zerogroupvelocity modes of insulatormetalinsulator and ...€¦ · outline lossless case:...

ZeroshyGroupshyVelocity Modes of InsulatorshyMetalshyInsulator and

InsulatorshyInsulatorshyMetal Waveguides

Dmitry Fedyanin Aleksey Arsenin Vladimir Leiman and Anantoly Gladun

Department of General Physics Moscow Institute of Physics and Technology (State University)

Dolgoprudny Russiaeshymail feddumailru

AcknowledgementVladimir Tarakanov the athor of PIC code KARAT

OUTLINE

Lossless caseDispersion relations and dispersion curves of SPPs

in IMI and IIM waveguide structuresExistence conditions of waves with zero group velocity

Lossy caseEffect of losses on dispersion curvesNumerical calculations of energy velocityAn analysis of points of zero energy velocity

Excitation of zeroshygroupshyvelocity modesResults of numerical simulationApplications

History

Backward waves and waves with zero group velocity are typically associated with periodic structures Nevertheless a periodicity is not the only way to obtain such waves - particularly investigations of slow wave propagation in plasma-dielectric structures showed that plasma waveguides with a specially designed circular crosssection allow the existence of backward waves so are some kinds of planar plasmawaveguides

Trivelpiece A W Gould R W 1959 Space charge waves in cylindrical plasma columns J Appl Phys 30 1784 ndash 1793

Paik S F 1962 A backward wave in plasma waveguide Proc IRE 50 462 ndash 463Allis W P Buchsbaum S J Bers A 1963 Waves in Anisotropic Plasma

(Cambridge MA MIT Press)Oliner A A Tamir T 1962 Backward waves on isotropic plasma slabs

J Appl Phys 33 231 ndash 233Schumann W O 1960 Z angew Phys 12 4 145

Lossless caseLossless case

Metal is described by DrudeshyZener model

Dispersion Relation

where

For IMI waveguide structures with ε2=ε3 dispersion relation can be easily simplified and rearranged as two branches

thκ1 a=minusκ2 ε1

κ1ε2

antishysymmetric mode

symmetric mode

and a is a half thickness of the film

thκ1a=minusκ1 ε2

κ2 ε1

IMI IIM

AS shy antishysymmetric mode S shy symmetric mode

Dispersion Curves (Lossless Case)

AS

S

AS

S

ε2gtε3

The decay constant κ2 may be complex while κ1 and κ3 are real

Existence Conditions of Waves with Zero Group Velocity

Equation for Poynting vector in a complex form

SPPs group velocity is negative if and only if net energy flux is opposite to the phase velocity direction that could be written as

and the group velocity is zero if

It will be recall that the only nonshyradiative modes interest us thus the projection of net power flow on z axis is equal to zero

ck ω= where and

[ Allis W P Buchsbaum S J Bers A 1963Waves in anisotropic plasma ]

Existence Conditions of Waves with Zero Group Velocity

IMI IIM

Re(Sx)x

Single metalshyinsulatorinterface

Existence Conditions of Waves with Zero Group Velocity

IMI IIMsmall kx

large kx

small kx

large kx

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

In order to satisfy the existence condition of waves with zero group velocity the power flow inside the metal must be equal (in absolute value) to the power flow in dielectric media Solving Maxwell equations we obtain that this requirement is equivalent to the following equality

This lengthy expression can be significantly simplified in case of antishysymmetric mode and ε2=ε3

[ Allis W P Buchsbaum S J Bers A 1963Waves in anisotropic plasma chapter 7 ]

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

Dispersion curves of SPP in the insulatorshysilvershyinsulator structures for various permittivities Nonshyradiative modes are presented The value of ε2 is fixed and is equal to 25 film thickness d=15 nm Permittivity ε3 takes values from 1 till 12 (1 2 25 3 4 5 7 12) The upper curve corresponds to ε3=1

According to [Palik E D 1998 Handbook of Optical Constants of Solids I (San Diego CA Academic Press) ]and DrudeshyZener modelin a wavelength range 300ndash600 nm

silver

[ Johnson P B Christy R W 1972Phys Rev B 6 4370 ]

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

silver

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametricfunctions of permittivity ε3 Permittivity ε3 takes values from 1 till 20

The value of ε2 is fixed (three materials are considered ZrO2 SiO2 Al2O3

with permittivities 55 22 and 284 correspondibly) film thickness d=15 nm

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

silver

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametricfunctions of permittivity ε3 Permittivity ε3 takes values from 1 till 20

The value of ε2 is fixed (three materials are considered ZrO2 SiO2 Al2O3

with permittivities 55 22 and 284 correspondibly) film thickness d=15 nm

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametric functions of permittivity ε3

Permittivity ε3 takes values from 1 till 20

The value of ε2 is fixed (three materials are

considered ZrO2 SiO2 Al2O3 with permittivities 55 22 and 284 correspondibly) film thickness d=15 nm

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametricfunctions of film thickness d which takes values from 3 nm till 50 nmThe values of ε2 and ε3 are fixed

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 3 nm till 50 nm The values of ε2 and ε3 are fixed

Existence Conditions of Waves with Zero Group Velocity (IMI structures)

d=10 nm d=15 nm

Resonance circular frequency as function of permittivites ofsurrounding insulator media for fixed value of film thickness d

