Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization
J. A. Dionne,
*
L. A. Sweatlock, and H. A. Atwater
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, MC 128-95, Pasadena, California 91125, USA
A. Polman
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, MC 128-95, Pasadena, California 91125, USA
and The Center for Nanophotonics, FOM-Institute AMOLF, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
共Received 9 July 2005; revised manuscript received 2 November 2005; published 5 January 2006
兲
We present a numerical analysis of surface plasmon waveguides exhibiting both long-range propagation and
spatial confinement of light with lateral dimensions of less than 10% of the free-space wavelength. Attention
is given to characterizing the dispersion relations, wavelength-dependent propagation, and energy density
decay in two-dimensional Ag/SiO
2
/Ag structures with waveguide thicknesses ranging from 12 nm to 250 nm.
As in conventional planar insulator-metal-insulator 共IMI兲 surface plasmon waveguides, analytic dispersion
results indicate a splitting of plasmon modes—corresponding to symmetric and antisymmetric electric field
distributions—as SiO
2
core thickness is decreased below 100 nm. However, unlike IMI structures, surface
plasmon momentum of the symmetric mode does not always exceed photon momentum, with thicker films
共d ⬃50 nm兲 achieving effective indices as low as n= 0.15. In addition, antisymmetric mode dispersion exhibits
a cutoff for films thinner than d = 20 nm, terminating at least 0.25 eV below resonance. From visible to near
infrared wavelengths, plasmon propagation exceeds tens of microns with fields confined to within 20 nm of the
structure. As the SiO
2
core thickness is increased, propagation distances also increase with localization remain-
ing constant. Conventional waveguiding modes of the structure are not observed until the core thickness
approaches 100 nm. At such thicknesses, both transverse magnetic and transverse electric modes can be
observed. Interestingly, for nonpropagating modes 共i.e., modes where propagation does not exceed the micron
scale兲, considerable field enhancement in the waveguide core is observed, rivaling the intensities reported in
resonantly excited metallic nanoparticle waveguides.
DOI: 10.1103/PhysRevB.73.035407 PACS number共s兲: 73.20.Mf
I. INTRODUCTION
Photonics has experienced marked development with the
emergence of nanoscale fabrication and characterization
techniques. This progress has brought with it a renewed in-
terest in surface plasmons 共SPs兲—electron oscillations that
allow electromagnetic energy to be localized, confined, and
guided on subwavelength scales. Waveguiding over distances
of 0.5
m has been demonstrated in linear chains of metal
nanoparticles,
1
and numerous theoretical and experimental
studies
2–5
indicate the possibility of multicentimeter plasmon
propagation in thin metallic films. Moreover, the locally en-
hanced field intensities observed in plasmonic structures
promise potential for molecular biosensing,
5–10
surface en-
hanced Raman spectroscopy,
11–13
and nonlinear optical de-
vice applications.
14–18
In planar metallodielectric geometries, surface plasmons
represent plane-wave solutions to Maxwell’s equations, with
the complex wave vector determining both field symmetry
and damping. For bound modes, field amplitudes decay ex-
ponentially away from the metal/dielectric interface with
field maxima occurring at the surface.
19
While the dispersion
properties of long-ranging SPs mimic those of a photon,
multicentimeter propagation is often accompanied by signifi-
cant field penetration into the surrounding dielectric. For thin
Ag films 共⬃10 nm兲 excited at telecommunications frequen-
cies, electric field skin depths can exceed 5
m.
3,4
In terms
of designing highly integrated photonic and plasmonic struc-
tures, a more favorable balance between localization and loss
is required.
While metals are characteristically lossy, the bound SP
modes of a single metal/dielectric interface can propagate
over several microns under optical illumination.
