a quick look to materials at hand ... - catalogue des cours

Cours Nanoph English– Benisty 20131

NANOPHOTONICS (Photonic crystals and Emitters)Sep 17th : Waveguide - Confinement, representations, wg garrays (3h)

Sep 24th : from 1D periodic to 2D periodic, complete Photonic band gap (3h) Defect and light line along a non periodic dimension.

Nov 5th : Defect in Photonic band gap (1h30) + C. Sauvan (Exercices)

nov 12th : C. Sauvan (Exercices) + HB : Nano-emitters (1h30)

Nov 19th : Emission issues Purcell effect, light extraction (3h)

Nov 26 th : Combinations – emitters and cavities, interplay of confinements

Dec 10th : Written exam in two parts(H B part will use ~3 Appl Phys Lett papers (available in early Nov.) as a basis

4 sessions till 22 Oct. included : Ph. Lalanne and C. Sauvan (artif Mat, Metamat, plasmons)


Waveguide - Confinement, representations

Confinement = boundary conditions

Example ‐ Perfect conductor E=0Magnetic wall H=0

Realistic ‐ Real metal, dielectric interfaces

Confinement shape

0 or 1or 2 remaining invariant dimensions (translation or rotation)

Dictates coordinate system and choice of field basisPlane waves or cylindrical waves or spherical waves


A quick look to materials at hand

Oxides Conductors Semiconductors organics

- glass SiO2+some othersn=1.5quartz- CaCO3? (calcite)n=1.6 to 1.8

- PbO, Ta2O5,...

- TiO2 n=2.2 to 2.8

Al203=aluminasapphire n=1.7

LiNbO3 n=2.2 ...

MgO n<1.5________________

MgF2 n=1.38

- Au, Ag, Cu, Al,...Drude modelindex is complex-valued

- Transparent conductor

Indium-tin-oxide (n~2.0)

?- organics (low conductivity)

? -Graphene ?

Si n=3.5 + more dispersive

Ge n=3.8 ?

GaAs n=3.5

GaN (blue LEDs) n=2.5

Carbon : Graphene ? Nanotubes ?(Graphene is a semi-metal)

water n=1.33

alcohol n=1.4PVC n=1.45-1.55

PMMA poly methyl metacrylate"Plexiglas"

C=C chemistryn=1.7-1.8


/ /k

nck ///

Slab waveguide with perfect conductor boundary...

E Eo f (x) exp(i z t) e y ;

Field takes the form

z

x

2 2 2

2 2 2( )y y yE E x Ex z c

2 22

2 2( ) ( ) ( ) ( )f x f x x f xx c

Helmholtz eqn becomes 1D eqn.

Present case 2( ) constant = x n

12( ) sin( ) ( ) exp( ) exp( )x xi

mf x x ik x ik xL

2 2 22 2 2 2

2 2xmk n

c L

L

22 2

21 m

c n L

2( ) ( )x n x

Dispersion diagram : family of hyperbola

m=1

m=3

m=6

m=1 m=3


/ /k

nck ///

Slab waveguide (perfect) ; group velocity

...

Group velocity

z

x

Here : group velocity inverse of phase velocityL / /

gd ddk d

v

Dispersion diagram

in general ( )g kv

2

21

gc cc

n n n

v

2g

cn

v2

gcn

v v

Low, High gv v

Both and c/ngv v

1/222 2

21 12

2g c mn L

v


Density of statesThere are two in‐plane directions (z and y for us)

k// has to be discretized in a big box Ly × Lz

Dispersion diagram

/ /k

nck ///

...

x

Ly

z

number of states in ring and per branch

2 2(2 )

y zL LdN k dk

Born‐van Karman periodic boundary cdts

2 2(2 )

y zL L dkdN k dd

2

2 22g g

dk k nk md c L

v v

So But

Hence2

2...(2 ) (2 )

dN S n S nd c c

PER BRANCH (m=1,2,...)


Density of states

DOS

Envelope of DOS is a parabola

2~dN Vd

in 3D limit

There are cut‐off frequencies

simply

These are relevant to Fabry‐Perot physics

m mc nL

2m

cnL

Adding branch contributions :


More general boundariesReflectivity can be introduced

R=1 : perfect 1‐R<<1 : Fabry‐Perot R<<1 : “usual” weak R

L

Less singularity ? guided and non‐guided (Fabry‐Perot) modes...

