arXiv:0808.1324v1 [physics.optics] 9 Aug 2008
Nanoconcentration of Terahertz Radiation in Plasmonic Waveguides
Anastasia Rusina,1Maxim Durach,1Keith A. Nelson,2and Mark I. Stockman1,3
1Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA
2Department of Chemistry, MIT, Cambridge, MA 02139, USA
3On sabbatical leave at Ecole Sup´ erieure de Physique et de Chimie Industrielle de la Ville de Paris,
10, rue Vauquelin, 75231 Paris, CEDEX 05, France
(Dated: August 9, 2008)
Recent years have seen an explosive research and development of nanoplasmonics in the visible and
near-infrared (near-ir) frequency regions.1One of the most fundamental effects in nanoplasmonics is
nano-concentration of optical energy. Plasmonic nanofocusing has been predicted2and experimen-
tally achieved.3,4,5Nanoconcentration of optical energy at nanoplasmonic probes made possible op-
tical ultramicroscopy with nanometer-scale resolution6,7,8and ultrasensitive Raman spectroscopy.9
It will be very beneficial for the fundamental science, engineering, environmental, and defense appli-
cations to be able to nano-concentrate terahertz radiation (frequency 1 − 10 THz or vacuum wave-
length λ0 = 300 − 30 µm). This will allow for the nanoscale spatial resolution for THz imaging10
and introduce the THz spectroscopy on the nanoscale, taking full advantage of the rich THz spec-
tra and submicron to nanoscale structures of many engineering, physical, and biological objects
of wide interest: electronic components (integrated circuits, etc.), bacteria, their spores, viruses,
macromolecules, carbon clusters and nanotubes, etc. In this Letter we establish the principal limits
for the nanoconcentration of the THz radiation in metal/dielectric waveguides and determine their
optimum shapes required for this nanoconcentration We predict that the adiabatic compression of
THz radiation from the initial spot size of R0 ∼ λ0 to the final size of R = 100 − 250 nm can be
achieved with the THz radiation intensity increased by a factor of ×10 to ×250. This THz energy
nanoconcentration will not only improve the spatial resolution and increase the signal/noise ratio
for the THz imaging and spectroscopy, but in combination with the recently developed sources of
powerful THz pulses11will allow the observation of nonlinear THz effects and a carrying out a
variety of nonlinear spectroscopies (such as two-dimensional spectroscopy), which are highly infor-
mative. This will find a wide spectrum of applications in science, engineering, biomedical research,
environmental monitoring, and defense.
There are existing approaches to deep subwave-
length THz imaging and probing based on sharp tips
irradiated by a THz source,12adiabatically-tapered
metal-dielectric waveguides13similar to optical adia-
batic concentrators,2,3,4,5and nonlinear microscopic THz
sources.14For the development of the THz nanotechnol-
ogy, it is extremely important to understand spatial lim-
its to which the THz radiation energy can be concen-
A major challenge for the nanoconcentration of the
electromagnetic energy in the THz region is the large ra-
diation wavelength in vacuum or conventional dielectrics,
λ0 = 30 µm − 300 µm, where the THz radiation can
only be focused to the relatively very large regions of
size ∼ λ0/2. The developed field of optical energy con-
centration, which is based on surface plasmon polaritons
(SPPs), suggests that one of the ways to solving this
problem is to employ the surface electromagnetic waves
(SEWs). In the far infrared (ir), the dielectric permit-
tivity of metals has large imaginary part which dom-
inates over its negative real part.15This implies that
SEWs propagating along a metal-dielectric flat interface
in this frequency range, known as Sommerfeld-Zenneck
waves,16,17are weakly bound to the surface18and can
hardly be used for the confinement of THz radiation.
It has been suggested that periodically perforating flat
surfaces of ideal metals with grooves or holes leads to the
appearance of SEWs, which mimic (“spoof”) SPPs to be
stronger bound to the surfaces19,20,21permitting a better
control over the THz fields. It has been predicted that
SPPs on an array of parallel grooves cut on the surface
of a perfect conductor wire can be localized by adiabatic
deepening of the grooves.22At a point, where grooves are
approximately a quarter of wavelength of light, the high-
est concentration is achieved to be on the order of tens
of micrometers. This method restricts the localization
point to a particular frequency, making the concentration
very narrow-band. Also, the depth of a groove should be
≈ λ0/4, i.e., in the tens to hundred micron range, which
precludes completely nanoscale devices.
