Engineering Optical Properties UsingLayered MetamaterialsbyMohammed Hashim Al ShakhsB.Sc. Hons., King Fahd University of Petroleum and Minerals, 2004M.Sc., Dalhousie University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)August 2017© Mohammed Hashim Al Shakhs, 2017The undersigned certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, a thesis entitled: Engineer-ing Optical Properties Using Layered Metamaterials submittedby Mohammed Hashim Al Shakhs in partial fulfilment of the require-ments of the degree of Doctor of PhilosophyDr. Kenneth Chau, School of EngineeringSupervisor, ProfessorDr. Lo¨ıc Markley, School of EngineeringSupervisory Committee Member, ProfessorDr. Andre´ Phillion, School of EngineeringSupervisory Committee Member, ProfessorDr. Murray Neuman, School of Arts and SciencesUniversity Examiner, Professor (please print name and faculty/school above the line)Dr. Lora Ramunno, Department of Physics, University of OttawaExternal Examiner, Professor (please print name and faculty/school above the line)(Date Submitted to Grad Studies)iiAbstractThis thesis explores the concept of metamaterials; a fairly recent conceptin the literature which has attracted the attention of researchers due to theirabnormal electromagnetic properties. We will particularly consider one di-mensional version of a metamaterial made of layers. At the first glance,layered metamaterials are simply multi-layer thin films. The distinguish-ing feature of layered metamaterials is that they usually incorporate metalswhereas most thin film structures in the past have only incorporated di-electrics. The immense interest in certain layered configurations of metalsand dielectrics, particularly when the thicknesses are really thin comparedto the wavelength, is due to their exhibition of seemingly counter-intuitive orimpossible properties such as refraction to the same side of normal (negativerefraction), evanescent wave amplification, or light focusing with a flat in-terface (flat lensing). The simple configuration of layered metamaterials andtheir interesting properties are the prime motivations of this work. In thisthesis, we first start with a very generic electromagnetic description of theoptical properties of layered structures. This general description appears tobe novel due to presenting theory in new form. We use this understanding toexplain how and why certain layered structures can exhibit negative refrac-tion or flat lensing. This investigation has also led to several new predictionsof new optical properties of layered metamaterial structures. We concludethis work by various experimental studies which validate the predictions ofthe work and also explore fabrication challenges in the making of layeredmetamaterials.iiPrefaceThis work has been done under the guidance of Dr. Kenneth Chau atthe School of Engineering in The University of British Columbia. Portionsof my thesis have been published in four journal articles:− Al Shakhs, Mohammed, Lucian Augusto, Lo¨ıc Markley, and KennethJ. Chau. ”Boosting the Transparency of Thin Layers by Coatings ofOpposing Susceptibility: How Metals Help See Through Dielectrics.”Scientific reports, vol. 6 (2016).− Ott, Peter, Mohammed H. Al Shakhs, Henri J. Lezec, and Kenneth J.Chau. ”Flat lens criterion by small-angle phase.” Optics express, vol.22, pp. 29340-29355 (2014).− Al Shakhs, Mohammed H., Peter Ott, and Kenneth J. Chau. ”Banddiagrams of layered plasmonic metamaterials.” Journal of AppliedPhysics, vol. 116, pp. 173101 (2014).− Kenneth J. Chau, Mohammed H. Al Shakhs, and Peter Ott, Fourier-Domain Electromagnetic Wave Theory for Layered Metamaterials ofFinite Extent, Progress In Electromagnetics Research M, vol. 40, pp.45-56 (2014).Portions of my thesis have also been presented at the following confer-ences:− Mohammed Al Shakhs, Lucian Augusto, Lo¨ıc Markley, and KennethChau, Light Transmission Enhancement through a Thin Layer by aCoating of Opposing Susceptibility: How Metals Help See ThroughDielectrics?, APS/IEEE International Symposium, Fajardo, PuertoRico (2016).− Mohammed H. Al Shakhs, Peter Ott, Henri J. Lezec, and Kenneth J.Chau, A General Flat-Lens Criterion, APS/IEEE International Sym-posium, Vancouver, Canada (2015).iiiPreface− Mohammed H. Al Shakhs, Peter Ott, Henri J. Lezec, and KennethJ. Chau, A General Flat-Lens Criterion, Photonics North, Ottawa,Canada (2015).ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Electromagnetic Wave Theory . . . . . . . . . . . . . . . . . . 11.2 Electromagnetic Wave Propagation . . . . . . . . . . . . . . . 31.3 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Layered Metamaterials . . . . . . . . . . . . . . . . . . . . . . 121.4.1 Flat Lens Imaging . . . . . . . . . . . . . . . . . . . . 121.4.2 Spectral Light Transmission . . . . . . . . . . . . . . . 141.4.3 Surface Plasmon Resonance Sensing . . . . . . . . . . 151.5 Homogenization Theory . . . . . . . . . . . . . . . . . . . . . 161.5.1 Floquet-Bloch Theory . . . . . . . . . . . . . . . . . . 171.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 19Chapter 2: Electromagnetic Fields in Layered Metamaterialsof Finite Extent . . . . . . . . . . . . . . . . . . . . . 202.1 Light Scattering at a Single Plane Boundary . . . . . . . . . . 212.2 Transfer-Matrix Representation of the Electromagnetic Fieldin a Layered Medium . . . . . . . . . . . . . . . . . . . . . . . 24vTABLE OF CONTENTS2.3 Isolating Floquet-Bloch Modes by Fourier Transformation . . 272.4 Analysis of Electromagnetic Fields in Layered Metamaterials 322.5 Elecromagnetic Field Band Diagrams in Layered Metamaterials 352.6 Band Diagram Analysis of Layered Metamaterials . . . . . . 392.6.1 SPP Mode as a Window into the Negative-Index World 402.6.2 Super-Resolving Silver Slab Lens and the Veselago Lens 432.6.3 Canalization in Multi-Layered Structures . . . . . . . 462.6.4 Far-Field Flat Lens Imaging . . . . . . . . . . . . . . . 482.6.5 Far-Field Flat Lens with Less Metal . . . . . . . . . . 492.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Chapter 3: Flat Lens Criterion by Small-angle Phase . . . . . 523.1 Flat Lens Criterion . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Comparison with Past Flat Lens Results . . . . . . . . . . . . 553.3.1 Pendry’s Silver Slab Lens . . . . . . . . . . . . . . . . 553.3.2 Near-field Imaging with Silver Layers . . . . . . . . . 573.3.3 Anisotropic Metamaterial Lenses . . . . . . . . . . . . 593.3.4 Negative-Index Metamaterial Lens . . . . . . . . . . . 603.4 Validation of Flat Lens Criterion by Full-Wave Simulations . 603.5 Flat Lens Condition for a Single Layer . . . . . . . . . . . . . 623.5.1 Flat Lens for TM Polarization . . . . . . . . . . . . . 633.5.2 Flat Lens for TE Polarization . . . . . . . . . . . . . . 653.6 Flat Lens Condition for Multi-layers . . . . . . . . . . . . . . 663.7 Broadband Flat Lens Designed by Small-angle Phase . . . . . 673.8 Far-field Immersion Flat Lens . . . . . . . . . . . . . . . . . . 683.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Chapter 4: Transparency of Thin Layers: Light TransparencyBoost by Opposite Susceptibility Coating . . . . . 714.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 764.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Chapter 5: Surface Plasmon Resonance: Copper as Good asGold . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 885.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93viTABLE OF CONTENTSChapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Appendices A:Surface Plasmon Polaritons . . . . . . . . . . . . . . 116Appendices B: Evanescent Wave Amplification with a Thin SilverLayer . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Appendices C:S-parameter Method . . . . . . . . . . . . . . . . . . 120Appendices D:Maxwell-Garnett Method . . . . . . . . . . . . . . . 123Appendices E: Finite Difference Frequency Domain MATLAB Code 125Appendices F: Quality Factor Tables . . . . . . . . . . . . . . . . . 135Appendices G:Experimental Tools . . . . . . . . . . . . . . . . . . 136viiList of TablesTable 1.1 Plasma frequency ωp and damping constant (γ) of se-lected noble metals [1] . . . . . . . . . . . . . . . . . . 6Table F.1 The dip properties of the reflected light intensitiesfrom 50 nm of silver sputtered using different depo-sition parameters at two different wavelengths . . . . . 135Table F.2 The dip properties of the reflected light intensitiesfrom 50 nm of gold sputtered using different depositionparameters at two different wavelengths . . . . . . . . 135Table F.3 The dip properties of the reflected light intensitiesfrom 40 nm of copper sputtered using different depo-sition parameters at two different wavelengths . . . . . 135Table G.1 The measured average surface roughness of silver, goldand copper films fabricated at different deposition andslew rates. . . . . . . . . . . . . . . . . . . . . . . . . . 137viiiList of FiguresFigure 1.1 Real and imaginary components of the permittivityvalues of silver predicted by the Drude model (blue)and obtained from measurements (red). This Figureis published by permission from [2]. . . . . . . . . . . 7Figure 1.2 Parameter space of and µ. . . . . . . . . . . . . . . 8Figure 1.3 Schematic of the charge density oscillations and asso-ciated electromagnetic fields in the SPR phenomenonat the interface of a metal and dielectric. . . . . . . . 9Figure 1.4 The schematic of negative refraction. . . . . . . . . . 9Figure 1.5 Different metamaterial structures. a) double-fishnetnegative-index metamaterial with several layers, b)stereo or chiral metamaterial fabricated through stackedelectron-beam lithography. c) chiral metamaterialmade using direct-laser writing and electroplating, d)hyperbolic (or indefinite) metamaterial, e) metaldi-electric layered metamaterial composed of coupledplasmonic waveguides, enabling angle-independent neg-ative n for particular frequencies, f) split ring res-onators oriented in all three dimensions, g) wide-anglevisible negative-index metamaterial based on a coax-ial design, h) connected cubic-symmetry negative-indexmetamaterial structure, i) metal cluster-of-clusters visible-frequency magnetic metamaterial, j) all-dielectric negative-index metamaterial composed of two sets of high-refractive-index dielectric spheres arranged on a simple-cubic lattice. This Figure is published by permissionfrom [3] . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 1.6 The schematic depiction of metamaterials with het-erogeneity along a) one, b) two, and c) three directions. 12ixLIST OF FIGURESFigure 1.7 Optical ray visualization of imaging in (a) a standardplano-convex lens, (b) a planar negative-index slab,and (c) a planar anisotropic slab where the perpindic-ular permittivity value is negative. The red lines andblue arrows respectively indicate the local power andphase flow. (d) Imaging in a thin silver layer byevanescent wave amplification. [4] . . . . . . . . . . . 14Figure 2.1 Light scattering of a TM wave at a plane boundary. . 22Figure 2.2 Geometry under consideration consisting of a one-dimensional periodic layered medium bounded by twosemi-infinite half spaces and composed of M repeatedunit cells, each consisting of J layers. The mediumis excited from one half-space by an incident planeinclined at an arbitrary angle θ in the xz plane. [5] . 25Figure 2.3 Decomposition of the wave solution in a metal-dielectricbi-layer system consisting of alternating layers of 30-nm-thick Ag and 30- nm-thick TiO2, assuming a nor-mally incident TM-polarized wave with a free-spacewavelength of λ0 = 365 nm. . . . . . . . . . . . . . . . 33Figure 2.4 Forward and backward components of the weightingmatrix versus the number of repetitions. Here, wehave assumed a metal-dielectric bi-layer system witha unit-cell made from a 30- nm-thick Ag layer and 30-nm-thick TiO2 layer, assuming a normally incidentTM-polarized wave with a free-space wavelength ofλ0 = 365 nm. . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.5 Light refraction at an interface described by EPCs.EPCs for finite, lossy slabs of thickness 2λ0 havingeither a predfined (a) positive refractive index or (b)negative refractive index. . . . . . . . . . . . . . . . . 38xLIST OF FIGURESFigure 2.6 Band diagrams reveal interesting propagation char-acteristics of the SPP mode. We first examine EPCsfor a single 40-nm-thick silver layer illuminated bya plane wave (λ0 = 400 nm) incident from a dielec-tric (n = 2) prism for either (a) TM or (b) TE po-larization. (c) depicts the electric (blue arrows) andmagnetic (contour lines) fields in the 40-nm-thick sil-ver layer at the surface plasmon resonance. UnderTM-polarized illumination at the same wavelength,we next examine EPCs for a 5-layer stack of silverand silicon nitride (n = 2) layers, where the silverlayer thickness is fixed at 40 nm and the thickness ofthe silicon nitride is either (d) 10 nm or (e) 40 nm. (f)depicts the electric (blue arrows) and magnetic (con-tour lines) fields in the multi-layer structure at thesurface plasmon resonance for the case of dielectricthickness 10 nm. . . . . . . . . . . . . . . . . . . . . . 42Figure 2.7 Band diagram analysis of Pendry’s super-resolvinglayer. EPCs for (a) Pendry’s 40-nm-thick silver slablens for TM-polarized illumination at the wavelengthof λ0 = 357 nm. The inset describes the geometricalconfiguration. The red arrows trace out the time-averaged Poynting vector along a frequency contour.Simulations of a test object imaged by Pendry’s sil-ver slab lens when the two point-like features of theobject are spaced by (b) λ0/2.5, (c) λ0/3.0 and (d)λ0/3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.8 Band diagram analysis of Veselago super-resolvinglayer. EPCs for (a) a Veselago lens ( = µ = −1) ofequivalent thickness for TM-polarized illumination atthe wavelength of λ0 = 357 nm. The inset describesthe geometrical configuration. The red arrows traceout the time-averaged Poynting vector along a fre-quency contour. Simulations of a test object imagedby the Veselago lens when the two point-like featuresof the object are spaced by (b) λ0/4, (c) λ0/5, and(d) λ0/6. . . . . . . . . . . . . . . . . . . . . . . . . . 45xiLIST OF FIGURESFigure 2.9 Band diagram for the metal-dielectric multi-layer sys-tems studied in Ref. [6] for the case of TM-polarizedplane-wave (λ0 = 600 nm) illumination from a high-index dielectric (n = 5) prism. The blue line depictsthe simplified EPC predicted from effective mediumtheory (namely, Maxwell-Garnett theory). FDFD sim-ulations of the multi-layers imaging a test object whenthe two point-like features of the object are spaced by(b) λ0/8, (c) λ0/9, and (d) λ0/10. . . . . . . . . . . . 47Figure 2.10 (a) EPC for the flat lens structure shown in Ref. [7] tobe capable of far-field imaging in the UV. The EPC isderived for TM-polarized plane-wave (λ0 = 365 nm)illumination from a dielectric (n = 5) prism. Theinset depicts the geometrical configuration of the flatlens. (b) shows the simplified EPCs derived using twocommon homogenization techniques: S-parameter method(blue solid) and Floquet-Bloch modes (red dashed). [8] 49Figure 2.11 (a) EPC of a new layered flat lens structure that iscapable of far-field imaging, yet possesses about halfthe metallic fill fraction of the flat lens presented inRef. [7]. The proposed structure consists of 8 repe-titions of a unit cell with the layer sequence Ag (7.5nm) - TiO2 (25 nm) - Ag (7.5 nm). The EPC is de-rived assuming TM-polarized plane-wave (λ0 = 330nm) illumination from a dielectric (n = 2) prism. (b)shows an FDFD simulation of the proposed flat lensimaging a test object placed on its surface. Taperingof the magnetic energy density in the image region ata location spaced about a wavelength from the exitsurface confirms that the structure is capable of form-ing real images in the far field. [8] . . . . . . . . . . . 50Figure 3.1 Curvature of the wavefronts exiting a planar slab forthe cases of (a) virtual and (b) real image formation.By plotting the phase Φz(z = 0) versus qt = 1−cos θt,the possibility of a flat lens can be determined byinspection from a positive slope. [4] . . . . . . . . . . 54xiiLIST OF FIGURESFigure 3.2 (a) PSF phase for Pendry’s silver slab lens consist-ing of a 40-nm-thick Ag layer at a wavelength ofλ0 = 356.3 nm where the object is 20 nm away fromthe entrance of the slab. The phase has been calcu-lated at the exit of the slab (z = 0 nm) and the parax-ial image location has been predicted at z = 36 nm.(b) FDFD-simulated profile of the magnetic energydensity at the image plane z = 36 nm for the caseswhere the near diffraction limit spaced objects areimaged without (red) and with (blue) the silver slablens. Simulated time-averaged magnetic energy den-sity distributions of the illuminated object are shown(c) without and (d) with the 40-nm-thick Ag layer.The yellow dashed lines in (d) show the positionswhere the PSF phase profiles have been calculatedin (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.3 Flat lens criterion applied to past implementations.(a) PSF phase at the exit surface of lenses based onthe 36-nm-thick silver layer studied in Fang et al.[9], the 50-nm-thick silver layer studied in Melville etal. [10], and the 120-nm-thick silver layer studied byMelville et al. in [11, 12], along with the control usedin [12] of a 120-nm-thick PMMA layer. (b) PSF phaseat the exit surface of lenses based on metal-dielectricmulti-layers studied by Belov et al. [6]. The inset in(b) shows a magnified view of the data near normalincidence. (c) Paraxial image location as a functionof unit cell repetition for the periodic metal-dielectriclayered system studied by Kotynski et al. [13]. (d)PSF phase at the exit surface of the geometry studiedby Xu et al. [7] where the flat lens composed of metaland dielectric layers. [4] . . . . . . . . . . . . . . . . . 58xiiiLIST OF FIGURESFigure 3.4 Comparison of paraxial image locations predicted byPSF phase and numerical simulations. FDFD-calculatedtime-averaged energy density distributions for flat lensesconsisting of (a) a 120-nm-thick silver layer studiedin [11, 12], (b) a 50-nm-thick silver layer studied in [10],(c) metal-dielectric multi-layers studied in [6], and(d) metal-dielectric multi-layers studied in [7]. Inall cases, we use near diffraction limit spaced objectsconsisting of two, λ0/10-wide openings spaced λ0/2.5apart in an opaque mask that is illuminated by a TM-polarized plane wave. The yellow dashed line in eachpanel shows the corresponding paraxial image loca-tion calculated from the slope of the output phase. [4] 61Figure 3.5 Paraxial image location versus thickness for an illu-minated object located at the entrance of the idealVeselago lens (red) and Pendry’s silver slab lens (blue)when illuminated at the wavelength of λ0 = 356.3 nm.For the ideal Veselago lens, the image location isequivalent to the slab thickness, s = d. [4] . . . . . . . 64Figure 3.6 Flat lens for TE polarization based on a 50- nm-thicklossless dielectric (n = 4) layer immersed in air andilluminated at a wavelength λ0 = 365 nm. (a) PSFphase at the paraxial image location s = 2 nm. (b)FDFD-simulated profile of the electric energy densityat the paraxial image location for the cases wherethe object is imaged without (blue) and with (red)the dielectric slab. Simulated time-averaged electricenergy density distributions of the illuminated objectare shown (c) without and (d) with the 50-nm-thickdielectric layer. The yellow dashed line in each panelshows the paraxial image location calculated by thePSF phase. [4] . . . . . . . . . . . . . . . . . . . . . . 66xivLIST OF FIGURESFigure 3.7 Engineering a broadband flat lens. (a) Paraxial imagelocation over the ultraviolet-blue spectrum for a bi-layer flat lens consisting of a 28- nm-thick silver layerand a 29- nm-thick gold layer immersed in air. (b)PSF phase (red) and amplitude (blue) at the imageplane location (s = 37 nm) of the bi-layer flat lensat the wavelengths of λ0 = 365 nm (solid lines) andλ0 = 455 nm (dashed lines). Time-averaged energydensity distributions for the bi-layer system underplane-wave illumination at (c) λ0 = 365 nm and (d)λ0 = 455 nm. The yellow dashed line in each panelshows the paraxial image location calculated by PSFphase. [4] . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 3.8 Enhancing the image plane location of the multi-layeredflat lens system previously studied in [7] by immer-sion of the image region in a dielectric. (a) PSF phase(red) and amplitude (blue) at the paraxial image lo-cation for the cases where the dielectric medium hasrefractive index n = 1.0, 1.3, 1.5, and 2.0. (b) Parax-ial image location versus the refractive index of thedielectric medium predicted by PSF phase (blue line)and FDFD simulations (red circles). (c), (d), and(e) show FDFD-calculated magnetic energy densitydistributions of the immersed flat lens system for n= 1.3, 1.5, and 2.0, respectively. The yellow dashedlines in panels (c)-(e) show the paraxial image loca-tion calculated by PSF phase. [4] . . . . . . . . . . . . 69Figure 4.1 (a) Ideal configuration of a bi-layer immersed in twohalf-spaces, illuminated at normal incidence from theleft half-space. (b) Predicted normal-incidence trans-mittance at a wavelength of 650 nm through a bilayercomposed of a 50-nm-thick base layer of silicon nitrideand a coating layer of silver of variable thickness. Apositive derivative of the transmittance in the limitof zero coating layer thickness can be used as an in-dicator of transmission enhancement. [14] . . . . . . . 73xvLIST OF FIGURESFigure 4.2 Changing the optical properties of semi-transparentsilver by sputtered TiO2 coatings. (a) Experimentaland (b) calculated normal-incidence transmittance spec-tra for 23-nm-thick silver layers that are either un-coated or coated with a TiO2 layer ranging in thick-ness from 18 nm to 57 nm. The experimental spectraare obtained from the average of 5 independent mea-surements, where each measurement is made from anaverage of 40 traces. Photographs of the samplesplaced on the printed UBC logo are shown at thetop of panel (a) to highlight the visible appearancechanges caused by the thin TiO2 layer. The leftmostphotograph is of uncoated silver and the adjacent im-ages are of coated silver (in order of increasing coatinglayer thickness from left to right). [14] . . . . . . . . . 77Figure 4.3 Changing the optical properties of semi-transparentsilver by various sputtered elemental semiconductorcoatings. Experimental normal-incidence transmit-tance spectra for (a) 23-nm-thick silver coated withsputtered silicon, (b) 23-nm-thick silver coated withsputtered p-type silicon, and (c) 18-nmthick silvercoated with sputtered germanium. The experimentalspectra are obtained from the average of 5 indepen-dent measurements, where each measurement is madefrom an average of 40 traces. Photographs of the sam-ples placed on the printed UBC logo are shown at thetop of each corresponding panel to highlight the visi-ble appearance changes caused by the thin layers. [14] 78xviLIST OF FIGURESFigure 4.4 Experimental measurements of the transmission en-hancement of a 50-nm-thick silicon nitride membraneconferred by coating the membrane with 10-nm-thicksilver layers in three different configurations: single-sided coating with silver, single-sided coating with sil-ver followed by a 10-nm-thick TiO2 passivating layer,and double-sided coating with passivated silver. (a)shows tabulated the average normalized transmittancevalues for the bare membrane and the three silver-coated membranes at the wavelengths of 400 nm, 420nm, 440 nm, and 460 nm. Cells in the table corre-sponding to transmission enhancement (beyond ex-periment error) are shaded green. (b) shows the av-erage transmittance spectra for the bare membranes(red dashed), the membrane that is coated on a sin-gle side by silver (green line), the membrane that iscoated on a single side by passivated silver (orangeline) and the membrane that is coated on both sidesby passivated silver (blue line). The error has a mag-nitude comparable to the line widths and has not beenexplicitly plotted for clarity of presentation. [14] . . . 79Figure 4.5 Experimental transmittance change over the entirevisible spectrum for a 50-nm-thick silicon nitride mem-brane coated with a 10-nm-thick silver layer that ispassivated by a 10-nm-thick TiO2 layer (red line witherror bars). Also shown are calculations of the trans-mittance change for the three-layer system assum-ing various silver layer thicknesses. The error barsin the experimental measurement represent one stan-dard deviation. [14] . . . . . . . . . . . . . . . . . . . 81xviiLIST OF FIGURESFigure 4.6 Microscope images of (left column) an uncoated 50-nm-thick Si3N4 membrane and (right column) an iden-tical membrane coated with 10-nm-thick Ag and 10-nm-thick TiO2 under laser illumination at wavelengthsof (a) 365 nm, (b) 470 nm, and (c) 590 nm. The im-ages were collected using a monochrome camera andhave been false-colored to reflect the color of laserillumination. The percentages on the images in theright column indicate the percent change in the aver-age image brightness relative to the adjacent imagesin the left column. [14] . . . . . . . . . . . . . . . . . 83Figure 5.1 Schematic of light coupling to surface plasmon wavesusing Kretschmann configuration. . . . . . . . . . . . 86Figure 5.2 Schematic of the sputtering process in the vacuumdeposition chamber. . . . . . . . . . . . . . . . . . . . 87Figure 5.3 Show the SPR measurements of the 50-nm-thick sub-strates deposited at low and high, deposition/slewrates, for a) silver, and b) gold, using a coherent redHe-Ne laser with free-space wavelength λo = 632.8 nm. 89Figure 5.4 Show how the SPR dips of the 40-nm-thick copper(blue lines) become sharper and deeper comparableto the 50-nm-thick gold SPR dips (red lines), whenthe deposition rate of copper is increased from low de-position rate (1.0 A˚/s, blue) to high deposition rate(10.0 A˚/s, green). SPR measurements using a coher-ent yellow λo = 594.0 nm a) b), and red λo = 632.8nm c) d) He-Ne laser. . . . . . . . . . . . . . . . . . . 90Figure 5.5 The quality factor bar charts with error bars for silver(blue), copper (green), and gold (yellow) at differentdeposition/slew rates using yellow a) b), and red c)d) lasers. The deposition rates I, II, III, and IV re-spectively correspond to 1.0 A˚/s, 7.0 A˚/s, 14.0 A˚/s,and 20.0 A˚/s for silver, and 1.0 A˚/s, 3.0 A˚/s, 7.0 A˚/s,and 10.0 A˚/s for copper and gold. . . . . . . . . . . . 91Figure 5.6 Photograph images at low and high deposition ratesfor the 50-nm-thick silver a) b), the 50-nm-thick goldc) d), and the 40-nm-thick copper e) f) thin filmsdeposited on glass substrates. . . . . . . . . . . . . . . 92xviiiLIST OF FIGURESFigure 5.7 Scanning electron microscope (SEM) images at lowand high deposition rates for the 50-nm-thick silvera) b), the 50-nm-thick gold c) d), and the 40-nm-thickcopper e) f) thin films deposited on glass substrates. . 92Figure 5.8 The atomic force microscopy (AFM) images at lowand high deposition rates for the 50-nm-thick silvera) b), the 50-nm-thick gold c) d), and the 40-nm-thickcopper e) f) thin films deposited on glass substrates. . 93xixAcknowledgementsIt was certainly the blessing of God to be surrounded by many wonderfulpeople throughout my Doctoral studies. I would like to take this opportunityto thank them all for their support. First of all, I would like to express mysincere gratitude to my PhD advisor, Dr. Kenneth Chau, for his unlimitedsupport, guidance and encouragement throughout my PhD studies. Withouthis continuous support and confidence in my abilities conducting this workwould not be possible. Working with him was a great opportunity. Hiswords and attitude will be forever a source of inspiration.I would like to sincerely thank Dr. Peter Ott from Heilbronn Universityand Dr. Lo¨ıc Markley from the University of British Columbia (UBC) fortheir significant contributions in this work. My sincere thanks also goes toDr. Andre Phillion, the PhD committee member, for his valuable commentsand suggestions that enriched my work. I am thankful to Dr. ThomasJohnson, Dr. Stephen OLeary and Cassidy Northway from UBC for theirhelpful discussions. I am grateful to Dr. Amit Agrawal from SyracuseUniversity, and Dr. Henri Lezec, Dr. Vladimir Aksyuk and Dr. Alex Liddlefrom the National Institute of Standards and Technology for their helpfulcomments.I would like to thank Dr. Deborah Roberts from UBC for giving meaccess to her microscope facility. I am grateful to Mark Nadeau and DavidZinz for assisting me working on the nano-film deposition station at UBC.I am thankful to David Arkinstall for assisting me working on the ScanningElectron Microscope at UBC. I would like to thank Mario Beaudoin forhis assistance with measurements made at the AMPEL Facility at UBC.I am also thankful to YaTung Cherng and Nathanael Sieb for taking theAtomic Force Microscopy images using the 4D LABS facilities at SimonFraser University (SFU).I would like to thank all my exceptional friends and research colleagues.Working beside Samuel Schaefer, Max Bethune-Waddell, Iman Aghanejad,Asif Alnoor, Masoud Ahmadi, and Vincent Loi was a great experience. I amespecially thankful to Reyad Mehfuz, Lucian Augusto, and Cailan Libby forall the wonderful moments throughout our collaborative works.xxAcknowledgementsFinally, I would like to thank each individual of my big family. Myspecial thanks and appreciation to my parents, in-laws, sister, wife and kidsfor their sincere love and continuous encouragement. Words cannot expressmy appreciation...xxiDedicationTo the hidden sun behind the cloudsxxiiChapter 1IntroductionControlling light requires rigorous understanding of the behaviour ofelectromagnetic waves in different media. In this thesis, we will study theinteraction between electromagnetic waves and layered metamaterials, a sim-ple thin film structure that uses thin metal layers to achieve abnormal lightbehaviour. The thickness of these layers is typically one tenth or less ofwavelength of light. In the first part of the thesis, we focus our attention tothe negative refraction property of such layered systems at optical frequen-cies. We start by examining the fundamental principles of electromagneticsto explain why and when a layered metamaterial can perform negative re-fraction. We develop explicit wave equations that describe electromagneticwave propagation through thin film structures. The derived explicit solu-tions provide clear physical insights and shed the light on the correlationbetween the intrinsic electromagnetic properties and the external light be-haviours. Based on this study, we develop two distinct methods to predictthe imaging capability of layered metamaterials. In the second part of thethesis, we conclude the research with two experimental works. The firstexplores spectral light filtering and light transmission enhancement