Coupling and Propagation of Sommerfeld Waves at 100 and 300 GHz

Abstract

The coupling between a linearly-polarized gaussian beam and a Sommerfeld
wave propagating on a circular metallic wire is obtained owing to a
differential phase element inserted in front of the metal wire. At
millimeter-wavelengths we calculate a theoretical maximum coupling
efficiency of 32% for this system in spite of the metal nature and
radius in the range of a few hundreds of microns. A detailed
experimental study of 100 and 300 GHz Sommerfeld waves propagating on
stainless steel and tungsten wires is reported. The measured field at
any distance from the wire compares well with theoretical predictions.

Full text

J Infrared Milli Terahz Waves (2012) 33:174–182
DOI 10.1007/s10762-011-9854-x
Coupling and Propagation of Sommerfeld Waves
at 100 and 300 GHz
Laurent Chusseau ·Jean-Paul Guillet
Received: 8 September 2011 / Accepted: 24 October 2011 /
Published online: 11 November 2011
© Springer Science+Business Media, LLC 2011
Abstract The coupling between a linearly-polarized gaussian beam and a
Sommerfeld wave propagating on a circular metallic wire is obtained owing to
a differential phase element inserted in front of the metal wire. At millimeter-
wavelengths we calculate a theoretical maximum coupling efficiency of 32%
for this system in spite of the metal nature and radius in the range of a few
hundreds of microns. A detailed experimental study of 100 and 300 GHz
Sommerfeld waves propagating on stainless steel and tungsten wires is re-
ported. The measured field at any distance from the wire compares well with
theoretical predictions.
Keywords Far infrared ·Waveguides ·Surface plasmons
1 Introduction
The development of terahertz technology requires, besides the sources and the
detectors, new components to handle beams appropriate to applications [1].
As is the case in the microwave and optical frequency ranges, waveguiding
technology is expected to be the most convenient way to distribute and
split signals between sources and detectors. The teraherz frequency range
being intermediate between the microwave and optics frequency ranges, one
may employ either dielectric or metallic waveguides. It seems that planar
dielectric waveguides [2], dielectric fibers [3,4], pipe waveguides [5], hollow
glass waveguides [6], or polymer tube [7] are not as frequently employed
L. Chusseau (B)·J.-P. Guillet
Institut d’Électronique du Sud, UMR 5214 CNRS, Université Montpellier 2,
Place E. Bataillon, 34095 Montpellier, France
e-mail: chusseau@univ-montp2.fr

J Infrared Milli Terahz Waves (2012) 33:174–182 175
than metallic waveguides. The latter are simply scaled down versions of clas-
sical waveguides, for example rectangular waveguides [8], circular waveguides
[9], two-wire waveguide [10], or slit waveguides [11]. One may also employ
structures that are more specific to the THz frequency range like parallel
plate waveguides [12,13], or the plasmonic metallic wire supporting cylindrical
waves, studied by Goubau in the microwave frequency range [14], and recently
shown to be of tremendous importance in the THz regime [15–25].
In this paper we study the propagation of cylindrical waves on metallic
wires at 100 and 300 GHz. An original set up is described which helps us to
evaluate the propagating field extension around the wire. We consider the
coupling of linearly-polarized beams to such guided modes, and measure the
propagating loss due to the metal non-zero resistivity. Comparison between
our experimental results and theory is made.
2 Sommerfeld Waves
Electromagnetic waves guided by cylindrical metallic surfaces were described
in [26]. Let zdenote the metallic wire axial coordinate and let us employ
cylindrical coordinates. Only the components Ez,Erand Hϕdo not vanish. For
frequencies in the THz range and the usual ohmic losses the most important
field component is the radial electric field Er. It propagates in the air (or
vacuum) surrounding the wire. We have Er∝jh
γH(1)
1(γ r)expj(ωt−hz),with H(1)
1
the Hankel function of the first kind, hthe propagation constant of the guided
wave and γ2=k2−h2where kdenotes the wavenumber in air. Its magnetic
counterpart Hϕ=ω0Er/h. The small longitudinal electric field component
Ez∝H(1)
0(γ r)expj(ωt−hz)usually does not exceed a few percent of Er[27]. The
propagation constant hfollows from the transcendental equation that derives
from the continuity of the Ezcomponent at the wire surface [14,26]
μγ
k2
H(1)
0(γ a)
H(1)
1(γ a)=μmγm
k2
m
J0(γma)
J1(γma)(1)
where ais the wire radius, γ2
m=k2
m−h2,andkmis the wavenumber in the
metal. μm/μ is the metal relative permeability that are always taken to 1 in this
work. H(1)
0(.) and H(1)
1(.) are Hankel functions and J0(.) and J1(.) are Bessel
functions of the first kind.
Equation 1(or its dual form using the In(.) and Kn(.) generalized Bessel
functions [27,28]) cannot be solved in terms of known functions. Most solu-
tions found in the literature characterize the metal only by its dc conductivity
[14] and assume that the wire radius is large compared with the skin depth
[14,19]. Asymptotic expressions of the Bessel functions [27] or Taylor series
expansions are used to obtain approximate solutions [28]. Here we directly
solve Eq. 1using arbitrary precision numerical facilities of Mathematica [29].
When pure metals are considered the Drude approximation of the permittivity
is used because of its accuracy in the far infrared and THz frequency range

