Evanescent Tunneling of an Effective Surface Plasmon Excited by Convection Electrons
Young-Min Shin,*Jin-Kyu So, Kyu-Ha Jang, Jong-Hyo Won, Anurag Srivastava, and Gun-Sik Park†
Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
(Received 10 October 2006; published 3 October 2007)
We introduce the subwavelength transmission of an effective surface plasmon beyond the light zone via
the proximity interaction of convection electrons with a metal grating. A comparative analysis of
dielectric homogenization and a finite-difference–time-domain simulation shows that out-of-phase-like
modes (? modes) have strong transmission below the cutoff frequency relying on the parametric condition
of structural dimension and electronic energy. The synchronous spatial field and charge distribution of the
? mode system confirms the evanescent tunneling effect of the electron-coupled plasmons.
DOI: 10.1103/PhysRevLett.99.147402 PACS numbers: 78.68.+m, 41.20.Jb, 42.25.Bs, 73.20.Mf
A great deal of attention has been focused on the ex-
traordinary optical transmission (EOT) of light, which is
remarkably greater than predicted by classical aperture
theory , since Ebbesen et al.  discovered it in the
near infrared experimentally. Despite intense controversy
over its physical origin, subsequent theoretical and experi-
mental studies [3–6] have explored the connection be-
tween the anomalous phenomena and surface plasmons
(SPs). The well-established hypothesis attributes the im-
mense subwavelength transmission to the tunneling effect
of surface plasmon polaritons (SPPs) coupling to the inci-
dent light. Interestingly,this elusive manifestation has been
observed in terahertz and microwave regions far below
metal plasma frequencies [7,8]. Associated with this broad
spectral emergence of the particular surface mode effect,
Pendry  proposed that the surface structuring gives rise
to an effective surface bound state, similar to a genuine
SPP, featured by a cutoff wavelength. In the low-frequency
regime, structured surfaces such as grooves, holes, dielec-
tric layers, or other geometric patterns constitute pseudo-
bound statesproducing SP-like
Subsequently, Hibbins et al. substantiated the existence
of a geometrically controllable surface-confined mode ex-
perimentally by measuring reflection  and transmission
 on illuminating brass waveguide arrays. This result
clearly demonstrated the relevant role of the effective
surface plasmon (ESP) in EOT within the light zone.
Nonetheless, further intriguing questions still remain con-
cerning the ingenious embodiment. How does a surface
plasmon exist beyond the light zone and how is it trans-
mitted below the cutoff? These fundamental questions
have focused our attention on the electronic excitation of
plasmonic modes buried below the light line. In fact, the
proximity interaction of the convection electrons to peri-
odic metal structures has long been studied in plasma and
microwave electronics dealing with the generation of an
intense electromagnetic wave [12–16]. Specifically, ex-
tending traditional electronics to near-field optics could
allow us to examine optically invisible channels in the
transmission spectrum. More precisely, beyond the light
zone the kinetic convection electrons can selectively in-
duce phasetransmissionofa specific eigenmode in the
plasmonic band through subwavelength hole arrays.
This Letter presents the beyond-light evanescent trans-
mission of electronically excited ESP modes, specifically
out-of-phase-like modes (? modes,kspd=2? ? 0:5).In this
system, a metal slab with a one-dimensional slit array is
positioned between two counterstreaming electron beams.
We present a parametric analysis of the evanescent trans-
mission along the SP dispersion curve with respect to the
structural dimensions and electronic energies. For this
analysis, the plasmonic dispersion relation derived from
plugged into the transmittance obtained from solving the
boundary-matching problem of Maxwell’s equation. By
scanning the electronic energies, the analytical predictions
are verified through a comparison with numerical simula-
tions, based on a finite-difference-time-domain (FDTD)
and particle-in-cell (PIC) algorithm. In addition, the syn-
chronous field and charge distributions of the ? mode
system show that convection electrons are momentum-
energetically coupled to the surface plasmon in the trans-
Figure 1(a) illustrates our system conceptually and the
simulation model considered here. A one-dimensionally
perforated metal slab is held between two electron beams
traveling in opposite directions. In the laboratory frame,
the counterstreaming electron beams allow transmitted
SPPs to confront the identical coupling circumstance in
regions I and III ("I? "IIIand ?I? ?III) symmetrically.