Outside medium now exists, so overall R,T can be defined(denoted TFP, RFP)


1=n1 n2

c

ox kk /

General representation for slab with two dielectric media

In each medium, local solution is propagative or is evanescent

1:evan2:prop

1:prop2:prop

1:evan2:evan

Critical angle for the more refringent mediumSnell‐Descartes ≡ conserva on of k//

/ / / ok k

Definition of EFFECTIVE INDEX / / / /eff

vacuum( / )k k

nc k

1 eff 2n n n eff 1n n


z

x

y


kz 1=n1 n2

c

n

1

1

slab (plaque) = thin film ....reflectivity r=r()

~0°

p

T

nnrrFP 2~

2

22

)2exp11Airy(p)

ip(-rt

1

(if r<<1)

Airy(p)2 t

)1(~ FPo rEE

otEE ~

ox kk /

oz kk /

)'cos(2nep


slab (plaque) ... guided modes ...effective index


kz

z

x

y

n

1

1

f(x)

f(x)

DEFINITION OF EFFECTIVE INDEX kz = neff ko

independent of the choice of a reference medium (≠angles)

n1 n2

c

neff

oz kk /

ox kk /

FUNDAMENTAL MODE highest eff & neff (bottom of well)

Guided modes


Energy(& potential)

Equivalence with potential (quantum) well

eff

x

0(bands ...)

localized resonances(guided modes)

2eff( )n

y or z


OVERLAP : CONFINEMENT FACTOR maxmin

2

layer 2

xx E dx

E dx

bottom_clad core top_clad 1

core /L

/L

1

0

~

field profile width(« mode volume »)

effn

1n

2n

core

(curve ~ valid for both scales)

Optimalconfinement

(vs. L)


INDEX CONTRAST size2

2eff2

( ) ( ) ( ) ( )f x n f x x f xx

2 2 2vacuum eff core/xk k n

2 2 2vacuum eff clad/K k n

core clad2eff 2

n

For a typical « mid-well »case2 2

core clad2 2vacuum vacuum2

xk Kk k

2eff( )n

~2xk L size (µm)

2 2core clad~

2 2 2L

1/2core clad

1/2core clad1/2

core clad

~4 2

( )( )

4 2

L

n nL n n

Best confinement vs. index contrast ? ?

« half » sine wave

10-3 10-2 10-1 100

10

1.0

0.1

So the typical size L of coreassociated to tightest mode

n

opt. fiber

SC heterostruct

membrane

Field profile

L


POLARISATION

TE and TM cases (Ey,Hx,Hz) &(Ez,Ex,Hy)(also called « s » and « p » or « E » and « H »)

TE and TM modes, their dispersion are interspersed

TE symmetric solution easiest to get : same graphics solution as quantum well

[ tan(kx*L/2) = some function of kx ]

symmetric waveguide : no cutoff for fdtl modes TE or TM

asymmetric waveguide : cutoff is possible also for fdtl mode,but different for TE and TM


GENERALISATION

Channel guiding

Rectangular, Circular, Elliptic,...

Field has all six em components of E and H !

Approximate solutions for rectangular « wire »

Bessel/Hankel basis for circular : case of optical fibers, V-number : should be =2.4 for monomode criterion ...

Still the concepts of overlap and optimal confinement of fundamtl mode hold

But cutoff conditions may be widely different


Special : « Capillary » guiding : low-index core !

n

1

1

n

r1

Never Total internal reflection, but still high enough to get tens, or hundreds, of reflexions, hence a macroscopic path of100's or 10,000's of capillary diameters


TYPICAL EXAMPLES

Small index contrast : - Silica/Ge-doped silica- LiNbO3/Ion-exchanged LiNbO3

Medium index contrast : -GaAs/AlGaAs ; InP/GaASInP ; -GaN/AlGaN ; ZnSe/CdTe-Silicon nitride/silica-silica/water

In-between : Silica/air

High index contrast :-Silicon/silica-Silicon air-Silicon nitride/air-Ga(Al)As/air-InP/air

« membranes »

« nanofibers »


HOW “NANOPHOTONICS” COMES IN WAVEGUIDES

Simplest periodicity : same symmetry, k// « good quantum number » periodicity allow kx (vertical)