It is well known from microwave technology that the
ideal-metal waveguides with smooth surfaces support
TEM waves, where the electric field lines are either in-
finitely extended or terminate at the metal surfaces nor-
mally to them. The latter case requires the waveguide
cross-section topology to be more than single-connected;
an example may be a coaxial waveguide (“coax”). Such
waveguides possess are very wide-band in frequency. The
THz waveguides can be adiabatically tapered to concen-
trate energy. The idea of adiabatic energy concentration
comes from ultramicroscopy23,24,25and nanoplasmonics,2
where it has been developed both theoretically and
experimentally3,4,5and used in ultrasensitive surface en-
hanced Raman spectroscopy.9Employing these ideas of
the adiabatic concentration and using a tapered metal-
dielectric waveguide, the THz spatial resolution achieved
is ∼ 20 µm across the entire THz spectrum.13
In this Letter, for the first time, we establish the
fundamental limits and find the principles of designing
the optimum and efficient metal/dielectric waveguides
suitable for the THz nanofocusing. The specific exam-
ples are for the wide-band concentrators: a plasmonic
metal wedge cavity and tapered coax waveguides, which
are terminated by funnel-type adiabatic tapers.
nano-concentrators along with the advent of high-power
sources11,26and sensitive detectors27of THz radiation,
will open up an extremely wide range of possible THz
applications, in particular, in material diagnostics, probe
nanoimaging, biomedical applications etc.
Note that an alternative approach to the THz energy
concentration using doped semiconductor tapers has also
been proposed.31However, the required heavy doping of
the semiconductors may cause fast electron relaxation
due to the collisions with the inflicted lattice defects and
bring about high losses.Therefore, in this Letter we
will pursue the adiabatic nanoconcentration of the THz
radiation using metal/dielectric structures.
Conventionally for THz and microwave regions, the
metals are considered as ideal which is equivalent to ne-
glecting their skin depth ls= λ0/Re√−εm, where εmis
the permittivity of the metal (we take into accoint that
in the THz region |εm| ≫ 1), and λ0= c/ω is the reduced
wavelength in vacuum. It is true that in the THz region
ls = 30 − 60 nm. i.e., ls ≪ λ0. However, as we show
below in this Letter, it is the finite skin depth, though
as small as it is, that principally limits the ultimum lo-
calization size of the THz fields. For larger waveguides,
the THz wave energy is localized mostly in the vacuum
(dielectric) and its losses, which occur in the metal’s skin
layer, are correspondingly small. The effective quality
factor (or, figure of merit) of the waveguide, which shows
how many periods the wave can propagate without sig-
nificantly loosing its energy, can be estimated as
– cf. Refs.
Q ∼ 2a/ls, (1)
where a is the characteristic minimum size of the waveg-
uide; this estimate becomes a good approximation for a
metal-dielectric-metal planar waveguide [see below Eq.
(2)]. When the waveguide size reduces to become on the
order of the skin depth, a ? ls, the THz field is pushed
into the metal, and the quality factor reduces to Q ? 1,
which implies strong losses.
lishes the limit to the nanoconcentration: for upper THz
region a ? ls ≈ 30 nm, while for the 1 THz frequency
a ? ls≈ 60 nm. These are the practical limits of the THz
nanoconcentration for the noble metals (silver, gold, and
platinum) and for aluminum.
If one pursues the goal of creating enhanced local fields
in a small region, but not necessarily to efficiently trans-
fer the THz energy from the far field to the near field,
then the apertureless SNOM approach, where a sharp
metal or dielectric tip is irradiated by THz radiation, can,
in principle, achieve even higher resolution.32However,
Qualitatively, this estab-
FIG. 1: Geometry and properties of the THz TM mode in
a parallel metal-slab waveguide. (a) Schematic of the waveg-
uide. The width of the dielectric gap a and the skin depth ls
are indicated. (b) An instantaneous distribution of the lon-
gitudinal electric field Ey along the propagation coordinate y
for a = 10 µm and frequency 1 THz in a silver-vacuum-silver
waveguide. (c) The same as in panel (b) but for a = 200
nm. (d) Modal refraction index n = k/k0 (Ren is denoted by
the red line and Imn by the blue line) as a function of the
waveguide width a. Dashed green line indicates the value of
n for the perfect conductor. Skin-depth value is shown by the
vertical dashed line.
the efficiency of utilizing the THz energy of the source in
this case will be extremely low; the stray, far-field THz
energy may create a significant parasitic background.