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

silver

vacuum

Dispersion curves of SPP in the vacuumshyinsulatorshysilver structures for various permittivities ε2 Film thickness d=8 nm

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

silver

vacuum

Zeroshygroupshyvelocity points ofthe dispersion curves of SPPs as parametric functions of permittivity ε2 Permittivity ε2 takes values from 1 till 15The value of ε3 is fixed (vacuum) Filmthickness is fixed too and three different values are considered

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametric functions of permittivity ε2 Permittivity ε2 takes values from 1 till 15The value of ε3 is fixed (vacuum) Filmthickness is fixed too and three different values are considered

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

silver

vacuum

Dispersion curves of SPP in the vacuumshyinsulatorshysilver structures for various film thicknesses Permittivity ε2=55 (ZrO2)

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

silver

vacuum

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 2nm till 20nm The value of ε3 is fixed (vacuum) Four different materials as medium ldquo2rdquo are considered dcoff is a cutoff thicknessie the maximum thickness thatallow the existence of zeroshygroupshyvelocity mode

=350 nmλ

=450 nmλ

Existence Conditions of Waves with Zero Group Velocity (IIM structures)

Zeroshygroupshyvelocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 2nm till 20nm The value of ε3 is fixed (vacuum) Four different materials as medium ldquo2rdquo are considered

Lossy caseLossy case

Metal is described by DrudeshyZener model

Silver

Γasymp7 10 13 sshy1

DrudeshyZener dielectric function with dumping factor

Effect of Losses on Dispersion Curves

IMIIMIsilver

Al2O3

Dispersion curves of SPP in the Al2O3shysilvershyinsulator structures for various permittivities ε2 Film thickness d=10 nmDashed lines showdispersion curves ina lossless systemOnly the region where |Im(kx)|lt|Re(kx)| is considered

Effect of Losses on Dispersion Curves

IMIIMI

Effect of Losses on Dispersion Curves

IMIIMIFilm thickness d=10 nmDashed lines showdependecies forlossless systemsOnly the region where |Re(kx)|gt( cω ) max( ε2 ε3) is consideredPoints A B C D correspond to crosspoints of dispersioncurves with light lines

A

B

C

D

Re( )

Effect of Losses on Dispersion Curves

IIMIIM

Dispersion curves of SPP in the vacuumshyinsulatorshysilver structures for various permittivities ε2 Film thickness d=8 nmDashed lines showdispersion curves ina lossless systemOnly the region where |Im(kx)|lt|Re(kx)| is considered

silver

vacuum

Re( )

Effect of Losses on Dispersion Curves

Re ( )

AB

C

AB

C

IIMIIM

Dispersion curves of SPP in the vacuumshyinsulatorshysilver Film thickness d=8 nm Negative values of Im(kx) correspond to backward waves

|Im(kx)|lt|Re(kx)|

=Γ Γ0+βω2

Modified DrudeshyZener model

Γ0=59 10 13 sshy1

=β 32 10 shy18 s[ Chen L Lynch D W 1987Phys Rev B 36 1425 ](red curve)

Effect of Losses on Dispersion Curves

(cmshy1)

Dispersion curves of SPP in the vacuumshyinsulatorshysilver structures for various permittivities ε2 Modified DrudeshyZener model is used Film thickness d=8 nm |Im(kx)|lt|Re(kx)|

Effect of Losses on Dispersion Curves

Comparison of dispersion curves with different dumping factors Γ Silver dielectric functionobeys DrudeshyZener model Silver is covered byultrashythin insulator film of Al2O3Film thickness d=8 nm |Im(kx)|lt|Re(kx)|

IIMIIM

silver

vacuum

Al2O3

Effect of Losses on Dispersion Curves

IIMIIM

silver

vacuum

Al2O3

Dispersion curves of SPP in the vacuumshyinsulatorshysilver structures for various dumping factors Γ Film thickness d=8 nm

Excitation of Excitation of ZeroshyGroupshyVelocity ModesZeroshyGroupshyVelocity Modes

Three-layer Kretschmann

geometry

εpr gt ε2 ε3

Excitation of ZeroshyGroupshyVelocity Modes

IMI

IIMThree-layer Kretschmann

geometry or Otto configuration

εpr gt ε2gt ε3

SimpleKretschmann

geometry

εpr gt ε2gt ε3

Excitation of ZeroshyGroupshyVelocity Modes

Results of NumericalResults of NumericalSimulationSimulation

ApplicationsApplications

andand

Stored Light in NanoshyScale Plasmonic Cavity

d=10 nm

L=180 nm

s=50 nm IMIIMI=Γ 7 10 12 sshy1

εpr=74

=ω 566 10 15 sshy1

=Γ 7 10 13 sshy1

=ϴ 45⁰

and

Stored Light in NanoshyScale Plasmonic Cavity

IMIIMI

Stored Light in NanoshyScale Plasmonic Cavity

IMIIMI=Γ 7 10 12 sshy1E=E0sin(ωt)

point A

point B

Stored Light in NanoshyScale Plasmonic Cavity

IMIIMI=Γ 7 10 12 sshy1E=E0sin(ωt)

V=V0sin(ωt)

point A

point B

Stored Light in NanoshyScale Plasmonic Cavity

IMIIMI=Γ 7 10 12 sshy1

Stored Light in NanoshyScale Plasmonic Cavity

IMIIMI=Γ 7 10 13 sshy1

FutureFuture