6. Disersion relation of surface lasons on ielectric-etal bounaries
Surface lasons (Gary Wieerrecht, Purue University) Definitions: collective ecitation of the free electrons in a etal Can be ecite by light: hoton-electron couling (olariton) Thin etal fils or etal nanoarticles Boun to the interface (eonentially ecaying along the noral) Longituinal surface wave in etal fils Proagates along the interface anywhere fro a few icrons to several illieters (long range lason) or can be etreely confine in nanostructures (localize lason) Note: SP is a TM wave!
Local fiel intensity eens on wavelength (sall roagation constant, ) (large roagation constant, )
Plasa oscillation ensity fluctuation of free electrons - - - - Bul lasons Plasons in the bul oscillate at eterine by the free electron ensity an effective ass Surface lason olaritons Plasons confine to surfaces that can interact with light to for roagating surface lason olaritons (SPP) Localize lasons Confineent effects result in resonant SPP oes in nanoarticles s rue c Ne 0 rue article 1 3 Ne 0
Disersion relation for bul lasons Bul lasons Disersion relation: ( )
surface lason laritons Disersion relation for surface lason olaritons Let s solve the curl equations for TE & TM oes with bounary conitions ji Hi i 0iE i Ei(, y, z) Ei( z) e : i ( z > 0) & i ( z < 0) ji Ei iμ0h i Hi(, y, z) Hi( z) e TE oe TM oe E ( z) (0, E,0), H ( z) ( H,0, H ) i yi i i zi E H y (0) E (0) y (0) H (0) E ( z) ( E,0, E ), H ( z) (0, H,0) i i zi i yi H E y (0) H (0) y (0) E (0)
TE oes : Ei( z) (0, Eyi,0) H ( z) ( H,0, H ) i i zi surface lason laritons H H H H i E i E i H i E z z E E zi yi Eyi Ei iμ0hi iμ0hi iμ0hi y z z i zi i i 0 i i 0 i yi i zi 0 i yi E E y yi i μ0 zi i H i E i yi iμ H 0 zi E yi ( 0 i i) Eyi 0 z We want wave solutions roagating in -irection, but confine to the interface with evanescent ecay in z-irection. Curl equation [ ] ji ± zi z E ( z) Ae e : ( i ), ( i ); Re > 0 yi i zi E z yi zi i ± z z iμ0hi Hi ( z) ± iai e e μ0 Bounary con. E (0) E (0) & H (0) H (0) y y A A & A ( ) 0 z z A A 0 No surface oes eist for TE olarization!
surface lason laritons TM oes : Ei( z) ( Ei,0, Ezi) H ( z) (0, H,0) i yi ( izih yi,0, iih yi ) ( i iei,0, i iezi ) zi H yi E i i z z H H y y E E E E z H H y y H z H y y z z
TM oes : surface lason laritons For any EM wave: i zi, where c SP Disersion Relation c
TM oes : surface lason laritons -irection: z-irection: i" c zi i For a boun SP oe: zi ust be iaginary: < 0 c 1/ i " i zi zi izi ± c 1/ zi ± i ± i > i i c c c ust be real: < 0 for z < 0 - for z > 0 So, <
( ) ( ) ( ) ( ) ( ) 1 " 4 " 1 " " 1 " 4 1 " ) ( ) ( e e e e c c ( ) ( ) ", e where 1/ " i c etals, in ost of an, 0, " >> > <, ( ) " 3/ " 1/ c c " i surface lason laritons
surface lason laritons Proagation length The length after which the intensity ecreases to 1/e : 3/ " " 1 " 1 1 i ( ), where c 1 ( 1) L
Plot of the isersion relation : For ieal free-electrons 1 ) ( c Plot of the ielectric constants: Plot of the isersion relation: s 1,, When ) (1 ) ( s c surface lason laritons