usingbi-layer thin films of metal and dielectrics, and the second examines the in-fluence of nano-film fabrication parameters on the surface optical resonancesof thin metal films. The outcomes of this thesis work are explicit analyticalsolutions of the electromagnetic fields propagating through layered meta-materials, new design methods for layered metamaterials, and experimentalinvestigations of enhancing light transmission and surface optical resonanceof layered metamaterials.1.1 Electromagnetic Wave TheoryThe electromagnetic wave theory of light was established in 1873 byJames Clerk Maxwell. In three dimensional vector notation, Maxwell’s equa-tions relate six real functions dependent on the position vector (~r) and thetime variable (t); the electric field, ~E, the magnetic field, ~H, the electric dis-placement, ~D, the magnetic flux density, ~B, the electric current density, ~J ,11.1. Electromagnetic Wave Theoryand the electric charge density, ρ. Maxwell’s equations in their differentialforms are given by~∇ · ~D(~r, t) = ρ, (1.1)~∇ · ~B(~r, t) = 0, (1.2)~∇ × ~E(~r, t) = −∂~B(~r, t)∂t, (1.3)and~∇ × ~H(~r, t) = ~J(~r, t) + ∂~D(~r, t)∂t. (1.4)To solve for Maxwell’s equations, typically we assume to have a chargefree region where ρ is zero. However, still we have two independent equationsand five unknowns. To overcome this problem, we need more relations. Thetwo electric field vectors, ~E and ~D, and the two magnetic field vectors, ~H and~B, can be often related to each other based on the electric and the magneticproperties, and µ, of the medium. Moreover, the electric field ( ~E) andthe electric current density ( ~J) can be related to each other based on theconductivity (σ) of the medium. These relations, known as the constitutiverelations, provide the required missing equations to solve for the electric andmagnetic fields in Maxwell’s equations.In linear, isotropic, homogeneous and dispersionless media, the consti-tutive relations are:~D(~r, t) = ro ~E(~r, t), (1.5)~B(~r, t) = µrµo ~H(~r, t), (1.6)and~J(~r, t) = σ ~E(~r, t), (1.7)where and σ are the constant electric properties of the medium, and µ rep-resents the medium’s constant magnetic property. These material propertieswill be discussed in details over the next section. Using the constitutive re-lation equations and assuming the region of interest is a charge free region(ρ = 0), Maxwell’s equations can be re-written as:~∇ · ~E(~r, t) = 0, (1.8)21.2. Electromagnetic Wave Propagation~∇ · ~H(~r, t) = 0, (1.9)~∇ × ~E(~r, t) = −µrµo∂~H(~r, t)∂t, (1.10)and~∇ × ~H(~r, t) = σ ~E(~r, t) + ro∂~E(~r, t)∂t. (1.11)The wave nature of light is not explicitly shown in Maxwell’s equations.After few lines of manipulation, it can be shown that the electric field obeysthe wave equation~∇2 ~E(~r, t)− µrrc2∂2 ~E(~r, t)∂t2= σµ0µr∂ ~E(~r, t)∂t, (1.12)where c = 1√µoo is the free-space speed of light. For complex electric fieldsunder the plane wave assumption, the general solution to the electric fieldis~E(~r, t) = E0 ei(~k.~r−ωt), (1.13)where ~k is the complex wave vector.The magnetic field wave equation can be derived using a similar approachwhere~∇2 ~H(~r, t)− µrrc2∂2 ~H(~r, t)∂t2= σµ0µr∂ ~H(~r, t)∂t. (1.14)1.2 Electromagnetic Wave PropagationThe propagation of electromagnetic waves through matter is governed bythree parameters: the conductivity, the permittivity, and the permeability.Conductivity (σ) is the measure of material’s ability to conduct electriccurrents and it can be approximated as a function of the current densityand the applied electric field ( ~J/ ~E). Materials are classified based on theirconductivity as metals, semiconductors or dielectrics. The conductivity ofa material usually depends on temperature and frequency.Permittivity () is the measure of material’s ability to resist an electricfield. It describes how much electric flux, electric field flow through a givenarea, is generated per unit charge of a medium. The absolute permittivityof a medium is defined as31.2. Electromagnetic Wave Propagation = 0r = 0(1 + χe), (1.15)where 0 is the permittivity in free-space, r is the relative permittivity ofthe medium, and χe is the electric susceptibility which indicates the degreeof polarization of the electric dipoles in response to an applied electric field.As opposed to the constant electric field response of free-space, materi-als are usually dispersive and their associated permittivities are dependanton the frequency of the applied electric field. The frequency dependency ofpermittivity indicates that the polarization response of dispersive materialsto an applied field is not instantaneous. To represent this response, permit-tivity is defined as a complex function of the angular frequency (ω) of theapplied field(ω) = ′(ω) + i′′(ω). (1.16)To simplify the atomic-scale description of permittivity in dispersive ma-terials, a classical kinetic model has shown to provide satisfactory results.This simple model is based on describing the electromagnetic properties ofa material by considering the motion of its constituent atoms, electrons andmolecules using a harmonic oscillator model. In dielectrics, the total forcesacting on electrons can be written asFtotal = Fbinding + Fdamping + Fdriving, (1.17)where Fbinding is the binding forces between electrons and molecules, Fdampingis the force associated with the radiation of oscillating charges, and Fdrivingis the force due to an applied electromagnetic wave.Eliminating the contributions of the binding forces on electrons yields avery similar model that can be used for metals. This model, known as Drudemodel, was originated in 1900 by Paul Drude [15, 16] and described theelectromagnetic properties of metals by considering a density of unboundedmotion-free electrons. Under the influence of a time-varying electric field,the equation of motion of free electrons can be written asqE¯0e−iωt = mγdx¯dt+md2x¯dt2, (1.18)where m and q are the effective mass and the charge of the electron, respec-tively, x¯ is the electron displacement, and γ is the damping constant. Theelectron displacement is a time harmonic vector (x¯ = x¯0e−iωt), and Eq. 1.1841.2. Electromagnetic Wave Propagationcan be re-written asqE¯0 = mγ(−iωx¯0) +m(−ω2x¯0). (1.19)Solving for x¯0, we getx¯0 =−qE¯0mω(ω + iγ). (1.20)Then, the electric dipole moment can be defined asp¯ = qx¯0 =−q2E¯0mω(ω + iγ). (1.21)The polarization density, which is the density of permanent or inducedelectric dipole moments, is defined in general asP¯ = Np¯ = lim∆v→0∑Nn=1Qndn∆v, (1.22)where d is the distance vector from negative to positive chargeQ of the dipoleand N is the total number of dipoles in a volume ∆v. Invoking the electricdipole moment definition (Eq. 1.21), the polarization can be re-written asP¯ =−Nq2E¯0mω(ω + iγ). (1.23)In linear, isotropic, and homogeneous media, the polarization is relatedto electric field byP¯ = 0(r − 1)E¯ = 0χeE¯. (1.24)Solving for r by substituting Eq. 1.23 in Eq. 1.24, we getr = 1−−ω2pω(ω + iγ), (1.25)where ωp is the plasma frequency and it is defined asωp =√−Nq2m0. (1.26)51.2. Electromagnetic Wave PropagationSolving for the real and imaginary components of permittivity, we get(ω) = 1− ω2p(ω2 + γ2)+ iω2pωγ(1 + ω2/γ2). (1.27)Assuming γ ' 0, the permittivity can be simplified to(ω) = 1− ω2pω2, (1.28)which will be negative when the frequency (ω) is less than the plasma fre-quency (ωp), like silver and gold at visible frequencies. The plasma frequency(ωp) and the damping constant (γ) of selected noble metals are shown in Ta-ble 1.1. A common feature of these metals is their relatively low loss, wheredamping constants are several orders of magnitude less than the plasmafrequency. Although Drude model showed to provide fairly accurate per-mittivity values for metals when compared to the experimentally-measuredvalues, its predictions are found to be limited especially for the imaginaryparts of permittivity. To demonstrate such limitations, the Drude modelof silver (based on the parameters in Table 1.1) is plotted against its ex-perimental data [17] (Fig. 1.1). These discrepancies in the Drude modelpredictions can be attributed to the interband transitions which are notconsidered in the Drude model.Table 1.1: Plasma frequency ωp and damping constant (γ) of selected noblemetals [1]Metal ωp(×1015Hz) γ (×1012s−1)Silver (Ag) 2.186 5.14Gold (Au) 2.15 17.14Copper (Cu) 2.12 23.09Permeability (µ) is the measure of material’s ability to support the mag-netic field formation within itself. In other words, it is the material’s degreeof magnetization in response to an applied magnetic field. The absolutepermeability of a medium is defined asµ = µ0 µr = µ0(1 + χm), (1.29)where µ0 is the permeability in free-space, µr is the relative permeability,and χm is the magnetic susceptibility which indicates whether a material isattracted into or repelled out of an external magnetic field.61.2. Electromagnetic Wave PropagationFigure 1.1: Real and imaginary components of the permittivity values ofsilver predicted by the Drude model (blue) and obtained from measurements(red). This Figure is published by permission from [2].The combination of relative permittivity and permeability yields therefractive indexn =√rµr, (1.30)which is a dimensionless number used in optics to describe how light propa-gates through different media. It determines how much light is reflected ortransmitted (Fresnel equations), and the angle of refraction (Snell’s law) atthe interface between two media of different refractive indices. It also relatesthe speed and the wavelength of light in vacuum (c, λ0) to the correspondingones in a different medium where v = c/n, and λ = λ0/n. For complex per-mittivity and permeability, the resultant refractive index will be a complexvalue where the sign of the refractive index can be determined according tothis equation [18]n = sgn(<[]|µ|+ <[µ]||)√µ. (1.31)Across the electromagnetic parameter space where the permittivity ()is the x-axis and the permeability (µ) is the y-axis (Fig. 1.2), each quad-rant represents a different set of materials based on their electromagneticresponses. Simultaneously positive and µ results in having positive refrac-tive index and consequently positive refraction. These materials are repre-sented by the first quadrant. At the plasma frequency which was discussedin the Drude model, the