Preparatory School to the Winter College on Micro and...

1931-4

Preparatory School to the Winter College on Micro and NanoPhotonics for Life Sciences

Marco CENTINI

4 - 8 February 2008

Universit degli Studi di Roma'La Sapienza'Rome, Italy

Photonic crystal basics.

1

Preparatory Schoolto the

Winter College on Micro and Nano Photonics for Life Sciences(4-8 February 2008)

Photonic Crystals Basics

M.CentiniUniversita di Roma, La Sapienza, Roma, Italy

OutlineIntroduction: Photonic Crystals: definition and History;

1D, 2D,3D Photonic Crystals: examples;

Propagation of e.m wavesin periodic media: Blochs Theorem, Band Diagrams;

Group velocity, band edge effects;

Analysis of 1D Ph. C. Calculation of photonic bands;Off axis propagation, isofrequency curves;Finite size structures;

2D and 3D structures: Reciprocal lattice;Brillouin zone, irreducible Brillouin zone;Optical properties of Bulk Ph. C.

Photonic Crystals withIntentional defects: High Q cavities; Control of spontaneous emission:

Bragg waveguides; Photonic crystal waveguides;Photonic crystal fibers

2

Photonic CrystalsPeriodic dielectric structures that can interact resonantly with radiation with wavelengths comparable to the periodicity length of the dielectric lattice.

WHY ARE THEY CALLED PHOTONIC CRYSTALS?Can affect the properties of photons in much the same way that ordinary semiconductor and conductor crystals affect the properties of electrons.Properties of electrons in ordinary crystals are affected by parameters like lattice size and defects, for example. Crystals are made of periodic arrays of atoms at atomic length scales.

HOW DO THEY LOOK?

1d 2d 3d

YablonovitePhotonic CrystalsCrystal

From Bragg Gratings to Photonic Crystals in 5 Steps

1913: Bragg formulation of X-ray diffraction by cristalline solids

1785: The first man-made diffraction grating was made by David Rittenhouse, who strung hairs between two finely threaded screws.

1976: A.Yariv and P. Yeh, study of dielectric multilayer stacks, waveguides and bragg fibers.

1987: Prediction of photonic crystals

S. John, Phys. Rev. Lett. 58,2486 (1987), Strong localization of photons in certain dielectric superlatticesE. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987), Inhibited spontaneous emission in solid state physics and electronics

1928: Blochs Theorem describe the conduction of electrons in crystalline solids. ( Math developed by Floquet in 1D case in 1883)

3

Websites:

http://www.pbglink.com for Extensive web listing of PBG informationhttp://ab-initio.mit.edu/mpb MIT photonic- band packagehttp://homepages.udayton.edu/~sarangan Dayton Univ. Lectures on Photonic crystals

Books:

Photonic Crystals: Towards Nanoscale Photonic Devices, J-M Lourtioz et al. Springer (2003)

Roadmap on Photonic Crystals by Susumu Noda (Editor), Toshihiko Baba (Editor), KluwerAcademic Publishers; (2003).

Nonlinear Photonic Crystals by R. E. Slusher and B. J. Eggleton Eds., (2003).

Photonic Crystals: The Road from Theory to Practice: S. G. Johnson and J. D.Joannopoulos, Kluwer (2002).

Optical Properties of Photonic Crystals K. Sakoda, Springer (2001).

Quantum Electronics 3edition A. Yariv, Wiley (1989).

Optical waves in Layered media Pochi Yeh, Wiley (1988).

Suggested readings

1-D Photonic Crystalsa closer look

x

a

1 2The dielectric constant is periodically modulated in one dimension:

( ) ( ),...2,1,0

;)(

=

+==

n

naxxr r

From cartoons to pictures.

A scanning electron microscopy image of a tiny dielectric mirror being cut from a larger piece of material.

The alternating layers of Ta2O5 and SiO2 which form the mirror are clearly visible.

Unit cell

4

Broadband dielectric mirrors used in optics experiments. The characteristic color of the mirrors is caused by their wavelength dependent reflectivity.


According to their definition, photonic crystals can interact resonantly with radiation with wavelengths comparable to the periodicity length of the dielectric lattice.


