Magneto-Plasmonic Gold & Nickel Core-Shell Structures

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TVE-F 19010

Examensarbete 15 hp 2019-06-11

Magneto-Plasmonic Gold & Nickel Core-Shell Structures

Max Brynolf

Rohini Sengupta

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Magneto-Plasmonic Gold & Nickel Core-Shell Structures

Max Brynolf, Rohini Sengupta

The presented project explores the optical properties of magneto- plasmonic Au/Ni core-shell structures. The work aims at controlling dimensions and parameters in order to influence and analyze the optical properties of the nanostructures. The softwares utilized for

the simulations were COMSOL Multiphysics 5.1 and MATLAB. Experimental results were acquired from labs done at Ångströms laboratory. From the research based study where the gold to nickel ratio was

influenced, it was observed that the transmissions for the

nanostructures at the differing wavelengths produced transmissions of similar bearings. Modes for certain wavelengths were found in correspondence with the transmissions and could potentially render explanations for influence on the optical properties of the

nanostructures. Conclusively, it can be stated that the optical properties of the nanostructures could be influenced and controlled by varying the dimensions and properties of the said structure.

Differing dimensions corresponded to noteworthy changes in the cross sections, the transmissions as well as the mode formations.

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1 Populärvetenskaplig sammanfattning

I projektet undersöks de optiska egenskaperna hos guld/nickel-nanostrukturer med varierande dimensionering. Undersökningen görs med hjälp av simuleringsprogram- met COMSOL. Egenskaper som undersöks är transmissioner samt olika former av tvärsnitt, såsom spridningstvärsnitt och absorptionstvärsnitt, vilka är mått på hur väl nanostrukturen sprider respektive absorberar inkommande ljus. Därutöver undersöks olika fältfördelningar över nanopartiklarnas yta, varmed olika så kallade modes kan identifieras.

Resultaten framhäver en väsentlig skillnad i optiska egenskaper mellan olika dimen- sioneringar av nanostrukturen. Dessutom syns flertalet modes i fältfördelningen över nanopartiklarnas ytor. Jämförelse med motsvarande grafer från laborationsexper- iment saknar styrkande effekt, då fåtal likheter är synliga, vilket kan undersökas vidare i framtida projekt.

Projektet utgör en mindre del av ett större syfte, där olika nanostrukturer undersöks och utvecklas i hopp om att finna nya tillämpningar. Forskningsområdet är relativt nytt och det finns mycket kvar att utforska. I framtiden skulle nanoteknologier exem- pelvis kunna utgöra en viktig del av elektroniken, där de optiska egenskaperna kan kontrolleras med hjälp av strukturen.

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2 Introduction

For many years researchers, scientists and engineers have been experimenting with the diverse properties and characteristics of electromagnetic radiation and optics in order to attain desired outcomes on a plethora of different nanostructures. Studies have recently grown to include research and development on a relatively new field, namely metasurfaces and plasmonics.

Conventional techniques relating to optics rely heavily on the propagation effect of light. Through the ages conventional methods of optics have been used to study light, including its behaviour and effects, but to shape the wavefronts to be studied, light propagation has had to be done over distances much larger than the wavelength itself.

Optical devices, such as lenses and polarizing filters etc. can modify the wavefront of light but only by changing properties like its amplitude, phase, polarization etc.

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Metasurfaces however, have opened up the possibility of studying the very same effects without any dependence on the propagation effect of light. These surfaces are often made of arrays of scatterers or “optical antennas” which can constitute a wide range of metallic or dielectric microparticles or nanopaticles. When utilizing metasurfaces, the optical properties on a surface can be controlled by designing interactions between light and the optical scatterers. (1)

The three main features that metasurfaces encompass which differ from conventional components and which therefore allow for studies independent of the propagation effects of light are the following: (i) wavefront shaping is achieved at much shorter distances than that of the wavelength from the interface after a light ray crosses or reflects from the metasurfaces (ii) amplitude, phase and polarization response can be designed and controlled by metasurfaces and its optical scatterers at subwavelength resolutions (iii) interaction of the nanoparticles with not only electric fields but also magnetic fields is possible. (1)

Plasmonics utilizes these optical properties of metasurfaces and nanostructures with metallic origins in order to manipulate the behaviour and effect of light at nanometer scales and investigating optical and magnetic nanoarrays employing light. The special characteristics of nanometallic structures come from their ability to achieve collective electron excitations and coupled charge oscillations, which results in surface plasmons.

(2) These are the properties and characteristics of nanometallic structures that have been studied and utilized in this project.

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2.1 Aim

The goal of this research based project is to examine the optical properties of specific magneto-plasmonic gold and nickel core-shell nanostructures. From the examina- tion, the aim is to investigate whether the optical properties of the said structures can be influenced and controlled by using different dimensions and properties of the structures.

2.2 Purpose

The purpose of this project is to gain further insight into the new branch of the ongoing research on metasurfaces and plasmonics and attain desired optical properties for the studied structures. Furthermore, this study can provide guidance in creating a computational model to help model structures with desired optical behaviour as well as aid in deciphering experimental data. It will even find its usefulness in technological development and advancement in this field and drive the study’s use in various types of nanophotonic technologies.

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3 Theory

3.1 The Electromagnetic Wave Equations

3.1.1 Electromagnetic Wave Equation in Vacuum

Maxwell’s differential equations describe every phenomenon involving electromag- netism. This includes optics and light interactions with matter. In their basic form, Maxwell’s equations can be written as:











∇ · EEE = ρ/₀

∇ × EEE = −∂BBB/∂t

∇ · BBB = 0

∇ × BBB = µ₀JJJ + (1/c²)∂EEE/∂t

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In a subset S ⊆ R³ where there are no charges and hence neither any currents, (1) simplifies to:











∇ · EEE = 0

∇ × EEE = −∂BBB/∂t

∇ · BBB = 0

∇ × BBB = (1/c²)∂EEE/∂t

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Next the second and fourth equation of (2) are differentiated using the curl operator ∇×. The differentiation order of the right hand side of these equations can be changed, assuming reasonable solution fields EEE and BBB:

(∇ × ∇ × EEE = −(∂/∂t)∇ × BBB

∇ × ∇ × BBB = (1/c²)(∂/∂t)∇ × EEE

Substituting the expressions for ∇ × EEE and ∇ × BBB from (2) the following is obtained:

(∇ × ∇ × EEE = −(1/c²)∂²EEE/∂t²

∇ × ∇ × BBB = −(1/c²)∂²BBB/∂t²

The vector relation ∇ × ∇ × fff = ∇(∇ · fff ) − ∇²fff along with the fact that ∇ · EEE =

∇ · BBB = 0 from (2), the result obtained is:

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((∇²− (1/c²)∂²/∂t²) EEE = 0

(∇²− (1/c²)∂²/∂t²) BBB = 0 (3) This is the wave equation for the electromagnetic field in vacuum, and shows the wave-like nature of light in areas absent from charges.

