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2009. 04. Hanjo Lim School of Electrical & Computer Engineering hanjolim @ajou.ac.kr. Lecture 2. Basic Theory of PhCs : EM waves in mixed dielectric media and Eigenvalue approach. Maxwell equations are given as; Constitutive relations ; relations between - PowerPoint PPT Presentation
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1
2009. 04.
Hanjo Lim
School of Electrical & Computer Engineering
hanjolim@ajou.ac.kr
Lecture 2. Basic Theory of PhCs : EM waves in mixed dielectric media and Eigenvalue approach
2
Maxwell equations are given as;
Constitutive relations ; relations between
in optical freq. range for most materials
and especially for dielectric materials.
∴ Maxwell equations are given as;
medialosslessandisotropiclinearifEOEEkEDj
kjijkj
jiji ,,)( 3
01,4
tB
cED
.014,0 unitscgsintD
cJ
cHB
)()(
1)()()(
),(),(),()()()(
rHrB
rHrrB
rErrDorrErrD
0),(1),(,0),()(
ttrH
ctrEtrEr
0),()(),(,0),(
ttrE
crtrHtrH
etcEJHBED ,&,&,&
3
Then ; complicated functions of time and space.
But Maxwell eqs. are linear => time dependence can be expressed
by harmonic modes.
Then mode profiles of a given frequency are given from Maxwell
equations. ; Nonexistence of source or sink
Field configurations are build up of transverse EM waves.
Transversality : If
titi erEtrEerHtrH )(),(,)(),(
HandE
0)()(0),(1),( rH
cirE
ttrH
ctrE
0)(,0)( rDrH
0),exp()( karkiarH
0)()()(0),()(),( rEr
cirH
ttrE
crtrH
4
Take main function as magnetic field
Master eq. with condition completely determines
* Schrodinger equation :
eigenvalue problem => eigenvalue and eigenfunction
For a given photonic crystal master equation => eigen modes.
If modes for a given
0)( rH
)()(0)()(
),(rH
cirErE
ci
rtrH
equationMasterrHc
rHr
;)()()(
12
)(rH
:)()()(2
22
rErrVm
)(r
),(r
)()(
)(, rHri
crEknownare
),( rH
E
5
Interpretation of Master equation ; Eigenvalue problem
operator eigenvalue eigenvector if is allowed.
with
operation on => eigenvector & eigenvalue
eigenvectors ; field patterns of the harmonic modes.
Note) operator ; linear operator wave eq.; linear differential eq.
∴ If and are two different solutions of the eq. with same
general solution of
)()()(
12
rHc
rHr
)(rH
)(
)(1)( rHr
rH
)()(2
rHc
rH
)(rH
2/c
)(rH
)(1 rH
)(2 rH
;)(/)( 2 rHcrH
)()()( 21 rHrHrH
,
)(rH
6
∴ Two field patterns that differ only by a multiplier ; same mode.
Hermitian property of
def) inner product of two vector fields
Note that
Proof :
Note that
If called normalized mode, Normalization of with
def) Hermitian matrix (self-adjoint)
adjoint Hermitian
)(
1
r
***** )()()()()()()()(),( rFrGrFrGrGrFrGrFrdGF
realalwaysrFrdrFrFrdFF ;)()()(),(2
*
;1),( FF
),()()()(),( ** rGrFrddvrGrFGF
.),(),( *FGGF
jiijijjiij AAAifAA ** ,
F
1),( FF
*** ,)()( FGrFrGrd
7
def) Hermitian operator
for arbitrary normalizable functions .
Properties)
1. If operator is Hermitian are Hermitian.
2. A linear combination of Hermitian operators is a Hermitian operator.
3. The eigenvalues of a Hermitian operator are all real.
Proof; Let
If Hermitian operator
4. Any operator associated with a physically measurable quantity is Hermitian (postulate).
dvQdvQ baba ***
HermitianQdvQdvQQdvQ bababa ;)( 2*2**2* ...,, 32 QQQ
Q
realqqq nnn ;*
.nnn qQ
dvqdvqdvQ
dvqdvqdvQ
nnnnnnnn
nnnnnnnn
****
***
)()(
8
def) Hermitian operator for vector fields and
If that is, the inner
product of –operated field is independent of which function is
operated, Hermitian operator.
