A One Dimensional Photonic Crystal - 台大電機系計算機中心

Chapter 4 

The Multilayer Film: A One 

Dimensional Photonic Crystal 

台大光電所暨電機系 

邱奕鵬 

Room 617, BL Building 

(02) 3366-3603 

ypchiou@cc.ee.ntu.edu.tw 

Photonic Modeling and Design Lab. 

Graduate Institute of Photonics and Optoelectronics & 

Department of Electrical Engineering 

National Taiwan University 

One Dimensional Structure - Quarter Wave Stack 

n − n 

r = 

n + n 

Dielectric mirror 1 2 

φ=kd+φ r 

R, T ? 

Destructive Interference 

Decay exponentially 

Theory built by Lord Rayleigh in 1887 

1 2 

r > 0, n > n 

r < 0, n < n 

1 2 

1 2 

tilt 2dcosθ=mλ 

λ varies => θ varies 

NTU GIPO & EE YPC Photonic Modeling and Design Lab. 

Block form: 

The Multilayer Film 

Bragg mirror Lord Rayleigh solved in 1887 and 1917 

: any value 

: restrict to a 

finite interval 

NTU GIPO & EE 2 YPC Photonic Modeling and Design Lab. 

1D Band Structure of A Homogeneous Material 

NTU GIPO & EE YPC Photonic Modeling and Design Lab.

Helmholtz eq. 

∂ ψ ω 

∂z 

c 

ω 

ψ1 = A1cosαx+ B1sinαx α = n1 

c 

ω 

ψ = A cos βx+ B sin βx β = n 

c 

2 2 2 2 

z = 0 

A1 = A2 

B1α= B2β 

' 

ψ1 =− A1αsinα0+ B1αcosα0 

' 

ψ =− A β sin β0+ B β cos β0 

2 2 2 

Kronig Penney Model 

2 2 

2 + 

2 

n ψ = 0 

2 

z = p (Bloch thm.) 

ika 

ψ1( x + a) = e ψ1( 

x) 

ika 

ψ1( p) = ψ1( − q+ a) = e ψ1( 

−q) 

' ika ' 

ψ ( p) = e ψ ( −q) 

1 1 

⎧⎪ ψ1 = ψ 2 

B.C. ⎨ ' ' 

⎪⎩ ψ1 = ψ 2 

across interfaces 

period a = p+q 

⎡ ika ika α ⎤ 

⎢ e cosαq−cos β p −e sinαq− sin β p A1 

β ⎥⎡ 

⎤ 

= 0 

⎢ ⎥⎢ ika ika 

B 

⎥ 

1 

⎢e αsinαq+ βsin β p e αcosαq−αcos β p⎥ 

⎣ ⎦ 

⎣ ⎦ 

1 2 

det ( M ) = 0 

HW#2.3 

2 2 

α + β 

cos ka =− sinαqsinβ p + cosαqcos β p 

2αβ 

2 2 

n1 + n2 

⎛ω ⎞ ⎛ω ⎞ ⎛ ω ⎞ ⎛ ω ⎞ 

=− sin nq 1 sin n2pcosn1 q cos n2 p 

2nn 

⎜ 

c 

⎟ ⎜ + 

c 

⎟ ⎜ 

c 

⎟ ⎜ 

c 

⎟ 

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 


The Physical Origin of Photonic Band Gaps 

Nearly Nearly-homogeneous homogeneous medium 

Smaller index difference 

⇒Narrow-band filter, e.g. in opt. comm. 

fiber Bragg grating, thin-film filter 

Wavelength 

3(a) or 3(b), otherwise 

violate symmetry 

More field in high-ε 

=> lower freq. 

more field in low-ε 

=> higher freq. 

air band 

dielectric band 



Photonic band gap: no allowed mode regardless of k 

Light lines 

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


Simple Plane Wave Expansion Approach (1D) 

Helmholtz Equation 

2 2 2 

c ∂ E ∂ E 1D 

ε 

= 

←⎯⎯ 

2 

1 

∂ E 

∇× 

∇× 

E = − 

2 2 

2 

( x) 

∂x 

∂t 

ε ( r ) ∂t 

( x a) 