3.1.2 Electromagnetic Wave Equation in Dielectric Medium

A dielectric is a medium which responds to external fields without conduction. By introducing the DDD- and HHH-fields, the Maxwell equations can be rewritten without any regard to the so-called bound charges. The bound charges are direct results of the structure of the dielectric. The equations reduce to:











∇ · DDD = ρ_f

∇ × EEE = −∂BBB/∂t

∇ · BBB = 0

∇ × HHH = µ₀JJJ_f_f_f + ∂DDD/∂t

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The index f denotes that the charge density ρ_f and the current density JJJ_f_f_f are from free charges. As per definition:

(DDD = ₀EEE + PPP H

H

H = (1/µ₀)BBB − MMM (5)

Here, PPP stands for the polarization, which is the dipole moment per volume. Hence, the total dipole moment ppp in a volume V ⊆ R³ should fulfill ppp =˝

V PPP dV . Similarly, the magnetization is defined through the equation mmm =˝

V MMM dV .

Using a similar approach as for the case in vacuum, the following form is achieved by differentiating (4) in areas where there are no free charges or free currents:

(∇ × ∇ × EEE = −(∂/∂t)∇ × BBB

∇ × ∇ × HHH = (∂/∂t)∇ × DDD

Using (5), the more general case results in, ∇ × BBB = µ₀∇ × HHH + ∇ × MMM and ∇ × DDD =

₀∇ × EEE + ∇ × PPP . Substituting the curls of the electric- and magnetic fields from equation (4), the following is obtained:

(∇ × ∇ × EEE = −µ₀(∂²/∂t²)DDD − (∂/∂t)∇ × MMM

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Lastly, using the definitions from (5) the following is obtained:

(∇ × ∇ × EEE = −0µ0(∂²/∂t²)EEE − µ0(∂²/∂t²)PPP − (∂/∂t)∇ × MMM

∇ × ∇ × HHH = −0µ0(∂²/∂t²)HH − H 0µ0(∂²/∂t²)MMM + (∂/∂t)∇ × PPP If we assume that PPP = χ_e₀EEE and MMM = χ_vHHH, this reduces to:

(∇ × ∇ × EEE = −₀(1 + χ_e)µ₀(1 + χ_v)(∂²/∂t²)EEE

∇ × ∇ × HHH = −₀(1 + χ_e)µ₀(1 + χ_v)(∂²/∂t²)HHH

Defining _r = 1 + χ_e, µ_r = 1 + χ_v as well as = ₀_r and µ = µ₀µ_r gives the simpler form:

(∇ × ∇ × EEE = −µ(∂²/∂t²)EEE

∇ × ∇ × HHH = −µ(∂²/∂t²)HHH

In this case, the vector identities can also be applied along with the fact that the phase velocity in the medium is defined as v = 1/√

µ to get, ((∇²− (1/v²)∂²/∂t²) EEE = 0

(∇²− (1/v²)∂²/∂t²) BBB = 0 (6)

3.1.3 Wave Equation in Frequency Domain

The fouriertransform of the solutions EEE and HHH are given by:

(EEE =´

REEEω_ωωe^iωtdω HHH =´

RHHHω_ωωe^iωtdω Substituting this into (6) gives:

(´

R(∇²+ ω²/v²) EEE_ω_ω_ωe^iωtdω = 0

´

R(∇²+ ω²/v²) BBB_ω_ω_ωe^iωtdω = 0 From this, the frequency-domain wave equation is obtained:

((∇²+ ω²/v²) EEE_ωω_ω = 0

(∇²+ ω²/v²) BBB_ωω_ω = 0 (7)

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3.1.4 Wave Equation in COMSOL

The wave equation has been solved numerically in the frequency domain in COMSOL.

The equation solved by COMSOL is similar to equation (7).

COMSOL uses the Finite Element Method to solve (7). The exact form of the equation may depend on the complexity of the materials used. For simple dielectric materials, the equation will simplify to (7).

3.2 Cross Sections

Three of the optical properties that have been measured in the simulations include the absorption cross section, the scattering cross section and the extinction cross section.

3.2.1 Absorption Cross Section

The absorption cross section is a measure of how much energy is absorbed by the nanoparticle. Since the lost energy depends on the energy of the incident wave, it has the intensity-independent unit, area squared, and thus is only related to the geometry of the nanoparticle.

If the power loss density Q is a known function of space, the absorption cross section is obtained by integrating Q over the nanoparticle volume V ⊂ R³ and normalizing by the incident intensity:

σ_abs = 1 I0

˚

V

QdV.

3.2.2 Scattering Cross Section

The scattering cross section is a measure of how much light is scattered by the nanoparticle. It is defined using the Poynting vector SSS := EEE × HHH. As a consequence of Poynting’s theorem, the integral‚

∂V (SSS · nnn) dS gives the energy flux per unit time out of the volume V ⊂ R³, where nnn is a normal vector to the surface ∂V .

By dividing the outgoing energy per unit time with the incident energy, a measure of the scattering cross section is obtained. This is a measure of how much energy is scattered per incident energy, which is a property of the nanoparticle, independent of the incident radiation. Thus:

σ = 1 ‹

(SSS · nnn) dS.

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Here, SSSsc_sc_sc refers to the Poynting vector of the scattered field solution, i.e. the solution superimposed on the background field solution (without nanoparticles present).

3.2.3 Extinction Cross Section

The extinction cross section gives information about how much light is "lost" when it passes the nanoparticle. Loss in this sense can either refer to scattered light in unwanted directions or light that has been absorbed and converted into other forms of energy, such as heat.

The extinction cross section is calculated by adding the absorption cross section and the scattering cross section, as follows:

σ_ext = σ_abs+ σ_sc.

3.3 Plasmons

Surface plasmons, can in simple terms be described as coherent delocalized electron wave oscillations that can be excited in certain materials (mostly metals) with certain conditions. These conditions include for example having the dielectric function, which has to do with the complex permittivity of the material, alter sign across the interface between the materials. (3)

The excitation of the electrons in the material take place due to the electromagnetic field associated with the light wave that is shone on the material. The electric field of light wave will make the electrons near the surface of the material start oscillating and usually this oscillation only takes place in the form of horizontal vibrations.

However, when placed under the right conditions, longitudinal oscillations of the electrons can be achieved by the light and these electron waves then produce their associated electromagnetic waves. It is this electron wave along with its associated electromagnetic field that gives rise to what is known as surface plasmons (SPs).