4,19
In such a
geometry, the field skin depth increases exponentially with
wavelength in the dielectric but remains approximately con-
stant 共⬃25 nm兲 in the metal for visible and near-infrared
excitation frequencies. This observation has inspired a new
class of plasmon waveguides that consist of an insulating
core and conducting cladding. Not unlike conventional
waveguides 共including dielectric slab waveguides at optical
frequencies, metallic slot waveguides at microwave frequen-
cies, and the recently proposed semiconductor slot
waveguides of Ref. 20兲, these metal-insulator-metal 共MIM兲
structures guide light via the refractive index differential be-
tween the core and cladding. However, unlike dielectric slot
waveguides, both plasmonic and conventional waveguiding
modes can be accessed, depending on transverse core dimen-
sions. MIM waveguides may thus allow optical mode vol-
umes to be reduced to subwavelength scales—with minimal
field decay out of the waveguide physical cross section—
even for frequencies far from the plasmon resonance.
Several theoretical studies have already investigated sur-
face plasmon propagation and confinement in MIM
structures.
21,22
However, few studies have investigated
wavelength-dependent MIM properties arising from realistic
models for the complex dielectric function of metals. The
critical dependence of waveguiding experiments on excita-
PHYSICAL REVIEW B 73, 035407 共2006兲
1098-0121/2006/73共3兲/035407共9兲/$23.00 ©2006 The American Physical Society035407-1
tion wavelength and surface plasmon frequency renders such
an analysis essential. In this paper, we discuss the surface
plasmon and conventional waveguiding modes of MIM
structures, characterizing the metal by the empirical optical
constants of Johnson and Christy
23
and numerically deter-
mining the dispersion, propagation, and localization for both
field symmetric and antisymmetric modes.
II. PLASMON SLOT WAVEGUIDE DISPERSION
The eigenmodes of planar multilayer structures may be
solved via the vector wave equation under constraint of tan-
gential E and normal D field continuity. Uniqueness of the
results is guaranteed by the Helmholtz theorem. For unpolar-
ized waves in a three-layer symmetric structure, the electro-
magnetic fields take the form
E共x,z,t兲 = 共E
x
x
ˆ
+ E
y
y
ˆ
+ E
z
z
ˆ
兲e
i共k
x
x−
t兲
B共x,z,t兲 = 共B
x
x
ˆ
+ B
y
y
ˆ
+ B
z
z
ˆ
兲e
i共k
x
x−
t兲
, 共1兲
with E
y
, B
x
, and B
z
identically zero for transverse magnetic
共TM兲 polarization and E
x
, E
z
, and B
y
identically zero for
transverse electric 共TE兲 polarization.
Inside the waveguide, the field components may be writ-
ten as:
E
x
in
= e
−ik
z1
z
± e
ik
z1
z
,
E
y
in
=0,
E
z
in
=
冉
k
x
k
z1
冊
共e
−ik
z1
z
⫿ e
ik
z1
z
兲,
B
x
in
=0,
B
y
in
=
冉
−
1
ck
x
冊
共e
−ik
z1
z
⫿ e
ik
z1
z
兲,
B
z
in
=0, 共2兲
for the TM polarization and as:
E
x
in
=0,
E
y
in
= e
ik
z1
z
± e
−ik
z1
z
,
E
z
in
=0,
B
x
in
=
冉
− k
z1
c
冊
共e
ik
z1
z
⫿ e
−ik
z1
z
兲,
B
y
in
=0,
B
z
in
=
冉
k
x
c
冊
共e
ik
z1
z
± e
−ik
z1
z
兲. 共3兲
for the TE polarization. Outside the guide, the components
are given by:
E
x
out
= 共e
−ik
z1
d/2
± e
ik
z1
d/2
兲e
ik
z2
共z−d/2兲
,
E
y
out
=0,
E
z
out
=
冉
1
k
x
2
k
z1
冊
共e
−ik
z1
d/2
⫿ e
ik
z1
d/2
兲e
ik
z2
共z−d/2兲
,
B
x
in
=0,
B
y
out
=
冉
−
1
ck
x
冊
共e
−ik
z1
d/2
⫿ e
ik
z1
d/2
兲e
ik
z2
共z−d/2兲
,
B
z
out
=0, 共4兲
for the TM polarization and as:
E
x
out
=0,
E
y
out
= 共e
ik
z1
d/2
± e
−ik
z1
d/2
兲e
ik
z2
共z−d/2兲
,
E
z
out
=0,
B
x
out
=
冉
− k
z1
c
冊
共e
ik
z1
d/2
⫿ e
−ik
z1
d/2
兲e
ik
z2
共z−d/2兲
,
B
y
out
=0,
B
z
out
=
冉
k
x
c
冊
共e
ik
z1
d/2
± e
−ik
z1
d/2
兲e
ik
z2
共z−d/2兲
. 共5兲
for the TE polarization. The in-plane wave vector k
x
is de-
fined by the dispersion relations:
L +:
冦
1
k
z2
+
2
k
z1
tanh
冉
− ik
z1
d
2
冊
=0TM
k
z2
+ k
z1
tanh
冉
− ik
z1
d
2
冊
=0TE
共6兲
L −:
冦
1
k
z2
+
2
k
z1
coth
冉
− ik
z1
d
2
冊
=0TM
k
z2
+ k
z1
coth
冉
− ik
z1
d
2
冊
=0TE
共7兲
with k
z
defined by momentum conservation:
k
z1,2
2
=
1,2
冉
c
冊
2
− k
x
2
. 共8兲
The analysis assumes the structure is centered at z =0 with
core thickness d and wave propagation occurring along the
positive x direction 共see Fig. 1兲. The core 共cladding兲 is com-
posed of material with complex dielectric constant
1
共
2
兲;
we assume all materials are nonmagnetic so that the mag-
netic permeability
has been taken equal to 1. Since surface
plasmons represent charge density oscillations, the disper-
sion relations of Eqs. 共6兲 and 共7兲 define tangential electric
field configurations that are either antisymmetric 共 L +兲 or
DIONNE et al. PHYSICAL REVIEW B 73, 035407 共2006兲
035407-2
symmetric 共L−兲 with respect to the waveguide median. TE
surface plasmon waves do not generally exist in planar met-
allodielectric structures, since continuity of E
y
forbids charge
accumulation at the interface.
When a plasma is excited at a metallodielectric interface,
electrons in the metal create a surface polarization that gives
rise to a localized electric field. In insulator-metal-insulator
共IMI兲 structures, electrons of the metallic core screen the
charge configuration at each interface and maintain a near-
zero 共or minimal兲 field within the waveguide. As a result, the
surface polarizations on either side of the metal film remain
in phase and a cutoff frequency is not observed for any trans-
verse waveguide dimension. In contrast, screening does not
occur within the dielectric core of MIM waveguides. At each
metal-dielectric interface, surface polarizations arise and
evolve independently of the other interface, and plasma os-
cillations need not be energy- or wave-vector-matched to
each other. Therefore, for certain MIM dielectric core thick-
nesses, interface SPs may not remain in phase but will ex-
hibit a beating frequency; as transverse core dimensions are
increased, “bands” of allowed energies or wave vectors and
“gaps” of forbidden energies will be observed.
This behavior is illustrated in Fig. 2, which plots the TM
dispersion relations for an MIM waveguide with core thick-
nesses of 250 nm 关Fig. 2共a兲兴 and 100 nm 关Fig. 2共b兲兴. Since
TE modes in MIM guides resemble those of a conventional
dielectric core, conducting cladding waveguide, their disper-
sion is not explicitly considered here. The waveguide con-
sists of a three-layer metallodielectric stack with a SiO
2
core
and a Ag cladding. The metal is defined by the empirical
optical constants of Johnson and Christy
23
and the dielectric
constant for the oxide is adopted from Palik’s handbook.
24
Solution of the dispersion relations 关Eqs. 共6兲 and 共7兲兴 was
achieved via application of a Nelder-Mead minimization rou-
tine in complex wave-vector space; details of implementa-
tion and convergence properties are described elsewhere.
