Transient behavior of surface plasmon polaritons scattered at a
  subwavelength groove
Transient behavior of surface plasmon polaritons scattered at a subwavelength groove
G. Lévêque and O. J. F. Martin
Laboratoire de Nanophotonique et Métrologie
Ecole Polytechnique Fédérale de Lausanne
CH1015-Lausanne, Switzerland∗
J. Weiner
IRSAMC/LCAR
Université Paul Sabatier, 118 route de Narbonne,
31062 Toulouse, France†
(Dated: October 24, 2018)
We present a numerical study and analytical model of the optical near-field diffracted in the
vicinity of subwavelength grooves milled in silver surfaces. The Green’s tensor approach permits
computation of the phase and amplitude dependence of the diffracted wave as a function of the
groove geometry. It is shown that the field diffracted along the interface by the groove is equivalent
to replacing the groove by an oscillating dipolar line source. An analytic expression is derived
from the Green’s function formalism, that reproduces well the asymptotic surface plasmon polariton
(SPP) wave as well as the transient surface wave in the near-zone close to the groove. The agreement
between this model and the full simulation is very good, showing that the transient “near-zone”
regime does not depend on the precise shape of the groove. Finally, it is shown that a composite
diffractive evanescent wave model that includes the asymptotic SPP can describe the wavelength
evolution in this transient near-zone. Such a semi-analytical model may be useful for the design and
optimization of more elaborate photonic circuits whose behavior in large part will be controlled by
surface waves.
Keywords: plasmon; surface wave; nanostructure
I. INTRODUCTION
The analysis of light diffraction and transmission through a slit has a long history in physical optics [1, 2]. As
discussed by Kowarz [3, 4] the analysis can be separated into two problems: the boundary value problem and the
propagation problem. The boundary value problem concerns the determination of the field immediately at the output
plane, and interest is usually concentrated on the boundary values in the vicinity of the slit. The propagation problem
involves determination of the field at points in the halfspace beyond the output plane in the near- and far-field. Recent
interest in light transmission through subwavelength periodic structures with subwavelength pitch [5, 6] has stimulated
some experimental [7, 8, 9, 10, 11] and many theoretical studies [12, 13, 14, 15, 16, 17, 18, 19, 20] with the aim of
better understanding the nature of the light field at the surface (the boundary value problem) and the influence this
surface field on light transmission (the propagation problem).
We report here numerical simulations of single groove and slit-groove structures using a Green’s tensor method to
solve the Maxwell field equations near the subwavelength structures on the metal/free-space interface. The simulations
are compared with recent experimental results on single slit-groove structures [8, 9, 10] and essentially confirm the
observed amplitude and phase evolution of the scattered waves as a function of groove geometry and groove-slit
distance. We then show that the results of the full numerical simulation can be recovered by replacing the groove
structure with a simple oscillating dipole source and again applying the Green’s tensor method to find the near- and
far-field along the surface. This oscillating dipole picture is consistent with recent charge and field distributions at
metal-slit and metal-groove boundaries found numerically by a finite-difference time domain (FDTD) technique [17]
and permits calculation of both the propagating and evanescent contributions to the scattered field. Finally we also
present a simple analytic aperture-in-opaque-screen model, in the same vein as earlier models [3, 4, 7], but with a
boundary condition that posits the SPP mode at the metal/free-space interface. Comparison of the Green’s tensor
simulations to the analytic model helps to physically interpret the numeric results in terms of surface-wave modes.
The oscillating dipole picture, however, provides deeper insight into the essential physics of surface wave generation
∗Electronic address: gaetan.leveque@epfl.ch
†Electronic address: jweiner@irsamc.ups-tlse.fr
http://arxiv.org/abs/0704.0703v1
mailto:gaetan.leveque@epfl.ch
mailto:jweiner@irsamc.ups-tlse.fr
FIG. 1: Geometry of the single groove system.
at the groove while overcoming the limitations of any fixed-boundary-condition model.