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SPs have several interesting and useful properties. Firstly, SPs have been found to have the ability to enhance fields. Since SPs have extremely localized electric fields, this charge density results in plasmonic field amplitudes greater than two orders of the original electric field that excited them. The phenomenon known as "Surface Enhanced Raman Scattering" (SERS) makes use of this property of SPs. (4)

Secondly, SPs have been found to propagate through not only short distances but even through long distances. As known, light shone onto a metal surface will penetrate only on a nanometer scale, but SPs however have the ability to traverse significantly longer distances, up to million times longer than the skin depth of the material. Due

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to this ability, SPs have made a breakthrough for research on electrical and optical circuits also known as plasmonics.

Thirdly, it has been found that for a certain frequency, the oscillations of the SPs have a wavenumber that is always greater than that of a photon. This implies that an SP always has shorter wavelength than light of the same frequency. This is of major importance in the field of optics since this implies that the resolution of imaging systems for example will no longer be restricted by the wavelength of the incident light ray, hence greatly improving system resolutions. (4)

3.4 Physical Setup

3.4.1 Simulated Environment

The schematic of the simulation is shown in Figure 1. As can be seen, the nanostructure is placed on a thin glass film. The film’s index of refraction is set to be 1.5.

Above the film is a thin layer of air. The unit element is set to be repeated infinitely in x and y, which is accounted for in COMSOL using periodic boundary conditions for the surfaces x = ±w/2 and y = ±w/2.

Figure 1: A visualization of the simulated environment, corresponding to a unit cell of the lattice structure.

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The light is excited in the top boundary z = hair as a plane wave. The angles of incidence φ and θ can be varied generally. The individual nanoparticles in the nanostructure consist of a nickel core with radius r_i and a gold shell with radius r_o. The index of refraction for these are complex, with the imaginary part, depending on sign, representing an attenuation of the wave.

3.4.2 Experimental Comparison

The lattice structure described above has also been manufactured. The nanostructures are prepared combining nanoimprint lithography and electroless Au deposition.

The process starts with first creating the nickel core, and then letting the gold atoms replace the outer nickel atoms in a chemical process. (5) By varying the time during which this replacement takes place, different shell thicknesses can be obtained, with a thicker shell corresponding to a longer time.

Scanning electron microscopy (SEM) micrographs of these structures are shown in Figure2. (5) As can be seen, their shape is nowhere as ideal as the simulated structure, however they can still be used for comparison with the simulated results.

As shown in Figure 3 (5), samples immersed in Au solution for 2.5 minutes have a resulting Ni diameter of 330 nm and a 10-20 nm thick Au shell. These dimensions are later recreated in COMSOL. Other possible dimensions are shown in Figure3for both 2.5 minutes and 3.0 minutes.

Figure 2: Microscopic picture of manufactured samples. The time refers to the time during which gold atoms replace the outer nickel atoms. The shell thickness is indicated in red. © NMDL, Korea University.

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Figure 3: Dimensions of two manufactured samples. The left one corresponds to 2.5 minutes and the right one to 3 minutes. © NMDL, Korea University.

3.5 Modes

From the simulated environment the linearly varied radii produce different outlooks of the nanoparticle due to different material behaviour after the incident radiation.

These differing outlooks can be studied to observe and attain different modes in the behaviour of the nanoparticle.

The nanoparticle, or the plasmonic nanostructure exhibits characteristic resonant optical modes in spectral ranges both high and low. These optical modes can be either symmetric or antisymmetric and this behaviour is called bimodal behaviour.

This behaviour of mode formation has been studied extensively and much focus has been placed on the trial to control the spectral positions of the modes. The control of the spectral positions of these mode formations, both symmetric and antisymmetric, can be done by controlling the morphological structure parameters i.e the radii and height of the shells. (6)

To get a clearer understanding of these mode formations in the structures, Figure 4 can be considered. This figure clearly shows how symmetrical modes can look in both high energy fields and low energy fields when considering optical dipoles or magneto-optical dipoles. The presented structures in the figure consist of rings whereas this study deals with discs but the same idea of mode formation remains applicable. (6)

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Figure 4: Mode formations for both optical and magneto-optical dipoles in high and low energies. Feng et. al.

3.6 Numerical Simulation

3.6.1 Types of Domains

The simulated environment can be split into two different categories: Physical Do- mains and Perfectly Matched Layers (PML:s). The physical domain is the space which exists in a laboratory experiment, while the PML is a numerical domain that functions like a boundary condition.

A Perfectly Matched Layer absorbs the incoming radiation perfectly. (7) While an ideal PML would do this without any reflection, computational PML:s implemented in practice are not ideal and will therefore reflect some of the incoming radiation back into the physical domain. PML:s are therefore used as open boundaries in a simulated environment.

3.6.2 Mesh

The solution to the wave equation is a continuous field in R³. The numerically obtained solution however is limited to a finite number of points in the domain. These points are determined using a mesh, which specifies which points the solution should

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be obtained for. COMSOL then uses a method called the Finite Element Method, sometimes along with other methods, to determine the solution in these points.

3.6.3 Ports

A port is a boundary that is specifically designed to excite or absorb radiation. The boundary at which the field is excited is a port with wave excitation activated. In the simulated scenario, where the incident radiation comes from the top, it is appropriate to create a port at the top boundary of the physical domain, with wave excitation activated. Moreover, since radiation is expected to reach the bottom of the physical domain, another port is created at the bottom of the domain, meant to receive some of the incoming radiation.

3.6.4 Periodic Boundary Conditions

The simulated environment is repeated infinitely in x and y. The unit, which is repeated, is called a unit cell. In order to determine the field for infinite repetitions, the field on the boundaries has to be periodic, for symmetric reasons. This condition is called a periodic boundary condition.

4 Method

Required equipment was COMSOL Multiphysics installed on a computer with required specifications to run the program. The version used was 5.1. MATLAB was used to generate the graphical data from the simulation results, though other softwares could have been used as well to accomplish the same task. The MATLAB version used here was version R2017b.

The first step was to create the experimental set up in COMSOL. The instructions for this can be found in Appendix 8.1. Once everything was set up, simulated and exported according to Appendix 8.1, the results were visualized in MATLAB. A new script file ”CoreShellTransmittance.m” was created in MATLAB with the following contents:

% Import data from exported file fileName = 'Filename.txt';

fileData = importdata(fileName);

wavelengths = fileData.data(:, 1);

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% Draw

[rows, cols] = size(fileData.data);

hold on;

for i = 3:cols

transmissions = fileData.data(:, i);

plot(wavelengths, transmissions);

end

xlabel('Wavelength (nm)');

ylabel('Transmission');

title('Transmission');

legend('ri = x1 nm, ro = y1 nm', ...

'ri = x2 nm, ro = y2 nm' ...

