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Visualization of the complex refractive index of a conductor by frustrated
total internal reflection
Yu. P. Bliokh,
a兲
R. Vander, S. G. Lipson, and J. Felsteiner
Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
共Received 12 February 2006; accepted 23 May 2006; published online 11 July 2006兲
A simple imaging geometry in which total internal reflection in a glass prism is frustrated by the
proximity of a metal surface is implemented for observation of surface plasmon resonance. At a
certain angle of incidence, the total internal reflection is completely suppressed at a certain distance
between the metal and the prism surfaces. Using planar metal and spherical prism surfaces, the
distance parameter is sampled in a single image. This allows a direct determination of the complex
refractive index in bulk samples as well as in thin films. Our experimental data are in good
agreement with previously published data. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2220540兴
Total internal reflection of light at a glass-air interface
can be spoilt by the proximity of a second optical medium
situated within about one wavelength of the interface. This is
a well-known phenomenon and can be described as a process
in which photons tunnel from the glass, through an air bar-
rier, into the second medium. We consider the case where the
second medium is a conductor, having complex refractive
index n
˜
⬅n+ik, which supports surface plasmon resonance
共SPR兲.
1
We use a simple imaging geometry often used to
demonstrate frustrated total internal reflection by a prism in
contact at one point with a convex spherical optical surface.
In contrast to previous investigations, this configuration pro-
vides continuous and accurate control of the thickness of the
barrier. A high contrast interference fringe is observed, which
indicates destructive interference between the totally re-
flected wave and that reflected from the metal. Since two
parameters 共incidence angle and barrier thickness兲 are re-
quired for maximum frustration, the experiment provides a
simple visual method of determining n
˜
directly, and has the
advantage that it can be carried out on the polished surface of
a bulk sample, as well as on thin films. It may also be appli-
cable to identifying left-handed materials.
2
Surface plasmons
3
are electronic excitations with p po-
larization which propagate on the interface between a con-
ductor and a conventional dielectric. They have many appli-
cations, including electronic devices and biosensors. When
Maxwell’s equations are solved for the interface between a
metal 共dielectric constant
⑀
=
⑀
r
+i
⑀
i
=n
2
−k
2
+2ink兲 and a
conventional dielectric 共
⑀
d
=n
d
2
⬎ 0兲, “surface plasmon” solu-
tions are found. These decay exponentially away from the
interface into both media and have component k
x
of the wave
vector parallel to the surface given by
4
k
x
2
= k
0
2
⑀⑀
d
⑀
+
⑀
d
, 共1兲
where k
0
is the value of the free space wave vector. When
⑀
r
⬍ −
⑀
d
, Re共k
x
兲⬎ k
0
, and so the wave has to be excited ei-
ther by a traveling wave in an optically dense medium
关Kretschman configuration 共Ref. 5兲, Fig. 1共a兲兴 or by a grating
coupler
3
关where the grating period provides the necessary
addition to k
0
, Fig. 1共b兲兴. In these cases the metal must be in
the form of a thin film, where the coupling is from one side
and the surface plasmon is on the other. Another method
1
couples the plasmon to an evanescent wave in air, produced
by total internal reflection in a glass prism 关Otto configura-
tion, Fig. 1共c兲兴. In this case the surface plasmon is excited on
the air-sample interface, and therefore a bulk conducting
sample can be used. This type of coupling has been em-
ployed in several laboratories 共for example, Ref. 6兲, but the
need for accurate control of the separation between the prism
and conductor surfaces makes for complicated mechanics
when used at visible wavelengths. It was also studied by
Bliokh et al.
7
using a dense gas plasma at microwave fre-
quencies, where diffraction by a subwavelength grating pro-
vided the evanescent wave. At optical frequencies, the eva-
nescent wave can be provided by an internally totally
reflecting glass surface, when the angle of incidence in the
glass
inc
is chosen appropriately, so that resonance occurs
when n
glass
k
0
sin
inc
=Re共k
x
兲 from 共1兲. The resonance has a
width determined by the damping, i.e., Im共k
x
兲, and so the
phase of the plasmon varies continuously with angle.
8
To
satisfy the boundary conditions between the plasmon and the
evanescent wave in the air gap there has to be a reflected
evanescent wave, which in turn creates an additional travel-
ing wave on its incidence on the glass. This wave interferes
with the totally reflected wave; both of them have phases
determined by
inc
, and for a lossy metal there exists at least
one angle of incidence at which the difference between them
is
.