II. NUMERICAL SIMULATIONS
The numerical simulations are performed with the Green’s tensor method [21, 22]. This method is very convenient
for the study of finite-size, two-dimensional (2D) or three-dimensional (3D) objects embedded in a multilayered
background. It relies on the resolution of the Lippmann-Schwinger equation of the electric field.
E(r) = E0(r) + k20
dr′G(r, r′)∆ǫ(r′)E(r′), (1)
where G(r, r′) is the Green’s tensor associated with the stratified background, k20 = ω
2ǫ0µ0 is the square of the
wave propagation parameter, and ∆ǫ(r) is the “dielectric contrast,” the relative permittivity difference between the
scatterer of volume V and the adjacent layer. The Green’s tensor itself is the solution of a vector wave equation with
a point dipole source,
∇×∇×G(r, r′)− k20ǫG(r, r
′) = 1δ(r− r′), (2)
where 1 is the unit tensor. The 3D Green’s tensor represents the electric field at r produced by three orthogonal unit
dipoles located at r′ in a layer of dielectric constant ǫ. An advantageous distinguishing property of the Green’s tensor
method is that only the objects of interest need be discretized. The boundary conditions at infinity are included in the
Green’s tensor of the multilayered background. In the present case, the calculation takes a very short time because
of the small size (some tens of nanometers) of the groove and slit. Details and extensive references concerning this
method can be found in a recent review [23].
The system initially studied is shown in Fig. 1. It is an empty groove milled into a metallic substrate with the
groove profile along x and extending invariantly along y. Outside the groove, the metal/free-space interface lies in
the 0xy plane with z axis in the vertical direction. The rectangular section of the groove is characterized by width
w and depth t. The incident free-space electromagnetic plane wave, with E-field polarized along x, impinges on the
groove and interface at normal incidence. Because of the reduced dimensionality of the problem, all scattered waves,
propagating and evanescent, are restricted to the 0xz plane.
The electric field is calculated along the line z = 0+, x > 0. All simulations are performed at the free-space
wavelength λ0 = 852 nm. In order to compare these results with previously published studies [8, 10, 20], the relative
permittivity (dielectric constant) of silver is taken to be ǫ = −33.22 + 1.17 i. The corresponding propagation length
against absorption is 680 µm, quite long compared to λ0, because the imaginary term in the dielectric constant is
small for silver at this wavelength. The results of the calculation are plotted in Fig. 2. The principal plot in Fig. 2
shows the evolution of the effective index neff = λ(x)/λ0 as a function of the distance from the center of the groove,
for both Ex and Ez components. The dotted line indicates the effective index of the SPP guided wave,
nspp = ℜ
1 + ǫm
)−1/2
= 1.0154
0 5 10 15 20
x (µm)
sp1 3 5 7
x (µm)
FIG. 2: Evolution of the Ez (solid curve) and Ex (dashed curve) effective index neff(x) = λ(x)/λ0 with the distance from the
center of the groove. The groove dimensions are: t = 100 nm, w = 100 nm. Inset: evolution of the real part of Ez with x,
along the z = 0+ line.
for the silver/free-space interface. The variation of the effective index is sightly different for the two components close
to the groove, but both curves converge rapidly after 3 µm. In both cases, the effective index is larger than nspp out
to ∼ 10 µm, but for greater distances, it converges to the expected nspp. The results of the simulation are consistent
with the measurements of Ref. [8], that reported a value of nsurf = 1.04± 0.01 over a distance of ∼ 6 µm. They are
also consistent with recent finite-difference time-domain (FDTD) simulations on silver surfaces [10] as well as similar
measurements and simulations on gold surfaces [11]. The inset of Fig. 2 shows the z-component of the electric field
diffracted by the groove along the interface. We can clearly identify two regimes. The first extends out to ≃ 3 µm along
x and is characterized by a relatively rapid decay of the amplitude. For further distances, the amplitude decreases
much more slowly (due to absorption) and appears constant over the displayed range. This two-step evolution is
characteristic of a transient regime within the first few micrometers from the groove. Since the incident wave is TM
polarized (E-field parallel to x), Ez belongs only to the scattered field, and does not interfere with the incident wave.