176 J Infrared Milli Terahz Waves (2012) 33:174–182
[30,31]. Otherwise we use the permittivity derived from the Ohm law, =
0−jσ/ω,with0the vacuum permittivity and σthe dc conductivity. This
procedure does not suffer from numerical approximations and is easy to
implement. The main features of the waves sketched in [27] are accurately
described by the present exact calculations. Namely, the large field extension
outside the wire, a quasi TEM behavior in air, a phase velocity very close to
the light speed c, and very low propagation losses due to the small penetration
of the fields into metals.
3 Coupling Between Linearly-polarized Gaussian Beams
and Sommerfeld Waves
The major difficulty when dealing with Sommerfeld waves is the low coupling
efficiency between the wire and the usual sources. The main reason is the
large difference between the radial polarization of wire modes and the usual
sources generating linearly-polarized beams propagating in free-space. The
first demonstrations of Sommerfeld waves employed an adiabatic transforma-
tion of coaxial modes [14,32]. This technique is efficient and almost lossless
but is practical only for microwave sources. With the renewed interest in
Sommerfeld waves for the THz range, the concern has been mostly to couple
to it the broadband impulse signals produced by photoconductive antennas.
Mode matching has been obtained by various methods: concentrating a freely-
propagating THz pulse using circular plasmonic grooves surrounding a circular
aperture with a wire at its center [33], modifying the photoconductive cell itself
so as to exhibit a radial symmetry [17,34], exciting only one side of the wire
with the near-field of a linearly polarized source [35], or using another metal
wire to serve as an intermediate antenna breaking the radial symmetry at the
focus point of a linearly polarized source (dual-wire coupler) [15,16]. All these
solutions operate well but with coupling efficiencies ranging from 0.4% for the
dual-wire coupler to more than 50% with dedicated radial photoconductors
[34]. Although highly satisfactory, the latter solution is restricted to photocon-
ductive switches and thus cannot be adapted to other sources like quantum
cascade lasers or frequency-multiplied microwave sources.
To solve the free-space to Sommerfeld mode coupling problem in the cw
case, we previously proposed the use of a differential phase plate that induces
a polarization reversal of one half-space as compared to the other before
launching the electromagnetic field on the wire [36]. This is a direct translation
of common practices in optics and its implementation to the THz range is
schematically drawn in Fig. 1a.
The peculiarities of the Sommerfeld waves recalled earlier are that the
propagating-wave phase velocity is close to cand that the reflection from wire
ends is negligible [27]. Besides, the very small longitudinal (axial) electrical
field renders the local field in air similar to a TEM wave. Since we are
interested only in the coupling of the wire mode with an x-polarized input
beam (such as an x-polarized gaussian beam with a half-space phase reversal)