ε( 

x) 

+ ( ) ∑ ∞ 

−1 

ε x = 

m= 

−∞ 

ε = 

ε : lossless → ε 

κ 

−m 

= κ 

* 

m 

κ 

κ 

is real 

−1 

→ κ ∝ ε 

m 

−m 

* 

m 

∝ 

∝ 

∫ 

( x) 

e 

2π 

−i 

mx 

a 

2π 

κ me 

dx 

i mx 

−1 

a 

∫ε 

( x) 

e dx 

2π 

i mx 

−1 

* 

a 

∫ [ ε ( x) 

] e dx 

From Bloch theorem 

ikx −i 

t 

E( 

x, 

t) 

= E ( x, 

t) 

= e u ( x) 

e = u ( x) 

k 

u 

k 

( x) 

∑ 

= m 

E 

k 

2π 

i mx 

a 

me 

2π 

i mx 

a 

( ) t kx i 

ωk −ωk 

k e 

( x, 

t) 


E 

k 

− 

ε 

= 

= 

( x) 

1 

∑ 

m 

∑ 

m 

E 

E 

2π 

i mx 

a ikx −iωt 

me 

e e 

⎛ 2π 

⎞ 

i⎜ 

k + m ⎟ x−iωt 

⎝ a ⎠ 

me 

small ε difference or 

sinusoidal distribution 

∞ 2π 

i mx 

a κ me 

m= 

−∞ 

= ∑ 

≅ κ + κ 

0 

2π 

i x 

a 

1e 

(Cf. 1D Equations) 

+ κ 

2π 

−i 

x 

a 

−1e 

π 

h≡k− a 

πc 

ac ⎛ ⎞ 

ω± = κ ± κ ± κ − 

a 

⎝ ⎠ 

2 

2 κ1 

2 

0 1 ⎜ 0 ⎟h 

πκ ⎜ 1 2 ⎟


Higher Higher dielectric contrast 

1ST band are more concentrated than 

2nd band in high–ε region 

Gap occurs at center 

or edge of Brillouin 

zone 

Gaps always appear 

in 1D PhC as 

air band 

dielectric band 


Gap max. as 

Quarter-wave stack (QWS) 

Reflective waves from each 

layer is exactly in phase. 

HW#3 

Dielectric contrast 13:1 

d1: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 @ k=0 HW#3 

The Size of the Band Gap 

HW#3 

HW#3 

(ε1=13, d1=0.2a, ε2=1, d2=08a) 


The Size of the Band Gap 

Scaling property 

Structure expanded by a factor s => gap size Δω/s 

Gap-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

No band gaps ( no scattering in y) 

No degeneracy (split: TMy & TEy ) 

Approximately linear at 

long wavelengths 

for all PhC 

effectively homogeneous 

lower modes concentrate 

electrical energy in the 

higher ε-region 

ω TM < ω TE 

Off-Axis Off Axis Propagation 


Localized Modes at Defects 

Defects + ω in band gap 

may exist localized modes 

0.2 a (ε=13) + 0.8a (ε=1) 

Like two parallel perfect mirrors 

discrete freq. (quantized) 

thicker defect => more states 

1.6a = 

2*0.8a 

E-field strength 

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

Defects by 

Increase low/high-ε thickness 

or lower/increase ε (same thick.) 

or combined 

Pull down/push up a sequence 

of discrete modes from the 

upper/lower bands 

Max. exp. decay in mid-gap 

Single peak associate with the defect 

Used in dielectric Fabry-Perot filter 

DOS 

= # states / Δω 


Short wavelength (high freq.) 

Freq. difference 

Below light line 

* index guided 

* decay exp. 

* negligible overlap 

* small coupling 

Every mode become 

a uncoupled 

guided mode 

Off-Axis Off Axis Propagation 

Blue: (0, k y,0) 

Green: (0, k y, π/a) 

Red: light line 


Localized Modes at Defects 

Defects by 

Increase low/high-ε thickness 

or lower/increase ε (same thick.) 

or combined 

Pull down/push up a sequence 

of discrete modes from the 

upper/lower bands 

Off-axis propagation 

Still localized in z-direction 

Can be guided in both 

High and low ε regions (even air) 

2D => photonic crystal fibers (PCF) 

NTU GIPO & EE 16 YPC Photonic Modeling and Design Lab.

