Thesis Submitted to the College of Engineering at Florida Institute of Technology in Partial Fulfillment of the Requirements for the Degree Of

Resonant Coupling of an Infrared Metasurface with PMMA

Michael Francis Finch, E.I.T. Bachelor of Science Electrical Engineering Florida Institute of Technology 2012

A thesis submitted to the College of Engineering at Florida Institute of Technology in partial fulfillment of the requirements for the degree of

Master of Science in Electrical Engineering

Melbourne, Florida May, 2014

The author grants permission to make single copies______We the undersigned committee hereby recommends the attached Thesis

“Resonant Coupling of an Infrared Metasurface with PMMA”

Michael Francis Finch, E.I.T.

______

Brian A. Lail, Ph. D. Joel A. Olson, Ph. D. Associate Professor Associate Professor of Chemistry Electrical and Computer Engineering Chemistry Major Advisor

______

Samuel P. Kozaitis, Ph. D. Professor Electrical and Computer Engineering Department Head

Abstract

Title:

Resonant Coupling of an Infrared Metasurface with PMMA

Author:

Michael Francis Finch, E.I.T.

Major Advisor: Brian A. Lail, Ph.D.

Resonance coupling between a metasurface or planar metamaterial and PMMA’s, polymethyl methacrylate, phonon resonance corresponding to the C=O molecular bond at 1733 cm-1 (52 THz) is investigated. Two metasurfaces, nano rod and nano slot, are modeled using ANSYS HFSS and measured material optical properties for gold and PMMA in inferred spectrum. Both metasurfaces exhibited normal mode splitting also known as avoidance crossing, quantum level repulsion, or vacuum

Rabi splitting. Normal mode splitting is shown to be a result of coupling between the PMMA phonon resonance and the resonance of the metasurface. Analytical models, which are tied to physical descriptions, are explored for normal mode splitting. The description varies from simple coupled pendulum oscillators to quantum electrodynamics descriptions like exciton-polaritons in microcavities. The analytical model produces an anti-crossing or avoidance crossing dispersion plot of hybrid modes from a system of differential Equations or uses a Hamiltonian matrix

iii for the coupled oscillator or Rabi description respectively. The Hamiltonian approach, in line with a quantum state description, was employed and solved for equations that describe the eigenfrequancies. A Fano resonance explanation tried together with a description of normal mode splitting is presented. A comparison is drawn between the complementary metasurface pairs with a discussion of

Babinet’s Principle. Couple resonates have applications in biosensing, surface- enhanced infrared absorption (SEIRA), and infrared imaging.

Table of Contents

Abstract ...... iii

Table of Contents ...... v

Keywords ...... viii

List of Figures ...... ix

List of Tables...... xi

List of Symbols ...... xii

List of Abbreviations...... xiv

Acknowledgment ...... xv

Dedication ...... xvi

Chapter 1: Introduction ...... 1

1.1 Motivation ...... 1

1.2 Design of a Metasurface Unit Cell ...... 2

1.3 Thesis Contribution ...... 6

Chapter 2: Background Material ...... 7

2.1 Classical Model Description ...... 8

2.2 Temporal Coupled Mode Theory ...... 10

2.3 Rabi Model Description ...... 11 v

2.4 Exciton Dynamics ...... 12

2.4.1: Plexciton ...... 12

2.4.2: Exciton-polaritons ...... 13

2.4.3: Fano Resonance ...... 14

2.5 Conclusion ...... 15

2.5.1: Summary of Analytical Models ...... 15

2.5.2: Analytical Model Description ...... 17

Chapter 3: Nano Rod Metasurface Design...... 21

3.1 Reflection Spectrum ...... 22

3.1.1: Uncoupled Modes ...... 22

3.1.2: Coupled Modes ...... 26

3.2 Normal Mode Splitting ...... 26

3.3 Conclusion ...... 28

Chapter 4: Nano Slot Metasurface Design ...... 29

4.1 Reflection Spectrum ...... 31

4.1.1: Uncoupled Modes ...... 31

4.1.2: Coupled Modes ...... 35

4.2 Normal Mode Splitting ...... 35

4.3 Conclusion ...... 37

Chapter 5: Conclusion ...... 38

5.1 Future Work ...... 38

5.1.1: Babinet’s Principle ...... 38

5.1.2: Perfectly Absorbing Metamaterial (PAM)...... 39

5.1.3: Fano-resonant Asymmetric Metamaterial (FRAMM) ...... 39

5.1.4: Self-complementary Metasurface ...... 40

5.1 Summary of Results ...... 40

References ...... 41

Appendix ...... 53

Appendix A: Classical Model Derivation ...... 54

A.1 Transfer Function Approach ...... 55

A.2 State Variable Approach ...... 56

Appendix B: Rabi Model Derivation ...... 58

vii

Keywords

Metasurface

Normal Mode Splitting

Metamaterial

Coupled Resonates

PMMA

Fano Resonates

Phonon Resonate

Surface-Enhanced Infrared Absorption

Vacuum Rabi Splitting

Babinet’s Principle

Plexciton

Exciton-polaritons

Temporal Coupled Mode Theory

Coupled Pendulum Oscillators

viii

List of Figures

Figure 1.1: (Top) Plot of the absorption k of the PMMA. (Bottom) Plot of the index of refractive of PMMA...... 4

Figure 1.2: HFSS metasurface unit cell with PMMA ...... 5

Figure 2.1: General dispersion plot describing normal mode splitting related to the resonance of the metasurface and PMMA...... 7

Figure 2.2: Classical Oscillator model where the upper figure is a coupled LCR circuit and the lower one is a mass spring system...... 9

Figure 2.3: Reflectance spectrum lineshape produced by the simulation...... 17

Figure 3.1: Nano rod metasurface unit cell designed in HFSS...... 21

Figure 3.2: Reflection spectrum for uncoupled mode in the nominal case for the shifted and unshifted cases ...... 23

Figure 3.3: The uncoupled shift normal modes of the metasurface for five different cavity thicknesses ...... 24

