Lecture 2. Basic Theory of PhCs : EM waves in mixed
dielectric media and Eigenvalue approach
2009. 04.
Hanjo Lim
School of Electrical & Computer Engineering
hanjolim@ajou.ac.kr
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