Surface magneto-plasmons in magnetic multilayers - Walther ...

TECHNISCHE
UNIVERSITÄT
MÜNCHEN
WALTHER - MEISSNER -
INSTITUT FÜR TIEF -
TEMPERATURFORSCHUNG
BAYERISCHE
AKADEMIE DER
WISSENSCHAFTEN
Surface magneto-plasmons in magnetic
multilayers
Diploma thesis
Themistoklis P. H. Sidiropoulos
Advisor: Prof. Dr. R. Gross
Garching, September 2010

Contents
1 Introduction 1
2 Theory 5
2.1 Surface Plasmon Polaritons . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Kretschmann-Raether configuration . . . . . . . . . . . . . . . 11
2.3 Reflectivity of surface plasmons . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Reflectivity of a single layer . . . . . . . . . . . . . . . . . . . 14
2.3.2 Surface plasmons at rough surfaces . . . . . . . . . . . . . . . 17
2.3.3 Reflectivity at a multilayer system . . . . . . . . . . . . . . . 17
2.4 Simulation of the reflectivity of surface plasmons . . . . . . . . . . . . 22
2.4.1 Surface plasmons in different materials . . . . . . . . . . . . . 25
2.5 Magnetic field dependence of surface plasmons . . . . . . . . . . . . . 27
2.5.1 Interaction of an external magnetic field with light in condensed
matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.2 Change of the dispersion relation of surface plasmons . . . . . 29
2.5.3 Surface plasmon enhanced electromagnetic field . . . . . . . . 31
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Excitation of surface plasmon polaritons 33
3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Alignment procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Experimental study of surface plasmons in thin metallic films 49
4.1 Surface Plasmons in thin silver films . . . . . . . . . . . . . . . . . . 49
4.1.1 Comparison of UA-B and UA . . . . . . . . . . . . . . . . . . . 51
4.1.2 Adjustment of nBK7 . . . . . . . . . . . . . . . . . . . . . . . . 55
I

II Contents
4.2 Reflectivity of multilayer thin films . . . . . . . . . . . . . . . . . . . 56
4.3 Beyond the ideal surface plasmon resonance curve . . . . . . . . . . . 60
4.3.1 Thin film limit . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.3 Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.4 Polarisation effects . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.5 Immersion oil . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Surface plasmons in magnetic environments 69
5.1 Surface plasmon resonance at constant magnetic field . . . . . . . . . 69
5.2 Surface plasmon resonance at constant angle . . . . . . . . . . . . . . 73
5.3 Surface plasmons in improved sample structure . . . . . . . . . . . . 76
5.4 Reflectivity at constant magnetic field . . . . . . . . . . . . . . . . . . 79
5.5 Transversal magneto-optical Kerr effect . . . . . . . . . . . . . . . . . 84
5.6 Magnetic anisotropy measurement with TMOKE . . . . . . . . . . . 88
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Conclusions and Outlook 93
A Simulation code for the surface plasmon resonance 101
B CAD sketches 105
C Prism correction 107
D Code for SPP simulation after Hermann et al. 111
Bibliography 122

Chapter 1
Introduction
Plasmons i.e., longitudinal density fluctuations of quasi-free electrons in a metal, are
studied since the early 20th century and can be directly derived from Maxwell’s equa-
tions [1, 2]. The first who theoretically investigated plasmons in thin metal films was
Ritchie [3], who started the field of surface plasmons with his pioneering work in 1957.
In the following years his work was verified by electron-loss spectroscopy experiments
and was theoretically extended [4, 5, 6, 7].
In 1968 Kretschmann and Raether introduced their method to excite surface plas-
mons by light, simply by bringing a metal film in contact with a (glass) prism and
coupling the light into the prism [8], allowing to record angle resolved surface plasmon
resonances [9]. Prior to the prism coupling method, it was thought no to be possible
to excite surface plasmons by light, as the dispersion relation of surface plasmons and
the dispersion relation of light do not intersect. But by coupling light into a prism
its momentum changes from ω/c to (ω/c) √ ε (0) (ε (0) is the dielectric constant of
the prism) and thus both dispersion relations, the one of light and the one of surface
plasmons, do cross. In another picture the excitation of surface plasmons is explained
by an evanescent wave, which occurs when the light is totally reflected at the prism-
metal interface. The evanescent wave has a phase velocity of v = c/( √ ε (0) sin θ0) (θ0
is the incident angle of the light with respect to the normal of the sample surface),
which fulfils the resonance condition of surface plasmons for θ0 = θSPP, the surface
plasmon resonance angle.
Today, the excitation of surface plasmons with light is a well established technique
and found its way into many applications, including biosensors [10, 11] or microscopy
[12]. Furthermore, with the possibility to pattern and characterise metals on the
nanometre scale a new field developed: plasmonics. In this field plasmons are coupled
1

2
Chapter 1
Introduction
via a grating into waveguides, metallic slabs embedded in a dielectric, what allows for
subwavelength optics [13].
Moreover, in recent years the interaction of an external applied magnetic field with
surface plasmons in multilayers, especially in Au/Co/Au multilayers, became another
focus of attention [14, 15, 16]. The corresponding experiments are based on theoretical
work by Chiu and Quinn [17] presented in 1972, where they theoretically described
the influence of a magnetic field on the dispersion relation of surface magneto-plasmon
waves in a metal. The influence of the magnetic field strongly depends on the orien-
tation of the field vector with respect to the incident plane of light and to the sample
surface. In this thesis, the transversal geometry is regarded, meaning that the mag-
netic field vector is parallel to the sample surface but perpendicular to the plane of
incidence of light, and hence the magnetic field vector is also perpendicular to the
propagation of the surface plasmon. In the presence of a magnetic field off-diagonal
components appear in the dielectric tensor, which depend on the so-called Voigt con-
stant [18]. Furthermore, this causes the surface plasmon dispersion relation to be
dependent on the propagation direction of the surface plasmon, what in turn affects
the surface plasmon resonance angle θSPP.
This magnetic field dependence of the surface plasmon resonance opens the way for
future applications such as magneto-plasmonic modulators [19], magneto-plasmonic
sensors [20], or optical isolators [21]. Due to this recent interest it is the aim of this
thesis to study angle resolved surface plasmon resonances in the Kretschmann-Raether
configuration in the presence of an external magnetic field. To this end, we assembled
a setup, to detect angle resolved surface plasmon resonances of single and multilayer
films. Using this setup we investigated the behaviour of surface plasmons in presence
of an external magnetic field.
The thesis is organised as follow:
In Chapter 2 the theoretical considerations of surface plasmons are briefly reviewed.
These considerations start with the Maxwell equations and yield the dispersion rela-
tion of surface plasmons. By comparing the dispersion relation of surface plasmons
with the dispersion relation of light, the condition for surface plasmon resonance as
function of light incident angle is obtained. With the then introduced concept of
the reflection of a multilayer structure, the surface plasmon resonance is simulated.
Simulations for different multilayers are shown and compared.

The setup to detect these angle dependent surface plasmon resonances and its com-
ponents are described in Chapter 3. Furthermore, in this chapter, calibrations of
the used detector and the alignment procedure are discussed.
Next, in Chapter 4 the experimental study of surface plasmon resonances in single-
and multilayer films is presented. The measured resonances are compared to the sim-
ulations from Chapter 2 and the differences between measurements and simulations
are critically discussed.
Then, the influence of a magnetic field on the surface plasmon resonance curves is
investigated for different samples in Chapter 5. The results obtained in this thesis
are compared with measurements from the literature.
In Chapter 6 a conclusion of the results is given and from this further experiments
are proposed.
3

4
Chapter 1
Introduction

Chapter 2
Theory
In this chapter a short overview of the theoretical basics of surface plasmons is given,
where the focus is on the aspects necessary to understand and interpret the experi-
mental results. At first, the nature of surface plasmons is discussed, leading to the
dispersion relation for surface plasmons. Next, the reflectivity of single and multi
layer systems is derived and simulations are presented and discussed. Finally, the
interactions of surface plasmons with an external magnetic field is considered.
2.1 Surface Plasmon Polaritons
Collective longitudinal oscillations of quasi-free electrons in a metal or semiconductor
are called plasmons. When these oscillations propagate along the interface of a metal
and a dielectric medium (or vacuum), they are called surface plasmons. For surface
plasmons the component of the electromagnetic wave perpendicular to the interface
decays exponentially, what can be described by an evanescent wave. Thus, the sur-
face plasmons are localised near the interface, giving rise to an enhancement of the
electromagnetic field [9, 22]. The enhancement can be explained and calculated by
regarding energy conservation [23].
2.2 Dispersion relation
Plasmons in general can be described by a plane wave.
A(r, t) = A0e i(kr − ωt) . (2.1)
Here, k = (kx, ky, kz) is the wave vector, r = (rx, ry, rz) the position vector, ω the
5

6
angular frequency, t the time, and A0 the amplitude of the plane wave A.
Chapter 2
Theory
Since surface plasmons are longitudinal oscillations propagating parallel to an inter-
face the wave vector can be divided into a part parallel to the interface (kx) and a
part perpendicular to the interface (kz), resulting in
A(r, t) = A0e i(kxx − ωt) e ikzz . (2.2)
Here, it is assumed that the interface lies in the x-y plane and the oscillations propa-
gate along the x-direction, see Fig. 2.1. Assuming a homogeneous, isotropic material
then without loss of generality the y-component of the wave vector (ky) can be treated
as zero. As surface plasmons are electromagnetic waves the plane wave expression for
Ei ~ e-|kz|
z
x
z
+
- -
+ + - - + + - - + + - -
Figure 2.1: Surface plasmon oscillations along an interface between a metal (ε (1) ) and a
dielectric (ε (2) ), where ε is the dielectric constant of each material. The interface
extends in the x-y plane at z = 0 and the oscillations are along the x-axis.
Further, the black arcs illustrate the electric component of the electromagnetic
wave perpendicular to the interface. On the left its exponential decay into both
media is indicated which is analogue to an evanescent wave. The penetration
depth of the evanescent wave into the metal depends on the Thomas-Fermi
screening length.
A has an electric character E and a magnetic character H. Since the surface plasma
oscillations propagate along the x-direction, and the evanescent part of the electric
field is along the z-direction the electric field can be written as
E (i) (r, t) =
⎛
⎜
⎝
E (i)
x
0
E (i)
z
⎞
z
ε1
ε2
⎟
⎠ e i(k(i)
x x − ωt) e ∓ik (i)
z z , (2.3)
y
x

Section 2.2
Dispersion relation 7
where i represents the layer, the metal (1) and the dielectric (2). In the situation
discussed here Ey can again without any loss of generality treated as zero.
Furthermore, the magnetic field must be perpendicular to the electric field, the mag-
netic field has only a component in y-direction
H (i) (r, t) =
⎛
⎜
⎝
0
H (i)
y
0
⎞
⎟
⎠ e i(k(i) x x − ωt) ∓ik
e (i)
z z
(2.4)
Here, the − sign in the exponent of the evanescent part (e ∓ik(i)
z z )is valid for the metal
(z < 0) and the + sign for the dielectric (z > 0). It has to be denoted that the
surface plasmons propagate in positive and negative x-directions. However, since
both directions are symmetrical only the positive direction is considered, i.e. kx > 0.
Moreover, as the interface lies at z = 0 it is assumed that kz > 0 for both directions
of z.
To get the dispersion relation for surface plasmons the fact is used that both the
electric (Eq. (2.3)) and magnetic field (Eq. (2.4)) have to fulfil the Maxwell equations
in vacuum [24].
∇ × E (i)
= − ∂B(i)
∂t
∇ × H (i) = ∂D(i)
∂t
= −µ0µ (i) ∂H(i)
∂t
(i) ∂E(i)
= ε0ε
∂t
(2.5)
(2.6)
ε0ε (i) ∇E (i) = ∇D (i) = 0 (2.7)
µ0µ (i) ∇H (i) = ∇B (i) = 0 (2.8)
with ε0 and µ0 being the dielectric and magnetic permeability in vacuum and ε (i) and
µ (i) being the dielectric and magnetic permeability in a medium i. The solutions have
to satisfy the continuity conditions at the interface.
E (1)
x = E (2)
x
H (1)
x = H (2)
x
(2.9)
(2.10)
ε (1) E (1)
z = ε (2) E (2)
z . (2.11)

8
Chapter 2
Theory
Here, µ (1) = µ (2) = 1 because without loss of generality a medium can always be
expressed by an effective electric permeability or an effective magnetic permeability
[25, 26].
Equation (2.9) and Eq. (2.10) yield
Hence, kx is continuous throughout the interface.
k (1)
x = k (2)
x = kx (2.12)
Substituting Eq. (2.3) and Eq. (2.4) into the Maxwell equations and using the con-
tinuity conditions besides the continuity of kx, the following two equations can be
obtained
k (1)
z k(2) z
+
ε (1) ε
(2) = 0 (2.13)
k (i) 2 2 (i)
z + kx = ε
ω
c
2
. (2.14)
Here, c = 1/ √ ε0µ0 is the speed of light in vacuum and in a medium i the speed of
light is cmedium = ε0ε (i) µ0µ (i) .
Using Eq. (2.14) and (2.14) the dispersion relation for surface plasmons can be derived
to
2 ε
kx = (1) ε (2)
ω 2
c
ε (2) + ε (1) . (2.15)
Now, regarding that in metals the dielectric functions are complex quantities, ε (1) will
read ε (1) = ε ′(1) + iε ′′(1) and the dispersion relation will be complex. Assuming a real
ω and ε ′(2) (ε ′′(2) = 0) and for the dielectric function in the metal ε ′′(1) < ε ′(1) , the
dispersion relation (Eq. (2.15)) can be approximated [27] by
kx = k ′ x + ik ′′
x = ω
ε
c
′(1) ε ′(2)
ε ′(1) ω
+
+ ε ′(2) c
ε ′(1) ε ′(2)
ε ′(1) + ε ′(2)
3
2 ε ′′(1)
. (2.16)
2ε ′(1)2
It can be seen that k ′ x becomes real for ε ′(1) < 0 and ε ′(1) > ε ′(2) . The imaginary
parts of ε (i) and kx represent the absorption of an electromagnetic wave or a surface
plasmon.
Since in metals the electric permeability depends on the frequency of the electromag-
netic wave, it has to be written as ε (1) (ω) = ε ′(1) (ω) + iε ′′(1) (ω).

Section 2.2
Dispersion relation 9
In metals the electrons can be treated as a free electron gas. The dielectric function
for a free electron gas can be calculated with the Drude formula[28, 5], giving
ε ′(1) (ω) = 1 − ne2
ε0m∗ω2 = 1 − ω2 p
ω
2 , (2.17)
with the electron density n, the elementary charge e, ε0 being here the vacuum per-
meability and m ∗ the effective mass of the electrons, and ωp is the plasma frequency
of metals,
ωp =
ne2 . (2.18)
ε0m∗ With nAu = 5, 9 × 10 28 m −3 [29] the plasma frequency becomes ωp = 1.36×10 16 rad/s.
Equations (2.17) and (2.15) can thus be rewritten
k ′ x = ω
c
ε′(2)
1 − ω2
p
ω
ε ′(2)
+ 1 − ω2 (2.19)
p
ω
This is the approximate dispersion relation for surface plasmons at a metal-dielectric
interface, where the complex part of the metal’s permeability is neglected.
In Fig. 2.2 the dispersion relation for an air-gold interface (ε ′(2) = 1) is plotted (black
line). It splits up into two branches with a forbidden gap in between. The upper
branch becomes for kx → 0 equal to the plasma frequency ωp (upper green line) in
Fig. 2.2(a) and increases for kx → ∞. Because ωp can be regarded as the plasma
frequency of volume plasmons the upper branch describes the dispersion relation for
volume plasmons [30]. The lower branch becomes zero for kx → 0. For kx → ∞ the
dispersion relation ω approaches the surface plasmon frequency (lower green line).
ωsp =
ωp
√ 1 + ε ′(2)
(2.20)
This is the limiting case when the real part of the metal’s dielectric function ap-
proaches the permeability of the dielectric ε ′(1) → −ε ′(2) . Or in other words the
damping part of the metal ε ′′(1) can be neglected. With Eq. (2.18) and E = ω the
energy of surface plasmons is in the order of Esp = 4.8 eV
So in the case of kx → ∞ the group velocity (∂ω/∂kx) and the phase velocity (ω/kx)
goto zero. Therefore, surface plasmon are localised longitudinal oscillations of the

10
electron density.
Chapter 2
Theory
Furthermore, from Eq. (2.20) it can be seen that ωSP < ωP and thus the energy of
surface plasmons is smaller than for volume plasmons. The shift of the resonance
frequency is due to depolarisation effects at the metal/air boundary [3, 31].
In principle a longitudinal oscillation like a surface plasmon can not be excited by a
transversal wave like light. This can be seen in Fig. 2.2(b), as the dispersion relation
for surface plasmons at a gold-air interface (black line) and the dispersion relation for
light in air (ε ′(0) = 1) with
ω =
ckx
√ ε ′(0) sin θ
(2.21)
(red solid line), do not cross. This does not change, even when then incident angle
(θ) is changed (red dashed line). When regarding surface plasmons their dispersion
relation lies right to the dispersion relation of light thus surface plasmons have a
larger wave vector than light. Therefore they are also labelled as "nonradiative"
surface plasmons [9].
However, this changes when using the so-called Kretschmann configuration [8] (see
Sect. 2.2.1). In this configuration the light does not directly illuminate the gold
surface. Instead, it is coupled through a glass prism (medium 0) and is reflected
at the prism-gold interface. Now, the dispersion relation of surface plasmons at the
gold-air interface and the dispersion relation of light, being coupled into a glass prism
(ε ′(0) = 1.514 2 ε ′′(0) = 0 [32]; blue line in Fig. 2.2(c), do cross 1 . And therefore it is
possible to excite surface plasmons with light. This does not change when the incident
angle θ is varied (dashed blue line). But since the surface plasmons are excited at the
gold-air interface and the light is reflected at the glass-gold interface, there must be a
coupling in between. This coupling is due to an evanescent wave which only appears
in the case of total internal reflection of light at the glass-gold interface. So surface
plasmons can only be excited in the Kretschmann configuration when the incident
angle of light is greater than the angle of total internal reflection θc (Eq. (2.23)) at
the glas-gold interface.
To show that no surface plasmons at the glass-gold interface are excited, the dispersion
relation for surface plasmons at a glass-gold interface is plotted in Fig. 2.2(d) (dashed
magenta line).
1 Since the surface plasmons are excited by an electromagnetic wave, they correctly should be called
surface plasmon polaritons for short SPP.

Section 2.2
Dispersion relation 11
Further, it is noted that surface plasmons can only be excited by p-polarised light
(electric field vector parallel to the incident plane). This is obvious since surface
plasmons are longitudinal oscillations and therefore only an electric field which has a
parallel component to the propagation direction, can excite those oscillations.
ω [s -1 ]
4x10 16
3x10 16
1x10 16
2x10 16
ω
p
ω
sp
0
4x10 16
3x10 16
2x10 16
1x10 16
k
(a)
(c)
k
εprism θ
εAu εdielectric kc/sinθ√ε prism
0 5x10 7
0
Au-Air
Au-Air
c sin(θ)
( )
/ √ε 0
( )
c/√ε 0
1x10 8
(b)
(d)
k
k
k
0 5x10 7
k x [m -1 ]
εAu εdielectric θ
ε θ
prism
εAu εdielectric kc/sinθ√ε prism
Au-Air
/
Au-Air
c sin(θ)
c sin(θ)
c
( )
/ √ε 0
( )
c/√ε 0
Au-Glass
1x10 8
4x10 16
3x10 16
2x10 16
1x10 16
0
4x10 16
3x10 16
2x10 16
1x10 16
Figure 2.2: Dispersion relation of surface plasmons. The black line ( )is the dispersion
relation for a gold-air interface. The red line ( ) is the dispersion relation for
light with k-vector parallel to the interface. The dotted red line ( ) is the
dispersion relation for light under the incident angle θ. The blue lines are the
dispersion relations for light coupled through a prism, parallel to the interface
( ) and under the incident angle θ ( ). And the magenta line ( ) is the
dispersion relation for a glas-gold interface. The inset in the upper left shows
the Kretschmann configuration with the k vectors for the incident light and
the surface plasmons.
2.2.1 Kretschmann-Raether configuration
As mentioned above it is possible to excite surface plasmons when the light is coupled
into a prism before it is reflected at the metal interface.
This method was first introduced by Erwin Kretschmann and Heinz Raether in 1968
0

12
Chapter 2
Theory
[8]. In this configuration a glass prism or glass half cylinder is brought into contact
with the metal interface (see Fig. 2.3).
When the light is travelling through the glass with ε ′(0) > 1 the momentum is
k
ε prism
ε Au
ε dielectric
θ
Figure 2.3: Kretschmann configuration. A prism is brought into contact with a metal
surface. The projection of the wave vector of light in the prism onto the surface
is kx. In the case of total reflection a evanescent wave propagates throughout
the metal interface and excites surface plasmons at the metal-air interface.
changed and the projection of the k-vector onto the surface is
k x
z
kx = ω √
ε ′(0) sin θ. (2.22)
c
θ is the incident angle with respect to the normal of the surface.
For angles greater than the angle of total internal reflection θc at the glass-metal in-
terface, an evanescent wave propagates through the metal. The angle of total internal
reflection can be calculated with Snell’s law [33]
θc = arcsin
n0
n1
x
(2.23)
with n0 the refractive index of the glass and n1 the real part of the complex refractive
index ñ of the metal. The complex refractive index ñ = n+iκ is linked to the dielectric
function by
ε ′ = n 2 − κ 2
and ε ′′ = 2nκ. (2.24)
For light coupled in a BK7 prism (n0 = nBK7 = 1.514 [32]) the total internal reflection
at an air interface occurs at θc = 41.3 ◦ . And the total internal reflection at an interface
BK7-gold (n1 = 0.2 [9]) occurs at θc = 7.6 ◦ .
The resonance angle θSPP of the surface plasmons can be calculated when Eq. (2.15)

Section 2.3
Reflectivity of surface plasmons 13
is equated with Eq. (2.22) what yields
θSPP = arcsin
1
ε ′(0)
ε (1) ε ′(2)
ε (1) . (2.25)
+ ε ′(2)
The imaginary part of the dielectric function ε ′′(1) describes the damping of light when
travelling through the metal. It depends on the thickness of the metal layer and it
is called radiation damping [34]. The radiation damping decrease exponentially with
the film thickness and therefore when assuming a thin film ε ′′(1) can be neglected.
With this assumption for a gold layer the resonance occurs at θSPP = 43.6 ◦ .
Since at resonance, surface plasmons are excited, there must be an attenuation of the
intensity of the totally reflected light in the prism. That is why the prism coupler is
also known as Attenuated Total Reflection coupler or short ATR coupler.
2.3 Reflectivity of surface plasmons
As mentioned in Sect. 2.2.1, the excitation of surface plasmons leads to a reduction
of the intensity of the reflected light. To calculate the angle dependency of the re-
flectivity of surface plasmons excited by an ATR device only the Fresnel equations
for p-polarised light with the wave vectors kx (Eq. (2.22)) and k (i)
z (Eq. (2.14)) are
needed [35].
Here, r p
ij
r p
ij
k(i) z /ε
= (i) − k (j)
z /ε (j)
k (i)
z /ε (i) + k (j)
z /ε (j)
and t p
ij = 1 + rp ij . (2.26)
is the reflection coefficient at an interface between media i and j, tp
ij
is the
transmission coefficient through medium j. The index i in tij says that the light
comes from medium i. For the sake of completeness the reflection and transmission
coefficients for s-polarised waves are
r s ij = k(i) z − k (j)
z
k (i)
z + k (j)
z
and t s ij = 2k(i) z
k (i)
z + k (j) . (2.27)
z