Figure 3.4: Normal mode splitting resulting from the coupled mode as shown for five different cavity thicknesses ...... 26

Figure 3.5: Dispersion curve that depicts normal mode splitting. The cyan and red dots are the HFSS model results ...... 27

Figure 4.1: Nano slot metasurface design in HFSS...... 29

Figure 4.2: Coupled resonance of a complementary slot metasurface with respect to the nanorod presented in Chapter 3...... 30

Figure 4.3: Reflection spectrum for the uncoupled mode in the normal case for the shifted and unshifted cases ...... 31

Figure 4.4: The uncoupled shift normal modes of the metasurface for five different cavity thicknesses ...... 32

Figure 4.5: Normal mode splitting resulting from couple mode as showed for five different cavity thicknesses ...... 35

Figure 4.6: Dispersion curves that depict normal mode splitting. The cyan and red dots are the HFSS model results...... 36

Figure A.1: (a) Circuit description for the coupled oscillator system. (b) Mechanical mass spring coupled oscillator description...... 544

Figure B.1: Visualization for a two-level coupled quantum system under perturbation from an external electrical field...... 58

List of Tables

Table 1: Analytical models used to describe normal mode splitting given different physical descriptions...... 16

Table 2: Approximate resonance frequency and losses of the uncoupled shifted modes ...... 25

Table 3: Approximate resonance frequency and losses of the uncoupled shifted modes ...... 33

List of Symbols

ħΩr [meV, J] Rabi energy

2Ωr [rad/sec, Hz] Rabi splitting d [μm] Distance between two mirrors in a cavity f’ms [Hz, THz]] Shifted (red or blue) uncoupled metasurface frequency f’pmma [Hz, THz] Shifted (red or blue) uncoupled PMMA frequency k [rad/sec, Hz] Coupling coefficient used in TCMT n Real part of the index of refraction

Rn Power reflectivity from a mirror in a cavity trod [μm] Distance between nano rod metasurface and PEC ground plane tslot [μm] Distance between nano slot metasurface and PEC ground plane

V[rad/sec, Hz, meV] Coupling coefficient used quantum state descriptions

β [rad/m] Phase constant

γ [rad/sec, Hz] Damping/Loss coefficient

δ [rad/sec, Hz] Detuning (ω1-ω2)

κ Imaginary part of the index of refraction

λ [nm, μm] Wavelength

ω [rad/sec] Radial frequency or radial eigenfrequency

th ωn [rad/sec] n uncoupled normal mode xii

th ωn’ [rad/sec] n shifted (red or blue) uncoupled normal mode

ωp [rad/sec] Plasmon frequency

th ωsn [rad/sec] a delta shift (red or blue) in radial frequency to the n uncoupled mode

xiii

List of Abbreviations

EIT Electromagnetic Induced Transparency

FEM Finite Element Method

FRAMM Fano Resonance Asymmetric Metamaterial(s)

FWHM Full Width Half Maximum

HFSS High Frequency Structural Simulator

IR Infrared

L Mode Longitudinal Mode

PAM Perfect Absorber Metamaterial(s)

PEC Perfect Electrical Conductor

PMMA Polymethyl methacrylate

SEIRA Surface Enhanced Infrared Absorption

SEIRS Surface Enhanced Infrared Spectroscopy

SERS Surface Enhanced Raman Spectroscopy

T Mode Transverse Mode

TCMT Temporal Coupled Mode Theory

xiv

Acknowledgment

I will like to acknowledge all of my supporters, directly and indirectly, in this grueling undertaking. May their efforts not be in vain.

Dedication

I would like to dedicate this work to all the people like me. May they see their inner genius and become greater than me.

“You should prefer a good scientist without literary abilities than a literate one without scientific skills.” - Leonardo da Vinci

xvi

Chapter 1: Introduction

1.1 Motivation

The stereotypical image that forms from a discussion of metamaterials [1] or metasurfaces, planar metamaterial, is either the Romulan cloaking device from Star

Trek or H. G. Wells’ “The Invisible Man”. However, beyond the realm of science fiction, metamaterials themselves have gained applications in cloaking, optical filters, optical antennas, lenses, photovoltaic, and sensors [2-9]. Metamaterials are made up of a sub-wavelength resonator unit cell referred to as either “meta-atoms” or “meta-molecules.” The effect of the contribution of each unit cell creates artificial or effective material properties that differ from the bulk material [10, 11].

A metasurface is a subcategory metamaterial with a patterned planar surface that can be symmetrical or asymmetrical as only desired for the meta-molecule. A metasurface is appealing for fabrication constraints where typical micron fabrication is done by layering material and still provides artificial optical properties. Another common metasurface design is a periodic surface with a ground plane, which is the design of interest (Figure 1.2) [12]. This metasurface then can be designed to resonate at a frequency band as well as have improved spectrum responses that natural materials do not possess.

The particular application of interest is coupling a tailored metasurface with a phonon resonance or vibrational moment, where coupling between materials like organic and/or polymer materials gives rise to a potential of application for biosensors [13-18].

Surface-enhanced Raman spectroscopy (SERS) and surface-enhanced infrared spectroscopy (SEIRS) are two methods used in biosensing. SERS employees

Raman scattering and nanoparticles for biosensing applications like cancer detection and treatment [19, 20]. Raman scattering is an inelastic scattering of light off of a material. The difference between SERS and SEIRS is how the molecules’ bonds are being bent or stretched. In the case of SEIRS, the resulting agitation of the bond results in a dipole moment. In case of both SERS [16-18, 21] and SEIRS

[5, 22-25], an anti-crossing or normal mode splitting dispersion relationship has been reported. Both techniques have exploited the vibration mode or phonon modes of a given material. This thesis focuses on Polymethyl methacrylate (PMMA) and

IR active phonon modes; therefore, SEIRS is more applicable in the discussion of applications. A SEIRS sensor that uses resonant coupling can be used to detect the vibrational footprint of a material and thus can be implemented in nondestructive identification applications.