Here and below in this Letter, we consider examples
of the THz adiabatic nanoconcentration quantitatively,
where the effect of the specific geometry will become ap-
parent. Consider first a parallel plate waveguide that
consists of a dielectric slab of thickness a with dielec-
tric permittivity εdsandwiched between two thick metal
plates (with thickness of at least a few ls, i.e., greater
than 200 nm in practical terms) [see Fig.1(a)]. The per-
mittivity of the metal εmin the THz region has a very
large (? 106) imaginary part that defines the very small
skin depth ls? 100 nm, which justifies the usual consid-
eration of the metals as perfect conductors.15However,
as we have already mentioned, for our purposes of the
THz nanoconcentration, we need to take into account
the field penetration into the metal, which makes the
problem plasmonic. In this case, the propagating modes
of the system are SPPs, which are TM modes character-
ized by the symmetry with respect to the reflection in
the center plane. We will orient the coordinate system
with its z axis normal to the plane and the y axis in the
direction of propagation. The symmetric (even) modes
have even field components Hxand Ezand odd Ey; the
parity of the antisymmetric (odd) modes is opposite.
From plasmonics it is known that the even modes have
a larger fraction of their energy localized in the dielec-
tric and the odd modes in the metal.
even modes have much smaller damping and are, there-
fore, most suitable for the THz energy concentrations.
The dispersion relation for the even modes is given in
the Methods section as Eq. (5). This equation can be
much simplified and solved in a closed analytical form
taking into account that we are interested in the sub-
wavelength focusing, i.e., a ? λ, where λ = λ0/εdis the
reduced wavelength in the dielectric; also, in the entire
THz region ls≪ λ. This shows that there exists a small
parameter in the problem lsa/λ2≪ 1 [see also Eq. (6)],
which allows one to solve analytically the dispersion rela-
tion (5) obtaining the modal refraction index n = k/k0,
where k is the THz wavevector, and k0= 1/λ0,
1 +ls(1 + i)
1 + ils
where the approximate equality is valid for not too tight
nanofocusing, i.e., for ls≪ a. From this, we can obtain
the quality factor of the waveguide Q = Ren/Imn =
2a/ls, giving a quantitative meaning to the estimate (1).
Plasmonic effects (i.e. those of the finite skin depth)
are illustrated in Fig. 1 for silver-vacuum-silver waveg-
uide and frequency of 1 THz. Panels (b) and (c) display
the longitudinal electric field Ey obtained by the exact
solution of the Maxwell equations. Note that this field
component is absent for the ideal conductor; here it is
relatively small: on the order of 10−3of the transverse
field. Panel (b) illustrates the case of a relatively wide
waveguide (a = 10 µm), where it is evident that the elec-
tric field is localized mostly in the dielectric region of the
waveguide, and the extinction of the wave is small. In a
sharp contrast, for a nanoscopic waveguide (a = 200 nm)
in panel (c), the electric field significantly penetrates the
metal. In accord with our arguments, there is a very
significant extinction of the fields as they propagate; the
retardation effects are also evident: the lines of equal
amplitude are at an angle relative to the normal (z) di-
rection. The dependence of the modal refraction index
on the thickness a of the waveguide obtained from Eq.
(2) is plotted in Fig. 1 (d). This index increases as a be-
comes comparable with the skin depth. While Ren and
Imn increase by the same absolute amount, the quality
factor Q, obviously, greatly decreases with decrease of a.
The mode described above can be used for broadband
energy concentration of THz waves.