J Infrared Milli Terahz Waves (2012) 33:174–182 177
wire mode
free-space
beam
0.32
0.30
0.28
0.26
0.32
0.05
0.10
0.15
0.20
0.25
0.30
0.30
0.28
0.26
0.5 1.0 1.5 2.00.5 1.0 1.5 2.0
12 12
f (THz)
f (THz)f (THz)
f (THz)
40
20
40
20
1 2 5 10 20 50 100 200
a
cd
b
Fig. 1 Calculated coupling efficiency Cbetween a linearly polarized gaussian beam of full-width at
half power and a Sommerfeld mode propagating along a metal wire of radius a.(a) Schematic of
the coupling scheme. (b) Coupling to a tungsten wire of radius a=250 μm versus :f=100 GHz,
dashed line;f=300 GHz, dot-dashed line;f=1,000 GHz, solid line.(c) Tungsten and (d) Gold:
Maximum coupling efficiency, Copt, as a function of frequency for various wire radius: a=250 μm,
solid line;a=500 μm, dashed line;a=1mm, dot-dashed line. Insets give the ratio of opt to the
wavelength, opt being the beam width that achieves the maximum coupling efficiency Copt.
we may express the wire field in terms of a real field ψB(x,y)=cos(θ)Er(r)
with r=x2+y2,tan(θ) =y/x,where:Er(r)=hH
(1)
1(γ r)/γ , leaving aside
normalization. The power coupling is [37]
C=SψAψBdS2
.(2)
where ψArepresents the incident beam.
We evaluate Canalytically for an incident gaussian field at its waist:
ψA(x,y)=exp(−(x2+y2)/(2σ2)). After the phase plate, we have: ψA=
exp(−(x2+y2)/(2σ2)), y>0,ψ
A=−exp(−(x2+y2)/(2σ2)), y<0.Insert-
ing ψAand ψBin Eq. 2yields the power coupling.
First, we have calculated the power coupling defined above for various
metals and parameters. Figure 1b gives calculated values of Cas a function of
≡2.35 σ, which is approximately the full width at half maximum (FWHM)
of the gaussian beam. A tungsten wire of radius a=250 μm and three fre-
quencies were considered. As shown, Cmay exceed 30% when the input
gaussian beam width is appropriately chosen. This occurs for beam sizes of
a few millimeters, which are easily generated using off-axis parabolic mirrors.
Moreover, the tolerance associated to these widths is great since more than

178 J Infrared Milli Terahz Waves (2012) 33:174–182
80% of the maximum coupling efficiency is obtained over a decade in -values
for the three frequencies considered.
Next, we investigate the influence of frequency and of the kind of metal
employed on the maximum available coupling Copt. This is illustrated in Fig. 1c
and d by plots of Copt versus frequency deduced from series of curves like that
of Fig. 1b. Results are given for three different wire radii, a=250 μm, 500 μm
and 1mm. Data in Fig. 1c relate to tungsten while those of Fig. 1d relate to
gold. For each plot an inset shows the beam width opt corresponding to Copt.
The main conclusion is that the maximum coupling efficiency can be optimized
up to ≈32% at any frequency in the range considered provided that the wire
radius is properly selected. The lower the frequency, the wider the radius a
should be to optimize the coupling. Only a slight difference occurs in opt (see
insets). The explanation is that the increase in the Sommerfeld mode extension
as a function of arequires an increase of to keep the coupling at its optimum
value. Finally, a comparison between the results in Fig. 1c and d shows a similar
behavior for other metals, even for very different conductivities. For similar
conductivities (aluminum and gold), separate calculations show plots that are
indistinguishable at the scale of Fig. 1d.
The analytic evaluation of the efficiency of the coupling scheme given in
Fig. 1a thus predicts a power-coupling efficiency exceeding 30% whatever
the frequency in the range considered and for the metals considered. This is
achieved through an uncritical selection of the incident beam size. The C-values
obtained here are close to the maximum limit of 50% that one would ideally
obtain while projecting a linearly polarized gaussian mode with a half-space
phase reversal on a radially polarized gaussian mode. As a consequence the
additional ≈20% loss calculated here most likely originates from wavefront
mismatch.
The method involved in Eq. 2is also suitable to calculate the coupling
between two distinct metal wires separated a distance dapart. This situation is
described in the inset of Fig. 2and corresponds to ψA(x,y)=hH
(1)
1(γ r)/γ and
ψB(x,y)=hH
(1)
1(γ r)/γ with r=x2+y2and r=(x+d)2+y2. We have
calculated Cat f=100 GHz numerically in the case of tungsten wires of radius
Fig. 2 Calculated coupling
efficiency Cbetween the ends
of two tungsten wires of
radius a=250 μm supporting
Sommerfeld propagating
modes at f=100 GHz and
separated laterally by the
distance d. The inset gives a
schematic description of the
system considered.
5 10 15 20 25
dmm
0.1
0.2
0.3
0.4
0.5
0.6
0.7

