Bachelor Thesis

Plasmonics with a twist:

Simulating Resonances in a Möbius Ring

Author : Floris Heu� (5935601)

Supervisor : Prof. Dr. L (Kobus) Kuipers

12 EC Bachelor project performed 1-3-2012 - 31-5-2012 at FOM Institute AMOLF

Abstract

Using FEM simulations in COMSOL Multiphysics v4.2 the plasmonic resonances of Möbius Rings havebeen simulated. In particular resonances with an odd number of nodes found in previous work have beenthoroughly investigated. This is done by calculating the extinction spectra and investigating the electric�eld near the ring for di�erent ring sizes, using di�erent linear polarizations and varying the wavelengthof the light. It is concluded that these odd mode resonances do not exist. Instead they are a resonancewith an antisymmetric charge distribution. This, combined with the special geometry of the ring, madeit look like a resonance with an odd mode resonance and can be interpreted as a resonance oscillating atthe edge of the Möbius ring. Due to the chirality of a Möbius ring simulations with right-handed andleft-handed circular polarized light have been run as well. Chirality does have an e�ect on the extinction,which increased more for left-handed than for right-handed polarized light.

Contents

1 Introduction 1

1.1 Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Plasmonics in rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Plasmonic resonances in rings 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Ring resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Investigation of the pentapole mode 7

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Circular polarized light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Odd mode resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Finding more odd modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Shedding light on odd modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Y-polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Discussion and Conclusion 17

4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A Phase evolution of resonances 19

A.1 8 Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.2 5 Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20A.3 7 Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.4 9 Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B Material De�nitions 23

B.1 Palik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23B.2 Johnson & Christy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

C COMSOL formula's 24

D Populaire samenvatting 25

Chapter 1

Introduction

1.1 Plasmonics

Plasmonics explores how light can be manipulated and con�ned over dimensions on the order of, or smallerthan, its wavelength by using metallic structures. It is based on interactions between electromagneticradiation and conduction electrons at metallic interfaces or in metallic nanostructures leading to anenhanced optical near-�eld of sub-wavelength dimensions. Plasmonic structures can be used for wave-guiding, chemical and biological sensing, and medical applications [1, 2], making them a major conceptin the �eld of nanophotonics. A very nice aspect of plasmonics is that it is �rmly grounded in classicalphysics and that it can be broadly understood by classical electrodynamics and the Drude free electronmodel [3].

Plasmons, Surface Plasmons and Surface Plasmon Polaritons

Plasmons are quasi-particles and are the quantization of plasma oscillations just like photons are thequantization of light and phonons the quantization of vibrations of molecules or atoms in condensedmatter. A plasma oscillation is one of the electron density in metals, ionized gases or other electricalconducting media. In such media if the electrons are pulled away from the positive ions the Coulombforce will pull them back, acting as a restoring force and an oscillation will occur. These are classicaloscillations and their properties can be derived from Maxwell's equations. An oscillating electric �eld,like light for example, can be used to drive these oscillations.

At the interface between a metal and a dielectric material it is possible to excite plasmons using light.These oscillations are con�ned the surface and referred to as surface plasmons. Surface plasmons have alower energy than their counterparts in the bulk of the metal, because of boundary e�ects. In nanoscalestructures the modes that can be excited arise naturally from the scattering problem in an oscillatingelectromagnetic �eld. In this case they are also referred to as localized surface plasmons [3, 4, 5].

Figure 1.1: Surface Plasmon Polaritons. Src: [6]

Under speci�c conditions, light can couple to the surface plasmons and create self-sustaining electro-magnetic waves known as surface plasmon polaritons (SPPs) which propagate along the surface of themetal. (Confusingly enough SPPs are referred to as surface plasmons) These coherent electron oscilla-tions are essentially light waves that are trapped because of their interaction with the free electrons ofthe metal. In this interaction, the free electrons respond collectively by oscillating in resonance with

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the light wave. The resonant interaction between the surface charge oscillation and the electromagnetic�eld of the light constitutes the SPP. What distinguishes SPPs from photons is that they have a muchsmaller wavelength at the same frequency and so light can be manipulated at scales smaller than its ownwavelength providing a signi�cant increase in spatial con�nement and local �eld intensity.

1.2 Plasmonics in rings

Within the �eld of plasmonics nanostructures with ring shapes have generated a lot of interest becauseof the inherent high degree of symmetry that they possess. Light can be con�ned within a ring andgenerate standing waves that directly depend on the structure of the ring and the wavelength of theincident light. Since the optical properties of the ring are determined by these plasmonic resonances theycan be tuned across the visible and infrared region of the electromagnetic spectrum creating a way tocontrol and manipulate the intensity of the light at a speci�c position.