The dielectric constant is periodically modulated in two dimensions:

Square lattice of cylindrical rods

( ) ( ),...2,1,0,

;,),(

=

++==

mn

maynaxyxr yxr

A cylinder is put at every lattice point. Any Bravais 2-D lattice is defined by 2 primitive vectors, for the square lattice they are:

y

x

ay

ax

;

;

2

1

yaa

xaa

y

x

=

=r

r

Thus every lattice point can be written as:

,...2,1,0,

;21=

+=

mn

amanRrr

r

u.c.

5


y

xa1

Obvious primitive cell can be defined starting from the primitive vectors:

Primitive unit cell: Unit Cell: a volume (if 3D) space that, when translated through all the vectors of a Bravais lattice, just fills all of space without either overlapping itself or leaving voids.

( ) ( ),...2,1,0,

;21=

++=

mn

amanrrrrrr

;

;

2

1

a

ar

r

There is no unique way of choosing a primitive cell for a given Bravais lattice.

a2With r belonging to the primitive cell

Wigner-Seitz primitive cell

Primitive cell with the full symmetry of the Bravais lattice. It is the region of space around a lattice point, closer to that point than to any other lattice point

Rhombic lattice or isosceles triangular lattice( triangular lattice if triangles are equilateral)


Living example of a 2D Photonic Crystal, the Sea Mouse spine.

6



The dielectric constant is periodically modulated in three dimensions.Lattice is defined by 3 primitive vectors:

( ) ( ),...2,1,0,,

;321=

+++=

lmn

alamanrrrrrrr

;

;

;

3

2

1

a

a

a

r

r

r

,...2,1,0,,

;321=

++=

lmn

alamanRrrr

r

With r belonging to the primitive cell

Reference: Ashcroft & Mermin, Solid State Physics, Saunders College Publishing for details on 3D Bravais lattices and crytal symmetries

7

Natural opalsNatural opals

At the micro scale precious opal is composed of silica(SiO2) spheres some 150 to 300 nm in diameter in a hexagonal or cubic closed packed lattice. These ordered silica spheres produce the internal colors by causing the interference and diffraction of light passing through the microstructure of opal


Propagation of e.m waves in periodic media:

8

Periodicity comparable to incident light wavelengths can be responsible for constructive interferences among scattered light by every lattice point. The phenomenon is exactly the same as it happens in diffraction gratings.

The tracks of a compact disc act as a diffraction grating, producing a separation of the colors of white light. The nominal track separation on a CD is 1.6 micrometers, corresponding to about 625 tracks per millimeter. This is in the range of ordinary laboratory diffraction gratings. For red light of wavelength 600 nm, this would give a first order diffraction maximum at about 22.

Scattering of light in periodically patterned media

http://my.fit.edu/~phsu/

Periodically patterned mediaVs.

Random media

Si wafer top side with devices - patterned roughness. wafer back side with random roughness

Resonant reflectivity at given incident angle and wavelengths

Light is scattered in all directions, there is no evidence of frequency or angle dependent response induced by random roughness

9

If the atoms were randomly placed the free electrons would experience strong scattering with the lattice atoms and a short mean free path is expected.

This is in contrast with the high value of measured conductivity for some crystals.

But crystal lattice are periodic with typical sizes of a few and electrons can be treated as waves (quantum mechanics) with energies corresponding to a typical wavelength comparable to lattice size.

Introduction to Blochs Theorem waves in a periodic medium can propagate without scattering

Electrons Photons

Photonic crystals are characterized by primitive cells of a size comparable to the wavelength of the photon. Resonant scattering can occur as a function of frequency and wavevector.

Light scattered from a random media: The effect can be weak or strong depending on density of scatterers. Size and shape of the scatterers produce wavelength and angular dependence on the efficiency of scattering

Laser scattered by dust

As a results electrons can propagate in a periodic lattice without experiencing scattering with a proper dispersive relation(energy as a function of the wave vector)

wavevector

elec

tron

ene

rgy

Band diagrams can be calculated using Bloch theorem

NOTE forbidden region called GAPs. Electrons with forbidden energies cannot propagate inside the crystals.