In addition, the cross interaction between the two electron
beams through field transmissions strongly circulates elec-
tromagnetic energy over the entire system. This reinforced
mutual interaction rapidly accelerates the excitation and
transmission of plasmons, lowering the SP excitation
threshold of the electron beam. In the slab, a, d, s, and L
denote the width, period, length, and thickness of the
rectangular hole array, respectively, and we normalize all
of the dimensions and physical parameters using d (lattice
constant) for universal spectral application. Likewise, the
electron beam with a kinetic energy of ?e? eVeis speci-
fied geometrically using the thickness, db, of the electron
PRL 99, 147402 (2007)
5 OCTOBER 2007
© 2007 The American Physical Society
beam and the impact parameter, b, of the spacing between
the beam center and the slab. The electrons interact with
evanescent surfacewaves exponentially decaying along the
z direction inducing TE waveguide modes in the holes. It is
postulated that the beams appear as sheets, sufficiently
wide to cover the entire planar array, and the slab is a
near-perfect conductor, which implies that no electromag-
netic mode exists inside the metal. Theoretically, periodic
gratings or slits have long been considered an effectively
homogenous medium [18–20]. With homogenization, a
one-dimensional slit array is usually described as an an-
isotropic dielectric medium with an effective permittivity,
"z! 1, "x? "y, and permeability, ?z! 1, ?x? ?y.
At the subwavelength scale, a < d ? ?0, consequently, an
average optical response in a unit area is the same at the
metal and the effective dielectric. Namely, the average
instantaneous transverse fields and power flows on the
unit area are the same at the surfaces of both media.
Modifying the dielectric conversion  to the surface of
the one-dimensionally pierced metal plate, we obtain the
dispersion relation of the effective SPP mode,
where f ? a=d is the filling factor, !pl? m?c=s is the
effective surface plasma frequency, corresponding to the
cutoff frequency, fc, of the hole-waveguide, m is the mode
number along the y direction in the hole, and c is the speed
of light. Using an umklapp momentum shift, the parallel
quasimomentum along the metal surface becomes kx?
2?p=d ? ksp, where p is the diffraction order. The plus
and minus signs stand for the positive and negative direc-
tional surface waves relative to the electron beam, respec-
tively. These plasmonic modes are excited when the
electrons satisfy the matching condition kx? ?e?
!=ve. The electron velocity is ve? ?c, where ? ?
are the rest mass and charge of the electron, respectively.
The electron-plasmon coupling condition from the con-
servation of energy and momentum provides the kinetic
1 ? f2641
1 ? ?1 ? ?2=2??2
Vn? mec2=e ? 5:11 ? 105V ? 0:5 MV, and meand e
is the relativistic factor, ?2? Ve=Vn,
energy of the electron beam,
?e? eVe? eVn
1 ? ?!=c?e?2
for exciting an SPP mode. Figure 1(b) shows the SP dis-
persion curves of the zeroth and first diffraction orders and
the momentum-shifted electron beam lines, which are de-
rived from Eqs. (1) and (2). The electron beams can couple
to the reflected first order (p ? 1) and transmitted zeroth
order (p ? 0) SPP modes simultaneously. And also they
couple to photons in the light zones, as well as to the SP
bands. It is well known  that the electron-plasmon cou-
pling frequency-locks the constructive interference, the so-
called ‘‘Smith-Purcell radiation (SPR)’’ , of the reflec-
ted waves generated from the electron-photon coupling.
Let us consider the boundary problem of the three-
dimensional Maxwell’s wave equation to derive the trans-
mittance of the electronically excited SPP modes through
the one-dimensional hole arrays on the metal slab.
