Feature Selection/Extraction for Classification Problems
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Transcript Feature Selection/Extraction for Classification Problems
Lecture 2. Basic Theory of PhCs : EM waves in mixed
dielectric media and Eigenvalue approach
2009. 04.
Hanjo Lim
School of Electrical & Computer Engineering
[email protected]
1
Maxwell equations are given as;
D 4 ,
1 B
E
0
c t
B 0,
4 1 D
H
J
0
c
c t
in cgs units .
Constitutive relations ; relations between
D & E , B & H , J & E , etc
D i ij E j k ijk E j E k O ( E ) if linear , isotropic , and lossless media
j
j
D ( r ) ( r ) E ( r ) or D ( r , ) ( r , ) E ( r , )
B ( r ) ( r ) H ( r ) 1 in optical freq. range for most materials
B (r ) H (r )
and especially for dielectric materials.
3
∴ Maxwell equations are given
as;
(r ) E (r , t ) 0,
H (r ,t) 0 ,
1 H (r , t )
E (r , t )
0
c t
(r ) E (r ,t)
H (r ,t)
0
c
t
2
E and H
Then
; complicated functions of time and space.
But Maxwell eqs. are linear => time dependence can be expressed
by harmonic modes.
i t
i t
H ( r ,t ) H ( r ) e , E ( r ,t ) E ( r ) e
Then mode profiles of a given frequency are given from Maxwell
equations. H ( r ) 0 , D ( r ) 0 ; Nonexistence of source or sink
Field configurations are build up of transverse EM waves.
Transversality : If H ( r ) a exp( i k r ), a k 0
i
1 H (r ,t)
E (r ,t)
0 E (r )
H (r ) 0
c
t
c
(r ) E (r , t )
i
H (r , t )
0 H (r )
(r )E (r ) 0
c
t
c
3
Take main function as magnetic field
H ( r , t ) i
i
E (r ) 0 E (r )
H (r )
(r )
c
c
2
1
H ( r ) H ( r ) ; Master equation
(r )
c
Master eq. with H ( r ) 0 condition completely determines H (r )
* Schrodinger equation :
2
2
V (r ) (r ) E (r )
2m
:
eigenfunction (r )
eigenvalue problem => eigenvalue E and
For a given photonic crystal (r ), master equation => eigen modes.
c
H (r )
If modes H ( r , ) for a given are known , E ( r )
i ( r )
4
Interpretation of Master equation ; Eigenvalue problem
2
1
H (r )
H (r )
(r )
c
operator
eigenvalue eigenvector if
2
H ( r ) H ( r )
c
with
operation on H (r ) =>
eigenvectors H (r ) ; field
H (r ) is
allowed.
1
H (r )
H (r )
( r )
H (r )
2
/
c
& eigenvalue
eigenvector
patterns of the harmonic modes.
Note) operator ; linear operator wave eq.; linear differential eq.
∴ If H 1 ( r ) and H 2 ( r )are two different solutions of the eq. with same ,
2
general solution of H ( r ) / c H ( r ) ; H ( r ) H 1 ( r ) H 2 ( r )
5
∴ Two field patterns that differ only by a multiplier ; same mode.
Hermitian property of
1
(r )
def) inner product of two vector fields
*
* Note that ( F , G ) ( G , F ) * .
( F , G ) F ( r ) G ( r ) dv d r F ( r ) G ( r ),
*
*
*
*
*
Proof :( F , G ) d r F ( r ) G ( r ) F ( r ) G ( r ) G ( r ) F ( r ) G ( r ) F ( r )
*
* *
dr G (r ) F (r ) G , F
*
2
Note that ( F , F ) d r F ( r ) F ( r ) d r F ( r ) ; always real
If ( F , F ) 1 ; called normalized mode, Normalization of F with ( F , F ) 1
def) Hermitian matrix (self-adjoint)
Aij A ji , if
*
adjoint
Aij Aij A ji
*
Hermitian
6
def) Hermitian operator
Q
a Q b dv Q a b dv
*
*
*
Properties)
1. If operator Q is Hermitian
for arbitrary normalizable functions .
2
3
Q , Q , ...
are Hermitian.
a Q b dv Q a Q b dv ( Q a ) b dv
*
2
*
*
2
*
Q ; Hermitian
2
2. A linear combination of Hermitian operators is a Hermitian operator.
3. The eigenvalues of a Hermitian operator are all real.
*
*
*
Proof; Let Q n q n n .
n Q n dv n q n n dv q n n n dv
*
*
*
*
If Hermitian operator
( Q n ) n dv ( q n n ) n dv q n n n dv
q n q n q n ; real
*
4. Any operator associated with a physically measurable quantity is
Hermitian (postulate).