permittivity sign of metals switch from positive tonegative. Negative permittivity and positive permeability materials whichhave complex refractive index are illustrated by the second quadrant. The71.2. Electromagnetic Wave PropagationFigure 1.2: Parameter space of and µ.plasma frequency of most metals reside over the ultraviolet and the visiblepart of the electromagnetic spectrum. Metals with negative permittivitycan sustain an interesting phenomenon known as surface plasmon resonance(SPR). The surface plasmon resonance occurs due to the rapid oscillationof the electron densities at the surface of metals when they are exited byelectromagnetic waves at their resonance frequency (Fig. 1.3). The detailedtheoretical concept of SPR will be discussed in the Appendix. This phe-nomenon allows to store energy and also manipulate the stored energy inthe form of surface waves, which has been proven to be the key ingredientfor realizing unusual optical properties.Usually natural materials have neutral magnetic responses. However,there are few materials that exhibit magnetic responses when they are ex-posed to electromagnetic waves as in the third quadrant ( > 0, µ < 0). Thefourth quadrant represents the most interesting and challenging case where and µ are both negative resulting in a medium with negative refractiveindex. In normal circumstances, most materials refract light positively (tothe opposite side of normal) and support forward waves defined by righthand rule, where the phase velocity (directed along the wave vector) andthe group velocity (directed along the time averaged Poynting vector) point81.2. Electromagnetic Wave PropagationFigure 1.3: Schematic of the charge density oscillations and associatedelectromagnetic fields in the SPR phenomenon at the interface of a metaland dielectric.to the same direction. In the double negative media ( < 0, µ < 0), on theother hand, it has been shown that light can be refracted negatively (to thesame side of normal) and backward waves described by left hand rule (phasevelocity and group velocity pointing in opposite directions) can be sustained(Fig. 1.4). Although such media possess interesting unusual properties [19],they are not naturally available. Therefore, artificial structures known asmetamaterials have been fabricated to imitate abnormal properties like dou-ble negative property.Figure 1.4: The schematic of negative refraction.91.3. Metamaterials1.3 MetamaterialsMetamaterials are generally defined as materials with properties that arenot achievable with natural materials. In the field of electromagnetics, meta-materials are engineered structures with inhomogeneity scale smaller thanthe operational wavelength (sub-wavelength). They are designed to realizeelectromagnetic behaviours that are not possible using bulk natural mate-rials [20–23]. For instance, metamaterials can imitate the electromagneticproperties of the double negative media and they can be effectively char-acterized by negative refractive index. Some of the explored metamaterialstructures in the literature are shown in Figure 1.5.Figure 1.5: Different metamaterial structures. a) double-fishnet negative-index metamaterial with several layers, b) stereo or chiral metamaterialfabricated through stacked electron-beam lithography. c) chiral metama-terial made using direct-laser writing and electroplating, d) hyperbolic (orindefinite) metamaterial, e) metaldielectric layered metamaterial composedof coupled plasmonic waveguides, enabling angle-independent negative n forparticular frequencies, f) split ring resonators oriented in all three dimen-sions, g) wide-angle visible negative-index metamaterial based on a coaxialdesign, h) connected cubic-symmetry negative-index metamaterial struc-ture, i) metal cluster-of-clusters visible-frequency magnetic metamaterial,j) all-dielectric negative-index metamaterial composed of two sets of high-refractive-index dielectric spheres arranged on a simple-cubic lattice. ThisFigure is published by permission from [3]Metamaterials were first proposed by Bose in 1898 to achieve an artificial101.3. Metamaterialschiral effect [24]. However, the term metamaterial first appeared in 1999 byRodger M. Walser [25], a physics professor from the University of Texas.Few months later, the term metamaterial became more popular throughthe pioneering work of Smith et al. [26] on materials with simultaneouslynegative permeability and permittivity at microwave frequencies.There are three ground-breaking works that can be counted as mile-stones in the area of modern metamaterials. The first one is the Veselago’spaper which conceptually laid down the theory of left-handed metamateri-als, materials in which the wave vector ~k and the field vectors ~E, ~H form aleft-handed system [19]. The next outstanding work is the first experimen-tal realization of Veselago’s left-handed medium by Smith et al. [26]. Thethird exceptional work is Pendry’s work on perfect lensing, perfect imagingbeyond the conventional resolution limit, with a layered metamaterial com-posed of a single layer of silver [27]. This single layer structure representsthe only route to realize artificial magnetism and left-handed behavior atvisible and UV frequencies.Based on the targeted frequency range, the size and the achievable elec-tromagnetic properties of metamaterials can be vastly different. While thesize of metamaterial structures will be centimeter-scale for microwave fre-quencies, their size can drop down to nanometer-scale for optical frequencies.In this thesis, we examine nano-scale metamterials that work over the ul-traviolet and visible frequencies. Metals are the main building elements formaking such optical metamaterials. This is due to the interesting propertiesassociated with metals around the optical range such as negative permittiv-ity and surface plasmon resonance. It has been shown that these propertiescan contribute significantly towards realizing the unusual properties of op-tical metamaterials [27–29].The heterogeneity in metamaterials could be along one [7, 30, 31], two[32, 33], or three [3, 34, 35] directions (Fig. 1.6), and it could be periodic[7, 36, 37], or non-periodic [38–40]. Making optical metamaterials with nano-scale inhomogeneity requires sophisticated nano-fabrication methods. Cur-rently, nano-fabrication methods are mainly limited to one dimensional andtwo dimensional fabrication techniques. Three dimensional nano-structurefabrication with heterogeneity in three directions is possible by methodslike focused ion beam (FIB) and electron beam lithography (EBL). How-ever, such 3-D fabrication methods require very delicate experimental setupand alignment, which make them quite difficult and costly [20, 41]. In thisthesis, we restricted our work to the one dimensional nano-structure be-cause it can be easily modeled, and its fabrication requires tools which arewell-controllable and accessible to our research group.111.4. Layered MetamaterialsFigure 1.6: The schematic depiction of metamaterials with heterogeneityalong a) one, b) two, and c) three directions.1.4 Layered MetamaterialsMaterials with sub-wavelength one-dimensional heterogeneity can beachieved by fabricating a structure made of a multitude of thin films. Thereis a lot of overlap between layered optical metamaterial and classical thinfilm structures [42]. Both systems are composed of thin films of metal anddielectric. However, the design purposes and the complexity in terms ofthickness and number of layers are different. Layered optical metamateri-als are usually more complex and designed to exhibit abnormal properties.Over the last two decades, a vast variety of applications for layered meta-materials have been proposed and implemented[20, 41]. In this thesis, weparticularly consider flat lens imaging [27], spectral light transmission [43],and surface plasmon resonance sensing [44] applications. Here, we discussthe recent achievements in the designated research fields, and we will thor-oughly examine different methods to engineer and improve the performanceof these applications in the next chapters.1.4.1 Flat Lens ImagingLenses are optical transmissive devices that are capable of reassemblingelectromagnetic fields to a focus by correcting the phase of each Fourier com-ponent of the fields. This phase correction feature is due to the curvatureshapes of the entrance and exit faces of conventional glass lenses (Fig. 1.7a)). Flat lenses, on the other hand, are perfectly flat slabs with imagingcapabilities as a result of the optical properties of their material composi-tion. Flat lenses can be made of materials with gradual variation of therefractive index (graded-index lenses) [45], or artificially structured sheetof materials with subwavelength-scaled patterns in the horizontal dimen-121.4. Layered Metamaterialssions (electromagnetic metasurfaces) [46], or ultra-thin metasurfaces withbalanced loss and gain [47], or layered metamaterials [11, 13, 27, 48]. Themain distinguishing feature of layered metamaterial flat lenses is their per-fect homogeneity along the front and back surfaces. This implies that theimaging process is not associated with a principle optical axis which resultsin the abnormal possibility of imaging with an infinite aperture if we assumeto have infinitely long flat lens [49].In theory, imaging with a planar homogeneous slab (Fig. 1.7) is possi-ble if the flat slab is constructed from either isotropic and negative indexmedium, first proposed by Veselago [19], or anisotropic medium with a con-stitutive tensor having diagonal components of opposite sign [23, 48, 49]. Inthe absence of naturally-occurring negative-index or anisotropic materials, athin layer of silver (Fig. 1.7(d)) has been suggested by Pendry to work as flatlens based on the electrostatic limit approximation. Pendry’s flat lens canform a real image at a distance less than a wavelength from the lens (near-field) using a transverse-magnetic (TM) polarized wave [27]. What makesPendry’s flat lens very interesting is the capability of this lens to performsuper-resolution imaging. The imaging resolution of traditional glass lensesis limited by the applied wavelength since the produced images compriseonly the propagating waves. This image resolution limit is known as thediffraction limit. To break this