14
2.3.1 Reflectivity of a single layer
Chapter 2
Theory
The reflected electric field E ′ can be calculated by multiplying the incident electric
field E with a reflectivity matrix ˆ R.
Es′ Ep′
=
r ss r sp
r ps r pp
Es
E p
(2.28)
Here, s and p denote s- and p-polarised light, and the r ij are the Fresnel coefficients
for i-polarised incoming and j-polarised reflected light.
The reflectivity for p-polarised light is then
R =
E p′
E p
2
, (2.29)
where E p′ is the the reflected part of the electric field vector of a p-wave, and E p
is the electric field vector of the incident wave. As shown in Fig. 2.4 the light path
t 10
t 10
r 01
E
0
r 10 e iα
r 10 e iα
t 01 e iα
1 2
d
r 12 e iα
r 12 e iα
Figure 2.4: The sketch represents a system like it is used in an ATR coupler. Medium 0 is
the prism, medium 1 the metal and medium 2 air or vacuum. E is the amplitude
of the electric field of the incident light wave. rij and tij are the reflection and
transmission coefficients, respectively (Eq. (2.26)), and e iα is the damping and
phase shift of light when propagating through medium 1. The transmission
into medium 2 is neglected since only the reflectivity is investigated.
through a thin film 1 in contact with two semi-infinite media 0 and 2, E p′ is the sum

Section 2.3
Reflectivity of surface plasmons 15
of all reflected and transmitted light paths.
When E is the electric field of the incident wave, one part of it is reflected at the
interface 01 which yields a term (Er01) and the other part is transmitted giving a
term (Et01e iα ). With
e ik′
k = k ′ + ik ′′
describes a phase shift and e ik′′
⇒ e iα ≡ e ikd = e ik′ d + e ik ′′ d
(2.30)
a damping of the light when propagating through
medium 1. The transmitted part is now reflected at the interface 12 (Et01e iα r12e iα ). As
only the reflected part is of interest, the transmitted part through 12 is omitted. Now,
repeating the above procedure shows that the nth transmitted light wave through 10
is E0t01e iα (r12e iα r10e iα ) n t10. Using
E p′ can be calculated, to
r p
ij = −rp ji and t p
ij = 1 + rp ij , (2.31)
E p′ = E p r p
01 + E p t p
01e iα r p
12e iα t p
10
∞
n=0
r p
12e iα r p
10e iα
= E p r p
01 + E p (1 + r p
01)e iα r p
12e iα (1 − r p
01)
=
Ep (r p
01 + r p
12e2iα )
1 + r p
01r p
12e2iα Substituting Eq. (2.32) into Eq. (2.29) gives
R p
=
2
r p
01 + r p
12e2ikz1d 1 + r p
01r p
12e2ikz1d ∞
n=0
r p
12e iα r10e iα
(2.32)
(2.33)
(2.34)
2
. (2.35)
Equation (2.35) is the reflectivity of an ATR coupler consisting of a single metal layer
in contact with the prism on one side and air on the other. Note that for Eq. (2.35)
the prism is assumed to be semi-infinite.
Since kx is continuous throughout all the interfaces (see Eq. (2.12)) only the evanescent
part k (i)
z is damped phase shifted when propagating through the metal film.
For light with λ = 650 nm and a silver layer with thickness dAg = 50 nm the reflectivity

16
Chapter 2
Theory
curve shown in Fig. 2.5 can be calculated with Eq. (2.35). For the dielectric functions
ε (1) = εAg = −17 + i1.51 [36] and ε ′(0) = εBK7 = 1.514 2 is used. As evident from
p
re fle c tiv ity R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0 2 0 4 0 6 0 8 0
θ
λ
ε
ε
Figure 2.5: Simulation of the angular dependence of the reflectivity for a 50 nm thick silver
film. At θ ≈ 43 ◦ a clear resonance of the surface plasmons can be seen.
Fig. 2.5, at a light incident angle of θ = 43 ◦ a clear dip in the reflectivity can be
seen. At this angle the dispersion relations of the light and of the surface plasmons
at the silver-air interface match (see Fig. 2.2). The strong decrease of the reflected
light intensity can be explained with Eq. (2.35) and Fig. 2.4. The light reflected
into medium 0 is minimal when all light paths reflected into medium 0 interfere
destructively, e.g. 2α = π = 2k (i)
z d. As mentioned in Sect. 2.2.1 at angles greater
than the angle of total internal reflection θc an evanescent wave propagates throughout
the silver film. At the silver-air the light is then scattered back into the silver film.
The back scattered light is out of phase with the incoming light and thus interferes
destructively at the metal-prism interface, so that the reflectivity decreases. Until
now the surface of the metal was treated as smooth, so that no light can be emitted
into the air or vacuum. But real surfaces have a finite roughnesses which will have
an effect on the reflectivity. Thus they will be discussed in the next chapter before
continuing with the reflectivity of multilayer systems in Sect. 2.3.3.

Section 2.3
Reflectivity of surface plasmons 17
2.3.2 Surface plasmons at rough surfaces
For smooth surfaces, surface plasmons, excited at the metal-air interface, cannot
radiate light into air or vacuum as it is depicted in Fig. 2.2(b). There it is shown that
surface plasmons at an air-metal interface have a greater wave vector than light and
thus are "nonradiative" surface plasmons. This changes when the surface has afinite
roughnesses.
To understand why surface plasmons can radiate light into air when the surface is
rough, it has to be mentioned that surface plasmons can also be excited by a grating
coupler [37, 9, 22]. Then to the wave vector kx a ∆kx is added which depends on the
grating constant. The sum of both can then fulfil the dispersion relation.
kx + ∆kx = ω
c sin θ ± ∆kx = kSPP
(2.36)
With a grating coupler not only surface plasmons can be excited, but also the inverse
can take place, surface plasmons propagating along a grating can transform into light.
Of course, rough surfaces are not periodic like a grating but rather show a statistical
distribution. Nevertheless, due to surface roughness light is radiated from the metal
surface into air by surface plasmons [38, 39]. In a simple picture this can be explained
by scattering of surface plasmons at roughnesses. Due to the scattering the wave
vector of the surface plasmons changes so that it matches the light wave vector and
surface plasmons can transform into light. In a more mathematical consideration
a Fourier transformation of the statistical distribution yields not only one ∆kx like
for a grating coupler but a continuum of ∆kx. With this Fourier transformation
a correlation function f(∆kx) is introduced which is then solved for the ∆kx by
perturbation calculations [40, 9, 22].
The effects of surface roughnesses on the reflectivity R p are a broadening of the
resonance dip as well as a shift of θSPP to larger angles [41]. Further effects on R p will
be regarded when the measurements are dicsussed.
2.3.3 Reflectivity at a multilayer system
As shown in Sect. 2.3.1 the reflectivity for a single layer can easily be derived by
summing up all light paths in the layer. To calculate the reflectivity of a system
consisting of n layers, Fig. 2.4 has to be extended as shown in Fig. 2.6. Now, the
reflectivity at an interface i is given not only by the reflection coefficient of the ith

18
t 10
t 10
t 10
r 01
E
0
r 10 e iα 1
t 01 e iα 1
1 2
d 1
r 12 e iα 1
t 12 e iα 2
d 2
r 23 e iα 2
t n-1n e iα n
n
r nn+1 e iα n
d n
Chapter 2
Theory
Figure 2.6: The sketch shows the light propagation through a system with n layers. E is
again the amplitude of the electric field of the incident light wave. rij and tij are
the reflection and transmission coefficients, respectively (Eq. (2.26)), and e iαi
describes the damping and phase shift of light when propagating through media
i. The transmission into media n + 1 is neglected since only the reflectivity is
investigated.
interface, but the transmission of the backscattered light at the interfaces j > i also
contributes. To calculate the reflectivity for a multilayer system this becomes tedious.
A very elegant, alternative method to derive the reflectivity of a multilayer system
is the 2 × 2 matrix method which was introduced in 1980 by Pochi Yeh [42, 43],
an extension of the 2 × 2 matrix method introduced by Abéles [44]. The 2 × 2
matrix method is quite similar to the method of calculating the tunneling of a wave
through a barrier in quantum mechanics. Instead of considering the light path through
the layers, the matrix method is based on the light waves which are reflected and
transmitted at the interface. Fig. 2.7 illustrates the matrix method concept.
The electric field of the light wave is written as
E = E(z)e i(ωt−kxx) . (2.37)
This equation is the same as Eq. (2.3) only that the z-component and the amplitude
are combined as E(z). Now, E(z) can be split into a right-travelling and a left-
travelling wave

Section 2.3
Reflectivity of surface plasmons 19
E(z) = Re −ik(i) z z ik
+ Le (i)
z z
≡ A(z) + B(z). (2.38)
R and L are constants in each layer. A(z) represents the amplitude of the right-
travelling and B(z) the amplitude of the left-travelling wave. Further, as shown in
0 1 2
n 0
A 0
B 0
A ’
1
B ’
1
n 1
d
A 1
B 1
Figure 2.7: Illustration of the matrix method after Yeh [43]. Ai represents the amplitude
of a right-travelling wave and Bi of a left-travelling wave on the left side of the
interface ij. And A ′ j and B′ j are the amplitudes of a right- and a left-travelling
wave on the right side of the interface ij. ni is the complex refractive index of
the medium i.
Fig. 2.7, it is not only differentiated between right- and left-travelling waves but also
between waves on the right (with prime) and left (without prim) side of a given
interface. Regarding A (′)
i
by a matrix.
Ai
Bi
and B(′)
i
= ˆ D −1
i ˆ Dj
n 2
A ’
2
B ’
2
x
as column vectors, the amplitudes are connected
A ′ j
B ′ j
≡ ˆ Dij
where the ˆ Di are the so-called dynamical matrices given by
Di =
⎧⎛
⎝
⎪⎨
⎛
1 1
√ ε (i) cos θi − √ ε (i) cos θi
√ ε (i) − √ ε (i)
⎞
⎞
z
A ′ jB ′
j , (2.39)
⎠ for s-polarised light
⎝
⎪⎩
cos θi cos θi
⎠ for p-polarised light .
(2.40)

20
Chapter 2
Theory
Hereby, i = 1, 2, 3 and θi is the angle of incidence with respect to the normal for each
layer i. 2
The matrix ˆ Dij is the so-called transmission matrix. It links the amplitudes of the
waves to the right and to the left of the interface and thus describes the transmission
of the light wave through the interface:
ˆDij = 1
tij
1
rij
rij
1
, (2.41)
where rij and tij are the reflection and transmission curves for p- or s-polarised waves
Eq. (2.26) or Eq. (2.27), respectively.
The amplitudes inside one medium A (′)
i and B(′) i are linked by the so-called propagation
matrix ˆ Pi:
A ′ i
B ′ i
= ˆ Pi
Ai
Bi
≡
e iαi 0
0 e −iαi
Ai
Bi
, (2.42)
with the damping term αi = k (i)
z di in medium i of thickness di. The propagation
matrix describes the damped propagation of the light wave through medium i.
With the transmission matrix ˆ Dij (Eq. (2.41)) and the propagation matrix ˆ Pi (Eq. (2.42))
the incident light waves A1, B1 and the transmitted light waves A ′ 3,B ′ 3 in Fig. 2.7 can
be connected:
A1
B1
= ˆ D12 ˆ P2 ˆ D23
A ′ 3
B ′ 3
(2.43)
In the matrix formalism, a light wave propagation through a thin film thus is described
by the multiplication of transmission and propagation matrices. This can easily be
extended to multilayer systems. Each interface ij is described by a transmissions
matrix ˆ Dij and every layer i by a propagation matrix ˆ Pi. For a multilayer system as
studied in Fig. 2.8, the incident and transmitted wave amplitudes are linked by
where
A0
B0
= ˆ D01
N
i=1,j=i+1
ˆPi ˆ Dij
A ′ N
B ′ N
, (2.44)
2 The dynamical matrix for s-waves is not needed to describe the reflectivity of surface plasmons
but it shown for the sake of completeness

Section 2.3
Reflectivity of surface plasmons 21
0 1 2 . . . N-1 N N+1
n 0
A 0
B 0
A ’
1
B ’
1
n 1
d 1
A 1
B 1
A ’
2
B ’
2
n 2
d 2
. . .
. . .
. . .
. . .
n N-1
d N-1
A N-1
B N-1
A ’
N
B ’
N
n N
d N
A N
B N
n N+1
A ’
N+1
B ’
N+1
Figure 2.8: Here a sketch of a multilayer system with n layers is shown. 0 and N +1 are not
regarded as layers since they are assumed to be semi-infinite half spaces which
cover the multilayer system. Ai represents the amplitude of a right-travelling
wave and Bi the amplitude of a left-travelling wave, both on the left side of
the interface. A ′ i and B′ i
are the amplitudes of right- and left-travelling waves
on the right side of the interface. And ni is the complex refractive index of the
medium i.
ˆM =
M11 M12
M21 M22
= ˆ D01
N
i=1,j=i+1
ˆPi ˆ Dij
x
z
(2.45)
is the so-called transfer matrix ˆ M. The reflection coefficient of a multilayer system
can then be written as the ratio of the reflected wave amplitude (B0) and the incident
amplitude (A0):
r =
B0
With Eq. (2.44) and Eq. (2.45) this yields the reflectivity of a multilayer system.
A0
R = |r| 2
=
M21
M11
B ′ =0
2
(2.46)
(2.47)

22
2.4 Simulation of the reflectivity of surface
plasmons
Chapter 2
Theory
With the described matrix formalism in the previous chapter, it is now possible to
study the impact of surface plasmons on the reflectivity in single and multilayer films.
To study the evolution of R p for a single silver film as a function of film thickness
Eq. (2.47) is used. The simulation (cf. Appendix A) is done for light at λ = 650 nm
and for ε ′(0) = εBK7 = 1.514 2 [32], and ε (1) = εAg = −17 + i1.15 [36] and ε ′(2) = 1
for air. The results are summarised in Fig. 2.9. The reflectivity for dAg = 0 nm, i.e.,
R p
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
θ
θ
k SPP
0 20 40 60 80
θ
θ
θ
k SPP
k SPP
λ
ε
ε
ε
0 20 40 60 80
Figure 2.9: Reflectivity R p for ATR with silver films with (a) 0 nm, (b) 20 nm, (c) 50 nm,
and (d) 100 nm.
for a plain BK7 prism, is shown in Fig. 2.9(a). At 0 ◦ , R p ≈ 0.05 and R p decreases
with increasing angle. At θB ≈ 33.4 ◦ the reflectivity goes to zero. θB is the so-called
Brewster angle [45] where the p-polarised light is totally transmitted. The angle can
be obtained with the use of Snell’s law and is given by:

Section 2.4
Simulation of the reflectivity of surface plasmons 23
θB = arctan
n2
n1
. (2.48)
n1 and n2 are the refractive indices of the two media. For BK7 the Brewster angle is
about 33.4 ◦ [46] what matches with the simulated value.
With increasing θ the reflectivity increases and reaches 1 at θ ≈ 41.4 ◦ what is quite
similar to the literature value of 41.34 ◦ (Eq. (2.23)). Since the simulation is done
with a step size of 0.1 ◦ the difference between both values can be explained by the
lack of resolution in the simulation.
When the prism is covered by a Ag film of dAg = 20 nm (Fig. 2.9(b)) then the
reflectivity for θ = 0 ◦ goes up to 0.67. With increasing θ, R p slowly decreases and
goes through a minimum at θB = 38.6 ◦ and then increases and reaches a maximum
of 0.98 at θc, i.e. for total internal reflection. At somewhat larger θ, a minimum in
R p develops at the resonance angle (θSPP = 44.2 ◦ ) of the surface plasmons. This can
be explained with destructive interference (Sect. 2.3.1).
For silver films with dAg = 50 nm the surface plasmon resonance dip gets sharper
and θSPP shifts to 43 ◦ (Fig. 2.9(c)). While for dAg = 100 nm, a vanishing resonance
appears at θSPP = 42.9 ◦ (Fig. 2.9(d)).
The change of R p at θSPP can be explained by radiation damping [9]. As explained
in Sect. 2.3.1 the light at the metal-air interface is radiated light into the metal
which interferes with the incident light at the prism-metal interface. Since, the metal
has a complex dielectric function, this back-radiated light is damped. This effect is
called radiation damping. On increasing the thickness of the metal films the radiation
damping increases, so that the destructive interference between the incident and back-
radiated light at the prism-metal interface decreases and thus R p increases. As the
surface plasmon dip in R p is due to interference the depth of the surface plasmon dip
(the minimal R p value) thus decreases. Now, decreasing the metal film thickness at a
certain metal film thickness dopt the amplitude of the back-reflected light equals the
amplitude of the incident light and the phase of the back-reflected light is shifted by
π, so that R p = 0. When the thickness is decreased further d < dopt the effect of the
radiation damping increases again which yields an increase of R p .
For a single layer the optimal layer thickness dopt for which the R p goes to zero at
θ = θSPP can be calculated with a first order approximation equation reported by
Kretschmann [34].

24
dopt =
|ε ′(1) | − 1
4π |ε ′(1)
ln
|
4ε ′(1)2
ε ′′(1) (|ε ′(1) 2ε
| + 1)
(2) |ε ′(1) | (ε (2) − 1) − ε (2)
ε (2)2 + |ε ′(1) | (ε (2) − 1) − ε (2)
Chapter 2
Theory
λ . (2.49)
Using εBK7 = 1.514 2 , ε (1) = −17 + i1.15, and λ = 650 nm this equations yields a
optimal thickness of dopt = 47 nm.
Since Eq. (2.49) is only valid for single layers for multilayer the optimal thickness is
determined numerically. To verify the numerical simulation, first the optimal layer
thickness of single Ag film is determined. To this end, the R p (θ) evolution is calculated
for a series of dAg. For each dAg, the minimum is determined numerically, which
yields the curves shown in Fig. 2.10. Figure 2.10(a) shows that the ideal silver layer
p (θS P P )
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0 2 0 4 0 6 0 8 0 1 0 0
λ
ε
ε
ε
0 2 0 4 0 6 0 8 0
Figure 2.10: In (a) the reflectivity minimum R p (θSPP) is plotted against the thickness of the
silver film. At dAg ≈ 46 nm, R p goes to zero. Panel (b) shows the reflectivity
of a silver film with optimum layer thickness dopt = 46 nm at λ = 650 nm.
thickness dopt = 46 nm for λ = 650 nm, is in good agreement with the calculated value
dopt = 47 nm.
θ
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
p
R

Section 2.4
Simulation of the reflectivity of surface plasmons 25
2.4.1 Surface plasmons in different materials
It is evident from Fig. 2.9 that there is a sharp surface plasmon resonance in silver.
A similar analysis of other metals yields optimal layer thicknesses for Au, Ni, Co
and are dAu,opt = 47 nm, dNi,opt = 15 nm, and dCo,opt = 12 nm as shown in Fig. 2.11.
It shows that the optimal layer thickness for the magnetic materials Ni, and Co
p (θS P P )
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
1 .0
0 .8
0 .6
0 .4
0 .2
(a ) ε A u = -1 2 .1 + 1 .4 2 i
(b )
1 5 n m
A u
4 7 n m
ε N i = -1 0 .4 + 1 5 .3 4 i
N i
0 .0
0 2 0 4 0 6 0 8 0 1 0 0
(c )
1 2 n m
la y e r th ic k n e s s [n m ]
ε C o = -1 2 .9 + 1 8 .9 5 i
C o
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0 2 0 4 0 6 0 8 0 1 0 0
Figure 2.11: Simulation results for the reflectivity at θSPP plotted versus the layer thickness
of (a) Au, (b) Ni , and (c) Co. The minimum corresponds to the optimal layer
thickness which is for gold 47 nm, for nickel15 nm, and for cobalt 12 nm.
(15 nm, and 12 nm) are much smaller than those for the noble metals Au and Ag
(47 nm, and 46 nm). Calculating the optimal layer thickness with Eq. (2.49) yields
dAg,opt = 47.9 nm, dNi,opt = 15.7 nm, and dCo,opt = 13.3 nm. The deviation between
calculated and numerically values can again be explained by the resolution limit of
the simulation. Like mentioned before the step size of the simulation is 0.1 ◦ . An
other reason for the deviation might be that Eq. (2.49) is only an approximation.
The R p curves of the three materials for the respective dopt are shown in Fig. 2.12. For
the noble metal gold, a clear surface plasmon resonance at θSPP = 43.8 ◦ can be seen.
The R p (θ) evolution shows a similar behaviour like for a silver layer. In contrast, the

26
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0 .8
0 .6
0 .4
0 .2
0 .0
ε
1 .0 ε
0 2 0 4 0 6 0 8 0
θ
ε
0 2 0 4 0 6 0 8 0
Chapter 2
Theory
Figure 2.12: R p (θ) for the optimal layer thickness in (a) 47 nm Au , (b) 15 nm Ni , and (c)
12 nm Co.
reflectivity of the magnetic materials nickel and cobalt, do not show a clear, sharp
surface plasmon resonance. Instead, for θ > θc R p exhibits a very broad dip.
Comparing the dielectric functions for gold, εAu = −12 + i1.42 [9], for nickel εNi =
−10 + i15.34 [36], and for cobalt εCo = −13 + i18.95 [47] the main difference is the
imaginary part. For the magnetic materials the imaginary parts of the dielectric func-
tions are one order of magnitude larger than the ones for the noble metals. Since the
Im(ε) determines the absorption of light in the material this explains the broadening
of the resonance peak [9, 22].
Since a magnetic material has no clear sharp surface plasmons resonance, now noble
metal/magnetic metal heterostructures are investigated. Figure 2.13(b) shows that
a sharp surface plasmon resonance is expected in a 10 nm thick nickel film combined
with a 40 nm thick gold film. This resonance is not as clear as for a pure gold film,
but much clearer than for a 50 nm thick nickel film (Fig. 2.13(a)) or a 15 nm thick
nickel film (Fig. 2.12(b)).
The surface plasmon resonance becomes even sharper in a trilayer: noble metal/magnetic
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0

Section 2.5
Magnetic field dependence of surface plasmons 27
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
1 .0
0 .8
0 .6
0 .4
0 .2
ε
0 .0
2 0 4 0 6 0 8 0
θ
ε
ε
2 0 4 0 6 0 8 0
Figure 2.13: R p for multilayers consisting of a noble and a magnetic metal. Panel (a)
corresponds to 50 nm Ni, (b) to a double layer of Au40 nmNi10 nm, (c) to a
trilayer of Au20 nmCo5 nmAu5 nm, and (d) to a trilayer Au12 nmCo8 nmAu12 nm.
metal/noble metal (Fig. 2.13(c) and (d)). This is to be expected because the surface
plasmons are excited at the metal air interface and therefore the resonance at a noble
metal-air interface is much sharper than for a magnetic metal-air interface. But, in
such a trilayer surface plasmons are also excited at the noble metal/magnetic metal
interfaces [9, 3] and thus they will be sensitive to the magnetisation of the magnetic
material.
2.5 Magnetic field dependence of surface plasmons
Before discussing the interaction of surface plasmons with an external applied mag-
netic field, the interaction with light and a magnetic field is regarded.
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0

