0. Introduction 1. Reminder: E-Dynamics in homogenous ...esperia.iesl.forth.gr/~soukouli/OFY/lectures/Dialexi_7.pdf2.9 Quasicrystals 2.10 Photonic Crystal Fibers – „Holey“ Fibers

0. Introduction

1. Reminder:E-Dynamics in homogenous media and at interfaces

2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals

CROW2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers

3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments

A. Yariv et al., Opt. Lett. 24, 711 (1999)

Coupled-resonator optical waveguide (CROW)

Unit cell

Defect cavity

R

x



“tunneling”

Waveguiding in the CROW is achieved through weak couplingbetween otherwise localized high-Q optical cavities!



We assume that the electromagnetic field distribution in one of the resonators is only slightly modified in the CROW-structure compared to an isolated defect (weak coupling).

The eigenmodes of the CROW-structure are Bloch modes.

)ˆ(),( 0 xn

inKRtiK enRrEeeEtrE

K −= Ω− ∑ r

rrr ω

Tight-binding ansatz for the eigenmode of the CROW-structure:

Eigenmode of an isolated resonator centered at x=nR

normalized to unity according to 1)()()(0 =⋅ ΩΩ∫ rErErrdrrrrrrε



The eigenmodes of the CROW-structure satisfy the following wave equation

( ) ),()(),(2

2

trEc

rtrE Kk

K

rrrvrrr ωε=×∇×∇

Dielectric function of the CROW

while the eigenmode of a single defect (centered at x=0 ) satisfies the wave equation

( ) ),()(),(2

2

0 trEcrtrE

rrrvrrrΩΩ

Ω=×∇×∇ ε

Dielectric function of a single defect



( )

)ˆ()(

)ˆ(

02

2

0

xn

inKRtik

xn

inKRti

enRrEeeEc

r

enRrEeeE

K

K

−=

−×∇×∇

Ω−

Ω−

∑

∑rrr

vrrr

ω

ω

ωε

After substituting the ansatz for into the wave equation we obtain:

),( trEKvr

Using the wave equation for the isolated defect leads to:

)ˆ()(

)ˆ()ˆ(

02

2

2

2

00

xn

inKRtik

xxn

inKRti

enRrEeeEc

r

enRrEc

enRreeE

K

K

−=

−Ω−

Ω−

Ω−

∑

∑rrr

vrr

ω

ω

ωε

ε



Next, we multiply this equation by and spatially integrate:

)(rEvr

Ω

)ˆ()()(

)ˆ()()ˆ(

2

02

xn

inKRk

xxn

inKR

enRrErErrde

enRrErEenRrrde

−⋅=

−⋅−Ω

ΩΩ

ΩΩ

∫∑

∫∑rrvrrr

vrvrrr

εω

ε

Solving for , we obtain:

∑∑

≠

≠

+∆+

+Ω=

0

022

1

1

nn

inKRn

ninKR

k e

e

αα

βω

2kω



nα nβ α∆, , and are defined as:

)ˆ()()( xn enRrErErrd −⋅= ΩΩ∫rrrrrrεα

)ˆ()()ˆ(0 xxn enRrErEenRrrd −⋅−= ΩΩ∫rrrrrrεβ

)()(])()([ 0 rErErrrdrrrrrrr

ΩΩ ⋅−=∆ ∫ εεα



If the coupling between the resonators is sufficiently weak, we can keep only the nearest neighbor coupling,

i.e and if n ≠ 1, -1.0=nα 0=nβ

From symmetry considerations, we also require α1=α-1and β1=β-1.

We assume α1, β1, and ∆α to be small.



Finally, we obtain the dispersion relation for the CROW

with .

+∆−Ω= )cos(

21 1 KRk κ

αω

111 αβκ −=



Example (∆α=0, κ1=-0.03) :

Very small group velocity possible!

0. Introduction



Examples2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers


•There are no band gaps for propagation in z-direction.

•Even for in plane propagation, we require a large aspectratio (height/period) in order to meet experimental constraints (beam diameter).

•Scattering losses in the 3rd dimension are responsible forlow transmittance in experiments with 2D PhotonicCrystals.

Some general problems with 2D Photonic Crystals:

•index guiding for the 3rd dimension=> Photonic Crystal Slabs

Strategies to overcome these problems:

•3D Photonic Crystals: 4 examples

3D Photonic Crystals - the Yablonovite

E. Yablonovitch, Phys. Rev. Lett. 67, 2295 (1991)

Wigner-Seitz real-space unitcell of an fcc lattice



1st Brillouin zone of anfcc lattice

Parameters: n = 3.6, d/a = 0.469

3D Photonic Crystals - the opal

abcabc: fcc-Struktur

Opals do not have a complete Photonic Band Gap!

