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

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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

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∴ 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

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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