Characterisation of metallic particle distributions by scanning near-field optical microscopy (SNOM) in simultaneous reflection and transmission mode
©2015
Textbook
43 Pages
Summary
Unlike conventional optics, scanning near-field optical microscopy (SNOM) overcomesthe Rayleigh criterion and can therefore achieve better resolutions than conventionaloptical microscopes. This feature is utilized to measure the optical propertiesof different silver particle distributions on a glass surface. This paper mainlylays focus on intensity correction of the optical data due to topographical artifacts,analysis of plasmonic behavior and a tentative representation of the optical data.The simple approach for optical artifact correction has been shown to yield qualitativesuccess, with necessity of improvement for quantitative results. Given theconditions of the experiment, it has also been observed that plasmonic couplingseems to have a greater impact on the small observed particles. The tentative representationof the optics suggests that the larger particles are able to emit light byabsorption of electromagnetic energy from their surrounding.
Excerpt
Table Of Contents
Acronyms
SNOM Scanning near-field optical microscope/microscopy
SEM Scanning electron microscope
Nd:YAG laser Neodymium-doped yttrium aluminum garnet laser
AC
Alternating Current
Q-Factor Quality Factor
USB Universal Serial Bus
CCD Charge-coupled device
Symbols and Constants
E
Electromagnetic Field Vector [V/m]
r
Position Vector [m]
t
Time [s]
Angular Wave Frequency [1/s]
c
Speed of Light [m/s]
,
Angle [rad]
R, r Radius [m]
D
Diameter [m]
Wavelength [m]
I
Intensity [W/m
2
]
H
Magnetic Field [T]
n
Refractive Index [ ]
h
Height [m]
IV
Abstract
Unlike conventional optics, scanning near-field optical microscopy (SNOM) over-
comes the Rayleigh criterion and can therefore achieve better resolutions than con-
ventional optical microscopes. This feature is utilized to measure the optical prop-
erties of different silver particle distributions on a glass surface. This paper mainly
lays focus on intensity correction of the optical data due to topographical artifacts,
analysis of plasmonic behavior and a tentative representation of the optical data.
The simple approach for optical artifact correction has been shown to yield qual-
itative success, with necessity of improvement for quantitative results. Given the
conditions of the experiment, it has also been observed that plasmonic coupling
seems to have a greater impact on the small observed particles. The tentative rep-
resentation of the optics suggests that the larger particles are able to emit light by
absorption of electromagnetic energy from their surrounding.
1
Introduction
SNOM is a form of scanning probe microscopy, which can produce optical pictures that
transcend the far field resolution limit given by the Rayleigh criterion. This technique is
used to analyze the optical properties of random silver particle distributions on a glass
substrate. Topography and optics are measured simultaneously with resolutions in the
magnitude of 100 nm [1]. Therefore, SNOM excels in optical images, but underperforms
in topographical resolutions when contrasted to methods like atomic force - or scanning
tunneling microscopy, which both can achieve atomic resolutions [2].
In order to theoretically understand and appreciate the SNOM, one must first look into
the - usually applicable - optical resolution limit given by the Rayleigh criterion.
The next step involves the comprehension of how the SNOM overcomes this limit by
detection of the near field. This detected field however, is measured in the far field -
demanding a comprehension of the near-field/far-field translation.
After the acquired optical pictures are freed from artifacts, they can be examined in terms
of their plasmonic properties.
1
2
Physical Principles
In the chapter, the near field and its advantages over the far field shall be examined. The
Rayleigh criterion is first needed to understand the reason in utilizing this near field. It
will then be gone over to a theoretical approach revealing the plausibility of near field
measurements via far field. After then showing the physical origin of the this field, the
functioning principle of the setup will be gone over.
2.1
Near and Far Field
The near and far field describe regions in distance to a source of radiation. The border
between near and far field lies at about one wavelength from the source [3].
Figure 1: Visualization of Near and Far Field intensity [4]
As illustrated above, electromagnetic waves directly behind a light source are very in-
tense and closely correlate with shape and dimensions thereof. This distance of about one
wavelength to a source is called near field.
After passing through a transition zone, the light reaches the far field which is located two
wavelengths from the source until infinity. This light does not contain all the information
needed to draw conclusions about the source.
The reason for this behavior is that the near field contains non-propagating waves which
exponentially decay leaving only propagating waves for the far field.
The field of an electromagnetic wave E(r, t) can be written as
E(r, t) = E
0
· e
i(k·r-t)
(1)
with wave vector
|k| = k
2
x
+ k
2
y
+ k
2
z
=
c
.
r
- point of evaluation
t
- time
- angular frequency of the wave
c
- speed of light
k
i
- wave vector in direction i
2
As mentioned, waves of the far field propagate requiring all components of k to be real. In
the near field however, non-propagating - evanescent - waves outweigh, describable with
at least one imaginary component of k.