Proof of is Hermitian operator.
,)(.,.),,(),( ** GFrdGFrdeiFGGF
)(rF
)(rG
)(
1r
:
**** )()(
1),( FFFG
rFdvGF
v s
ndsadvtheoremdivergenceFdvFdv
)()( **
let
0)(
1
)(
1* **
dsaFG
rFG
rdv n
v s
9
Note ; 1) zeros at large distances due to dependence
2) periodic fields in the region of integr. (∵ harmonics)
After integration,
)(
)(
1),( *FG
rdvGF
32 /1,/1 rr
*
*
)(
1
)(
1)(
F
rdvG
rFdv
GF
&
G
rFdv
)(
1)( *
)()()(
1)(
1)()(
1 **
* GFr
Fr
GGFr
)()(
1)(
)(
1 *
*
Gr
FGFr
operatorHermitianaisr
)(
1
),( GFG
10
Note ;
since is not a constant.
General properties of harmonic modes
1) Hermitian operator eigenvalue must be real.
Proof)
Note that for any operator
G
rFFG
rG
rF
)(
1)(
)(
1
)(
1)( ***
*
**
)(
1)(
)(
1
)(
1
F
rGF
rGFG
r
Gr
Gr
)(
1
)(
1
)(
1
r
),()(2
rHc
rH
2
c
)()(),(),( *22
rHrHrdc
HHc
HH
),(),(),(*
2
2*
*
2
2* HH
cHH
cHH
*),(),( HHHH
11
Proof)
Hermitian operator ;
Then
Note) ; is actually positive =>
If becomes imaginary in some frequency range, what dose it mean?
Proof) From
****** )()(),( HHrdHHrdHHrdHH
),(),(),( ** HHHHHHHHrd
),()/(),(),( 2 HHcHHHH
),)(/(),(),(),(),(),,()/(),( 22****22* HHcHHHHHHHHHHcHH
realeicc ;,.,)/()/( 2*2222*22
operatoranyfor),)(/( 22 HHc
operatorHermitian
2 .; real
HGFLetGr
FrdGr
FrdGF
)(1)(
)(1),( **
12
Then
positive positive
positive
If is negative in some frequency range, ; imaginary. Meaning?
2) Operator is Hermitian means that and with different frequencies and are orthogonal.
Proof) let than
Hermitian
2
*
)(1)(
)(1)(),( H
rrdH
rHrdHH
realpositivepositivec ;;;)/( 22
)(1 rH
)(r
),,(),,( 2211 rHrH
),()/( 2 HHc
)(2 rH
1 2
).()/()( 22
22 rHcrH
),()/()( 12
11 rHcrH
),(),(; 1212 HHHH
),(),(),(),( 122212
212
212
21 HHHHcHHcHH
0),(,0),)(( 12211222
21 HHifHH
13
Orthogonality & modes
Meaning of orthogonality of the scalar functions => normal modes.
Meaning of orthogonal vector fields :
Meaning of orthogonal vector modes. Degeneracy : related to the rotational symmetry of the modes.
Electromagnetic energy & variational principle => qualitative features
Def) EM energy functional
if is a normal mode.
The EM modes are distributed so that the field pattern minimizes the
EM energy functional
Proof) When
0),( GF
),,(2/),()( HHHHHE f
)(rH
fE
0;)()(,)()( HEHHEEHrHrH fff
),(2/),()/()( 2 HHHHcHE f
14
Binomial (or Tayler) expansion
),(
),(
2
1)(,
),(
),(
2
1),(
HH
HHHE
HHHH
HHHHHHE ff
)()(),( * HHHHdvHHHH
),(),(2),(
),(),(),(),(
2
1
HHHHHH
HHHHHHHH
Hermitian
HH
HH
HH
let),(
),(
0),(
),(
),(21),(
),(2),(
2
1
HH
HHHH
HHHH
),(
),(21
),(
),(21
1
HH
HH
HH
HH
),(
),(
),)(,(4
),(
),(),(2),(2),(
2
1
HH
HH
HHHH
HH
HHHHHHHH
15
If is an eigenvector of with an eigenvalue of
: stationary with respect to the variations of
when is a harmonic mode
Lowest EM eigenmode ; minimizes Then next lowest EM
eigenmode ; minimizes in the subspace orthogonal to etc.