1.2 Design of a Metasurface Unit Cell

Prior to fabrication, numerical simulation and modeling of complex structures using commercially available software is the norm. ANSYS HFSS [26] is the 2 industry standard numerical tool used to solve full 3D solutions to Maxwell’s equations. HFSS uses the finite-element method (FEM) to solve for the electric and magnetic fields. Ref [27] is an excellent source on numerical techniques for solving

Laplace’s Equation for electromagnetic applications.

HFSS provides the feature that allows the use of frequency-dependent material properties in the simulation of the designed structure. Figure 1.1 is plot of the material data used in the model for PMMA. Gold exhibits a dispersive complex refractive index in the infrared that must be accounted for by importing measured optical properties into the finite element models. The PMMA properties are crucial in this investigation as can be seen in Figure 1.1. There are IR active resonance peaks in the absorption spectrum. The peaks in the spectrum correlate to vibration stretching of molecular bond in the compound chain [24]. The peak of interest in this thesis is the double bond stretching of carbon and oxygen (C=O) located at

52THz (1733 cm-1) highlighted in red in Figure 1.1.

Wavenumber (cm-1) 1000 1500 2000 2500 3000 3500 4000 0.5

0.4

0.3 k 0.2

0.1

0 20 30 40 50 60 70 80 90 100 110 120 Frequency (THz) Wavenumber (cm-1) 1000 1500 2000 2500 3000 3500 4000 1.6

1.5

1.4 n 1.3

1.2

1.1 20 30 40 50 60 70 80 90 100 110 120 Frequency (THz)

Figure 1.1: (Top) Plot of the absorption k of the PMMA. (Bottom) Plot of the index of refractive of PMMA. Two different metasurfaces were simulated using HFSS based on periodically- placed unit cells containing different elements: a nano rod or nanowire as depicted in Figure 1.2 and the complementary nano slot structure. The unit cells were set up using Floquet ports and master/slave boundaries. Under these conditions the structure is simulated as an infinite array in the x-y plane [26, 28, 29]. The model is set up to have two linear polarized modes. An E-field along the rod or across the slot and a mode normal to the previous are designated in accordance with the literature as follows: the longitudinal mode (L mode) and transverse mode (T 4 mode) respectively [5, 14, 30]. The L mode is the resonance mode of the metasurface that results in normal mode splitting. The T mode is cross polarized and only the uncoupled normal mode would exist. Therefore; for the investigation in this thesis the L mode is the mode of interest as it would produce the sought after

“coupling event” or normal mode splitting.

Air Box

PMMA

Metasurface

PEC Ground Plane Si Cavity

Figure 1.2: HFSS metasurface unit cell with PMMA From the model, the reflection coefficient or scattering parameters are quantities of interest, as becomes clearer in the following discussion of normal mode splitting.

The design contains a reflective ground plane, which reduces the transmission throughput to zero. The power can only be reflected, absorbed, or stored via the design. 5

1.3 Thesis Contribution

The thesis presents a nano rod/nanowire metasurface/metascreen coupled to the

1733cm-1 PMMA resonance with results that show evidence of normal mode splitting. The results are formulated via an ANSYS HFSS model. The self- complementary structure of the metasurface is also presented. This thesis investigates the normal mode splitting between the stated metasurfaces coupled to a polymer PMMA (polymethyl methacrylate). PMMA has an absorption peak at 52

THz which corresponds to the C=O molecular bond present in PMMA. The complementary metasurface coupling results are also compared with a discussion related to electromagnetic duality and Babinet’s Principle.

Chapter 2: Background Material

Normal mode splitting which is analogous to quantum level repulsion, level repulsion, or level anti-crossing has been described in various manners, from simple coupled mass spring oscillators [31-33] to light-matter coupling in quantum electrodynamics [34-36]. This section provides a survey of model descriptions and resulting Equations that have been used by scholars apply in this description of the coupling event or normal mode splitting.

Normal Mode Splitting

0.2 0.15

m) 0.1  0.05 0 -0.05

-0.1 Peak Wavelength ( -0.15 -0.2 Hybrid + Hybrid - -0.25 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Metamaterial Resonance Wavelength (m) Figure 2.1: General dispersion plot describing normal mode splitting related to the resonance of the metasurface and PMMA.

The dispersion, avoidance crossing, curves of normal mode splitting are shown in

Figure 2.1. The horizontal axis describes the uncoupled metamaterial resonance wavelength or frequency. The vertical axis of Figure 2.1 is the eigenfrequencies of the coupled system. The red line is the light line of the uncoupled metamaterial resonance which is a 1:1 relationship. The cyan horizontal line is the uncoupled phonon or vibrational resonance’s light line, which is constant over all frequencies of the metamaterial. The blue and black solid lines are the hybrid ± modes that result from the coupling of the metamaterial and phonon uncoupled resonances. If there were no coupling, then the blue and black mode lines would follow the cyan and red line and cross at the origin in Figure 2.1. The form of the graph leads to its names: avoidance crossing, level repulsion, or level anti-crossing.

2.1 Classical Model Description

The classical model that could describe the system contains two mass spring systems coupled together with a coupling spring or modeled with two LCR systems coupled with a capacitor as can be seen in Figure 2.2. A system of second order differential equations can be formulated where either mesh current or mass displacement is the parameter of interest. Therefore, as seen in Appendix A, an equation expressing displacement or mesh current as a function of frequency can be used to determine the characteristic equation of the system, i.e., the denominator of the equation. With this description, Figure 2.1 contains a plot of the eigenwavelength or eigenfrequency of the characteristic equation. An analogy of

8 the eigenfrequency would be a cut-off frequency of a filter from circuit theory or control theory.

R2 R1

VS2(t) VS1(t) AC I2(t) I (t) AC CC 1

L2 C2 L1 C1

F2(t) F1(t) k2

kc M2 M1

B2 k1 B1

x2(t) x1(t)

Figure 2.2: Classical Oscillator model where the upper figure is a coupled LCR circuit and the lower one is a mass spring system.