To introduce the THz nanoconcentration, consider a
metal-dielectric-metal waveguide that is slowly (adiabat-
ically) tapered off as a wedge, as illustrated in Fig. 2
(a). Because of the adiabatic change of the parameters,
a wave propagating in such a waveguide will adjust to
it without reflection or scattering, just as it takes place
in nanoplasmonic waveguides.2As a result, propagating
it will concentrate its energy, conforming to the taper-
ing of the waveguide. The corresponding solution can be
obtained from the Maxwell equations using the Wentzel-
Kramers-Brillouin (WKB) approximation, similarly to
the nanoplasmonic case in the visible,2as described in
the Methods section. The WKB approximation is appli-
cable under the conditions that
δ =??d?Rek−1?/dy??≪ 1 ,
where δ is the well-known adiabatic parameter describing
how slowly the modal wavelength changes on a distance
of its own, and |da/dy| is a parameter describing how
adiabatically the transverse size of the confined mode
changes along the propagation coordinate.
In the WKB approximation, the behavior of the domi-
nating transverse field component Ezas a function of the
coordinate y along the propagation direction is shown for
the last 6 µm of the propagation toward the edge in Fig.
2 (b). There is a clearly seen spatial concentration of the
energy and increase of the field as the wave is guided into
the taper. The predicted behavior of the two components
of electric field and the magnetic field for the last micron
of the propagation is shown in panels (c)-(e). It appar-
ently indicates the adiabatic concentration, without an
appreciable loss of the intensity. The THz wave follows
the waveguide up to the nanometric size.
The red line in Fig. 2 (f) indicates that the local inten-
sity I as the function of the thickness a of the waveguide
for a < 4 µm increases significantly with 1/a, in qual-
itative accord with the behavior expected for the neg-
ligibly low losses. This intensity reaches its maximum
for a = 1.6 µm and then starts to decrease as the losses
overcome the adiabatic concentration. At smaller thick-
nesses, a ? 400 nm, the intensity in Fig. 2 (f) starts
to increase again, which is unphysical.
revealed by the behavior of the adiabatic parameter δ
shown by the blue line: for a ? 400 nm, δ becomes rel-
atively large (comparable with 1), i.e., the adiabaticity
is violated. This is due to the fact that the fraction of
the THz field energy propagating in the metal is dramat-
ically increased for a ? 400 nm due to the constricted
transverse extension of the dielectric in the waveguide.
This causes a significant loss per wavelength λ, leading
to a rapid change of the wave vector k, breaking down
the adiabaticity. This constitutes a fundamental differ-
ence from the nanoplasmonic adiabatic concentration in
the optical region where the adiabatic parameter is con-
stant, and the adiabaticity holds everywhere including
the vicinity of the tip.2
To provide for the optimum guiding of the THz wave
and its concentration on the nanoscale, the terminat-
ing (nanoscopic) part of the waveguide should be ta-
pered slower, in a funnel-like manner.
needs to decrease the grading da/dy of the waveguide
near the edge in order to keep the adiabaticity parame-
ter δ =??d?Rek−1?/da × da/dy??approximately constant
cause for the adiabatic grading (tapering), the derivative
d?Rek−1?/da does not depend on the grading (it is the
same as for the plane waveguide) and is only a function
of a, the equation δ = δ(y) is a differential equation for
the shape of the waveguide that can be easily integrated.
This results in the dependence of the thickness a on the
|da/dy| ≪ 1 , (3)
The reason is
That is, one
and small enough to prevent the back-reflection. Be-
FIG. 2: Adiabatic concentration of THz field energy in a
graded waveguide, where a dielectric wedge is surrounded by
the thick silver layer. (a) Schematic of energy concentration,
where θ is the wedge opening angle, the arrow indicates the
direction of propagation of the THz wave, and the red high-
lights the area of the adiabatic concentration. The orienta-
tion of the coordinate system is shown in the inset. (b) An
instantaneous distribution of the transverse electric field Ez
of the THz wave propagating and concentrating along the
wedge waveguide for the last 6 mm of the propagation to-
ward the edge. Note the difference in scales in the z and y
directions. (c) An instantaneous spatial distribution of the
transverse electric field Ez close to the edge of the wedge,
for the last 640 µm of the propagation. (d) The same as (c)
but for the longitudinal (with respect to the propagation di-
rection) component of the field Ey. (e) The same as (c) but
for the transverse component of the magnetic field Hx. The
units of these field components are arbitrary but consistent
between the panels. (f) Dependence of THz field intensity in
the middle of waveguide on the dielectric gap width a (the
red line). The blue curve displays the dependence on a of the
adiabatic parameter δ, scaled by a factor of 5. The values of
a indicated at the successive horizontal axis ticks differ by a
factor of 10−1/2, i.e., by 5 dB.
longitudinal coordinate y determined by a simple integral
Ren−1(a) = k0
where n(a) is the modal index defined in this case by Eq.