... % Proceed );

% Add limits if necessary

%xlim([x1 x2])

%ylim([y1 y2])

The file was saved in the same directory as the exported files from COMSOL. For each exported textfile in Appendix 8.1, corresponding to a recreated sample, the script was executed with "Filename.txt" replaced with the name of the file. The legends were added according to the format in thelegend-function. Limits were added if a specific part of the spectrum was to be focused on. The graphs from each execution were saved.

For the linearly varied nanoparticles, the following MATLAB-file was created, with the name "CoreShellTransmittanceLin.m":

% Draw hold on;

for r = 50:50:150

selectedIndices = (fileData.data(:, 1) == r);

wavelengths = fileData.data(selectedIndices, 2);

transmissions = fileData.data(selectedIndices, 4);

plot(wavelengths, transmissions);

end

ylabel('Transmission');

title('Transmission');

legend('ri = 50 nm', 'ri = 100 nm', 'ri = 150 nm');

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%xlim([x1 x2])

%ylim([y1 y2])

The file was saved in the same folder as the exported file for the linearly varied case, and the resulting graph was saved.

Lastly, a MATLAB-file called "CoreShellOpt.m" was created for generating graphs for the cross sections, with the following contents:

% Draw

wavelengths = fileData.data(:, 2);

abs = fileData.data(:, 4);

sc = fileData.data(:, 5);

ext = fileData.data(:, 6);

hold on;

plot(wavelengths, abs*10^15);

plot(wavelengths, sc*10^15);

plot(wavelengths, ext*10^15);

ylabel('Cross Section (fm)');

title('Cross Sections');

legend('Absorption Cross Section', 'Scattering Cross Section', 'Extinction Cross Section');

%xlim([x1 x2])

%ylim([y1 y2])

The file was saved in the same location as the exported files, with "Filename.txt"

replaced with the actual filename of each file. To generate the cross section graphs, the file was then executed for each exported COMSOL-file containing cross section data.

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5 Results and Discussion

5.1 Optical Properties

5.1.1 Simulated Linearly Varied Radius

In Figure 5, Figure6 and Figure 7, the absorption-, scattering- and extinction cross sections can be seen for the nanoparticle with linearly varied inner radius between 50 nm and 150 nm.

Figure 5: Absorption-, scattering- and extinction cross section for nanoparticle with inner radius 50 nm and outer radius 200 nm.

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From all three cross section graphs presented, the most noticeable peak is seen at 1400 nm wavelength. At this peak it can also be observed that as the inner radius of the nanoparticle in increased, it scatters less of the incoming light, and absorbs more of the incoming light. The decrease in scattering is greater than the increase in absorption, hence the extinction cross section decreases with inner radius at this wavelength.

From this, it can be said that the ratio between the different materials present in the nanoparticle seems to produce a strong effect on its total absorption and scattering ability. In this case, as the total amount of nickel is increased by increasing the inner radius of the nanoparticle, it achieves a stronger retention possibility of the incoming light. Hence it also seems to imply that a greater percentage of nickel (and smaller percentage of gold) in the nanoparticle allows for less of the light to be scattered.

When the other wavelengths of the cross section plots are analyzed it can be stated that no extreme behaviour is observed but a general trend of increasing extinction cross section can be seen. In general it can hence be stated that with increasing percentage of nickel in the nanoparticle, there seems to be a faster rate of absorption but a slower rate of scattering.

5.2 Transmission Graphs

5.2.1 Simulated Linearly Varied Radius

Figure 8 presents the simulated plot of the transmissions concerned with the three linearly varied radii of the nanoparticle. As can be seen, the figure presents the transmission for wavelengths between 500 nm and 1500 nm for the different dimensions of the nanoparticle. The general trend of the plot can be compared to that of the cross sections described earlier to conclude that a similar outlook is acquired. It can also be seen that the main peak of the transmission graph is at about 1400 nm which is close to the most dominant peaks of the cross section Figures 5, 6and 7.

When specifically looking into the transmissions for the different dimensions of the nanoparticle above 750 nm it can be said that when the nanoparticle is composed of the least amount of nickel i.e at an inner radius of 50 nm, the nanoparticle seems to have the greatest amount of transmission for most wavelengths. As the amount of nickel is increased in the nanoparticle, i.e the inner radius is increased to 100 nm and then to 150 nm, the transmission decreases successively for the said wavelengths.

It can also be noted that there seems to be certain differences in the appearance between the cross section graphs and the transmission graph when considering lower wavelengths. These lower wavelengths, i.e the wavelengths less than 750 nm, present no distinct behaviour but give rise to rather irregular behaviour. However from the general outlook of the transmission graph, the peak at 1400 nm can be interesting for further investigation and possible mode identification.

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Figure 8: The transmission for wavelengths between 500 nm and 1500 nm for the three different nanoparticle dimensions.

To get a closer look at the lower part of the transmission, with a smaller step size, the interval can be changed from 400 nm to 900 nm for the incident light. The resulting transmission graph for this simulation is shown in Figure 9. The observed rapid peaks are assumed to be due to numerical errors. This can be further confirmed by examining the solution at these wavelengths. Moreover, since 500 nm is the width of the physical domain, it is reasonable to assume that this leads to some possible numerical instability in the solution. For this reason, Figure 9 has been zoomed in such a way that the scale does not get affected by the peaks.

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Figure 9: The transmission for wavelengths between 400 nm and 900 nm for the three different nanoparticle dimensions.

5.2.2 Experimentally Developed

Figure 10 presents the experimentally developed transmission graph for wavelengths between 400 nm and 900 nm. The number in the legend represents the time it takes in minutes for the chemical process that replaces nickel atoms with gold atoms to occur, giving an approximate shell thickness. From this graph it can be seen how the transmission varies with the wavelength.

5.2.3 Simulated version of Experimentally Developed Structures

From Figure11the transmission graph for the simulated version of the experimentally developed transmission graphs can be seen. This plot presents the transmission for the same wavelength range for a nanoparticle with an outer radius of 180 nm and an inner radius of 165 nm, corresponding to the dimensions of the laboratory sample.

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Figure 10: The experimentally developed transmission graph for wavelengths between 400 nm and 900 nm for the three different nanoparticle dimensions.

Figure 11: The simulated version of the experimentally developed transmission graph for wavelengths between 400 nm and 900 nm for the three different nanoparticle dimensions.

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5.2.4 Comparison of Experimentally Developed and Simulated version When comparing the graphs from Figure 10 and 11 it can be said that they seem rather different and that the only resembling outlook seem to be around 550 nm where both transmission graphs seem to have a dip. However it must be noted that the transmission graphs do not have the same scales. The simulated plot hence varies much more than the experimentally developed plot.

