arXiv:0807.1537v1 [physics.optics] 9 Jul 2008
Resonance-like Goos-Hänchen Shift induced by
R. Gruschinski+, G. Nimtz∗, and A.A. Stahlhofen+
*II. Physikalisches Institut, Universität zu Köln and+Institut für Integrierte Naturwissenschaften,
Abstract. The influence of nano-metalfilms on the Goos-Hänchenshift (GHS) is investigated.The
films deposited at the total reflecting surface of a perspex prism/air have a sheet resistance varying
between Z?= 25 and 3 000 Ω?. A resonance-like enhancement of the shift and of the absorption
is found for TE polarized waves, when the sheet resistance approaches the value of the vacuum
impedance. For TM waves the influence of the metal films on the GHS is comparatively weak. The
experiments are carried out with microwaves.
Keywords: Goos-Hänchen shift; nano-metallic films, microwaves
PACS: 42.25.Bs, 42.25.Gy, 42.50.-p, 73.40.Gk
A shift of the reflected electromagnetic wave against the incident beam in the case of
total reflection was conjectured by Newton 300 years ago. Goos and Hänchen have mea-
sured the shift around 1946  for the first time. A quantitative experimental analysis
of the shift was performed by Haibel et al. with microwaves . In the study the shift
was measured depending on the wave polarization, on the angle of incidence, and on the
beam diameter. The result was that the reflected TM waves are much more shifted than
the TE waves. The GHS was found to be of the order of magnitude of the wavelength.
In addition the shift was observed to increase with decreasing beam diameter.
For the time being there is no exact theoretical description of the GHS available.
Moreover, the theoretical and some experimental investigations predict a discontinuity
of the GHS at the critical angle and no shift at angles below the critical one, see for
instance Refs.. Recently, however, it was shown by Müller et al.  that there is a
continuous transition from the angles of total reflection to angles of partial reflection.
The novel result is, also partial reflection shows a significant beam shift between the
incident and the reflected beam.
In this study we present data of the total reflection influenced by nano metal films
deposited on the total reflecting surface. The Al-films were between 10 and 100 nm
thick and were vapor deposited on polyethylene films of 10 µm thickness. In addition
we have used conducting films with higher sheet resistance values (≥ 1000 Ω) based on
an organic metal (). The organic material having a higher resistivity was sprayed also
on polyethylene films of 10 µm thickness. We applied single films as well as stacks of
several sheet layers. The latter behaved as a parallel circuit with a corresponding lower
total sheet resistance.
Incidentally, Woltersdorff  has shown for thin metal films at low frequencies that
the absorption A has a maximum with A = 0.5 and T = R = 0.25 hold at the thickness
where Z0= 377 Ω and σ are thevacuumimpedanceand theconductivity,respectively.
The measured sheet resistance Z?is given by
Z?= 1/(d σ)
The thickness of the Al-films produced by vapor deposition on 10µm polyethylene
films could not be measured. It was estimated to be between 10 and 100 nm. Only the
sheet resistance Z?was precisely measured. It is assumed that the conductivity of the
nano films is one to two orders of magnitude smaller than the bulk conductivity .
In addition to Z?we have measured the reflection R and the transmission T of the
microwave at normal incidence. The following relations are valid for the low frequency
R+T +A = 1 with
R = (1+
The relations with experimental values are presented in Fig.1. The following study
was carried out with Al and metallic organic films near the absorption maximum
A,R,and T vs sheet resistance.
beam of microwave pulses of 8 ns half width. Details are the same as in Ref.. The receiver was a horn
antenna and a fast rectifier.
Sketch of the experimental set-up. The perspex prism. A parabolic antenna transmitted a
The total reflection was investigated at a perspex prism of 400 mm height and side
length. The refractive index was 1.605 corresponding to a critical angle of 38.5o. The
wavelength and the frequency of the microwaves are 32.8 mm and 9.15 GHz. The
parabolicantennatransmittedparallel waves.Thebeam widthwaslimitedbyan aperture
of 120 mm diameter. The angle of incidence was 45o. As shown in the sketch of Fig.2
the metal films were placed on the reflecting surface. The metal films were produced by
aluminum evaporated on 10 µm polyethylene substrates. Alternatively we used organic
conducting poly-aniline films sprayed also on 10 µm polyethylene substrates . Layer
thickness including substrate of all the applied films was ≪ than λ. The aluminum films
are about 20 nm thick. The sheet resistances were between 50 and 2746 Ω?.
In the case of TE polarized waves the reflection decreases with increasing impedance
with a minimumof 20 dB near 377 Ω?, however,at higher sheet resitances the reflection
increases approaching the GHS and the mirror value finally as shown in Fig.3. At the
reflection minimum (i.e. the maximum of signal amplification in Fig.3) the power is
absorbed and not transmitted. More important is that at the reflection minimum the GHS
has a strong maximum: three times the normal GHS.