The goal of this work is to study plasmonic resonances in these metallic ring structures, but with asmall twist. As well as investigating "normal" rings, we will also consider the case of a Möbius Ring (�g.1). We will employ simulations of Maxwell's equations in order to calculate the electromagnetic �eldsand transmission spectra in order to determine the optical properties in both cases. A Möbius ring is aring but with a twist. It only has one edge and one surface. If one would imagine themselves walkingon its surface one would have to traverse the ring twice before one would end up at his or her startingposition. It is not hard to imagine such a special geometry having interesting e�ects on the plasmonicbehavior of the ring like a wave having to travel around twice before biting itself in its tail.

Figure 1.2: The Möbius Ring as used in this model.

The research done is a follow up on a project done earlier where the plasmonic resonances of Möbiusrings have been compared to normal rings of the same diameter and cross section[7]. The most remarkableresult was the discovery of a resonance with an odd number of nodes dubbed a pentapole. This type ofresonance has been more thoroughly investigated and the results will be discussed here. Also, because ofthe chiral nature of the Möbius ring, simulations with circular polarized light will be presented.

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Chapter 2

Plasmonic resonances in rings

2.1 Introduction

The next chapter consists of two sections. The �rst section aims to explain the nature of the ring resonanceand how to excite it. In the second section an introduction to the work this project is based upon willbe given.

2.2 Ring resonances

Interestingly plasmonic resonances in a ring are practically independent of the geometric cross sectionof the ring. There is a shift of a couple of nanometers in the resonance wavelength when a rectangularcross-section is compared to a cylindrical cross-section of similar area [1, 8]. In this work rings with acylindrical cross-section have been modeled. For thin metal rings the plasmon resonances, usually referredto as modes, can be described as standing waves in an in�nite metal waveguide with a cylindrical cross-section and have wave-vectors such that an integer number of wavelengths are equal to the circumferenceof the ring (�g. 2.1). The plasmonic properties of a thin metal wire are well known and easily calculated,so treating the ring as an in�nite wire provides an easy way to describe the resonances.

Figure 2.1: Schematic illustration of a plasmon resonance in a torus. The resonance can be describedby standing waves in a in�nite metallic waveguide. A plasmon mode can �t into the ring if an integernumber of wavelengths �ts in the ring. Src: [1]

Di�erent modes can be excited depending on the polarization, frequency and direction of the incidentlight. For example, for di�erent frequencies dipole, quadrupole and higher order modes can be excited.Dipole modes can be recognized by their two nodes, whereas a quadrupole as four nodes. The modescan also be symmetric or antisymmetric with respect to the cross-section of the ring. The di�erence canbe recognized by the charge distribution and is best explained by looking at �gure 2.2 where both thesymmetric mode and antisymmetric mode are shown [2].

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Figure 2.2: Left the symmetric and right the antisymmetric mode as calculated by COMSOL using its2D mode solver for an in�nite waveguide. The plot shows the charge density inside the ring. Theantisymmetric mode has negative charges concentrated on one side and positive charges on the other.Src: [7]

Exciting modes

The method of excitement used in this research is quite intuitive. The ring is excited by light polarizedand propagating in plane with the ring. The result of this con�guration is a coupling mechanism wherethe electric �eld of the electromagnetic wave drives the oscillation in the ring. When for instance thewavelength is about twice the diameter of the ring the electric �eld pushes the electrons in the samedirection. Di�erent wavelengths '�t' and can cause a mode to be excited [8]. These wavelengths can bedetermined by looking at the extinction cross-section, where peaks indicate a resonance.

Figure 2.3: Incident light generating a plasmon in the ring. The horizontal arrows represent the oscillatingelectric �eld. The circular arrows indicate the direction of oscillation of the charge density inside the ring.

2.3 Previous work

The work presented here naturally follows from a previous project at AMOLF [7]. The model used andresults found in [7] will be brie�y discussed here.

[7] compared the resonances in a silver torus in air to those of a silver Möbius ring. For both rings thecross-section is an ellipse with a short semi axis of 20 nm and long semi axis of 60 nm and of radius 150 nm.The rings were excited by light with a wavelength ranging from 200nm to 1400nm linearly polarized in theplane with the ring. Afterwards the extinction cross sections were calculated. Due to small discrepanciesin the optical constants, both simulations were done twice, once for the Palik de�nitions for the refractiveindex of silver and one for the Johnson and Christy de�nitions [9]. The results are shown in �gure 2.4.

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Figure 2.4: Extinction spectra of a normal ring and Möbius ring for both material de�nitions.

Comparing the extinction cross sections showed that for longer wavelengths the peaks of the Möbiusring are slightly red-shifted, but otherwise comparable to those of the normal ring. In both cases thespectra with Palik de�nitions are slightly red-shifted compared to spectra with JC de�nitions. For shorterwavelengths however, new peaks were found. Plotting the norm of the electric �eld at a peak showedthe expected resonances, i.e. a dipole at the longest wavelength (1100 something nm) and higher orderresonances for shorter wavelengths and are in agreement with similar simulations (�g. 2.5).