2nd « good quantum number » only in 1st Brillouin Zone

Simplest technology : breaks invariance

k// « good quantum number » only in 1st Brillouin zone


DISPERSION GENERALIZED (SLAB) Perfect Conductor

waveguide

/ /k

nck ///

...

x

L

Light line definition

/ / clad/k c n FP modes ≡ "fuzzy" modes, "finite Q" Above light line:

//k

nck ///

f(x)

Dielectric slab Waveguide

Finite Q

/ / clad/k c n

Below light line : // clad/k c n

nclad is index of cladding, nclad =1, or more

Discrete Guided modes ≡ "Q= FP modes"

(Q ∞ )

(Q finite )

/ / clad/k c n


DOS GENERALIZED (SLAB)

Light line definition

/ / clad/k c n FP modes ≡ "fuzzy" modes, "finite Q" Above light line:

//k

nck ///

f(x)

Dielectric slab Waveguide

Finite Q

/ / clad/k c n

Below light line : // clad/k c n

nclad is index of cladding, nclad =1, or more

Discrete Guided modes ≡ "Q= FP modes"

(Q ∞ )

(Q finite )

/ / clad/k c n

DOS

overall DOS is smoother

But an important issue is : What is DOS of modes of both types !And the ratio of them ! We will come back to this for the

emission control / extraction issue


BASIC OF PERIODIC SYSTEM

- SLAB ARRAY

- LIGHT LINE


NEXT SCOPE : slab array

zkk //

xk

?

/a

xk

zkk //


BAND GAP OPENING

r ~0.2, N~5

order=phase/2

N1

'1N

~ 2 ~ 4 ~ 6

~ reflection

round-tripphase

1 2 3

N1

~ normal incidence , kz=0


NAIVE ORIGIN OF GAP SIZE

)/(25.02 Np

)/(2 Np

(1) phasor of successive reflections

N reflections needed to decay by ~1/eN ~1/r where r=reflection of one period


(2) Reflections in one period

1r 2r{

Depends on internal round-trip phase(fraction of 1-period round-trip phase)Can be low if rather destructiveCan be high if etc.

NAIVE ORIGIN OF GAP SIZE


EXAMPLE8 unit cells of dielectric ( = 5 and = 1)

embedded in air of filling ratio 0.4

Transmission Re(ka) - - -2 2

Im(ka) Tr( [T] )

frequ

ency

a/

2c

average medium

(case kz=0)

0 1

0 1

[T]=1period transmission matrix Cours Nanoph English– Benisty 2013

29

k Periodicity Brillouin zone

/a

-/a

(Un)folding + symmetry

3/a

Conventional choice

2/a

2/a2/a

1st Brillouin Zone:modulus of k minimal

reducedBrillouin Zone

(takes symmetriesinto account)


π/a

-π/a 3π/a2π/a

2π/a2π/a

k

modes are plane wave, but it is allowed (conventional/...)to represent them shifted by 2/a.

The "empty lattice"


Dispersion along invariant direction

x

?//k

nck ///

/ / clad/k c n

Single slabSlab ARRAY


Dispersion along invariant direction

x

Now r can be large Bigger gaps, smaller room between

f(x)

or T by evanescence can occur Narrow T windows open insteadof « evanescent death » exp(-Kx)


Slab array in « potential well picture »

0

core

clad

2t

core

clad

1

bands

bands

The whole spectrum becomes a BAND spectrumThe nature of waves ? Spans from "very localised" to "quasi plane waves"

reflection=1

forbiddenbands(around half-integer valuesof )

Examples

p

paroundinteger

effectif

effectif

T modulation becomes the « band & gap » landscape with abrupt cliffs

Discrete modes couple together and form bands


Generality of « tight binding »

Bloch modes

k=0

k=/ak=2/a

T

FSR

k

n

1period ~FP

double FP

triple FP

« PARTICLES » « MODES »


0*00*...

0...*0...0*

*00

A

vv A )( EH

N1-N1

1-N2-NN

213

1N2

vv*vvv*v

......vv*vvv*v

Nnqi

Nne /2

n1...0v(q)v~

All the eqns then become :

Nnqi

qe /2

n (q)v~v

(q)v~(q)ev~*(q)ev~ /i2-i2 Nqq/N

Hence eigenvalues : )2cos(2

eee*e i2-i2i2-i2

q/Nq

q/Nq/Nq/Nq/Nq

There are only N eigenvalues !!