2.1.1
Limitation of Far-Field Measurement
The Rayleigh criterion declares the smallest resolvable distance between two light-sources
in the Far-Field.
To derive the criterion, one can use an illuminated pinhole as a light-source. The pattern
emerging from this pinhole is called an Airy disk and is shown below [5].
Figure 2: Airy Disks with logarithmic intensities. The middle shows two Airy Disks at
resolution limit. Left and Right show fully resolvable and resolvable Disks. [6]
The criterion asserts, that two light sources are indistinguishable, when the center of one
airy disk lies on the first minimum of the second disk. Figure (2b) shows two Airy Disks
at resolution limit.
In order to mathematically derive the Rayleigh criterion, one must formulate the 2-
dimensional Fourier transform of a an illuminated pinhole - merely using propagating
waves - and then determine the first minimum of the emerging Airy disk in the far field.
The 2-dimensional Fourier transform is given by:
F (u, v) =
f (x, y)e
-i(ux+vy)
dxdy .
(2)
3
The illuminated pinhole with radius R can be described by
x
y
= r
cos()
sin()
, by integrating
r from 0 to R and from 0 to 2
The emitted propagating waves with wave vector - traveling in the direction determined
by - can be described by
u
v
=
cos()
sin()
.
Using polar coordinates and normalizing the field intensity f(x,y) to 1 results in
F (u, v) =
R
0
2
0
e
-ir(cos()cos()+sin()sin())
rdrd =
R
0
2
0
e
-ir·cos(-)
rdrd
(3)
for the occurrent field. The trigonometric addition theorem
cos(a - b) = sin(a)sin(b) + cos(a)cos(b)
(4)
was used for simplification.
The Bessel function
J
1
(s) =
m=0
(
-1)
m
m!(m + 2)
s
2
2m+1
(5)
can be used to solve formula (3), resulting in
F (, ) = R
2
2J
1
(R)
R
.
(6)
The minimum intensity is given for R = 3.83, keeping in mind that the intensity is the
square amplitude of F.
When the geometrical relation
k
0
· sin() =
(7)
is reorganized and R = 3.83 is inserted, this finally gives the Rayleigh criterion:
sin() =
k
0
=
3.83
2R
=
1.22
2R
=
1.22
D
(8)
where is the angular resolution, D the minimum resolvable distance and the wave-
length of the used light. D is now minimized by maximizing sin() which results in:
D = 1.22
(9)
Therefore, the best obtainable resolution is approximately 1.22, meaning the lower the
frequency of the used light, the better the resolution.
4
2.1.2
Evanescent Wave Measurement via Slit
This section shall show the plausibility of near-field detection via far field with a mathe-
matical approach. It is necessary as our detectors are far over two wavelengths away from
the illuminated particles thus only measure light in the far field.
Figure 3: Abstraction of two distinct particles by a double slit. Detection via classical
microscope (a) and via SNOM (b).
As seen in figure (3), two separate particles with radius L, which shall be seen as sources
of electromagnetic radiation, will be represented by a double slit with distance 2d. The
double slit is illuminated by a monochromatic field E
0
(x, z). In first approximation, the
field just behind the aperture (z=0) is then given by [7]:
E
1
(x, z = 0) = E
0
(x, z = 0) · [C(x, d - L, d + L) + C(x, -d - L, -d + L)]
(10)
with the rectangular function C(x, a, b) defined to be equal to 1 in the interval [a,b].
The Fourier transform of formula (10) yields:
E
1
(k
x
, z = 0) = 4 · E
0
· cos(k
x
d)
sin(k
x
L)
k
x
(11)
where k
x
is the wave vector in x direction.