(2), and δ(y) is the desired dependence of the adiabatic
parameter along the waveguide, which is an arbitrary
function of y satisfying the adiabaticity conditions (3).
FIG. 3: Terahertz energy concentration in adiabatically ta-
pered curved-wedge waveguide. (a) Instantaneous distribu-
tion of the transverse component of the THz electric field Ez
(in the central plane z = 0) as a function of the coordinate
y along the propagation direction for the last 400 µm of the
propagation. (b) The same as in panel (a) but for the longi-
tudinal electric field component Ey. (c) The same as panel
(a) but for the transverse magnetic field Hx. The units for
the fields are arbitrary but consistent between the panels. (d)
The THz field intensity I (relative to the intensity I0 at the
entrance of the waveguide) as a function of the dielectric gap
thickness a is shown by the red line. The adiabatic parame-
ter scaled by a factor of 10 as a function of a is indicated by
the blue line. The values of a indicated at the horizontal axis
ticks correspond to the values of y at the ticks of panels (a)-
The geometry of an adiabatically-tapered end of the
silver/vacuum waveguide found from Eq. (4) and satis-
fying Eq. (3) and the corresponding WKB solutions for
the 1 THz fields are shown in Fig. 3 (a)-(c). The opti-
mum shape of the waveguide in this case is funnel-like,
greatly elongated toward the edge. The nanoconcentra-
tion of the field is evident on panels (a)-(c), as well as
its penetration into the metal for a ? 100 nm. As these
panels show quantitatively and the red curve on panel
(d) qualitatively, the field intensity reaches its maximum
at a ≈ 300 nm where it is enhanced with respect to the
field at the entrance to the funnel waveguide by a modest
factor of 1.2. At the same time, the adiabatic parameter
δ decreases toward the tip from 0.07 to 0.05, indicated
the applicability of the WKB approximation everywhere.
Note that the this funnel-shaped wedge, indeed, contin-
ues the linearly-graded wedge waveguide shown in Fig. 2,
which yields the enhancement factor of ≈ 8 at a = 2 µm.
Sequentially, these two waveguides provide the intensity
enhancement by approximately ×10 while compressing
the THz wave to the thickness of a = 300 nm and the
enhancement by a factor of 3 for a = 100 nm.
Thus, true nanolocalization of THz radiation in one
dimension (1d) is possible. The minimum transverse size
of this nanolocalization is determined by the skin depth,
as we have already discussed qualitatively in the intro-
ductory part of this Letter. The obtained 1d beam of the
nanoconcentrated THz radiation may be used for differ-
ent purposes, in particular as a source for the diffraction
elements including the nanofocusing zone plates of the
type introduced in Ref. 33.
The two-dimensional (2d) concentration of the THz ra-
diation can be achieved by using an adiabatically-tapered
conical coax waveguide, whose geometry is illustrated in
Fig. 4 (a). The central metal wire of radius r is sur-
rounded by a dielectric gap of the radial thickness a,
which is enclosed by a thick (∼ 200 nm or thicker) outer
metal shell. Both r and a are smooth functions of the
longitudinal coordinate y, which describes the tapering
of the coax toward the apex (tip) at y = 0. The THz
waves propagate from the wide end of the coax toward
the apex, adiabatically following the tapering. In the
spirit of WKB, for any particular y the wave behavior
for the tapered coax is the same as for a cylindrical coax
with the values of r and a equal to the local values r(y)
The dispersion relation for the coax waveguide that
takes into account the plasmonic effects (i.e., the pen-
etration of radiation into the metal and the concurrent
losses) is obtained in the Methods section as Eq. (9).
Calculated from this expression, the dependence of the
modal refractive index n = k(a,r)/k0 on the dielectric
gap a is displayed in Fig. 4 (b) for the frequency of 1
THz, silver as a metal, and vacuum in the dielectric gap.