An equation for the eigenfrequency can be obtained by solving the characteristic equation. The derivation for this model is performed in Appendix A. The equation for the eigenfrequency is as follows:

( ) √( ) (2.1)

Here ωn are the uncoupled resonance frequencies of the individual systems, Ωr is related to the coupling strength of the two resonances as such the literature in general refers to double this quantity as Rabi splitting [35]. Equation 2.1 is well documented and as such has been used to describe Lorentz oscillators and electromagnetic induced transparency (EIT) [22, 23, 31-33, 37].

2.2 Temporal Coupled Mode Theory

Typical coupled mode theory has applications in fiber optics with optical switches and 3dB couplers, which in turn make up interferometers and optical modulators. It can be shown that spatial coupling takes the form of two coupled first-order differential equations [38]:

(2.2a)

(2.2b)

where an is the field amplitude at a given position, knm are the coupling coefficients, and βn is the propagation constant. Equation 2.2 assumes weak lossless coupling and it is assumed that orthogonal modes with positive energy are present; then k21 = k12* = k resulting from conservation of energy [39]. Next, rewriting the spatial mode coupling to temporal mode coupling only results in changing z to t and β to ω in Equation 2.2.

(2.3a)

(2.3b)

Equation 2.3 can be solved similar to the method presented in Appendix A to obtain an expression for the eigenfrequencies with either method. It can be shown that the solution is [39, 40]:

√( ) | | (2.4)

It the case of lossy resonances, then in Equation 2.3, jan(t)ωn can be replaced with jan(t)[ωn –jγn] [13].

2.3 Rabi Model Description

By the turn of the twentieth century, new theories using quantum dynamics were developed that could be used to explain the nature of matter and light. One of the simple models for two level atomic systems was developed by Isidor Rabi. Rabi neglected losses like lifetime and gain like pumping to the system with the only perturbations being a time harmonic field [35]. Rabi dynamics is considered to be a semi-classical approach as a result of these simplifications; however, Rabi’s model is the foundation for subsequent models like the Jaynes-Cumming model.

Appendix B provides a derivation of hybrid eigenfrequencies given a Rabi-like approach to coupling. The Equation derived is as follows:

√( ) (2.5)

Equation 2.5 has been documented by several sources [14, 22, 24, 41, 42] in their description of normal mode coupling. As in the case of Equation 2.5, ωn are the uncoupled resonances and 2Ωr is referred to as Rabi splitting.

2.4 Exciton Dynamics

2.4.1: Plexciton

Excitons are quasi-neutral particles resulting from photon interaction with matter

[34]. Excitons can be thought of as a bound state or a simple bounded electron-hole pair by Coulombic binding potential [34]. Plexciton is the coined term used in the study of plasmonic coupling with excitons [42, 43]. The concepts of plasmonic, metal optics, is coupled with the concepts of metamaterials. Classically metals are thought of as a “sea,” “gas,” or “cloud” of electrons within a positive nucleus lattice. The Drude-Sommerfeld model expresses the frequency dependent dielectric function as:

(2.6)

where ωp is the plasma frequency and γ is a damping or loss term [22, 23]. When an external field is applied to a metal, the plasmon induced can be visualized as an oscillation of charge densities along the surface for a surface plasmon. The plasmon

12 itself is treated with wave mechanics; therefore, simulation packets using numerical techniques like FEM used by HFSS are valid [23].

The coupling between plasmonics and excitons has been shown to result in normal mode splitting or hybridization with nanostructures and a nanoparticle with J- aggregates\H-aggregates [17, 43]. The resulting Equation to describe the splitting is derived from Rabi splitting [14, 24, 42] and is the solution derived in Appendix B.

√( ) (2.7)

In Equation 2.7, ωn are the uncoupled excitons and the localized surface plasmon resonance (LSPR), ħΩr is the Rabi spitting energy.

2.4.2: Exciton-polaritons

Exciton-polaritons are quasi-particles that describe light-matter coupling [34].

Applications of exciton-polaritons include superfluidity and Bose-Einstein condensation (BEC) [35, 44, 45]. The discussion of exciton-polaritons is limited to coupling in a microcavity, which then results in normal mode splitting. The two modes that couples are a bare cavity mode, ω1, and the exciton state, ω2, which then produce upper and lower polaritons. These then produce anti-crossing behavior or normal mode splitting [46]. Equation 2.8 is referred to as the coupled mode

Equation for exciton-polaritons in a microcavity [22, 23, 35].

(2.8)

The quantities γ1 and γ2 are the finesse of the cavity mode and inhomogeneous broadening or damping of the excitonic state respectively.

2.4.3: Fano Resonance

Fano in his 1961 paper described the phenomenon that bears his name as follows:

“The interference of a discrete autoionized state with a continuum gives rise to characteristically asymmetric peaks in excitation spectra.” [47] Here the discrete autoionized state is a bound state, which can be thought of as an electron in the valance band. In the terms of electrodynamics, Fano resonance can be described as the interference between superradiant (bright) and subradiant (dark) modes [13, 48,

49]. The concept of superradiant and subradiant plasmonic and nanostructures can be designed to have Fano responses, which have applications in biosensors [30, 50,

51]. The design system in this thesis does exhibit a Fano-like asymmetric peak. The two modes that create the Fano-like response are the PMMA and metamaterial uncoupled resonances.

2.5 Conclusion

2.5.1: Summary of Analytical Models

Table 1 documents the resulting eigenfrequencies expressed for the different descriptions of normal mode coupling. It can be seen that the descriptions produce similar results for the expression. Table 1 shows that the equations used to describe normal mode splitting are very similar and fundamentally formulated from the conservation of energy.

Table 1: Analytical models used to describe normal mode splitting given different physical descriptions.