Figure 2.5: a) Dipole resonance (peak A), b) quadrupole (peak B) and c) hexapole (peak C) for a torususing the J&C de�nitions. The top plot is a slice of the ring through the zx-plane, bottom left through thexy-plane and the bottom right of the yz-plane.

For the Möbius ring a very interesting resonances at 431 nm in the Palik simulation was found having�ve nodes. This �pentapole� can be explained by an antisymmetric waveguide mode having to traversethe ring twice before biting itself in the tail. Lower order modes were not found, because [7] calculated

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that the antisymmetric mode does not exist for wavelengths longer than 550 nm. At wavelengths shorterthan 400 nm the simulations failed to �nd a solution or gave unreliable results, so these had to be ex-cluded. For the Johnson & Christy de�nitions no �odd-pole� resonance was found, but the tail of a peakthat might be the pentapole was seen just above 400nm.

Figure 2.6: Pentapole resonance at 431 nm for the Möbius ring using the Palik de�nitions.

This follow-up project set out to investigate this �pentapole� resonance more thoroughly. A �rst stepis to look for the pentapole using the Johnson & Christy de�nitions as they should be present there also.Then checking whether higher order modes indeed exist and calculating the charge density to check forthe antisymmetric mode are the next steps into better understanding such a resonance.

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Chapter 3

Investigation of the pentapole mode

3.1 Introduction

The next sections will discuss the di�erent steps taken to investigate the pentapole resonance as wellas simulations done with circular polarized light. The �rst section will explain the simulation softwareand the model that was built using it. The next section will discuss the simulations done with circularpolarized light. The �nal sections show the results of the pentapole investigation.

3.2 The Model

3.2.1 COMSOL

COMSOLMultiphysics Finite Element Analysis Simulation Software [10] was used to model the propertiesof a Möbius ring. COMSOL can solve complicated 3D problems without requiring the user to build themodel from the ground up. Model set-up is quick, thanks to a number of prede�ned physics interfacesfor applications ranging from �uid �ow and heat transfer to structural mechanics and electromagneticanalysis. For this research their RF module was especially useful, allowing to solve the propagation ofelectromagnetic waves in and around structures that can be metallic, dielectric, gyromagnetic or havearti�cial properties.

COMSOL de�nes a mesh of points where the di�erential equations determining the physics are solved.Mesh points are denser where the electromagnetic �elds are concentrated or where the structure gets amore complicated geometry. A �ner mesh means more points and therefore a longer calculation time.COMSOL will then iterate the calculations until it �nds a steady-state solution.

3.2.2 Simulation parameters

In �gure 3.1 both the model for the torus and Möbius ring are shown. The silver ring is placed in thexz-plane in the middle of a sphere of air embedded in perfectly matched layer (PML), which acts as theboundary condition absorbing all incident energy. In the picture shown, the PML is the outer sphere, thesmaller sphere shown is used for calculating the scattering cross section. The ring size and shape remainthe same as in [7], namely with a cross-section that is an ellipse with a short semi axis of 20 nm and longsemi axis of 60 nm and with a ring radius of 150 nm.

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Figure 3.1: The Torus (a) and Möbius (b) ring as modeled in COMSOL.

In the simulations the wavelength or frequency of the incident light is set, and the scattering cross-section for that wavelength is calculated as well as the electric �eld. To build up the spectra, simulationsare done from 400 to 1400 nm in steps of about 3.3 nm for each 100nm interval.

Again, both the Palik and JC optical properties of silver are used because both have speci�c problems.Appendix C shows the real and imaginary part of the refractive index for both de�nitions. Althoughroughly the same in shape, the real part of the refractive index of Palik is slightly higher, especiallyfor the longer wavelengths. Also, the imaginary part shows discontinuities above 1200 nm. However,between 200nm and 400nm , it is much smoother and has more data points than the data set of JC has.Because of this COMSOL does not �nd any solutions below 400nm using the JC data set. However, usingthe Palik de�nitions gave computational problems as well. Ultimately this resulted in not running anysimulations below 400nm at all.

The incident light is de�ned as a background �eld, which means that the source of the light is farenough away that the incident light can be treated as plane waves and that the source can be left out ofthe model.

3.3 Circular polarized light

A Möbius ring is a chiral structure so exciting it with left-handed or right-handed circular polarized lightis expected to give di�erent results. An object is chiral when it is not identical to its mirror image.Or to put it into mathematical terms, its mirror image cannot be mapped onto itself by rotations andtranslations alone. The human hand is the prime example of chirality. No matter how you twist or turnyour left hand it will not �t on your right hand. Another example is a helix.