1 2 N...

0oE

(q)v~vN)(qv~ 2/2n

1...0

niNnqi

Nnee

Let us define a Fourier transform

« Simple » ... because math is the same(wonkish)


zkk //

xk

/a

xk

zkk //

So the overall issue ...


...can be sketched ...

zkk //xk

? ? ?

how are thesetwo plots compatible

zkk //

bandsalong the periodic direction

bandsalonginvariant directions

xk

coupling of guided modes


zkk //

xkk //

Chigrin & Sotomayor-Torres, NATO workshop Proc. 2001

zkk //

xk

very rarely shown (unfortunately in my opinion)...can be shown ! ...


... and brings a third diagram : WAVEVECTOR diagram(for k-space geeks)

zkk //

cut in the planes=cte

Also called "iso-frequency curves"(analogue to Fermi surfaces)

zkk //Difficulties : several folds !(plusieurs nappes)(i.e., multivalued )

xk

xk

Locus of vector k at constant


zkk //

The basic point :Structure of k-space...

xka

xka

xk


Invariance and periodicity vs. our 3D

Invariance × Invariance × Invariance BULK

Invariance × Invariance × Periodicity ARRAY OF SLABS

x × y × z

Invariance × Periodicity × Periodicity INIFINITE 2D PH.CRYSTAL

Periodicity × Periodicity × Periodicity 3D PH.CRYSTAL

/ a


Invariance and periodicity and « FINITE » vs. our 3DInvariance × Invariance × Finite

Invariance × Finite × Finite

Finite × Finite × Finite

Invariance × Periodicity × Finite

Periodicity × Periodicity × Finite

Periodicity × Finite × Finite

SLAB

CHANNEL

SPHERE, RESONATOR, TORE, PILLAR

(WAVEGUIDE) (RESONANT) GRATING

(WAVEGUIDE) CROSSED GRATINGFINITE-HEIGHT 2D PH. CRYSTAL

PERIODIC (CHANNEL) WG


Invariance × Periodicity × Finite

« Resonant waveguide grating »

zkk //

xk

or


« Finite 2D Photonic crystal »

,x yk k

zk Confinement

Periodicity × Periodicity × Finite


PERIODIC (CHANNEL) WG

xk

,z yk k Confinement²

Periodicity × Finite × Finite


LIGHT LINE GENERAL IDEAPeriodicity

Plane Waves of surrounding media are on finite circle.

How can their k component along periodic axis match k of periodic system ?

k along the periodic dimension (« and in the periodicsystem ») goes to as high as desired because it is [2/a]

So k component of outside plane wave could fit such k ?

?? need more drawing...


LIGHT LINE from WG Perspective

z

Gui

ded

mod

e

radi

atin

gm

ode

"leak

y" m

ode

G

"cladding circle"k=nclad/c

• In presence of periodicity kx -component is modulo G=2/a (≡Fourier)

• Bloch mode : Nature of wave weight of the various G components

Leakage ~ ratio of wave amplitudes

z-evanescent

nclad/c

weight of kx –component <nclad/c

weight of kx –components >nclad/c

zk zk zk


LIGHT LINE in k- diagram

• kin-plane> or < nclad/c"being OVER/UNDER the LIGHT CONE"

G GG

π/akx

"cladding light line"k=nclad/c

Gkx

G

(just)

4 cases ofincreasing

frequency

zk


Leakage and resonance dampingQuite general phenomenology. For instance :

‐ surface plasmon resonance ("SPP » : will be seen with metal)‐microcavity resonance :

Photon Lifetime in a mode periods

"Res

onan

t " m

ode

G

incident plane wave

emergent plane wave

mode

π/a

kx

k

2 /Q T Q

"fuzzy" dispersion line~FP ~Airy resonance

G


Grating laws in k- diagram

π/akx

No-diffraction

G

One order allowed

Two orders allowed

G

Transm.grating

Refl.grating

Ideal Grating in air, supporting no other mode


Grating laws in k- diagram

π/akx

air to air

Grating at an interfaceindex "1" and index "n"

G

air to "n"

"n" to "n"

Of course, many more cut-off situations


π/a

kx

k

"fuzzy" dispersion line~FP ~Airy resonance

Resonant waveguide grating

A long history, a challenging topic

hint : « Wood anomalies of gratings »(1902) revisited by Rayleigh

Good papers by Ugo Fano (1940...196x) relation to « Fano resonance »

Then branched to : optics : narrow filters

(R,T outside) and optoelectronics in- & out-couplers

(wg to outside, wg to wg...)