28
2.5.1 Interaction of an external magnetic field with light in
condensed matter
Chapter 2
Theory
Until now the dielectric function ε(ω) was treated as a scalar. This is only true
for isotropic media with no applied magnetic field, as then the dielectric tensor ˆε is
symmetric. The off-diagonal elements are zero and the diagonal elements are identical
[48, 49]. In a magnetised material (M = 0), ˆε becomes asymmetric with none-
vanishing off-diagonal elements. For a material with cubic symmetry ˆε is given to
first order in mi by [50]
⎛
1
⎜
ˆε = ε ⎝ iQmz
−iQmz
1
⎞
iQmy
⎟
−iQmx⎠
, (2.50)
−iQmy iQmx 1
with the projections mi = Mi/ |M|, of the components of the magnetisation M along
the x−, y−, and z−directions. ε is the dielectric constant for vanishing magnetisation,
and Q is the so-called Voigt constant [18] or gyroelectric constant [51]. . The latter
is given by [52, 53]
Q = i εxy
, (2.51)
and is a material constant which describes the magneto-optical rotation of the light
polarisation and is proportional to the induced magnetisation of the medium [54].
Using the dielectric tensor, Eq. (2.50), the optical behaviour of a magnetic material
can be described. This is exploited in magneto-optical experiments, i.e. Kerr- or
Faraday measurements [55, 56]. As only the so-called transversal geometry is of
interest in experiments related to surface plasmons, here it is only stated that there
εxx
are also a polar and longitudinal geometry, see Fig. 2.14.
In the transversal geometry (Fig. 2.14(c)) the Fresnel coefficients become [58, 59, 57]
r ss
t = n1 cos θ1 − n2 cos θ2
n1 cos θ1 + n2 cos θ2
(2.52)
r pp
t = n2 cos θ1 − n1 cos θ2
n2 cos θ1 + n1 cos θ2
+ i 2n1n2Q cos θ1 sin θ2
(n2 cos θ1 + n1 cos θ2) 2 (2.53)
r sp
t = r ps
t = 0. (2.54)

Section 2.5
Magnetic field dependence of surface plasmons 29
(a) (b)
(c)
M
polar longitudinal transversal
M
Figure 2.14: Illustration of the Kerr effect geometries with respect to the plane of incidence.
(a) In the polar Kerr effect the magnetization is oriented perpendicular to the
sample surface. (b) The magnetization is parallel to the the sample surface
and parallel to the plane of incidence in the longitudinal Kerr effect. (c)
In the transversal geometry the magnetization is within the film plane, but
perpendicular to the plane of incidence[57].
These equations show that in the transversal geometry there is no polarisation rotation
like in the polar and longitudinal configuration [59, 57]. The magnetisation (Q) only
influences the amplitudes of the light waves. Furthermore, magnetisation or magnetic
field effects will only be seen in the p-polarised component, because it is the only
component that depends on Q. In this component the magnetic field of the light
wave is parallel (antiparallel) to M.
2.5.2 Change of the dispersion relation of surface plasmons
Like the Kerr and Faraday effect show, electromagnetic waves interact with the mag-
netisation in a magnetic material. Since surface plasmons are also electromagnetic
waves there should also be an interaction with the magnetisation or an applied mag-
netic field.
The interaction of surface plasmons with an external magnetic field was first reported
by Chiu and Quinn [17] and will here be shortly reviewed.
When a magnetic field is applied kx is no longer symmetrical, regarding positive and
negative propagation directions. Thus, the lower branch of the dispersion relation
shown in Fig. 2.2(b) splits up into two branches, each for every propagation direction
(Fig. 2.15). To calculate the splitting of the dispersion relation ωB(kx) under an ap-
plied magnetic field parallel to the sample surface, the Maxwell equations with the
dielectric tensor have to be solved. Since Chiu and Quinn use a different notation for
their dielectric tensor, its components for the transversal geometry will be shown.
M

30
ω
|k x |
Chapter 2
Theory
Figure 2.15: The sketch shows the splitting of the dispersion relation under applied external
magnetic field (two red lines) against |kx|. The dashed curve in the middle is
the dispersion relation without an applied magnetic field.
ˆεChiu =
⎛
⎜
⎝
εxx εxy εxz
εyx εyy εyz
εzx εzy εzz
⎞
⎛
⎟ ⎜
⎠ = ⎜
⎝
1 − ω2 p
ω2 B
0 1 − ω2 p
ω2 B−ω2 c
−i ωc
ωB
ω 2 p
ω 2 B −ω2 c
0 0
i ωc
ωB
ω 2 p
ω 2 B −ω2 c
0 1 − ω2 p
ω 2 B −ω2 c
⎞
⎟
⎠
(2.55)
Here, ωc is the cyclotron frequency and εxx is equal to the dielectric function of a free
electron gas (Eq. (2.17)).
With ˆεChiu and the Maxwell equations, ωB is obtained.
εzz
k 2 x − ω 2 B (ε2 zz + ε 2 yz)ε −1
zz + ε (2) k2 x − ω 2 B εzz
k 2 x − ω 2 B ε(2) − kx|εyz| = 0 (2.56)
This yields the dispersion relation shown in Fig. 2.15. For kx = 0 both branches
are equal to zero. For small |kx| they both follow ωB = kx/ √ ε (2) , and split up with
increasing |kx|. In the limit |kx| ≫ ωp the dispersion relation becomes
ωB =
ω2 p
+
1 + ε (2)
ωc
2
∓
2
ωc
, (2.57)
2
where ∓ denotes both propagation directions. As the external field goes to zero ωc
also becomes zero and ωB(kx) in Eq. (2.57) becomes equal to ωsp (Eq. (2.20)).
The dispersion relation ωB(kx) derived by Chiu and Quinn [17] is only valid for single
layers. Up to now no complete theory for surface plasmons in multilayer with applied
magnetic field could be found. Only Gonzalez-Diaz [60] reported a formula for the
surface plasmon wave vector under applied field.

Section 2.5
Magnetic field dependence of surface plasmons 31
kx = k 0 x
1 + 2i
ε2
ε
− 1 ε + 1 ωdFQmy
. (2.58)
Here, k 0 x is the wave vector for surface plasmons in non magnetic media (Eq. (2.15))
and dF is the thickness of the magnetic layer. The equation shows that for surface
plasmons in a magnetic material a imaginary term in Q must be added to the non
magnetic wave vector k 0 x.
But Eq. (2.58) is only a rough estimation, since it is assumed that the dielectric
constant is constant throughout the multilayer. But as discussed in Sect. 2.4.1 the
dielectric functions for noble metals and magnetic metals differ. Especially the imagi-
nary parts show huge differences, which have a strong effect on R p . Thus, the equation
reported by [60] will have a non negligible error.
As explained in Sect. 2.1 surface plasmons are localised oscillations and due to this
localisation the electric and magnetic field of the incoming light wave are enhanced
[22]. That is why with surface plasmons it is expected to see an enhancement of the
Kerr effect [15]. The field enhancement due to surface plasmons is calulated in the
next section.
2.5.3 Surface plasmon enhanced electromagnetic field
The electromagnetic field reaches it’s maximum when the reflectivity reaches is low-
est value what is a direct consequence of energy conservation [23]. The maximum
enhancement can be described by dividing the field at the metal-air interface by the
incoming field for d = dopt [9, 27, 61].
|H 21
y | 2
|H 01
y | 2
max
= ε′(2)
ε ′(0)
21 2 |E |
|E 01 | 2
max
= ε′(2)
ε ′(0)
1
ε ′(2)
2|ε ′(1) | 2
ε ′′(1)
|ε ′(1) |(ε ′(0) − 1) − ε ′(0)
1 + |ε ′(1) |
≡ ε′(2) el
T
ε ′(0) max (2.59)
T el
max is the maximum transmitted electric field at the metal-air interface.
For ε (2) = 1 (air), ε (0) = 1.5142 (BK7), and λ = 650 nm, T el
max becomes T el
max, Ag
T el
max, Au
= 57, T el
max, Ni
= 4, and T el
max, Co
= 124,
= 5. The greatest enhancement is reached
for noble metals. But a field enhancement by a factor of 4 to 5 is still present also
in magnetic materials. When combining a noble metal with a magnetic metal, the

32
Chapter 2
Theory
enhancement will be in between the noble metal and magnetic material enhancement.
The enhancement for a combined structure can only be calculated with Eq. (2.59)
when a mean ε for the structure is defined. Or when the thickness of the magnetic
material is small compared to the noble metal so that the dielectric function of the
magnetic material can be neglected. Thus the dielectric function of the hole structure
is equal to the noble metal dielectric function [60]. In this case the enhancement in a
combined structure should be same like for a pure noble metal.
2.6 Summary
In this chapter a basic idea of surface plasmons in single noble metal layer as well as in
multilayer with noble metal/magnetic metal/noble metal is conveyed. It is discussed
how to excited surface plasmons optically by using the Kretschmann configuration.
Further, a method to calculate the reflectivity of single layer is described. Since
this method could not easily be extended to multilayer, the Yeh’s matrix method is
introduced. Then this method is applied to show and discuss simulations of different
single layer and multilayer systems, consisting out of noble metals, magnetic metals
and combinations of both. With these simulation a way how to obtain an optimal
layer thickness, so that the R p (θSPP) goes to zero, is explained. Thus the behaviour
of R p can be explained in a macroscopic picture by destructive interference of back-
scattered light, and in a mesoscopic picture by excitation of surface plasmons.
Next, the interaction of light and condensed matter with magnetisation or under
applied magnetic field is discussed, resulting in the Faraday or Kerr effect. Then,
some ideas how a magnetisation or an applied field affects surface plasmons are given.
One is the calculation done by Chiu and Quinn which results in a splitting of the
dispersion relation. And the other is given by Gonzalez-Diaz, where only a linear
magnetic field dependent term is added to the normal wave vector.
Finally, the field enhancement due to surface plasmons is discussed.

Chapter 3
Excitation of surface plasmon
polaritons
In the beginning of this chapter the setup is explained and then a description of
the components used in the experiments is given. Next, some calibrations of the
photo-detector and laser are shown. Finally, the alignment of the setup is described.
3.1 Experimental setup
In this thesis, surface plasmons are studied in the Kretschmann-Raether configuration
[9]. The experimental setup is sketched in Fig. 3.1 and a photograph of the setup is
shown in Fig. 3.2.
The light emitted by a laser diode (λ = 650 nm, output power = 5 mW) is attenuated
by a neutral density (ND = 1) filter. The laser beam then is split into two beams:
a weak beam and a main beam. The weak beam is detected by the photodiode de-
tector (detector 1) in order to monitor the light intensity. The main beam is linearly
polarised by a Glan Thompson polariser, and then is incident onto the prism, where
most of it is reflected. The reflected light passes a second Glan Thompson system,
which allows for polarisation analysis. Finally, the light is detected in the detector
2. In front of each detector a bandpass filter (central wavelength (CWL) = 650 nm,
FWHM = 10 nm) is mounted.
In the following a few more details concerning the individual components are given :
33

34
LASER
lter
beam
splitter polariser
detector 1
Chapter 3
Excitation of surface plasmon polaritons
band-pass lter
θ
θ
magnet
θ - 2θ table
detector 2
analyser
Figure 3.1: Sketch of the surface plasmon setup built in this thesis.
magnet
detector
analyser
hall probe
stepper motor
θ - 2θ table
polariser
bandpass lter
detector
ND lter
laser diode
beam splitter
Figure 3.2: Photograph of the setup which was assembled during this thesis.
Light source
The light source is a laser diode module from Roithner Lasertechnik with a wavelength
of 650 nm and 5 mW output power. The diode is supplied by an Agilent E3630A power
supply and is operated with 5 V. A ND = 1 neutral density filter is mounted in front
of the diode, so that only 0.5 mW passes the ND filter.

Section 3.1
Experimental setup 35
Polariser/Analyser
Calcite Glan Thompson polarisers are used as polariser and analyser. They consist
of two calcite prisms which are glued at the base side. Since calcite is a birefringent
material, in this configuration only one polarisation direction is transmitted. The
used polarisers are from Thorlabs and have an extinction ratio of ε = 10 −6 .
Sample preparation
The metal films investigated are deposited onto 150 µm thick glass microscope slides
made of D263M glass from Menzel with an refractive index of nD263M ≈ nBK7 = 1.514 1
[62]. Prior to the thin film deposition, these were cleaned with isopropanol in an
ultrasonic bath for a few minutes. The metal films were deposited via electron beam
evaporation (E-VAP, described in [63]) at room temperature with a base pressure of
1 × 10 −8 mbar and with deposition rates of 1 − 2 Å/s. The deposited film thickness is
controlled by an oscillating crystal which has an accuracy of ≈ 0.5% [64].
Prism
For the ATR coupler in this thesis, a BK7 (εBK7 = 1.514 2 [32]) right angle prism is
used. Two different sizes are used, a prism with a side length of 20 mm and a prism
with a side length of 5 mm. The main measurements were done with the 5 mm prism,
since it is much easier to handle than the 20 mm prism. It also allows a smaller gap
size in the used electromagnet. A picture of the 5 mm prism is shown in Fig. 3.3.
To be able to excite surface plasmons, the metal films have to be in contact with the
prism. This is done by using an index matching immersion oil from Richard-Allan
Scientific with nd = 1.515 ≈ nd, BK7 = 1.516 2 . To remove the immersion oil from the
prism, it is cleaned with isopropanol in an ultrasonic bath.
Rotation stage
As a rotation stage an old x-ray θ − 2θ table from Rich. Seifert & Co. is used. It is
made out of two independently turnable stages, one inner stage on which the prism
is mounted and an outer stage on which the detector is mounted. Thus the θ − 2θ
condition required can be achieved. To drive the rotation stages two stepper motors
1n refers to λ = 650 nm. If no other wavelength is stated, then n and ε will always refer to
λ = 650 nm.
2The index d in n denotes λ = 587.6 nm

36
(a)
εsubstrate εCo ε prism
ε Au
εAu εair 5 mm
Chapter 3
Excitation of surface plasmon polaritons
(b)
5 mm
Figure 3.3: (a) Sketch of the 5mm prism with a trilayer sample (the dimensions are not
to scale). (b) Photograph of the 5 mm prism and a trilayer sample which is
brought into contact with the prism by an immersion oil.
(AS411-E) with position controllers (SMCI33) from Nanotec were used. The stepper
motors are 1.8 ◦ stepper motors operated in the full step mode and since 200 steps
are needed for turning the table 1 ◦ , a mechanical angle resolution of 0.005 ◦ should
be possible. When assuming an error of ±1 steps a resolution of 0.01 ◦ should still be
feasible. In this thesis the smallest angle step size used is 0.1 ◦ .
To place the prism in the optical path a prism stage was designed, which consists of
two parts. One part is designed to fit directly onto the inner rotation stage. And the
other part is designed to adjust the prism’s position. The second part has to meet
two requirements: one is to simplify the adjustment of the prism in the optical path
and the other is that the prism is positioned on the rotation axis. To simplify the
positioning of the prism in the optical path, there is a mark running in radial direc-
tion from the rotation axis, allowing for the alignment of the prism (see Fig. 3.4 and
Appendix B). To fulfil the second requirement, perpendicular to the mark there is a
step over the centre of rotation at which the prism can be aligned 3 . The two parts
of the prism stage are connected by distance pieces, which are mounted in between
of both stages.
Further, the rotation stage has three support structures which can be rotated indi-
vidually. With these the height and the decline of the table can be adjusted.
3 Actually it is 1 mm off-axis, because in the beginning it was assumed that the sample, which is
mounted on the back of the prism, has a thickness of 1 mm

Section 3.1
Experimental setup 37
prism
centre of
rotation
incident laser
beam
step
mark
Figure 3.4: The figure shows a sketch of the prism stage and how the prism can be adjusted
with the help of the mark and the step. Further, the black vertical line
represents the centre of rotation and the red line the incident laser beam.
Magnetic field
The magnetic field is produced by a home-built electromagnet.
The requirements for the magnet are a homogeneous magnetic field of µ0H > 20 mT,
as the coercive field of thin Co film is µ0HC ≈ 1 − 20 mT 4 [65, 66, 67].
To calculate the maximum magnetic field in the air gap of the electromagnet, the
concept of a magnetic circuit is used [68]. To this end we start from Ampère’s law
s
Hds = nI. (3.1)
Here H is the magnetic field, n the number of turns, I the current through the coil,
and s the integration path (cf. Fig. 3.5(a)). This yields with H = B/(µ0µr)
B =
lF e
µ0µr
nI
+ gair
µ0
, (3.2)
where µ0 = 4π × 10 −7 As/Vm is the vacuum permeability and µr ≈ 50 5 [71] the per-
meability in a medium, in our case steel.
4 µ0HC, strongly depends on the film thickness
5 µr can vary for different iron alloys over some orders of magnitude [69, 70]. Here µr of a steel with
0.9 % C is chosen, since this is a common used steel type.

38
Chapter 3
Excitation of surface plasmon polaritons
As the inner part of the rotation stage has a diameter of ≈ 10 cm, the total length
lFe of the yoke is lFe ∼ = 3 × 10 cm = 30 cm. Moreover, for the 5 mm prism, an air gap
of 5...10 mm is mandatory. Thus, lFe = 0.3 m and gair = 0.01 m are determined by
the setup, hence only the product nI is a free parameter. For B = 20 mT, Eq. (3.2)
yields nI = 255. In this calculation losses due to stray fields are neglected. These
losses can be approximated with the ratio of the air gap length (gair) to the iron yoke
thickness (d) and are here ≈ 50 % (gair = 10 mm and dyoke = 20 mm) [68]. Further, a
field µ0H > µ0HC is desired, which is arbitrarily chosen to be 30 mT. Thus Eq. (3.2)
with B = 60 mT yields nI ≈ 820, with a current of I ∼ = 2 A roughly 410 turns are
needed.
Following this rough estimation, 500 turns with a 0.5 mm diameter copper wire, are
used (Fig. 3.5(b)). To prevent the wire from Joule heating and allowing a continu-
ous operation the VDE 0100 − 430 [72] recommends a maximum current density of
6 A/mm 2 for a copper wire with 1 mm diameter. With I = 2 A only 4 A/mm 2 are
reached and thus even with 0.5 mm diameter copper wire there should not be any
problem with Joule heating.
The electromagnet yields a maximum field of about 55 mT at the yoke and about
30 mT in the middle of the gap (for I = 2 A). The inhomogeneity can be explained
by the gap size. To achieve a homogeneous field in the gap, the gap should be much
smaller than the thickness of the yoke (gair ≪ d). For the yoke a steel of 20 mm×20 mm
thickness is used. This is about size of the air gap (gair = 10 mm). But for this thesis
the homogeneity is acceptable.
The magnet is driven by bipolar BOP 20 − 10 power supply from Kepco. The mag-
netic field is measured and controlled with a 475 DSP gaussmeter from Lakeshore.
During measurements the hall probe is positioned behind the sample and is roughly
adjusted on same height as the laser. Via an analog output the 475 DSP controls the
output current of the BOP 20 − 10. Thus, the magnetic field desired is always indeed
applied, and the magnetic hysteresis of the Fe yoke is "automatically" compensated.
Table 3.1 summarises the most important specifications of the built electromagnet.
Photo-detectors
The photo-detectors used in this thesis are built according to a design which was
developed by Reinhard Roßner during his student work a the WMI [73].
They consist of a photodiode (BP104S from Osram Opto Semiconductors) and an

Section 3.1
Experimental setup 39
l 1,Fe
I
l 2,Fe
(a) 59 mm
s
d
d
l 3,Fe
g air
100 mm
20 mm
(b)
11.2 mm
Figure 3.5: (a) Sketch of the electromagnet to estimate the number of turns. li,Fe are the
lengths of the segments of the iron yoke and gair is the gap size and d = 20 mm
is the lateral dimension of the quadratic iron yoke. The red curve depicts the
integration path s.
(b) Photograph of the electromagnet with 500 turns.
length of the yoke lFe = l1,Fe + 2l2,Fe + 2l3,Fe 30.9 cm
length of the gap gair
1.1 cm
number of turns 500
maximum field at the yoke (I = 2 A) 55 mT
maximum field in the middle (I = 2 A) 30 mT
Table 3.1: Specifications of the electromagnet
operational amplifier (OPA627B from Burr - Brown). Tables 3.2 and 3.3 give the
relevant parameters of the photodiode and the operational amplifier.
The photo-detector is driven in the low gain mode. This means that R2 = 1 MΩ is
used (see the wiring diagram in [73]). Again the E3630A power supply from Agilent
is used which also allows to supply the photo-detector with ±10 V, what should be
the optimal voltage [73].

40
Chapter 3
Excitation of surface plasmon polaritons
radiant sensitive area 4.84 mm 2
spectral sensitivity S470 nm
0.42 A/W
noise equivalent power NEP, V = 10 V, λ = 850 nm 3.6 × 10 −14 W/ √ Hz
dark current IR
Table 3.2: Specifications of the photodiode (BP104S)[74]
input bias current 1 pA
input offset voltage 40 - 130 µV
noise density, 10 kHz 4.5 nV/ √ Hz
voltage noise, BW= 0.1 to 10 Hz 0.6 µV
current noise, BW= 0.1 to 10 Hz 30 fA
2 nA
Table 3.3: Specifications of the operational amplifier (OPA627B) [75]
Other components
Besides the above mentioned components bandpass filters, neutral density (ND) filter,
and a beam splitter are used. The bandpass filters are specified for 650 nm with
F W HM = 10 nm. They are mounted in front of the detectors, so that only the light
of the laser diode is detected, thus the photo-detectors are insensitive to the light in
the lab (except for λ ≈ 650 nm).
The ND filter is mounted in front of the diode to attenuate the emitted laser light.
The beam splitter used in this setup is just a piece of glass, which is mounted in a
lens holder and turned 45 ◦ to the optical path. The light intensity is about 0.1 % of
the laser intensity after the ND filter 6 .
Measurement software
The stepper motors are controlled by a software, written in LabVIEW, which was
developed with the help of M. Althammer, during the thesis. With the software it is
6 The value was determined by calculating the difference in the detector output signal with and
without the beam splitter being in the optical path, assuming that the photodiode signal scales
linearly with the incoming laser intensity.

