Download Light confinement in ordered plasmonic metallic arrays and in microcavity structures

Ben-Gurion University of the Negev
Faculty of Natural Sciences
Physics Department
Light confinement in ordered plasmonic metallic arrays and in
microcavity structures
Submitted by: Daniel Marx
Supervised by: Mr. Rotem Kupfer and Prof. Ilana Bar
Abstract
The behavior of light in silver nanoparticle arrays and microcavities was studied using detailed finitedifference time-domain (FDTD) simulations. We examined the relation between shape and size of the
nano-structures, the space (the dielectric) between them and the electromagnetic field inside the array
with possible implications on surface enhanced Raman spectroscopy (SERS). Next, we determined the
optimal conditions for confinement of a steady second electromagnetic mode (the Gaussian-Hermite
modes: 10 and 01) inside a microcavity at the center of a structure with nanoscale circles.
Introduction
Plasmonics
A plasmon is a quasiparticle resulting from the quantization of free electron oscillations in metals, similar
to how phonons result from a quantization of mechanical vibrations. We distinguish between oscillations
at the metal bulk (bulk plasmon) and the coupled electromagnetic-charge oscillations at the metaldielectric surface [surface plasmon (SP)]. This phenomenon can be interpreted as a macroscopic dielectric
function that is obtained by a resonant absorption at the plasmon frequency. Surface plasmons have
different dispersion relations, as a result of the dielectric function of the metal; when phase matching
conditions are met, energy can be transferred from the electromagnetic wave to propagation of a SP at the
metal-dielectric surface. When a laser is exciting a SP at a frequency near its resonance, the SP will have
a shorter wavelength than the photon exciting it (at the same frequency), the wavelength will usually be at
the nanoscale [1].
Surface plasmon-based photonics - or plasmonics - is in essence the combination of photonics and
electronics at the nano scale, and has some rather interesting properties such as overcoming diffraction
limit and using microcavity structures to trap electromagnetic waves (e.g., visible light). Both may have
several applications, for example by overcoming the diffraction limit, one is able to build photonic
circuits at a smaller scale thus creating faster computer chips. In addition, it is possible to create a perfect
lens through a negative refraction index and to make certain materials transparent [1] [2] [3].
Objectives and Significance
At this stage, our purpose is:
1) To check the influence of the array parameters (distance between elements, their shape, etc.) on
the localization and enhancement of the incident electromagnetic field.
2) To trap light, and to obtain the conditions for achieving the steady transverse modes 01 or 10.
The results that will be obtained will contribute to our understanding on the factors that lead to
localization and enhancement. This understanding may have implications on surface-enhanced Raman
spectroscopy (SERS), where electromagnetic enhancement plays a great role. [4].
Methods
The approach we took was somewhat different than the usual approach, as the structure itself was
composed of an array of silver particles of different shapes, with vacuum in between rather than the other
way around; this was intentional - in order to see if this kind of structure would work. Furthermore, the
possibility to trap light in unique structures was tested.
The Finite-Difference Time-Domain method (FDTD)
The FDTD is a numerical method for direct time integration of Maxwell’s curl equations by a finite
difference scheme on a discrete electromagnetic mesh. A finite-difference scheme for solving Maxwell's
equations was used as early as 1928, the algorithm's foundations were laid by Yee [5] and the algorithm
was further refined by Taflove [6], and its uses are expanded and continued to be researched today.
This method enables us to specify the permittivity, permeability, and incident fields (by setting
appropriate boundary conditions) and to model dispersive materials by auxiliary differential equation
(ADE) formulation [7,8].
The basic scheme is given in appendix A.
We used a temporal and spatial Gaussian source, introduced on the simulation boundary by the total-fieldscattering-field (TF-SF) formulation:
*
+
√
(
)
)1(
where
is the amplitude,
is the discrete time step, X – the size of the xaxis from the computational domain in meters (actually greater than the physical space of the simulation),
is the pulse duration, and
waist size of the Gaussian beam.
The computational domain is discretized using a grid (a multiple-dimension array), which was a square in
our case, the dielectric properties are specified for each intersection of the grid - or “node” -which let us
solve Maxwell's equations in the time-domain at the node. In order to get a viable solution in each cell,
Courant's condition [6,7] must be met:
)2(
In our case we used
Drude model
In this project, we modeled the response of metallic materials that have frequency dependent dispersions
and so in the time domain they do not hold a constant value for the permittivity. For correct working of
the algorithm, we needed to use the permittivity/susceptibility function for which there are three different
models: Lorentz, Debye and Drude model. The Debye and Drude models are simpler than the Lorentz
model, but they hold only at long wavelength response (> 300 nm) [9] .
