Surface polaritons inSurface polaritons in layeredlayered semiconductor structuressemiconductor structures

M. Duracz, A. Rusina.

Saint-Petersburg State Polytechnical University, Saint-Petersburg, Russia.

Surface polaritonSurface polariton

A polariton is an electromagnetic wave that is linearly coupled to an electric or magnetic dipole active elementary excitation in a condensed medium, i.e. it is a photon coupled to a plasmon, phonon, exciton, etc.

A surface polariton is a polariton whose associated electromagnetic field is localized at the surface of the medium.

ContentsContents

Brief review of the surface polaritons

Surface polaritons at interface

Experiments with surface polaritons

Surface polaritons in a layer

Surface electromagnetic wavesSurface electromagnetic wavesZenneck modes

radio frequency surface electromagnetic wavesthat occur at the surface of absorbent medium

Brewster modes

damping brings ‘Brewster case’ rays into twoexponentially decaying away from the interface waves

Fano modes

the only surface normal modes that existat the surface in absence of damping

Negative dielectric functionNegative dielectric function occursoccursin conductors

in insulators

the nearly free electron picture of simple metals gives,1)( 22 ωωωε p ep mneπ ω 22 4

surface polaritons (called surface plasmons) can propagate

in the vicinity of natural frequency of the medium

,)(

)(22

0

200

ωω

ωεεεωε

0ω

(0),0 εε

is the plasma frequency

condition for surface polariton propagation is realized in

dielectrics almost always just above an absorption line

)( εε

(surface phonon, exciton polaritons)

Planar wave hits the interfacePlanar wave hits the interfaceincidence of p-polarized wave

electric fields

)(111

11 ωtxkiziαziαy

xeeBeAH

x

z

1A 1B

B)(- ωtxkiziα

yxeBeH

)(11

1

11

11

11 ωtxkiziαziαyx

xeeBeAωε

cα

z

H

ωε

icE

)( ωtxkiziαx

xeBeωε

cαE

21

2221 -)( xkεсωα

2222 -)( xkεсωα ε

01ε

Boundary conditionsBoundary conditionsfor magnetic field

BBA 11

Bε

αBA

ε

α 11

1

1

for electric field

,11 11

11

1

1 Aε

α

α

εB

ε

α

α

ε

1

1

1 ABα

ε

ε

α21

after the transformation

001 zyzy HH or

001 zyzy EE or

Fresnel formulaeFresnel formulaeequations for reflected and refracted waves

,

1

1

1

1

11

1

1

εα

αε

Aεα

αε

B

1

2

1

1

1

αε

εα

AB

01

1

1

ε

α

α

εif there’s no incident wave

and 01 BB

Fano, 1941

Surface polaritonsSurface polaritonscondition for field to exist

01

1

1

ε

α

α

εαα ,1together with definitions of

lead to εε

εε

с

ωkx

1

12

22

and ,εε

ε

с

ωα

1

21

2

221 εε

ε

с

ωα

1

2

2

22

restrictions on permittivities

ε01- ε

,011 ααεε and for wave to propagate along the

interface ,02 xk so 1εε

Localized fieldLocalized fieldwave vector

magnetic field distribution

,Rεε

εε

с

ωkx

1

1 ,11

21

1 iβεε

ε

с

ωiα

iβ

εε

ε

с

ωiα

1

2

)(1

1 ωtxkizβy

xeBeH )( ωtxkizβ

yxeBeH

x z

yH

z

yH1ε

ε

1εε

Dispersion curveDispersion curveSP at the media with the resonance

220

200 )(

)(ωω

ωεεεωε

xkωc )( 0

0ωω

0ω

ωsp

1

1εсkω x

)(

)(

1

1

ωεε

ωεε

с

ωkx

ω

)(ωε

0ε

εspω0ω

1- ε

Exciting of SP on a line gratingExciting of SP on a line gratingconservation law

)(2)( 11

1 tπnsinφεсωεε

εε

с

ωkx

t

φ

ε

01ε

Beaglehole, 1969

Prism coupling. Otto geometryPrism coupling. Otto geometryattenuated total reflection

sinφεсωεε

εε

с

ωkx 2

1

1 )(

φ

2ε

1ε

ε

Otto, 1968

Kretschmann geometryKretschmann geometryattenuated total reflection

φ

2ε

1ε

ε

sinφεсωεε

εε

с

ωkx 2

1

1 )(

Kretschmann, 1971

Two-prism methodTwo-prism methodcoupling-decoupling of light & surface waves

coupling decoupling

Edge coupling techniqueEdge coupling techniquesurface polariton frustration on the edge

inverse process

diffractionpattern

Agranovich, 1975

Chabal,Sievers, 1978

From edge to edge “jumping”From edge to edge “jumping”frustrated SP transforms into another one