Section 3.2
Calibration 41
possible to control each motor separately or to move both stages simultaneously in
a θ − 2θ mode. The photo-detectors are connected to voltmeters which then can be
read out by the software.
This software then is implemented into an existing program which allows to control
the magnetic field and to read out multiple volt- and sourcemeters, simultaneously.
3.2 Calibration
Photodiode
To test whether the output signal of the photodiode changes linearly as a function of
the light intensity, the output voltage of the photo-detector is measured for different
laser intensities, obtained by bringing different ND filters into the light path. Fig-
ure 3.6 shows a double logarithmic plot of the measured photo-detector signal Uout
against the laser power (). Comparing the data with the linear fit (red line), it can
be seen that for a laser power ≥ 0.05 mW Uout scales linearly with the laser power
and for a laser power < 0.05 mW Uout slightly deviates from the linear behaviour.
But from the table in the upper left corner in Fig. 3.6 it can be seen that the error of
the linear fit is only about 10% for the intercept and negligible ≈ 0.5% for the slope.
In the experiments the used laser power will be 0.5 mW and thus it can be assumed
that the photo-detector output voltage Uout scales linearly with the laser intensity.
With this assumption only in the area of the surface plasmon resonance where the
reflectivity will go close to zero there will be a small error. This will be important,
later, when measurements are analysed and compared to simulations.
Photo-detector noise
Before measuring the noise floor of the built photo-detector, the theoretical noise
floor will be calculated with the in Tab. 3.2 and Tab. 3.3 given parameters. For this
purpose the important parameter is the noise density.
As the the photo-detector is operated with 10 V in the low gain mode (R2 = 1 MΩ),
then the NEP = 3.6 × 10 −14 W/ √ Hz of the photodiode is equal to a noise density
of 3.6 nV/ √ Hz. This is smaller than the noise density of the operational amplifier
4.5 nV/ √ Hz, and thus will be the limiting element in the circuit. To circumvent this,
the photo-detector could be operated in the high gain mode (R2 = 10 MΩ), hence the
noise density would be 36 nV/ √ Hz. But then for great signal intensities the photo-

42
p h o to -d e te c to r s ig n a l U o u t [m V ]
1 0
1
0 .1
Equation y = a + b*x
Chapter 3
Excitation of surface plasmon polaritons
Value Standard Error
a 6.39962E-5 6.77783E-6
b 0.0022 1.3905E-5
m e a s u re m e n t
lin e a r fit
1 E -3 0 .0 1 0 .1 1 1 0
la s e r p o w e r [m W ]
Figure 3.6: The output voltage of the photodetector is measured as a function of the incoming
laser power. The laser power is changed by changing the ND filters in
the optical path. The black squares () are the experimental data, the red line
is a linear fit.
detector would be close to the saturation region. Therefore, will the photo-detector be
driven in the low gain mode, accepting that the operational amplifier is the limiting
element.
Now, the noise level of the photo-detector and the laser diode are measured. For ac-
curate calibrations of the photo-detector it is referred to the work of Reinhard Roßner
[73].
But prior, to understand the measured signal, the operation amplifier is shortly dis-
cussed (for a detailed discussion see [73]). As the here used operation amplifier con-
verts a photocurrent into a voltage it is called current-to-voltage converter and the
output voltage Uout linearly depends on the input current Iin
Uout = −RIin. (3.3)
R is the feedback resistance in the amplifier circuit, see Fig. 3.7. The output current
Iout of the photodiode is linked to the spectral sensitivity Sλ and the input power P

Section 3.2
Calibration 43
I in
-
+
R
Figure 3.7: The circuit diagram shows an operation amplifier operated as a current-tovoltage
converter
via
U out
Iout = SλP. (3.4)
The sensitivity is limited by dark noise which is equal to the noise when no light
illuminates the photodiode. To quantify the dark noise, a cover is mounted in front
of the detector, so that no light can reach the photodiode. Figure 3.8 shows the
corresponding output voltage of the photo-detector as a function of time. The graph
shows a background signal of Uout = 74.9 µV with a standard deviation of ∆Uout =
±3.20 µV. Additional to the above mentioned noise sources the power supply with
Urms = 350 µV [76] and the voltmeter with Urms = 1 µV [77] will contributed to the
background signal. The noise of the power supply will have an effect on both the
measured noise floor and the background signal as the power supply is connected to
the voltage inputs of the amplifier and to the photodiode. The noise of the voltmeter
can be neglected, since it is much smaller than the background signal.
To detect a change in laser power this change must yield an output signal greater
than the noise floor (Uout > ±3.20µV). With Eq. (3.3) and R = 1 MΩ it follows that
the change in input current must be Iin > ∓3.20 × 10 −12 A. Assuming that the input
current is equal to the output current of the photodiode, the minimal change in laser
power that can be detected is P > ±0.008 nW, considering that Sλ = 0.42 A/W (Tab.
3.3).
With the smallest detectable change in laser power of P > ±0.008 nW a noise density
for the setup can be calculated. With the bandwidth of the SIM 970 of BW = 3.6 Hz,

44
9 0
8 5
8 0
7 5
7 0
6 5
6 0
Chapter 3
Excitation of surface plasmon polaritons
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
tim e [s ]
Figure 3.8: Dark signal of the photo-detector over time.
an applied voltage of 10 V, and R2 = 10 MΩ this will yield ≈ 0.4 µV/ √ Hz.
Laser stability
Having established the noise floor of the photo-detector it is now possible to test the
stability of the laser diode. For this purpose the laser diode is focused on the reference
photo-detector (detector 1 in Fig. 3.1). The detected Uout as a function of time for
the illuminated and the dark photo-detector is shown in Fig. 3.9. The figure shows
for the illuminated photo-detector a photovoltage of about 1.6 mV with a standard
deviation of ±2.9 µV. This is about the same noise floor as for the dark detector,
hence the laser diode does hardly contribute to the noise floor.
The noise of about ±2.9 µV corresponds to ≈ ±0.2% of the background signal of
about 1.6 mV. As mentioned before, Uout scales linearly with the laser intensity.
Thus, a ±0.2% change in Uout is acceptable, compared to the change in reflectivity of
the simulated curves in Sect. 2.4 which is at least 1%.

Section 3.2
Calibration 45
U o u t [m V ]
1 .6 2
1 .6 1
1 .6 0
1 .5 9
la s e r
d a rk
2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
tim e [s ]
Figure 3.9: The black curve shows Uout as a function of time, when the laser diode illuminates
the reference photodiode. The red curve shows the dark signal of the
photo-detector.
Resolution limit of the voltmeter
So far, measurements are discussed for which the output voltage of the photo-detector
was recorded directly by a 195A voltmeter from Keithley. However, the surface plas-
mons experiments typically require the measurement of small intensity changes on a
large constant background intensity. To circumvent limitations due to the finite digi-
talisation depth a differential (A-B) technique is used. To this end, a potentiometer
is connected in parallel to the power supply of the laser diode. This allows to source
a constant voltage of 0 ≤ UB ≤ 5 V to the input B of a JFET preamplifier (SIM 910)
from Stanford Research Systems. The output of the photo-detector is connected to
input A, and the output of the preamplifier is connected to a SIM 970 voltmeter from
Stanford Research Systems, as shown in Fig. 3.10. This technique allows to subtract a
constant level from the detected signal, so that only the smaller difference Uout − Uref
is measured. Because this difference is much smaller than the signal itself it is possible
to increase the signal resolution and circumvent digitalisation issues.
0 .0 9
0 .0 8
0 .0 7
0 .0 6
U o u t [m V ]

46
5 V
0 V
U out
Chapter 3
Excitation of surface plasmon polaritons
SIM 910
A
B
SIM 970
V
Figure 3.10: Parallel to the power supply of the laser diode a potentiometer is connected.
The output of the potentiometer is connected to the input B of the JFET
preamplifier SIM 910 via a coaxial cable. Uout from the detector is connected
to the input A of the preamplifier.
3.3 Alignment procedure
To measure the reflectivity of the metal films in the Kretschmann-Raether configura-
tion over a wide range of angles θ it is important that the setup is adjusted properly.
In particular, the centre of the rotation table and the prism must be aligned to the
laser beam.
As the setup is assembled on an optical table, the optical components can be mounted
and aligned easily. To ensure that the laser beam goes through the centre of rotation
of the θ − 2θ, stage pinholes are mounted onto the rotation stage, and then both pin-
holes and rotation stage are adjusted so that the laser beam goes through all pinholes,
see Fig. 3.11. Next, the prism stage is mounted on the rotation table. To be able to
define an incident angle, a reference position for the stage has to be defined. To this
end, the stage is rotated until the laser beam is parallel the mark of the prism stage,
see Sect. 3.1 and Fig. 3.4. This will be the θtheo = 0 ◦ position for the prism.
When the prism stage is adjusted the position of the detector can be defined. For this,
the detector arm is rotated until the laser beam hits the middle of the photodiode.
This will be the θtheo = 180 ◦ position for the detector.
After the θtheo positions of the prism stage and the detector are defined, the prism
itself is aligned parallel to the step on the prism stage, in such a way that the laser
beam hits the apex of the prism. When this is the case, behind the prism two equally
bright spots will appear. If the prism is slightly tilted a backscattered reflection in
the direction of the incoming beam appears.
Now, the prism stage is rotated by 45 ◦ so that the laser beam is perpendicular to the

Section 3.3
Alignment procedure 47
Figure 3.11: Picture of the configuration which is used to align the rotation stage. This is
done by using a set of pinholes. The red line symbolises the laser beam.
side of the prism. In this case neither the incoming nor the outgoing laser beam will
be refracted at the prism-air interface. When rotating the detector by 90 ◦ the outgo-
ing laser beam hits the photo diode. This will be the θmes = 0 ◦ position for the prism
and the detector during the measurement, see also Fig. 3.2. For the measurement
angle θmes applies
θ ≡ θtheo = θmes + 45 ◦ , (3.5)
where θtheo is the angle shown in the simulations. For the following graphs only the
θtheo will be shown.
Now, to adjust the polarisers, the polariser is rotated such that the transmission line
is parallel to the plane of incident, this is equivalent to p-polarised light. The analyser
is rotated until no light will pass through, this is equivalent to the s-polarised position.
This is done iteratively until a minimum in the detected Uout is reached.
Prism correction
During the measurement the prism is rotated, and therefore the laser beam will not
be perpendicular to the side of the prism anymore. Thus, the laser beam will be
refracted at the prism-air interface, so that its reflection point on the hypotenuse will
not be in the middle of the hypotenuse anymore (cf. Fig. 3.12(b)). The reflected

48
Chapter 3
Excitation of surface plasmon polaritons
beam then is not along 2θ, but instead is shifted, see red line in Fig. 3.12(b). With
increasing incident angle θ the reflected laser beam will be shifted so far that it does no
longer hit the detector anymore, see Fig. 3.12. Thus, the detector position has to be
corrected. The offset can be calculated and implemented into the software (Appendix
(a) (b)
Figure 3.12: The figure shows the light path when the prism is rotated about θ. In (a) the
bluish prism is the normal case, when the light is perpendicular to the side of
the prism. In (b) the grayish prism is rotated about θ. Where the green line
shows the real light path after rotating the prism. The red line is the light
path when refraction is neglected and thus the beam is reflected under 2θ.
C). In the experiment, it turned out to be easier and more reliable to do a calibration
measurement: For a series of angles θ the difference position between the normal 2θ
angle and the angle for which the laser is centered on the photo-detector is measured.
This angular deviation is fitted with a polynomial which in turn is implemented in
the software. Thus, the detector position is automatically corrected.
Another approach would be to move the prism perpendicular to the hypotenuse, see
[78].
θ
2θ

Chapter 4
Experimental study of surface
plasmons in thin metallic films
In this chapter, it is shown that with the in this thesis assembled setup it is possible to
detect surface plasmon resonances. In the beginning measurement problems and the
transformation from the measured voltage signal into the reflectivity R p are discussed
on the basis of a 50 nm Ag sample. Then the measured reflectivity for multilayer
samples will be discussed and critically compared to the simulations presented in
Chap. 2.
4.1 Surface Plasmons in thin silver films
To demonstrate that the surface plasmon setup assembled as a part of this thesis
indeed allows the observation of surface plasmons (cf. Chap. 2), Fig. 4.1 shows a
measurement of a 50 nm silver film. The figure shows the raw output signal UA-B(θ)
of the preamplifier versus the incident angle θ measured at zero field (µ0H = 0 mT).
Recalling that Uout(θ) ∝ R p (the sample reflectivity), the shape of the experimen-
tally observed reflectivity curve is similar to the surface plasmon simulations shown
in Chap. 2 (cf. Fig. 2.5). For small θ there is hardly any change in UA-B(θ), while
UA-B(θ) increases and goes through a maximum, when θ reaches the angle (θC ≈ 40 ◦ )
of total internal reflection, at the prism-silver interface. Then for θ > θC, UA-B(θ)
shows the characteristic, clear dip when the resonance condition for surface plasmon
excitation at θ = θSPP ≈ 42 ◦ is fulfiled. Upon further increasing θ there is again
hardly any change in UA-B(θ).
As Uout(θ) ∝ R p > 0 the signal UA-B(θ) < 0 can only be explained by the used mea-
49

50
U A-B (θ) [V]
0.00
-0.05
-0.10
-0.15
Chapter 4
Experimental study of surface plasmons in thin metallic films
θ ≈ 40°
θ ≈
30 40 50 60 70 80 90
θ
k
ε Ag
ε dielectric
Figure 4.1: Reflectivity of a 50 nm thick Ag film measured in the Raether-Kretschmann
configuration in 0.1 ◦ steps at zero applied field (µ0H = 0 mT). The raw
voltage output signal UA-B(θ) of the preamplifier is plotted as a function of the
incident angle θ. At θ = 0 ◦ the incident light is perpendicular to the sample
surface and at θ = 90 ◦ it is parallel to the sample surface, as depicted in the
sketch on the right side (cf. Sect. 3.3). At θC ≈ 40 ◦ the total internal reflection
of the prims can be seen, as UA-B(θ) reaches a maximum. And at θSPP ≈ 42 ◦
the surface plasmon resonance can be seen, as UA-B(θ) shows a minimum.
surement technique. Since UA-B(θ) is measured with the differential A-B technique,
it is possible to set the differential signal UA-B(θ) = 0, at the first measured angle
(here θ = 20 ◦ ) by applying an appropriate voltage to input B, so that UB becomes
equal to UA. Thus when the reflectivity is decreased the output voltage of the de-
tector decreases and the measured voltage becomes negative. To get the true voltage
signal at input A (Uout = UA, cf. Chap. 3.2) of the detector, the output mode of the
preamplifier is changed from A-B to A, before recording the first and after recording
the last data point (here θinitial = 20 ◦ and θfinal = 90 ◦ ). With these two values of
UA the subtracted voltage UB can be determined, by subtracting the first or the last
measured UA-B from the first or the last recorded UA. And thus, the true output volt-
age UA of the detector for the whole measurement can be calculated from UA-B(θ),
see Fig. ??. In this measurement the initial subtracted voltage UB, initial = 94 mV and
ε prism
θ
λ
k SPP

Section 4.1
Surface Plasmons in thin silver films 51
[V ]
U (θ) A in itia l
0 .1 2
0 .0 8
0 .0 4
0 .0 0
-0 .0 4
4 0 6 0 8 0
θ
4 0 6 0 8 0
θ
0 .1 2
0 .0 8
0 .0 4
0 .0 0
-0 .0 4
Figure 4.2: The graphs show the calculated UA of the detector versus θ, for the two obtained
values of UB. In (a) UA, initial corresponds to UA when using UB, initial
to calculate the true output voltage. In (b) UA, final is determined by using
UB, finial.
the final subtracted voltage UB, final = 75 mV are different. Thus it can be assumed
that UB is not stable during the measurement or it has great noise floor as it varies
≈ 19 mV. As the variation (drift) in the reference voltage UB is comparable to the
total change in UA-B, the data must be considered as questionable or even invalid.
4.1.1 Comparison of UA-B and UA
Therefore, the above measurement is repeated, but now instead of only recording
UA-B at each angle θ the output voltage of the photo-detector UA is also measured,
by switching the input mode of the preamplifier form A-B to A at each θ. The thus
obtained UA-B is shown in Fig. 4.3(c). Furthermore, a measurement is done where
only the output voltage of the photo-detector UA is measured (Fig. 4.3(b)). The thus
obtained UA(θ) and UA-B(θ) are compared to the measurement where only UA-B(θ)
is recorde, shown in Fig. 4.3(a). Comparing these three different measured signal,
[V ]
U (θ) A fin a l

52
U A -B (θ) [V ]
U A (θ) [V ]
U A -B (θ) [V ]
0 .0
-0 .1
0 .2
0 .1
0 .0
0 .0
-0 .1
Chapter 4
Experimental study of surface plasmons in thin metallic films
3 0 4 0 5 0 6 0 7 0 8 0 9 0
Figure 4.3: Panel (a) shows UA-B(θ) of the 50 nm Ag sample, which is measured as described
in Sect. 4.1. Panel (b) shows UA(θ) which is measured by changing
the input mode of the preamplifier from A-B to A. And panel (c) shows again
UA-B(θ), but here this signal the preamplifier is switched at every θ to also
measure UA.
no difference in the shape of the reflectivity signal of the Ag sample can be seen.
They all show a increasing signal at θ = θC ≈ 40 ◦ and clear dip at the surface
plasmon resonance at θ = θSPP ≈ 42 ◦ . And also the signal level of UA-B(θ) for the
measurements in Fig. 4.3(a) and (c) is similar. Now, to see if there is a qualitative
difference between the signals, UA-B(θ) and UA(θ) are transformed into a reflectivity
and compared to the simulations of R p for a 50 nm Ag film, discussed in Chap. 2.
θ
4.1.1.1 Comparison of simulation and measurement
As shown in Sect. 3.2 the output signal of the detector Uout changes linearly with the
reflectivity, therefore it is possible to normalise UA-B and UA to a given R p (θ) of the
simulated curve. R p (θC) is used to match the experiment and simulation, because
this point only depends on the prism material and hence is a good reference point.

Section 4.1
Surface Plasmons in thin silver films 53
The detected voltage U(θ) (UA-B(θ) or UA(θ)) is normalised by
R p mes = U(θ)Rp (θC, theo)
. (4.1)
U(θC,mes)
In Fig. 4.4 the experimental data of the silver sample (Fig. 4.1) is normalised and
compared to the simulated reflectivity. The figure shows that there is no difference
p (θ)
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
θ
θ
θ
3 0 4 0 5 0 6 0 7 0 8 0 9 0
θ
λ
ε
ε
Figure 4.4: The graph shows the measurements from Fig. 4.3 (, , and ) normalised
to the total internal reflection at θ = θC of the simulation (red curve). The
simulation parameters are the same like the ones used in Fig. 2.9.
between the two UA-B measurements ( and ) but R p (θSPP) is for both measurements
larger than for the UA measurement () and the simulation (red curve). Furthermore,
a clear mismatch in θC and θSPP between the simulated and the measured data is
evident. More precisely, the measured curves are all shifted by ∆θ ≈ 2 ◦ to smaller
θ compared to the simulation. Since θC only depends on the prism material, this
suggests that the refractive index of the prism used for the simulation is not the same
as the real refractive index. This will be further regarded in Sect. 4.1.2. Prior to that,
the difference of R p (θSPP) between the UA-B measurements and the UA measurement

54
will be discussed.
4.1.1.2 Determination of UB
Chapter 4
Experimental study of surface plasmons in thin metallic films
With the UA-B measurement, where also at each angle UA was measured, it is possible
to determine UB with
UB = UA − UA-B. (4.2)
The result UB and the in this measurement recorded UA is shown in Fig. 4.5. Regarding
U A (θ) [V ]
U B (θ) [V ]
0 .1 5
0 .1 0
0 .0 5
0 .0 0
-0 .0 5
0 .1 2
0 .1 0
0 .0 8
3 0 4 0 5 0 6 0 7 0 8 0 9 0
Figure 4.5: The upper panel shows UA(θ) obtained when switching the preamplifier mode
from A-B to A at each measured angle θ. The lower panel shows UB(θ) obtained
when subtracting UA-B(θ) from UA(θ).
the upper panel in the figure "four shifted" UA(θ) curves can be seen. For all "four"
curves UA increases at θ = θC and decreases at θ = θSPP. The "shift" of the signals
can also be seen in UB, shown in the lower panel. There "four" different UB are recog-
nisable with a spacing between two UB of ≈ 15 mV. As at about each measured θ, the
measured UB jumps from one level to another, it can be assumed that these "four"
θ

Section 4.1
Surface Plasmons in thin silver films 55
different UB and the "shift" in UA are due to the switching of the preamplifier. Fur-
thermore, some measured UB are between the four levels and therefore a noise floor of
UB ≤ 15 mV can be assumed. This is about the difference between UB, initial = 94 mV
and UB, final = 75 mV shown in Fig. 4.2 1 . As the noise floor of UB is in the the order
of the differential signal UA-B for further shown surface plasmon resonances only UA
is measured.
4.1.2 Adjustment of nBK7
As shown in Fig. 4.4 there is shift in θC and θSPP between measurement and sim-
ulation. Therefore, the value of nBK7 in the simulation is changed, thus simulation
and measurement fit. A discussion on the mismatch of R p (θSPP) will then follow in
Sect. 4.2 and Sect. 4.3.
To determine the real refractive index of the prism only the BK7 prism is measured
and then the parameters for the simulation are adapted to the measurement, see Fig.
4.6. The figure shows measurements () and simulations (red curve) for the reflec-
tivity of a BK7 prism. For the simulation in panel (c), the refractive index of BK7
nBK7 = 1.514 is used. This is the value given by the manufacturer of the prism [32],
while the shape of the measurement and the simulation, is quite similar, a clear shift
in θC is observed. To match the simulation to the measurement, the refractive index
in the simulation is adapted until θC is equal for both curves . The adapted simulation
with nBK7 = 1.58 is shown in Fig. 4.6(b). Now, both curves lie on top of each other
and the brewster angle θB as well as θC can be seen. Therefore, in later simulations
nBK7 = 1.58 will be used, instead of nBK7, manuf. = 1.514 as the refractive index for
the BK7 prism.
Now, with nBK7 = 1.58 the simulation for the 50 nm Ag film is repeated and com-
pared to the simulation with nBK7, manuf. = 1.514, see Fig. 4.7. The figure indicates
that now with nBK7 = 1.58 the simulated value R p (θC) does match with the measured
R p (θC) for the 50 nm Ag film. Further, Fig. 4.7(b) shows that there is still a difference
between the measured and simulated values of R p (θSPP) and θSPP. R p (θSPP) for the
measured reflectivity is much greater than for the simulated one and θSPP for the
measured reflectivity of the Ag film is shifted to greater θ compared to the simulation
of the reflectivity of the Ag film. Different possible explanations for this difference
will be discussed in the next section and in Sect. 4.3.
1 But there were also measurements where this change was in the order of 100 mV

56
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
Chapter 4
Experimental study of surface plasmons in thin metallic films
λ
0 2 0 4 0 6 0 8 0
θ
λ
0 2 0 4 0 6 0 8 0
Figure 4.6: In both panels the measured reflectivity of a BK7 prism is shown as open
squares (). In (a) the red line shows the simulation for a BK7 prism with
nBK7 = 1.514. In (b) the blue line shows the simulation for a BK7 prims with
nBK7 = 1.58
4.2 Reflectivity of multilayer thin films
Having established that the surface plasmon resonance of Ag, well established in the
literature, can be reproduced, it is now turned to more complex multilayer thin film
samples. Figure 4.8 shows a compilation of experiments and simulations for a 50 nm
Ni, a 40 nm Au 10 nm Ni doublelayer, a 20 nm Au 5 nm Co 5 nm Au trilayer, and a
12 nm Au 8 nm Co 12 nm Au trilayer sample. Comparing the measurements with the
simulations in Fig. 4.8, it can be observed that the shape of the experimental curves
is similar to the one expected from simulations. However, for θ ≈ θSPP the measured
R p is larger and broadened compared to the simulated R p , especially for the AuNi
sample, and θSPP itself is shifted compared to the simulated curves. Also for θ around
90 ◦ the simulated reflectivity goes up to R p = 1, whereas the measured reflectivity
stays more or less constant for θ ≫ θSPP.
Like it is shown in Fig. 2.9 for a silver layer, the position and depth of R p (θSPP)
θ