J Infrared Milli Terahz Waves (2012) 33:174–182 179
a=250 μm assuming the two wire ends close enough so that it is unnecessary
to account for free-space propagation [36,38]. The result given in Fig. 2shows
a rapid fall out of the coupled intensity as a function of d. As an example, when
d=2a(the first point in Fig. 2) a drop of 25% of the transmission is obtained.
This calculation explains quantitatively well the experiment described in [36]
where the coupling between two such wires was shown to exhibit a FWHM
≈±2mm.
4 Propagation of Sommerfeld Waves
With continuous-wave frequency-multiplied electronic sources, and the pre-
viously designed coupler, a preliminary experiment was conducted to verify
that the propagating mode is a true Sommerfeld mode and to evaluate the
radial extension of the field around the wire. This kind of experiment has
already been performed in pulsed mode with a direct field detection by a
photoconductive antenna [16,17]. A good agreement with the known field
profile has been found except for a reduced accuracy far from the metal wire.
On the other hand, the field emission (radiation pattern) from the wire end
has been measured [36,38,39]. It resembles the emission of traveling wave
broadband antennas [40].
We propose here another kind of experiment involving an iris diaphragm
of variable clear aperture around the metal wire supporting the Sommerfeld
mode. By closing the iris the open space around the wire gets reduced
and the mode transmission decreases (see Fig. 3a). As a major consequence,
the power transmission of the system is reduced and easily measured. The
experiment was conducted with various wires of length ≈20 cm at two different
frequencies. At 100 GHz the detection was obtained with a Schottky diode
and at 300 GHz a 4K Si-bolometer was employed. Results are given as dots
in Fig. 3b and show very similar behavior whatever the frequency and the
metal. Attenuations of ≈10 dB are observed when the iris is nearly completely
closed and in contact with the wire. The transmitted power increases rapidly
when the diaphragm is opened progressively. A nearly complete transmission
is achieved at full opening. The propagating modes are experimentally found
to be confined to cylinder volumes of radii between 5×aand 9×a, depending
on the frequency and wire radii.
Using the known theoretical field from Section 2, the experimental results
can be compared to the calculated transmitted power obtained by
P(r)=r
a
ErHϕ2πrdr∝r
a2πr|Er(r)|2dr(3)
where the field inside the metal and the Ezcomponent in air have been ne-
glected. Data of Fig. 3are plotted with the theoretical attenuations P(r)/P(∞),
the propagation constant hhaving been previously determined by Eq. 1
for the a=250 μm tungsten wire and the a=400 μm stainless steel wire
of bulk resistivity 1.32 106−1m−1. Comparisons between measurements and