Introduction to Blochs Theorem

wavevector

phot

on f

requ

ency

For certain geometries it exists a range of frequencies called GAP for which light is forbidden to exist inside the crystal

Periodic modulation of the dielectric constant can affect the properties of photons in much the same way that ordinary semiconductor and conductor crystals affect the properties of electrons.

Electrons Photons

10

Blochs Theoremfor e.m waves in photonic crystals

Ashcroft & Mermin, Solid State Physics, Saunders College Publishing for a proof of the theorem

Starting point: Maxwell equations and constitutive relations.

.0

,

,

,

=

=

+

=

=

B

D

jt

DH

t

BE

r

r

r

r

r

r

r

.,, 00 HBEDEj rrrrrrrr

===

If the e.m field is time-harmonic and if the bodies are at rest or in very slow motion relative to each other:

Neutral Insulators or dielectrics: =0(conductivity) is negligibly small; non magnetically active r=1;

If we seek solutions of the form: ;)(),(;)(),( titi erHtrHerEtrE ==r

r

r

r

r

r

r

r

We have.


the following eigenvalue equations:

{ } ),()()(

1)( 2

2

rEc

rEr

rEr

E

r

r

r

rrr

r

r

r

==

),()()(

1)( 2

2

rHc

rHr

rHr

H

r

r

r

rr

r

r

r

r

=

=

If r is a periodic function of the spatial coordinate Blochs Theorem states that fields solutions are characterized by a Bloch wave vector K, a band index n and have the form:

;)()( rKinK

erErEr

r

r

r

r

r

r

= ;)()( rKinK erHrHr

r

r

r

r

r

r

=

with:

);()( inKnK arHrHrr

r

r

r

rr +=

);()( inKnK arErErr

r

v

r

rr += i =1,2,3.

;

;

;

3

2

1

a

a

a

r

r

r

);()( irr arrrrv

+= Primitive vectors of the periodic lattice

11


Substituting Blochs solution into eigenvalue equations it is possible to calculate:

Eigen-angular frequencies: ;nKv

Bloch modes (eigenvectors):

);(rEnK

v

r

r );(rHnK

v

r

r

If we consider infinite periodic structures, real values of (i.e lossless case), Eand H are Hermitian eigen-operators:

RnK v

Are a complete set of orthogonal eigen-functions.

);(rEnK

v

r

r );(rHnK

v

r

r

Solving equations for several values of K and w it is possible to calculate and plot band diagrams ( dispersion relations)

Dispersive properties:

Group velocity=Energy velocity ( proof: Yarivs Optical waves in crystals)

Band DiagramsSome Computational tools:

Plane Wave Expansion (PWE) Method -CPU time demanding and poor convergence

Koringa-Kohn-Rostker (KKR) Method - spherical waves expansion

Transfer matrix method - Developed by Pendrys group at Imperial College. It also provides amplitude and phase information.

- Rigorous Coupled-wave analysis RCWA

0

)(KK

g Kv rrrr

r

==

Phase velocity CANNOT be defined appropriately in photonic crystals because eigenfunctions are superposition of plane waves and equiphase surfaces cannot be defined properly. Nevertheless, in the effective medium approximation, long wavelengths with respect to lattice periodicity the effective phase velocity can be defined as:

00

0 )( KK

Kv

r

r =

12

Analysis of 1D photonic crystals:

Analysis of 1D Photonic crystals

For 1D systems we can write the field as:

iKxK exExE )()(

rr

= );()( maxExE KK +=rr

with m = 0, 1, 2,

1 2 1 2 1 2 1 2 1 2 1 2

(x) = (x+a)a

Also called Bragg gratings since 1887

But what happens if we consider the Bloch mode with a wavevector K=K+2*/a?

iKxa

xi

K eexExE2

' )()(rr

= Periodic function satisfying the same conditions as

);()( maxExE KK +=rr

13


band gap

k

0 /a/a

irreducible Brillouin zone

First Brillouin zone

Almost linear behaviour:

Slow light close to the band edge

Effective medium regime

k is periodic:

k + 2/a equivalent to kquasi-phase-matching


1 2 1 2 1 2 1 2 1 2 1 2

(x) = (x+a)a

xa11D lattice has only one primitive vector

;1 xaa =r

Being a periodic function, the dielectric constant can be expanded as a Fourier serie

,...2,1,0 ,)(~)(/2

== =

nGexanG

iGx

The set of values of G are called the reciprocal lattice vectors:

NOTE: the primitive cell of the reciprocal lattice is g=2/a

Dispersion curves are periodic with a unit cell = 2/a called Brillouin zone

The Brillouin zone is a primitive cell of the reciprocal lattice

14

For simplicity we consider only TE polarized waves.In the l-th layer the field at angular frequency w can be written as:

[ ] tiyixxilxxil eeebeatyx yllxl

l +=k)(k)(k

zx),,(E

( ) ;kkk 2y2

x =ll

Where ky is a conserved quantity in every layer and ;k ll n

c

=

z z z

n0

a1

b1

z z zn-2

b(n-1)

a(n-1) an

bn

0 1 2 n-1 n

1n n n nnn-12

z zN-1 N

a(N+1)

n n(N+1)N

a0

b0

y

zx

x0 x1 x2 xn-2 xn-1 xn xN-1 xN

Analysis of 1D Photonic crystalsCalculating photonic bands

Thus the field in xl can be described by the following vector:

Nlb

a

l

l ,...,1,0 =

With this fields representation and imposing continuity of Ez and Hy at every interface it can be shown that: ( see Optical waves in layered media by Pochi Yeh edited by Wiley & Sons)

=

+

+=

l

l

l

l

di

di

lx

lx

lx

lx

lx

lx

lx

lx

l

l

b

a

DC

BA

b

a

e

e

kkkk

kkkk

b

al

l

ll

x

x

k

k

11

11

1

1

0

0

11

11

2

1

Back propagation through the l-layer

Jump at the interface from l to l-1


15

For periodic structures with a unit cell composed of 2 layers, the transfer matrix of a unit cell, relating the fields at the beginning of a unit cell with the field at the next cell is:

=

=

n

n

n

n

n

n

b

aM

b

a

DC

BA

DC

BA

b

a 22

22

11

11

1

1

* The index n refers to the unit cell number

Transfer matrix of the layer composed by material 1

Transfer matrix of the layer composed by material 2

Transfer matrix of the unit cell


Next step: Use Blochs theorem to calculate bands!

According to Blochs Theorem the field in the (n)-th unit cell can be written as:

aniKiKxK eexEanxE

)1()())1(( =+rr

ax

16

Thus we have an eigenvalue problem:

,2221

1211

=

n

niKa

n

n

b

ae

b

a

MM

MM

So we can calculate eigen values and eigenvectors:

=

11

12

Me

Me

b

aiKa

inKa

n

n

and

( )

+= 22112

1arccos

1),k( MM

aK y

If:|(M11+M22)|2; K=m/a + iKi evanescent Bloch wave


-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5x 10

15

K (units of /a)

angu

lar

freq

uenc

y (r

ad/s

)


Matlab code developed using the model previously described. Codes are available.

Onedimsimple.m

17

Off axis propagation in one dimensional photonic crystals

Consider the plane x,y. and e.m. wave propagating with an angle with respect to the direction normal to interfaces.

Ky component is conserved because it is parallel to the interfaces.

Electric field can be written as:

1 2 1 2 1 2 1 2 1 2 1 2

(x) = (x+a)a


y

x

yiKxiKK

yx

xeexEyxE )(),(

rr

=

);sin()sin( 2211

nc

nc

K y ==

Bloch mode for 1D PhC Bloch wavevector

( );, yx KKK =v

( );, yx KKK =v

-15-10

-50

510

15

-8

-6

-4

-2

0

2

4

6

80

0.5

1

1.5

2

2.5

x 1015

Ky ( 1/m)Kx ( 1/m)

ang

ular

freq

uen

cy (

rad/

s)


( );, yx KK =

Brillouin zoneNo periodicity in y direction

n1=1.5n2=2d1=250 nmd2=200 nm

18

The group velocity ( or energy velocity) is orthogonal to the isofrequency curves

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Kx (units of /a)

Ky

(uni

ts o

f /a

)

w=1.32e+015


Isofrequency curves: slices of (Kx,Ky)=cost.