Experimental data of the reflection and the GHS for TM polarized waves is plotted in
Fig.4. In turn the zero impedance values are the values measured with a bulk aluminum
plate as reference, the GHS without films is also presented. At small resistances a GHS
is not observed, with increasing resistance of the metal films the data approaches the
represents the total reflected GHS of 26.9 mm without a metal film on the reflecting surface. The GHS
increased up to 82 mm and the reflection decreased up to -20 dB at a sheet resistance near 377 Ω. The
values at zero resistance correspond to the data of a bulk aluminum plate at the reflecting surface. The
errors are ± 1 dB and ± 5 mm.
Experimentaldata of GHS and reflection vs sheet resistance for TE waves. The broken line
Thereis no exact theory of theGHS availablefor thetimebeing.Besides thequantitative
deviations between experimental and theoretical data two main problems are not solved:
The GHS diverges to large values when approaching the critical angle and the observed
dependence on the beam diameter. As mentioned above there is a continuous transition
from total to partial reflection as is shown in Ref.. There is one agreement of the
experimental results and the theoretical data since Artman’s study: the GHS is larger for
TM than for TE polarized waves. The relationship for the GHS D of the reflected beam
is given by the proportionality
D ∝ −1
where k1is the wave number of the incident wave, n1and n2are the refractive indices
of the first and the second optical media with n1> n2. θinand φ are the angles of
incidence and φ the phase shift between the incident and the reflected wave.
It is interesting that the anomalous behavior of R and GHS for TE polarization
happens when the metal film layer has a resistance equal to the vacuum (air) impedance.
resistance corresponds to a bulk aluminum mirror. The dotted line represents the GHS of 46.7 mm for the
prism-air configuration. The errors are ± 1 dB and ± 5 mm.
Reflection and GHS vs sheet resistance Z?for TM polarized waves. The data at zero sheet
Obviously, at this impedance the thin metal film matches the air medium resulting in a
long surface beat mode with a high absorption of the TE mode. The electric field is now
in the plane of the conducting layer. In the case of TM polarization the electric field is
perpendicularly orientated to the metal film and is less phase shifted and less absorbed.
The resonance like anomaly can not be related to a surface plasmon resonance be-
cause this plasmon enhancement is only expected and observed for transverse magnetic
excitation e.g. Ref..
The conductivities σ of the films result in a low frequency complex refractive index,
where both components are of equal value
n = n1−i n2
n1 ≈ n2≈
100 ≥ n1 ≈ n2≤ 1000,
where ε0is the vacuum permittivity and ν the electromagnetic wave frequency. This
extremely strong step of the refractive index makes the observed maximum of the GHS
for TE polarized waves plausible, having in mind Eq.7 and dφ ∝ dn. A quantitative
theoretical approach is not available yet.
Experiments on total reflection and GHS under the influence of nano-metal films with
a thickness d ≪ λ the wavelength and near the Woltersdorff thickness dW show a
resonance like behavior for the TE polarized wave near the vacuum impedance. The
reflection has a strong minimum of -20 dB compared to total reflection or reflection by
a bulk aluminum mirror. At the same time the GHS shows a large peak triple the value
of the normal GHS. The TM polarized wave is not much effected since the electric field
is normal to the metal films.
We gratefully acknowledge the technical support by U. Blancke, A. Enders, and U.
I. Newton, Opticks (Dover, New York, 1952)
F. Goos and H. Hänchen, Ann. Phys. (Leipzig), 1, 333 (1947)
A. Haibel, G. Nimtz, and A. A. Stahlhofen, Phys. Rev. E, 63, 047601 (2001)
K. Artman, Ann. Phys. (Leipzig), 6, 87 (1948); J. Cowan and B. Anicin, J. Opt. Soc. Am., 67, 1307
(1977); J. Agudin and A. Platzeck, J. Opt. (Paris), 9, 187 (1987); A. Ghatak, M. Shenoy, I. Goyal,
and K. Thyagarajan,Opt. Commun., 56, 313 (1986); B. Hecht, H. Bielefeld, Y. Inouye, D. Pohl, and
L. Novotny, J. Appl. Phys., 81, 2492 (1997)
D. Müller, D. Tharanga, A.A. Stahlhofen, and G. Nimtz, Eur. Phys. Lett., 73, 526 (2006)
W. Woltersdorff, Z. Phys., 91, 230 (1934)
G. Dumpich, Festkörperprobleme,31, 59 (1991)
X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett., 85, 372 (2004)