Surface plasmons

Basics :
Dispersion relation of surface plasmons
References
Surface plasmons
“Metal Optics”
Prof. Vlad Shalaev, Purdue Univ., ECE Department,
http://shay.ecn.purdue.edu/~ece695s/
“Surface plasmons on smooth and rough surfaces and on gratings”
Heinz Raether(Univ Hamburg), 1988, Springer-Verlag
“Surface-plasmon-polariton waveguides”
Hyongsik Won, Ph.D Thesis, Hanyang Univ, 2005.
50nm

Metal Optics: An introduction
Photonic functionality based on metals?!

What is a plasmon?
Note: SP is a TM wave!

Plasmon-Polaritons
Plasma oscillation = density fluctuation of free electrons
+ + +
- - -
k
+ - +
Plasmons in the bulk oscillate at ω p determined by
the free electron density and effective mass
2
Ne
mε
Plasmons confined to surfaces that can interact
with light to form propagating “surface plasmon
polaritons (SPP)”
Confinement effects result in resonant SPP modes
in nanoparticles
2
drude 1 Ne
ω particle
3 mε
=
ω
drude
p
=
0
0

Local field intensity depends on wavelength
(small propagation constant, k) (large propagation constant, k)

Dielectric constant of metal : Drude model
σω ( )
εEFF ( ω) = εB+
i
εω o
= ε + i
1 ⎛ σ ⎞
= ε + i
σ (1 + iωτ)
B
o
⎜ ⎟
ε oω⎝1 − iωτ
⎠
B
o
2 2
εoω(1 + ω τ )
B
2 2
ωτ p
2 2 i
2 2
ωτ p
3 3
⎛ ⎞ ⎛ ⎞
= ⎜ε− +
⎜
⎟ ⎜ ⎟
1+
ωτ ⎟ ⎜ωτ+ ωτ ⎟
⎝ ⎠ ⎝ ⎠
σ
ne
2
2 o
where, ω p = = : bulk plasma frequency ( ∼10
eV for metal)
ετ o εome
Plasma frequency

Ideal case : metal as a free-electron gas
• no decay (γ 0 : infinite relaxation time)
• no interband transitions (ε B = 1)
σ ( ω)
εEFF ( ω) = εB+
i
εω
o
2 2 2 2 2
⎛ ω pτ ⎞ ⎛ ωpτ ⎞ ⎛ ω ⎞
τ →∞
p
= ⎜εB − + i
⎯⎯⎯→ ε 1
2 2 3 3 ε B = 1 EFF = −
⎜
⎟ ⎜ ⎟ ⎜ 2 ⎟
1+
ω τ ⎟ ⎜ωτ + ω τ ⎟ ⎜ ω ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
ε
r
ω
2
p
2
= 1−
0
ω

Surface-plasmons
on dielectric-metal boundaries

Dispersion relation for EM waves in electron gas (bulk plasmons)
• Dispersion relation:
ω = ω(
k)

Dispersion relation for surface plasmons
Exm Exd
ε d
ε m
Z > 0
Z < 0
TM wave
• At the boundary (continuity of the tangential E x , H y , and the normal D z ):
= Hym = Hyd
ε mEzm = ε dEzd

Dispersion relation for surface plasmon polaritons
k H = ωε E
zi
yi
i
xi
k H = ωε E
zm
k
ε
ym
m
k H =
ωε E
zd
yd
xm
Exm = Exd
Hym = Hyd
k
zm zd H ym = H yd
m ε d
( −ikziH
yi,
0,
ikxiH
yi )
d
xd
( −iωε
iE
xi,
0,
−iωε
iE
zi )
k k
=
ε ε
zm zd
m d

Dispersion relation for surface plasmon polaritons
• For any EM wave:
2
2 ⎛ω⎞ 2 2
= εi = x + zi x ≡ xm = xd
k ⎜ ⎟ k k , where k k k
⎝ c ⎠
SP Dispersion Relation
k
x
ω
=
c
ε mεd ε + ε
m d

Surface plasmon dispersion relation: relation
ω
p
1+
ε
ω
ω p
d
ω = ω + ck
2 2 2 2
p x
ck x
z
ε
d
Dielectric:
ε d
x
Metal: εm = ε '
m +
ε "
m
k
x
ω ⎛ ε ⎞ mε
d =
⎜
⎟
c ⎝ ε m + ε d ⎠
1/
2
Radiative modes
(ε'm > 0)
Quasi-bound modes
(−εd < ε'm < 0)
Bound modes
(ε' m < −ε d )
k
zi
Re k x
⎛ 2
εi
1/2
⎞
⎝εm + εd
⎠
ω
= ⎜ ⎟
c
real k x
real k z
imaginary k x
real k z
real k x
imaginary k z

