Surface plasmonics 중간고사 (2008. 5. 6 화)
1. Dispersion relation of surface plasmons
The dispersion relation for the surface palsmon excited on the metal-dielectric interface is given by
1/ 2
' " m d
x x x
m d
k k ik
c ω ε ε
ε ε
⎛ ⎞
= + = ⎜ ⎝ + ⎟ ⎠
where the metal has the dielectric onstant' "
m m
i
mε = ε + ε Questions:
1) Show that the cuttoff frequency of the surface plasmon is
sp p
/ 1
dω = ω + ε
.2) Show that the penetration depth in the metal is
(
') ( )
' 20
1 /
m m d m
z = k − ε + ε ε
3) Show that under the conditions that the propagation length is
( )
" 1 " '1 2' 3 / 2 1"' 21 2 1
2 , where
i x x
2( )
L k k
c
ε ε ε
ω
ε ε ε
−
⎛ ⎞
= = ⎜ ⎝ + ⎟ ⎠
4) Explain the principle of the total external reflection of a surface plasmon propagating on a dielectric loaded metal surface. (Use the dispersion-relation curve for the surface plasmon)
2. Metallic nanoparticles (Prof. Shalaev)
We would like to study the optical properties of a piece of SiO2 glass with a refractive index of nSiO2 = 1.5 that has a low concentration of Ag nanoparticles (radius R ~ 10 nm) embedded in it (See the figure). We illuminate the glass on one side with a white light source and notice a beautiful yellow color in transmission.
To understand the transmission properties let us first investigate the propagation in the glass of a monochromatic light wave at an angular frequency ω of the form:
,
E
o=
In this equation β and α are related to the real and imaginary part of the refractive index in the following way: β = kon' and α = −2kon''.
The incident light produces a dipolar excitation in the particle in the y-direction whose dipole moment is
p
y(t)
. The polarization is uniform and parallel to the driving field. When we choose a coordinate system that has its origin in the center of a nanoparticle, the potential inside that nanoparticle is given by:in which Ei is the field inside the nanoparticle. Outside (but close to) the nanoparticle we find that the potential has contributions from the photons and the induced dipole:
Questions:
1) Explain the role of α and β in the expression for a monochromatic wave 2) Explain why the excitation of the nanoparticle is dipolar. (and not for example
quadrupolar)
3) Using the Maxwell’ equations and the well-known boundary conditions stating that the normal components of the electric flux, D, and the tangential components of the electric field, E, are continuous across an interface, show that the field inside the nanoparticle,
E
i, and the induced dipole moment, Py, are given by:and
where
ε
p is the dielectric constant of particle and εh is the dielectric constant of the host material (SiO2). (Hint: use that in general: E = −∇ϕ)4) Does the frequency of maximum absorption shift to higher or lower frequency if the particles where embedded in a higher index glass? Explain your answer.
(SOLUTION #2)
4) Maximum absorption occurs at a frequency for which the polarizability of the metallic nanoparticle is largest (when py/E is largest).