The results are shown for two values of the radius of the
central wire: r = 10 µm and r = 60 nm. As one can
see, the real part of the modal index practically does not
depend on r; it starts growing when a decreases. The
imaginary part of the index n increases when the central
wire thickness r decreases, but this dependence is very
weak. Both Ren and Imn grow dramatically for r ? ls.
This is due to the penetration of the THz field into the
metal, i.e., it is a plasmonic effect.
The WKB solution for the radial field Eρin the cross
section of this coax waveguide is shown for the last 3 µm
of the propagation toward the tip in Fig. 4 (c). The adi-
abatic following and energy concentration are evident in
this panel. The penetration into the metal of the tan-
gential (to the metal surface) field components Ey and
Hϕis noticeable in Figs. 4 (d) and (e). The intensity I
of the THz field (relative to the intensity I0 at the en-
trance of the waveguide) as a function of the waveguide
outer radius R = r +a is shown by the red line in Fig. 4
(f). Dramatically, it shows the adiabatic nanoconcentra-
tion and the intensity increase by more than two orders
of magnitude for the nanoconcentration from the waveg-
uide radius R = 300 µm, where the THz radiation can
be focused, toward R = 300 nm. However, the dramatic
increase of the adiabatic parameter δ [plotted by the blue
line in Fig. 4 (f)] for R ? 1 µm shows that these results
can only be trusted for R ? 1 µm.
For the true 2d nanoconcentration of the THz radia-
tion below this micron-scale radius, similar to the 1d case
FIG. 4: Geometry, modal index of refraction, and THz energy
concentration in conically-tapered metal-dielectric waveguide.
(a) Schematic of geometry and energy concentration. The
central wire and the coax shell are shown along with the
schematic of the THz energy concentration. (b) Dependence
of modal refraction index n in coaxial waveguide on the di-
electric gap width a for two central wire radii: r = 10 µm
and r = 60 nm. The color coding of the lines is indicated.
The dielectric in the gap is vacuum. (c) Instantaneous dis-
tribution of the radial THz electric field amplitude Eρ in the
cross section of the coax for the last 3 mm of the propagation
toward the tip. The amplitude of the field is color coded by
the bar at the top of the panel. (d) Instantaneous distribu-
tion of the longitudinal THz electric field amplitude Ey on the
coordinate y for the last 620 µm of the propagation. (e) The
same as (d) but for the transverse magnetic field Hϕ. The
units of these field components are arbitrary but consistent
between the panels. (f) Dependence of THz field intensity in
the middle of waveguide gap on the waveguide outer radius
R = r + a is shown in red. The blue curve displays the adi-
abatic parameter δ as a function of R, scaled by a factor of
102. The values of R indicated at the successive horizontal
axis ticks differ by a factor of 10−1/2, i.e., by 5 dB.
of the wedge, to preserve the adiabaticity, a funnel-like
tapering is necessary. Generally, the tapering of the cen-
tral wire and that of the outer metal shell do not need
to be the same. However, we found that better results
are obtained when it is the case, i.e., the waveguide is
tapered-off self-similarly. In specific calculations, as ev-
erywhere in this Letter, we assume that the metal of
the waveguide is silver, the dielectric is vacuum, and the
frequency is 1 THz. Doing so, we have found the cor-
FIG. 5: Adiabatic terahertz energy concentration in a self-
similarly curved, funnel-shaped coaxial waveguide, where the
metal is silver, and the dielectric in the gap is vacuum. The
dielectric gap is between the pair of the neighboring curved
lines, and the metal is everywhere else. (a) Instantaneous
distribution of the radial (transverse) component Eρ of the
electric field of the guided THz wave as a function of the
propagation coordinate along the wedge y for the last 600 µm
of the propagation. (b) The same for the longitudinal electric
field component Ey. (c) The same for transverse magnetic
field Hϕ, whose lines form circles around the central metal
wire. The units of these field components are arbitrary but
consistent between the panels. (d) The THz intensity I as a
function of the waveguide radius R, displayed relative to the
intensity I0 at the beginning of the waveguide (red line). Adi-
abatic parameter δ multiplied by a factor of 10 as a function
of R (blue line). The values of the radius R shown at the ticks
correspond to those of y shown in panels (a)-(c).