The most general equation for a linearly polarized wave is described by equation 3.1:

f(z, t) = (Acosθ)ei(kz−ωt)x+ (Asinθ)ei(kz−ωt)y (3.1)

If the two components are of equal amplitude, but out of phase by 90 degrees, the result is a circularpolarized wave. Whether it's called right-handed/clockwise or left-handed/counterclockwise is a matterof convention, depending whether the point of view is at the source or at the receiver. In this thesis aphase delay of 90 degrees is referred to as the left-handed polarization and a phase advance referred toas the right-handed polarization.

The simulations are done for both polarizations with light propagating in the minus z-direction andunit amplitude, see equations 3.2 and 3.3. Note that there is no time dependence. Only the Johnson &Christy de�nitions for silver were used.

E(z) = ei(kz)x+ ei(kz+0.5π)y (3.2)

E(z) = ei(kz)x+ ei(kz−0.5π)y (3.3)

To calculate the extinction cross-section Poynting's Theorem is used, which states that the work done

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on the charges by the electromagnetic force is equal to the decrease in energy stored in the �eld, less theenergy that �owed out through the surface [11].

σext =

ˆV

E · Jdτ − 1

µ0

˛S

(E × B) · da (3.4)

The extinction cross-section is found by the integrating the resistive losses, the �rst term on the rightside of formula (3.4), over the volume of the ring and integrating the Poynting vector, the second term,of the scattered �eld over a spherical surface around the ring. Both quantities are normalized with theimpedance of free space, Z.

The results are plotted in �gure 3.2.

Comparing the right-handed spectrum to the left-handed spectrum shows no di�erences, but compar-ing them to the spectrum [7] found reveals 2 new peaks at 420 nm and 455 nm. The absolute value of theelectric �eld is plotted for both these peaks and show a very di�erent type of resonance, where the electric�eld is much stronger near the arc of the ring and less strong further away from the ring compared to theother resonances. Since the resonances for linearly polarized light are present it seemed likely that thesenew resonances were caused by the y-polarization of the incident light. Indeed, running the simulationagain for light polarized in the y-direction the resonances (peak E and F) were again found (�g. 3.3 and3.4).

Comparing the Möbius spectra of circular polarized light to linear polarized light doesn't reveal anynew peaks, but does show an increase in extinction for the right-handed polarization for the shorterwavelengths and an even bigger increase for the left-handed polarization. Plotting the absolute valueof the electric �eld shows slight changes in the distribution of the electric �eld, but the order of theresonance has stayed the same.

Figure 3.2: Results for the simulations with circular polarized light. The extinction spectra for the Möbiusrings show are raised compared to the linear results, for left handed light higher than for right handedlight. The torus also shows an increase, but is the same for both the right and left handedness. However,two new peaks are found, E and F. The absolute value of the electric �eld is also plotted for each peak,which is concentrated more near the arc of the ring than for the normal resonances.

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Figure 3.3: Cross-section of torus using y-polarized light.

Figure 3.4: Plots of peak A (left) and B (right).

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3.4 Odd mode resonances

3.4.1 Finding more odd modes

After successfully running the simulations with circular polarized light the next step is to look for the �odd-pole� resonance using the Johnson & Christy de�nitions with linear polarized light. Previous researchtells that by increasing the radius of the ring the resonance peaks will move to longer wavelengths [12]. As[7] found a hint of a peak near 400nm, increasing the ring radius should bring this into the range whereit can be investigated. Simulations are done for rings with radii of 175nm and 200nm. The incidentlight propagates in the minus z-direction, polarized in the x-direction and unit amplitude (3.5). Forcomparison the extinction for normal rings (torus) has been plotted also.

f(z) = ei(kz)x (3.5)

Changing the radius of the ring to 175 nm did bring out the peak. However, plotting the electric�eld did not reveal the expected pentapole, but a resonance with 7 lobes, a �heptapole� (Fig 3.5). Tomake sure the radius wasn't increased too much and the pentapole was overshot, new simulations weredone for smaller radii. However, the pentapole was not found. Increasing the ring radius even furtherto 200nm brings out another peak, showing nine lobes. This is then an "enneapole". Curiously enough,peak F, the "heptapole", seems to be showing 9 lobes now as well. An obvious di�erence between peakF and G is that most of the electric �eld is concentrated inside the ring at peak G, but is concentratedoutside the ring for peak F. More subtle di�erences can also be seen; inside the ring it seems to onlyhave 5 lobes and it's hard to tell whether the lobe on the z-axis to the right of the ring is just one lobeor actually two lobes.The next section will discuss e�orts made to explain these di�erences by looking atcharge distributions at these resonances.

Figure 3.5: The extinction plots for Möbius and Torus silver rings with radii of 175nm (a) and b)) and200nm (c) and d)) using the J&C de�nitions. Note that each peak gets red-shifted for larger radii.PeakE is a decapole. Bottom: e) Resonance with 7 lobes (heptapole). f) and g) Resonances with 9 lobes(enneapole). Peak F has 7 lobes for a radius of 175nm, while it has 9 for a radius of 200nm.