7
def)
Hermitian operator for vector fields F (r ) and G (r )
*
*
If ( F , G ) ( G , F ), i.e., d r F G d r ( F ) G , that
is, the inner
product of –operated field is independent of which function is
operated, : Hermitian operator.
1
Proof of ( r ) is Hermitian operator.
* 1
*
* *
( F , G ) dv F G ( F ) F F
(r )
let
*
*
dv ( F ) dv ( F ) divergence
theorem dv a n ds
v
1 *
* dv
G F
( r )
v
s
1 *
G F a n ds 0
(r )
s
8
Note
;F &G
1) zeros at large distances due to 1 / r 2 , 1 / r 3 dependence
2) periodic fields in the region of integr. (∵ harmonics)
* 1
1 *
( F , G ) dv G ( F ) dv ( F ) G
(r )
(r )
1 * 1 *
1 *
( F ) G G F ( F ) ( G )
(r )
(r )
(r )
*
* 1
1
F G ( F ) ( G )
(r )
(r )
*
After integration,
* 1
1
dv ( F ) G dv F G ( F , G )
(r )
(r )
1
is a Hermitian
(r )
operator
9
Note
* 1
* 1
1
; ( F ) G G F ( F ) * G
(r )
(r ) (r )
*
1 * 1 * 1
G F G ( F ) G F
(r )
(r )
(r )
1
1
1
since is not a constant.
G G
(r )
(r ) (r )
General properties of harmonic modes
2
2
1) Hermitian operator H ( r ) H ( r ), eigenvalue must be real.
c
c
2 2 *
Proof)( H , H ) ( H , H ) d r H ( r ) H ( r )
c
c
*
*
* 2 * 2
( H , H ) 2 ( H , H ) 2 ( H , H )
c
c
Note that
( H , H ) ( H , H )
*
for any
operator
10
* *
* *
*
*
Proof) ( H , H ) d r ( H ) H d r ( H ) H d r H H
*
*
d r H H ( H , H ) ( H , H ) ( H , H )
2
Hermitian operator ; ( H , H ) ( H , H ) ( / c ) ( H , H )
*
*
*
2
2 *
*
2
2
Then ( H , H ) ( / c ) ( H , H ), ( H , H ) ( H , H ) ( H , H ) ( H , H ) ( / c )( H , H )
Hermitian operator
for any operator
( / c )( H , H )
2
( / c ) ( / c ) , i.e ,
2
2 *
2
2
2
2*
2
2
; real
Note) ; 2 is actually positive => ; real .
If becomes imaginary in some frequency range, what dose it mean?
Proof)
*
* 1
1
From ( F , G ) d r F G d r ( F ) G Let F G H
(r )
(r )
11
* 1
1
Then ( H , H ) d r ( H ) ( r ) ( H ) d r ( r ) H
2
( / c ) ( H , H )
positive
( / c )
2
2
positive
positive
; positive
; positive
2
; real
If (r ) is negative in some frequency range, ; imaginary. Meaning?
2) Operator is Hermitian means that H 1 ( r ) and H 2 ( r ) with different
frequencies 1 and are orthogonal.
2
2
Proof) let H 1 ( r , 1 ), H 2 ( r , 2 ), than H 1 ( r ) ( 1 / c ) H 1 ( r ), H 2 ( r ) ( 2 / c ) H 2 ( r ).
Hermitian ; ( H 2 , H 1 ) ( H 2 , H 1 )
2
2
2
2
( H 2 , H 1 ) c ( H 2 , H 1 ) c ( H 2 , H 1 ) 2 ( H 2 , H 1 )
2
2
( 1 2 )( H 2 , H 1 ) 0 if 1 2 , ( H 2 , H 1 ) 0
2
1
12
Orthogonality & modes
Meaning of orthogonality of the scalar functions => normal modes.
Meaning of orthogonal vector fields : ( F , G ) 0
Meaning of orthogonal vector modes. Degeneracy : related to the
rotational symmetry of the modes.
Electromagnetic energy & variational principle => qualitative features
Def) EM energy functional E f ( H ) ( H , H ) / 2 ( H , H ),
2
E f ( H ) ( / c ) ( H , H ) / 2 ( H , H ) if H (r ) is a normal mode.