Surface lason isersion relation: surface lason laritons c 1/ zi i c 1/ c c Raiative oes ( > 0) real real z Quasi-boun oes ( < < 0) iaginary real z 1 z Dielectric: Metal: " Boun oes ( < ) real iaginary z Re
Disersion relation for bul an surface lasons surface lason laritons c 1/ τ τ 1 i 1 3 3 τ τ τ Cut-off frequency of SP When 1, 1 s 1
Ag/air, Ag/glass surface lason laritons τ " τ i B i 3 3 1 τ τ τ
For noble etals : J&C easure constants Silver(Ag) isersion 5 4 SP Ag/air light line air SP Ag/glass light line glass 300 E [ev] 3 600 λ [n] Gol(Au) isersion 1 0 10 0 30 40 50 60 5 SP Au/air light line air 4 [u -1 ] 0.1 1 10 100 L [u] 900 100 1500 300 E [ev] 3 SP Au/glass light line glass 600 λ [n] Coer(Cu) isersion 1 0 5 10 15 0 5 30 35 40 [u -1 ] 5 0.1 1 10 100 L [u] 900 100 1500 4 SP Cu/air light line air 300 SP Cu/glass E [ev] 3 light line glass 600 λ [n] 1 0 10 0 30 40 50 60 [u -1 ] 0.1 1 10 100 L [u] 900 100 1500
surface lason laritons X-ray wavelengths at otical frequencies Very sall SP wavelength λ vac 360 n SiO Ag
Penetration eth surface lason laritons 1 At large ( 1 ), z i. Strong concentration near the surface in both eia. E ± ie ( air : i, etal :- i) At low ( 1 >> 1), Ez i 1 E in air : Larger E z coonent Ez 1 i E in etal : 1 z Saller E z coonent Goooo waveguie!
surface lason laritons
Another reresentation of SP isersion relation surface lason laritons ν ( ) " / c
Generalization : Surface Electric Polaritons an Surface Magnetic Polaritons : Energy quanta of surface localize oscillation of electric or agnetic ioles in coherent anner Surface Electric Polariton (SEP) E Surface Magnetic Polariton (SMP) H q -q q -q N S N S Couling to TM olarize EM wave Couling to TE olarize EM wave Coon Features - Non-raiative oes scale own of control eleents - Saller grou velocity than light couling to SP - Enhanceent of fiel an surface hoton DOS
Generalization : Surface Electric Polaritons an Surface Magnetic Polaritons Disersion Relation & Decay Constants ( "/, "/ ) { For μ μ << (1,1), β ( μ μ ) " ", μ μ 1 1 1 SEP 0 O 1 ( μ μ ) " μ" γ γ γ SEP i O,, i 1 1 SEP,1 SEP,, 0 1 μ 1 J. Yoon, et al., Ot E 005.
In Suary Perittivity of a etal ( ) 1 i 1 / γ γ γ Disersion relations SPP c 1/
Tye-A : low Tye-A - Low frequency region (IR) - Wea fiel-confineent H. Won, APL 88, 011110 (006). - Most of energy is guie in cla - Low roagation loss cla sensitive alications SPP waveguies alications
Tye-B : ile Tye-B - Visible-light frequency region - Couling of localize fiel an roagation fiel Nano-hole - Moerate fiel enhanceent Sensors, islay alications Etraorinary transission of light
Tye-C : high Ag (0n) -GaN (0n, 10n) Tye-C Λ QW n-gan - UV frequency region - Strong fiel confineent - Very-low grou velocity Nano-focusing, Nano-lithograhy SP-enhance LEDs Light eission QW SE Rate : 1 1 R f i ρ( ) τ ( ) E 0 Electric fiel strength of half hoton (vacuu fluctuation) Photon DOS (Density of States)
Iortance of unerstaning the isersion relation : Broaban slow an subwavelength light in air
Iortance of unerstaning the isersion relation : Negative grou velocity < 0 SiO 0 1 SiO Si 3 N 4 SiO Si 3 N 4 Al Re 1 Si3N 4