responding grading of the waveguide using Eq. (4) and
setting δ = 0.05, which is small enough to satisfy the
adiabaticity very well. In this case, indeed, we have used
the corresponding dispersion relation (9). The obtained
shape of the waveguide is a strongly-elongated funnel, as
shown in Fig. 5 (a)-(c). These figures display the THz
fields that we have calculated in the WKB approximation
for this waveguide. As one can see from these figures,
within the last half micron of the propagation, the elec-
tric and magnetic fields of the THz wave efficiently fol-
low the adiabatically curved waveguide. The penetration
into the metal of the tangential (to the metal-dielectric
interfaces) field components for y < 400 µm is evident in
panels (b) and (c). The longitudinal electric field com-
ponent Ey is significantly localized in the central metal
wire [panel(b)], which is a plasmonic effect.
The dependence of the THz field intensity in the gap
(relative to the intensity I0 at the entrance of this fun-
nel) on the total radius of the waveguide R is shown in
Fig. 5 (d) by the red line. In this case, the adiabatic
concentration is very efficient. The intensity of the THz
radiation increases by a factor of ×5 when it is com-
pressed from the initial radius of R = 1 µm to the radius
R ≈ 250 nm. The penetration of the fields into the metal
for smaller values of the radius R (tighter confinement)
causes losses that dominate over the effect of the concen-
tration. Again, we remind that this funnel waveguide is
a continuation and termination for the straigt cone that
yields the field enhancement by ×50 for R = 1 µm [see
Fig. 4 (f)]. Consecutively, these two waveguides (the ini-
tial cone continued and terminated by the funnel) are
very efficient, adiabatically compressing the THz radia-
tion from the initial radius R = 300 µm to the radius
R = 250 nm increasing its intensity by a factor ×250.
Even for the final radius R = 100 nm, the total THz
intensity is increased by a factor of ×10 (which is the
products of factors ×50 for the cone part and ×0.2 for
the funnel. Thus, the optimally graded plasmonic-metal
2d waveguide is very efficient in the concentration and
guidance of the THz fields with the transverse radius of
confinement R ? 100 nm.
To discuss the results, we have shown that the THz ra-
diation can be concentrated to the ∼ 100 nm transverse
size in adiabatically graded plasmonic (metal/dielectric)
waveguides. In the optimum adiabatically-graded, coax-
ial waveguide, which consists of the initial cone termi-
nated with a funnel, the radiation of a 1 THz frequency
whose wavelength is 300 µm, can be compressed to a
spot of 250 nm radius, where its intensity increases by a
factor of ×250. Even in the case of the extreme compres-
sion to a spot of the 100 nm radius, the THz intensity is
enhanced by one order of magnitude with respect to the
initial intensity of the 300 µm spot at the entrance of the
waveguide. The physical process that limits the extent of
this spatial concentration is the skin effect, i.e., penetra-
tion of the radiation into the metal that causes the losses:
the THz field penetrates the depth of ls= 30−60 nm of
the metal, which by the order of magnitude determines
the ultimum localization radius.
The THz nanoconcentration predicted in this Let-
ter for optimally-graded adiabatic plasmonic waveguides
provides unique opportunities for THz science and tech-
nology, of which we will mention below just a few. The
nanoconcentration of the THz radiation will provide the
THz ultramicroscopy with a THz source of unprece-
dented spatial resolution and brightness. The increase
of the THz intensity by two orders of magnitude along
with the novel high-power THz sources11would allow
the observation of a wide range of electronic and vibra-
tional nonlinear effects in metal, semiconductors, insula-
tors, and molecules.
These nonlinear THz phenomena can be used to inves-
tigate behavior of various materials in ultrastrong fields,
for nonlinear spectroscopy (including the multidimen-
sional spectroscopy), and for monitoring and detection of
various environmental, biological, and chemical objects
and threats such as single bacterial spores and viruses.
Such applications will certainly be helped by very large
absorption cross sections of various materials in the THz
region. A distinct and significant advantage of the adia-
batic nanofocusing is that the THz energy is mostly con-
centrated in the hollow region of the waveguide, whose
size can be made comparable with the size of the objects
of interest: in the range from 1 micron to 70 nm, which is
a typical range for bacteria and their spores, and viruses.