9
At that angle, for a certain value of the air-gap thick-
ness, the amplitudes of the two waves are equal, and com-
pletely destructive interference occurs. Because the wave
a兲
Electronic mail: bliokh@physics.technion.ac.il
FIG. 1. Three methods of coupling to surface plasmons. 共a兲 Prism coupling
共Kretschman兲; 共b兲 grating coupling; 共c兲 evanescent wave coupling 共Otto兲.SP
indicates the surface on which surface plasmons are excited.
APPLIED PHYSICS LETTERS 89, 021908 共2006兲
0003-6951/2006/89共2兲/021908/3/$23.00 © 2006 American Institute of Physics89, 021908-1
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propagation in the gap is evanescent, no further phase differ-
ences are induced there.
The solution of Maxwell’s equations for this problem is
conveniently expressed in terms of two equations.
1,9
The first
expresses the p-wave Fresnel reflection coefficient at the in-
terface between media i and j:
r
ij
=
n
j
cos
i
− n
i
cos
j
n
j
cos
i
+ n
i
cos
j
, 共2兲
in which
j
is the 共complex兲 angle of refraction in the me-
dium, calculated by Snell’s law. The reflectance r共
兲 of a
three-layer system 关glass 共1兲/air gap 共2兲/metal 共3兲兴 is given
by the second equation:
r共
兲 =
r
12
+ r
23
exp共−2d兩k
z
兩兲
1+r
12
r
23
exp共−2d兩k
z
兩兲
, 共3兲
where k
z
2
=k
0
2
−k
x
2
⬍ 0 and d is the gap thickness. The surface
plasmon energy loss in the metal leads to an appearance of a
deep minimum at the resonant angle of incidence,
res
, when
the plasmon is excited. It was shown in Ref. 9 that the re-
flectance is exactly zero for a certain value d
res
of the gap
thickness. The two experimentally measured values,
res
and
d
res
, allow us to determine the two optical characteristics, n
and k, of the metal under investigation. A useful way of
representing the relationship between n, k and
res
, d
res
is the
series of curves shown in Fig. 2. Any particular conductor
can be identified in the plane of the diagram by means of its
values of n and k. The ordinate and abscissa then show,
respectively, the thickness of the air gap d and the resonance
angle
inc
at which the reflected light has zero intensity.
In order to display these two parameters visually, we use
the proximity of a slightly convex glass-air interface to a
planar air-metal interface 共or vice versa兲, the two touching at
one point. The former is made by optically contacting a
0.5 m focal length planoconvex singlet lens to the hypot-
enuse of a right-angled prism. Samples of various metals, in
the form of thick 共d ⬎兲 evaporated layers on flame-polished
plane glass slides or polished bulk metal plates, were then
lightly pressed onto the convex glass surface using a foam-
rubber plug. The interfaces were illuminated with plane par-
allel p-polarized He–Ne laser light 共 =633 nm兲 through the
prism, at an angle greater than the critical angle, and the
interface was imaged on a charge-coupled device 共CCD兲
camera using the reflected light 共Fig. 3兲. The imaging lens
had a sufficiently large aperture that the prism could be ro-
tated through about 2° without image vignetting, for deter-
mination of the angle of minimum reflected intensity. The
angles could be accurately measured with respect to the criti-
cal angle, at which Newton’s rings appear in the field of
view. Measurement of the radius of curvature R of the con-
vex surface using an autocollimator determined the relation-
ship between air-gap thickness d and radius r in the image
field, d = r
2
/2R. Since the observation is at an aspect angle
close to 45°, a particular value of d corresponds in the cam-
era plane to an ellipse with axes ratio approximately
冑
2.
Destructive interference occurring at a certain thickness d
min
then gives rise to a dark elliptical fringe 共Figs. 3 and 4兲.As
inc
is varied, the contrast of the fringe changes, becoming
close to unity at the abscissa angle
in Fig. 2. The breadth of
the fringe can be related to Im共k
x
兲.
Preliminary experiments to evaluate the technique were
carried out on the metals Cu, Au, Ag, Al, and In, which
satisfy the condition
⑀
r
⬍ −1 required for support of surface
plasmons, and also on Fe to confirm that the dark fringe is
absent for a metal with
⑀
r
⬎ −1. The results are indicated in
Fig. 2, together with literature values for these metals at
=633 nm.