For this reason, the mean value of the real part of Ez in the total field along the interface must be zero. This is not
the case for the x-component, as the incident field is polarized along the x direction.
t (nm)
w = 20 nm
w = 40 nm
w = 60 nm
w = 80 nm
w = 100 nm
20 40 60 80 100 120 140
FIG. 3: (a) Evolution of the amplitude of the z–component of the electric field, 18 µm away from the groove ; (b) Evolution
of the intrinsic phase ϕint of the z–component of the electric field diffracted by the groove (color online).
The simulations also show that the phase and the amplitude of the diffracted wave are sensitive to the groove
dimensions, as reported in experiments [9]. In Fig. 3(a) and (b) are plotted the amplitude and phase evolution of the
surface wave, at large distance (18 µm). The phase ϕint has been determined by comparison with a cosine function
representing the SPP guided surface wave. It corresponds to the asymptotic phase of the diffracted wave, called the
“intrinsic phase” in Ref. [9]. The evolution of the scattered wave phase and amplitude as a function of groove depth is
typical of a resonant phenomenon. Here, the resonance concerns standing-wave modes created inside the groove. The
w and t dimensions of the groove may be varied so as to produce a cavity that resonates when excited by an incident
surface wave. At resonance a vertical standing mode dominates the field distribution inside the groove. Because
of the boundary conditions, the electric field must be almost null at the bottom of the groove. Near a resonance,
the phase of the diffracted wave varies rapidly and passes through an inflection point, while the amplitude reaches a
maximum. As can be observed in Fig. 3, the amplitude is maximal when the phase is almost π. The experimental
results [9] reported an intrinsic phase value of π/2 at the resonance groove depth. This difference of π/2 between the
experiment and the simulation arises from the fact that x– and z– components of the surface wave E-field oscillate
in quadrature. The experiment essentially measures the intrinsic phase difference in a far-field interference pattern
between two oscillating dipoles oriented along x: one localized at the corners of a slit and the other localized at the
groove (see inset of Fig. 4). Thus the experiment is sensitive to the intrinsic phase of the x–component of the surface
E-field while the simulation calculates the intrinsic phase of the z–component. After taking this quadrature phase
difference into account, we see that the simulations are consistent with the measurements. The groove depth for
0 1 2 3 4 5 6
x (µm)
FIG. 4: Comparison between the experimental results of Ref [8] and the numerical simulations performed with a groove of
width w = 100 and depth of t = 120 nm. The silver slab is 400-nm thick, and the slit 100-nm wide. The electric-field intensity
was computed at the exit of the slit (black cross).
which the resonance occurs increases with the width. For a width of w = 100 nm, the simulations yield an optimal
depth t ≈ 110 nm. For shallow depths, some tens of nanometers, the amplitude and phase depend weakly on the
width. At greater depths, the amplitude becomes quite sensitive to the groove width, but the phase does not change
dramatically for widths w > 40 nm. Clearly the phase and the amplitude of the diffracted wave are very sensitive to
the groove geometry. As the absolute groove depth is difficult to determine experimentally, the simulations results
for ideal geometries may differ somewhat from the nominal parameters of fabricated structures.
In the case of the slit-groove experiments reported in [8] the Green’s tensor simulations produced the best agreement
with the experimental points by considering a depth of t = 120 nm, rather than the nominal experimental depth of
100 nm. A comparison between the simulation and measurement is plotted in Fig. 4; only the initial amplitude has
been normalized to the experimental curve. Although the experimental intensity derives from a far-field interference
fringe and the simulated field is evaluated at the output-side plane, it is legitimate to compare the two curves because
the far-field signal is proportional to the calculated field intensity at the output-side slit exit. We note that the same
slit-groove calculation performed with t = 100 nm is in excellent agreement with the simulation of Lalanne et al. [20]
using an entirely different simulation technique.