Considering the p-polarized waves (the magnetic fields
pointing in the y direction) above and below the metal
slab and the TE-waveguide mode in the rectangular holes,
the respective normalized tangential electric fields of the
pthdiffraction order in regions I, II, and III are
EIx?x;y;z? ? ei?0zeikxx?
x?x;y;z? ? ?Am?x?eiqz
?Bm?x?e?iqz?sinkyy for jx?mdj ? a=2;
momentums in regions I and III, and II of Fig. 1(a), re-
spectively, and ky? m?=s is the wave vector of the wave-
guide mode in the y direction. The tangential magnetic
fields could be derived from Hy? ?1=i?!?@Ex=@z and
the tangential fields are continuous at the boundaries be-
tween the regions. In this theoretical approach, in region II
we consider a single mode model , which describes the
and q ?
model (bottom) of the electronic excita-
tion system for the effective surface plas-
ductor,which has no internal bound state.
bands and electron beam lines in the ze-
roth (p ? 0) and first (p ? 1) diffraction
orders. The gray areas represent the SPR
zones (dark gray), arising from photon-
electron couplings, in the diffracted light
zones (gray). (a=d?0:37, s=d?4:3,
L=d?0:63, and ?e?13:8keV).
PRL 99, 147402 (2007)
PHYSICAL REVIEW LETTERS
5 OCTOBER 2007
superposition of two forward and backward fundamental
waves. In particular, for slit widths typically smaller than
?0=2, this assumption is valid because the fundamental
waveguide mode is the least strongly decaying in the
subwavelength air slit. By matching the boundary condi-
tions  with periodic integral manipulations, the trans-
mittance is given as
phase accumulation factor across a metal slab of thickness
L. Recalling the pthdispersion relation of Eq. (1) and the
corresponding electronic energyofEq.(2),fromEq.(4)we
can derive the transmittance along the forward and back-
ward plasmonic bands in terms of the electronic energies,
?e. In the transmittance 3D plot in Figs. 2(a) and 2(c), the
filling ratio, a=d, is 0.37, the normalized surface plasma
frequency, d=?pl, calculated from s=d ? 4:3, is about
0.117, as depicted in Fig. 1(b), and the slab thickness,
L=d, is 0.63. In Fig. 2(a), after a steep increase along the
zeroth dispersion curve, the transmittance gradually falls
off from a maximum of 0.656 above !d=2?c ? 0:107 and
kxd=2? ? 0:252. The transmission feature on the SP band
is connected to the corresponding dispersive configuration
of the SPP modes. Under the condition in which L equals
the lattice thickness, d ? a, approaching !pl, the confine-
ment of the surface mode becomes stronger because the
evanescent wave vector, ?0, in region III is divergent, i.e.,
teraction becomes weaker with increasing kx, as the trans-
mittance in Eq. (4) is proportional to the squared sinc
?1 ? ’?2? ?1 ? ’?2u2
q, gp? sinc?kxa=2?, and is the
where ’ ?P1
0? ?!=c?2? k2
x! ?1, whereas the dipole-dipole in-
function of kxa=2. In Fig. 2(a), the two factors appear to
be superimposed. As ?sp??e? is close to ?pl? 2s, enhance-
ment of the resonant mode at the boundaries weakens the
transmission of the SPPs. Similarly, the first order trans-
mittance in Fig. 2(c) also decreases rapidly along the
dispersion curve away from the p ? 1 light zone. In par-
ticular, summation up to p ? 1 in ? increases the denomi-
nator of Eq. (4), so that the first order has roughly 10 times
smaller transmittance than does the zeroth order. This
analysis implies that the electron beams are more strongly
coupled to the transmitted SPmodes.Complementarily,we
see an effect of thickness variation on the transmission
spectrum. In Figs. 2(b) and 2(d), the difference between
the maxima and minima decreases gradually at L=d ? 0:4
or L=d ? 0:8. The second term in the denominator of
Eq. (4), which determines the variation in the amplitude
of transmission, is the sinusoidal function of cos?2qL?, so
that the maximum-minimum gap decreases as qL ap-
proaches 0 or ?. In the case L ? d ? a, the same momen-
tum of kx? q leads to coupling between the Bloch state in
the x direction and the eigenstate in the z direction, which
attenuates the evanescent tunneling of SPPs. However, as
L=d deviates from 1 ? f ? 0:63, an increase in momen-
tum mismatching between kxand q causes a large portion
of the electromagnetic energy to transfer in the z direction,
which decreases thegapbetween the maximal andminimal
transmittances. In addition, note that electron beams of
about 14 keV simultaneously excite out-of-phase-like
modes (? modes), which have sequential field-phasevaria-
tion along the slit array, around kxd=2? ? 0:5 in the two
SP bands, but those are indistinguishable.