barrier and achieve super-resolution, evanes-cent waves, which exponentially decay, should be restored alongside propa-gating waves at the image location. Unlike propagating waves restoration,which is possible by phase correction, evanescent waves restoration requiresamplitude amplification. Pendry showed that a solution to Maxwell’s equa-tions demonstrates that ideal negative index slabs (Veselago lens) as well asthin silver slabs can amplify evanescent waves by restoring wave amplitudesin the image region and enable super-resolution imaging. He interestinglyshowed that the condition required for super-resolution imaging ( < 0) isexactly the same condition needed for the existence of surface plasmon po-laritons. This implies that these two phenomena are inherently interrelated.The mathematical proof of evanescent wave amplification using a thin layerof silver is discussed in the Appendix.Since the first practical flat lens proposal (Pendry’s thin silver layer) [27],several research works have been published investigating this phenomenaboth numerically [6, 9] and experimentally [10, 50]. These works includeresearch on near-field [28] and far-field [7] flat lens imaging, where the far-field flat lens can project the image apart from the lens at a distance greaterthan the wavelength in the hosting medium. The introduced flat lenses arecomposed of a single metallic layer [9–12, 50–54] and metal-dielectric multi-131.4. Layered Metamaterialsq A AA Aa) Imaging with conventionalpositive-index convex lensb) Imaging with negative-indexVeselago flat lensn = - 1n > 1c) Imaging with positive-indexanisotropic flat lense|| e < 0AAn = 1 n = 1d) Imaging with a thinsilver layerA Ae = - 1Figure 1.7: Optical ray visualization of imaging in (a) a standard plano-convex lens, (b) a planar negative-index slab, and (c) a planar anisotropicslab where the perpindicular permittivity value is negative. The red linesand blue arrows respectively indicate the local power and phase flow. (d)Imaging in a thin silver layer by evanescent wave amplification. [4]layers [6, 7, 13, 28, 29, 55–63]. A variety of methods have been used todesign these systems. Some past flat lens designs are based on imitatingphysical processes, such as evanescent wave amplification [10–12, 50, 54],or electromagnetic properties, such as anisotropy [6, 28, 29, 56] or negativeindex [7]. Some others employ transfer function calculations [57, 58, 60, 61]or simulations [13, 59, 62]. In this thesis, we postulate a new criterion todesign flat lenses based on transfer function calculations and ray optics, andthen we investigate the proposed criterion numerically.1.4.2 Spectral Light TransmissionControlling and systematically manipulating light transmission throughtransparent, semi-transparent, and naturally opaque lossy materials was asubject of research since the early days of science. After the revolutionaryadvances in science and technology over the last century, the significanceof engineering light transmission became more prominent due to its highpotential in various energy and optical applications. Metals are the basicbuilding block in making layered metamaterials, flat lenses and many light-transmission-based implementations. Though metals have valuable proper-ties such as conductivity, they are naturally opaque when they are thickand semi-transparent when they are sufficiently thin. Therefore, researcherssince 1950s examined the possibility of boosting light transmission throughlow-loss metallic films and they reported that such a light transparencyenhancement can be possible by coating the metallic film with a thin, high-141.4. Layered Metamaterialsindex dielectric layer [64–68].The investigations of thin film coating led to the next research on energy-efficient heat-reflecting windows [69–73], cheap flexible transparent conduc-tors [74–77] which are useful in making photovoltaics and displays, andnanophotonic devices such as spectrally selective transmission filters [36, 78–80], and flat lenses [6, 7, 81–83]. The applicability of the dielectric-coatingmethod in flat lens fabrication steered our research towards several lighttransmission experiments that led to very interesting conclusions which willbe discussed in Chapter 4.1.4.3 Surface Plasmon Resonance SensingThe surface plasmon resonance phenomenon is sensitive to the refractiveindex of the medium attached to the metal surface, which makes it appealingfor a wide range of physio-chemical sensing applications [44, 84–89]. Thebest metals for SPR sensors are noble metals with low losses, such as silver,gold, and copper [90, 91]. Of these three, silver attains the least optical lossesand the best SPR coupling [92]. Gold has an acceptable optical resonanceand copper has the worst [93]. Though gold is the most expensive, goldSPR sensors are commercially preferred over silver ones due to the chemicalinertness of gold [94].To enhance SPR coupling, usually the thickness of metal or the thicknessof the coating dielectric layer is optimized [95]. In more sophisticated works,silver islands were deposited on copper films [96], silver thin films are de-posited on a glass substrate with some attached SiO2 droplet residuals [97],gold nano-particles were prepared on a glass slide [98], microhole arrays withvariable periodicity and diameter were fabricated [99], nano-textured metalwas incorporated into the cathode structure of solar cells [100], or a polymerchannel waveguide structure was coated by a passivated layer of copper [101]for the purpose of SPR enhancement.In a different approach, depositing at various deposition rates using e-beam evaporation technique, the measured surface roughness and dielectricvalues of very thin Au films show that dielectric values, which play the majorrole in the SPR wave confinement and longer range SPR wave propagation,are dependent on their deposition rates [102]. A recent work emphasizeson the importance of deposition at the lowest possible vacuum pressureand the fastest deposition rates to avoid metal-oxide contamination [103].Varying the deposition rate and the base vacuum pressure in the depositionchamber, they have been able to deposit high-quality plasmonic films ofaluminum, copper, gold, and silver using thermal evaporation. Investigating151.5. Homogenization Theorythe influence of different deposition parameters on the SPR couplings of thinmetal films, we will introduce a new deposition method using the sputterdeposition in Chapter 5.1.5 Homogenization TheoryThroughput the past years of metamaterial studies, two distinguishableresearch fields had the major impact on the advances in metamaterials. Thefirst research field provided the technological platform for fabrication of com-plex nanostructure systems and the second approach theoretically character-ized the consequences of having bulk materials with abnormal electromag-netic properties. Assigning effective material parameters to metamaterialscan bridge the gap between both fields where theory can be linked to the realworld implementations [104]. The process of defining the electromagneticproperties of inherently heterogeneous systems by invoking the properties ofideally homogeneous metamaterials, when the heterogeneity scales are onetenth or less than a wavelength of light, is known as homogenization.Homogenization of artificially fabricated heterogeneous structures wasa subject of research for several years and various approaches have beenproposed and investigated in the literature. Mainly working with periodicsystems, it has been shown that the elctromagnetic properties of the build-ing blocks of metamaterials, the sub-wavelength periodic structures, can behomogenized to express the electromagnetic responses of the entire structurein terms of effective parameters [105].The Nicholson-Ross-Weir (NRW) retrieval method or the scattering pa-rameter extraction method (S-parameter) is one of the earliest and simplesteffective parameter retrieval methods, where the metamaterial structure ishomogenized by extracting its effective parameters from the experimentallymeasured external light scattering of the system [106–111]. This simple ho-mogenization method yields an infinite branch of solutions which raises theconcept of branch ambiguity as a clear disadvantage of this solution [112].Other simple homogenization methods include assigning effective nega-tive index parameter to a single thin layer of metal based on electrostatic ap-proximation [27] and assigning the negative index parameter to a multi-layermetal-dielectric system based on the geometric optic visualization of lightrefraction and bulk propagation in the layered system [7]. These methodsare based on unrealistic assumptions and are only applicable to particularlayered metamaterials.The more sophisticated homogenization methods are mainly based on161.5. Homogenization Theorytheoretical calculations and a variety of averaging techniques, such as field-averaging approaches [113–118], volumetric averaging of the local permittiv-ity values based on the effective medium theory usually known as Maxwell-Garnett method [119–121], and averaging the energy densities [122]. Thecurve-fitting approach [123], and the dispersion equation method [33] havebeen also reported for homogenizing metamterials. These methods are notuniversal and they can lead to different solutions for the same heteroge-neous system [124]. The detailed derivations of two common homogeniza-tion methods, the S-parameter and Maxwell-Garnett methods, are providedin the appendices.One of the most rigorous homogenization methods is based on the Floquet-Bloch theorem, usually used in crystals, where the electromagnetic fields ina metamaterial is described by the dominant Floquet-Bloch harmonic [124–129]. Although, this method was used based on unpractical assumptionsand suffers from branch ambiguities [31], we managed to circumvent thedrawbacks and invoke this theorem in our work. Here, we provide a generalintroduction to this homogenization method and we will discuss the detailsof our work in the next chapter.1.5.1 Floquet-Bloch TheoryIn 1928, Felix Bloch studied the quantum mechanics of electrons in threedimensional periodic media [130] in which he unknowingly extended theone dimensional theorem of Gaston Floquet published in 1883 [131]. Blochproved that