May exist localized modes 

propagating along surface 

(e.g. And x-polarized ) 

Extended (E): propagating 

Decaying (D): evanescent 

Air region: light line 

upper left => 

lower right=> 

PhC region: 

w/ gap: 

w/o gap: 

Surface modes for some choice 

of termination (e.g. 0.1 a) 

Surface states 

0.1 a 

air + dielectric 


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 modes 

i.e. propagating 

transmitted 

Exceptions: smoothly curved 

and symmetry conserved 

Continuous rotational and 

spherical symmetry 

Omnidirectional Multilayer Mirrors 

Size of the omnidirectional gap 

ε a < ε 1 < ε 2 



Can reflect light wave from any 

angle with any polarization 

is conserved (far source) 

from air 

(above light line, not decay 

light from source can reach) 

Necessary conditions 

ω U > ω L 

large enough to open 

gap within light cone 

ε a < ε 1 < ε 2 

point B below light line 

ky = kasinθ a = k1sinθ1 = k2sinθ2 

nasinθa = n1sinθ1 = n2sinθ2 

n1 

tanθ 

B = 

n 

a 

ambient ε a 

Brewster’s angle line 

TE, yz-polarized 

TM, x-polarized 

a quarter-wave stack with ε = 13 & 2 



Translational Symmetry 

⇒Cylindrical & Spherical cases 

Continuous rotational and 

spherical symmetry 

Brag fiber 

Brag onion 

OL 28, 2144 (2003) 

(See Chap. 9 PCF) 

NTU GIPO & EE 20 YPC Photonic Modeling and Design Lab.

Periodic Hermitian Eigenproblems in 1d 

H(x) = e ikx H k (x) 

Consider k+2π/a: e 

ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 

a 

2π 

i(k + ) x 

a H 2π 

k + 

a 

k is periodic: 

k + 2π/a equivalent to k 

“quasi-phase-matching” 

ε(x) = ε(x+a) 

(x) = e ikx ⎡ 

⎢ e 

⎣ ⎢ 

2π 

i 

a x 

H 2π 

k + 

a 

⎤ 

(x) ⎥ 

⎦ ⎥ 

periodic! 

satisfies same 

equation as H k 

= H k 


Any 1d Periodic System has a Gap 

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of 

waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] 

Start with 

a uniform (1d) medium: 

ω 

0 


ε 1 

ω = k 

ε 1 

k 

Periodic Hermitian Eigenproblems in 1d 

k is periodic: 

k + 2π/a equivalent to k 

“quasi-phase-matching” 

–π/a 

ω 

ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 

band gap 

ε(x) = ε(x+a) 


a 

0 π/a 

irreducible Brillouin zone 




Treat it as 

“artificially” periodic 

bands are “folded” 

by 2π/a equivalence 

–π/a 

ω 


a 

ε 1 

ε(x) = ε(x+a) 

k 

π 

+ 

a e x 

π 

− 

a , e x 

→ cos 

0 π/a 

k 

π 

a x 

⎛ ⎞ 

⎜ ⎟ , sin 

⎝ ⎠ 

π 

a x 

⎛ ⎞ 

⎜ 

⎝ ⎠

ω 




Treat it as 

“artificially” periodic 

0 π/a 

ω 

sin π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

cos π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

ε 1 

ε(x) = ε(x+a) 


band gap 

a 

x = 0 




Add a small 

“real” periodicity 

ε 2 = ε 1 + Δε 

0 π/a 

sin π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

cos π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

Splitting of degeneracy: 

state concentrated in higher index (ε 2) 

has lower frequency 


a 

ε(x) = ε(x+a) 

ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 

x = 0 

ω 




Add a small 

“real” periodicity 

ε 2 = ε 1 + Δε 

0 π/a 

sin π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

cos π 

a x 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

ε(x) = ε(x+a) 


a 

ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 ε 1 ε 2 

x = 0

A One Dimensional Photonic Crystal - 台大電機系計算機中心

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