4
For reference, the figures include the waveguide TM disper-
sion curve in the limit of infinite core thickness, plotted in
black. Allowed wave vectors are seen to exist for all free-
space wavelengths 共energies兲 and exhibit exact agreement
with the dispersion relation for a single Ag/SiO
2
interface
SP.
Figure 2共a兲 plots the “bound” modes 共here, modes occur-
ring at frequencies below the SP resonance, see Ref. 25兲 of a
Ag/ SiO
2
/Ag waveguide with core thickness d = 250 nm. The
asymmetric bound 共a
b
兲 modes correspond to solution of L+
and are plotted in light gray; the symmetric bound 共s
b
兲
modes correspond to solution of L− and are plotted in dark
gray. As seen, multiple bands of allowed and forbidden fre-
quencies are observed. The allowed a
b
modes follow the
light line for energies below ⬃1 eV and resemble conven-
tional dielectric core, conducting cladding waveguide modes
for energies above ⬃2.8 eV. Tangential electric fields E
x
in
each a
b
regime are plotted in the inset and highlight the
distinction in mode localization: At 410 nm, the mode is lo-
calized within the waveguide core and resembles a conven-
tional TM waveguide profile. At 1.7
m, dispersion is more
plasmonlike and field maxima of the mode occur at each
metal-dielectric interface.
In contrast to the a
b
modes, the s
b
modes are only ob-
served for energies between 1.5 and 3.2 eV. Dispersion for
this mode is reminiscent of conventional dielectric core, di-
electric cladding waveguides, with end-point asymptotes
corresponding to tangent line slopes 共effective indices兲 of n
=8.33 at 1.5 eV and n = 4.29 at 3.2 eV. For energies exceed-
ing ⬃ 2.8 eV, wave vectors of the s
b
mode are matched with
those of the SP and the tangential electric field transits from
a core mode to an interface mode 关see the 1st 共⬃3eV兲 and
3rd 共⬃1.9 eV兲 panels of the inset兴. As the core layer thick-
ness is increased through 1
m 共data not shown兲, the number
of a
b
and s
b
bands increases with the a
b
modes generally
lying at higher energies. In analogy with conventional
waveguides, larger 共but bounded兲 core dimensions increase
the number of modes supported by the structure.
Figure 2共b兲 plots the bound mode dispersion curves for an
MIM waveguide with SiO
2
core thickness d= 100 nm.
Again, the allowed a
b
modes are plotted in light gray while
the allowed s
b
modes are dark gray. Although the s
b
mode
resembles conventional waveguide dispersion, the a
b
mode is
seen to exhibit plasmonlike behavior. Accordingly, the con-
FIG. 1. 共Color online兲 Geometry and charac-
teristic tangential 共x兲 electric field profiles for
MIM slot waveguides. The waveguide is centered
at z = 0 with core thickness d and wave propaga-
tion occurring along the positive x direction. 共a兲
Field antisymmetric mode, 共b兲 Field symmetric
mode.
PLASMON SLOT WAVEGUIDES: TOWARDS CHIP-… PHYSICAL REVIEW B 73, 035407 共2006兲
035407-3
ventional waveguiding modes are found only at higher ener-
gies 共over a range of ⬃1eV兲, where photon wavelengths are
small enough to be guided by the structure. The inset shows
snapshots of the tangential electric field for both modes at a
free-space wavelength =410 nm 共⬃3eV兲. As seen, the s
b
field is concentrated in the waveguide core with minimal
penetration into the conducting cladding. In contrast, the a
b
field is highly localized at the surface, with field penetration
approximately symmetric on each side of the metal-dielectric
interface. The presence of both conventional and SP
waveguiding modes represents a transition to
subwavelength-scale photonics. Provided momentum can be
matched between the photon and the SP, and energy will be
guided in a polariton mode along the metal-dielectric inter-
face. Otherwise, the structure will support a conventional
waveguide mode, but propagation will only occur over a
narrow frequency band.
As MIM core thickness is reduced below 100 nm, the
structure can no longer serve as a conventional waveguide.