Model/Description Equation Ref

[31-33]

Classical ( ) √( )

Couple Mode [13, 38-

√( ) | | Theory 40]

[32, 35,

Rabi √( ) 41, 42]

[14, 17,

Plexciton 42, 52-

√( ) 54]

[34, 35,

Exciton-polaritons 44, 45]

[16, 17,

Mie/Nanoparticle 52]

√

2.5.2: Analytical Model Description

The goal is now to develop an analytical model that can be used to describe normal mode splitting for the results in the next section. It can be seen from Figure 2.3 that there is an asymmetrical lineshape that is a signature of Fano resonance and has been called Fano lineshape [55]. Therefore, it can be said there are two states: a continuum of states (ω1) and a bound state (ω2). It is assumed that there are losses in the systems for both states (γ1 & γ2),Where quantity γn contributions are, but not limited to, Ohmic loss the material especially the metal in the IR, from lifetime of the states, and/or spontaneous emission. In general, the laying of material may cause red shifting or blue shifting of the resonance of the system; therefore, ωn’ =

ωn + ωsn.

Wavenumber (cm-1) 1700 1710 1720 1730 1740 1750 1760 1770 1

0.9

0.8

0.7

2Ωr

Reflectance 0.6

0.5

0.4 5.9 5.85 5.8 5.75 5.7 5.65 5.6 Wavelength (m)

Figure 2.3: Reflectance spectrum lineshape produced by the simulation.

Using a similar approach as in Appendix B as well as in [32, 43], a Hamiltonian can be used to describe the damped resonance system as follows:.

[ ] [ ] [ ] [ ] (2.9)

The eigenvalues of H are the eigenfrequencies of the hybrid modes, which can be determined using standard process, i.e, det (H – ħωI) = 0.

(2.10)

It can be seen that Equation 2.10 is in the same form as Equation 2.8, which describe coupling of exciton-polaritons in microcavities. Equation 2.10 can be solved for the eigenfrequencies [45].

(2.11)

√( ) ( )

From Equation 2.8, the eigenfrequencies are complex and the real component is of interest. The complex component relates physically as damping or loss. The damping of the eigenfrequencies implies a “bandwidth” or a regime of strong- coupling [44, 45]. This “bandwidth” in practice would be a fabrication tolerance or would dictate which molecules could be detected by the designed system.

Therefore, the b2-4ac term of Equation 2.11 must be a positive real number to have strong-coupling. It is also assumed that the difference between the state energy is

negligible, i.e., (the term is referred to as detuning

[45]). Equation 2.12 relates coupling strength and loss of the system.

| | (2.12)

If the condition of equation 2.12 holds, then there is strong-coupling between the two uncoupled eigenstates. Due to the coupling, hybrid modes or mode splitting can be seen in the reflection or transmission spectrum that is similar to vacuum- field Rabi splitting [44, 45]. A special case of the condition in 2.12 would be where

the resonances are “well-matched”, i.e., | | . The full width haft maximum

(FWHM) can be defined as γ; therefore, the uncoupled resonances would overlap each other perfectly as well as the coupled resonances would approach from asymmetrical to more symmetrical lineshapes. The condition of 2.12 also places a fundamental constraint on coupling due to the fact that γpmma is a constant dictated by the nature of the material.

A process similar to the Appendix B solution can be implemented to solve for the coupling strength. First, assume the minimum difference in eigenfrequencies is related to maximum coupling, defined by 2Ωr as depicted in Figure 2.3, i.e., ω+ - ω-

= 2Ωr.

√( ) ( ) (2.13)

Again, δ ≈ 0; therefore, Equation 2.13 simplifies and can easily be solved for V. 19

( ) (2.14)

Comparing Equations 2.14 and 2.12 implies that strong-coupling and splitting are one and the same. Using the information from Equations 2.12, 2.13, and 2.14, equation 2.11 is approximated as:

√( ) (2.15)

Equation 2.15 is the same Equation as in Appendix B used to describe vacuum- field Rabi splitting and has a similar result as those used in references [14, 24, 32,

41-43].

Chapter 3: Nano Rod Metasurface Design

The analytical model developed is now applied to a nano rod metasurface as seen in

Figure 3.1. Equation 2.15 is rewritten in the form of the resonated or eigen states of interest as well as in terms of frequency in hertz. The two states are metasurface or metamaterial and the phonon resonated state in the PMMA can be written as

√( ) (3.1)

It is evidence that the nano rod metasurface exhibits normal mode splitting and can be described by Equation 3.1. The next sections of this chapter use the model as seen in Figure 3.1 to gather data that then is used in Equation 3.1.

trod [μm]

Figure 3.1: Nano rod metasurface unit cell designed in HFSS.

In the present investigation of the coupling and decoupling of the metasurface with the phonon resonance of PMMA, the quantity trod, shown in Figure 3.1, is tuned.

The parameter is tuned so that the uncouple metasurface’s resonance is located at

52 THz which then results in normal mode splitting. Once the optimum case of normal mode splitting is obtained, then the parameter can be detuned to investigate the decoupling of the system. The resulting effects are shown in Figures 3.3 and 3.4 for uncoupled and coupled cases.

3.1 Reflection Spectrum

3.1.1: Uncoupled Modes

It is assumed that there is negligible shifting of the PMMA resonance. Therefore, only the metasurface frequency shift is considered. Two auxiliary models are used: one with a dispersionless PMMA or dummy material, and another with no PMMA modeled. The nominal case, where maximum coupling is observed with both dipersionless and no PMMA is shown in in Figure 3.2.

0.9

0.8

0.7 Reflection 0.6

0.5 Unshifted Metasurface Resonance Red Shift Metasurface Resonance 0.4 51 51.5 52 52.5 53 53.5 54 Frequency (THz)

Figure 3.2: Reflection spectrum for uncoupled mode in the nominal case for the shifted and unshifted cases From Figure 3.2, it can be seen that there is a 0.2 THz red shift induced due to the

PMMA coating. The shift is taken into account when computing the normal mode splitting with Equation 3.1.

0.9

0.8

0.7 Reflection 0.6 0.96m 0.98m 0.5 0.995m (Nominal Case) 1.0m 1.02m 0.4 51 51.5 52 52.5 53 53.5 54 Frequency (THz)

Figure 3.3: The uncoupled shift normal modes of the metasurface for five different cavity thicknesses The model with the “dummy” material was swept for various cavity thicknesses, and the resulting reflection curves are shown in Figure 3.3. The nominal case is precisely at 52 THz or the PMMA resonance, which results in maximum coupling of the two eigen states. Next the PMMA with the dispersive material is considered.