In our simulation we used the Drude model:
Drude model is named after Paul Drude who first proposed it in 1900, the model was first used to explain
transport properties of electrons in metals, and was developed using the classical approach of kinetic
theory for electrons - treating them as an ideal gas [10].
Although the model derivation from kinetic theory is conceptually false, its results were found to be
reasonably accurate in describing the dielectric response of metals at the low frequency limit.
The dielectric function according to the Drude model is (for a full derivation see appendix B):
)3(
Where
frequency and
is the inverse of the relaxation time,
is the unit step function.
is the plasma
By incorporating the ADE derived from Drude model, we get the following numerical scheme.
)4(
)5(
)6(
)7(
An explicit expression for
and
can be found in the derivation of the ADE (appendix B).
Local Field Enhancement in Metallic Nanoparticle Array
Verification of the numerical model
To show that the enhancement is consistent with previous results, we will refer to the work of Hao and
Schatz [4]- and try to replicate some of the results. In their work, they used the discrete dipole
approximation - not the FDTD method – albeit, as will be shown, similar results were obtained.
The structures that will be tested are those of dimers; for our case: two isosceles triangles with parallel
bases and two circles.
Four patterns were tried, two with a space of 10 nm between them (Figure 1.a) and two additional with
320 nm between them (Figure 1.b) – this was done for the purpose of showing that the amplification of
the electric field using the dimer as opposed to two monomers that are relatively far away from each other
is higher in the former.
Taking results from the above cited publication - where the peak for the circles was at a wavelength of
410 nm and the peak for the triangles was at at a wavelength 650 nm – we used the peak wavelength for
each dimer. The enchancement can be seen as a sharp peak: for the triangles (Figure 1.c) the enhancement
at a distance of 320nm was larger than the enhancement in the circle monomers' field (Figure 1.d), this is
most likely because of enhancement around sharp corners occurring for each triangle monomer.
Figure 1.a: The shapes of the dimmers with space between the nanoparticles of 10 nm.
Figure 1.b: The shapes of the monomers, where the nanoparticles are separated.
Figure 1.c: A comparison between the maximum electric field when the triangles are 'far-away' (Left), and when the
shapes are 10nm apart (Right)
Triangles
Distance between monomers: 10 nm
Distance between monomers: 320 nm
Electric field (V/m)
6.3127
3.6885
Figure 1.d: A comparison between the maximum electric field when the circles are 'far-away' (Left), and when the shapes
are 10 nm apart (Right), notice there are two spikes.
Circle
Distance between monomers: 10 nm
Distance between monomers: 320 nm
Electric field (V/m)
5.9375
2.1945
Results and Discussion
Circle Arrays
We simulated the propagation of a light beam with a wavelength of 700 nm – which is in the
neighborhood of the resonance wavelength for this array – by propagating a spatial and temporal
Gaussian pulse through a nanoparticle array of bulk circles. The size of the circles ranged from 60 to 130
nm in radius, while the electromagnetic wave propagated approached from the side of the array (Figure
2.a).
Figure 2.a: An array of circles, each with a radius of 60 nm with a 700 nm Gaussian beam propagating from the left side
Two types of parameters have been changed: the effective distance - the width of the vacuum area
between one nanodot and the other; and the distance from their centers - the distance between one
nanodot and the other. While one parameter was changed the other stayed constant.
The arrays used had 5x5,4x4 and 3x3 circles - almost no difference was observed between them, thus the
results shown are from the 5x5 array.
In the first simulation, the distance between the centers of two circles was (vertically and horizontally, but
not diagonally) 275 nm. The peak electric field amplitude as a function of radius and effective space are
presented in Figures 2.b and 2.c, respectively. We found that a larger a radius or rather a tighter space increases the electric field exponentially, therefore by using a fit; the maximum value (for the minimum
distance between circles) for the electric field is:
.
Figure 2.b: The electric
field as a function of the
effective radius of the
nanodots.
Figure 2.c: Electric field as
a function of the distance
between two circles – with
a fit for exponential decay
This result is not enough to determine the effect of radius change, and therefore we need to make another
measurement with a constant effective space between two circles.