Light impinging the structure will diffract and decay evanes-
cently, unless it is coupled into a SP mode. Figure 3 illus-
trates the TM dispersion of such subwavelength structures as
core thickness is varied from 50 nm down to 12 nm. As in
Fig. 2, the dispersion curve in the limit of infinite core thick-
ness is also included 共black curve兲.
Figure 3共a兲 plots the TM dispersion relation for the sym-
metric electric-field bound modes with d =50, 35, 20, and
12 nm. Insets plot the tangential electric field profiles for d
=50 nm and d =12 nm at free-space wavelengths of
=1.55
m 共⬃0.8 eV兲 and =350 nm 共⬃3.5 eV兲, respec-
tively. Functionally, the dispersion behaves like the thin-film
s
b
modes of the IMI guide, with larger wave vectors
achieved at lower energies for thinner films. As free-space
energies approach SP resonance, the wave vector reaches its
maximum value before cycling through the higher energy
“quasibound” modes 共not shown; see Ref. 4兲. A 12-nm-thick
oxide can reach wave vectors as high as k
x
=0.2 nm
−1
共
sp
=31 nm兲, comparable to the resonant wave vectors observed
in 12 nm Ag IMI waveguides. However, unlike IMI struc-
tures, the dispersion curve does not lie entirely below the
single-interface 共thick-film兲 limit. Over a finite energy band-
width, SP momentum exceeds photon momentum both in
SiO
2
and in vacuum. The 50-nm-thick oxide provides the
most striking example of this behavior: dispersion lies com-
pletely to the left of the decoupled SP mode. In addition, the
low-energy asymptotic behavior follows a light line corre-
sponding to a refractive index of n = 0.15. This low effective
index suggests that polariton modes of MIM structures more
highly sample the imaginary component of the metal dielec-
tric function than the core dielectric function. In the low-
energy limit, the s
b
SP truly represents a photon trapped on
the metal surface.
Figure 3共b兲 plots the TM dispersion relation for the anti-
symmetric electric-field bound modes, again with d =50, 35,
20, and 12 nm. The tangential electric field profile 共see inset,
plotted for d =50 nm at =1.55
m兲 confirms the purely
plasmonic nature of the mode. In contrast with IMI struc-
tures, where a
b
dispersion approaches the light line for thin-
ner films, wave vectors of the MIM structure achieve larger
values at lower energies. For energies well below SP reso-
nance 共
sp
兲, the observed behavior is not unlike the s
b
modes
of IMI guides. However, at higher energies, bound mode
dispersion does not always terminate at
sp
. In fact, cutoff
frequencies occur at least 0.25 eV below SP resonance for
core thicknesses of less than 20 nm. And, while the maxi-
mum wave vector always exceeds the decoupled SP reso-
nance wave vector, the cutoff k
x
for d 艋 20 nm remains es-
FIG. 2. 共Color online兲 Transverse magnetic dispersion relations
and tangential electric field 共E
x
兲 profiles for MIM planar
waveguides with a SiO
2
core and a Ag cladding. Dispersion of an
infinitely thick core is plotted in black and is in exact agreement
with results for a single Ag/SiO
2
interface plasmon. 共a兲 For oxide
thicknesses of 250 nm, the structure supports conventional
waveguiding modes with cutoff wave vectors observed for both the
symmetric 共s
b
, dark gray兲 and antisymmetric 共a
b
, light gray兲 field
configurations. 共b兲 As oxide thickness is reduced to 100 nm, both
conventional and plasmon waveguiding modes are supported. Ac-
cordingly, tangential electric fields 共E
x
兲 are localized within the core
for conventional modes but propagate along the metal-dielectric
interface for plasmon modes 关inset, plotted in 共a兲 at free-space
wavelengths of = 410 nm 共⬃3eV兲共top two panels兲, = 650 nm
共⬃1.9 eV兲, and =1.7
m 共⬃0.73 eV兲, and in 共b兲 at = 410 nm
共⬃3eV兲兴.