10–12
Apart from indium, the results are in good
agreement with the published values. Observed ellipse im-
ages are shown in Fig. 4 for several of the metals; the dif-
FIG. 2. Map of n and k. Contours of n and k on the plane representing the
thickness of the air gap, d /, and the angle
at which minimum 共ideally
zero兲 reflectance is obtained for thick samples. This map can be used to
derive n
˜
from experimental data. Points representing literature values 共Refs.
10–12兲 for several metals are shown, together with our measured values. Al,
Au, Cu, and In samples were prepared by vacuum evaporation 共thickness
⬎兲, and Cu and Ag samples from bulk 共⬎ 99.99% purity兲.
FIG. 3. Experimental setup. A typical image is shown on the left, and details
of the tunnelling region are shown on the right.
FIG. 4. Elliptical fringes observed for six samples. Fe does not exhibit SPR.
All films have d ⬎. The differences between the radii and the sharpness of
the fringes can be seen clearly. In general, the larger and sharper the ellipse,
the greater the SPR sensitivity of the sample.
021908-2 Bliokh et al. Appl. Phys. Lett. 89, 021908 共2006兲
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ferences between them are very clear. Experiments on Cu
samples, for example, showed considerable dependence on
the method of preparation of the sample 共evaporated film or
bulk metal兲. In general, a large sharp ellipse indicates a high
SPR sensitivity. Intensity profiles as a function of air space
thickness were deduced from these images 共Fig. 5兲.
The simplicity of the experiment, and the fact that it is
applicable to samples prepared by different methods, makes
it a convenient method to determine n
˜
for conductors satis-
fying n
2
−k
2
⬍ −1. The experiments showed clearly the dif-
ference between surface treatments 共chemical, electropolish-
ing, and abrasive polishing兲, and we believe that the
anomalous behavior of In films is due to surface oxidation,
which may also explain the large variation in published val-
ues for n and k for this metal.
11,12
In addition, the experi-
ments showed that for materials with large k, for which the
angle
is close to critical, the resonance ellipse was visible
below the critical angle, indicating that SPR could be in-
duced by light incident from air. This was also observed by
Cairns et al.
6
Furthermore, the method may be useful in
identifying left-handed 共negative refractive index兲 materials,
whose SPR properties are discussed by Ruppin
2
and may
include an s-polarized mode. We also used it to compare the
quality of spoons made of nominally pure silver; the convex
shape of a spoon allows contact with a plane prism surface to
be made, since the curvature is provided by the spoons them-
selves. The surface has to be well polished only over a very
small area, and further study should allow a quantitative re-
lationship between the metal quality and its complex refrac-
tive index to be established.
1
A. Otto, Z. Phys. 216, 398 共1968兲.
2
R. Ruppin, J. Phys.: Condens. Matter 13, 1811 共2001兲.
3
H. Raether, in Springer Tracts in Modern Physics, edited by G. Hohler and
E. A. Niekisch 共Springer, Berlin, 1988兲, Vol. 111.
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W. Knoll, Annu. Rev. Phys. Chem. 49, 569 共1998兲.
5
E. Kretschman, Opt. Commun. 6, 185 共1972兲.
6
G. F. Cairns, S. M. O’Prey, and P. Dawson, Rev. Sci. Instrum. 71,4213
共2000兲.
7
Yu. P. Bliokh, J. Felsteiner, and Ya. Z. Slutsker, Phys. Rev. Lett. 95,
165003 共2005兲.
8
A. V. Kabashin and V. I. Nikitin, Opt. Commun. 50,5共1998兲.
9
W. Lukosz and H. Wahlen, Opt. Lett. 3,88共1978兲.
10
P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 共1972兲.
11
G. J. Kovacs, Thin Solid Films 60,33共1979兲.
12
H. E. de Bruijn, R. P. H. Kooyman, and J. Greve, Appl. Opt. 31,440
共1992兲.
FIG. 5. Radial intensity profiles of images as a function of d/. The profiles
are measured at the angle of incidence where highest contrast was obtained.
The minimum intensity is not zero as predicted by the theory, probably
because of surface scattering. For each profile, the intensity shown at a
particular value of d is an average of the values measured around an ellip-
tical ring with the appropriate axis ratio.
021908-3 Bliokh et al. Appl. Phys. Lett. 89, 021908 共2006兲
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