3D Photonic Crystals - the inverse opal

K. Busch et al., Phys. Rev. E 58, 3896 (1998)

3D Photonic Crystals - the inverse opal

A. Blanco et al., Nature 405, 437 (2000)

Band structure of silicon inverse opal with an 88% infiltration of Si into the available opal template voids.

Complete photonicband gap between8th and 9th band

Very sensitive to disorder!

3D Photonic Crystals - the Woodpile (Layer-by-Layer structure)

Proposal: C.M. Soukoulis et al., Solid State Commun. 89, 413 (1994)



fcc for (c/a)2=2, full gap for index contrast > 1.9, 25% gap for holes in Si



Band structure of a woodpile composed of Si-rods

0. Introduction



Refraction at Photonic Crystal interfaces2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers


Refraction at an interface

air/vacuum photonic crystal

?0||

cK

=rω

Kr

Sr

Br

Er

Martin Wegener



?0||

cK

=rω

Kr

Sr

Br

Er

• tangential component of thewavevector is conserved

• frequency is conserved• look at corresponding

iso-frequency curve(analogy: Fermi surface)

Martin Wegener

Snell‘s law of refraction


0||c

K=rω 0||

ccK


0||c

K=rω Sv

rr||group

ω group Kv rrr ∇=


Er

Br

Kr

Kr

Martin Wegener

Martin Wegener


Svrr

||group

ω group Kv rrr ∇=

Er

Br

Kr

Kr

0||c

K=rω


S. Kosaka et al., Phys. Rev. B 58, R10096 (1998)

The result is negative refraction, i.e., refractionthat looks as if the refractive index in Snell’s law would be negative.

The angle inside the medium can be a very sensitive function of the incident (vacuum) angle.

The angle inside the PC also sensitively depends on the frequency via the dependence of the shape of theiso-frequency curve on frequency.

The latter effect can be used as a “superprism“.

n=)(sin

)(sin

med

vac

αα

TM polarization

0. Introduction



CROW2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers


Two ways to numerical success

• Time domain techniques:

FDTD, finite difference time domain

• Frequency domain techniques

“expansion in a basis”

• multiple multipole method (MMP)

• plane wave expansion

0. Introduction


2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods

2.4.1 FDTD2.4.2 Plane-Wave Expansion2.4.3 T-Matrix, Scalar-Wave-Approximation, S-Matrix

2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers


Finite Difference Time Domain - FDTD

Goal: solve Maxwell’s equation in the time domain

Approach:

•Establish a finite computational domain (space where the simulation will be performed)

•Define the material of each cell within the computational domain

•Specify source (e.g. plane wave or gaussian pulse impinging on the boundary of the computational domain)

•Define the boundary conditions (important issue in FDTD!)

•Solve Maxwell’s equations in a leap-frog manner

A. Taflove, Computational electrodyamics: The finite-difference time-domain method


Assume 1D Photonic Crystal:

1ε 2ε 1ε 2ε 1ε 2ε 1ε 2εz

Maxwell’s equations:

t

H

z

E yx∂

∂−=

∂∂

0µ

Ex

Hy kz

t

D

z

Hxy

∂∂−=

∂∂


In FDTD literature one often uses the following notation for functions of space and time:

)(),( iFtzF nni =

where zi= i ∆z and tn= n ∆t.

∆z is the grid separation and ∆t is the time increment.


The spatial and temporal derivatives of F n (i) are written using central finite difference approximations as:

z

iFiF

z

iF nnn

∆−−+=

∂∂ )2/1()2/1()(

t

iFiF

t

iF nnn

∆−=

∂∂ −+ )()()( 2/12/1

and


The stability condition relating the spatial and temporal step size is

To yield accurate results, the grid spacing ∆z in the finite difference simulation must be less than the wavelength,

usually less than λ/10.

ztv ∆=∆max

where vmax is the maximum velocity of the wave.


In FDTD one uses a leap-frog algorithm:

• The grid of the magnetic field is shifted by ∆z/2 with respect to the grid of the electric field

• The electric field is calculated at times t=n*∆t while the magnetic field is calculated at times t=(n+1/2)*∆t.

z

i∆z (i+1)∆z (i+2)∆z (i+3)∆z

Ex


In FDTD one uses a leap-frog algorithm:

• The grid of the magnetic field is shifted by ∆z/2 with respect to the grid of the electric field

• The electric field is calculated at times t=n*∆t while the magnetic field is calculated at times t=(n+1/2)*∆t.

z

(i+1/2)∆z (i+3/2)∆z (i+5/2)∆z (i+7/2)∆z

Hy


( ))1()1()2/1()2/1(0

2/12/1 −−+∆

∆−+=+ −+ iEiEz

tiHiH nx

nx

ny

ny µ

In order to advance the algorithm to the next time step

(n+1)∆t we start with

Magnetic field at the same position but two adjacent time steps.