The field at distance Z is determined by back-transformation into the position space with
z = Z:
E
near
(x, z = Z) =
1
2
+
-
dk
x
e
-ik
x
x
E
1
(k
x
, z = 0)e
-i
k
2
-k
2
x
Z
(12)
As shown in (2.1), the following statements are true for the integration interval considering
a 2-dimensional wave vector k:
k
2
- k
2
x
= k
z
C |k
x
| >
c
k
2
- k
2
x
= k
z
R |k
x
| <
c
5
As mentioned in (2.1), (k
z
C) implies evanescent waves. The exponentially decreasing
amplitude of these waves cause them not to be detected if the distance Z is larger than
the wavelength . In this case the integration interval only includes propagating parts:
E
far
(x, z = Z) =
1
2
+
c
-
c
dk
x
e
-ik
x
x
E
1
(k
x
, z = 0)e
-i
k
2
-k
2
x
Z
(13)
Now a slit of size 2l, representing the SNOM-tip, will be placed in front of the double
slit. The slit should be smaller than the wavelength of the light as this will overcome the
Rayleigh criterion shown in chapter (2.1.1). It shall be centered at the point (z= ,x=X),
in very small distance . (
)
By using the same approximation as above, the field just behind the slit is given by:
E
2
(x, z = ) = E
near
· C(x, X - l, X + l)
(14)
The Fourier transform right behind the slit is:
E
2
(x, z = ) =
1
2
+
-
dk
x
E
1
(k
x
, z = 0)e
-i
k
2
-k
2
x
2
· sin(k
x
- k
x
)l
(k
x
- k
x
)
e
i(k
x
-k
x
)X
(15)
After propagation to a detector in the far field at z = Z, the field in the position space is:
E
far2
(x, z = Z) =
1
(2)
2
+
c
-
c
dk
x
e
-ik
x
x
e
-i
k
2
-k
2
x
(Z
- )
×
+
-
dk
x
E
1
(k
x
, z = 0)e
-i
k
2
-k
2
x
2
· sin(k
x
- k
x
)
(k
x
- k
x
)
e
i(k
x
-k
x
)X
(16)
If now the equations (13) and (16) are compared, it can be seen that the slit - or SNOM
tip - causes an integration from
- to + to appear in the equation and therefore
including complex values for
k
2
- k
2
x
. In other words containing evanescent waves.
2.2
Evanescent Waves and Plasmons
In contrary to the mathematical approach in (2.1.2), this section will focus on the physical
origin and properties of the near field.
Similar to quantum tunneling - which results from the continuous solutions to the Schrödinger
equation - evanescent waves are the consequence of the continuous electric and magnetic
fields at a boundary [8]. Naturally, evanescent waves are always present but only gain
detectable amplitudes when generated by Plasmons.
6
Figure 4:
Simulated Near-Field for an illuminated Silver Particle using COMSOL
( = 532 nm) [9]
Plasmons are charge density oscillations which can be driven by light and described math-
ematically as a dipole p(t), with an electromagnetic field given by [8]:
E
R
= 2
p(t)
R
3
+
p (t)
cR
2
cos()
(17)
E
=
p(t)
R
3
+
p (t)
cR
2
+
p (t)
c
2
R
sin()
(18)
H
=
p (t)
cR
2
+
p (t)
c
2
R
sin()
(19)
with p(t) = p
0
cos(t -
R
c
)
p(t) - dipole moment
R
- distance of dipole to point of evaluation
Figure 5: Schematic representation of evanescent waves propagating along a metal-
dielectric surface. The exponential dependence of the electromagnetic field intensity is
shown on the right. [10]
The intensity decline after leaving the surface as seen in figure (5) can be formulated as
follows [11]:
I(Z) = I(0)e
-Z
4
n
2
s
sin
2
()
-1
(20)
7
The formula reflects the exponentially falling intensity when in distance Z from the surface.
It also shows a dependency of the wavelength , the average angle of the surface and
the reflective index n
s
of the substrate.
2.3
Plasmon Resonance from Theory
Plasmon resonance is dependent on many factors [12] such as shape, size or the dielectric
properties of an object and its surrounding. Even though the shape of the particles in
this experiment is spherical, the plasmon resonance is nontrivial and must be calculated
numerically.
Figure 6: (a),(b) multipolar resonance wavelengths of silver spheres for
free space (n
out
= 1) and in suspension (n
out
= 1.5). The used laser frequency of 532 nm
is plotted horizontally. [13]
The plasmon resonance frequency of silver spheres in relation to the radius thereof can
be seen in figure (6). On the left, the relation in air with a refractive index of 1 can be
seen in contrast to glass with a refractive index of approximately 1.5 seen on the right.
The used laser frequency of 532 nm is plotted horizontally. The average particle size will
be shown in chapter (4) but has already been plotted as a dashed line.
Most surface area of the silver particles is surrounded by air, only some of it is in contact
with the glass surface suggesting the actual resonance frequency to be between the two
graphs but strongly leaning towards the left graph with a refractive index of 1. As can be
seen in the figure, dipole plasmons (l=1) and quadrupole plasmons (l=2) are of greatest
interest, as small particles will hit the dipole resonance and large particle the quadrupole
resonance.
8
Details
- Pages
- Type of Edition
- Erstausgabe
- Year
- 2015
- ISBN (eBook)
- 9783954898763
- ISBN (Softcover)
- 9783954893768
- File size
- 10.7 MB
- Language
- English
- Publication date
- 2015 (March)
- Keywords
- characterisation snom