III. FIELD SCATTERED BY A DIPOLE ALONG THE INTERFACE
In this section we consider the 2D field radiated by a line dipole p0 (rather than the 3D field radiated by a point
dipole) located just above the metal/free-space interface, as indicated in Fig. 5. This approach has been applied by
Lalanne and Hugonin in [20] to study the amplitude evolution of the scattered magnetic field. The choice of placing
the dipole just above the surface may seem arbitrary, but it is shown in the appendix that placing the dipole just
under the interface leads to the same conclusions. The dipole is aligned parallel to the x-axis, consistent with the
previous groove calculations of section II. For the same reasons discussed there, we only calculate the expression of
the z component of the electric field. The dipole oscillation wavelength is 852 nm. We will use the Green’s tensor
formalism to extract a simple expression for the field just above the interface.
FIG. 5: The source is a dipole line located along the y axis at z = 0+, and oriented parallel to the x axis. The field is computed
along (x, z = 0+).
The field radiated by the dipole at a point just above the interface at a distance x is given by the equation:
E(r, r0) =
[G0(r, r0) +GS(r, r0)]p0,
with r(x, z) = (x, 0+) and r0 = (0, 0
+). We denote the couple (r, r0) as (x, 0). The 3× 3 tensors G0 and GS are the
dyadic Green’s functions associated with free space and the metal/free-space surface at the considered wavelength.
Thus, the first term represents the field directly radiated by the dipole to the observation point through free space. The
second term represents the field radiated to the observation point after reflection from the surface. The observation
point is displaced along x, on a line just above and parallel to the surface, running through the dipole. Due to
symmetry of the dipole radiation pattern, the z-component of the directly radiated term along the line of observation
points is 0, and we have:
Ez(x) =
[G0(x, 0)p0 +GS(x, 0)p0] · ez
[GS(x, 0)p0] · ez
GzxS (x, 0)p0, (3)
where ez is the unit vector of the z axis and G
S is the zx component of the surface Green’s function, i.e. the
z-component of the Green’s function produced by a dipole aligned parallel to x.
Although there exist approximate expressions for x small compared to the wavelength (electrostatic approximation),
these expressions are not appropriate here because the line of observation points extends far beyond a wavelength.
The exact expression of the surface Green’s function cannot be written in closed form in direct space, but we can find
an expression susceptible to numerical evaluation by standard methods. The Green’s tensor is analytically defined in
the half Fourier space (q, z), where q is the spatial frequency parallel to the x axis. The general expression can be
found in reference [22]. The GzxS component is given by:
GzxS (r, r
′) = −
4πk20
dq q R(q)eiqx,
where R is the Fresnel reflection coefficient for TM (transverse magnetic) polarization:
R(q) =
ǫ(ω)k20 − q
2 − ǫ(ω)
k20 − q
ǫ(ω)k20 − q
2 + ǫ(ω)
k20 − q
. (4)
Then, Eq. (3) becomes:
Ez(x) = −
dq q R(q)eiqx. (5)
In order to interpret this last equation, consider the value of the integral without the reflection coefficient:
0 1 2 3 4 5 6 7 8 9
x (µm)
dipolar model
full simulation
FIG. 6: Comparison between the field obtained from Eq. (5) and with the full Green’s tensor numeric simulation.
I(x) =
dq q eiqx.
This function I is the derivative of the function J :
J(x) = −i
dq eiqx,
proportional to a Dirac delta function located at x = 0. So I is the derivative of a Dirac delta function, that in fact
represents a point dipole located at x = 0: the z component of the electric field in the direction of the dipole is 0
everywhere, except at two points infinitly close where it is not defined. Thus, Eq. (5) simply states that plane waves
diffracted by the dipole in the x direction are reflected by the surface with a factor given by the Fresnel reflection
coefficient.
Because qR(q) is an odd function of q:
Ez(x) =
dq q R(q) sin qx. (6)
This integral cannot be expressed in closed form but can be computed by conventional numerical techniques. In Fig. 6
are compared the z components of the electric field computed with the groove simulation (for t = w = 100 nm) and
with Eq. (6). The two curves, after proper normalization, agree very well. It might appear surprising that the phase
of the dipolar model does not need to be adjusted compared to the groove calculation, but Fig. 3 shows that the phase
of the wave diffracted by the groove is precisely π for this groove geometry. The overall conclusion is that it is the
accumulation of oscillating charges at the corners of the groove, rather than details of the groove profile itself, that
plays a key role in the global shape of the diffracted wave a few hundreds of nanometers away from the groove center.