The effect of tuning the metasurface resonance on the coupling or hybridization of the two resonance states can be seen from Figure 3.4.

Table 2: Approximate resonance frequency and losses of the uncoupled shifted modes

| | fms trod [μm] [meV] [THz] [meV] [THz]

0.96 218.4 52.8 0.931 0.225

0.98 216.3 52.3 0.931 0.225

0.995 215.1 52 1.03 0.25 (Nominal Case) 1.0 214.6 51.9 1.03 0.25

1.2 212.6 51.4 0.931 0.225

Table 2 tabulates the shift uncoupled modes and losses of the system, which are calculated by the full width half maximum (FWHM) of the normalized metasurface and PMMA [22, 56]. The values calculated by Equation 2.12 are used to investigate if the condition for normal mode splitting is valid.

3.1.2: Coupled Modes

0.8

0.6

Reflection 0.4 0.96m 0.98m 0.2 0.995m (Nominal Case) 1.0m 1.02m 0 51 51.5 52 52.5 53 53.5 54 Frequency (THz)

Figure 3.4: Normal mode splitting resulting from the coupled mode as shown for five different cavity thicknesses From Figure 3.4, it can be seen that the two resonances couple the most strongly at the red curve where the metamaterial red shifted uncoupled resonance is at 52 THz.

It can be seen that as the metamaterial mode moves close to maximum coupling, the mode resonance frequency blue shifts then red shifts after the maximum is reached; therefore, it could be said that the resonance is being repelled from 52

THz, and thus the name “level repulsion” for normal mode splitting follows.

3.2 Normal Mode Splitting

The coupling strength can be calculated from the nominal case in Figure 3.4 under the assumption that the coupling strength is only related to the Rabi energy. Then

Equation 3.1 can be used with the calculated coupling strength. The coupling 26 strength is approximately 0.83 meV (6.67 cm-1). The calculated coupling is smaller than the ones given in references [17, 22, 24, 57, 58]. The smaller coupling energy may be due to approximate expressions and, in the case of perfect metamaterial absorber, PMA, which is used the uncoupled metamaterial mode is approximately

80% to well over 90% absorbed [11, 24, 59].

Metamaterial Resonance Frequency (THz) ) 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 -1 1770 53 1760

1750 52.5

1740 52 1730

1720 51.5 1710 Hybrid + Hybrid - 1700 51 Coupled Resonances Frequency (THz) Coupled Resonances Wavenumber (cm 1700 1710 1720 1730 1740 1750 1760 1770 Metamaterial Resonance Wavenumber (cm-1)

Figure 3.5: Dispersion curve that depicts normal mode splitting. The cyan and red dots are the HFSS model results Using the results from the previous section in Figures 3.2, 3.3, and 3.4 as well as in

Equation 3.1, the dispersion plot or anti-crossing plot can be calculated. This plot is shown in Figure 3.5. The horizontal axes are the unshifted and uncoupled or

PMMA-less metasurface resonance. The red line is the unshifted metasurface light line, the magenta is the red shift metasurface light line, and green is the PMMA uncoupled resonance line. The cyan and red dots represent the HFSS model results

27 for the resonance dips from Figure 3.4. The blue and black lines labeled hybrid +/- are the analytical model resulting curve.

3.3 Conclusion

It can be seen that there is modest agreement with the analytical and HFSS results.

| | Recalling Equation 2.14 and the tabulated results from Table 2 for , the

| | two qualities, and Ωr, are comparable; therefore, the coupling strength

is closer to the results of Equation 2.14, which is 1.28 meV (10.4 cm-1). For the case of a nano rod metasurface, FEM techniques applied to electromagnetics were applied to show some agreement with mathematical model predictions for normal mode splitting. The following section evaluates the self-complementary metasurface to the nano rod, i.e., a nano slot with similar arguments to those for the nano rod.

Chapter 4: Nano Slot Metasurface Design

The nano slot is the self-complementary, or simply complementary, metasurface to the nano rod presented in the previous section. A similar process that was applied to the nano rod is applied to the nano slot design. Equation 3.1 is also used with different specific values for the case of the nano slot. Figure 4.1 shows the designed nano slot metasurface in HFSS where tslot is the parameter that is swept to detune the metasurface.

tslot [μm]

Figure 4.1: Nano slot metasurface design in HFSS.

The nominal case of the nano rod is first investigated because the complementary metasurfaces have electromagnetic duality. The electrical field is linearly polarized across the slot, longitudinal mode, to form the magnetic dipole. If Babinet’s

Principle holds for this system, then the design from the nominal case of the nano rod should produce evidence of normal mode splitting.

0.95

0.9

Reflection 0.85

0.8

0.75 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 Frequency (THz)

Figure 4.2: Coupled resonance of a complementary slot metasurface with respect to the nanorod presented in Chapter 3. Figure 4.2 shows the reflection spectrum for the nano rod’s self-complementary metasurface. It can be seen that the slot metasurface resonance is blue shifted compared to the rod metasurface. Therefore, the system cavity thickness, tslot, needs to be tuned. The concluding chapter discusses the implications of the apparent violation of the self-complementary metasurface.

4.1 Reflection Spectrum

In a similar nature to that of the nano rod, the reflection is used to gather information about the coupled and uncoupled resonance in order to plot the dispersion curve described by Equation 3.1.

4.1.1: Uncoupled Modes

Similarly, the PMMA mode is assumed to have no shift in its resonance. After the metasurface is tuned back to 52 THz using the “dummy” PMMA procedure, the resonances of the system red shifts by 0.2 THz due to the presence of the PMMA.

This should be no surprise given that the shifting only depends on the presence of the dispersionless component of PMMA and not the metasurface itself.

