chapter 4 the multilayer film - a one dimensional...

Photonic Modeling and Design Lab.Graduate Institute of Photonics and Optoelectronics &Department of Electrical EngineeringNational Taiwan University

Chapter 4Chapter 4The Multilayer Film: A One The Multilayer Film: A One

Dimensional Photonic CrystalDimensional Photonic Crystal台大光電所暨電機系

邱奕鵬Room 617, BL Building

(02) [email protected]

NTU GIPO & EE Photonic Modeling and Design Lab.YPC2

The Multilayer FilmThe Multilayer Film

Block form:

Bragg mirror Lord Rayleigh solved in 1887 and 1917

: restrict to a finite interval

: any value

NTU GIPO & EE Photonic Modeling and Design Lab.YPC

One Dimensional Structure - Quarter Wave Stack

Dielectric mirror 1 2

1 2

n nrn n

−=

+

Destructive Interference

tilt 2dcosθ=mλλ varies => θ varies

1 2

1 2

0, 0,

r n nr n n

> >< <

φ=kd+φr

R, T ?

Theory built by Lord Rayleigh in 1887

Decay exponentially


1D Band Structure of A Homogeneous Material


KronigKronig Penney ModelPenney Model

1 1 1 1cos sinA x B x ncωψ α α α= + =

2 2 2 2cos sinA x B x ncωψ β β β= + =

1 2' '1 2

ψ ψ

ψ ψ

=⎧⎪⎨

=⎪⎩

0z =

'1 1 1sin 0 cos 0A Bψ α α α α= − +

'2 2 2sin 0 cos 0A Bψ β β β β= − +

1 2A A= 1 2B Bα β= 1

1

cos cos sin sin0

sin sin cos cos

ika ika

ika ika

e q p e q p AB

e q p e q p

αα β α ββ

α α β β α α α β

⎡ ⎤− − − ⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥ ⎣ ⎦+ −⎢ ⎥⎣ ⎦

( )det 0M =

2 22

2 2 0nz cψ ω ψ∂

+ =∂

Helmholtz eq.

2 2

2 21 2

1 2 1 21 2

cos sin sin cos cos2

sin sin cos cos2

ka q p q p

n nn q n p n q n p

n n c c c c

α β α β α βαβ

ω ω ω ω

+= − +

+ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

B.C.

z p=( ) ( )1 1

ikax a e xψ ψ+ =

( ) ( ) ( )1 1 1ikap q a e qψ ψ ψ= − + = −

( ) ( )' '1 1

ikap e qψ ψ= −

(Bloch thm.)

across interfacesperiod a = p+q

HW#2.3


The Physical Origin of Photonic Band GapsThe Physical Origin of Photonic Band Gaps

Light lines

ε= 13 & 13 ε= 13 (GaAs) & 12 (AlGaAs) ε= 13 & 1

Photonic band gap: no allowed mode regardless of k


air band

dielectric band

7

The Physical Origin of Photonic Band GapsThe Physical Origin of Photonic Band GapsNearlyNearly--homogeneous mediumhomogeneous mediumSmaller index difference

⇒Narrow-band filter, e.g. in opt. comm.fiber Bragg grating, thin-film filter

3(a) or 3(b), otherwise violate symmetry

Wavelength

More field in high-ε=> lower freq.more field in low-ε=> higher freq.


Simple Plane Wave Expansion Approach (1D)Simple Plane Wave Expansion Approach (1D)

( ) ( ) 2

21

2

2

2

22 1tEE

rtE

xE

xc D

∂∂

−=×∇×∇⎯⎯←∂∂

=∂∂

εε

( ) ( )xax εε =+ ( ) ∑∞

−∞=

− =m

mxa

i

mexπ

κε2

1

realislossless εε →:

( )

( )

( )[ ] dxex

dxex

dxex

mxa

im

mxa

i

m

mxa

i

mmm

π

π

π

εκ

εκ

εκκκ

2*1*

21

21*

∫

∫

∫

−

−−

−−−

∝

∝

∝→=

( ) ( ) ( ) ( ) ( )tkxik

tik

ikxk

kk exuexuetxEtxE ωω −− === ,,

( ) ∑=m

mxa

i

mk eExuπ2

( )

∑

∑−⎟

⎠⎞

⎜⎝⎛ +

−

=

=

m

tixma

ki

m

tiikx

m

mxa

i

mk

eE

eeeEtxE

ωπ

ωπ

2

2

,

( )

xa

ixa

i

m

mxa

i

m

ee

ex

ππ

π

κκκ

κε

2

1

2

10

21

−

−

∞

−∞=

−

++≅

= ∑

Helmholtz Equation

From Bloch theorem

small ε difference or sinusoidal distribution

212 2

0 1 01 2

c ac ha

κπω κ κ κπ κ±

⎛ ⎞⎜ ⎟= ± ± −⎜ ⎟⎝ ⎠

h kaπ

≡ −(Cf. 1D Equations)