wave functions of electrons in periodic media, e.g. crystals, aregoverned by a periodic envelop function multiplied by a plane wave. Inter-estingly, the first study of electromagnetic propagation in one dimensionalperiodic structures was presented by Lord Realeigh before Bloch’s work in1887, where he postulated the possibility of controlling light propagatingthrough periodic media [132]. In 1972, Bykov proposed that periodic struc-tures can be particularly used to control the spontaneous emission while therelation between the electromagnetic properties of periodic media and theFloquet-Bloch modes have not been addressed yet [133]. It was not untilwork by Pochi Yeh et. al. in 1977 that bridged the gap between classicalelectromagnetism and solid-state physics and showed that Floquet-Blochmethod can be applied to electromagnetic waves propagating through crys-tals [37].Although Floquet-Bloch’s theorem is a solid state theorem for describ-ing the energy states of electrons in natural crystals, it has been commonlyused for characterizing the electromagnetic properties of artificial electro-171.5. Homogenization Theorymagnetic crystals, known as photonic crystals, since the photon behaviourin a photonic crystal is quite similar to electron and hole behaviour in anatomic lattice [134]. The term photonic crystal was first introduced in 1987by Yablonovitch and John, where they expanded the concept of photonicband gaps in two and three dimensions [135, 136].Beside photonic crystals, Floquet-Bloch’s theorem has been also used toanalyze the electromagnetic properties of periodic stratified metamaterials.Under the implicit assumptions of translational symmetry (infinite extent)and no loss, the discrete Floquet-Bloch modes of electromagnetic waves inperiodic media were derived [37]. For lossy media, the Floquet-Bloch modesare complex-valued [137–139] and no longer discrete [125].Periodic thin films became popular after the developments in the crystal-growing field in 1970s, especially with the advent of molecular beam tech-nology. These fabrication methods enabled multilayer growing of very thinlayers, down to 10 A˚, with well-controlled periodicity [37, 140]. Periodic lay-ered metamaterials and photonic crystals are both periodic thin films withabnormal electromagnetic properties, but they are fairly different. Pho-tonic crystals are only made of dielectric materials with lattice periodicitycomparable to the operation wavelength, while periodic layered metamate-rials can be composed of metal or dielectric materials with lattice constantsmaller than the wavelength. Over and above, the practical significance ofphotonic crystals is mainly due to their associated photonic bandgaps, fre-quency ranges where light cannot propagate through. This implies that theabnormal electromagnetic properties of photonic crystals can be attributedto the unusual refraction of light within their periodic lattices rather thanthe microscopic electromagnetic resonance effects which are associated withlayered metamaterials. [134, 141, 142]Considering a general periodic thin film system, the dielectric constant depends on the spatial coordinate ~r and the angular frequency ω. Since isa periodic function, Bloch’s theorem will be applicable for the electromag-netic eigenmodes similar to electronic eigenstates in crystals. Hence, theeigenfunctions of electric and magnetic wave equations can be expressed as~Ek(~r) = uk(~r) ei(~k.~r), (1.32)~Hk(~r) = vk(~r) ei(~k.~r), (1.33)where uk(~r) and vk(~r) are the periodic envelope functions of electric andmagnetic fields respectively, and satisfy the following conditions:181.6. Thesis Outlineuk(~r + ~a) = uk(~r), (1.34)vk(~r + ~a) = vk(~r), (1.35)where ~a is the elementary lattice vector and ~k is the Bloch wave vector. Ifthe periodicity disappears, the periodic envelope functions uk(~r) and vk(~r)become constant and the Bloch wave vector equals to the wave vector of theplane wave of propagation. [140, 143]1.6 Thesis OutlineThe thesis presents rigorous electromagnetic solutions for practical im-plementations of layered metamaterials for the purpose of engineering theirexotic electromagnetic properties. Chapter 1 discussed the fundamental con-cepts of the electromagnetic wave theory. We also presented some more ad-vanced concepts like metamaterials, homogenization theory, and the Floquet-Bloch theory. Following the rapid pace of research on metamaterials, wehighlighted some of the most influential research in the field of layered meta-materials focusing on three influential applications. Chapter 2 discusses thederived analytical solutions and the corresponding band diagrams as a prac-tical design tool for engineering the refractive properties of layered meta-materials. The next three chapters then discuss three layered metamaterialapplications. A flat lens criterion is proposed and validated in Chapter 3.The concept of spectral light transmission in bi-layer thin films and thegeneral conditions for light transmission boost with a series of experimen-tal investigations will be presented in Chapter 4. Chapter 5 discusses anew nano-film fabrication method for surface plasmon coupling enhance-ment. We conclude the thesis by summarizing the contributions, sheddingthe light on the limitations, and discussing the possible directions for futurework.19Chapter 2Electromagnetic Fields inLayered Metamaterials ofFinite ExtentLayered structures are one of the simplest forms of a metamaterial, as ithas heterogeneity along just one direction. Solving for the electromagneticfields within and about a layered system is straightforward using methodslike the transfer matrix method, a method used to analyze the propagationof electromagnetic waves through a stratified (layered) medium by account-ing for all transmitted and reflected waves within the layers and formalizingthe problem in matrix form. However, analyzing the intrinsic electromag-netic field interaction, which can provide physical insights and explain theassociated abnormal optical properties such as super-resolution imaging, isfairly complex due to the contribution of surface plasmon modes in these sys-tems. To provide a simple solution to this problem and to characterize theproperties of layered metamaterials, various homogenization methods havebeen introduced. An overall view of these homogenization methods has beenprovided in the introduction, and it has been shown that they are generallyeither based on unpractical assumptions or encounter non-uniqueness. Eventhe Floquet-Bloch method, which is a widely used homogenization methodin the literature, is limited to applications for lossless layered systems ofinfinite extent.Numerical simulations of the electromagnetic fields can directly calcu-late the spatial fields and energy distributions based on Maxwell’s equations.This is a commonly used approach to study light behavior in layered meta-materials [6, 9, 51]. The numerical simulation approach can be adapted tosimulate the electromagnetic behaviour of any system. These simulationsprovide data on field and energy distributions of almost any configurationand graphically demonstrate the collective electromagnetic wave behaviourin complex structures. The major drawback of the numerical simulationmethods in analyzing mutual multi-mode electromagnetic wave behaviours202.1. Light Scattering at a Single Plane Boundaryis their incapability of accurately identifying the constituting modes and theinfluence of each mode.As a compliment to these methods, we introduce a bottom-up approachthat rigorously analyzes the intrinsic electromagnetic fields of lossy finitelayered media in closed form. We start with the transfer matrix methodusing the fundamental Maxwell’s equations with the least possible assump-tions and without invoking any homogenization methods. Then, we applyFourier transformation to find the Fourier domain representations of thetransfer matrix method solutions. Through a series of mathematical manip-ulations, we manage to derive a field expression which is a product of threeterms: a term explicitly dependent on the Floquet-Bloch modes, a term gov-erned by reflections from the medium boundaries, and another term which isdependent on layer composition. We use the new field expression to decom-pose and analyze the wave function in single layer, bi-layer, and multi-layersystems.Knowing the spatial-frequency representations of the electric and mag-netic fields in a layered medium, the spectral time-average Poynting vector(the directional energy transfer rate of electromagnetic fields per unit area)and its associated equipotential contours (the real wave-vector coordinatecontours at which the potential energy of the electromagnetic fields is con-stant) can be directly calculated. Plotting the derived equipotential contours(EPCs), we graphically describe a wide range of refractive properties asso-ciated with layered metamaterials. In this chapter, we will illustrate theEPC band diagrams of a wide range of layered metamaterials from simplestconfiguration of a thin silver layer to complex periodic systems composed ofmulti-layered unit cells. We will study some frequently reported abnormaloptical properties associated with layered metamaterials such as negativephase velocity, super-resolution, canalization, and far-field imaging. We alsoutilize EPC band diagrams as standard gauge to validate some conventionalhomogenization methods.2.1 Light Scattering at a Single Plane BoundaryWe start with the simplest configuration of single interface where a planeboundary surface separating two isotropic homogeneous semi-infinite media.Assuming the boundary is at z = 0 and that it is infinite along positiveand negative lateral directions, a plane wave is obliquely incident upon theboundary from the first medium (z > 0). The incident light will be par-tially reflected back to the first medium and partially transmitted through212.1. Light Scattering at a Single Plane Boundarythe second medium (z < 0). The incident, reflected and transmitted wavevectors are all lie in the plane of incidence which is determined by the inci-dent wave vector and the normal to the boundary surface (Fig. 2.1). In suchconfiguration, the problem of solving for the reflected and the transmittedfield coefficients is a boundary value problem which is solvable