0

)(KK

g Kv rrrr

r

==

parallelnot are and Kvgr

r

Anisotropy induced by stratification in one direction. For 2D cases strong curvatures of dispersion curves leads to superprismeffects and negative refraction.


Onedimband.m


0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

15

Ky ( 1/ m)

ang

ula

r fr

equ

ency

(ra

d/s)

n1=1.5n2=3d1=250 nmd2=200 nm

Omnidirectional mirror: Consider a finite structure embedded in air,lightfrom outside is reflected at any angles of incidence.

If there is some fluorescence inside the structure at this frequency range, it will be guided inside the layers

NOTE: a True omnidirectional mirror should work for both TE and TM polarization. Dispersion curves for TM polarized field should be calculated and overlapped to the dispersion curves for TE pol.

air light cone

19


0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

15

Ky ( 1/ m)

ang

ula

r fr

equ

ency

(ra

d/s)

TETM

Wavelength range for omnidirectional mirror

Onedimguide.m

Consider a structure made of N periods embedded in air:

For a a given incident plane monochromatic wave we want to calculate transmitted field and reflected field.

WE use the transfer matrix method:

Finite Size 1D Photonic Crystals

1 2 12 1 2 12 1 2 12

(x) = (x+a)a

=

n

n

n

n

b

aM

b

a 1

1

In this case we cannot invoke Blochs Theorem because the translational symmetry is broken. Nevertheless we can take advantage of the following relation:

+

+=

011

11

2

122

22out

xoutxx

outx

xoutxx

outx

N

N a

kkkk

kkkk

b

a

At the end of the structure there is only the transmitted field

20


+

+

+

+=

011

11

2

111

11

2

1122

22

11

111

2211

t

kkkk

kkkkMPDP

kkkk

kkkk

r xoutxx

outx

xoutxx

outxN

inxx

inxx

inxx

inxx

Going back to the input:

Unitary incident field

Reflected fieldTransmitted field

1 2 1 2 1 2 1 2 1 2 1 2

(x) = (x+a)a

outn2 material

Transfer matrix of the unit cell applied N-1 times

First unit celln1 materialinput

Finite Size 1D Photonic CrystalsGap for the infinite structureTransmittance for N=5 and N=10 periodsNormal incidence

n1=1.5; nin=1;n2=2.5; nout=1;d1=250 nmd2=400 nm


0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 . 4 1 . 6 1 .8 2

x 101 5

0

0 .2

0 .4

0 .6

0 .8

1

a n g u la r fr e e q ue n c y ( r a d / s )

T

0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2

x 1 01 5

0

0 . 2

0 . 4

0 . 6

0 . 8

1

a n g u la r f r e e q u e n c y ( r a d / s )

K u

nits

of

/a

;in

out

I

IT =

=

0

1 t

DC

BA

r

;1A

t = ;A

Cr =

;)cos(

)cos( 2t

n

nT

inin

outout

=

Onedimfinite.m

21


DOS for infinite lattice

DOS for 15 period lattice

ddk= Density of States

Calculation of DOS for 2D and 3D requires a complex procedure. Nevertheless a s ageneralproperty of photonic crystals, DOS is maximum at band edges

/

ho

mo

gen

eou

s m

ediu

m

Finite Size 1D Photonic CrystalsFields inside the structure can be calculated starting from the value of transmitted field and applying the transfer matrix backward, layer by layer:

=

0

2

2

2 tDb

aout

N

N Interface from output to n2 material

+

+=

N

N

di

di

Nx

lx

Nx

Nx

Nx

Nx

Nx

Nx

N

N

b

a

e

e

kkkk

kkkk

b

aN

N

NN

2

2

k

k

12122

122122

12

12

22x

22x

0

0

11

11

2

1

And so on.

Localization of field is related to the bandwidth of the resonance. Maximum field localization is achieved at the band edge.