The observed dissimilarities can be due to possible errors in the simulation setup or maybe due to simulation errors. There is a possibility of error in the experimentally developed graph as well. This difference could also be a source of something interesting to further investigate and doesn’t necessarily have to hint at errors.

5.3 Mode Identification

5.3.1 Simulated Linearly Varied Structures

When considering the mode formations on the nanoparticle within the specified range, i.e between wavelengths 500 nm and 1500 nm, the distinct peaks in the transmission graph was considered of interest and the respective mode formations studied. These modal formations were studied through the plotted electric field on the nanoparticle surface, or so called, the distribution graphs. A stronger electric field is given by the darker red colour and a weaker electric field is given by the darker blue colour. This study was done for differing inner radii since it is the change in radii that changes the material constitution of the nanoparticle and hence gives rise to the different possible mode formations.

From Figure 8 that presents the transmission graph, a maximum point can be seen at 1400 nm wavelength, as mentioned before. The electric field on the nanoparticle surface for the three different radius dimensions for this exact wavelength can be seen in Figures12,13and 14in increasing radius dimension from 50 nm to 150 nm.

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Figure 12: Visualization of electric field on the nanoparticle for ri = 50 nm at wavelength 1400 nm. The field strength |E| is shown to the left and the y-component Ey to the right.

Figure 13: Visualization of electric field on the nanoparticle for ri = 100nm at wavelength 1400 nm. The field strength |E| is shown to the left and the y-component Ey to the right.

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When focusing on Figure 12 the distribution graph for the nanoparticle at 50 nm inner radius, a very clear pattern can be seen across the surface. The image to the left presents the norm of the electric field whereas the image to the right presents the y-component of the electric field. When analyzing the y-component of the electric field, it can be seen that there is a presence of a stronger field to the left and right whereas there is a weaker field at the top and bottom of the nanoparticle.

When continuing by looking at Figure 13, the field distribution for the same wavelength but for an inner radius of the nanoparticle at 100 nm, it can be seen that the same pattern persists but that the y-component of the field seems to get weaker to the left and right. As the inner radius in sill further increased, the field to the left and right seem to get even weaker as is shown in Figure 14.

When analyzing the electric field patterns as observed from these three field distribution graphs, it can be concluded that there seems to be certain modal formations at this wavelength on the nanoparticle. These mode formations could possibly be caused due to the gold shell of the nanoparticle as the patterns have their strongest presence in the shell structure. This could hence possibly be a resonant peak for the nanoparticle. While the field distribution pattern is not clear enough to conclude whether gold is the main cause of the peak, Figure8shows that increasing the amount of gold gives a stronger resonance peak, strengthening this observation.

When considering the transmission graph again from Figure 8, the other point of interest is more difficult to find. Looking at Figure 9, wavelengths around 600 nm and 675 nm, as well as 490 nm could be of interest. By plotting the field for different wavelengths in this area however, it could be found that 600 nm produced the clearest, most interesting pattern. The electric field on the nanoparticle surface for the three linearly varied radius dimensions for this exact wavelength can be seen in Figures 15, 16and 17in increasing radius dimension from 50 nm to 150 nm.

Figure 15: Visualization of electric field on the nanoparticle for ri = 50 nm at wavelength 600 nm. The field strength |E| is shown to the left and the y-component Ey to the right.

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From Figure 15, the field distribution for the nanoparticle at an inner radius of 50 nm can be seen, and this also presents a clear pattern across the surface. As can be seen from the y-component of the distribution graph, the field is strongest at the nickel core and gets weaker as it moves towards the boundaries. The field seems to get weaker faster for the top and bottom of the nanoparticle at this wavelength. The pattern therefore clearly shows two maximum points and one strong minimum point in the middle.

When considering Figure 16, a similar pattern can be observed, but the field seems to lose strength overall. Here again, a relative stronger field can be seen at the nickel core and a weaker field to the top and bottom of the nanoparticle surface.

Finally, when looking at Figure17, which also presents the wavelength of 600 nm but at an inner radius of 150 nm, similar distribution graphs can be observed but again with overall weakening field strength as compared to the previous two radii at this

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From all the three distribution graphs at the 600 nm wavelength, it can be concluded that there seems to be modal formation at this wavelength on the nanoparticle. The modal pattern seems to be mostly connected to the gold shell, suggesting that the mode is associated with the gold material. This is further strengthened by Figure 17, where the pattern is much more difficult to see, due to the high amount of nickel and hence reducing amount of gold.

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6 Conclusions

This project was based on investigating the optical properties of specific magneto- plasmonic gold and nickel core-shell nanostructures. Through simulations with the aid of COMSOL, the nanostructures were modelled and thereafter the optical properties were graphically visualized. These visualizations included cross sections, transmissions and field distributions.

From the study of the cross sections for the different dimensions of the nanostructures, it can be concluded that changes in the inner radius had an impact on the different cross sections. Further it was noted that the cross sections were impacted by the inner radial changes which corresponded to different material ratios. The effect was mostly seen as changes in peak heights.

When considering the transmissions, it was noted that alterations of the inner radius had an impact on the total transmission as was seen for the analysis of the cross sections. The observed peaks from the transmissions corresponded, to a large extent, to the peaks as seen from the cross sections. The clearest peak from both the cross sections and the transmissions were observed at about 1400 nm wavelength. In comparison with the experimentally developed transmission graph, few similarities could be found which leaves room for further investigation.

For specific wavelengths, the electric field strength was graphically represented on the nanoparticle surface. From this study, modal formations were discovered at certain wavelengths. For 1400 nm, a clear pattern was observed which could possibly correspond to a mode. For 600 nm, two distinct maxima and one minimum was found, which strongly suggests that there is a presence of an optical mode. For both 1400 nm and 600 nm the mode formations seemed to be associated with the gold shell rather than the nickel core. The nickel core however, seemed to have certain impacts on the behaviour on the passing light.

Conclusively, it can be said that the optical properties of the nanostructures can be influenced and controlled by varying the dimensions and properties. This work, although being a small part of a much bigger study, sheds light on several interesting properties concerning magneto-plasmonic gold and nickel nanostructures. Having said that, there is big scope for further research, which presents the possibility to spark several technological advancements within this field.

Conclusively, it can be stated that the optical properties of the nanostructures could be influenced and controlled by varying the dimensions and properties of the said structure. Differing dimensions corresponded to noteworthy changes in the cross sections, the transmissions as well as the mode formations.

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7 References

(1) Yu, Nanfang, and Federico Capasso. “Flat Optics with Designer Metasurfaces.”

Nature Materials, vol. 13, no. 2, 2014, pp. 139–150., doi:10.1038/nmat3839.

(2) Schuller, Jon A., et al. “Plasmonics for Extreme Light Concentration and Manip- ulation.” Nature Materials, vol. 9, no. 3, 2010, pp. 193–204., doi:10.1038/nmat2630.