This will assure high sensitivity and low background for
the objects that are confined inside these waveguides.
Consider as a specific example the spectroscopy or de-
tection of single particles, such as, e.g., anthrax spores, in
the air. A sample containing the suspected nanoparticles
in a gas, which can be air for the frequencies in the trans-
parency windows, can be pumped through a THz waveg-
uide, and the detection can be made for each particle in
the gas separately on the basis of the two-dimensional
nonlinear THz spectra that are expected to be highly in-
formative for the detection and elimination of the false-
positive alarms. Likewise, many other scientific, tech-
nological, environmental, and defense applications may
A.Terahertz TM Wave in Finite-Conductivity
Parallel Plate Waveguide
A parallel plate waveguide supports an even TM mode
with wavenumber k, which satisfies the dispersion rela-
where κd=?k2− εdk2
terahertz range, εmis mainly imaginary, where Imεm≫
1. Therefore, κm ≈ k0√−εm = l−1
?|εm|) is the metal skin depth, which
is on the order of tens of nanometers. We also assume
that κda ≪ 1, which is always the case for the mode
under consideration because either this mode is close to
the TEM mode where k = k0εd, or the gap a is thin
enough. This leads to a closed expression for the index
of refraction of the mode, which is Eq. (2). Using this,
one can check that κda ≈?εdals/λ2?1/2. Consequently,
the applicability condition of the approximation used is
?εdals/λ2?1/2≪ 1 .
This condition is satisfied for the realistic parameters of
the problem. For instance, for the frequency f = 1 THz,
the skin depth for metals is ls ≈ 60 nm, while reduced
wavelength is λ = 75 µm. The condition (6) is well sat-
isfied for a ≪ 100 µm, i.e., in the entire range of interest
?1/2, κm=?k2− εmk2
?1/2. In the
s(1 − i), where ls =
B.Terahertz TM Wave in Finite-Conductivity
Consider a coaxial waveguide (coax) with the inner
wire radius r and the outer radius R = r + a, where a is
the dielectric gap width. The characteristic relation for
the TM modes of this waveguide has has the following
where Iν(x) and Kν(x) are modified Bessel functions,
and ξ =
εmκd. This equation is quadratic with respect
to ξ and can be written in the form αξ2+ βξ + γ = 0,
where the coefficients α, β, and γ can be easily found
by comparison to Eq. (7) as combinations of the Bessel
functions. It can obviously be resolved for ξ yielding
In the THz region, only the mode with the minus sign
in Eq. (8) propagates. It can be treated in a manner
similar to the mode described by Eq. (5). The equation
(8) can be expanded over the small parameter κda ≪ 1,
and the explicit form of the modal refraction index can
be readily obtained as
where κm= l−1
case, the applicability condition of this solution is given
by Eq. (6).
s(1 − i). Similar to the wedge waveguide
C.WKB Solution for the TM Wave in Graded
In the WKB approximation, a solution of the Maxwell
equations can be represented as a wave with amplitude
and phase that are slowly varying functions of y on the
scale of local wavelength. The behavior in the transverse
direction z is the same as for the non-graded system. The
WKB solution is valid if the adiabatic parameter is small
δ = |d
k(y)| ≪ 1 .(10)
In the WKB approximation, the phase of the mode
(eikonal) is given by an integral
φ(y) = k0
where n(y) is the local refraction index of the mode. The
behavior of wave amplitude as a function of the propa-
gation coordinate y is found from the condition of flux
W(y,z)dz = const ,(12)
8 Download full-text
where vg(y) = ∂ω/∂k is the wave local group velocity,
and W(y,z) is energy density in the mode.
This work was supported by grants from the Chemi-
cal Sciences, Biosciences and Geosciences Division of the
Office of Basic Energy Sciences, Office of Science, U.S.
Department of Energy, a grant CHE-0507147 from NSF,
and a grant from the US-Israel BSF. MIS is grateful to
S. Gresillon for helpful remarks.
Correspondence and requests for materials should be
addressed to MIS (email: firstname.lastname@example.org)
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