Electric field at the same time stepbut two adjacent grid points.

Assume that the magnetic and the electric field are given on

the whole computational domain for the time steps (n-1/2)∆tand (n)∆t , respectively.


( ))2/1()2/1()()( 2/12/11 −−+∆∆−= +++ iHiH

z

tiDiD ny

ny

nx

nx

Dielectric displacement at the same position but two adjacent time steps.

Magnetic field at the same time stepbut two adjacent grid points.

Next, we calculate the dielectric displacement at the

time (n+1)∆t


)(

)()(

11

i

iDiE

nxn

x ε

++ =

For the third and final step of the algorithm we need a

relation between Dx and Ex.

For nondispersive materials we simply obtain:

We are now in a position to start the algorithm once more and

to propagate the fields to the time step (n+2)∆t .


Because of the finite computational domain, the values of the fields on the boundaries must be defined so that the solution region appears to extend infinitely in all directions.

It is important to avoid artificially reflected waves at the boundaries leading to inaccurate results.

The effective implementation of “absorbing boundary conditions” is still an active field of research.

A good summary can be found in:

A. Taflove, Computational electrodyamics: The finite-difference time-domain method


• Note, that the FDTD algorithm does not take advantage of the periodicity of the Photonic Crystal => “brute force approach”.

Some remarks:

• FDTD is often the method of choice when dealing with defect structures in Photonic Crystals.

FDTD – example of a 2D Photonic Crystal with a linear defect

A. Mekis et al., Phys. Rev. Lett. 77, 3787 (1996)

Fourier

transformation

0. Introduction






∑=i

kii

k bbHrr rr

φα

Expand the “unkown” fields into a finite set of basis functions:

basis functions

(complex) expansion coefficients

multiindex, to be specified

As for all good basis functions, these ones should be orthonormal:

mnk

mk

nbb

,δφφ =rr rr

With the definition of the scalar or inner product:

∫=V

km

kn

km

kn r

bbbbrrrrr rrrr

d*φφφφ

∑=i

kii

k bbHrr rr

φα

Expand the “unkown” fields into a finite set of basis functions:

∑∑

=

j

kjj

ki

j

kjjH

ki rc

rrLr bbbb )()()()(2

rrrrrrtrr rrrr φαωφφαφ

… and project the basis functions from the left:

Now, let us introduce this expansion for the fields here …

bb kkH Hc

HLrr rrt 2

= ω

∑∑

=

jjji

jjji Bc

A αωα ,2

,

Some transformations:

∑∑

=

jj

kj

kij

j

kjH

ki rrc

rLr bbbb αφφωαφφ )()()()(2

rrrrrrtrr rrrr

∑∑

=

j

kjj

ki

j

kjjH

ki rc

rrLr bbbb )()()()(2


bb kjH

kiji LA

rr rtrtφφ=, bb

kj

kijiB

rr rrtφφ=,

=> eigenvalues and expansion coefficients can be computed

Now: choose plane waves as basis

( ) rGkik

kG

b

b

b eGpV

rrrr

r

r

vrrrr ⋅+= )(

,,

1)( σσφ

Reciprocal lattice vector

Polarization vector

Volume of the primitive cell

Multiindex i,j run now over reciprocal lattice vectors and polarization directions.

Additional constraint (Maxwell equations): 0=⋅∇ Hrr

Therefore, we choose for each G the proper base to fulfill the constraints right away.

∑∑

=

jjji

jjji Bc

A αωα ,2

,

Some transformations:

∑∑

=

jj

kj

kij

j

kjH

ki rrc

rLr bbbb αφφωαφφ )()()()(2

rrrrrrtrr rrrr

∑∑

=

j

kjj

ki

j

kjjH

ki rc

rrLr bbbb )()()()(2


bb kjH

kiji LA

rr rtrtφφ=, 1,

trrt rr== bb kj

kijiB φφ

=> eigenvalues and expansion coefficients can be computed

( ) ( ) rGkikbb

rGki

k

kjH

ki

b

b

b

b

bb

eGpV

kikieGpV

L

rrr

rrrr

r

rr

rrrrtrrrr

rtr

⋅+−⋅+− ×∇+×∇+

=

)(2,

1)(1,

211

)()(1

σσ ε

φφ

)()( 2,1,1,21,;, 22112211

GpGpGkGkAbb kGGkbbGG

rrtrrrrrrtrrrrrr

σσσσ ε−++=

bb kjH

kiji LA

rr rtrtφφ=,

×∇+×∇+= − )()( 1rrtrrt

bbH kikiL ε( ) rGkikkG bbb eGpVrrrr

r

r

vrrrr ⋅+= )(

,,

1)( σσφ

)()( 2,1,1,21,;, 22112211

GpGpGkGkAbb kGGkbbGG

rrtrrrrrrtrrrrrr

σσσσ ε−++=

1122

22

2211 ,

2

,,

2,1,1,21

)()( σσσ

σσ αωαε

GGG

kGGkbb cGpGpGkGk

bb

rrr

rrrrtrrrrrr

=++∑ −

31

2

3221 ,,1, GG

GGGGG

rrr

rrrrtt δεε =⋅∑ −

rGGi

VGG

erV

rrr

rrtrt ⋅−−∫= )(, 2121 d

1 εε

rGGi

VGG

erV

rrr

rrtrt ⋅−−−− ∫= )(11, 2121 d

1 εε

Furthermore, we know:

Therefore, compute real-space distribution of material:

1122

22

2211 ,

2

,,

2,1,1,21

)()( σσσ

σσ αωαε

GGG

kGGkbb cGpGpGkGk

bb

rrr

rrrrtrrrrrr

=++∑ −

Some remarks:

• Partial differential equation -> algebraic eigenvalue equation with an infinite number of eigenvalues (sum over all G)

• Numerical computation: truncate after sufficiently large number N of reciprocal lattice vectors G

• CPU time is proportional to N3

• Problem: sometimes poor convergence due to discontinuities in dielectric function

After calculation of eigenvalues (band structure) and coefficients,only the calculation of the fields is required:

rGi

GG

rki

eGpV

erH

b rr

rr

r

rrrr ⋅∑=σ

σα,

,)()(

)()( 1

0

rHi

rErrrtrr ×∇= −ε

ωε

rGki

GbG

beGpGkV

rrr

rr

rrrrt ⋅+− ∑ += )(,

,1

0

)(11

σσσαεωε

Plane wave expansion calculates fields and bandstructure forinfinitely extended systems with fixed epsilon!

Band structure of an empty 2D square-lattice for TE polarization, calculated with plane-wave expansion

choose k point

choose appro-priate base with respect to G

calculate the eigenvalues of the matrix

choose next k point …

How to implement frequency dependent dielectrics in plane wave expansion?

How to implement frequency dependent dielectrics in plane wave expansion?

Cutting surface method, O. Toader and S. John, Phys. Rev. E 70, 046605 (2004)

0. Introduction






How to calculate spectra?

• Finite 1D Photonic Crystal sandwiched between two halfspaces

a

1ε 2ε 1ε 2ε 1ε 2ε 1ε 2ε

E0

rt

trateair/supersε ateair/substrε

Reflection and transmission at an interface

n1 n2

x0 x

E0

r

t

E1

Field and first derivative have to be continuous across interface:

02020101 10xikxikxikxik eEetereE −− +=+

02020101 122101xikxikxikxik eEiketikerikeEik −− −=−

Transfer Matrix

−=

− −−

−

−

122

0

110202

0202

0101

0101

E

t

ekek

ee

r

E

ekek

eexikxik

xikxik

xikxik

xikxik

=

−1

trixTransferma

21

10

E

tMM

r

E 48476 tt

02020101 10xikxikxikxik eEetereE −− +=+

02020101 122101xikxikxikxik eEiketikerikeEik −− −=−

Multiple interfaces

• Finite 1D Photonic Crystal with N-1 layers and N interfaces

a

1ε 2ε 1ε 2ε 1ε 2ε 1ε 2ε

E0

rt

trateair/supersε ateair/substrε

=

+

11)-N(

0

00 E

t

r

Exax MM

tL

t

Transfer Matrix

=

12221

12110

E

t

mm

mm

r

E

0;

0

222111

0

22110

mtmrm

Et

mtmE

+==⇒

+=

Finally, t and then r can be calculated (E1=0!):

Repeat calculation for each frequency to obtain spectrum.

How many periods do we need to obtain a “Photonic Crystal”?

Live numerical experiments for 1d Photonic Crystals with and without defect.

Documents

0. Introduction 1. Reminder: E-Dynamics in homogenous ...esperia.iesl.forth.gr/~soukouli/OFY/lectures/Dialexi_7.pdf2.9 Quasicrystals 2.10 Photonic Crystal Fibers – „Holey“ Fibers