Section 4.2
Reflectivity of multilayer thin films 57
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
λ
ε
ε
4 0 6 0 8 0
θ
λ
ε
ε
4 0 6 0 8 0
Figure 4.7: The left panel shows the same graph as Fig. ??. In the right panel the simulation
of R p with nBK7 = 1.58 is shown (blue line) and compared to the
measurement of R p of the 50 nm Ag film ().
depends on the thickness of the used film. Thus one explanation for the difference in
R p between measurement and simulation maybe that the real sample thickness and
the thickness measured by the oscillating crystal differs. This difference can only be
a few nanometres as the deposition rates are small (1 − 2 Å/s) and the accuracy of
the oscillating crystal is given by ≈ 0.5% [64]. Like mentioned in Sect. 3.1 the sample
thickness corresponds to the one indicated by the oscillating crystal used during evap-
oration and was not verified after deposition the samples. Hence, it is possible that
real thickness differs from the one indicated by the oscillating crystal. To verify if the
mismatch between simulation and measurement is only due to a incorrect thickness of
the samples, the thickness parameter of the samples can be changed in the simulation.
Exemplarily this is done for the silver sample. In Fig. 4.9(a) the thickness parameter
in the simulation is changed, so that the simulation of R p (θ) matches for θ < θC the
measured R p (θ). This is the case for a silver film thickness of dAg = 43 nm. But this
thickness seems not realistic as it is ≈ 14% smaller than the measured one by the

58
p
R
0 .8
0 .4
0 .0
0 .8
0 .4
Chapter 4
Experimental study of surface plasmons in thin metallic films
ε
0 .0
2 0 4 0 6 0 8 0
ε
ε
ε
λ
2 0 4 0 6 0 8 0
θ
Figure 4.8: Reflectivity Rp of a Ni, AuNi, and two AuCoAu samples, measured in Raether-
Kretschmann configuration, as a function of incident angle θ. The four graphs
show a comparison between the simulation of Rp (colored lines) and the measurement
of Rp (). In (a) Rp of a film with 50 nm Ni, in (b) of a film with
40 nm Au and 10 nm Ni on top, in (c) of a film with 20 nm Au, 5 nm Co, and
5 nm Au on top, and in (d) of a film with 12 nm Au, 8 nm Co and 12 nm Au
on top is shown.
oscillating crystal and as explained only a difference of a few nanometres is expected.
Furthermore, the simulated and measured R p (θSPP) does not match. When chang-
ing the thickness dAg in the simulation so that R p (θSPP) matches the measured one
(Fig. 4.9(b)), which is the case for dAg = 52 nm, the simulation of R p (θ) for θ < θC
does not match the measured R p (θ) anymore. Nevertheless, in this case, R p (θSPP)
and most of the measured R p (θ) > R p (θSPP) fit the simulated curve. This suggests
that the real layer thickness of the Ag film is between 50 nm < dAg < 52 nm, but it
also shows that there are other effects which cause a discrepancy between simulation
and measurement, especially a shift of θSPP.
Like mentioned in Sect. 2.4 the dielectric function ε has a great effect on R p . With
0 .8
0 .4
0 .0
0 .8
0 .4
0 .0

Section 4.2
Reflectivity of multilayer thin films 59
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
λ
ε
ε
4 0 6 0 8 0
θ
λ
ε
ε
4 0 6 0 8 0
Figure 4.9: Upon changing the nominal Ag film thickness in the simulation, the agreement
between experimentally measured R p () and the calculated one ( , )
can be improved. Panel (a) shows the simulation for a silver film thickness of
dAg = 43 nm and in (b) for a thickness of dAg = 52 nm.
increasing imaginary part ε ′′ the surface plasmon resonance broadens, as evident from
the measurements, compare Fig. 4.8(a) and Fig. 4.9. In Fig. 4.8(a) R p of Ni, which
has a ≈ 13 times greater imaginary dielectric component ε ′′ than Ag, is shown. Com-
pared to Fig. 4.9, where R p of Ag is shown, the surface plasmon resonance of Ni is
much more broadened than the Ag resonance. Clearly, the "right" value for ε is a
difficult matter as the values in the literature vary from ε = −17.03 + i1.15 [36] to,
ε = −20.6 + i0.97 [79], and ε = −19.06 + i0.47 [80].
Besides the dielectric constant, there are other effects which have a great influence on
the surface plasmon resonance curve, e.g. film thickness, surface roughnesses, oxida-
tion, or polarisation effects. These effects will be explained in more detail in the next
section.
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
p
R

60
Chapter 4
Experimental study of surface plasmons in thin metallic films
4.3 Beyond the ideal surface plasmon resonance
curve
4.3.1 Thin film limit
The surface plasmon theory presented so far applies to "thick" films. In films with
a thickness d which fulfils the condition 2d/λ ≤ 1, this theory has to be modified to
thin films [81, 82, 83]. Since a laser diode with λ = 650 nm is used in the experiments
discussed here and the film thicknesses are of about d = 50 nm, 2d/λ ≈ 0.15 ≤ 1 and
thus the thin film limit does apply.
The thin film limit will have an effect on R p in layered structures, when for example
in a trilayer structure the middle layer is in the thin film limit. Then the dispersion
relation ω splits into a high-frequency mode ω + and a low-frequency mode ω − [81,
82, 83]. The splitting can be explained by two surface plasmon waves which travel
along the interfaces of the middle material. When the film thickness decreases these
two waves interact and the dispersion relation splits into a symmetric (low-frequency)
mode and an antisymmetric (high-frequency) mode. The splitting of the dispersion
relation has two major effects on the reflectivity. First, because of the two modes ω +
and ω − , two surface plasmon resonances are observed, one at θ < θSPP and a smaller
one at θ > θSPP [83]. Second, the resonance sharpens for decreasing film thickness
[83].
When taking a closer look at Fig. 4.8(c) and (d) a small decrease in reflectivity can
be seen at about 82 ◦ or 83 ◦ , respectively. A close-up view is shown in Fig. 4.10.
However, Fig. 4.8 also shows that the surface plasmon resonance is not shifted to
larger angles, instead it is shifted to greater angles. Thus the measured resonance
curve cannot be completely described by the thin film limit, further there may be an
other effect which suppresses the effect due to the thin film limit.
4.3.2 Surface roughness
Until now, it is assumed that the sample surfaces were ideal smooth surfaces. In
reality the surfaces will have a finite roughness. As already mentioned in Sect. 2.3.2,
this roughness affects the measured reflectivity R p . The effect of roughness on the
reflectivity can be calculated, see Refs. [9, 27, 84, 85], but here just the results will
be discussed.

Section 4.3
Beyond the ideal surface plasmon resonance curve 61
p
R
0 .5 0
0 .4 5
0 .4 0
0 .3 5
0 .3 0
6 0 7 0 8 0 9 0
θ
0 .5 0
0 .4 5
0 .4 0
0 .3 5
0 .3 0
6 0 7 0 8 0 9 0
Figure 4.10: Close-up of Fig. 4.8. Panel (a) shows the reflectivity of the
Au20 nmCo5 nmAu5 nm sample. At θ ≈ 75 ◦ a small decrease of the reflectivity
is observed. In (b) a close-up of R p (θ) of the Au12 nmCo8 nmAu12 nm sample is
shown. Here, the small decrease is at θ ≈ 75 ◦ .
Due to statistical roughness of the surface, the surface plasmons do not only radiate
light back into the metal, they also radiate light into the dielectric [84], what leads to
a damping of the back radiated light. In Sect. 2.4 it was discussed that the minimum
of R p is due to interference of the incident light with the back scattered light at the
metal-air interface. If the surface is not smooth, there will be phase shifts between
different back scattered light waves and thus the interference will be less strong, which
yields a greater reflectivity at θSPP [9]. Surface roughness do not only contribute to
the damping of surface plasmons, they also shift the surface plasmon resonance to
greater angles [86, 87] and is explained by Braundmeier et al. as follow:
According to Eq. (2.22)
kSPP = ω √
ε ′(0) sin θ, (4.3)
c
a shift to greater θ can also be seen as a shift to greater wave vectors kSPP, i.e. smaller

62
Chapter 4
Experimental study of surface plasmons in thin metallic films
phase velocities ω/kSPP. This can be explained with the dispersion relation for surface
plasmons at a smooth metal-air interface given by Eq. (2.15)
k = ω
c
ε ′(1)
ε ′(1) . (4.4)
+ 1
As mentioned in Sect. 2.2 the dielectric function for metals can be expressed with the
dielectric function of a free electron gas
ε ′(1) (ω) = 1 − ω2 p
ω2 , (4.5)
+ γ
where γ is the damping due to the surface roughness. Considering that the here used
dielectric function ε ′(1) ≤ −1. With these equations for the change of k with γ it
follows
∂k
∂γ
1 ω
=
2 c
1
ε1(ε1 + 1)
1/2 ∂ε1
∂γ
≥ 0. (4.6)
∂k/∂γ ≥ 0 is valid because [ε1(ε1+1)] −1 and ∂ε1/∂γ are both positive. Thus it follows
that for increasing damping constant due to surface roughness the wave vector kSPP
increases and therefore also the resonance angle θSPP increases.
Following these arguments, the experimentally observed shift of θSPP to larger values
and the small dip magnitude, evident from Fig. 4.8, most probably are due to surface
roughness.
Regarding the broadening of R p (θSPP) this can also deduced to surface roughness,
as with increased damping the resonance broadens [86]. From θSPP the energy and
from the full width at half maximum (FWHM) the energy distribution of the surface
plasmons can be calculated with Eq. (2.22) and E = ω. For the Au20 nmCo5 nmAu5 nm
sample θSPP = 44.5 ◦ and FWHM = 7.6 ◦ is observed, what corresponds to E =
2.1 ± 0.21 eV. For the simulated curve θSPP = 42 ◦ and FWHM =3.2 ◦ , what it is
equal to E = 3 ± 0.09 eV. This is difference of ∆E = 0.9 ± 0.12 eV and compared
to the theoretical value for a gold layer from Sect. 2.2 of EAu = 4.6 eV the difference
to the measured energy is ∆E = 2.5 eV. This decreased energy yields less localised
surface plasmons as the confinement to the interface is 1/kz [30].

Section 4.3
Beyond the ideal surface plasmon resonance curve 63
4.3.3 Oxidation
In general when a dielectric is brought into contact with the metal surface, the plasmon
resonance is only shifted and not damped. This is not the case anymore when the
material has a complex dielectric constant: in this case, the resonance is also damped
[9, 5]. Since silver is a noble metal it is not expected that it oxidises when exposed
to air. Instead, silver reacts with the hydrogen sulfide in the air and a thin layer of
silver sulfide (Ag 2S) will cover the silver surface. But this process is very slow and
has no substantial effect on the resonance curve [87].
This argument does not hold anymore for nickel and cobalt. When these metals are
exposed to air, they will be covered by an oxide layer [88, 89, 90] which will influence
the surface plasmon resonance. In Fig. 4.11 two measurements of the Au40 nmNi10 nm
sample are shown, which have been taken with a temporal separation of one month.
The figure shows a clear difference in the minimums of both curves corresponding
p
R
0 .9 2
0 .9 0
0 .8 8
0 .8 6
0 .8 4
3 0 4 0 5 0 6 0
θ
Figure 4.11: Reflectivity R p (θ) of a 40 nm Au covered with 10 nm Ni sample, taken one
month apart. First the red curve was measured and then about one month
later the black curve. Between both R p (θSPP) there is a change of about
0.04 % in the reflectivity 2 .
to a change of 0.04 % in the reflectivity. Furthermore, a broadening of the resonance
as well as a slight shift of θSPP can be seen. This is an indication that oxidation of
2 Measured with UA-B, see Summary.

64
Chapter 4
Experimental study of surface plasmons in thin metallic films
the metal changes the resonance curve. To prevent the metals from oxidation they
can be covered with a thin dielectric film like LiF [9], or with gold. Since gold is a
noble metal, it is much less prone to oxidise or react with anything else in the air.
This might explain why the experimental R p (θ) curves of the Au12 nmCo8 nmAu12 nm
and the sample Au20 nmCo5 nmAu5 nm do best fit with the simulations.
4.3.4 Polarisation effects
So far, the shift and the damping of the surface plasmon resonance has been addressed.
The corresponding arguments do not explain the difference between reflectivity mea-
surements and simulation for angles θ close to 90 ◦ . There the simulation goes up to
R p = 1 and the measurement stays at a certain level smaller than R p = 1. The polar-
isations of the incoming and outgoing light might give an explanation here. Therefore
the polariser and analyser are adjusted such that e.g. only p-polarised light is incident
on the sample and only s-polarised light is detected, i.e. a so-called p-s configuration.
The other configurations are p-p, s-s, s-p, p-s, and no polarisation, where the po-
lariser/analyser are removed. The five different measurements are shown in Fig. 4.12.
The graph confirms that only p-polarised light can excite surface plasmons. But
more importantly it shows that when no polarisation filters are mounted (turquoise
line) the reflectivity increases instead of staying at a constant level like for the p-p
polarised curve (black line). This is the same behaviour shown by the simulated re-
flectivity, where for θ > θSPP R p increases to R p (θ = 90 ◦ ) = 1. Furthermore, in the no
polarisation measurement the second resonance is stronger than for the p-p polarised,
and even the s-s polarised (green line) has a more distinct second resonance than
the p-p polarised curve. When comparing the no polarisation measurement with the
p-s polarised one (red curve) then for θ > θSPP for both R p (θ) increases. Especially
the p-s polarised measurement may imply that there is an effect on the polarisation
direction by the surface plasmons, because when only p-polarised light incidents then
it would not be expected to detect any s-polarised light (cf. Sect 2.5.1). A similar
behavior is described by Clavero et al. [15].
3 Measured with UA-B, see Summary.

Section 4.3
Beyond the ideal surface plasmon resonance curve 65
θ θ
3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0
θ
Figure 4.12: The figure shows the reflectivity R p (θ) for different polarisation configurations.
To compare the measurements the first value (R p (θ = 30 ◦ )) of every
measurement is subtracted from R p (θ). The turquoise curve is the reflectivity
when no polarisation filters are mounted. The black curve shows R p (θ) for
p-p, the blue for s-p, the green for s-s, and the red for p-s polarisation 3 .
4.3.5 Immersion oil
There is another minor effect which has a small influence on the R p (θ) and thus
is mentioned for completeness. This effect is due to the immersion oil. When the
refractive index of the prism and the immersion oil do not match exactly, then this
affects the resonance by broadening it [83]. But since the refractive index of the
immersion oil is nd = 1.515 ≈ nd, BK7 = 1.516 4 the effect due to the immersion oil is
small.
4 Even nBK7 = 1.58 is used for the simulations, the effect due to the difference between noil and
nBK7 can be neglected.

66
4.4 Summary
Chapter 4
Experimental study of surface plasmons in thin metallic films
In the beginning of this chapter the surface plasmon resonance of a 50 nm Ag sample
was shown (Fig. 4.1). On this basis the measured voltage signal UA-B was discussed
and it was shown that the subtracted voltage UB was no stable or at least had a great
noise floor (Fig. 4.2).
Unfortunately, the full extent of this problem became clear only during writing of
this thesis, so that it was not possible to repeat all measurements with an appropri-
ately stable UB. Only the surface plasmon resonances shown in this chapter (except
Fig. 4.11 and Fig. 4.12) could be remeasured by measuring the output voltage of the
photo-detector UA without using the differential method. However, as UA-B is quali-
tatively very similar to R p expected from simulations, in lack of a better alternative
in Chap. 5 simply UA-B is used to calculated R p and UB,initial to calculate UA.
To avoid problems with UB it would be necessary to ensure that the noise floor of UB
is much smaller than UA-B for every measured angle θ, but this is not possible with
the used setup. Measuring UB requires that the input mode of the preamplifier is
switched from A-B to A, but as shown in Fig. 4.5 the switching of the input mode
affects the detected signal UA. Clearly however, UB = const. is mandatory to obtain
sensible data.
Following the discussion on UB the measured reflectivity R p of the Ag films was com-
pared to the simulations of R p from Chap. 2 and the mismatch in the θSPP and θC was
regarded (Sect. 4.1.2). Since the mismatch in θC was due to an incorrect refractive
index of the glass prism, the refractive index has been adapted from nBK7 = 1.514 to
nBK7 = 1.58, so that the measurement and the simulation of the prism were equal,
see Fig. 4.6.
Next, the simulations of multilayer, which were shown in Chap. 2, were in Sect. 4.2
verified with measurements of R p . After a discussion on the differences between the
measurements and the simulations, different effects were discussed and how these
affect the resonance (Sect. 4.3). By discussing these effects it got clear that the dif-
ferences are most likely due to roughnesses at the sample surface, which cause θSPP
to be shifted to greater angles and R p (θSPP) to be greater for the measurement than
for the simulation. Furthermore, it is evident from Fig. 4.8 that the surface plasmon
resonance best fits for the Au/Co/Au trilayer samples, which could be explained by
the protective Au layer, which prevents the samples from oxidisation. In opposite
to the trilayer samples, the AuNi, with Ni exposed to air, and the Ag sample, both

Section 4.4
Summary 67
react with the air and thus a dielectric layer evolves on the sample surface, which
then affects the surface plasmon resonance, as discussed in Sect. 4.3.3.
Conclusively, it can be said that with the during this thesis assembled setup it was
possible to detect surface plasmon resonances with the during the thesis grown sam-
ples.

68
Chapter 4
Experimental study of surface plasmons in thin metallic films

Chapter 5
Surface plasmons in magnetic
environments
In this chapter the effect of an external applied magnetic field on surface plasmon
resonance will be investigated. This will be done for the measurements shown in
the previous chapter, where the angle is changed, but here also for measurements
were instead of the angle the magnetic field is changed. Further, the enhancement of
surface plasmons to the transversal magneto-optical Kerr effect is investigated. And
last, magnetic anisotropy measurements are done.
5.1 Surface plasmon resonance at constant
magnetic field
As discussed in Sect. 2.5 an external magnetic field will induce nonzero, off-diagonal
elements in the dielectric tensor ˆε (cf. Eq. (2.50)). In the transversal geometry this
will have an effect on r pp
t , as only in this term Q appears. Q is the Voigt constant,
which is proportional to the induced magnetisation of the medium. Furthermore, it
was discussed that the dispersion relation splits into two branches when an external
magnetic field is applied (cf. Fig. 2.15) [17]. This splitting corresponds to a positive
(kx > 0) and a negative (kx < 0) propagation direction. Analogous, regarding only
one propagation direction, this splitting can correspond to two opposite directions of
the external applied magnetic field µ0H [91]. According to [54], an inversion of the
magnetic field direction has an effect on R p , and can be seen in a change of R p (θSPP)
and position of the resonance of the surface plasmons θSPP. And as described with
Eq. (2.22) this corresponds to a change in kSPP
69

70
Chapter 5
Surface plasmons in magnetic environments
To observe this change the reflectivity of the sample is measured while perpendicular
to the plane of incident a magnetic field is applied, as depicted in Fig. 3.1. This
Figure 5.1: Sketched in this figure is the magnetic field configuration used in this thesis.
The red line illustrates the incoming light. As the figure shows the applied
magnetic field is perpendicular to the plane of incident of light and parallel to
prism/sample surface.
is then done for different magnetic fields. Because only two opposite magnetic field
directions are of interest in Fig. 5.2 only the reflectivity for −15 mT, 15 mT, again
−15 mT, and for 0 mT are shown. In the figure the surface plasmon resonance for a
AuNi film (dAu = 40 nm, dNi = 10 nm), with the Ag site in contact to the prism, can
be observed. The surface plasmon resonance without any applied field (µ0H = 0 mT)
is shown as . Subsequently −15 mT (), +15 mT (), and −15 mT () are applied
and UA-B is measured as a function of the incident angle θ between 25 ◦ and 60 ◦ .
Despite the applied magnetic field, the curves do not change significantly in amplitude
and shape. The two insets highlight small changes in the steep part (θC < θ < θSPP)
of UA-B(θ) and at UA-B(θSPP).
As obvious from Fig. 5.2 the focus is on small changes in the reflectivity. Therefore,
the first −15 mT () is used as reference and is subtracted from the following curves
(∆U = U − U,,). As UA-B ∝ R p (θ) (cf. Sect. 3.2) this procedure is analogue to
the method described in Ref. [54] and [14]. There ∆R = R + − R − is shown, with
R + denoting the reflectivity for positive magnetic field and R − for negative magnetic
field.
Figure 5.3 displays the reference and the field dependent residual signature of the

Section 5.1
Surface plasmon resonance at constant magnetic field 71
U A -B (θ) [V ]
0 .0 4
0 .0 2
0 .0 0
-0 .0 2
-0 .0 4
-0 .0 6
4 1 4 2
0 .0 0
-0 .0 2
-0 .0 4
2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0
θ
-0 .0 5 4
-0 .0 5 6
-0 .0 5 8
4 2 4 4
Figure 5.2: Reflected intensity as a function of the incident angle θ for four magnetic fields:
for −15 mT, for 0 mT, for 15mT, and for −15mT. The inset in the
lower left corner shows a close-up view UA-B(θ) at θ ≈ 41 ◦ . And the inset in
the upper right corner shows a close-up view of UA-B(θSPP) at θSPP ≈ 43 ◦ .
surface plasmon resonance. The left axis is the same as Fig. 5.2 and shows the
reference curve UA-B,−15 mT(θ) and the right axis displays the difference signal ∆U. The
red, turquoise and blue curve show the difference from the reference (first −15 mT)
for the 15 mT, 0 mT, and second −15 mT measurement, respectively. In all three
subtracted datasets show smaller voltages indicating that the measurement signal
drifts during this experiments, which is an artifact due to the great noise floor of UB
(cf. Sect. 4.1.1.2). Nevertheless, at θ ≈ 41 ◦ , corresponding to the steep part of the R p
curve (vertical dashed line in Fig. 5.2), a clear peak structure is observed. This peak
corresponds to a small upward shift to greater voltages for the blue curve compared
to the red curve, evident from Fig. 5.2, see also the inset in Fig. 5.2.
When neglecting the drift in the signals, the main characteristics of ∆U, mainly the
peak, look similar to the expected ∆R curve [54, 15], where a clear dip in the ∆R
curve at the steep part of R p can be seen.

