf
xM
rM x
)( rMfM xx
./)(/,0,22
nkxnkxnkxnkxM HMckHcMHMHMOx
nkxHM
.nk
H
),()( 1 nkRMxnkxRHTMHMT
x
).( 1xx MMnote
RMRMxxxRxxx
TTMMMTM 11
11
kM x
1
22
Since transforms to we may take
in 2D space. Let then
Thus is the Bloch state with the reflected wave vector
Note that we can always take a mirror plane so that since our
dielectric has CTS in -direction. But only for a certain and
If the Bloch wave propagates in the
from obeys similar eq.
∴ Both and must be either even or odd under the operation.
But, is a vector, is pseudovector. Thus -even mode must be
and while the –odd modes must have the components and
xM ),,( zyx ),,,(),,( zyxzyx
10
01xM
,
b
aR
y
xT
b
aM
y
x
by
axM
y
xMTM
RMxxxRxx
1
111
kM HOx
.kM x
).()()( )()( 11
1 kxRkMi
kxRMki
k
RMkixkRMxkxR
HMeHMeHeMHTMHMT xxx
x
).()()( rHrMHMrHOkxkxkM x
:)(rE
k
kE
kH
xMO
H
,rrM x
x rrM y
r
.yM
E
xMO yx EH ,
,zE yx HE , .zHxMO
.,., eikkM x
,planeyz )()( )( ruerH
k
rki
k
,rrM x
23
Difference of behaviors between vector and pseudovector under inversion
operation (coordinate transform) and mirror operation (world transform).
In general, for a given mirror operator such that mode
separation is possible at the position where for according to
the polarization depending on whether or is parallel to mirror
(ex: TE and TM modes in 2D PhCs).
But this mode separation concept is not so useful for 3D PhCs. Time-Reversal Invariance
Since is Hermitian and is real, complex conjugate of master eq.
is given by
satisfies the same eq. as with same
is just the Bloch state at and thus,
rrM
,0, MO
,kkM
kH
kE
*
2
2**
2
2* )(
.,.,)(
)(nk
nnknk
nnk
Hc
kHeiH
c
kH
k )(2 kn
*
nkH
nk
H
).(2 kn
*)(*)( ).()(nkk
rki
nkk
rki
nkHrueHrueH
),,( nk
M
M
24
holds independent of the photonic crystal structure.
Note)
If we take such that
Thus taking is equivalent to taking time as is a
consequence of the time-reversal symmetry of the Maxwell eqs. Electrodynamics in PhCs and electrons in crystals
Formation of energy bands and energy gap Eg in semiconductors: related
to the periodicity of crystals.
Schroedinger eq. is with for any
translational vector Hamiltonian has the
translational property such that since the kinetic term is
invariant under any translation.
).( t*
nkH
)()(** )()()(),( trki
k
ti
k
rkiti
nkeruerueerHtrH
)()( kk nn
)()( kk nn
)()( )()()(),( trki
k
ti
k
rkiti eruerueerHtrH
*
nkH
)()()()2/( 22 rErrVm
)()( rVRrV
)()2/()( 22 rVmrH
.R
),()( rHRrH
1)()()( )( TablerereRr RkiRi