The Physical Origin of Photonic Band GapsThe Physical Origin of Photonic Band Gaps

air band

dielectric band

Higher dielectric contrast Higher dielectric contrast

1ST band are more concentrated than 2nd band in high–ε region

Gap occurs at centeror edge of Brillouinzone

Gaps always appearin 1D PhC as


The Size of the Band GapThe Size of the Band Gap

10

Scaling propertyStructure expanded by a factor s => gap size Δω/sGap-midgap ratio Δω/ωm, ωm: freq. at the middle of the gap, generally in %

Normalized (dimensionless) wavevector: ka/2π , freq. ωa/2πc = a/λ

Weak periodicity Δε/ ε <<1 or d/a <<1for dielectric const.: ε & ε + Δε thickness: a-d & d

HW#2.2


The Size of the Band GapThe Size of the Band GapGap max. as

(ε1=13, d1=0.2a, ε2=1, d2=08a)

Quarter-wave stack (QWS)Reflective waves from eachlayer is exactly in phase.

Dielectric contrast 13:1d1:d2 =0.5: 0.5 => Δω/ωm=51.9%d1=0.217=> Δω/ωm=76.6% (QWS)d1:d2 =0.2: 0.8 => Δω/ωm=76.3%

QWS => no gap in k=0 HW#2.2


Evanescent Modes in Photonic Band GapsEvanescent Modes in Photonic Band Gaps2

12 20 1 0

1 2c ac h

aκπω κ κ κ

π κ±

⎛ ⎞⎜ ⎟= ± ± −⎜ ⎟⎝ ⎠

Weak modulation

ω in gap => no extended solution, evanescent complex wavevector

Time reversal => no odd power or k

(within gap)

Largest κ in mid gapLarger gap, larger κ in mid gap

No solution in gap for infinitely perfect PhCSolutions may exist for PhC with defects (defect states) or terminations (surface states).

penetrat. depth 1/κ


OffOff--Axis PropagationAxis Propagation

13

No band gaps ( no scattering in y)No degeneracy (split: TMy & TEy )

lower modes concentrateelectrical energy in thehigher ε-region

Approximately linear atlong wavelengths

effectively homogeneous

ωTM < ωTE

for all PhC


OffOff--Axis PropagationAxis Propagation

Blue: (0, ky,0)Green: (0, ky, π/a)Red: light line

Short wavelength (high freq.)

Freq. difference Below light line* index guided* decay exp.* negligible overlap* small couplingEvery mode becomea uncoupledguided mode


Localized Modes at DefectsLocalized Modes at Defects0.2 a (ε=13) + 0.8a (ε=1)

1.6a =2*0.8a

E-field strength

Defects + ω in band gapmay exist localized modes

Like two parallel perfect mirrorsdiscrete freq. (quantized)thicker defect => more states

Defects by Increase low/high-ε thicknessor lower/increase ε (same thick.)or combinedPull down/push up a sequenceof discrete modes from the upper/lower bands

Max. exp. decay in mid-gap

Single peak associate with the defectUsed in dielectric Fabry-Perot filter

Quantized like particle in a box (QM) or metallic cavity (μ-wave)

DOS= # states / Δω


Localized Modes at DefectsLocalized Modes at Defects

16

Defects by Increase low/high-ε thicknessor lower/increase ε (same thick.)or combinedPull down/push up a sequenceof discrete modes from the upper/lower bands

Off-axis propagationStill localized in z-directionCan be guided in bothHigh and low ε regions (even air)

2D => photonic crystal fibers (PCF)


Surface statesSurface states0.1 a

air + dielectricSurface modes for some choice of termination (e.g. 0.1 a)

Air region: light lineupper left =>lower right=>

PhC region:w/ gap: w/o gap:

Extended (E): propagatingDecaying (D): evanescent

May exist localized modes propagating along surface (e.g. And x-polarized )


OmnidirectionalOmnidirectional Multilayer MirrorsMultilayer Mirrors

18

a quarter-wave stack with ε= 13 & 2

TM, x-polarizedTE, yz-polarized

Brewster’s angle line

ambient εa

is conserved (far source)inc. from air (above light line, not decaylight from source can reach)

Necessary conditionsωU > ωL

large enough to open gap within light cone

εa < ε1 < ε2point B below light line

1 1 2 2

1 1 2 2

1

sin sin sin

sin sin sin

tan

y a a

a a

Ba

k k k k

n n nnn

θ θ θ

θ θ θ

θ

= = =

= =

=

Can reflect light wave from anyangle with any polarization



εa <ε1 < ε2

Size of the omnidirectional gap

Optimal: not quarter wave stacks but close to

Reflective property dep. on translational symmetry

=> Not confine a mode in 3D

If the interface is not flat or an object close to, then is not conserved.

=> couple to extended modesi.e. propagating

transmittedExceptions: smoothly curved

and symmetry conserved



20

OL 28, 2144 (2003)

Brag onion

Brag fiber

Translational Symmetry⇒Cylindrical & Spherical cases

(See Chap. 9 PCF)

chapter 4 the multilayer film - a one dimensional...

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