The field is essentially the field diffracted by a dipole placed near the surface, its structure is determined by the fact
that the source has a broad-band spatial frequency spectrum, and that the surface supports a long-lived, guided SPP
mode.
An interesting point is the role of constituent propagative and evanescent modes in the creation of the surface
wave. In particular the composite diffracted evanescent wave (CDEW) model [7], previously invoked to interpret
similar phenomena, considers explicitly only the evanescent part. The two contributions are difficult to extract from
full numerical simulations but can be easily carried out with the dipole approach. The scattered field of Eq. (6) is
separated in its propagative and evanescent components:
Ez(x) = E
z (x) + E
z (x), (7)
with:
Eprz (x) =
dq q R(q) sin qx
Eevz (x) =
dq q R(q) sin qx.
Figure 7 compares the real part of these two contributions. The propagative term represents a substantial fraction,
evanescent
propagative
total
x (µm)
0 2 4 6 8 10
FIG. 7: Contribution to the field of the radiative and evanescent components to the total wave calculated from Eq. 7 (color
online).
its amplitude being around 50% of the total wave a few hundreds of nanometers from the dipole location. However,
the amplitude of the propagative component decreases much faster than the evanescent term; and the wavelength of
the propagative part is clearly longer than the wavelength of the evanescent component. The reason is that for all
evanescent modes, including the SPP mode, q > k0, λ < λ0, and neff > 1. For the propagative modes q ≤ k0, λ ≥ λ0,
neff ≤ 1.
These trends appear clearly in Fig. 8, which represents the evolution with distance from the dipole of the effective
indices of refraction for the different contributions. The index of the evanescent part is initially larger than the
SPP index and decreases to this value asymptotically. The index of the propagative part, initially less than unity,
approaches n = 1 from below. This is a consequence of the fact that the propagative wave is dominated by grazing
plane waves, for which the reflection coefficient is almost equal to 1, implying q ≃ k0 and therefore n ≃ 1. When
these two contributions are summed, the effective index follows the curve computed from the groove simulation and
is in good agreement with measurement [8].
IV. ANALYTICAL MODEL
The dipole-on-surface model can be further simplified in order to express the diffracted wave in closed form. In
the following, we present a simple opaque-screen analytic model, similar to that of Kowarz [4], but in which the SPP
wave is introduced as a boundary condition on the surface. In the Kowarz model the evanescent part of the diffracted
wave is computed assuming the presence of a slit in an infinitely thin opaque screen, and taking into account only the
evanescent
propagative
total
full simulation
x (µm)
1 2 3 4 5 6 7 8 9 10
FIG. 8: Contribution to the effective index of the radiative and evanescent components to the total wave, and comparison with
the effective index calculated with the groove full simulation (color online).