X-ray wavelengths at optical frequencies !!
λ vac=360 nm
SiO2 Ag
Very small
SP wavelength

Excitation of surface plasmons

Surface-plasmon-polariton waveguides
Ssurface plasmon polaritons
propagating along very-thin(~ 10 nm) metal strips
Dielectric – ε 3
Metal – ε 2
Dielectric – ε 1

When the film thickness becomes finite.
mode
overlap

Possibility of Propagation Range Extension
frequency
in-plane wavevector
Long-Range SP:
weak surface confinement, low loss
Short-Range SP:
strong surface confinement, high loss

Several 1 cm long, 15 nm thin and 8 micron wide gold stripes guiding LRSPPs
3-6 mm long control electrodes
low driving powers (approx. 100 mW) and high extinction ratios (approx. 30 dB)
response times (approx. 0.5 ms)
total (fiber-to-fiber) insertion loss of approx. 8 dB when using single-mode fibers

Fundamental symmetric mode of a metal stripe : thickness (T)
14nm
W=10um
16nm
T
LR-SP WG
18nm 20nm

P. Berini, PhotonicWest 2005.

(Spectalis Co.)

Tunable Wavelength filter by direct heating a metal wire
Polymer 2
INPUT OUTPUT
+ -
Polymer 1
Substrate
Transmittance (dB)
-40
-45
-50
-55
-60
-65
-70
1544.1
1540 1545 1550 1555 1560
W avelength (nm )
1558.3

Vertical directional couplers
H. Won, APL vol.88, 011110 (2006)

Variable optical attenuator based on LR-SPP
Submitted to EL, S. Park & S. Song

Localized Surface Plasmons :
Nanofocusing and Nanolithography
ω (10 15 s -1 )
8
6
4
2
ω=c k x
λ=337 nm; ε 1 = -1
Broad
dispersion
0
0 20 40 60 80 100
k x (μm -1 )
'
1
At large kx
( ε1 →−ε2), z i ≈ .
kx
E = ± iE ( air : + i, metal :- i)
z x
Strong confinement at the interface
Nano focusing

Propose metal nanowires.
Beam radius -> zero!
Propagation Loss
(asymmetric mode)
High

Asymmetric mode : field enhancement at a metallic tip
Er Er Ez M. I. Stockman, “Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides,” Phys. Rev. Lett. 93, 137404 (2004)]
50nm
E z
* See MOVIES : SPP propagation through a metallic tip

2007/5/1 ~
an optical range resonator based on single mode metal-insulator-metal plasmonic gap waveguides.
A small bridge between the resonator and the input waveguide can be used to tune the resonance frequency.
FDTD with the perfectly matched layer boundary conditions

Plasmonic Crystal Demultiplexer and Multiports
the realization of two-dimensional optical wavelength demultiplexers and multiports for surface plasmons
polaritons (SPPs) based on plasmonic crystals, i.e., photonic crystals for SPPs.

Slow Propagation, Anomalous Absorption, and Total External Reflection
of Surface Plasmon Polaritons in Nanolayer Systems
n=2
n=1
n=0

we show how the dispersion relation of surface plasmon polaritons (SPPs) propagating along a perfectly conducting wire can be tailored by
corrugating its surface with a periodic array of radial grooves.
Importantly, the propagation characteristics of these spoof SPPs can be controlled by the surface geometry, opening the way to important
applications such as energy concentration on cylindrical wires and superfocusing using conical structures.

Summary : Plasmonic Photonics
Plasmonics: the next chip-scale technology
Plasmonics is an exciting new device technology that has recently emerged.
A tremendous synergy can be attained by integrating plasmonic, electronic, and conventional dielectric photonic devices
on the same chip and taking advantage of the strengths of each technology.
Plasmonic devices,
therefore, might interface naturally with similar speed photonic devices
and similar size electronic components. For these reasons, plasmonics
may well serve as the missing link between the two device
technologies that currently have a difficult time communicating. By
increasing the synergy between these technologies, plasmonics may be
able to unleash the full potential of nanoscale functionality and
become the next wave of chip-scale technology.