),(
),(
),)(,(),(
)()(HH
HH
HHHHHH
HEHHEE fff
H
HH
HHH
HHH
HE f
),(
),(
),(
1)(
H
.,
22
Hc
Hc
0),(
),(
H
HH
HHH
fE
H
H
0H
.fE
1H
fE ,0H
16
Another property of variational theorem on EM energy functional
(why?)
is minimized when the displacement field is concentrated in the regions of high dielectric constant (due to with continuous ).
fE
BAABBAHr
HdvHHHH
HHHE
v
f
)(
)(1),(,
),(
),(21)( *
vv
HHr
dvHHr
dvHH )()()(
1)(
)(
1),( **
v
n
s
HHr
dvdsaHHr
** )()()(
1)(
)(
1
)()()()(
1
),(2
1 2
rDc
irEr
c
iHH
rdv
HHE f
fE
2
)(
1
),(2
1D
crdv
HH
)(/1 r nDD
17
Physical energies stored in the electric and magnetic fields
Harmonic magnetic field electric field
Our approach ; master eq.
then and
Question ; Can we make up another master eq. for and
then calculate from or
From
dvrHEdvrDr
E HD
22
)(8
1,)(
)(
1
8
1
),(rH
)(rD
)(
1)()(
2
rwithrH
crH
),()(1
)( rErc
i
t
D
crH
)()(
)( rHr
icrE
),()( rErD
)()( rHcirE
)(rH
?)()( rEicrH
)()()(,1
)(2
rDc
rHc
irEH
c
i
t
H
crE
18
∴ Master eq. should be with the operator
and eigenvalue But operator is not Hermitian.
Proof)
Since is not a constant, is not Hermitian.
)(/)()(/1 2 rDcrDr
)(/1 rZ
./ 2c
BAABBAGr
FdvGZF
)()(
1),( *
**
)(1
)(1 FG
rdvFG
rdv
G
rFdv
)(
1)( *
** )(
)(
1)(
)(
1FG
rdvFG
rdv
Gr
FdvGZF
)(
1)(),( *
),()(
1*
GFZGFr
dv
)(/1 r )(/1 rZ
Z
19
If we take instead of is equivalent
to with and Even if
Hermitian operators, it is a numerically difficult task to solve.
Scaling properties of the Maxwell eqs./Contra. or expansion of PhCs.
Assume an eigenmode in a dielectric configuration
then What if we have another configuration
of dielectric with a scalling parameter s?
If we transform as then
∴ Let’s transform the position vector and operator as
).()()(
12
rHc
rHr
),( rH
),(r
srr /)(
,rsr
.s
az
ay
ax zyx
./,/ ssrr
Dr
)(
1
Drc
Drr
D
)(1
)(1
)(1,
2
DZcDZ
22
1 )/(
)(1
)(1
1 rrZ
;, 21 ZZ
r
.)(
12 r
Z
r
20
Then,
But
This is just another master eq. with
Likewise if dielectric constant is changed by factor of as
∴ Harmonic modes of the new system are unchanged but the mode frequencies are changed so that
Electrodynamics in PhCs and Quantum Mechanics in Solids
).()()(
1)()(
1 2
rsHc
rsHsrss
rHr
).()()(
1)()(2
rsHc
srsHr
rrs
.)()( sandrsHrH
,)(
)(2s
rr
).()()(
12
2 rHc
rHrs
.)()(
)(1
2
rHc
srHr
s
2s
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