evanescent components of the field just above the metal. With these assumptions, the amplitude of the evanescent
wave along the interface is given by:
E(x) = E0
|q|>|k0|
sin(qw/2)
eiqxdq (8)
k0(x+
k0(x−
, |x| > w/2,
where Si is the sine integral defined as:
Si(α) =
sin t
The interpretation is straightforward: the slit diffracts the incident wave into a sum of evanescent waves of spatial
frequency q whose amplitudes are weighted by the Fourier amplitudes of the slit. The Fourier spectrum of the slit is
a cardinal sine (sinc) function. Moreover, when x → ∞, the solution of Eq. 8 is correctly approximated by:
E(x) ≈
This model however does not reproduce correctly the result of Ref. [8]. One of the reasons is that the finite conductivity
of the screen is not included in the Kowarz approach. If we consider again a TM–polarized wave incident on the groove,
the SPP mode of complex wave vector qspp is excited along the interface. In that case, the corresponding wave vector is
created by the diffraction of the incident wave. The dispersion relation of the SPP guided mode, qspp = k0
ǫ/(ǫ+ 1),
can be retrieved by calculating the pole of the reflection coefficient of the metal/free-space interface R(q). At λ0 = 852
nm and value of ǫ for silver, qspp = (1.0154+i 5.54 10
−4)k0. For an evanescent wave whose wave vector is near qspp, the
reflection coefficient can be approximated by A/(q − qspp), where A is a constant. We can say that the incident wave
impinging on the groove is at first diffracted with amplitude corresponding to the Fourier spectrum of the groove,
and is then reflected along the metallic interface with a coefficient R(q). Hence, an approximate expression of the
evanescent wave propagating along the interface is obtained by replacing Eq. (8) with:
E(x) = E0
∫ −k0
sin(qw/2)
q + qspp
eiqxdq
sin(qw/2)
q − qspp
eiqxdq. (9)
Here two poles must be inserted because the SPP wave is excited in both ±x directions. This expression can be
simplified using the fact that the width of the groove Fourier spectrum is of the order of 1/w, whereas the width of
the “spectral line” of the Fourier SPP mode is of the order of ℑ(qspp), a thousand times narrower than the Fourier
0 0.2 0.4 0.6 0.8
Re{K}
FIG. 9: Parametric curve representing the function i − Ei[i(1 − α)t]/2π + Ei[−i(1 + α)t]/2π in the complex plane. In this
example, α = qspp/k0 = 1.02.
spectrum of the groove. Hence, in q−space the groove structure is essentially a constant over the width of the SPP
response. Changing the width of the groove will only modify the amplitude of the plane waves of wave vector q ≈ qspp,
and thus the amplitude of the diffracted wave. Hence we have:
E(x) ≈ E0
∫ −k0
q + qspp
eiqxdq + E0
q − qspp
eiqxdq. (10)
For x > w/2, this expression reads:
E(x) = 2π E0 e
iqsppx K(x), (11)
with:
K(x) = i−
Ei (i(k0 − qspp)x)
Ei (i(−k0 − qspp)x)
The function Ei is called exponential integral and is defined by:
Ei(z) = −
It appears that the amplitude of the SPP wave propagating along x, exp(iqsppx), is multiplied by an envelope of
complex value K(x). This function is represented in the complex plane for typical parameters of SPP wave vector
in Fig. 9(a). The low values of x correspond to the right of the curve. When x goes toward infinity, K(x) whirls
toward zlim = i. The strong oscillation at the beginning of the curve is due to a beating between the Ei(−qspp) and
the Ei(+qspp) term. As this function has a varying phase, it will affect the wavelength of the surface wave. Figure
10 compares the evolution of the surface index for x– and z–components using the index computed from the previous
formula. This analytical model predicts the same trends as the numerical and the dipole approach: the effective index
of the wave generated near the groove is greater than nspp but decreases and converges toward the expected SPP
value within a range of about 10 µm. In fact, the effective index oscillates slightly around nspp at larger distances.
There is a good qualitative agreement with the dipole model, because the evanescent waves play the main role in the
creation of the surface wave. The fact that the reflection coefficient is replaced by a simple pole in qspp modifies the
time evolution of the wave amplitude (not shown), but the wavelength evolution is unchanged. The effective index is
overestimated in the first micrometers near the groove because the radiative part is not taken into account.
V. SUMMARY
We have studied in detail the structure of the wave diffracted by a groove or slit milled in a metallic surface,
illuminated by a monochromatic plane wave. First, the Green’s tensor method has been used to analyze the amplitude,
0 5 10 15 20
x (µm)
analytic model
groove simulation
FIG. 10: Comparison between the Ez, effective index and the effective index computed from the analytical model. The groove’s
dimensions are: t = 100 nm, w = 100 nm.