0.9

0.8

0.7

0.6

Reflection 0.5

0.4

0.3 Unshifted Metasurface Resonance Red Shift Metasurface Resonance 0.2 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 Frequency (THz)

Figure 4.3: Reflection spectrum for the uncoupled mode in the normal case for the shifted and unshifted cases

In a similar fashion to that used in the case of the nano rod (the model with the

“dummy” PMMA), the cavity was swept to find the shifted uncoupled resonance of the metasurface that lay at the C=O resonance point of 52 THz. This can be seen in

Figure 4.4 for the nominal case of tslot = 1.40μm.

0.8

1.36m 0.6 1.38m 1.40m (Nominal Case) 1.42m

Reflection 0.4 1.44m

0.2

0 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 Frequency (THz)

Figure 4.4: The uncoupled shift normal modes of the metasurface for five different cavity thicknesses

What can be seen from Table 3 compared to Table 2 is that, in general, the FWHM of the uncoupled nano slot resonances match better to the resonance of the C=O vibrational bond. The better match would cause a stronger coupling as described in

Equation 2.12.

Table 3: Approximate resonance frequency and losses of the uncoupled shifted modes

| | fms tslot [μm] [meV] [THz] [meV] [THz]

1.36 216.7 52.4 1.14 0.275

1.38 215.9 52.2 0.724 0.175

1.40 215.1 52 0.517 0.125 (Nominal Case) 1.42 213.8 51.7 0.724 0.175

1.44 213.0 51.5 0.517 0.125

The nano slot FWHM is in general wider than the nano rod FWHM. If metasurface

PEC ground plane is thought of as a resonance optical cavity, then the equation for the optical cavity FWHM follows [60]:

(4.1) [ ]

where Rn is the power reflectivity of the PEC ground plane (i.e. RPEC = 1) and metasurface c is the speed of light, n is the index of refraction, and d is the cavity

33 length, in this case tslot. From equation 4.1, the dependence of the Rms and d are the variable of interest as there the tunable variables of the model.

(4.2.a)

[ ] (4.2.b)

From equation 4.2.a and 4.2.b the FWHM would decrease with increasing d and

Rms this contradicts the results of Figure 3.3 and 4.4. Therefore, the likely culprit of the wider FWHM in the case of the nano slot would stem from the extra metallization needed for the slot geometry. Metals in the IR wavelength are lossy and the FWHM of the resonance is related to the loss in the system, so it follows that the metal loss are counteracted any benefits from the optical cavity.

4.1.2: Coupled Modes

0.9

0.8

0.7

0.6

0.5 Reflection 0.4 1.36m 1.38m 0.3 1.40m (Nominal Case) 0.2 1.42m 1.44m 0.1 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 Frequency (THz)

Figure 4.5: Normal mode splitting resulting from couple mode as showed for five different cavity thicknesses In Figure 4.5 the nominal case can be analyzed in the same way as was done in the discussion of Figure 3.4. The cavity is tuned through the vibration resonance, which then shows the coupling event for the nominal case where the shifted uncoupled mode is located at 52 THz.

4.2 Normal Mode Splitting

The couple strength can be calculated via Figure 4.5 at the nominal case. The coupling strength is simply Ωr. Therefore, the Rabi energy is approximately 1.24 meV (10 cm-1). The slot Rabi energy is slightly larger than the rod’s case. The larger Rabi energy follows well with the conclusions drawn from comparing Table

2 and Table 3, i.e., that the slot shows evidence of stronger coupling compared to the rod case.

Metamaterial Resonance Frequency (THz) )

-1 51 51.2 51.4 51.6 51.8 52 52.2 52.4 52.6 52.8 53 1770 53 1760

1750 52.5

1740 52 1730

1720 51.5 1710 Hybrid + Hybrid - 1700 51 1700 1710 1720 1730 1740 1750 1760 1770 Coupled Resonances Frequency (THz) Coupled Resonances Wavenumber (cm -1 Metamaterial Resonance Wavenumber (cm )

Figure 4.6: Dispersion curves that depict normal mode splitting. The cyan and red dots are the HFSS model results.

As in Chapter 3, Figure 4.6 displays the dispersion curve that results from Equation

3.1. The horizontal and vertical axes are the same as in Figure 3.5. The only difference is that it is in the case of the nano slot. The cyan and red dots, which are the HFSS modeled results comprised from Figure 4.5, and the uncoupled unshift resonance modestly agree with the mathematical model.

4.3 Conclusion

As already stated, there is modest agreement between the analytical model and

HFSS results. For the points at 51.8 THz, the system may be decoupling from the vibrational mode which results in the cyan dot to falling onto the PMMA light line.

It can be also simple quantization error due to the step size of 0.1 THz in the spectrum analysis. Also in the same vain as in the case of the nano rod, the spectral

| | matching term, , is on the order of the Rabi energy; therefore, the

coupling strength is slightly larger than the approximation used in Figure 4.6.

Chapter 5: Conclusion

5.1 Future Work

The chief application for coupled resonances would reside in SEIRS or SERS. As can be seen hinted through figure 3.4 and 4.5 there is a modest enhancement of the

PMMA’s uncoupled resonance. Areas of investigation would be in the perfect absorbing metamaterial (PAM) [11, 24, 59] and Fano-resonant asymmetric metamaterial (FRAMM) [11, 13, 50] where dark modes interacted with bright modes to form the Fano lineshape.

5.1.1: Babinet’s Principle

As already discussed, Figure 4.2 is the coupled mode spectrum for the case that tslot

= trod. It can be seen that nano slot system resonance has been blue shifted compared to the nano rod system resonance. The classical Babinet Principle considers infinitely thin PEC screens embedded in one index of material [61-64].

Therefore, the principle breaks down for compound systems containing lossy IR metals. In a discussion of designing self-complementary compound systems it would be advantageous to derive an empirical mathematical model that corrects for the shifted uncoupled resonance.

5.1.2: Perfectly Absorbing Metamaterial (PAM)

A PAM itself has applications with thin-film photovoltaic devices due to the absorption nature of the system. An uncoupled PAM mode spectrum provides from

20% to less than 10% reflection signal; thus it is named the “prefect absorber”. The effect of coupling the vibrational resonance to PAM results is that the coupled resonance dips below 20% in the reflection spectrum. An impedance matching condition is used to achieve 100% absorption [11, 24, 59].