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3.4.2 Shedding light on odd modes

As COMSOL cannot plot charge densities for 3D structures the electric �eld normal to the surface andits amplitude are plotted instead (�g 3.6). The top picture is the hexapole, which is a symmetric mode.The odd modes which are also plotted are antisymmetric modes as [7] predicted. By changing a phase itis possible to evolve the resonance over one period. See appendix A. The plots shows the oscillation overone period divided into twenty frames. For the even mode shown in the appendix the electric �eld onlychanges in amplitude and sign, indicating a standing wave. Frame eleven has the opposite electric �eldof frame one, but is otherwise the same. This is the case for every pair, frame two and twelve, three andthirteen, etc. The odd mode resonances, however, are harder to interpret. The pentapole seems to bepropagating through the ring, but the other odd mode resonances do not. Looking at absolute values ofthe surface E-�eld con�rms that they are not propagating, but seen to be oscillating at the edge of thering.

Figure 3.6: Plots using JC for a) the hexapole (r=150nm), b) peak F: heptapole (r=175nm), c) Peak F:heptapole/enneapole (r=200nm), d) Peak G: enneapole (r=200m) Left is the absolute value of the E-�eld,middle is the E-�eld normal to the surface and right is its absolute value. For the absolute value plots thenumber of lobes is counted, red numbers indicate lobes on the inside and back of the ring, black numbersindicate lobes on the outside and front of the ring.

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Figure 3.7: Extinction plots for the Möbius ring (left) and Torus (right) for the Palik de�nitions.

Figure 3.8: Plots using Palik for a) the hexapole (r=150nm), b) pentapole (r=150nm), c) Peak F(r=175nm), d) Peak G (r=175nm). Left is the absolute value of the E-�eld, middle is the E-�eld normalto the surface and right is its absolute value. For the absolute value plots the number of lobes is counted,red numbers indicate lobes on the inside and back of the ring, black numbers indicate lobes on the outsideand front of the ring..

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To complete the picture the rings were increased to 175nm for the Palik de�nitions too. Due to timeconstraints simulations for a radius of 200nm could not be done. The results are shown in �gure 3.7 and3.8.

As already noted, peak F for the JC de�nitions is a heptapole for a radius of 175nm, but an enneapolefor 200nm. Comparing the electric �eld amplitude of these two resonances and counting the number oflobes reveals that the number stays the same on the inside of the ring, namely seven, but changes fromseven to nine on the outside of the ring. See �gure 3.6. Moving up to peak G displays something similar,only this time on the inside of the ring. The number of lobes goes up from seven to nine. Notice thatadding up all the lobes the total number of lobes is 14, 16 and 18.

The Palik de�nitions show similar behavior, although in a slightly di�erent manner. The lobes of thepentapole seem to split up going from 150 to 175 nm, going up from �ve to nine. Then moving frompeak F to peak the number of lobes increases from �ve to seven on the inside, but decreases from nineto seven on the outside. Adding all the lobes gives a total number of 10, 14 and 14.

Another observation is the that concentration of the E-�eld moves from the inside to outside and backagain for both cases.

3.5 Y-polarization

Because it is now possible to plot the surface electric �eld, a plot was made for the y-polarizationsimulation of the torus. See �gure 3.9. The �gure clearly shows that by using y-polarized light it ispossible to excite the asymmetric mode in a normal ring.

Figure 3.9: E-�eld normal to the surface of peak F for the JC de�nitions using y-polarized light.

This result inspired one last simulation for the 150 nm Möbius ring (JC de�nitions) using y-polarizedlight. The results are shown in �gure 3.10. The x-polarized plots are very similar to the y-polarizedplots, which is not surprising result, as the results for circular polarized light already showed this. Whatis really surprising is that y-polarized light still excited a symmetric mode at peak D. Peak F showed theexpected heptapole mode.

The electric �eld normal to the surface tells a similar story (�g. 3.11), but with one surprisingdi�erence for Peak E. The plot for the y-polarization is almost identical to the pentapole found using thePalik de�nitions! The absolute value of the electric �eld doesn't look anything like the pentapole (�g 2.6)equivalent though. Instead, it is only slightly di�erent from the one for x-polarized light (�g. 3.12).

For wavelengths longer than 600nm no resonances are found for either type of rings. This is explainableas [7] calculated that antisymmetric modes can't exist for wavelengths longer than 550nm. Symmetricmodes not showing up might be caused by the way of exciting the modes. The manner to excite themmust happen di�erently than explained in chapter 2, because the incident light is no longer polarizedin-plane with the ring. Since the resonances only appear for shorter wavelengths and not for longerwavelengths the driving frequency seems to play an important role.