72
U A -B , -1 5 m T (θ) [V ]
0 .0 4
0 .0 2
0 .0 0
-0 .0 2
-0 .0 4
-0 .0 6
Chapter 5
Surface plasmons in magnetic environments
θ
Δ θ
Δ θ
Δ θ
2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0
θ
Figure 5.3: The figure monitors the reference curve UA-B,−15 mT(θ) (left axis) and the subtracted
curves ∆U(θ) (right axis). The red curve displays the difference of the
15 mT measurement (∆U15 mT(θ)), the turquoise curve displays ∆U0 mT(θ), and
the blue curve displays ∆U−15 mT(θ).
Furthermore, due to the splitting of the dispersion relation of surface plasmons when
a magnetic field is applied, the surface plasmon wave vector kSPP also shifts, yielding
∆k = kH,SPP − kSPP (kH,SPP is the wave vector when a magnetic field is applied). As
discussed in Sect. 4.3.2 a variation in kSPP is analogue to variation in θSPP and thus also
in R p (θ). And hence, ∆k yields a change in the observed reflectivity ∆R p (θ). The shift
in the resonance angel θSPP is reported by Hickernell [54] to be only ∆θ ≈ 0.001 ◦ .
This change is smaller than the mechanical angle resolution of the rotation stage
(∆θ = 0.005 ◦ , cf. Sec. 3.1) and thus not possible to detect with this setup. The
change in the reflectivity ∆R p given in the litertautre is in the order of 10 −4 [54] to
10 −3 [15]. The here measured ∆U is in the order of 10 −3 V but with a poor signal-
to-noise ratio of only 0.6. To improve the signal-to-noise ratio it would be necessary
to modulate the applied field and then measure the detected signal with a Lock-In
technique [92].
2
0
-2
ΔU (θ) [m V ]

Section 5.2
Surface plasmon resonance at constant angle 73
As an alternative for varying the incident angle θ and keeping the magnetic field fixed
is to vary the field at each θ.
5.2 Surface plasmon resonance at constant angle
Figure 5.4 shows magnetic field scans (panel (b), (c), (d), (e)) for four selected angles
as indicated in (a) by circles. Position (b) is chosen because at θ = 20 ◦ no influence of
surface plasmons is expected. Points (c) (θ = 41.8 ◦ ) and (d) (θ = 49.8 ◦ ) are chosen
because a large change in the reflection is expected in the vicinity of the surface
plasmons. The reason why no point at the minimum of the reflectivity R p is chosen
will get clear later. And at point (e) (θ = 90 ◦ ) again one would at first not expect an
influence of the surface plasmons. The magnetic field is swept from −20 mT to 20 mT
p (θ)
R
0 .9 0
0 .8 5
0 .8 0
0 .7 5
0 .7 0
4 0 6 0 8 0
θ
-2 0 0 2 0
-1 5 .8
-1 6 .0
-3 2 .8
-3 3 .0
-4 8 .8
-4 9 .0
-8 5 .4
-8 5 .6
-8 5 .8
Figure 5.4: Field dependence of the surface plasmon curve. (a) shows the normal plasmon
resonance curve from Fig. 4.8. The four circles on the graph denote the position
where the reflectivity under varying magnetic field is measured. In the right
panels the detected signal U(H) versus the magnetic field µ0H is shown. The
magnetic field goes from −20 mT to 20 mT and again to −20 mT in 0.2 mT
steps. The signal for all four measurements is shown in a range of 0.4 mV.
U (H ) [m V ]

74
Chapter 5
Surface plasmons in magnetic environments
and back to −20 mT in 0.2 mT steps. Figure 5.4(b) shows hardly any drift in the
signal. In particular the first and last point of the measurement, lie close together.
The signal range shown is for all four graphs 0.4 mV and the signal-to-noise ratio of
about 1.6.
Comparing (b) (θ = 41.8 ◦ ) and (c) (θ = 49.8 ◦ ) reveals that hysteresis loops are visible
in the data. The coercive fields are in both cases µ0HC ≈ ±5 mT, the fields expected
for Ni is µ0HC ≈ ±1 mT [65].
In contrast to the red curve, where the switching of the magnetisation is identified
with a decrease of the detected voltage, in the blue curve the same change in the
magnetisation direction corresponds to a increase of the voltage. For panel (b) and
(e), corresponding to θ = 30 ◦ and θ = 90 ◦ no hysteretical behaviour is observed,
instead U(H) stays constant during the whole field sweep.
Now, subtracting the up and down sweep from each other the change between both
curves is highlighted. Hereby, the focus is on the difference between the up and down
sweep, which allows easy identification of the coercive fields, as shown in Fig. 5.5.
This particular way of plotting also highlights the sense of rotation of the hysteresis
loop. E.g. at θ = 41.8 ◦ the measurement shows a peak in the subtracted signals
U(H + ) − U(H − ), while for θ = 49.8 ◦ reveals a dip in the subtracted signal, with
U(H + ) being the signal for the up sweep and U(H − ) the signal for the down sweep.
But due to the great noise for the measurement at 49.8 ◦ the change in the signal can
be more clearly seen in Fig. 5.4 than in Fig. 5.5.
Conclusive it can be said that for small angles θ around 30 ◦ no hysteresis can be seen,
which changes for increasing θ. At θ = 41.8 degree a hysteresis occurs, which switches
for further increased angles θ = 49.8 ◦ and then at θ = 90 ◦ this hysteresis vanishes.
Now, to investigate if the hysteresis evolves slowly or occurs abrupt at a certain θ,
and if the same yields for the switching and vanishing of the hysteresis at greater
θ, the magnetic field sweep is not only measured at four certain angles, instead in
0.2 ◦ steps. Subtracting the so measured down sweep from the up sweep (what will
be further denoted as subtracted hysteresis), as shown in Fig. 5.5, the thus obtained
data is displayed as a colour plot in Fig. 5.6(a). The black curve in the background
is just shown to guide the eye, so it is easier to see which point on R p corresponds to
the actual measured angle.
In the angular range of θ = 40 ◦ - 42.8 ◦ the subtracted hysteresis results in a positive
response (red), which evolves and vanishes abrupt, while a negative response, i.e. a

Section 5.2
Surface plasmon resonance at constant angle 75
p (θ)
R
0 .9 0
0 .8 5
0 .8 0
0 .7 5
0 .7 0
4 0 6 0 8 0
θ
-2 0 0 2 0
Figure 5.5: Subtracted hysteresis curves.(a) displays R p for AuNi (cf. Fig. 4.8) and the
circles mark the points where U(H) is measured. In (b) the change in the
detected signal between the up and down sweep (U(H + ) − U(H − )) against
µ0H is shown. The signal range is again for all four measurements the same
and is 0.6 mT.
switched hysteresis loop, is recorded for θ ≥ 44 ◦ , which extends but with decreasing
signal up to grazing incidence of θ = 90 ◦ . In between, 42.8 ◦ < θ < 44 ◦ , the subtracted
hysteresis disappears corresponding to white colour in the colour code. Taking a cut
of this colour plot for vanishing external magnetic field µ0H = 0 mT this results in
Fig. 5.6(b). Here, a clear response of magnetic dependencies are observed for the
steep slope of the reflectivity curve. This can also be observed from Fig. 5.6(c),
which corresponds to a vertical cut through the colour plot (red line in Fig. 5.6(a)) at
θ = 40 ◦ . In particular it is emphasized that this response is proportional to the field
dependence recorded in Fig. 5.3, as it corresponds to a measurement at constant field,
corroborating the observed field dependence there. Because the curves in Fig. 5.6(b)
and Fig. 5.3 are denoted with ∆U the one obtained from constant field measurements
will from now on be denoted as ∆Uµ0H=const and the other one obtained from constant
angle measurements will be denoted as ∆Uθ=const. A further explanation will be given
0 .2
0 .0
-0 .2
0 .2
0 .0
-0 .2
0 .2
0 .0
-0 .2
0 .2
0 .0
-0 .2
- ) [m V ]
+ ) - U (H
U (H

76
Δ θ
0
2 0
1 5
1 0
5
0
-5
-1 0
-1 5
-2 0
-0 .2 8
-0 .1 6
Chapter 5
Surface plasmons in magnetic environments
3 0 4 0 5 0 6 0 7 0 8 0 9 0
θ
0 .0 6 0
0 .2 8
0 .2
0 .0
-0 .2
0 .0 0 .2
Figure 5.6: Magnetic field sweep of the surface plasmon resonance. The central graph (a)
shows the magnetic field sweep against the incident angle. µ0H is swept from
−20 mT to 20 mT and again to −20 mT in 0.2 mT steps and θ form 30 ◦ to
90 ◦ in 0.2 ◦ steps. The colour code equates the subtracted sweep. The black
curve in the central graph shows the reflectivity just to guide the eye on which
point of the reflectivity curve the actual point is measured. Panel (b) displays
a horizontal cut at 0 mT through the central graph. Indicated by the green
line in the central graph. (c) shows a vertical cut at 41 ◦ , denoted by a red line
in the central graph.
later, but prior other samples are regarded.
5.3 Surface plasmons in improved sample structure
The shown curves in the previous section displayed a weak signal with a small signal-
to-noise ratio (cf. Fig. 5.5). To improve the signal quality a better sample structure
can considered. As discussed in Sect. 4.2 and observable from Fig. 4.8, the AuCoAu
samples show a much more clearer resonance than the AuNi sample. This is due to
the covering Au layer, which increases the surface plasmon resonance, but also covers

Section 5.3
Surface plasmons in improved sample structure 77
the cobalt from oxidising.
Furthermore, from FerroMagnetic Resonance measurements (also done during this
thesis [93]) it is known that Co yields a stronger magnetical response than Ni. There-
fore when repeating the measurements from the previous section above with the 20 nm
Au 5 nm Co 5 nm Au sample, it is expected that the signal-to-noise ratio is much bet-
ter.
Starting again with a measurement of the reflectivity at a constant magnetic field,
and comparing the curves of two opposite magnetic fields (−15 mT and 15 mT) the
following graphs are obtained, as displayed in Fig. 5.7. Again on the scale of the
θ
0 .1
0 .0
-0 .1
-0 .2
0 .0 0
-0 .0 4
-0 .0 8
4 1 .5 4 2 .0
Δ
-3
3 0 4 0 5 0 6 0
θ
-0 .1 6
-0 .1 7
-0 .1 8
4 3 4 4 4 5
1
0
Δ
Figure 5.7: Surface plasmon resonance for AuCoAu at constant magnetic field. The left
axis displays the detected signal UA-B(θ). The reflected signal is shown for
four magnetic fields, −15 mT (), 0 mT (), 15 mT (), and again −15 mT
(). On the right axis ∆Uµ0H=const is plotted. The curve is the difference
between the first −15 mT and 15 mT curve.
reflectivity measurements the difference between the magnetic fields −15 mT, 0 mT,
+15 mT, and −15 mT is only minor. Subtracted from the reference curve () the
difference in the reflection for constant magnetic field (∆Uµ0H=const) is observed as a
peak structure (θ ≈ 41.6 ◦ ). As shown in Fig. 5.2 and Fig. 5.6(b) this peak structure
-1
-2

78
Chapter 5
Surface plasmons in magnetic environments
coincident with the section of the reflectivity curve with the steepest slope. The in
Fig. 5.7 shown signal has an improved signal-to-noise ratio for 4.8, what is 8 times
better than the signal-to-noise ratio obtained for Au40 nmNi10 nm. This corroborates
the above statement regarding the improved surface plasmon resonance and increased
magnetic response.
To grasp the full magnet information of the sample it is now turned again to field
scans with constant angles of incidence. Like in Sec. 5.2 this is first done at four
certain points of the surface plasmon reflectivity curve, as depicted in Fig. 5.8. The
p (θ)
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
4 0 6 0 8 0
θ
-1 5 0 1 5
Figure 5.8: Hysteresis curves at constant incident angle. In (a) R p of Au20 nmCo5 nmAu5 nm
is shown, with the four encircled measurement points. In (b) the detected
signal against the applied magnetic field, varied from −20 mT to 20 mT back
to −20 mT in 0.2 mT steps, is shown. The signal range for all four angles is
displayed in a 5 mV range.
detected signal U(H) against the µ0H is shown in the right panel of Fig. 5.8. The
displayed signal range for all four points is limited to 5 mV. Comparing these mea-
surements with those of AuNi it can immediately be seen that the signal-to-noise ratio
for Au20 nmCo5 nmAu5 nm is much better, which is in Fig. 5.8(c) (θ = 42.6 ◦ ) ≈ 10.8 this
is ≈ 7 times better than for AuNi.
4
3
2
-5 9
-6 0
-6 1
-1 6 0
-1 6 1
-1 6 2
-9 8
-9 9
-1 0 0
U (H ) [m V ]

Section 5.4
Reflectivity at constant magnetic field 79
The signal behaviour for Au20 nmCo5 nmAu5 nm is comparable to Au40 nmNi10 nm. For
θ = 30 ◦ there is no change in U(H) when the magnetic field is swept. This changes
again when measuring at θ = 42.6 ◦ , there a clear hysteresis with a coercive field of
µ0HC ≈ 7 mT, can be observed. At θ = 47.4 ◦ the measurement shows a rotated
hysteresis loop, like for AuNi. The detected coercive field of µ0HC ≈ 7 mT is the
same order of magnitude as reported ones which is in the order of µ0HC = 1 − 20 mT
[65, 66, 67]. Now, in contrast to the AuNi sample for Au20 nmCo5 nmAu5 nm at θ = 90 ◦
there is still a hysteresis.
For this sample also an angle sweep from 30 ◦ to 90 ◦ in 0.2 ◦ steps is done, while
sweeping the magnetic field at each angle. The colour coded subtracted hysteresis of
this measurements is plotted in Fig. 5.9(a). The structure of the figure is the same
like in Fig. 5.6. The central panel shows the difference of the signal between the up
and down sweep which is represented by the color code. Again at angles θ ≥ 30 ◦ no
hysteresis can be seen. Then at θ ≈ 40 ◦ to θ ≈ 45 ◦ abruptly a hysteresis occurs which
then also abruptly disappears. Until at θ ≈ 46 ◦ the switched hysteresis appears and
then slowly gets smaller for increasing θ. From the central panel also a clear switching
of the magnetisation at µ0H ≈ 7 mT can be seen. Compared to AuNi this AuCoAu
sample shows a much clearer and less noisy signal, what can also seen in the right
(red graph) and top (green graph) panel of Fig. 5.9. Now, in the next section a closer
look on the graph Fig. 5.9(b) is taken.
5.4 Reflectivity at constant magnetic field
Like mentioned in Sect. 5.1 ∆U(θ) = U(θ) + −U(θ) − is equivalent to ∆R(θ) = R + (θ)−
R − (θ). This variation ∆R(θ) noramlised to the reflectivity R p is often identified with
the MOKE signal [60, 94, 14, 16]. Furthermore, Gonzalez-Diaz et al. [60] report that
∆R(θ)/R p is equal to the derivative of R p with respect to θ times a scaling factor.
∆U
U(θ)
= ∆R(θ)
R p (θ)
∂R
= p (θ) ∆θ
∂θ Rp (θ)
G.−D.
≡ ∂U(θ)
∂θ
∆θ
. (5.1)
U(θ)
Where, ∆θ = θSPP(H + ) − θSPP(H − ) is the shift of the resonance angle θSPP for the
two different field directions.
From Eq. (??) it is obvious that the measured ∆U(θ) from Fig. 5.7, obtained from con-
stant field and the one from Fig. 5.9(b), obtained from constant angle measurements,
are equal to the derivative of the measured reflectivity U(θ) shown in Fig. 4.8(c).

80
ΔU θ = c o n s t (θ) [m V ]
μ 0 H [m T ]
2 0
1 5
1 0
5
0
-5
-1 0
-1 5
-2 0
-1 .8
-1 .1
-0 .3 6
Chapter 5
Surface plasmons in magnetic environments
3 0 4 0 5 0 6 0 7 0 8 0 9 0
θ
0 .3 6
1 .1
1 .8
1 .5
0 .0
0 1 2
Figure 5.9: Magnetic field sweep at constant angle of AuCoAu. In the center panel (a) the
magnetic field µ0H is plotted against the incident angle θ. The field is swept
from −20 mT to 20 mT to −20 mT in 0.2 mT steps and the angle from 30 ◦ to
90 ◦ in 0.2 ◦ steps. The color code displays the subtracted hysteresis U(H + ) −
U(H − ), where red is a positive difference and blue a negative difference. The
black curve shows R p and is just shown to guide the eye. The green line
indicates a horizontal cut through the measurement and the graph is shown in
the upper panel (b). The vertical red line indicates a vertical cut and is shown
in the right panel (b).
As mentioned in Sect. 5.1 ∆θSPP ≈ 0.001 or equivalent ∆kSPP0.4 × 10 −18 /m and can
thus hardly be directly determined by the experiment, instead it can be obtained by
fitting the measured data with a method described in [94]. Before normalising to
R p both ∆U(θ) (∆Uµ0H=const(θ) and ∆Uθ=const(θ)), are compared to (∂U(θ)/∂θ)∆θ.
Here, ∆θ = 0.001 is chosen because this is about the expected magnitude of order
for ∆θ [94, 54]. To determine the value of ∆θ by fitting, it would be necessary to
transform the ∆Uθ=const into a reflectivity, but this is not possible with this setup.
Furthermore, a simulation which is based on calculations done by Hermann et al. [14]
is also verified (Fig. 5.10). This simulation is based on Eq. (18) in [14]

Section 5.4
Reflectivity at constant magnetic field 81
∆RI = 2Re(R 0p ∆R pp ) (5.2)
Here, R 0p the plasmon resonance curve when no field is applied and ∆R pp is the
magneto-optical component of the reflectivity for p-polarised light (see Eq. (10)
and (15) in [14]). The simulation code is attached to this thesis Appendix D and
is done with following parameters: For the wavelength λ = 650 nm, magnetic field
µ0H = 7 mT, Voigt constant of cobalt Q = −0.02011+i0.0051 [95], and for the dielec-
tric constant the same as in Sec. 2.4.1 εCo = −13 + i18.95 [47]. Because ∆Uθ=const(θ)
p
R
1 .0
0 .8
1 .0
0 .8
Δ θ θ
0.001 δ δθ e t a l.
Δ θ
0.001 δ δθ e t a l.
3 0 4 0 5 0 6 0 7 0 8 0 9 0
θ
Figure 5.10: In the upper panel the black curve shows R p measured at µ0H = 0 mT. The
red curve is the derivative of R p where ∆θ = 0.001. The green curve is ∆U
from the constant angle measurement (upper panel in Fig. 5.9). And the
blue curve is the simulation after Hermann et al. [14]. In the lower panel
the same curves as in the upper panel are shown, except for ∆U from the
constant angle measurement. This curve is substituted by ∆U(θ) obtained
by a constant magnetisation measurement (dark yellow curve) which is the
same as the red curve in Fig. 5.3.
cannot easily transformed into R p , the curves are normalised to ∆Uθ=const(θ = θSPP)
and the maximum at ∆Uθ=const(θ ≈ 41 ◦ ) of ∂R p (θ)/∂θ. By normalising the curves
0 .8
0 .0
-0 .8
0 .8
0 .0
-0 .8
p /δθ x 1 0 -4
δR

82
Chapter 5
Surface plasmons in magnetic environments
the size of the value is changed and not the form of the curve, which is only of in-
terest here. Furthermore, small shifts in the resonance position, which occur due to
misalignments during different measurements, haven been corrected by shifting the
curve along θ. Now, comparing the curves in the upper panel it can be seen, that
the derivative ∂R p (θ)/∂θ (red curve) fits ∆Uθ=const(θ) (green curve) for θ ≥ 41 ◦ . For
θ < 41 ◦ ∂R p (θ)/∂θ shows a clear dip at θ ≈ θC, which is only a small in dip in the
measurement. The simulation (blue curve) instead shows a quite opposite behaviour
to ∂R p (θ)/∂θ. For θ ≤ 41 ◦ it fits ∆Uθ=const(θ). For θ > 41 ◦ it has a smaller reso-
nance width than ∆Uθ=const(θ) and then shows a more developed dip at θ ≈ θSPP than
∆Uθ=const(θ) and for increasing θ it then approaches again ∆Uθ=const(θ). The effect of
the parameters µ0H and Q in the simulation after Hermann et al. is the level of the
simulated values and ˜ε affects on the broadening of the peak. Thus the simulation
could be optimised but here it is emphasised that the same values are used as in the
simulations for R p in Chap. 4.
Continuing with the lower panel, here ∂R p (θ)/∂θ and the simulation are compared to
∆Uµ0H=const(θ) (dark yellow curve). Again the curves are normalised to ∂R p (θ)/∂θ in
same way like in the upper panel. Here, the simulation fits quite well to ∆Uµ0H=const
for θ ≈ 41 ◦ , whereas the resonance of ∂R p (θ)/∂θ is slightly broadened compared
to ∆Uµ0H=const(θ). For angles smaller and greater than θ = 41 ◦ the curves can-
not be compared, since there is drift in the signal of ∆Uµ0H=const(θ) as explained in
Sect. sec:sppmag.
Comparing ∆Uθ=const(θ) and ∆Uµ0H=const(θ) it can be said that both show a similar
behaviour what is expected since both curves show the change of the reflected signal
at two opposite applied magnetic fields. The broadening of ∆Uθ=constsec : sppmag
can be explained by a change of sample quality, as explained in Sec. 4.1.1.2. Further,
from the peak at θ ≈ 41 ◦ it can be concluded that there the greatest magneto-optical
response can be measured, what will be further discussed at the end of this section.
Comparing the here shown curves with curves shown in literature [15, 94, 14] and
Fig. 5.11, the here shown curves have approximately the same order of magnitude in
∆R p and show a similar behaviour, thus the here chosen ∆θ = 0.001 does seem to fit
well with the here measured data.
Now, in Fig. 5.12(a) the above shown curves are divide by R p . The curves are quite
similar to those shown in Fig. 5.10, they show a peak at the steep part of R p and
1 The label on the left axis has a typing error as it should say 10 −3 , what is obvious when reading
the paper.