For a constant distance of 15 nm the maximum electric field lies at 121 nm (Figure 2.d), which is larger
than the previous maximum value, implying that the radius does indeed matter. The second result make it
clear that the tighter space between the dots affects the electric field more efficiently, which is consistent
with previous results [1,4,9]. Also, it is apparent that the radius has an effect too, although it is unclear
how the radius truly effects the change of electric field.
Figure 2.d: The electric field as a function of the nanodot radius, for a fixed distance of 15 nm between the centers of two
nanodots.
Square Array
To have another perspective on the effect of the size of the elements in the array and the space between
them, there is a need to try another shape for each element. Therefore, a square array (Figure 3.a), with
edge sizes ranging from 150 to 260 nm and a 285 nm constant distance between two centers was tested.
As can be seen in Figure 3.b, the array of squares had a much stronger peak electric field than the circle
array - having a minimum value approximately the same as the maximum value of the circle array, which
is consistent with the strong localization of surface-plasmons around sharp corners [11], [12]).
Figure 3.a: An array of squares, each with an edge of 150 nm with a 700 nm Gaussian beam propagating from the left
side.
Figure 3.b: The amplification of the electric field, due to surface plasmons created at the inter-surface of air-silver square
particles.
Like previously, larger squares/smaller distance between the particles means a stronger electric field,
although not in all cases (Figure 3.c). Using interpolation the maximum intensity for a distance of 10 nm
between the nanosquares:
Figure 3.c: The Electric field as a function of each square's edge size (left) and as a function of the distance between
squares (right).
Again, further examination is needed to determine which of the parameters matter most: using a constant
distance between squares and changing their sizes.
Contrary to the result before, small squares increase the electric field, and thus we can rule out the trend
implied by the left graph in figure 6.
The results (Figure 3.d) imply that, given a constant distance of 15 nm between squares the arrays with
smaller squares produce a greater electric field, up until a point - 240 nm.
Maximal value:
Figure 3.d: Electric field as a function of the edge size for each square in the array, while using a constant distance
between squares.
To summarize, first the enhancement for dimers was tested to show that the results are consistent with
previous publications. Then, we simulated the electric field propagation inside arrays of particles of two
shapes: circles and squares. We looked for the relation between the intensity of the electric field and the
layout of the array, the size of the objects inside it and the distance between these objects.
The results clearly show the relation between the proximity of the elements in the array and the
amplification of the electric field, but no obvious trend with regard to the size of the elements was
obtained.
Stable EM Modes in Microcavity Arrays
Next, we examined a microcavity structure, similar in effect to photonic crystals (PCs) [13].
To determine how efficiently the light was trapped, the maximum intensity of the magnetic field was
determined at points outside and inside of the desired trapping area, using the conversion
(8)
,
And while compensating for Courant condition.
For the second transverse mode, 3 properties were checked: lifetime of second mode, maximum intensity
and symmetry of mode.
Trapping
First, light trapping was attempted. The structure itself was heavily inspired by the traditional approach,
such as the shape of the structure [14] [15], albeit the structure itself was different - having the dielectric
(vacuum) and metal switching places.
The entire size of the pattern changed drastically (
, in both axes) as the size of the dots and
spaces was altered; the beam propagated circularly
The structure used was probably not optimal – although the 00 mode was mainly present it was not steady
– which is likely due to improper matching between the structure and the light incident wavelength (thus,
not being in resonance), although a few combinations have been tried.
Figure 4.a: An example of the used pattern.
The simulations where performed using a point source located in the middle of the grid (5 μm for each
axis) of the microcavity, which itself was in the range of 200 - 400 nm width, with a distance of 20 - 50
nm between the circles, which were of 250-350 nm size. In all of the simulations we had near-field
pumping as a result of the source being very close to (less than a wavelength), and inside the silver array.
Figure 4.b: The electric field at onset (Left) and after about 10 femtoseconds.
We found that smaller space between the circles (with the same cavity) leads to a stronger intensity at the
cavity center and through its vicinity.
“Leakage” refers to the average intensity outside the cavity, a minimal leakage is desired - and is again
achieved by minimal space.
Hermite-Gaussian modes 01 and 10
Trapping of a higher mode, which has a smaller transverse wavelength than the original wavelength used,
requires a tighter formation with maximum distance of 30 nm between two circles. To achieve a second
mode inside the cavity, a certain asymmetry had to be introduced, and modes in an asymmetric cavity are
often called Hermite-Gaussian modes.