(3) “Surface Plasmon.” Wikipedia, Wikimedia Foundation, 8 May 2019.

(4) Gbur, Gregory. “Optics Basics: Surface Plasmons.” Skulls in the Stars, 21 Sept.

2010.

(5) Korea University, Nano Materials and Devices lab (NMDL).

(6) Feng, Hua Yu, et al. “ Magnetoplasmonic Nanorings as Novel Architectures with Tunable Magneto-Optical Activity in Wide Wavelength Ranges.” Advanced Optical Materials, 2014.

(7) "Perfectly matched layer." Wikipedia, Wikimedia Foundation, 10 June 2019.

(8) Scatterer on Substrate, https://www.comsol.com/model/scatterer-on-substrate- 14699, 10 June 2019.

(9) Hillborg, J., Laurell, H., "Towards a nanometer thick flat lens", Uppsala Uni- versity.

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8 Appendix

8.1 COMSOL Instructions

Following is a brief overview of how COMSOL is setup. It is based on the tutorial in Scatter on a Substrate (8) and the earlier project Towards a Nanometer Thick Flat Lens (9). Before setting up the environment, a project file has to be created. This was done as following:

Creating a New Project

1. To create a new project, press Model Wizard in the window that appears on startup.

2. Select 3D

3. In the Select Physics list, select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd).

4. Press Add.

5. Select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd) again.

6. Press Add.

7. Press the Study-button at the bottom of the sidebar.

8. From the Select Study list, select Empty Study.

9. Press Done at the bottom of the sidebar. This will create a basic project with the necessary components.

After this, the user will be taken to the main interface, containing the following three sections: Model Builder Window, Settings Window and Graphics Window. These are marked in Figure 18.

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Figure 18: The structure of the main interface in COMSOL.

The Model Builder Window shows all the components, such as parameters, materials and result objects. These can be modified in the Settings Window, by left-clicking them in the Model Builder Window. The simulated environment is visualized in the Graphics Window.

The Model Builder Window contains a hierarchy of items, with the main item having the same name as the project name, in this case "CoreShell.mph". Within this main item are four different subitems: Global Definitions, Component 1, Study 1 and Results. In the instructions below, this hierarchy is visualized through headings and subheadings. An example of this hierarchy is shown in Figure 19.

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Figure 19: Hierarchy of the following tutorial. The main headings refer to one of the four descending objects of CoreShell.mph (in the picture above, "Untitled.mph"). Subheadings are descendant one level deeper.

When it is stated in the instructions that something is to be added to an item in the Model Builder Window (such as "Add Cylinder to Geometry 1"), the user is to right-click on the Model Builder-item (Geometry 1) and select the item to be added (Cylinder). This procedure will generate a new object under Geometry 1 called Cylin- der. This can also be done using the toolbar at the top.

Global Definitions

Parameters

1. Left-click "Parameters" to define the global parameters to be used in the project.

In the rest of the instructions, this will be referred to as opening the setting window.

2. Enter the following data into the table (the value will be updated automatically):

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Name Expression Value Description

w 500 [nm] 1E-6 m Width of physical domain

t_pml 150 [nm] 1.5E-7 m Thickness of PML

h_air 400 [nm] 4E-7 m Air domain height

h_subs 250 [nm] 2.5E-7 m Substrate domain height

na 1 1 Refractive index in air

nb 1.5 1.5 Refractive index in substrate

phi 0 0 Azimuthal angle of incidence in both media

theta 0 0 Polar angle of incidence in air

I0 1 [MW/m2] 1E6 W/m2 Intensity of incident eld

wl 450 [nm] 4.5E-7 m Wavelength of incident light

ri 100 [nm] 1E-7 m Inner radius of cylinder

ro 200 [nm] 2.5E-7 m Outer radius of cylinder

h_cyl 20 [nm] 2E-8 m Height of cylinder

thetab asin(na/nb*sin(theta)) 0 rad Polar angle in substrate

dz 30 [nm] 3E-8 m

Component 1

Definitions

1. Add "Variables" to "Definitions". This will be used to define variables for the k-vector, later to be used on the top boundary to define the incoming wave.

2. In the setting window for Variables, change the label to "Wave vector variables".

3. Add the following data to the table (unit is updated automatically):

Name Expression Unit Description

ka ewfd.k0*na rad/m Wave number in air

kx ka*cos(phi)*sin(theta)rad/m x-component of wave vector ky ka*sin(phi)*sin(theta)rad/m y-component of wave vector kaz -ka*cos(theta) rad/m z-component of wave vector in air

kb ewfd.k0*nb rad/m Wave number in substrate

kbz -kb*cos(thetab) rad/m z-component of wave vector in substrate

Geometry 1

1. Add "Cylinder" to Geometry 1.

2. Open the setting window for the newly created cylinder and change the label to "Core". This geometric object will correspond to the nickel-core.

3. In the Radius field, type ri.

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4. In the Height field, type h_cyl.

5. Make sure that the position is set to x = y = z = 0 in the Position-fields.

6. Add "Cylinder" to Geometry 1.

7. Open the setting window for the created cylinder, and change the label to

"Shell". This object will correspond to the gold-shell.

8. In the Radius field, type ro 9. In the Height field, type h_cyl.

10. Make sure that the position is set to x = y = z = 0 in the Position field.

11. Add "Block" to Geometry 1.

12. Open the setting window for the created block, and change the label to "Air".

The created block will correspond to the air domain.

13. In the Width field, type w + 2*t_pml 14. In the Depth field, type w + 2*t_pml 15. In the Height field, type h_air + t_pml

16. In the z-field in the Position-section, type (h_air + t_pml)/2, to set the vertical offset of the air block.

17. In the Layers section, add a layer with the name "Layer 1" and thickness t_pml.

18. In the Layer Position section, check all boxes except for Bottom. This will add a layer around the air that will later be defined as a Perfectly Matched Layer.

19. Right-click the Air-object in the model builder window and press duplicate.

20. Open the setting window for the duplicated object and change the label to

"Dielectric". This block will correspond to most of the dielectric material under the nanoparticle. The reason that the dielectric is divided into two blocks is to allow COMSOL to evaluate the outward energy flux of the system, which is evaluated at the surface between the blocks. This will later be used to calculate the transmission.

21. Change the Height field to h_subs - dz.

22. Change the z-field in the Positions-section to -(h_subs - dz)/2.

23. In the Layer Position section, check all boxes except for Top and Bottom. This will, in the same way as for air, add a Perfectly Matched Layer around this part of the substrate.

24. Duplicate Dielectric in the model builder window.

25. Change the label of the duplicated object to "Dielectric 2". This will correspond

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26. In the Height field, type dz + t_pml.