0

0.2

0.4

0.6

0.8

1.0

0.56 0.57 0.58 0.59 0.60 0.61

III

0

4

8

12

0 4 8 12z(m)

| |2

T

22

0 2 4 6 8 10 120

2

4

6

8

10

12

14

x ( m)

|E|2

(arb

. u.

)

0 2 4 6 8 10 120

2

4

6

8

10

12

14

x ( m)

|E|2

(arb

. u.

)

0 0.5 1 1.5 2 2.5

x 1015

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

angular freequency (rad/s)

T

Field localized in the low index material

Field localized in the high index material


Finite Size 1D Photonic CrystalsGuided modes

n1n2n1n2

(x) = (x+a)a

Condition for guided modes can be set using the transfer matrix method in the following way:

Guided mode must fulfill the condition to be evanescent waves out of the guiding structure, thus:

=

+0

0 1

0

NaAb

;

;

suby

clady

nc

k

nc

k

>

>

nclad nsub

Moreover, if A is the matrix that relates the fields in the substrate to the field in the cladding, the following relation must be fulfilled

A11=0

Values of Ky fulfilling the condition A11=0 are the propagation constants of guided modes

23

Finite Size 1D Photonic CrystalsGuided modes

Infinite structure bands N=5 period structure guided modes

0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

15

Ky ( 1/ m)

ang

ula

r fr

eque

ncy

(rad

/s)

0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

15

N modes per bandTE40TE30TE20TE10TE00

Fields profile of guided modes can be calculated the same way as before using transfer matrix.

Analysis of 2D and 3D structures:

24

Analysis of 2D and 3D Photonic crystals2D and 3D structures have many properties in common with 1D structures but they offer the opportunity to tailor localization properties in 3D.

WE start by giving a general definition of the reciprocal lattice, already introduced for the one dimensional case.

A function of the 3d space with a given periodicity can be written as a function of a vector rbelonging to the primitive cell:

1,,0 321 ++= aaarrrrr

Primitive vectors

..,...;2,1,0,, )()( 321 curlmnrfalamanrf ==+++rrrrrr

A periodic function can be expanded according to Fourier series:

;)()( =G

rGi GFerfr

r

rr

r

1=RGierr

Equivalent to:

,...2,1,0 ;2 == ppRG rr

The reciprocal lattice is defined as the set of vectors G generated by its three reciprocal primitive vectors, through the formula :

( )

( )

( );2

;2

;2

213

213

132

132

321

321

aaa

aag

aaa

aag

aaa

aag

rrr

rr

r

rrr

rr

r

rrr

rr

r

=

=

=

Ther set of G vectors are selected in order to have a periodic oscillating function over the unit cell. This means that in a lattice point identified by vector R:

Reminding that every lattice point can be obtained by linear combination of primitive vectors:

,...2,1,0,,

;321=

++=

lmn

alamanRrrr

r

Any Bravais lattice has a reciprocal lattice

Analysis of 2D and 3D Photonic crystals

25

Exercise: Verify for 1D the reciprocal lattice primitive cell corresponds to the Brillouin zone.

This statement is always true and it can be used as a definition:

The Brillouin zone for a Bravais lattice is the primitive unit cell of the reciprocal lattice.As a consequence, for a given lattice, band diagrams are periodic functions over its reciprocal lattice.

Example

Square latticeReciprocal lattice

Brillouin zone

Analysis of 2D and 3D Photonic crystalsExamples in 2D

1ar

2ar

a3 would have infinite modulus and orthogonal to a1 and a2

;2

;2

22

2

11

1

aa

g

aa

g

=

=

r

r

x

yKy

Kx2g

r

2gr


Irreducible Brillouin Zone for the square lattice: first Brillouin zone reduced by all thhe symmetries in the point group of the lattice.

Ky

Kx

2gr

2gr

X

It is enough to calculate band diagrams in the irreducible Brillouin zone and then use the symmetry of the lattice to extend the diagrams to the first Brillouin zone.