Never ending ! General enough (2D gratings, LEDs...)

Why ? Mix of two dimensionalities !Cours Nanoph English– Benisty 2013

53

Resonant waveguide grating fishing (1)


Resonant waveguide grating fishing (2)


A tutorial paper for those interested


TOWARD2D STRUCTURES

- REMINDER ON DOS

- BANDGAP IN 2D :OMNIDIRECTIONNALITY


g()

scales as (–j)–13D 2D 1D 0D

L

~c/nL

L >>

L´ L

L´ L´ L

Directional control"total" control

extraction

Kleppner 1981

Demonstrated with Rydberg atoms (~1980) @ µwave em frequencies.

Confinement in 0,1,2,3 dimensions


Concept of photonic crystal and 3D forbidden gap (Yablonovitch 1987, John 1987)

zkk //

g()≠ !

NO « Constant Energy Surface"in a band of frequency

zk

xkyk


? Control of spontaneous emission by a 1D DBR ? (1988...1995)

g()

3D 2D

L

~c/nL

L >>

Directional control

extraction

Kleppner 1981

2D +1D periodic

zkk //

0

inhibition (directional)

coupling of guided modes


Density-of-states : A useful guide

Array of slabsInfinite along 2 dimensions

Array of teeth/groovesconfinement in third direction

infinite along one direction(e.g. grating,

or resonant-wg grating)


2D PHOTONIC BANDGAP


Concept of photonic crystal with forbidden bands in 2D

+ =

xy x y

"Tartan effect"3 distinctepsilon values!!!

1st Brillouin zoneof square lattice System invariant in the third direction

( Photonic crystal fiber, ...)

System confined inthe third direction( "PC membrane")


Omnidirectionnality Role of Lattice and of index contrast

Let us look at the Photonic Gaps along symmetry axis

k

BAND GAP

(Yablonovitch 1987)Two large gaps

have to overlap,

Each gap reflectsthe periodicity along its associated axis

(different period and strength)

The difference on |kmax| modulusbetween the two axis is therefore critical

It is 1.41 for the square lattice It is MUCH LESS for the triangular lattice : 1.15

1st Brillouin zoneof square lattice

/ a 2 / a


Simple and important

"Tartan effect" means less x smaller gaps in a given direction

More than 1D means mean-gap frequency depends on direction Roundest BZ preferred for coincidence ! Hexagonal lattice in 2D[fcc in 3D]

x y max0

small x

large x


Gap E or "TM" Gap H or "TE"

Key : Topology allows the first excited mode to change effectively change medium, while being orthogonal to the fundamental (see Joannopoulos et al. book, 1995)

E H

k generic 2D structure

low index is connex high index is connex

THE POLARISATION ISSUE


2D : triangular lattice of holes

real space reciprocal space2 2

3a

b

2M3a

22 4K33a a


Fig.5.21

Example : just the free photon : k = n/c(frequency ≡ distance to origin !)

"empty lattice"

Reading 2D dispersion relations (with care)


K lies on a symmetry axis• lin. comb. of two plane waves

stationary waves vg artificially low !!

Origin of Pseudo flat bands


Fig.5.22

Almost free photons (TE)K

M

Gaps at M and K : no overlap

Overlap of gapsat M and Kfr

éque

nce

norm

alis

ée

vecteur d'onde vecteur d'onde

2D dispersion for increasing "crystal strength"

=11


Fig.5.23

M gapOmnidirectional gap

van Hove singularity @ M

norm

alis

ed fr

eque

ncy

Density of states of triangular PhCs

Number of states per unit frequency (DOS)Cours Nanoph English– Benisty 2013

71

K

M

0.1

0.2

0.3

0.4

M K M

f = 30%

No TMgap

u =

a /

BAND STRUCTURE

dielectric neff2 = 11.3

TE gap

Around a=0.25, @ =1,56 µm, a=390 nm, hole diameter, ca. 200 nm

0 0.3 0.6 0.80

0.1

0.2

0.3

0.4

0.5

N=127

gap TE

Air Fraction

Gap TM

GAP MAP: GAP vs. Constituent Fraction (here air)