27. In the z field in the Position-section, type -h_subs + dz/2 - t_pml/2.

28. In the Layer Position section, check all boxes except for Top.

29. To render the geometry and see what has been created, press Build All Ob- jects in the setting window for Dielectric (or the setting window for any other geometric object).

30. In the graphics window, click Zoom Extents.

31. Click on Wireframe Rendering to toggle between a wireframe view.

Definitions

1. Add "Selections > Explicit" to Definitions. A selection is a generally defined volume or surface that will be used later in the code to refer to said region.

2. Open the setting window for the created selection and change the label field to

"Physical domains". This selection will refer to all regions that are not Perfectly Matched Layers.

3. Select all domains that are not Perfectly Matched Layers. This includes the air, nanoparticle and dielectric. To select domains, go to the Input Entities section in the setting window and make sure the switch is set to Active. Also make sure that the "Geometry entity level" is set to Domain, to specify that a volume is to be specified. Then use the mouse in the graphics window to select said regions, and the mouse wheel to change depth. Below is a Figure of a properly selected region.

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4. Add another "Selections > Explicit" to Definitions.

5. Open the setting window for said selection and change the label field to "PML domains". This selection will refer to all the Perfectly Matched Layers.

6. In the Input Entities section, add "Physical domains" to the list of sections to invert.

8. Open the setting window and change the label field to "Core domains". This selection refers to the nickel region in the nanoparticle.

9. Using the same method as before, select the nanoparticle core, as shown in the figure below.

11. Open the setting window and change the label field to "Shell domains". This selection refers to the gold shell in the nanoparticle.

12. Using the same method as before, select the nanoparticle core, as shown in the figure below.

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13. Add "Selections > Union" to Definitions.

14. Open the setting window for the added union and change the label field to

"Nanoparticle". This selection refers to the union of the core- and shell domains.

15. Add Core domains and Shell domains to the list in the Input Entities section.

16. Add "Selection > Explicit" to Definitions.

17. Open the setting window for the added selection and change the label field to

"Nanoparticle surface".

18. Keep the Geometric entity level on domain, in spite of the selection being a surface, and select the Nanoparticle domain (shell- and core domains).

19. In the Output Entities section, select "Adjacent boundaries" in the dropdown menu and make sure that the "Exterior boundaries" checkbox is checked and the "Interior boundaries" checkbox is cleared.

20. Add another "Selections > Explicit".

21. Open the setting window, change the label field to "Internal PML surface".

This selection refers to the inner surface of the PML.

22. Choose Boundary in the Geometric entity level dropdown menu.

23. Select the interior of the PML, as shown in the figure below.

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24. Add "Perfectly Matched Layer" to Definitions.

25. Open the setting window for the created Perfectly Matched Layer, and change the Domain Selection to PML domains.

26. In the Scaling section, locate Physics and select Electromagnetic Waves, Fre- quency Domain 2 (ewfd2) from the dropdown menu.

27. Add "Variables" to Definition.

28. Open the setting window for the created variables object, and change the label to PML variables.

29. In the Geometric entity level dropdown menu, choose Domain, and then choose PML domains from the Selection list.

30. Add the following to the Variables table:

ewfd.Ex 0

ewfd.Ey 0

ewfd.Ez 0

31. This will define the electric field for the background field solution to 0 in the PML domain. The difference between ewfd and ewfd2 is explained later.

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Materials

1. Add "Blank Material" to Materials.

2. Open the settings window for the newly added material. In the label, type

"Air".

3. Make everything air by default (unless overridden) by making sure that Geo- metric entity level is set to Domain, and the Selection is set to "All Domains".

4. In the Material Contents section, change the table into the following by editing the Value-columns. This will set the refractive index to 1 in all materials that are air.

Property Name Value Unit Property Group

Refractive index n na 1 Refractive index

Refractive index, imaginary part ki 0 1 Refractive index

5. Add another "Blank Material" to Materials.

6. Open the settings window for the newly added material. In the label, type

"Dielectric".

7. Set Geometric entity level to domain and choose "Dielectric Domains" in the selection menu.

8. In the Material Contents section, set the value of the refractive index to nb, as shown below.

Property Name Value Unit Property Group

Refractive index n nb 1 Refractive index

Refractive index, imaginary part ki 0 1 Refractive index

9. Right click on Materials in the model builder window and choose "Add Mate- rial".

10. In the new window, titled "Add Material", select "Optical > Inorganic Materials

> Ni (Rakic)".

11. Click "Add to Component". Doing this adds a new material to Material.

12. Open the setting window for the new material and select "Core domains" from the selection list.

13. Repeat the above process but for the gold shell by pressing "Add Material", choosing "Optical > Inorganic Materials > Au (Rakic)", and pressing "Add to Component". Open the settings window and change the selection to "Shell domains".

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Definitions

1. Add "Variables" to Definitions.

2. Open the setting window for the new variables and change the label to Port variables. These variables will only be defined on the ports (yet to be created), where the light wave is excited and absorbed.

3. In the Variables section, add the following:

E0x -sin(phi)*exp(-i*(kx*x + ky*y)) E0y cos(phi)*exp(-i*(kx*x + ky*y))

Electromagnetic Waves, Frequency Domain (ewfd )

1. Open the setting window for Electromagnetic Waves, Frequency Domain (ewfd ).

This object is used to define the actual physics. The interface ewfd is used to find a background field solution, without the nanoparticle cylinders present. The interface ewfd2 will then be used to find a scattered field solution, introducing the nanoparticles as a form of perturbation to the system. The solution for ewfd2 is the full solution, and is based on ewfd.

2. Change the selection to Physical Domains in the Domain Selection section. The PML is not included for ewfd. Instead, its components are defined to be 0 in the PML area, as done in an earlier step.

3. Add "Wave Equation, Electric" to Electromagnetic Waves, Frequency Domain (ewfd ).

4. Open the setting window for the newly created Wave Equation object, and change the selection to Nanoparticle.

5. In the Electric Displacement Field section, set n to 1 and k to 0 by changing them to user defined, rather than from material. This will redefine the physics in the nanoparticle area as that of air for ewfd.

6. Add "Port" to Electromagnetic Waves, Frequency Domain (ewfd ).

7. Manually select the area that the incoming light wave is excited in, just below the top PML, as shown in the image below.

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8. In the Port Properties section, make sure that "Wave excitation at this port" is set to On, and type P in the Port input power field. In the Port Mode Settings, make sure that the input quantity is set to Electric field, and set the x and y component to E0x and E0y respectively. The z component is set to 0.

9. Set the Propagation constant to abs(kaz) in the Propagation constant field.

10. Add another Port to Electromagnetic Waves, Frequency Domain (ewfd ).

11. In the Boundary Selection section, select the bottom part of the physical domain, just before the bottom PML, as shown in the figure below.