The asymmetry was present in three parameters: cavity size, position of beam (Figure 5.b) and size of
circles/spaces. The cavity was wider in one axis (the y axis), and so it was shaped like a diamond (Figure
5.a) with max width: 1080 nm and max length 1780 nm .
The dots marked Vdots and Hdots were different and the ratio Vdots/Hdots was always greater than 1.
Figure 5.a: The structure used, showing which circles are Vdots, Hdots, and the unchanged circles.
Figure 5.b: The beam is not centered but lays about at 200 nm vertically from the center, and the source starts as a single
point.
This was not the smallest cavity tried; however, beyond this point, a steady mode was not achieved.
The position of the beam was not constant, and was never in the center of the cavity:
the beam had to be in a range of 150 to 300 nm from center, too close to the center and mode 00 was
prevalent, too far - and the results were inconsistent.
The asymmetry in the size of the circles was done so that better reflection will be present in one axis and
was inspired by the work of Painter et al. [14] this was not crucial, but better assisted in having a steady
mode.
The size of circles and space between them varied; however the better reflection (tighter circles) was at
the end of the length of the cavity, otherwise the mode was even less steady than with symmetry in this
parameter.
The asymmetry in the mode itself was at the cost of the asymmetries introduced in the parameters (Figure
5.c)
Figure 5.c: The asymmetry of the modes for vertical orientation (Left), and horizontal orientation (Right)
Trying to fix this by making the cavity larger in one side resulted in mode 00. The results varied between
horizontal and vertical patterns - the duration of the desired mode and the average magnetic field peaked
at different places (Figure 5.d)
Figure 1.d: The average magnetic field and the duration of the mode as a function of the distance of the source from the
center for vertical and horizontal orientations.
The duration of the modes for both orientations, i.e., vertical and horizontal is at 18 nm from the center,
while the peak average magnetic field was at 18 and 20 nm for the horizontal and vertical modes
respectively.
Another parameter left to check was the ratio between the size of the dots. The size of the vertical dots did
not change and had a diameter of 350 nm, while the size of the horizontal dots was smaller (Figures 5.e,
5.f)
Figure 5.e: The duration of modes (in fs) as a function Vdots/Hdots ratios, red represents the horizontal layout and black
represents the vertical layout.
Figure 5.f: Results shown for vertical (Left) and horizontal (Right) layouts: the average magnetic field, as a function of
the Vdots/Hdots ratio.
So the most stable mode depends on the factors, for the vertical case the longest duration does not imply
a stronger average magnetic field.
To summarize, as mentioned above, our main objective was to show a steady Gauss-Hermite mode 10
or 01, and determine the parameters which allow obtaining these steady modes; but first, we have shown
that a simpler mode (00) can be achieved, implying that the structure works as intended.
The parameters required for obtaining the 01 and 10 steady modes are shown in Table I and Table II,
respectively (in both layouts Vdots was constant 350 nm).
Table I. The parameters of the structure, required for obtaining the 01 steady mode
Distance of beam from center
(nm)
Size of Hdots (nm)
Max. Duration
18
336
Max. Average Intensity
20
350
Table II. The parameters of the structure, required for obtaining the 10 steady mode
Distance of beam from center
(nm)
Size of Hdots (nm)
Max. Duration
18
339
Max. Average Intensity
18
336
Conclusions
In this project we have used a computational electrodynamics approach to study the optical characteristics
of silver nano-particles, in dimers, monomers, arrays, and optical microcavities. It was demonstrated that
a small distance (~10 nm) between the nanoparticles results in a major enhancement of the local electric
field and the influence of various array parameters on this enhancement was determined. Additionally, we
were able to mimic the effects of PC microcavities using a special array of silver nanoparticles, and it was
shown that more than one transverse mode in the optical microcavity is possible. The conditions for a
the stability of the 01 and 10 Gaussian modes were found.
Appendix A – The Finite-Difference Time-Domain method
A finite difference scheme uses an approximation of derivatives in order to get a discrete solution while
still express the change in a function.
Take:
*
where
(
(
+ , expressing the function as a Taylor series gives us:
)
( )
)
(
)
If we subtract the second equation from the first:
(
)
(
)
( )
Dividing by :
(
For sufficiently small
)
(
)
(exact conditions vary, but in our case we used Courant's condition):
(
)
(
)
This approximation is sometimes called a second order accuracy finite difference – due to the neglect of
the second order term - although higher orders of accuracy are also used.