phase and frequency behavior of the surface wave in the vicinity of the groove. In this zone, the surface wave has
a transient regime characterized by a rapid variation of the amplitude within the first 2 to 3 micrometers, and an
increase of the surface wavelength up to the value of the SPP-wavelength in the first 15 micrometers. The phase
and the amplitude of the scattered wave depends strongly on the groove geometry, as the incident wave excites an
“organ-pipe” mode inside the groove. Best agreement with the experimental results of Gay et al. [8] is obtained
assuming a somewhat deeper groove (120 nm) as in the experiment (100 nm). This value is within the uncertainty of
the actual milled depth using focused ion beam (FIB) fabrication. We have also presented a simplified model in which
the surface wave is excited by a line dipole parallel to and just above a flat surface without groove. This approach
permits the extraction of an analytical expression for the z component of the electric field along the interface. The
agreement between this model and the full simulation is very good, showing that the transient “near-zone” regime
does not depend on the precise shape of the groove. Indeed the details of groove depth and width only influences
the amplitude and the phase of the generated wave. The overall form results from a line dipole source with a broad
q spectrum interacting with a surface that supports a mode. Moreover, we have studied the influence on the wave
structure of the propagative and the evanescent contributions. The propagative waves contribute importantly in
the first few micrometers from the source, but their amplitude decay is faster and their wavelength is longer than
the evanescent contribution. The wavelength of the propagative contribution decreases with the distance down to
the excitation wavelength, whereas the effective wavelength of the evanescent contribution increases up to the SPP
effective wavelength. Finally, we have studied a simplified model of the diffraction process, in which the reflection
coefficient is replaced by a pole located at the SPP wave vector in q space. The scattered field can be then expressed
in closed form. This “minimal model” correctly reproduces the SPP excitation and the wavelength evolution with
distance. Such a semi-analytical model may be useful for the design and optimization of more elaborate photonic
circuits whose behavior in large part will be controlled by surface waves.
VI. ACKNOWLEDGMENTS
G.L. and O.J.F.M. acknowledge funding from the IST Network of Excellence Plasmonanodevices (FP6-2002-IST-1-
507879). J.W. acknowledges Support from the Ministère délégué à l’Enseignement supérieur et à la Recherche under
the programme ACI-“Nanosciences-Nanotechnologies,” the Région Midi-Pyrénées [SFC/CR 02/22], and FASTNet
[HPRN-CT-2002-00304]EU Research Training Network.
APPENDIX
The purpose of this appendix is to show location of the dipole, just above or below the metal/free-space interface
is independent of the results obtained in section III. We begin with Eq. 6:
Ez(x) =
2πk20
dq q [R(q)−R(∞) +R(∞)] sin qx (12)
The integrand diverges when q → +∞. However, the integral is defined for x > 0. We can write write Eq. (12) as:
Ez(x) =
2πk20
dq qR̃(q) sin qx+R(∞)
dq q sin qx,
where
R(∞) =
1− ǫ(ω)
1 + ǫ(ω)
and R̃ ≡ R(q)−R(∞) (14)
so that from Eq. 4:
R̃(q) =
2ǫ(ω)
1 + ǫ(ω)
ǫ(ω)k20 − q
k20 − q
ǫ(ω)k20 − q
2 + ǫ(ω)
k20 − q
The first part of Eq. 13 converges because:
qR̃(q) −−−−−→
, with γ ∈ C.
The second part is equal to the derivative of a Dirac function. Hence:
Ez(x) =
2πk20
dq qR̃(q) sin qx+ αδ′(x) (15)
The right term of the sum is only a term located at x = 0. For numerical integration, it is more convenient to use
this last expression.
If the dipole is located just under the interface:
E−z (x) =
2πk20
dq qT (q) sin qx,
with:
T (q) =
ǫ(ω)k20 − q
ǫ(ω)k20 − q
2 + ǫ(ω)
k20 − q
Here,
T (q) −−−−−→
T (∞) =
1 + ǫ(ω)
Hence:
E−z (x) =
2πk20
dq q T̃ (q) sin qx
+T (∞)
dq q sin qx
2πk20
dq q T̃ (q) sin qx+ β δ′(x) (16)
T̃ (q) = R̃(q)
Hence the field diffracted by two dipoles located just above or just under the vaccum/metal interface differs only in
x = 0.
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	Introduction
	Numerical simulations
	Field scattered by a dipole along the interface
	Analytical model
	Summary
	Acknowledgments
	APPENDIX
	References