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Figure 3.10: Top: Extinction spectra for x-polarized and y-polarized light. Bottom: |Enorm| for peak D,E and F for both the x-polarization and y-polarization.

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Figure 3.11: Enorm plots for x-polarized and y-polarized light. Note that peak E for Y plots is almostidentical to the pentapole.

Figure 3.12: |E| for peak E.

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Chapter 4

Discussion and Conclusion

4.1 Discussion

The start of with a general remark, as already noted in [7], the resonances found in the normal ringcorrespond very well with previously performed research and con�rms that the model produces reliableresults. Another thing to note is that the same peaks have di�erent maxima for di�erent radii, whichcould be explained by remembering that the step size used to sweep the wavelengths was approximately3nm, so the actual maximum might have been missed. The sharpness and the height of the peaks for theshorter wavelengths is something that might be worth looking into, because for this type of resonancethe shape of, for example peak C, is to be more expected. This might invalidate some of the conclusionsdrawn from the resonances found at these peaks. However, I have moved through the peaks checkingwhether the nature of the resonance changes and it does not, suggesting that this research produced atleast qualitative results.

I �nd the most remarkable result that it is possible to excite the same modes, be it an antisymmetric ora symmetric mode, in the Möbius ring using either x-polarized light or y-polarized light in the wavelengthregion below 600nm. This is not possible for the normal ring, where new resonance peaks with anantisymmetric mode were found. What I �nd interesting is that the way of exciting the modes is di�erentfor y-polarized light than it is for x-polarized, because the polarization is in-plane with the ring forx-polarized light, but perpendicular to the ring for y-polarized light.

I suspected I might �nd the same antisymmetric resonances using y-polarized light after I found theantisymmetric mode in the normal ring, but to �nd a symmetric mode as well surprised me. Then again,it makes sense that when it is possible to excite antisymmetric modes in a Möbius ring with x-polarizedlight it should be possible to excite the symmetric modes with y-polarized light as well. Apparently itis possible to excite both modes for a Möbius ring regardless of the polarization, whereas for the normalrings x-polarized light is required to excite symmetric modes and y-polarized light to excite antisymmetricmodes.

Finding higher order odd modes seemed to bring more questions than answers at �rst. However,looking at all the results, I see a trend. First of all the antisymmetric mode resonances (in Möbius rings)always appear after the symmetric mode resonances. Dipole, quadrupole, hexapole, octupole, pentapole,heptapole for the Palik de�nitions and dipole, ... , octupole, decapole, heptapole, enneapole for the JCde�nitions. The plots made of the antisymmetric resonances show that the number of lobes �t rather wellafter the the even mode resonances, namely, 10 (deca), 14, 14 and 14, 16, 18, as if the next even modefolds unto itself and becomes an antisymmetric mode. The pentapole is the prime example; for the JCde�nitions it's a symmetric mode and a decapole, but for the Palik de�nitions it's antisymmetric and gotcalled a pentapole. I think it should still be called a decapole, primarily because the same mode could beexcited in the JC ring by using y-polarized light. This didn't change the absolute value of electric �eldthough, which was used to determine the order of the resonance in the �rst place.

The 12 pole is missing in both cases. A possible explanation is that changing the radius and red-shifting the peaks has an e�ect on the con�gurations of the lobes. It was already noted in �gure 3.4 thatchanging the radius from 175nm to 200nm caused the heptapole to become an enneapole. Changing theradius created more "space" for the lobes and caused them to split up. The 12 pole might have had asimilar fate when increasing the radius to 175nm. Decreasing the radius might bring the 12 pole out. AMöbius ring size with all the resonances present would be ideal obviously, so solving the problems for the

17

simulations below 400nm to about 350nm might help in those regards. Going lower than 350nm is notrecommended, because the refractive index of silver starts behaving rather di�erently there.

4.2 Conclusion

In summary, plasmonic resonances in Möbius rings have been more thoroughly investigated. The projectwas a follow-up of [7] that found an "odd-pole" resonance using the Palik optical de�nitions for silver. Ithas been con�rmed that this resonance has an antisymmetric mode pro�le and higher order resonanceshave been found. Plotting the charge distributions directly was not possible, but plotting the electric �eldnormal to the surface made it possible to investigate the distribution. These plots showed the symmetricmodes for the even resonances and antisymmetric mode for the odd resonances.

Investigating not just the norm of the electric �eld, but also the surface electric �elds, showed thatthe odd mode in the Möbius ring is actually an even mode resonance on the edge of the ring and that it'sbest to steer away from calling these resonances odd modes. Odd modes show up at the point no moreeven modes are present, as if the �rst odd mode is just the next even mode in a di�erent con�guration.I referred to this as the mode folding unto itself, but the physics behind this is unknown.

