kx1=kx2=kx(A)由（1）和（2）可得Ak=2 = 0(B)S.由传播方程a?DI+k(c)c?V?E8Ot?由以上（A），（B）和（C），化简可得dispersion relation881/2C+828111Surfaceplasmonsonsmoothandroughsurfacesandongratings,Chapter2,HeinsRaether,1988
11 由（1）和（2）可得 1 2 1 2 1 2 0 xx x z z kkk k k ε ε = = + = （A） （B） 由传播方程 2 2 2 2 D c E t ∂ = ∇ ∂ 222 ( ) i k k x zi c ω ε = + （C） Surface plasmons on smooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 1988. 由以上（A），（B）和（C），化简可得 dispersion relation 1 2 1/2 1 2 ( ) kx c ω ε ε ε ε = +
08,82)1/2知道了dispersionrealtionkXC8.+8~讨论SPP出现的条件 , = +i则k=k.+ik&2和都是实的DielectricAAMetalP我们有17281k★spC +82818623/22(s))2C +6212Surfaceplasmonsonsmoothandroughsurfacesandongratings,Chapter2,HeinsRaether,1988
12 Surface plasmons on smooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 1988. 知道了dispersion realtion 1 2 1/2 1 2 ( ) kx c ω ε ε ε ε = + ε1 ε2 讨论SPP出现的条件 令 ε2和 ω都是实的 则 我们有 ' " 11 1 εε ε = + i ' " xx x k k ik = + ' ' 1 2 1/2 ' 1 2 ' " " 1 2 3/2 1 ' ' 2 12 1 ( ) ( ) 2( ) x sp x k k c k c ω ε ε ε ε ω εε ε εε ε = = + = + （★）
, <0,1, >62SP出现的条件由于k是实数，可以得到在以上条件下，由（★）可得下图C0=ck可以看到：SP的波失大于光在介质中波失，直接用光不能激0发出表面等离激元。kkspk最后，将kxk"代入以上（c），得到k_2<O，即z方向是候逝波。13Surface plasmons on smooth and rough surfacesand on gratings,Chapter2, Heins Raether,1988.SurfaceplasmonsubwavelengthopticsT.W.Ebbesenetal,NATURE/VOL424|824|2003
13 在以上条件下，由（★）可得下图 Surface plasmons on smooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 1988. Surface plasmon subwavelength optics，T. W. Ebbesen et al, NATURE | VOL 424 | 824| 2003 由于kx’ 是实数，可以得到 ' ' 1 12 ε εε < > 0,| | SP出现的条件 最后，将kx’ kx ”代入以上（C），得到 ，即z方 向是倏逝波。 2 kzi < 0 可以看到：SP的波失大于光在 介质中波失，直接用光不能激 发出表面等离激元
B.Properties of SPPSpatialextensionofSPPfieldsHerekz,andk,are imaginary, skin effect in dielectric and metalcanbecalculated1WhenEdecaysintoEe-1,ZIk.i1/2九8.+8InmetalZ22元[Ez]81/2元8+6IndielectricZ222元62Forexample,2=600nm,forsilver,Z=23nm,Z2=371nmforgold,z,=29nm,z2=281nmForexample,2=1000nm,forsilver,z,=22nm,z,=1122nmforgold,z=24nm,z2=1020nm14Surface plasmons on smooth and rough surfaces and on gratings,Chapter 2, Heins Raether, 1988.Numerical calculationsByL.J.Wang
14 B. Properties of SPP Spatial extension of SPP fields Here kz1 and kz2 are imaginary, skin effect in dielectric and metal can be calculated. 1 | | i zi z k When E decays into Ee-1 , = In metal 1/2 ' 1 2 2 2 2 2 z λ ε ε π ε + = In dielectric 1/2 ' 1 2 1 '2 2 1 z λ ε ε π ε + = For example, λ=600nm, for silver, z1= 23 nm, z2= 371 nm for gold, z1= 29 nm, z2= 281 nm For example, λ=1000nm, for silver, z1= 22 nm, z2= 1122 nm for gold, z1= 24 nm, z2= 1020 nm Surface plasmons on smooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 1988. Numerical calculations By L. J. Wang
PropagationlengthofSPPAlongasmoothsurface,Sp'sintensitydecreasesase1L; =ThepropagationisdefinedasL,>E2e-1,2kForexamples2=600nm,forsilver,L=50.7umforgold,L,=4.9umForexamples,2=1000nm,forsilverL,=698.1umforgold,L,=91.7umSurace plasmons onsmooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 98Numerical calculationsByL.J.Wang
15 Surface plasmons on smooth and rough surfaces and on gratings, Chapter 2, Heins Raether, 1988. Numerical calculations By L. J. Wang Propagation length of SPP Along a smooth surface, SP’s intensity decreases as " 2k xx e− For examples, λ=600nm, for silver, Li = 50.7 um for gold, Li = 4.9 um The propagation is defined as Li E2e-1, " 1 2 i x L k = For examples, λ=1000nm, for silver, Li = 698.1 um for gold, Li = 91.7 um