surface polaritons in layered semiconductor structures m. duracz, a. rusina. saint-petersburg state...
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Surface polaritons inSurface polaritons in layeredlayered semiconductor structuressemiconductor structures
M. Duracz, A. Rusina.
Saint-Petersburg State Polytechnical University, Saint-Petersburg, Russia.
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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.
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ContentsContents
Brief review of the surface polaritons
Surface polaritons at interface
Experiments with surface polaritons
Surface polaritons in a layer
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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Prism coupling. Otto geometryPrism coupling. Otto geometryattenuated total reflection
sinφεсωεε
εε
с
ωkx 2
1
1 )(
φ
2ε
1ε
ε
Otto, 1968
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Kretschmann geometryKretschmann geometryattenuated total reflection
φ
2ε
1ε
ε
sinφεсωεε
εε
с
ωkx 2
1
1 )(
Kretschmann, 1971
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Two-prism methodTwo-prism methodcoupling-decoupling of light & surface waves
coupling decoupling
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Edge coupling techniqueEdge coupling techniquesurface polariton frustration on the edge
inverse process
diffractionpattern
Agranovich, 1975
Chabal,Sievers, 1978
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From edge to edge “jumping”From edge to edge “jumping”frustrated SP transforms into another one
Zhizhin, 1982
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Insertion of second interfaceInsertion of second interfacealteration of the field
1εε
01ε
ε
01ε
02ε
h
x
z
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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
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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
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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
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““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
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““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
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““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
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““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
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““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 )(
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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
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Dispersion curvesDispersion curves“slow” & “fast” double-interface polaritons
1h
2h
2h1h
21 hh
εε0
0ωω
1
xkωc )( 0
dissipation
dissipation
SM
FM
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Frequency region shiftFrequency region shiftthe thickness of the slab varies
31 2εε
1
SM
FM
hcω )( 0
dissipation
dissipation
εε0
0ωω
1
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Excitonic polaritons in lasersExcitonic polaritons in lasersfrom volume to surface polaritons
1ε
2ε
21 , εεωε )(
1ε
2ε
0)(ωε
2ε1ε
)(ωεLedentsov, 1998
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Thank you!Thank you!