Our theoretical interpretation describing transmission
dispersive properties is corroborated by data from a nu-
merical electromagnetic simulator, MAGnetric Insulation
Code (MAGIC) , as illustrated schematically in
Fig. 1(a). In the computational framework, surrounded by
perfectly matched layers (PMLs) absorbing the recoiled
electromagnetic waves, two electron beams are specified
with b=d ? 0:16, db=d ? 0:17, and Pe(beam power) ?
200 W. The computational data at the top and bottom in
Fig. 3(a) illustrate the in-plane electric field distribution
and the charge distribution in the electron beam with ?e?
14 keV at the x-z plane, respectively. In this condition, the
contour plot shows the out-of-phase polarization of
kxd=2? ? 0:5 (? mode), denoted by the lines with arrows,
which has a fairly large amount of the localized EM
energies in region II with an evanescent profile in regions
I and III. Note that the modulated electron beams have a
repetition frequency of d=?e? 0:114 (?e=d ? 8:78;
bunching distance), which is slightly less than the d=?pl?
0:117 of the rectangular hole with s=d ? 4:3. This is very
consistent with the analytical prediction of Fig. 1(b).
Sweeping the electronic energies, we map the transmission
spectrum, which is obtained from a fast Fourier-transform
(FFT) of the monitored time-dependent field component in
the hole. In this mapping, the electronic energies, ?e, span
and their variants, (b) and (d), according to the slab thickness of
the zeroth and first orders (a=d ? 0:37, s=d ? 4:3, and L=d ?
Analytical transmittance dispersion curves, (a) and (c),
PRL 99, 147402 (2007)
5 OCTOBER 2007
from 0.63 to 837 keV with a momentum-resolution of
kxd=2? ? 0:05. In Fig. 3(b), very bright transmission
spectra appear at the normalized frequency region of 0:1 ?
d=? ? 0:125, which agrees well with the theoretical dis-
persion curves of the zeroth (dashed line) and first (dotted
line) orders. Only below d=? ? 0:1 do the numerical
spectra tend to diverge from the theoretical lines, vanishing
gradually owing to the PML boundaries of the computa-
tional building block in the y direction. Extending the
dimension to infinity, one would expect the spectra to
keep following the theoretical lines.
In summary, we havepresented the evanescent tunneling
properties of electronically excited ESPs along the plas-
monic band beyond the light zone. The transmission spec-
trum obtained numerically, which is very consistent with
the theory, shows that in practice a large number of the
effective SPPs are transmitted through the subwavelength
holes according to their energy and momentum state.
Below the cutoff frequency, the amplitude and phase of
the evanescent fields are determined by the kinetic energy
of the electron beam matching to the geometrically de-
signed SP modes. This exotic plasmon-controlling scheme
exhibits the peculiar case of an ESP interacting with elec-
trons and opens various potential applications in surface
plasmon diagnostics, photonic and electronic active de-
vices, and tabletop plasmonic accelerators [24,25].
This work is supported by the Ministry of Science and
Technology of the Republic of Korea through the National
Research Laboratory Program.
University of California, Davis, California 95616, USA.
 H.A. Bethe, Phys. Rev. 66, 163 (1944).
 T.W. Ebbesen et al., Nature (London) 391, 667 (1998).