Section 5.4
Reflectivity at constant magnetic field 83
R pp x 10 3
8
4
0
-4
10 nm Co
6 nm Co
4 nm Co
3 nm Co
2.8 nm Co
2.5 nm Co
40 45
Incidence angle (deg)
50
Figure 5.11: ∆R p (θ) measured by Clavero et al. [15]. The measurements are done for
varying thickness of Co in the trilayer Au/Co/Au. With decreasing Co thickness
the peak at θC < θ < θSPP shifts to greater θ and R p (θ) increases for
θ > θSPP 1 .
then a dip around θ = θSPP. But due to dividing by R p here the extrema are more
distinct than in Fig. 5.10. This is more clearly illustrated in Fig. 5.12(b), where the
simulation after Hermann et al. (turquoise curve) once divided by the measured R p
(blue curve) and the simulated Rp (magenta curve). As the change in the measured
Rp is small compared to the change in the simulated R p
simu , ∆Rp /Rp is suppressed
compared to ∆Rp /R p
simu . ∆Rp /R p
◦
simu shows a clear increased maximum at θ = 42
and small minimum at θ = 43 ◦ , this is similar to the behaviour of ∆Rp /Rp shown in
[15].
With Fig. 5.10 and Fig. 5.12 it gets clear that the greatest magneto optical effect on
R p can be seen in the range θC < θ < θSP P . In this range the surface plasmons are
excited which enhance the electric field component of the light wave (cf. Sect. 2.5.3)
and yield the greatest magneto optical response (peak in Fig. 5.12).
Furthermore, as mentioned in the beginning of this chapter ∆R p /R p is equal to the
transversal MOKE (TMOKE) signal normalised to R p [60, 94, 14, 16]. With this argu-
ment Fig. 5.9 can be understood as an angle dependent TMOKE measurement which
is enhanced by surface plasmons. To see the effect of the plasmons on the TMOKE
signal directly the TMOKE could be measured in a geometry where no plasmons are

84
p x 1 0 -4
p /δθ 1 /R
δR
0 .5
0 .0
Chapter 5
Surface plasmons in magnetic environments
1 .0 δ δθ ∗
4 0
2 0
0
Δ θ θ
e t a l.
Δ θ
e t a l.
e t a l.
e t a l.
4 0 4 2 4 4 4 6 4 8
Figure 5.12: In (a) the curves show the derivative of ∆R p with respect to θ normalised to
R p . The red curve is the normalised derivative, the green curve the normalised
of the measurement at constant angle, the blue curve the normalisation of the
simulation, and dark yellow curve the normalisation of the measurement at
constant field. On the right axis R p () is shown.
In (b) the simulation (turquoise curve) is compared to the simulation divided
by R p and to the simulation divided by the simulation of R p (magenta cuve).
excited (see Fig. 5.13) and then be compared to Fig. 5.9. Unfortunately, due to lack
of time this could not be done during this thesis, only certain angles were verified.
5.5 Transversal magneto-optical Kerr effect
To measure TMOKE the light illuminates the metal structure directly without pass-
ing through the prism before. With the existing setup this is done by rotating the
prism by 180 ◦ , as depicted in Fig. 5.13 (cf. Fig. 3.1). In the TMOKE configuration
no polarisation rotation is expected due to the Kerr effect (cf. Sect. 2.5), instead it
is only an intensity effect [14]. With Eq. 2.53 the angle dependence of the reflection
due the Kerr effect for the Au20 nmCo5 nmAu5 nm can be estimated. For this follow-
1 .0
0 .9
0 .8
0 .7
p
R

Section 5.5
Transversal magneto-optical Kerr effect 85
LASER
lter
beam
splitter polariser
detector 1
band-pass lter
θ
θ
magnet
θ - 2θ table
detector 2
analyser
Figure 5.13: Setup configuration to measure TMOKE. The setup is the same as shown in
Fig. 3.1, but here the prism is rotated by 180 ◦ . Thus the light is not coupled
into the prism, instead it hits the sample surface directly. Therefore no surface
plasmons are excited and only TMOKE can be measured.
ing assumptions are made: θ2 = 2θ1, QCo = −0.0201 − i0.0051 [95], n1 = 1, and
n2 = nCo = 2.41. Using these parameters the maximum intensity change is expected
between 70 ◦ to 90 ◦ .
With the configuration depicted in Fig. 5.13 TMOKE is measured at θ = 80 ◦ and
compared to the signal obtained in the surface plasmons configuration at θ = 80 ◦ , see
Fig. 5.14. The signals measured in the surface plasmons configuration will be referred
as SPPMOKE. To highlight the signal change of U(H) more clearly, the signal, nor-
malised to the background signal of the surface plasmon curve at the corresponding
measured θ, is plotted. To determine this background signal from the measured data
the difference between the maximum values of the up and down sweep is calculated,
denoted as U0. The normalised values are obtained with
Unorm(H) =
U(H) − U0
U(H)
δU
= . (5.3)
U(H)
As it can be obtained from Fig. 5.14, the coercive field for both measurement is

86
δU /U (H ) x 1 0 -3
2 .5
2 .0
1 .5
1 .0
0 .5
0 .0
-0 .5
-1 .0
-1 .5
-2 .0
-2 .5
Chapter 5
Surface plasmons in magnetic environments
-2 0 -1 0 0 1 0 2 0
Figure 5.14: Comparison of TMOKE and SPMOKE at θ = 80 ◦ . The figure shows
δU/U(M) against µ0H for a TMOKE measurement at θ = 80 ◦ (black curve)
and a surface plasmon measurement at θ = 80 ◦ (red curve). Also shown is
the approximate maximal δU/U(M).
µ0HC ≈ 7 mT. As both graphs show about the same order of magnitude in δU/U(H),
it can be concluded that at θ = 80 ◦ SPPMOKE (red curve), which is only about 1.5
times larger than the TMOKE signal (black curve), is equal to TMOKE and thus
there is hardly any enhancement of signal in the SPPMOKE configuration. This can
be explained by the fact that surface plasmons are not expected to be excited outside
of θC < θ < θSP P .
The greatest enhancement of the signal due to surface plasmons is in the range of
θC < θ < θSP P (cf. Sect. 5.4), therefore the TMOKE signal is measured at θ = 42.6 ◦
and 47.4 ◦ and is compared in Fig. 5.15 to the signals obtained in the SPPMOKE
configuration at these angles (cf. Fig. 5.8 and Fig. 5.9). The dip in the θ = 47.4 ◦
measurement (dark red) at µ0H ≈ 3 mT is just a measurement artifact and thus has
now physical meaning.
The figure shows for the measurements a coercive field of µ0HC ≈ 7 mT which is

Section 5.5
Transversal magneto-optical Kerr effect 87
δU /U (M ) x 1 0 -3
1 6 0
1 4 0
1 2 0
1 0 0
8 0
6 0
4 0
2 0
0
-2 0
-4 0
-6 0
-8 0
-1 0 0
-1 2 0
-1 4 0
-1 6 0
-1 0 0 1 0
0 .1 5
0 .0 0
-0 .1 5
-2 0 -1 0 0 1 0 2 0
Figure 5.15: Comparison of TMOKE and SPPMOKE in the surface plasmon enhancement
range. The figure shows δU/U(H) against µ0H for four different configurations.
The black curve is measured in the TMOKE configuration with an incident
angle of θ = 42.6 ◦ . The red curve is the hysteresis obtained with surface
plasmons at an incident angle of 42.6 ◦ , which is mirrored at µ0H = 0 mT.
And the dark red curve is obtained with surface plasmons at an incident angle
of 47.4 ◦ .
in good agreement with the coercive field from Fig. 5.14. For the normal TMOKE
measurements δU/U(H) ≈ 0.0001 what can only be seen in the inset. And for
SPPMOKE at θ = 47.4 ◦ the greatest δU/U(H) ≈ 0.071 is obtained, which is ≈ 9
times greater than for θ = 42.6 ◦ . Comparing the maximum SPPMOKE signal, which
is at θ = 47.4 ◦ , with the maximum TMOKE signal at θ = 80 ◦ , than SPPMOKE is
≈ 71 times greater than TMOKE. This suggests a field enhancement due to surface
plasmons of T el
max, AuCoAu ≈ 71. In Sect. 2.5.3 the enhancement due to surface plasmons
in a Au film was determined with Eq. (2.59)
T el
max = 1
ε ′(2)
2|ε ′(1) | 2
ε ′′(1)
|ε ′(1) |(ε′(0) − 1) − ε ′(0)
1 + |ε ′(1) .
|

88
Chapter 5
Surface plasmons in magnetic environments
Here, using ε (0) = εBK7 = 1.58 2 , and ε (1) = εAu = −12.1 + i1.42 or ε (1) = εCo =
−12.9 + i19.0 a field enhancement of T el
el
max, Au = 62 or Tmax, Co = 5 is obtained. This
is in good agreement with the experimental obtained value of T el
max, AuCoAu
garding that Eq. (2.59) is only valid for a single layer.
≈ 71, re-
By comparing Fig. 5.14 and Fig. 5.15, it can conclusively be said that for θ ≫ θSPP,
where no excitation of surface plasmons is expected, in the SPPMOKE configuration
directly the not enhanced TMOKE signal can be measured.
5.6 Magnetic anisotropy measurement with
TMOKE
From FMR measurements on Co (shown in Fig. 5.16) it is known that Co has uniaxial
magnetic anisotropy. Magnetic anisotropy means that there is a hard magnetisation
axis and perpendicular this a easy axis, and uniaxial refers to a 180 ◦ symmetry of the
anisotropy. For further readings on magnetic anisotropy it is referred to the diploma
thesis of Brandlmeier [96] and for further readings on the used FMR setup see the
diploma thesis of Schwarz [97]. The magnetic anisotropy of Co can be seen in Fig. 5.16
as a shifting of the resonance field what yields the "wave" like signal in panel (b) with
a periodicity of 180 ◦ . Because of this periodicity it can be said that cobalt has a
uniaxial anisotropy.
From simulating these measured FMR signals two anisotropy constants Ku,ip =
4.7 mT and Ku,oop = 0.5 mT are obtained and with the saturation magnetisation
of Co Ms = 1167 kA/m the TMOKE signal can be simulated. This simulation is
based on the Stoner-Wolfrath model [98] and minimises the total free energy to the
angle of magnetisation. With this a equilibrium orientation for each applied field is
obtained. And finally the projection of the magnetisation onto the magnetic field is
determined as the cosine of the angel between magnetisation and external applied
field, see [97]. The thus obtained hysteresis loops are shown for 0 ◦ , 45 ◦ and 90 ◦
in Fig. 5.17. Whereas in the simulation the hard axis is chosen to be along the 90 ◦
position. Due to the uniaxial anisotropy perpendicular to the hard axis there is a easy
axis (0 ◦ ). This can be seen in the simulations as the area enclosed in the hysteresis
loop tends to zero (panel (a)).
Therefore, when measuring TMOKE and rotating the AuCoAu sample by 90 ◦ a

Section 5.6
Magnetic anisotropy measurement with TMOKE 89
FMR Signal (w. E.)
(a)
60 80 100
µ 0 H (mT)
µ 0 H res (mT)
(b)
CoFeB: K uni, ip /M= 0,7 mT
Co: K uni, ip /M= 4,7 mT
in-plane-Orientierung α
Figure 5.16: FMR measurement of a glas/Co/Cu/CoFeB/Nb sample. The picture shows
a FMR measurement done by M. Schwarz [97]. In the left panel the raw
FMR measurement is shown. As the signal is measured with a Lock-In,
instead of peak in the signal a peak-dip can be seen. Each line in the left
panel corresponds to one angle position in the right panel. In (a) only the
resonances at smaller µ0H, what corresponds to the black curve in right panel,
is from interest as this is the resonance due to the Co. The resonance at larger
µ0H is due to the CoFeB.
change in the hysteresis loop could be expected. In principle this is with the in this
thesis used setup possible but not very accurate, but as the rotation of the sample has
to be done by hand it is not very accurate. Thus, only the three angles of 0 ◦ , 45 ◦ ,
and 90 ◦ are measured. The rotation is done in plane, this means that the magnetic
field vector is always parallel to the sample surface 2 Furthermore, it has to be added
that with this AuCoAu or AuNi sample used in this thesis no real change in the hys-
teresis curve can be expected, since the samples grown here are all polycrystalline and
thus should have no hard or easy axis. This is verified by the measurement shown in
Fig. 5.18. The figure shows the three measurements for 0 ◦ , 45 ◦ , and 90 ◦ , where 0 ◦ is
chosen arbitrarily. The three curves have all the same form as it is expected, thus the
sample shows no anisotropy. In the case of an anisotropy the hysteresis loop, for the
field applied along the easy axis, would have the form shown in Fig. 5.17(a). Whereas
for the field along the hard axis the hysteresis loop would look like Fig. 5.17(c). In
the simulated hysteresis loop for the 45 ◦ position (Fig. 5.17(b)) the field is applied
between the easy and hard axis. And because the measured hysteresis loops are all
2 It has to be said that this setup is not optimised to measured magnetic anisotropies, and this
measurement is only done to verify if it is possible to see a difference in the hysteresis curves.

90
s
M /M
1 .5
1 .0
0 .5
0 .0
-0 .5
-1 .0
-1 .5
-2 0 0 2 0
Chapter 5
Surface plasmons in magnetic environments
-2 0 0 2 0
-2 0 0 2 0
Figure 5.17: Simulation of the MOKE signal for Co. For the simulation Ku,ip = 4.7 mT,
Ku,oop = 0.5 mT, and Ms = 1167 kA/m is used and the solutions for 0 ◦ ,
45 ◦ and 90 ◦ are presented. These three angles refer to the angle between
magnetisation and applied field.
similar to Fig. 5.17(b) it follows, that independent of its orientation, the applied field
at the Co film is always orientated between an easy and a hard axis. This is due to
the polycrystalline structure of the Co film.
5.7 Summary
All in all, an influence of an applied magnetic field on the surface plasmon resonance
was confirmed. This influence can hardly be seen in constant field measurements (cf.
Fig. 5.2 and Fig. 5.7), but in constant angle measurements (cf. Fig. 5.6 and Fig. 5.9).
In constant angle measurements the influences were observable as hysteretical be-
hviour of the measured signal. These hysteresis loops, as expected from the surface
plasmons resonances shown in Chap. 4, were more clearly in AuCoAu than AuNi,
what is attributed to the covering Au layer. From comparison of simulations with
the measurements, it could be concluded that the influence of the magnetic field on
the surface plasmon resonance is the greatest in the range of θC < θ < θSPP and
then decreases with θ ≥ θSPP. For θ < θC and θ ≈ θSPP no influence could be seen.

Section 5.7
Summary 91
p
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
→
-2 0 -1 0 0 1 0 2 0
Figure 5.18: The graph shows R p against µ0H for three different sample orientations. The
black squares () show the arbitrary defined 0 ◦ position. The blue circles
() show the 45 ◦ position and the red triangles (△) the 90 ◦ position.
Furthermore, between θC < θ < θSPP and θ ≥ θSPP a switching of the hysteresis loop
could be observed.
As the simulations are based on ∆R p (θ)/R p (θ), what is equal to the magneto-optical
Kerr effect, it got clear that the constant angle measurements are equal to SPPTMOKE.
Therefore, constant angle measurements (SPPMOKE) were compared to TMOKE,
where no surface plasmons are excited. This was done by turning the prims by
180 ◦ . From comparison of SPPMOKE and TMOKE it followed that in the range
of θC < θ < θSPP, SPPMOKE is for AuCoAu ≈ 71 larger than for TMOKE, and
for θ ≥ θSPP SPPMOKE ≈ TMOKE. This is explained by the field enhancement of
surface plasmons in the range of θC < θ < θSPP and the vanishing excitation of surface
plasmons for θ ≥ θSPP.

92
Chapter 5
Surface plasmons in magnetic environments

Chapter 6
Conclusions and Outlook
Motivated by the recent fundamental and technological interest in surface magneto-
plasmons and their applications for integrated circuits, the aim of this thesis was to
study the influence of a magnetic field on the surface plasmon resonance. To this
end a setup was assembled and with this indeed an influence has been detected, as
summarised in the following.
Assembly to excite surface plasmons
The assembly of a setup to excite and detect surface plasmons in thin metal films
was one aim of this thesis. As discussed in Chapter 3, we started with an old x-ray
rotation table with two independent rotation stages, enabling a θ − 2θ rotation. This
rotation allowed us to measure an angle resolved surface plasmon resonance (R p (θ))
with a mechanical angle resolution of 0.005 ◦ .
As we used the Kretschmann-Raether configuration, a prism is in contact with the
metal films. To be able to align the prism in the optical path a prism stage was
designed, which on one side can easily be mounted onto the rotation stage and on the
other side allows a simple alignment of the prism.
To excite the surface plasmons a laser diode with a wavelength of 650 nm and 5 mW
output power, reduced by a ND = 1 filter to 0.5 mW, is used. A Glan Thompson po-
lariser p-polarises the incident light, as only in this polarisation surface plasmons are
excited. The light reflected at the sample surface then, passes a second Glan Thomp-
son polarsier, used as an anaylser, and is detected by a home-built photo-detector.
The analyser is adjusted, such that only p-polarised light can pass. Calibration mea-
surements for the photo-detector showed a linear scaling of the output voltage of the
93

94
Chapter 6
Conclusions and Outlook
photo-detector UA with the incoming laser light intensity and a noise voltage of the
setup of ≈ 0.5 µV/ √ Hz.
The magnetic field applied during the thesis is produced by a self-built electromagnet,
which is designed to fit on to the prism stage and further to yield a magnetic field of
≤ 30 mT, in the middle of its air gap. The whole setup is assembled in the transversal
geometry, which means that the applied magnetic field is perpendicular to the plane
of incidence of the light and parallel to the sample surface.
Excitation of surface plasmons in thin metallic films
In Chapter 4 we show that with the assembled setup, surface plasmons can be excited
and detected, by measuring the angle dependent reflectivity R p (θ) of a Ag film. The
excitation of surface plasmons can be seen as a clear dip in R p (θ) at θ = θSPP, the
resonance angle of the surface plasmons. We also compared, the measured R p exp(θ)
to the simulated R p
theo (θ), described in Chapter 2 (cf. Appendix A). The comparison
yield that measurement and simulation showed a similar behaviour in Rp (θ): A peak
at θ = θC the angle of total internal reflection at the glass-metal interface, followed
by a clear dip, the surface plasmon resonance, at θ = θSPP.
As single Ag layers have been excessively investigated before [9], we then moved
on to Ni, AuNi, AuCoAu samples. Comparing the surface plasmon measurements
with the simulations of these samples, they all show a peak at R p (θC) and a dip
at R p (θSPP). But also differences between measurement and simulations could be
observed for all samples: (i) A shift of the measured surface plasmon resonance
θSPP to larger θ, (ii) a reduced reflectivity at resonance as compared to simulation
R p exp(θSPP) < R p
theo (θSPP), (iii) a broadening of R p (θSPP), (iv) a small resonance at
large θ, (v) and R p exp(90 ◦ ) = R p
theo (90 ◦ ) = 1. A discussion of these observations led
us to propose following explanations for the discrepancies between measurement and
simulation: (i) The shift of θSPP to larger θ is most likely induced by surface roughness.
As surface roughness can act as a grating, an increase in the surface roughness will
increase the surface plasmon wave vector kSPP, which is analogous to a shift of θSPP.
Furthermore, due to this grating effect of the surface roughness, light can radiate in
the air half-space, which will reduce and broaden R p (θSPP), see (ii) and (iii).
The second resonance at greater θ (iv) can be explained by regarding the thin film
limit. In thin films two surface plasmons are propagating at the two surfaces of the

film. They interact, and cause a splitting of the dispersion relation into a high- and
a low-frequency mode. This leads to a second surface plasmon resonance at θ > θSPP
and a small shift of θSPP to smaller θ. But as the shift of θSPP is small compared to
the shift of θSPP due to surface roughness, only the second resonance is observable.
To address (v), different polarisation combinations for the incident and the reflected
light are compared. As in the p-s-configuration (p-polarised incident, s-polarised
reflected light) an increase of R ps (θ) for θ ≥ θSPP is observed. This indicates a po-
larisation rotation of the light from p- to s-polarised light, and the consequence is
R p (θ) < 1 for θ ≥ θSPP. Furthermore, R p exp(90 ◦ ) = 1 is also an indication for surface
roughness, as well. At θ = 90 ◦ the incident light is parallel to the surface of the
sample and the prism. It will thus not be refracted and hence all light is reflected
in an ideal sample. At finite roughness some of the light will be refracted, yielding
R p (90 ◦ ) < 1.
Despite these differences a broadening of the surface plasmon resonance and an in-
crease of R p expθSPP) is observed in samples which contain a magnetic material. This is
explained by the imaginary part of the dielectric function, which describes the absorp-
tion of light in a material and which is for magnetic materials greater than for noble
metals. Thus in magnetic materials the surface plasmon resonance is damped, causing
R p (θ) to be broaden. Overall, we find the best agreement between measurements and
simulations in AuCoAu. We attribute this to the particular properties of Au, which is
not prone to oxidisation. In contrast, Ag or Ni are covered by a thin dielectric layer,
when exposed to air. This dielectric layer, leading to a strong reduction of R p (θSPP).
Finally, from the resonance angle θSPP the energy of the surface plasmons can be
calculated and from the surface plasmon FWHM the energy uncertainty. For the
Au20 nmCo5 nmAu5 nm sample the energy of the surface plasmons is EAuCoAu = 2.1 ±
0.2 eV which is 2 times smaller than the theoretically obtained value for surface plas-
mons in Au EAu = 4.6 eV. The factor 2 is on one side due to the Co and on the other
side due to the roughness. Because of this energy decrease the surface plasmons in
AuCoAu are less localised.
Magnetic field dependence
In a finite magnetic field, the dispersion relation of surface plasmons is not symmetrical
regarding the propagation direction. For one surface plasmon propagation direction
95

96
Chapter 6
Conclusions and Outlook
the resonance frequency ωsp will decrease and for the other propagation direction it
will increase. This is analog to a switching of the magnetic field direction and will
cause a shift of θSPP. But this shift is only about ∆θSPP ≈ 0.001 ◦ and thus not
directly detectable with our setup.
Nevertheless, the impact of an external applied magnetic field (µ0H) on R p (θ) can be
measured, as shown in Chapter 5. On the one hand we recorded R p (θ) at an constant
µ0H. At first sight, no effect of µ0H is evident from this experiments. But when
measurements, taken at different values of µ0H, are subtracted from one another, a
change in R p (θ) of ∆R p (θ) ≈ 10 −3 at θC < θ < θSPP is resolved as a peak structure.
In a second measurement protocol, the reflectivity is recorded at a constant angle
θ, and the magnetic field is swept. The R p (H) thus obtained shows a hysteretical
behaviour. We observe a hysteresis of ∆U/U ≈ 0.05 1 in the range of θC < θ < θSPP,
no change at θSPP, and then again a large, but inverse hysteresis for θ > θSPP which
slowly decreases for θ ≫ θSPP.
Comparing the constant field measurements with constant angle measurements, both
show the same peak at θC < θ < θSPP at a given constant field. Furthermore, we
find that the magnetic field dependence of surface plasmons resembles transversal
magneto-optical Kerr effect (TMOKE) measurements as also reported in literature
[15]. A comparison of our experimental data with simulations and the literature
showed a good agreement in shape and in order of magnitude in signal level. To
investigate the influence of surface plasmons on the TMOKE signal (SPPMOKE),
the prism is rotated about 180 ◦ , allowing us to measure TMOKE and SPPMOKE
separately. A comparison shows that the largest SPPMOKE signal (in the range of
θC < θ < θSPP) is ≈ 71 times larger than the largest TMOKE signal (at θ ≫ θSPP).
This is because of the enhancement of the electric field component of the light wave,
which is due to surface plasmons. For a pure Au film the theoretical enhancement
is T el
max, Au
= 62 and for a Co film the enhancement is T el
max, Co
= 5 (cf. Sect. 5.5).
Furthermore, a direct comparison of SPPMOKE and TMOKE at the same angle
θ ≫ θSPP shows SPPMOKE ≈ TMOKE. Thus, it can be concluded, that in the
surface plasmon configuration, in constant angle measurements an enhanced TMOKE
and the true TMOKE can be recorded in one measurement.
In summary, the largest change in the magneto-optical signal is observed in the range
of θC < θ < θSPP. In this range surface plasmons are excited, which enhance the
1 Here, the measured voltage is given, as it is not possible to determine a reflectivity in these
measurements with this setup.

electromagnetic field due confinement and in turn increasing the magneto-optical
signal. For θ ≫ θSPP the excitation condition for surface plasmons is not fulfilled,
thus the signal is hardly enhanced. That is why, for θ ≫ θSPP, the "true" TMOKE
signal can be measured in the surface plasmon configuration, enabling to measure and
compare SPPMOKE and TMOKE without changing the setup configuration.
As described in Sect. 5.3, the magneto-optical signal can be described by the derivative
of R p (θ) [60]. With this, the vanishing at θ = θSPP and the rotation at θ > θSPP of
the hysteresis loop is explained, as the derivative of R p (θ) becomes zero for θ = θSPP
and changes the sign for θ > θSPP.
To date, angle resolved surface magneto-plasmon resonance curves, shown in Fig. 5.9,
have not been reported. The literature only shows magneto-plasmon resonances and
their differences for two opposite magnetic fields [15, 60] but never for a magnetic
field and angle sweep at the same time. Hence, the direct comparison of SPPMOKE
and TMOKE in one measurement, as shown in this thesis, is not reported to date.
The great enhancement of the surface plasmons on the magneto-optical signal makes
SPPMOKE to a great tool to characterise magnetic properties of ferromagnetic films
especially those which show a small TMOKE signal.
Outlook
From the above described experiments some new interesting question evolved which
can be investigated in further experiments. But prior to going into detail, some im-
provements to the experimental setup are proposed.
One major improvement would be to exchange the built prism stage by a x-y-z-stage,
simplifying and making the adjustment of the prism in the optical path more accu-
rate.
During experiments, it turned out that the photo-diode of the home-built detector is
slightly off-axis, thus the detector has to be somewhat rotated, so that the reflected
light strikes the photo-diode. Rebuilding the detector more accuratly is therefore ad-
vised. Furthermore, by mounting a lens into the reflected light path, the spot size can
be reduced significantly, which makes the position correction for the photo-detector
obsolete, as it has been tested to the end of this thesis.
The magnetic field experiments showed, that the shift of θSPP is in the order of 0.001 ◦ ,
thus it would be of great interest to improve the angle resolution. One option would
97