Center of the Brillouin zone

Center of an edgeM

Center of a faceX

26


y

x

Triangular lattice

Ky

Reciprocal lattice

Brillouin zone:

a 2

a 1

g2

g1 Kx

;3

4

;2

2

3

3

4

2

1

ya

g

yx

ag

=

=

r

r

Irreducible Brillouin zone

Middle of an edge joining two hexagonal facesK

=12:1

E

HTM

a

freq

uenc

y

(2

c/a)

= a

/

X

M

X M irreducible Brillouin zone

r

k

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Photonic Band Gap

TM bands

gap forn > ~1.75:1


http://ab-initio.mit.edu/photons/tutorial/

27

EH

E

HTM TE

a

freq

uenc

y

(2

c/a)

= a

/

X

M

X M irreducible Brillouin zone

r

k

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Photonic Band Gap

TM bandsTE bands


http://ab-initio.mit.edu/photons/tutorial/


Comparison between 2D free space dispersion diagram and 2D Photonic crystal dispersion diagram.

;22 yxb

kkc

+=

X; ky=0;

X M; kx=cost

M ; kx=ky

28


Triangular lattice

TM band gaps tend to occur in lattices formed by isolated high-permittivity regions; TE gaps in connected lattices.

High degree of compactness of triangular lattice make the connected lattice keep some of the properties of the isolated lattice.

Analysis of 2D and 3D Photonic crystals3D example

Yablonovite

I: rod layer II: hole layer

29

0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.42

1.44

1.46

1.48

1.50

520nm1560nm

Effe

ctiv

e re

frac

tive

inde

xNormalized frequency

Complex band structure Internal field enhancement- Low threshold lasing- Enhanced nonlinear optical effects

Spatial dispersion (Superprism effect)- Negative refraction- Large angle deflection 500x- Self-collimation

Dispersive refractive index dispersion- Control of light propagation- Phase-matching for harmonic generation

d/dk 0: slow light(e.g. DFB lasers)

backwards slope:negative refraction

strong curvature:super-prisms,

(+ negative refraction)

Review on Bulk Photonic crystals properties

Photonic crystals withintentional defects

30

Control of Electromagnetic WavesUsing Photonic Crystals with intentional defects

Photonic GAPs can be used to confine light in a region of space

WAVEGUIDES

Light cannot propagate inside the crystal and it is confined in a finite region of space.

Mode of electromagnetic waves inside the cavity must fulfill proper boundary conditions(discrete eigenfrequency spectrum)

RESONANT CAVITIES

conserved k!conserved k!

1d periodicity

Light tuned in the gap cannot exit form the line defect which acts as a waveguide. Propagation constants of guided modes will appear in the gap of the perfectly periodic crystals band diagram.

Control of Electromagnetic Waves1D Micro-cavity

Consider an infinite stack:n1=1.5; d1=250 nm;n2=2.0; d2=150 nm;

We introduce a defect, by adding a layer of material 2. The structure is no longer periodic. We can calculate the new dispersion relation defining a supercell. NOTE: in order to get good results distance between periodic defects in the supercell must be large with respect to wavelength ( weak coupling)

defect

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

15

K (units of /a)

ang

ula

r fr

equ

ency

(ra

d/s)

s.c=8periods-defect-8periods

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

15

K (units of /a)

ang

ula

r fr

equ

ency

(ra

d/s)

31


In we consider finite structures:

Symmetric structure,N=14 periods (HL)+H

0 0.5 1 1.5 2 2.5

x 1015

0

0.2

0.4

0.6

0.8

1

angular freequency (rad/s)

T

Structure with central defect:

7 periods (HL)+H+ 7periods (LH)

0 0.5 1 1.5 2 2.5

x 1015

0

0.2

0.4

0.6

0.8

1

angular freequency (rad/s)T

Micro-cavity sandwiched between two Bragg mirrors.High Q factors, high fields intensity enhancement inside the structure.

wavelength (m)x (m)

T|E|2(arb.u.)