180 J Infrared Milli Terahz Waves (2012) 33:174–182
Fig. 3 (a) Schematic of the
transmission experiment of a
Sommerfeld mode on a wire
of radius awith an iris as a
variable attenuator. (b)
Measured (dots)and
calculated (solid lines) power
transmissions as a function of
the iris opening radius.
Abscissa origin is taken at the
wire center. Upper curve:
stainless steel a=400 μm
f=100 GHz. Middle curve:
tungsten a=250 μm
f=100 GHz. Lower curve:
tungsten a=250 μm
f=300 GHz.
0 10 15 mm
5
dB
51010 mm
5
0 10 15 mm
15
10
5
a
b
Emitter
Detector
wire mode iris
calculations are in excellent agreement on the whole range of the diaphragm
opening. Calculations also show that the modes are more confined at higher
frequency and that the metal nature has little influence provided that the
conductivity is high. These features are known to scale down up to the μm
size [21,22,41], thus allowing a high field concentration at the wire end that
can serve as an excellent near-field probe mostly sensitive to the longitudinal
Ezcomponent [25]. The perfect agreement between measurement and theory
shows that Sommerfeld waves are indeed excited and that propagation lengths
of ≈20 cm prohibit the propagation of other low-loss mode.
Using the same experimental setup and with the iris diagram fully opened,
the propagation losses are easily estimated by reducing step by step the wire
length at the detection side. We already proposed this procedure [25]for
wire lengths decreasing from 65 to 40 cm but the previously estimated losses
were unexpectedly high. Our new experiments at f=100 GHz give losses of
0.13 ±0.02 dB cm−1. Measured results are however still high as compared to
other measured data [15,18]. They are estimated to be about three times larger

J Infrared Milli Terahz Waves (2012) 33:174–182 181
than the calculated value 0.043 dB cm−1. A similar discrepancy was found in
the earliest loss measurements of wire modes [32] and attributed to radiation
losses at surface imperfections. Because of the excellent surface aspect of
the wire used, we believe that the wire sag under its weight is the cause of
the excess losses. A large, unobservable, radius of curvature of the wire was
unavoidable during our experiments. This may have resulted into bending or
mismatch losses being added to linear losses.
As a consequence, from the full balance of power in the experiment (i. e.
18 dB of attenuation with the two couplers and a 40cm steel wire) one deduces
a revised experimental efficiency of 23 ±2% per coupler in reasonably good
agreement with previous theoretical estimates. Discrepancy is attributed to
the imperfect size matching of the gaussian input beam, and to a possible
misalignment between that beam and the wire axis that was not controlled
with precision.
5 Conclusion
We have efficiently coupled and propagated continuous-wave Sommerfeld
modes on various metallic wires at millimeter-wavelengths. Using the theo-
retical form of the electric field guided along the wire, we have calculated a
32% optimum cw coupling efficiency with a simple differential phase element
inserted in front of the metallic wire. Efficiencies exceeding 30% are theoreti-
cally obtained irrespectively of the metals investigated from a proper selection
of the input gaussian beam waist. From the known experimental propagation
losses we deduced a 23% coupling efficiency in experimental conditions. This
result is in reasonably good agreement with calculations since no special
care was taken to match the input gaussian beam size. With a simple set-up
involving an iris surrounding the metal wire, propagating modes are shown
experimentally to be in very good agreement with their known theoretical
form, emphasizing that only this mode is guided along the wire.
Acknowledgements The authors acknowledge the French National Research Agency for fund-
ing under grant TERASCOPE, #ANR-06-BLAN-0073. Authors are also highly grateful to
Jacques Arnaud for fruitful discussions and careful reading of the manuscript.
References
1. M. Tonouchi, Nature Photonics 197 (2007).
2. R. Mendis and D. Grischkowsky, Journal of Applied Physics 88, 4449 (2000).
3. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, Optics Letters 31, 308 (2006).
4. C. Jördens, K. L. Chee, I. A. I. Al-Naib, I. Pupeza, S. Peik, G. Wenke, and M. Koch, J. Infrared
Millim. Terahertz Waves 31, 214 (2010).
5. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, Opt. Lett. 34,
3457 (2009).
6. B. Bowden, J. A. Harrington, and O. Mitrofanov, Opt. Lett. 32, 2945 (2007).
7. D. Chen and H. Chen, Opt. Express 18, 3762 (2010).