Running simulations with circular polarized light to investigate the e�ect of the chirality of the Möbiusring showed an increase in extinction for both orientations, more for left-handed than right-handed lightand motivated running simulations with y-polarized light instead of x-polarized light. This showed thatfor the Möbius ring using either polarization excites the same modes for wavelengths below 600nm. Ringsize seems to have an e�ect on the con�guration of an antisymmetric mode as well. For symmetric modesthe distribution remains the same for bigger radii, but for antisymmetric modes a resonance can forexample change from a 12 pole to a 14 pole. It has also been shown that the resonances in both ringsare standing waves.

Because all the results are interpreted qualitatively and not quantitatively, a continuation of thisproject could focus on producing more quantitatively sound results. Other possible follow-ups are inves-tigating the change from symmetric to antisymmetric modes or the change of a resonance to a higherorder antisymmetric mode when the radius is increased.

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Appendix A

Phase evolution of resonances

The next pictures illustrate the oscillation of a resonance over one period. The phase step-size is 0.1 π.

A.1 8 Pole

Figure A.1: Octupole

19

A.2 5 Pole

Figure A.2: Pentapole

20

A.3 7 Pole

Figure A.3: Heptapole

21

A.4 9 Pole

Figure A.4: Enneapole

22

Appendix B

Material De�nitions

B.1 Palik

Figure B.1: Plots of the real (n) and imaginary (k) parts of the refractive index as given by Palik [9]

B.2 Johnson & Christy

Figure B.2: Plots of the real (n) and imaginary (k) parts of the refractive index as given by Johnson &Christy [9].

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Appendix C

COMSOL formula's

In this appendix the formula's used will be displayed.

• E = the total �eld

• Eb = background �eld

To calculate the relative �eld the background �eld has to be subtracted from the total (measurable) �eld.The relative �eld is the �eld generated by the ring. Same holds for the magnetic �eld.

Scattering Cross Section

(1/(1.327e-3[W/(m^2)]))*(real(conj(emw.Ey-emw.Eby)*(emw.Hz-emw.Hbz)-conj(emw.Ez-emw.Ebz)*(emw.Hy-emw.Hby))*emw.nx+real(conj(emw.Ez-emw.Ebz)*(emw.Hx-emw.Hbx)-conj(emw.Ex-emw.Ebx)*(emw.Hz-emw.Hbz))*emw.ny+real(conj(emw.Ex-emw.Ebx)*(emw.Hy-emw.Hby)-conj(emw.Ey-emw.Eby)*(emw.Hx-emw.Hbx))*emw.nz)

Absorption Cross Section

(1/(1.327e-3 [W/(m^2)]))*emw.Qrh

Norm of E-�eld

emw.normE

E-�eld normal to surface

(nx*(emw.Ex-emw.Ebx)+ny*(emw.Ey-emw.Eby)+nz*(emw.Ez-emw.Ebz)+j*(nx*(imag(emw.Ex)-imag(emw.Ebx))+ny*(imag(emw.Ey)-imag(emw.Eby))+nz*(imag(emw.Ez)-imag(emw.Ebz))))

Norm of surface E-�eld

Sqrt((nx*(emw.Ex-emw.Ebx)+ny*(emw.Ey-emw.Eby)+nz*(emw.Ez-emw.Ebz)+j*(nx*(imag(emw.Ex)-imag(emw.Ebx))+ny*(imag(emw.Ey)-imag(emw.Eby))+nz*(imag(emw.Ez)-imag(emw.Ebz))))*conj((nx*(emw.Ex-emw.Ebx)+ny*(emw.Ey-emw.Eby)+nz*(emw.Ez-emw.Ebz)+j*(nx*(imag(emw.Ex)-imag(emw.Ebx))+ny*(imag(emw.Ey)-imag(emw.Eby))+nz*(imag(emw.Ez)-imag(emw.Ebz))))

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Appendix D

Populaire samenvatting

Het onderzoekgebied Nanophotonics houdt zich bezig met het manipuleren van licht op schalen zo grootals, of kleiner dan de gol�engte van het licht. Een mogelijke toepassing van dit onderzoek is het maken vancomputers waarvan de schakelingetjes werken op licht in plaats van elektriciteit (elektronen). Hiervoormoet licht eerst goed begrepen worden en lopen er verschillende projecten die licht beter proberen tebegrijpen. Eén zo'n project is het proberen licht te laten lopen over kleine metalen ringen. Het ismogelijk licht te koppelen aan de elektronen die zich bevinden aan het oppervlakte van een metaal. Ditgebeurt wanneer het licht aan bepaalde voorwaarden voldoet en het wordt dan als het ware gevangenaan het oppervlakte. Zo'n golf wordt een surface plasmon polariton genoemd of SPP. Het bijzondereaan zo'n SPP is dat de frequentie hetzelfde blijft, maar dat de gol�engte veel kleiner is dan dat van hetoorspronkelijke licht.