0

002

==

lossPower

Energy

cicleperlostEnergy

EnergyStoredQ

32

Control of Electromagnetic Waves2D single mode Micro Cavity

Radius of Defect (r/ a)

0.1 0.2 0.3 0.40.2

0.3

0.4

0.5

Air Defect Dielect ric Defectair bands

dielect ric b ands

0

Ez:

monopole dipole

X

Mr

k X

Mr

k

X

Control of Electromagnetic WavesSpontaneous emission suppression

33

Control of Electromagnetic WavesPhotonic Crystal waveguides

Consider a finite stack as in the picture:n1=1.5; d1=250 nm;n2=2.0; d2=150 nm;

N=4 (HL)+1H; embedded in air 0 5 10 150

0.5

1

1.5

2

2.5x 10

15

Ky (1/ m)

ang

ula

r fr

eeq

uen

cy (

rad/

s)

Bands of the infinite structure

Guided modes for the finite structure

No guided modes in the gap

Control of Electromagnetic WavesPhotonic Crystal waveguides

We add a defect:

0 5 10 15 200

0.5

1

1.5

2

2.5x 10

15

Ky (1/ m)

angu

lar

free

quen

cy (

rad/

s)

guided mode in the gapCombination of index guiding andphotonic band gap guiding

34

air defect

Control of Electromagnetic WavesHollow Bragg waveguides

Guided light in low index material, ( for example air) is possible taking advntage of the photonic band gap.

For infinite structures guided modes are lossless ( in principle). Finite number of layers is responsible for leaky guided modes.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5x 10

15

Ky (1/ m)

ang

ula

r fr

eeq

uenc

y (r

ad/s

)

Air core guided modesEven and odd modes

Surface modes

Air light cone

Bragg waveguides (Yeh 1978) are the basic principle for the OmniGuide:

B. Temelkuran et al.,Nature 420, 650 (2002)

Control of Electromagnetic Waves2D waveguides

J

J JJ

JJ

JJ J

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.050.1 0.150.2 0.250.3 0.350.4 0.450.5

freq

uenc

y (c

/a)

wavenumber k (2/a)

states of the

bulk crystal

band gap

any state in the gap cannot couple to bulk crystal > localized

(2/a)

35

Control of Electromagnetic Waves2D waveguides

Lossless Bends

symmetry + single-mode + 1d = resonances of 100% transmission

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

Photonic Crystal FibersGuiding light: Conventional Optical Fibres

Cladding

Core

nCore>nCladding

nCladding

36

Photonic Crystal Fibers Bragg reflection

Very low losses

Bandwidth ? Angle of incidence ?

Index contrast ?

Fabrication ?

Bragg fibres, OmniGuide fibres

Burak Temelkuran et alNature 420, 650-653, December 2002.

dHoles

Silica (or other)

Core : - hollow- solid

Photonic Crystal

Birks, Roberts, Russel, Atkin, Shepherd, Electron. Lett. 31, 1941-1942 (1995)

Photonic Crystal Fibers

http://www.physics.usyd.edu.au/cudos/

37

Hollow core and solid core PCFs

Mangan et al, OFC 2004(1.7dB/km Loss@1550nm)

Photonic Crystal FibersHoley Silica Cladding

2r

a

n=1.46

(2/a)

r = 0.45a

(2

c/a)

light cone

airlig

htlin

e =

c

above air line:

guiding in air coreis possible

below air line: surface states of air core[ figs: West et al,

Opt. Express 12 (8), 1485 (2004) ]

Photonic Crystal FibersHollow Core and Holey Silica Cladding

38

Effective index vs PBG guidance

Hole/pitch

Core

Bandwidth

Periodicity

Mechanism

CrucialNot necessary

/10%Unlimited

Large (typ. d/>0.9

Small or large

Hollow or solidSolid

coherent scattering

TIR averaged index

Photonic Crystal Fibers

Photonic Crystals (1 hour)

- Natural Photonic crystal- Fabrication technologies- Properties of bulk photonic crystals- super prism, super lens- enhancement of nonlinear effects

Photonic Crystals- applications ( 1hour)

- Photonic crystal cavities sensors and detectors- Photonic crystal waveguides data processing - Photonic crystal fibers high intensity laser delivery

- supercontinuum generation for spectroscopy

Winter college lectures on Photonic Crystals

Preparatory School to the Winter College on Micro and...

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