DIONNE et al. PHYSICAL REVIEW B 73, 035407 共2006兲
035407-4
sentially unchanged. An explanation is provided by the
Goos-Hanchen effect. As a signal propagates along the
guide, it undergoes total internal reflection with slight phase
shifts induced by field penetration into and out of the metal
cladding. For thinner films, waveguide dimensions are com-
parable to the field skin depth 共
兲 in the metal. Moreover, the
a
b
E
x
field distribution exhibits a node at the waveguide me-
dian so that energy densities are more highly concentrated at
the metal surface. As waveguide dimensions are decreased,
this enhanced field in the metal magnifies Goos-Hanchen
contributions significantly. In the limit of d 艋
, complete SP
dephasing could result.
III. MODE PROPAGATION AND SKIN DEPTH
Surface plasmon dispersion and propagation are governed
by the real and imaginary components, respectively, of the
in-plane wave vector. Generally, propagation is high in re-
gimes of near-linear dispersion where high signal velocities
overcome internal loss mechanisms. In IMI structures, mul-
ticentimeter propagation is observed for near-infrared wave-
lengths where dispersion follows the light line. However, this
long-range propagation is achieved at the expense of con-
finement: transverse field penetration typically exceeds mi-
crons in the surrounding dielectric. In MIM structures, SP
penetration into the cladding will be limited by the skin
depth of optical fields in the metal. This restriction motivates
the question of how skin depth affects propagation, particu-
larly for thin films.
Propagation for the transverse magnetic polarization
Figures 4 and 5 illustrate the interdependence of skin
depth and propagation in MIM structures for film thicknesses
from 12– 250 nm. The top panels plot propagation of the TM
modes for the structure as a function of free-space wave-
length; the bottom panels plot the corresponding skin depth.
Figure 4 plots propagation and skin depth for a 250 nm oxide
layer. In accordance with the dispersion relations, wave
propagation exhibits allowed and forbidden bands for the
symmetric and antisymmetric modes. The s
b
mode is seen to
propagate for wavelengths between 400 and 850 nm, with
maximum propagation distances of ⬃15
m. The skin depth
for this mode is approximately constant over all wave-
lengths, never exceeding 22 nm in the metal.
26
In contrast,
the a
b
mode is seen to propagate distances of 80
m for
wavelengths greater than 1250 nm. For wavelengths below
450 nm, a smaller band of propagation is also observed,
FIG. 3. 共Color online兲 Transverse magnetic dispersion relations
and tangential electric field 共E
x
兲 snapshots of MIM 共Ag/ SiO
2
/Ag兲
structures as oxide thickness d is varied between 12 nm, 20 nm,
35 nm, and 50 nm. As in Fig. 1, dispersion for an infinitely thick
oxide core is plotted in black. While both the field symmetric 共a兲
and antisymmetric 共b兲 modes exhibit plasmonlike dispersion, the
results do not parallel the behavior observed in IMI waveguides. In
共a兲, the symmetric mode does not lie entirely below the thick-film
limit, and SP momentum can exceed photon momentum both in
SiO
2
and in vacuum. In 共b兲, the antisymmetric mode is seen to
exhibit dispersion similar to the symmetric mode of IMI guides.
However, as d decreases below 20 nm, dispersion no longer termi-
nates on resonance, a result of skin-depth effects related to the
Goos-Hanchen shift.
FIG. 4. MIM 共Ag/SiO
2
/Ag兲 TM-polarized propagation and
skin depth plotted as a function of wavelength for a core thickness
of d =250 nm. Both s
b
and a
b
modes are observed and exhibit cutoff
in accordance with the dispersion curve of Fig. 2. Propagation
lengths of conventional 共as opposed to plasmonic兲 waveguiding
modes are recovered and correlated with skin depth.
PLASMON SLOT WAVEGUIDES: TOWARDS CHIP-… PHYSICAL REVIEW B 73, 035407 共2006兲
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