12. Make sure that "Wave excitation at this port" is set to Off.

13. In the Port Mode Settings, define E0 as (E0x, E0y, 0), in the same way as for

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Port 1.

14. Set the propagation constant to abs(kbz).

15. Add "Periodic Conditions" to Electromagnetic Waves, Frequency Domain (ewfd ).

16. Open the setting window for the Periodic Conditions. In the Boundary Selection section, select the two adjacent boundaries x = ±w/2, right before the PML- region, as shown in the figure below.

17. In the Periodicity Settings, set the type of periodicity to Floquet periodicity and x- and y-components of the k-vector to kx and ky respectively. The z- component can be kept at 0. This will add periodic boundary conditions to these boundaries, equivalent to repeating the structure infinitely in the x-direction.

18. Add another Periodic Condition to Electromagnetic Waves, Frequency Domain (ewfd ).

19. Open the setting window and, as before, select the adjacent boundaries y =

±w/2 as shown in the figure below.

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20. As for the previous periodic conditions, set the type of periodicity to Floquet and the k-vector to (kx, ky, 0).

Electromagnetic Waves, Frequency Domain 2 (ewfd2 )

1. Open the setting window for Electromagnetic Waves, Frequency Domain 2 (ewfd2 ).

2. In the Settings section, set "Solve for" to Scattered field.

3. Make sure that Background wave type is set to User defined, and specify the Background electric field as ewfd.Ex, ewfd.Ey and ewfd.Ez for x, y and z respectively. This will make the solution ewfd2 depend on the solution ewfd, in the previously described manner.

Mesh 1

1. Add "Free Tetrahedral" to Mesh 1. Doing this will create both a Free Tetrahedral- object and a Size-object. These objects are used to define the discretization of the domain for the numerical solution.

2. Open the setting window for Size and check the Custom-button in the Element Size-section.

3. Set the Maximum element size to wl/4. This will be the maximum size of each mesh element for air.

4. Add "Swept" to Mesh 1.

5. Move the created swept window in the model builder so that it appears above the created tetrahedral, but below size. This can be done by either dragging

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and dropping or right clicking. This is done so that everything is overridden in the right order.

6. Open the setting window for the newly created swept, and set the Selection to PML domains (make sure that Geometric entity level is set to domain). This will make the swept discretization, corresponding to a specific discretization pattern, apply only to the PML area.

7. Add "Distribution" to the created swept.

8. Open the setting window for the created distribution object and set the Number of elements to 8, making sure that Distribution properties is set to "Fixed number of elements".

9. Add "More Operations > Free Triangular" to Mesh 1.

10. Add "Size" to the created Free Triangular-object.

11. Open the setting window for the Free Triangular-object and set the Geometric entity level to Boundary, and the Selection to Inner PML surface.

12. Open the setting window for the created Size-object (which is inside Free Tri- angular), and locate the Element Size section. Check the Custom-button and set the Maximum element size to wl/4.

13. Lastly move the Free Triangular object so that it appears above the Swept but still below Size. The final order should look like in the figure below.

14. Open the setting window for any object inside Mesh, and click the Build All- button. This will show the discretization of the entire domain in the graphics window. It should look similar to the figure below.

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Definitions

1. Add "Component Couplings > Integration" to Definitions.

2. Open the setting window for the created integral-object and change the label to

"Nanoparticle volume integral". This will represent a general integration over the nanoparticle domain.

3. Set the Operator name to "int_nanovol".

4. Set the Geometric entity level to Domain, and the Selection to Nanoparticle.

5. Add another "Component Couplings > Integration" to Definitions.

6. Open the setting window and change the label to "Nanoparticle surface integral", and the operator name to "int_nanosurf". This will represent a general integral over the nanoparticle surface.

7. Set the Geometric entity level to Boundary, and Selection to Nanoparticle surface.

8. Add another "Component Couplings > Integration" to Definitions.

9. Open the setting window and change the label to "Bottom surface integral", and the Operator name to "int_bottom".

10. Set the Geometric entity level to Boundary, and manually select the surface between the dielectric blocks, as shown in the figure below.

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11. Add "Variables" to Definitions.

12. Open the setting window for the newly created Variables-object and change the label to "Optical Properties". These variables will represent the cross sections as well as the total transmission.

13. In the Variables-section, add the following table:

nrelPoav nx*ewfd2.relPoavx+

ny*ewfd2.relPoavy+

nz*ewfd2.relPoavz

W/m² Relative normal poynting flux

sigma_sc int_nanosurf(nrelPoav)/I0 m² Scattering cross section

sigma_abs int_nanovol(ewfd2.Qh)/I0 m² Absorption cross section

sigma_ext sigma_sc + sigma_abs m² Extinction cross section

power_bottom int_bottom(nrelPoav) W Power on bottom surface

transmission power_bottom/(I0*wˆ2/cos(theta)) Transmission from top to bottom surface

Study 1

1. Open the setting window for Study 1.

2. In the Study Settings, clear "Generate default plots", to prevent COMSOL from generating default plots.

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4. Open the setting window for the newly created sweep and change the label to

"Wavelength sweep".

5. In the Study Settings, press Add, and add the table below. This will sweep the wavelength in 50 nm steps between 500 nm and 1500 nm.

Parameter name Parameter value list Parameter unit

wl range(500, 50, 1500) nm

6. Add another "Parametric Sweep" to Study 1.

7. Open the setting window for the added sweep and change the label to "Radius sweep".

8. In the Study Settings, press Add, and add the table below.

Parameter name Parameter value list Parameter unit

ri range(50, 50, 150) nm

9. Add "Study Steps > Frequency Domain > Wavelength Domain" to Study 1.

10. Open the setting window for the created Wavelength Domain-object and Wave- length to wl in the Study Settings-section.

11. Go to the Physics and Variables Selection-section and clear "Electromagnetic Waves, Frequency Domain 2 (ewfd2)". Make sure that ewfd is still checked.

12. Add another "Study Steps > Frequency Domain > Wavelength Domain" to Study 1.

13. As before, set the wavelength to wl in the Wavelength-field.

14. Under the Physics and Variables Selection-section, clear ewfd and keep ewfd2 checked.

15. Right-click on Study 1 in the model builder window and choose "Show Default Solver".

16. In the model builder window, find Solver Configurations in Study 1 and expand it. Further expand Solution 1.

17. Expand the Stationary Solver 1-object in Solution 1.

18. Make sure that the Direct-object under Stationary Solver 1 is enabled. If not, it can be enabled by right-clicking and choosing "Enable".

19. Open the setting window for Direct and set the Solver to PARADISO. Other methods can be used as well.