Wat nu als je zo'n SPP zou proberen te laten lopen over een ring. Geen gewone ring, maar eenMöbius ring. Dit is een ring met een draai erin. Hierdoor heeft de ring maar één vlak en één rand. Alsje over het oppervlak van de ring zou lopen, loop je eerst aan de buitenkant van de ring, ga je vanzelfdoor naar de binnenkant van de ring en kom je uiteindelijk weer terug waar je begonnen bent. Het iseenvoudig om zelf een Möbius ring te maken: neem een strook papier, breng de uiteinden bij elkaar endraai een van de uiteinden een halve slag. Plak de einden vervolgens op elkaar. Omdat een SPP aanallerlei (rand)voorwaarden moet voldoen en de Möbius ring zo'n bijzonder structuur is, is het doel van ditproject te onderzoeken wat er gebeurt als een SPP loopt over het oppervlak van zo'n ring. Dit is gedaanaan de hand van computer simulaties. Met het computerprogramma COMSOL is het mogelijk een modelvan de ring te maken en deze vervolgens te beschijnen met licht. Hierdoor is het heel makkelijk bepaaldevariabelen te veranderen, zoals de grootte van de ring, de gol�engte van het licht of de richting van hetlicht. In dit onderzoek is gebruik gemaakt van zilveren ringen met een diameter van 300nm, 350nm en400nm. Ter vergelijking zijn ook simulaties gedaan met een normale ring. De ringen zijn beschenenmet licht met gol�engtes van 400nm tot 1200nm. Simulaties met kortere gol�engtes konden niet wordengedaan, omdat COMSOL vastliep. Voor elke gol�engte heeft COMSOL berekend wat er met het lichtgebeurt als dit op de ring valt. Door al deze berekeningen in een gra�ek te zetten is het mogelijk te kijkenwelke gol�engtes het grootste e�ect hebben op de ring. Een gol�engte die goed koppelt aan de ring isterug te zien als een piek in deze gra�ek. Er is dan resonantie. Resonantie wil zeggen dat de SPP dieaan het oppervlak is ontstaan precies een aantal keer in de omtrek van de ring past, waardoor hij zichzelfversterkt en er een staande golf ontstaat. Dit gebeurt bij invallend licht dat aan de juiste voorwaardenvoldoet, waardoor het de resonantie kan aandrijven. In dit geval is dat licht met een gol�engte dateen aantal keer de diameter van de ring is. Op een normale ring met een diameter van 300nm pastbijvoorbeeld licht met een gol�engte rond de 450nm, 600nm en 1200nm, wat terug te zien is in de gra�ek.Bij elk van deze pieken is er gekeken naar de gol�engte van de SPP. Bij 1200nm is de gol�engte één keerde omtrek, bij 600nm 1/2 keer en bij 450nm 1/3 keer. Bij een resonatie is het gemakkelijker om de buikenvan de staande golf te tellen. Eén gol�engte heeft twee buiken, twee gol�engtes hebben vier buiken, etc.Dus, de eerste resonantie heeft twee buiken, de tweede resonantie vier en de derde resonantie zes.

Bij de Möbius ring is hetzelfde waargenomen, maar met een extra piek rond de 400nm. Dit bleekeen bijzonder resonantie te zijn met 5 buiken. Deze SPP zou dan twee keer door de ring moeten lopenvoordat hij zichzelf in zijn staart kan bijten, precies zoals verwacht. Als dit soort resonanties werkelijkbestaan zouden er ook resonanties mogelijk moeten zijn met 7 buiken of 9 buiken.

Om deze resonanties te vinden is de diameter groter gemaakt, zodat alle resonanties "rood" ver-schuiven. Hierdoor verschuiven de pieken naar langere gol�engtes en zullen pieken die zich lager dan400nm bevonden nu wel zichtbaar zijn. Beide resonanties zijn ook daadwerkelijk gevonden.

De conclusie is dat bij de normale ring de golf in het midden van de ring loopt en geen onderscheidt

25

maakt tussen de binnen- of buitenkant. De eerste resonantie heeft twee buiken, de tweede resonantie vier,de derde resonantie zes, etc. De ring is niet dik genoeg, waardoor de SPP beide kanten "raakt".

Bij de Möbius ring gaat dit in het begin precies hetzelfde, tot de vijfde resonantie. In dat geval gebeurter iets geks, de SPP loopt niet meer dwars door de ring heen, maar is verplaats naar de rand van deMöbius ring. Het is nog wel steeds een SSP met tien buiken maar loopt nu eerst langs de "buitenrand"en vervolgens langs de "binnenrand". Dit kan alleen door de unieke geometrie van de Möbius ring endaardoor lijkt het op een SPP met vijf buiken. Hoe dit precies gebeurt is echter onduidelijk en is eenonderwerp voor een volgend onderzoek.

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