2 11.3n

The most common tool : gap map


Only two practical kinds of 2D PhCare "infinite": macroporous silicon

and propous alumina

Limitations :‐ vertical guiding : none !‐ no large "hole‐free" zones (pores then coalesce)

limits the control of defects

The case of macroporous-Si


(z)

Substrate" approach"++ integrability++ electrical injection‐ ‐ etching (gravure)

[deep]

" approachMembrane"+(++) some modes are genuinely lossles

(usefulness in real‐world?)++ shallower etching requirement‐‐ intégration/interfaçage

Preferred approach for optoelectronic integration

2D Photonic crystals on waveguides


40°30°

Superprism

vg = k

diffraction~0 !!

iso-Contours

Supercollimator

1 µm

)(sin)(sin 22 ykxk yxo

(This simple dispersion relation is also that of atom vibrations (phonons) in a squareideal lattice when their interaction is limited to the next nearest neighbour)

ALLOWED BANDS

ALLOWED BANDS

Using 2D bands of triangular PhCs


Kosaka et al.NEC, NTT

Superprism Effect

iso-contours

)(sin)(sin 22 ykxk yxo


2D PHOTONIC BANDGAP : practical implementation


light-line issue : 1D Example

nclad

• kin-plane# nclad/c

π/a

Lowest possible nclad : air, a/

Intermediate nclad (SiO2, AlOx), a/

Highest nclad (AlAs, InP) , a/

a/ = 1/(2nclad)

Mid-gap : a/ = 1/(2nav) [~0.17 for nav=3]

nclad

nav

norm

alis

ed fr

eque

ncy.

a/

kx


Example on a 2D structure

f = 30%

0.1

0.2

0.3

0.4

M K M

u =

a /

matrix neff2 = 11.3=3.421) Example with nclad=3 (AlAs)• light line @M u = 0.192

(=3-3/2)• light line @K u = 0.222

(=2/9) Almost 100 % of the gap

is "leaky"!!

1) Example with nclad=3 (AlAs)• light line @M u = 0.192

(=3-3/2)• light line @K u = 0.222

(=2/9) Almost 100 % of the gap

is "leaky"!!

2) Membrane nclad=1• light line @M u = 0.577

(=3-1/2)• light line @K u = 0.667

(=2/3) Fair fraction of gap is OK !!(if neff =3.4 .... However, for an InP membrane, neff ~2.7 )

2) Membrane nclad=1• light line @M u = 0.577

(=3-1/2)• light line @K u = 0.667

(=2/3) Fair fraction of gap is OK !!(if neff =3.4 .... However, for an InP membrane, neff ~2.7 )

BAND STRUCTURE

membrane


Measuring the leaky part of band diagram is easy

R(

°°°

R(

kx = (/c) sin π/a

Find "anomalies"in reflection spectra takenat various angles

( old topic : Wood, Phil. Mag. 1902 !)


A measurement method…

several periods a a/ variable

cleaved edge

(

excitationlaser

d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

TE t

rans

mis

sion

alo

ng

K

u=a /

TE K

300nm280

nm

260nm

Stitching of 7 spectravisualisation of the whole gap

nm nm200 240

180nm

220nm

valence band edge

photonicgap

I2())I1

PL frontale

Lateral PL

1100

InAsQDs

ou QWs

PL

sign

al (

a.u.

)

900 1000Wavelenth (nm)

QDs

modest spectral width0.20 0.28 0.300.18

Tcristalref.

coll. EPFL, Lausannecoll. NanoElectronics, Glasgow

I2()


Defects in photonic crystal structure

Rationale : - Defect levels should be in gaps

- Bandgap cladding would guarantee absence of leakage : would act as

OPTICAL INSULATOR

- Very desirable becauseMETALS ARE LOSSY

Next : dimensionality of defecte.g. in 2D PhC : 1D or 0D 1 D = waveguide 0 D = (nano)cavity ...

a quick look to materials at hand ... - catalogue des cours

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