 L. Martin-Moreno, F.J. Garcia-Vidal, H.J. Lezec, K.M.
Pellerin, T. Thio, J.B. Pendry, and T.W. Ebbesen, Phys.
Rev. Lett. 86, 1114 (2001).
 L. Salomon, F. Grillot, A.V. Zayats, and F. de Fornel,
Phys. Rev. Lett. 86, 1110 (2001).
address:Department of Applied Science,
 E. Popov, M. Neviere, S. Enoch, and R. Reinisch, Phys.
Rev. B 62, 16100 (2000).
 W.L. Barnes, W.A. Murray, J. Dintinger, E. Devaux, and
T.W. Ebbesen, Phys. Rev. Lett. 92, 107401 (2004).
 J.W. Lee, M.A. Seo, S.C. Jeoung, Ch. Lieunau, D.S.
Kim, J.H. Kang, and Q.-Han Park, Appl. Phys. Lett. 88,
 J.W. Lee, M.A. Seo, J.Y. Shon, Y.H. Ahn, D.S. Kim,
S.C. Jeung, Ch. Lienau, and Q.-Han Park, Opt. Express
13, 10681 (2005).
 J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal,
Science 305, 847 (2004).
 A.P. Hibbins, B.R. Evans, and J.R. Sambles, Science 308,
 A.P. Hibbins, M.J. Lockyear, I.R. Hooper, and J.R.
Sambles, Phys. Rev. Lett. 96, 073904 (2006).
 A. Yariv and D.R. Armstrong, J. Appl. Phys. 44, 1664
 S.L. Chuang and J.A. Jong, J. Opt. Soc. Am. A 1, 672
 J. Urata, M. Goldstein, M.F. Kimmitt, A. Naumov,
C. Platt, and J.E. Walsh, Phys. Rev. Lett. 80, 516 (1998).
 S. Smith and E. Purcell, Phys. Rev. 92, 1069 (1953).
 K. Yamamoto, R. Sakakibara, S. Yano, Y. Segawa,
Y. Shibata, K. Ishi, T. Ohsaka, T. Hara, Y. Kondo,
H. Miyazaki, F. Hinode, T. Matsuyama, S. Yamaguchi,
and K. Ohtaka, Phys. Rev. E 69, 045601 (2004).
 A.P. Hibbins, I.R. Hooper, M.J. Lockyear, and J.R.
Sambles, Phys. Rev. Lett. 96, 257402 (2006).
 R.C. McPhedran, L.C. Botten, M.S. Craig, M. Nevie `re,
and D. Maystre, J. Mod. Opt. 29, 289 (1982).
 J.M. Bell, G.H. Derrick, and R.C. McPhedran, J. Mod.
Opt. 29, 1475 (1982).
 R.C. McPhedran and N.A. Nicorovici, J. Electro. Wave
Appl. 11, 981 (1997).
 Ph. Lalanne, J.P. Hugonin, S. Astilean, M. Palamaru, and
K.D. Mo ¨ller, J. Opt. A Pure Appl. Opt. 2, 48 (2000).
 J.T. Shen, P.B. Catrysse, and S. Fan, Phys. Rev. Lett. 94,
 B. Goplen, L. Ludeking, D. Smithe, and G. Warren,
Comput. Phys. Commun. 87, 54 (1995).
 N.K. Sherman, AIP Conf. Proc. 156, 161 (1987).
 Naoko Saito and Atsushi Ogata, Phys. Plasmas 10, 3358
simulation, MAGIC, results (a=d ? 0:37,
s=d ? 4:3, and L=d ? 0:63, and Pe?
200 W). (a) Field distribution of the
transverse electric field, Ex, with an ar-
polarization, and charge distribution in
the two electron beams. (b) Transmit-
tance dispersion spectrum, plotted nu-
merically with a momentum resolution
of kxd=2? ? 0:05, compared with the
analytic dispersion curves of the zeroth
(dashed line) and first (dotted line) or-
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5 OCTOBER 2007