The EM modes are distributed so that the field pattern minimizes the
EM energy functional E f
Proof) When H ( r ) H ( r ) H , E f E f ( H H ) E f ( H ) ; 0
13
1 ( H H , H H )
1 ( H , H )
, E f (H )
E f ( H , H )
2 ( H H , H H )
2 (H ,H )
*
( H H , H H ) dv (H H ) ( H H )
( H , H ) 0
1 ( H , H ) ( H , H ) ( H , H ) ( H , H )
( H , H )
let
2
( H , H ) 2 ( H , H ) ( H , H )
( H , H )
Hermitian
1
2 ( H , H )
1 ( H , H ) 2 ( H , H )
2 ( H , H )
1
1
2
2 ( H , H )
(H ,H )
(H ,H )
( H , H ) 1
(
H
,
H
)
Binomial (or Tayler) expansion
( H , H )
( H , H )( H , H )
( H , H ) 2 ( H , H ) 2 ( H , H ) 4
1
(H ,H )
(H ,H )
2
(H ,H )
14
( H , H )( H , H )
( H , H )
(H ,H )
E f E f ( H H ) E f ( H )
(H ,H )
If
E f ( H )
1
( H , H )
H
H
H
(H ,H )
(H ,H )
H is an eigenvector of with an eigenvalue
( H , H )
H
H
(H ,H )
E f
0
2
of
2
, H H .
c
c
: stationary with respect to the variations of
H when H is a harmonic mode
H0
Lowest EM eigenmode ; minimizes E f . Then next lowest EM
eigenmode H 1 ; minimizes E f in the subspace orthogonal to H 0 , etc.
15
Another property of variational theorem on EM energy functional
)
1 ( H, H
, ( H , H )
E f (H )
2 (H , H )
v
Ef
*
1
dv H H ( A B ) B A A B
(r )
1
*
*
1
( H , H ) dv ( H ) H dv
( H ) ( H )
(
r
)
(
r
)
v
v
*
*
1
1
( H ) H a n ds dv
( H ) ( H )
(r )
(r )
s
v
(why?)
1
1
dv H
Ef
(r )
2( H , H )
1
1
dv
D
(r ) c
2( H ,H )
2
i i
H (r ) E (r )
D (r )
c
c
2
E f is minimized when the displacement field D
regions of high dielectric constant (due to 1 / ( r )
is concentrated in the
with continuous D n ).
16
Physical energies stored in the electric and magnetic fields
ED
Harmonic
2
1
D ( r ) dv ,
8 ( r )
magnetic field H (r ),
1
EH
1
8
2
H ( r ) dv
electric field
D (r )
2
1
Our approach ; master eq. H ( r ) H ( r ) with
c
(r )
1 D i
ic
( r ) E ( r ), and E ( r )
then H ( r )
H
(
r
)
c t
c
( r )
Question ; Can we make up another master eq. for D ( r ) E ( r ),
and
i
ic
then calculate H (r ) from E ( r )
H ( r ) or H ( r ) E ( r ) ?
c
1 H i i 2
H , E (r )
H (r ) D (r )
From E ( r )
c t
c
c
c
17
2
1 / ( r ) D ( r ) / c D ( r )
∴
Master eq. should be
with the operator
2
Z 1 / ( r ) and eigenvalue / c . But operator Z is not Hermitian.
Proof)
*
1
( F , Z G ) dv F G
(r )
( A B ) B A A B
*
*
1
1
dv G F dv G F
(r )
(r )
*
1 *
1
1 *
dv ( F ) G dv G ( F ) dv G ( F )
(r )
(r )
(r )
*
* 1
1
( F , Z G ) dv ( F ) G dv F G ( Z F , G )
(r )
( r )
Z
1
/
(
r
) is not Hermitian.
1
/
(
r
)
Since
is not a constant,
18
1
If we take D
(r )
2
to Z 1 D ( / c ) Z 2 D
D,
2
1
1
1
instead of
D
D is equivalent
(r )
(r ) c (r )
1
1
1
with Z 1 ( r ) ( r ) and Z 2 . Even if Z 1 , Z 2 ;
(r )
Hermitian operators, it is a numerically difficult task to solve.
Scaling properties of the Maxwell eqs./Contra. or expansion of PhCs.
(r
Assume an eigenmode H ( r , ) in a dielectric configuration ),
1 2
then H ( r ) H ( r ). What if we have another
(r )
c
of dielectric ( r ) r / s with a scalling parameter s?
r
If we transform as
∴ Let’s transform the
configuration
then a x a y a z .
x
y
z
s
position vector r and operator as r r / s , / s .
r sr ,
19
1
1
Then, ( r ) H ( r ) s ( s r ) s H ( s r ) c
But
2
H ( s r ).
2
1
s
( s r ) ( r ) H ( s r )
H ( s r ).
( r )
c
This is just another master eq. with
H ( r ) H ( s r ) and
s .
(
r
)
Likewise if dielectric constant is changed by factor of s 2 as ( r ) 2 ,
s
2
2
1
s
H ( r ) .
1
H
(
r
)
2 H ( r ) H ( r ).
( r )
c
c
s (r )
∴
Harmonic modes of the new system are unchanged but the mode
frequencies are changed so that s
Electrodynamics in PhCs and Quantum Mechanics in Solids
20