98
Chapter 6
Conclusions and Outlook
be to change the stepper motor mode from full step to eighth step mode, what should
enable a mechanical angle resolution of 0.0006 ◦ . So it would be possible to determine
∆θSPP experimentally. This has not be done before in literature. To date this is only
done by adapting the simulation to the experimentally observed data.
From the experimental point of view it would be very interesting to measure the
magnetic anisotropy with SPPMOKE, as discussed and attempted in Sect. 5.6. But
the in this thesis used sample were polycrystalline, thus no anisotropy could be seen.
To this end, first of all a crystalline sample has to be grown. Magnetite (Fe3O4) on
MgO appears as an ideal candidate, as FMR measurements on this material showed
a clear anisotropy and substantial knowledge in growing Fe3O4 films is available at
the WMI [96]. Figure 6.1 shows the surface plasmon spectrum expected in a Fe3O4
film. As MgO has a refractive index of nMgO = 1.734 [47] the BK7 prism needs to be
p (θ)
R
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
ε
ε
ε
ε
λ
θ
0 2 0 4 0 6 0 8 0
θ
Figure 6.1: Simulation of Rp (θ) for Fe3O4/Au. The red curve shows a simulated Rp (θ) of
a 20 nmFe3O4 film covered with 5 nm Au. When a 1µm MgO layer is added
beneath the Fe3O4 film, the black curve is obtained. The simulation parameters
are as follow: εSF10 = 1.722 [99, 100], εMgO = 1.732 [47], εFe3O4 = 5 + i3.4 [99],
εAu = −12.1 + i1.4, and λ = 650 nm.
θ

exchanged with a SF10 prism with nSF10 = 1.721 and of course the index matching
fluid has to be exchanged, too. Because of the slight mismatch of nSF10 and nMgO, in
the simulation of MgO/Fe3O4/Au for θ > θSPP small steps can be seen, which are not
present when only Fe3O4 is simulated (cf. black and red curve in Fig. 6.1). The rest
of the curve is similar to simulations shown in this thesis (cf. Fig.4.8). It shows an
increase of R p (θ) at θ = θC = 35 ◦ and a broad resonance at θ = θSPP = 50 ◦ .
In such a sample, it should be possible to detect an anisotropy in the hysteresis curves
as depicted in Fig. 5.17. The thus obtained anisotropy could then be compared to
those obtained from FMR measurements.
Furthermore, the fact, that Fe3O4 shows a great temperature dependent electrical con-
ductivity [96] a strong temperature dependent surface plasmon resonance in Fe3O4
can be expected. Since, surface plasmons are oscillations of quasi free-electrons, the
surface plasmons resonance should vanish with decreasing temperature. Because, then
the electrical conductivity and thus the number of free charge carriers decreases, es-
pecially below the Verwey transition at 121 K [101], where Fe3O4 becomes an isolator
[96].
One interesting aspect of surface plasmons is the possibility to channel light in sub-
wavelength structures, which then could lead to integrated circuits. But to built such
a circuit it requires components such as switches.
As shown in Chap. 5, a small magnetic field shifts the surface plasmon resonance an-
gle what is equivalent to a change of the surface plasmon wave vector. Hence, when
coupling light via a grating coupler into Au/Co/Au slab waveguide, an influence of
a magnetic field on the surface plasmon wave vector can be expected. Therefore,
it would be of great interest for future applications like a magneto-optical switch, to
investigate the propagation of the surface plasmons along a waveguide in the presence
of a magnetic field.
In conclusion, the effect of an external magnetic field on the surface plasmons reso-
nance offers a great potential for further fundamental physics experiments, as well as
for future applications.
99

100
Chapter 6
Conclusions and Outlook

Appendix A
Simulation code for the surface
plasmon resonance
Clear["Global`*"];
(*Numbers of layers*)
n = 3;
(*Espilon metal*)
er = 0.1392844036 2 − 4.12853211 2 ; (*Ag*)
ei = 2 ∗ 0.1392844036 ∗ 4.12853211;
e1 = er + I* ei;
e2r= 2.013809523809524 2 − 3.8076190476190477 2 ; (*Ni*)
e2i = 2 ∗ 2.013809523809524 ∗ 3.8076190476190477;
e2 = e2r + I* e2i;
e3 = −12.1 + I ∗ 1.42; (*Au*)
e6r = 2.237 2 − 4.236 2 ; (*Co*)
e6i = 2 ∗ 2.237 ∗ 4.236;
e6 = e6r + I*e6i;
e7 = 1.7338; (*MgO*)
eps = List[1.58 2 , e3, e6, e3, 1];
101

102
(*Thickness of the layers in Angstrom*)
d = List[120, 80, 120];
(*mu of every material*)
mu = List[1, 1, 1, 1, 1];
(*k-vectors*)
omega[lambda_] := (2*Pi*c)/lambda
c = 3 ∗ 10 8 ;
(*kx incident beam in prism *)
Appendix A
Simulation code for the surface plasmon resonance
kx[lambda_, theta_] := omega[lambda]/c*Sqrt[eps[[1]]]*Sin[theta ◦ ] (*kzi, Raether 2.3*)
kzi[lambda_, theta_, i_] := Sqrt[eps[[i]]*(omega[lambda]/c ) 2 -kx[lambda, theta] 2 ]
(* for p- waves *)
t[lambda_, theta_, i_] := (2*eps[[i]]*mu[[i]]*kzi[lambda, theta, i+1])/(eps[[i]]*mu[[i]]*
kzi[lambda, theta, i+1]+eps[[i+1]]*mu[[i+1]]*kzi[lambda, theta, i]) (*Yeh 5.1-14*)
r[lambda_, theta_, i_] := (Sqrt[eps[[i]]*mu[[i]]] 2 *kzi[lambda, theta,i+1]-Sqrt[eps[[i+1]]*
mu[[i+1]]] 2 *kzi[lambda, theta,i])/(Sqrt[eps[[i]]*mu[[i]]] 2 *kzi[lambda, theta, i+1]+
Sqrt[eps[[i+1]]*mu[[i+1]]] 2 *kzi[lambda, theta,i]) (*Yeh 5.1-13*)
Di[lambda_, theta_,i_] := Chop[1/t[lambda, theta, i]]*{{1,r[lambda, theta, i]},
{r[lambda, theta,i], 1}}(*transmission matrix; Yeh 5.1-12*)
Districh[lambda_, theta_,i_] := 1/t[lambda, theta,i-1]*{{1, r[lambda, theta, i-1]},
{r[lambda, theta, i-1], 1}}(*transmission matrix for 1st interface; Yeh 5.1-12*)
P[lambda_, theta_,i_] := {{Exp[-I*kzi[lambda, theta, i]*d[[i-1]]], 0},
{0, Exp[I*kzi[lambda, theta, i]*d[[i-1]]]}} (*propagation matrix*)
(* for s-waves *)
ts[lambda_,theta_, i_] := 2*kzi[lambda, theta, i]/(kzi[lambda, theta, i]+
kzi[lambda, theta, i+1])

Appendix A
Simulation code for the surface plasmon resonance 103
rs[lambda_,theta_, i_] := (kzi[lambda, theta, i]-kzi[lambda, theta, i+1])
/(kzi[lambda, theta, i]+kzi[lambda, theta, i+1])
Dis[lambda_, theta_, i_] := Chop[1/ts[lambda, theta, i]]*
{{1, rs[lambda, theta, i]}, {rs[lambda, theta, i],1}}
Disstrich[lambda_, theta_, i_] := 1/ts[lambda, theta, i-1]*
{{1, rs[lambda, theta, i-1]}, {rs[lambda, theta, i-1], 1}}
(*transfer matrix for p-waves*)
M1[lambda_, theta_ ] :=
{
}
Matrix1[lambda, theta] = {{1,0},{0,1}};
Do[
]
Matrix1[lambda, theta] = Matrix1[lambda, theta].
P[lambda, theta, i].Di[lambda, theta, i];
, {i, 2, n+1}
Matrix1[lambda, theta]
M[lambda_, theta_] := Districh[lambda, theta, 2].M1[lambda, theta][[1]]
(*transfer matrix for s-waves*)
M1s[lambda_, theta_ ] := {{1,0},{0,1}};
Do[M1s[lambda_, theta_] = M1s[lambda, theta].P[lambda, theta, i].
Dis[lambda, theta, i], {i, 2, n+1}]
Ms[lambda_, theta_] := Disstrich[lambda, theta, 2].M1s[lambda, theta](*Yeh 5.1-27*)
(*Reflectivity*)
(*p-wave*)
R[lambda_, theta_] := Abs[Extract[M[lambda, theta], {2,1}]
/Extract[M[lambda, theta], {1,1}]] 2 (*s-wave*)
Rs[lambda_, theta_] := Abs[Extract[Ms[lambda, theta], {2,1}]

104
/Extract[Ms[lambda, theta], {1,1}]] 2
(*Find optimal thicknes, to this end FindMinimum of R,
start at reasonable Theta*)
Optthick = {};
Do[
Appendix A
Simulation code for the surface plasmon resonance
d = ReplacePart[d, 1-> d1]; (*chose material to be optimised*)
Optthick = Append[
]
ListPlot[Optthick]
(*Field enhancement*)
Optthick,
{d1,z= First[
}
]
FindMinimum[
]; ,{d1, 80, 300, 10}
]
{R[6500, theta], 404
Tmax[eps_] := 1/1*(2* Abs[Re[eps]] 2 )/(Im[eps])*Sqrt[Abs[Re[eps]](1.58 2 -1)
-1.58 2 ]/(1+Abs[Re[eps]])
Print["T_max: ", Tmax[e3]]

Appendix B
CAD sketches
5 mm
50 mm
Figure B.1: Prism stage
105
50 mm

106
40 mm
10 mm
130 mm
Figure B.2: Stage holder
Appendix B
CAD sketches

Appendix C
Prism correction
To determine the correct position of the detector when the light is not reflected at the
middle of the hypotenuse, it is important to know the angle about which the detector
has to be turned. This angle will be called γ, and will be calculated in the following
paragraph. The difficulty in determining γ is that only the two angles θ and β, the
hypotenuse length of the prism h, and the distance between the middle of the prism
and the detector r are known, see Fig. C.2.
First of all the distance a of which the incident beam is shifted when the incident
beam is not perpendicular to the adjacent is calculated.
θ
45°
h/2
Figure C.1: The sketch illustrates how the beam is shifted away from the middle of the
prism
β
d
θ-β
Thereforre, starting with the law of sines, we can write.
107
a

108
and
d
sin 45 =
h/2
sin(90 + θ)
⇒ d = h sin 45
2 sin(90 + θ)
a
sin =
⇒ a =
Substituting the first into the second equation yields
a = h
2
Now, knowing a the angle γ can be calculated.
45°
h/2
Appendix C
Prism correction
(C.1)
(C.2)
d
sin(135 + β)
(C.3)
d sin(θ − β)
.
sin(135 + β)
(C.4)
sin(θ − β)
. (C.5)
cos θ(cos β − sin β)
a
θ
β
45 -θ
γ
θ
r
s’’ b
90 +θ
Figure C.2: Sketch to illustrate the angles and paths needed to calculate γ.
Starting again with the law of sines and s = s ′ + s ′′ , the following two relations are
obtained:
r
s’
Δr
ψ
c

Appendix C
Prism correction 109
h/2
sin(90 + θ) =
h/2 + a
sin(90 + β) =
With those two equations s ′′ can be calculated.
s ′′ = s − s ′ =
(h/2 + a) sin(45 − β)
sin(90 + β)
s ′
sin(45 − θ)
s
sin(45 − β)
Substituting a into s ′′ and with the cos this yields b.
−
h/2 sin(45 − θ)
sin(90 + θ)
b = s ′′ cos θ = h
sin(45 − β) cos θ
+
2 cos β
sin(θ − β) sin(45 − β)
− sin(45 − θ)
cos β(cos β − sin β)
(C.6)
(C.7)
(C.8)
(C.9)
Now, because the detector moves on a circle around the prism, the radius can be
expressed with r 2 = x 2 + y 2 . With y = b and x the projection onto the red line.
x = √ r 2 − b 2 ⇒ ∆r = r − √ r 2 − b 2 (C.10)
Again with the law of sines and ψ = cos
−1 c
b
c
sin γ =
r
sin(90 + ψ)
c cos ψ
⇒ sin γ =
r
−1 c cos γ
⇒ γ = sin
r
= sin −1
, γ is obtained.
√ √
∆r2 − b2 ∆r2 + b2 r
= sin −1 ∆r2 + b 2
rb
= sin −1 r2 − 2r √ r 2 − b 2 + r 2 − b 2 + b 2
rb
b
(C.11)
When substituting Eq. (C.9) into a Eq. (C.11), an equation for γ is obtained which
depends only on measurable quantities.

110
Appendix C
Prism correction

Appendix D
Code for SPP simulation after
Hermann et al.
Clear["Global`*"];
e3 = -12.1+I*1.42; (*Au Raether p .126 Tab. A.2*)
e6r = 2.41 2 − 4.27 2 ; (*Co, (www.luxpop.com)*)
e6i = 2 ∗ 2.41 ∗ 4.27;
e6 = e6r+I*e6i;
eps = List[1.541 2 , e3, e6, e3, 1];
(*Thickness of the layers in Angstrom*)
d = List[200, 50, 50];
dges = Sum[d[[i]], i,Length[d]];
zc = d[[1]]+d[[2]]/2;
(*Voigt Numbers *)
Q = −0.0201 − I ∗ 0.0051; (*Co: http://www.msd.anl.gov/groups/mf/jmkerrcalc.php*)
(*k vectors*)
omega[lambda_] := (2*Pi*c)/lambda
c = 3*10 8 ;
kx[lambda_, theta_] := omega[lambda]/c*Sqrt[eps[[1]]]*Sin[theta ◦ ] (*kx incident beam
in prism*)
kzi[lambda_, theta_, i_] := Sqrt[eps[[i]]*(omega[lambda]/c ) 2 -
kx[lambda, theta] 2 ] (*Raether 2.3 *)
as[lambda_, theta_, i_] := kzi[lambda, theta,i]/(omega[lambda]/c) (*Gl.3*)
111

112
Appendix D
Code for SPP simulation after Hermann et al.
a[lambda_, theta_, i_] := -kzi[lambda, theta, i]/(omega[lambda]/c*eps[[i]]) (*Gl.3*)
ni[i_] := Sqrt[1/2*(Sqrt[Re[eps[[i]]] 2 +Im[eps[[i]]] 2 ]+Re[eps[[i]]])] (*Gl.3*)
gamma[i_,j_] := ni[i]/ni[j] (*Gl.3*)
(*p-waves*)
rij[lambda theta_ ,i_, j_] := (a[lambda, theta, i]-a[lambda, theta, j])/
(a[lambda, theta, i]+a[lambda, theta, j]) (*Gl .3*)
tij[lambda_,theta_,i_,j_] := 1+rij[lambda, theta, i, j] (*Raether A .16*)
tijh[lambda_, theta_, i_, j_] := 2*gamma[i,j]*a[lambda, theta, i]/
(a[lambda, theta, i]+a[lambda, theta, j]) (*Gl .3*)
e21x[lambda_,theta_, z_] := tij[lambda, theta, 1, 2]*Exp[I*kzi[lambda, theta, 2]*z]*
((1-rij[lambda, heta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*(dges-z)])/(1+
rij[lambda, theta, 1, 2]*rij[lambda, theta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*
dges]))*kzi[lambda, theta, 2]/(eps[[2]]*(omega[lambda]/c)) (*Gl .8b*)
e21z[lambda_,theta_,z_] := tij[lambda, theta, 1, 2]*Exp[I*kzi[lambda, theta, 2]*z]*
((1+rij[lambda, theta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*(dges-z)])/(1+
rij[lambda, theta, 1, 2]*rij[lambda, theta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*
dges]))*kx[lambda, theta]/(eps[[2]]*(omega[lambda]/c)) (*Gl .8b*)
(*s-waves*)
rijs[lambda_, theta_ , i_, j_] := (as[lambda, theta, i]-as[lambda, theta, j])/
(as[lambda, theta, i]+as[lambda, theta, j]) (*Gl .3*)
tijs[lambda_, theta_, i_, j_] := 2*as[lambda, theta, i]/(as[lambda,theta,i]+
as[lambda, theta, j]) (*Gl .3*)
e21s[lambda_, theta_, z_] := tijs[lambda, theta, 1, 2]*Exp[I*kzi[lambda, theta, 2]*z]*
((1+rijs[lambda, theta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*(dges-z)])/(1-
rijs[lambda, theta, 1, 2]*rijs[lambda, theta, 2, 5]*Exp[2*I*
kzi[lambda, theta, 2]*dges])) (*Gl .8a*)
(*Reflection pure Au*)
(*p-waves*)
Ra[lambda_, theta_] := (rij[lambda, theta, 1, 2] + rij[lambda,theta, 2, 5]*Exp[2*I*
kzi[lambda, theta, 2]*dges])/(1+rij[lambda, theta, 1, 2]*rij[lambda, theta, 2, 5]*
Exp[2*I*kzi[lambda, theta, 2]*dges]) (*Gl .7*)
(*s-waves*)
Ras[lambda_, theta_] := (rijs[lambda, theta, 1, 2] + rijs[lambda, theta, 2, 5]*
Exp[2*I*kzi[lambda, theta, 2]*dges])/(1+rijs[lambda, theta, 1, 2]*
rijs[lambda, theta, 2, 5]*Exp[2*I*kzi[lambda, theta, 2]*dges]) (*Gl .7*)

Appendix D
Code for SPP simulation after Hermann et al. 113
(*∆ R/R*)
(*p-waves*)
dRR[lambda_, theta_] := ((I*(omega[lambda]/c) 2 *d[[2]]/(2*kzi[lambda, theta,1]))*
(-e21x[lambda, theta, zc]*e21x[lambda, theta, zc]*(eps[[3]]-eps[[2]])-
e21z[lambda, theta, zc]*e21z[lambda, theta, zc]*(eps[[3]]-eps[[2]])*
eps[[3]]/eps[[2]]))/Ra[lambda,theta] (*Gl.A2a*)
(*s-waves*)
dRRs[lambda_, theta_] := (I*(omega[lambda]/c) 2 *d[[2]]/(2*kzi[lambda, theta, 1]))*
(-e21s[lambda, theta, zc]*e21s[lambda, theta, zc]*(eps[[3]]-eps[[2]]))
Ras[lambda, theta] (*Gl.A2a*)
(*∆ T/T*)
(*p-waves*)
dTT[lambda_,theta_] := (I*(omega[lambda]/c) 2 *d[[2]]*(eps[[3]]-eps[[2]])/
(2*kzi[lambda, theta, 2]*tij[lambda, theta, 1, 2]))*
(
(Exp[I*kzi[lambda, theta, 2]zc]-rij[lambda, theta, 2, 1]Exp[-I*kzi[lambda, theta,2]*
zc])*kzi[lambda, theta, 2]/(eps[[2]]*(omega[lambda]/c))*(eps[[3]]-eps[[2]])*
e21x[lambda, theta, zc]-(Exp[I*kzi[lambda, theta, 2]*zc]+rij[lambda, theta, 2, 1]*
Exp[-I*kzi[lambda, theta, 2]*zc])*kx[lambda, theta]/(eps[[2]]*(omega[lambda]/c))*
(eps[[3]]-eps[[2]])*eps[[3]]/eps[[2]]*e21z[lambda, theta, zc] ) (*Gl.A2b*)
(*s-waves*)
dTTs[lambda_,theta_] := (I*(omega[lambda]/c) 2 *d[[2]]*(eps[[3]]-eps[[2]])/
(2*kzi[lambda, theta, 2]*tijs[lambda, theta, 1, 2]))*(Exp[I*kzi[lambda, theta, 2]*
zc]+rijs[lambda, theta, 2, 1]*Exp[-I*kzi[lambda, theta, 2]*zc])*
e21s[lambda, theta, zc] (*Gl.A2b*)
(*Reflection trilayer no M*)
(*p-waves*)
R0a[lambda_, theta_] := Ra[lambda, theta]*(1+(dRR[lambda, theta]/
(1-dTT[lambda, theta]))) (*Gl.A1*)
(*s-waves*)
R0as[lambda_, theta_] := Ras[lambda, theta]*(1+(dRRs[lambda, theta]/
(1-dTTs[lambda, theta]))) (*Gl.A1*)
(*Reflectivity*)
Ri[lambda_, theta_] := Abs[R0a[lambda,theta]-R0as[lambda,theta]] 2 /4 (*Gl .19*)

114
(*MO Reflectivity*)
Appendix D
Code for SPP simulation after Hermann et al.
dRpp[lambda_, theta_, M_] := I*(omega[lambda]/c) 2 *d[[2]]/(2*kzi[lambda,theta,1])*
Q*M*e21z[lambda, theta, zc]*e21x[lambda, theta, zc]/(1-dTT[lambda, theta]) 2 (*Gl.15*)
(*Difference of Reflectivity*)
DRi[lambda_, theta_, M_] := 2*Re[R0a[lambda, theta]Conjugate[dRpp[lambda, theta, M]]]
(*Gl.18*)

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Acknowledgement
Finally, I want to thank all those people who made this thesis possible and supported
me during my time at the Walther-Meissner-Insitut. In particular:
Prof. Dr. Rudolf Gross, for the great opportunity to accomplish my diploma thesis at
the WMI, for supervising this thesis and for his support finding a PhD position. And
not to forget his descriptive explanations of physics and his great solid state script.
Dr. Sebastian T. B. Gönnenwein, for his great mentoring and support over the last
year. For his never-ending enthusiasm to find these plasmons, his discussion on mea-
surement results, and for his great ideas improving the setup. And also for the patient
review of this thesis and his support finding a PhD.
Dr. Hans Hübel, for his discussion on measurement results and his review of parts of
the thesis.
Andreas Brandlmaier, for his great support on Mathematica, which saved me hours
of work. He and Stephan Geprägs, who were always willing to listen to my questions
and always had a good advice.
Mathias Weiler, for his great support on LabVIEW and helping me to make the
FMR setup and software running.
Matthias Althammer, for his help on programming the LabVIEW software to control
the stepper motors.
Franz Czeschka, for his help and support with the FMR measurements.
Manuel Schwarz, for his support on the FMR setup.
123

Robert Müller, Helmut Thies and his crew for their time and support manufacturing
the components of the setup. And also the lab assistants for their nice atmosphere at
work.
And all the other people from the WMI who supported me during the last year. Espe-
cially the guys from 042, among others Christian Heeg, Alexander Krupp, Johannes
Manz, and Martin Radlmeier, for the great time at the office and their discussions on
physics and other topics.
Furthermore, I want to thank all my friends who supported me not only during the
last year but also during my whole studies in Göttingen and Munich. Who helped
me to forget physics from time to time.
Most of all, I want to thank my siblings and my parents for the great support in
every possible aspect and their great encouragement during my studies, which were
not possible without their help. Thank you!
124