182 J Infrared Milli Terahz Waves (2012) 33:174–182
8. D. Grischkowsky, IEEE Journal of Selected Topics In Quantum Electronics 6, 1122 (2000).
9. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, J. Opt. Soc. Am. B 17, 851
(2000).
10. M. Mbonye, R. Mendis, and D. M. Mittleman, Applied Physics Letters 95, 233506 (2009).
11. M. Wächter, M. Nagel, and H. Kurz, Appl. Phys. Lett. 92, 161102 (2008).
12. S.-H. Kim, E. S. Lee, Y. B. Ji, and T.-I. Jeon, Opt. Express 18, 1289 (2010).
13. H. Zhan, R. Mendis, and D. M. Mittleman, Optics Express 18, 9643 (2010).
14. G. Goubau, J. Appl. Phys. 21, 1119 (1950).
15. K. Wang and D. Mittleman, Nature 432, 376 (2004).
16. K. Wang and D. Mittleman, J. Opt. Soc. Am. B 22, 2001 (2005).
17. T. Jeon, J. Zhang, and D. Grischkowsky, Appl. Phys. Lett. 86, 161904 (2005).
18. M. Wächter, M. Nagel, and H. Kurz, Optics Express 13, 10815 (2005).
19. K. Wang and D. Mittleman, Phys. Rev. Lett. 96, 157401 (2006).
20. T. Akalin, A. Treizebre, and B. Bocquet, IEEE Trans. Microwave Theo. Techn. 54, 2762
(2006).
21. Y. B. Ji, E. S. Lee, J. S. Jang, and T.-I. Jeon, Opt. Express 16, 271 (2008).
22. X.-Y. He, J. Opt. Soc. Am. B 26, A23 (2009).
23. M. Awad, M. Nagel, and H. Kurz, Appl. Phys. Lett. 94, 051107 (2009).
24. V. Astley, J. Scheiman, R. Mendis, and D. M. Mittleman, Opt. Lett. 35, 553 (2010).
25. J.-P. Guillet, L. Chusseau, R. Adam, T. Grosjean, A. Penarier, F. Baida, and D. Charraut,
Microwave and Optical Technology Letters 53, 580 (2011).
26. A. Sommerfeld, Electrodynamics (Academic Press, New York, 1952).
27. Q. Cao and J. Jahns, Opt. Express 13, 511 (2005).
28. J. Yang, Q. Cao, and C. Zhou, Optics Express 17, 20806 (2009).
29. Mathematica 8, Wolfram Research Inc. (2011), http://www.wolfram.com/mathematica/.
30. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, Applied Optics 24,
4493 (1985).
31. M. A. Ordal, R. J. Bell, R. W. Alexander, L. A. Newquist, and M. R. Querry, Applied Optics
27, 1203 (1988).
32. F. Sobel, F. L. Wentworth, and J. C. Wiltse, IRE Trans. Microwave Theory Tech. 9, 512 (1961).
33. A. Agrawal and A. Nahata, Opt. Express 15, 9022 (2007).
34. J. Deibel, K. Wang, M. Escarra, and D. Mittleman, Opt. Express 14, 279 (2006).
35. M. Walther, M. Freeman, and F. Hegmann, Appl. Phys. Lett. 87, 261107 (2005).
36. R. Adam, L. Chusseau, T. Grosjean, A. Penarier, J.-P. Guillet, and D. Charraut, Journal of
Applied Physics 106, 073107 (2009).
37. J. A. Arnaud, Beam and Fiber Optics (Academic Press, 1976).
38. J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M.
Planken, Opt. Express 14, 8772 (2006).
39. M. Walther, G. S. Chambers, Z. G. Liu, M. R. Freeman, and F. A. Hegmann, J. Opt. Soc. Am.
B22, 2357 (2005).
40. C. A. Balanis, Antenna Theory - Analysis and Design (John Wiley & Sons Inc., 1982), 2nd ed.
41. H. Liang, S. Ruan, and M. Zhang, Opt. Express 16, 18241 (2008).