# Evanescent Tunneling of an Effective Surface Plasmon Excited by Convection Electrons

**Abstract**

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 (pi 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 pi mode system confirms the evanescent tunneling effect of the electron-coupled plasmons.

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 [1], since Ebbesen et al. [2] 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 [9] 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 states producing SP-like optical responses.

Subsequently, Hibbins et al. substantiated the existence

of a geometrically controllable surface-conﬁned mode ex-

perimentally by measuring reﬂection [10] and transmission

[11] 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 phase transmission [17] of a 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, k

sp

d=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

the three-dimensional dielectric homogenization is

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-

mission process.

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

"

III

and

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

eV

e

is speci-

ﬁed geometrically using the thickness, d

b

, of the electron

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beam and the impact parameter, b, of the spacing between

the beam center and the slab. The electrons interact with

evanescent surface waves 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 [9] to the surface of

the one-dimensionally pierced metal plate, we obtain the

dispersion relation of the effective SPP mode,

k

sp

!

c

1 f

2

64

4

1

!

pl

=!

2

1

s

; (1)

where f a=d is the ﬁlling factor, !

pl

mc=s is the

effective surface plasma frequency, corresponding to the

cutoff frequency, f

c

, 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 k

x

2p=d k

sp

, 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 k

x

e

!=v

e

. The electron velocity is v

e

c, where

1 1

2

=2

2

p

is the relativistic factor,

2

V

e

=V

n

,

V

n

m

e

c

2

=e 5:11 10

5

V 0:5MV, and m

e

and e

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

energy of the electron beam,

e

eV

e

eV

n

1

1 !=c

e

2

p

; (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 [14] that the electron-plasmon cou-

pling frequency-locks the constructive interference, the so-

called ‘‘Smith-Purcell radiation (SPR)’’ [15], 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

p

th

diffraction order in regions I, II, and III are

E

I

x

x; y; ze

i

0

z

e

ik

x

x

X

1

p1

r

p

e

i

p

z

e

ik

p

x

;

E

II

x

x; y; zA

m

xe

iqz

B

m

xe

iqz

sink

y

y for jx mdja=2;

E

III

x

x; y; z

X

1

p1

t

p

e

i

p

zL

e

ik

p

x

; (3)

where

p

!=c

2

k

2

x

p

and q

!=c

2

k

2

y

q

are the

momentums in regions I and III, and II of Fig. 1(a), re-

spectively, and k

y

m=s is the wave vector of the wave-

guide mode in the y direction. The tangential magnetic

ﬁelds could be derived from H

y

1=i!@E

x

=@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 [21], which describes the

FIG. 1. (a) Conceptual illustration

(top) and computational simulation

model (bottom) of the electronic excita-

tion system for the effective surface plas-

mon. The metal slab is a near-perfect con-

ductor,which has no internal bound state.

(b) Umklapp momentum-shifted SP

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

13:8 keV).

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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 [22] with periodic integral manipulations, the trans-

mittance is given as

T

p

p

q

g

2

p

4fu

1 ’

2

1 ’

2

u

2

2

; (4)

where ’

P

1

p1

fg

2

p

p

q

, g

p

sinck

x

a=2, and is the

phase accumulation factor across a metal slab of thickness

L. Recalling the p

th

dispersion relation of Eq. (1) and the

corresponding electronic energy of Eq. (2), from Eq. (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=2c 0:107 and

k

x

d=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.,

2

0

!=c

2

k

2

x

!1, whereas the dipole-dipole in-

teraction becomes weaker with increasing k

x

, as the trans-

mittance in Eq. (4) is proportional to the squared sinc

function of k

x

a=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 SP modes. 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 cos2qL,so

that the maximum-minimum gap decreases as qL ap-

proaches 0 or . In the case L d a, the same momen-

tum of k

x

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 k

x

and q causes a large portion

of the electromagnetic energy to transfer in the z direction,

which decreases the gap between the maximal and minimal

transmittances. In addition, note that electron beams of

about 14 keV simultaneously excite out-of-phase-like

modes ( modes), which have sequential ﬁeld-phase varia-

tion along the slit array, around k

x

d=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)[23], 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, d

b

=d 0:17, and P

e

(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

k

x

d=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

FIG. 2. Analytical transmittance dispersion curves, (a) and (c),

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

0:63).

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147402-3

from 0.63 to 837 keV with a momentum-resolution of

k

x

d=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 have presented 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.

*Present address: Department of Applied Science,

University of California, Davis, California 95616, USA.

†

gunsik@snu.ac.kr

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FIG. 3. Computational FDTD-PIC

simulation,

MAGIC, results (a=d 0:37,

s=d 4:3, and L=d 0:63, and P

e

200 W). (a) Field distribution of the

transverse electric ﬁeld, E

x

, with an ar-

row plot, which denotes the SP-

polarization, and charge distribution in

the two electron beams. (b) Transmit-

tance dispersion spectrum, plotted nu-

merically with a momentum resolution

of k

x

d=2 0:05, compared with the

analytic dispersion curves of the zeroth

(dashed line) and ﬁrst (dotted line) or-

ders.

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PHYSICAL REVIEW LETTERS

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- CitationsCitations51
- ReferencesReferences35

- "Due to short pitches of geometric patterns these modes are scattered and couple to higher order modes diffractively [21]. Under synchronous condition, electrons with high kinetic energies which are moving in the proximity of a sub-wavelength periodic structured surface metal can selectively stimulate phase transmission [20], [21] for an exact Eigen mode which is located beyond the light cone in the plasmonic band [9], [22] . The synchronous condition requires that electron beam velocity and phase velocity of evanescent surface wave be synced with each other. "

[Show abstract] [Hide abstract]**ABSTRACT:**at lower frequency regime, altering the surface structure modifies the characteristics of propagating effective surface plasmons (ESPs), particularly their interaction with light. Movement of an electron beam in the proximity of the surface of a structured metallic grating excite these ESPs, which due to existence of a velocity phase match, they are able to emit terahertz (THz) waves known as Smith-Purcell (SP) radiation. By introducing a femtosecond perturbation (a single electron bunch), the important characteristics of a desired grating is revealed. Through our 3-D investigations, we find the incoherent THz radiation frequency span, and moreover we confirm that the frequency of the ESP is always less than the minimum SP frequency. Additionally, we find that the maximum of signal amplitude is distributed around 90°, only if the grating width is comparable to the longest SP wavelength. Also we learned that by increasing the grating length, the magnitude and spectral resolution of the radiation increases too. In order to study the coherent (superradiant) radiation, we use a train of electron bunches with variable bunch-to-bunch distances, and we calculate the radiation angle of the coherent SP signal. Simulations on the generation of SP radiation at THz frequencies are performed with the help of the 3D particle-in-cell (PIC) finite integral method, in which the results agree very well with previously reported 2D simulations.- "[13] "

[Show abstract] [Hide abstract]**ABSTRACT:**The wakefield acceleration in a dual grating waveguide slow wave structure (DGWSWS) is proposed. Analytical expression of the wakefield amplitude and frequency is calculated. The wakefield amplitude is analyzed in terms of the grating waveguide parameters, i.e., grating width and grating slot height. The results indicate that the grating wakefield accelera-tors with shallower slots can operate at terahertz with competitive accelerating gradients when compared with a dielectric wakefield accelerator.- "[13] "

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