5.1.3: Fano-resonant Asymmetric Metamaterial (FRAMM)

In FRAMM, a Fano-resonant line-shape arises from interference between “bright” and “dark” modes of the metamaterial unit cell. An asymmetrical metamaterial unit is employed and described using TCMT and EIT [54]. The FRAMM provides an enhancement to the peak reflectivity of the molecule resonance [11, 13]. Future investigation may be modeling by describing the effect as - three coupled resonances, i.e., a “dark” mode coupled to a “bright” mode and a vibrational mode.

A similar process as described in Chapter two can be expanded and is similarly described in references [14, 23]. Here, the Hamiltonian can be written as

[ ] (5.1)

where ωn –jγn are the uncoupled resonated frequencies and Vnm are the coupling strengths between n onto m. With the expanded matrix the eigenfrequencies then could be solved for in similar process. 39

5.1.4: Self-complementary Metasurface

In the same vain as in FRAMMs, it would be interesting to investigate the coupling on the combined model of both the nano slot and nano rod in a layered structure.

Both the nano rod and slot couple individually with the resonant mode of the

PMMA, so a similar description may be used as in Equation 5.1.

5.1 Summary of Results

An analytical model was developed to approximate normal mode splitting that then was applied to model data obtained from HFSS results. A nano rod and a self- complementary nano slot structure were modeled and have shown evidence of coupling via normal mode splitting. Coupling between plasmonic and phonons have applications in surface-enhanced infrared spectroscopy. SEIRS applications can be inferred from Figures 3.4 and 4.5 where it can be seen that the PMMA vibrational mode absorbs more of the incoming energy. Thus a deeper notch in the reflection spectrum is observed. A goal for future designs would be to enhance the coupling, which in turn would enhance the sensitivity of the system. However, the present modeled results and analytical discussions have deepened our understanding of coupled resonant systems.

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Appendix

Appendix A: Classical Model Derivation

The derivation of the characteristic Equation of a dual driven coupled oscillator system is given. Figure A.1 gives two models for a coupled system to which a system of difference equations can be formulated. The system is a MIMO, Multiple

Input-Signal Output, system where the input is the input voltage or driving force onto the mass of the spring. The outputs are the mesh currents or the position of the masses with time or frequency.

R2 R1

VS2(t) VS1(t) AC I2(t) I (t) AC CC 1

L2 C2 L1 C1

F2(t) F1(t) k2

kc M2 M1

B2 k1 B1

x (t) x (t) 2 1

Figure A.1: (a) Circuit description for the coupled oscillator system. (b) Mechanical mass spring coupled oscillator description.

A.1 Transfer Function Approach

Simple circuit theory can be applied to the circuit in Figure A.1 (a) to form two mesh current equations:

( ) (A.1)

( ) (A.2)

Using algebra, Equations A.1 and A.2 can be used to solve the mesh currents and :

[ ( )]

(A.3)

[( ) ( ) ]

[ ( )]

(A.4)

[( ) ( ) ]

The characteristic Equation is the same for both outputs and can be written as follows when the complex frequency, , is set to :

(A.5)

where , , and in the characteristic Equation. It √

follows that is related to the coupling strength of the two systems under excitation. It follows that the eigenfrequencies can be solved from Equation A.5

(note that the eigenfrequencies are taken to be only real).

( ) √( ) (A.6)

A.2 State Variable Approach

The eigenfrequency can also be calculated by taking the system of differential equations and writing the state Equation. It already has been seen that the particular solution does not affect the characteristic equation. Therefore, only the system matrix A is required.

̈ ̇ (A.7)

̈ ̇

Equation A.7 is the standard equation for motion for coupled oscillators depicted in

Figure A.1. The system state matrix is as follows:

̇ ̇ [ ] [ ] [ ] (A.8) ̇

56 where χ1= , χ2= ̇ , χ3= , and χ4= ̇ respectively. From Matrix A the eigenfrequencies can be solved for by det (H – sI) = 0, where s is a complex eigenfrequency.

(A.8) where replacing the complex eigenfrequency s with jω, Equation A.9 is the same as Equation A.5; thus it is the solution to the eigenfrequency A.10 with Equation

A.6 (note that the eigenfrequencies are taken to be only real)..

(A.9)

( ) √( ) (A.6)

Appendix B: Rabi Model Derivation

A semi-classical approach using a two-level atom system with the Rabi model to study quantum resonance behavior in the system is shown in this section. Assume a two quantum energy level/state with energies ħω1 and ħω2. The system is perturbed by an external time harmonic field, electromagnetic dipole, which causes coupling between the state and thus Rabi splitting/state hybridization. Figure B.1 is a visualization using an energy band diagram of a coupled two level quantum system.

Energy (eV)

2ħΩr |±> E |2>

field |1>

| ± <

ħω1 ħω2

|g>

Figure B.1: Visualization for a two-level coupled quantum system under perturbation from an external electrical field. To solve for the eigenfrequencies of the system, a corresponding Hamiltonian that describes the system is developed.

| | | | (B.1) | | | | where ħωn are shown in Figure B.1 as state energy and ħV is the couple energy between energy states 1 and 2. It is shown later in this section that there is a relationship between ħV and 2ħΩr. The resulting Hamiltonian is the result of the sum of H0 and Hp and in a matrix is written as follows:

[ ] (B.2)

The eigenvalues of H are the desired eigenfrequencies of the system.. B.2 can then be solved for eigenfrequencies with the standard approach for solving eigenvalues, i.e, det (H – ħωI) = 0.

( [ ] [ ]) (B.3)

The solution to Equation B.3 is simple to obtain:

(B.4)

√( ) (B.5)

The difference between the two hybrid states at the minimum of the difference, i.e., where the anti-crossing curves are the closest together, is 2Ωr. Therefore, using

Equation B.5, we obtain

√( ) (B.6)

In the case where the state energies are very close together as in the case of this thesis then Equation B.6 simplifies.

(B.7)

Thus has been shown that the Rabi energy, ħΩr, is the coupling energy of the system.