When using
, the finite-difference is often called an (
) time step.
The basic FDTD scheme first proposed by Kane Yee [5,7] employs second order accuracy central
differences, and its layout is:
1. Replace derivatives in Maxwell's equations with finite differences, for discretized space and time
2. Solve the difference equations to get new equations for the fields after the time step (using the
known fields before the time step)
3. Evaluate the magnetic and electric field (one at a time) from the equations in 2. Now the fields
are known, and another time step can be made.
4. Repeat 2,3 for the duration of the simulation.
Appendix B – Drude model derivation
The permittivity and basic Drude model derivation
(Note: time-domain vectors will be bold and frequency-domain vectors will be added a hat)
In Drude model we assume electric charges are particles that are in a force field (electric field) and also
are influenced by a damping force.
The equation of motion for a charge is:
)9(
Where q is the charge, m is the mass,
is the electric field , is the damping coefficient
Using Fourier transform we can get the equation for the frequency domain:
̂
̂
̂
And we get
̂
̂
)11(
For a polarization vector with N dipoles per unit volume
̂
̂
)11(
And so:
̂
We note
̂
̂
)12(
which is the plasma frequency, and thus
̂
̂
)13(
And the permittivity is
)14(
Because we are interested in the time domain, we apply a Fourier transform again, to get equation (3):
where
is the unit step function.
The derivation of the Auxiliary Differential Equation
Writing the susceptibility as
We get
̂
̂
̂
̂
And so
̂
̂
̂
Using Fourier transform to express the above result in time-domain will yield:
For a time step of
:
Or:
(
)
(
)
Appendix C – Transverse Modes
When an electromagnetic beam is propagated, its intensity profile will generally change during
propagation; however, as a result from boundary conditions imposed upon it from the waveguide (usually
an optical resonator, the electromagnetic field creates several patterns of constant amplitude profile in the
plane perpendicular to the direction of propagation – these patterns are called transverse modes.
For cylindrical symmetry, the modes are denoted by
(Transverse electromagnetic modes of order
pl), or sometimes just , where p is the radial mode order, and l is the angular mode order.
Intensity of the mode at point
is given by the equation:
(
Where
)
)15(
is the Laguerre polynomial of order p and index l
Gaussian beam and Hermite-Gaussian Modes
When asymmetry between the horizontal and vertical measurements of a laser cavity is introduced, the
Intensity/amplitude description is given by a product of a Gaussian and a Hermite polynomial (hence the
name), which is a solution to the. These modes are denoted by
where m is the horizontal mode,
and n is the vertical mode.
Hermite-Gaussian modes are usually derived by solving the paraxial approximation for the Helmholtz
equation:
)16(
Where
is the wave number, and
is the transverse Laplace operator.
The way to describe the mode is given by the electric field:
*
Setting
(√
)
(
)
+*
(√
)
gives us a Gaussian beam which we used as source.
(
)
+
Figure C.1 : Some Hermite-Gaussian modes

In this simulation the only magnitude of the amplitude was given, instead of a signed value, and
so the graphs will show the absolute value of the function.
References
[1]
E. Ozbay, Science 311, 189 (2006).
[2]
A. Alù and N. Engheta, Phys. Rev. E 72, 016623 (2005).
[3]
J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[4]
E. Hao and G. C. Schatz, J. Chem. Phys. 120, 357 (2004).
[5]
K. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966).
[6]
A. Taflove and M. E. Brodwin, IEEE Trans. Microw. Theory Tech. 23, 623 (1975).
[7]
J. Schneider, Understanding the Finite-Difference Time-Domain Method,
www.eecs.wsu.edu/~schneidj/ufdtd , (2010)
[8]
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference
Time-Domain Method (Artech House, Norwood, MA 2005).
[9]
S. Gray and T. Kupka, Phys. Rev. B 68, 045415 (2003).
[10]
N. Mermin and N. Ashcroft, Solid State Physics , Saunders College, (1976).
[11]
B. Sturman, E. Podivilov, and M. Gorkunov, Phys. Rev. B 87, 115406 (2013).
[12]
-
94, 2837 (2003).
[13]
P. Russell, Science 299, 358 (2003).
[14]
J. Vuckovic, O. Painter, Y. X. Y. Xu, A. Yariv, and A. Scherer, IEEE J. Quantum Electron.
35, 1168 (1999).
[15]
V
vić
L
H
i
S
v
65, 016608 (2002).