Zhizhin, 1982

Insertion of second interfaceInsertion of second interfacealteration of the field

1εε

01ε

ε

01ε

02ε

h

x

z

Double-interface polaritonsDouble-interface polaritons

)(- ωtxkizβzβy

xeBeAeH

)(-22

2 ωtxkizβy

xeeBH

)(11

1 ωtxkizβy

xeeAH

22222

2 )( εсωkβ x

field associated with a new mode

ε

02ε

01ε1

22221 )( εсωkβ x

εсωkβ x )( 2222 h

Characteristic equationCharacteristic equationusing Fresnel formulae

1

11

1

1

1

1

ε

β

β

ε

ε

β

β

εAB

1

11

2

2

2

2-

ε

β

β

ε

ε

β

β

εBeAe hβhβ

hβeε

β

β

ε

ε

β

β

ε

ε

β

β

ε

ε

β

β

ε 1111 2-

2

2

1

1

2

2

1

1

these equations are consistent if

Maradudin, 1981

Two branches of the modesTwo branches of the modescharacteristic equation

hβeε

β

β

ε

ε

β

β

ε

ε

β

β

ε

ε

β

β

ε 1111 2-

2

2

1

1

2

2

1

1

for positive 21 εε , resolves only if 0εthis means left side of the equation is positive or null

so there’s two eventualities

1 ,1

2

2

1

1

ε

β

β

ε

ε

β

β

εare both positive or negative

““Slow” double-interface modesSlow” double-interface modesin case of negative brackets

characteristic equation transforms to

2

2

1

1 ArcthArcthε

β

β

ε

ε

β

β

εhβ

assuming 21 εε this equation is solvable if 1εε

ε01- ε 2- ε

““Slow” modes’ fieldSlow” modes’ field

2

2

1

1 ArcthArcthε

β

β

ε

ε

β

β

εhβ

εεεε

hxk 1

1

cω

one-interface limit

21

ArcthArcth1

ε

ε

ε

ε

hkx

asymptotic behaviour for small h

3

2 1

xkωc )(

hcω )(

yH

z

1

2

3

yH

z

z

yH

2ε 1εε

““Fast” double-interface modesFast” double-interface modesin case of positive brackets

2

2

1

1 ArthArthε

β

β

ε

ε

β

β

εhβ that is solvable if

21

21

- εε

εεε

characteristic equation transforms to

ε0

1- ε2- ε

21

21

--

εε

εε

ε0

1- ε

2- ε

21

21

--

εε

εε

0hh maxhh

maxhh 0hh

21 2εε

21 2εε

““Fast” modes’ field. Typical caseFast” modes’ field. Typical case

2

2

1

1 ArthArthε

β

β

ε

ε

β

β

εhβ

32

1

εεεε

hxk 2

2

cω

1ε

xkωc )(

hcω )(

one-interface limit

0)( hcω

1

2

3

z

z

z

yH

yH

yH

2ε 1εε

ε02- ε

21

21

--

εε

εε

““Fast” modes’ field. Unusual caseFast” modes’ field. Unusual case

1ε

0cω h maxc

ω hhc

ω

21

ArthArth1

ε

ε

ε

ε

hkx

2

2

1

1 ArthArthε

β

β

ε

ε

β

β

εhβ

ε02- ε

21

21

--

εε

εε

0 εε2

non-typical range

asymptotic behaviour for small h

xkωc )(

transparency

εε0

0ωω

11ωγ 0

dissipation

0ωγ

Influence of dampingInfluence of dampingchanges of dielectric function

220

200 )(

)(ωω

ωεεεωε

γiωωω

ωεεεγωε

,

22

0

200 )(

)(

γ - damping constant

transparency of the medium criterion

)( )( γω,Reγω,Im εε

Dispersion curvesDispersion curves“slow” & “fast” double-interface polaritons

1h

2h

2h1h

21 hh

εε0

0ωω

1

xkωc )( 0

dissipation

dissipation

SM

FM

Frequency region shiftFrequency region shiftthe thickness of the slab varies

31 2εε

1

SM

FM

hcω )( 0

dissipation

dissipation

εε0

0ωω

1

Excitonic polaritons in lasersExcitonic polaritons in lasersfrom volume to surface polaritons

1ε

2ε

21 , εεωε )(

1ε

2ε

0)(ωε

2ε1ε

)(ωεLedentsov, 1998

Thank you!Thank you!