based on aset of constraints called boundary conditions. At z = 0, the boundary con-ditions insist on the continuity of the tangential electric ( ~E) and magnetic( ~H) fields for all x and y.Figure 2.1: Light scattering of a TM wave at a plane boundary.Invoking Maxwell’s equations 1.10 and 1.11, we calculate the total elec-tric and magnetic fields in the first medium as the sum of the reflectedand incident components. Considering a TM-polarized plane wave inci-dent at an angle θ from the upper semi-infinite half space onto the bound-ary surface, the incident time-harmonic magnetic field can be written as~H = H0ei(kx,0x+kz,0z)e−iωt yˆ, where H0 is the magnetic field amplitude, ωis the angular frequency, and the real-valued vectors kx,0 and kz,0 are thewave-vector components of k0 along the x- and z-axes respectively, wherek0 =√k2x,0 + k2z,0. Solving for the fields at t = 0, the time-harmonicterm can be suppressed and the magnetic field can be expressed as ~H =H0ei(kx,0x+kz,0z)yˆ. Here, we are discussing the time-harmonic independentplane wave equations with TM polarization in frequency domain. In case ofTE polarization, the principle of duality (~E → −~H, ~H → ~E, µ) can beapplied to solve for the complimentary equations.222.1. Light Scattering at a Single Plane BoundaryFor a TM wave, where the magnetic field vector is normal to the plane ofincidence (Fig. 2.1), the total magnetic and electric fields in the first mediumcan be written asH1y = (H0e−ik1zz +RH0eik1zz)eikxx, (2.1)E1x =−k1zω1(H0e−ik1zz −RH0eik1zz)eikxx, (2.2)E1z =−kxω1(H0e−ik1zz +RH0eik1zz)eikxx, (2.3)where R is the reflection coefficient, and k1z and 1 are the wave-vector com-ponent along z-axes and the permittivity in the first medium respectively.The opposite sign of the z component is an indication of opposite propaga-tion in zˆ direction. In the second medium, the electric and magnetic fieldscorrespond only to the transmitted wave and they can be written asH2y = TH0e−ik2zzeikxx, (2.4)E2x =−k2zω2TH0e−ik2zzeikxx, (2.5)E2z =−kxω2TH0e−ik2zzeikxx, (2.6)where T is the transmission coefficient of the electric field. Based on theboundary conditions of E and H fields, we can obtain the relations betweenreflection and transmission coefficients asR+ 1 = T, (2.7)andk1z1(R− 1) = −k2z2T. (2.8)Solving for R and T coefficients, we getR =1− p1 + p, (2.9)andT =21 + p, (2.10)232.2. Transfer-Matrix Representation of the Electromagnetic Field in a Layered Mediumwhere p = 1k2z2k1z . For TE wave, where the electric field vector is normal to theplane of incidence, we have the same solutions for R and T coefficients wherep should be replaced by p = µ1k2zµ2k1z . The derived reflection and transmissioncoefficients are equivalent to the Fresnel coefficients for normal incidence.2.2 Transfer-Matrix Representation of theElectromagnetic Field in a Layered MediumWe consider the general one-dimensional configuration of a layered pe-riodic media bounded by two semi-infinite half spaces (Figure 2.2). Theinfinitely long layered system is aligned parallel to the xy plane with het-erogeneity along z direction. It is composed of M repeated unit cells of Jlayers where the total number of layers is MJ . The layers within a unit cellare referenced by the small letter j, where j = 1, ..., J , and the unit cells arereferenced by the small letter m, where m = 0, ...,M − 1. Consequently, thelabel reference of any layer ` will be a function of m and j, where ` = mJ+j.We label the half space before the first layer as ` = 0 and the half spaceafter the last layer as ` = MJ + 1. With layer thickness of d` for layer `, thetotal thickness of the layered medium will be L =∑MJ`=1 d`. The interfaceplane location between layer ` and ` + 1 is denoted by z`. Accordingly, ifthe plane of the first interface is at z0 = 0, the plane of the last interfacewill be at zMJ = L.The associated dispersive electromagnetic properties of each layer are thecomplex-valued relative permittivity `, the complex-valued relative perme-ability µ`, and consequently the complex-valued refractive index n`. Forthe defined multi-layer structure where the heterogeneity is just along onedirection, the electromagnetic fields in each layer are the result of the inter-ference between the infinite reflected and transmitted waves bouncing backand forth between the two consecutive interfaces. The convenient way toaccount for all reflected and transmitted waves and the boundary condi-tions is through the matrix form representation of the aggregated reflectedand transmitted coefficients. The transfer matrix representation was firstintroduced in optics by Jones’ and Abels in 1950 [144]. Later, this methodhas been extensively used to solve for light propagation in stratified media[81, 119, 140, 145–149].Here, we consider a TM-polarized plane wave incident at an angle θ fromthe semi-infinite left half space (` = 0) onto the layered medium. Using thetransfer matrix method, the magnetic field in an arbitrary layer ` is the sumof two counter-propagating waves, a forward wave and a backward wave.242.2. Transfer-Matrix Representation of the Electromagnetic Field in a Layered MediumUnit cell index, :mLayer index, :l0 M-1zx{Incident waveReflected waveTransmitted wave0 1 J... (M 1)J+1 ... MJ- MJ+1D DL...{...Unit cell layer index, :j 1 ... J 1 ... J...qFigure 2.2: Geometry under consideration consisting of a one-dimensionalperiodic layered medium bounded by two semi-infinite half spaces and com-posed of M repeated unit cells, each consisting of J layers. The mediumis excited from one half-space by an incident plane inclined at an arbitraryangle θ in the xz plane. [5]The matrix form representation of the magnetic field in an arbitrary layer `can be written as~H`(x, z) = H`(x, z)yˆ= eikx,0x(eikz,`(z−z`)e−ikz,`(z−z`))T (A`B`)yˆ,(2.11)where A` and B` are the wave coefficients of the forward and backwardwaves in layer `. The wave-vector component kz,` in layer ` can be relatedto the layer refractive index n`, and the tangential wave-vector componentkx,0 bykz,` = n`√k20 −(kx,0n`)2. (2.12)Using the boundary conditions at the interfaces, the wave coefficients A`and B` in layer ` and the wave coefficients A`+1 and B`+1 in the adjacent252.2. Transfer-Matrix Representation of the Electromagnetic Field in a Layered Mediumlayer `+ 1 can be related by(A`+1B`+1)= T`P`(A`B`), (2.13)where P` is the propagation matrix corresponding to layer `, and T` is thetransmission matrix corresponding to the interface between layer ` and `+1.The propagation matrix P` is given byP` =(eikz,`d` 00 e−ikz,`d`), (2.14)and the transmission matrix T` is given byT` =12(1 + p`1− p`1− p`1 + p`), (2.15)with p`= (`+1kz,`)/(`kz,`+1). Eq. 2.13 implies that the wave coefficientswithin a layer can be calculated knowing the wave coefficients of the previouslayer.Wave coefficients across a single layer can be related by(t0)= T1P1T0(1r), (2.16)and wave coefficients across an arbitrary multi-layer system can be relatedby (t0)= T`P`T`−1 · · · T1P1T0(1r). (2.17)This matrix yields two equations and two unknowns (t, r), since the normalwave-vector component kz, the layer thickness d, and the permittivity ofeach layer are known. This enable us to solve for the magnetic field ~H` ineach layer ` of the layered system. The total field distribution in the spatialdomain can be expressed as a piece-wise function subdivided into spatialintervals corresponding to the layer regions~H(x, z) = H(x, z)yˆ =MJ∑`=1rect(z − zc,`d`)H`(x, z)yˆ, (2.18)where zc,` is the location of the center of layer `. The rect function is defined262.3. Isolating Floquet-Bloch Modes by Fourier Transformationasrect(z − zc,`d`)={1 zc,` − d`/2 ≤ z ≤ zc,` + d`/20 otherwise.Equation (2.18) works well solving for sets of linear equations using nu-merical routines, but there are at least two disadvantages that should betaken in consideration. First, this form of solution does not provide physi-cal insight into the overall behavior of the repeated sets of layers. Second,the right and left-handed propagating modes cannot be distinctly identifiedand analyzed using this time domain solution. To account for these con-cerns, Fourier transformation is applied to the piece-wise wave solution inthe next section.2.3 Isolating Floquet-Bloch Modes by FourierTransformationWe solved for the electromagnetic fields everywhere using the transfermatrix method. Since we are interested in analyzing the electromagneticmodes within the layered medium, here we apply Fourier transformationto the obtained electromagnetic solutions within the layers only. Apply-ing Fourier transformation, the magnetic field distribution in the spatial-frequency domain will beH(κx, κz) =∫ zMJ0∫ ∞−∞H(x, z)e−iκxxe−iκzzdxdz, (2.19)where κx and κz are the spatial-frequency variables along the respective xand z directions. Using the well-known Fourier theorems and substituting(2.18) into (2.19), the integrand yieldsH(κx, κz) =(2pi)2δ(κx − kx,0)MJ∑`=1d` sinc(κzd`2pi)e−iκzzc,`∗(e−ikz,`z`−1δ(κz − kz,`)eikz,`z`−1δ(κz + kz,`))T (A`B`),(2.20)where δ is the Dirac delta function, the symbol ∗ indicates having a convo-lution, and the matrix superscript T denotes matrix transposition.To produce a physically insightful Fourier-domain wave solution, we ap-ply a series of mathematical manipulations to Equation (2.20). We start by272.3. Isolating Floquet-Bloch Modes by Fourier Transformationre-writing Equation (2.20) as a nested double summation over the numberof layers in a unit cell and the number of unit cells in the layered systemH(κx, κz) = (2pi)2δ(κx − kx,0)M−1∑m=0J∑j=1dj sinc(κzdj2pi)e−iκzzc,mJ+j∗(e−ikz,jzmJ+j−1δ(κz − kz,j)eikz,jzmJ+j−1δ(κz + kz,j))T (AmJ+jBmJ+j).(2.21)In equation (2.21), we substitute the variables d` with dj , and kz,` with kz,j ,and replace the index ` by mJ + j. Solving the convolution yieldsH(κx, κz) =(2pi)2δ(κx − kx,0)M−1∑m=0J∑j=1dje−iκzzc,mJ+j(eikz,jdj/2 sinc[(κz − kz,j)dj/2pi]e−ikz,jdj/2 sinc[(κz + kz,j)dj/2pi])T (AmJ+jBmJ+j),(2.22)where zc,mJ+j − zmJ+j−1 = dj/2. If the discrete summation form for thecentral position within the layer mJ + j is given byzc,mJ+j = mD + zj−1 + dj/2, (2.23)where D =∑Jj=1 dj is the thickness of the unit cell and zj−1 is the positionof the interface between layer j − 1 and j within unit cell m = 0, Equation(2.22) can be simplified asH(κx, κz) = (2pi)2δ(κx − kx,0)J∑j=1dje−iκzzj−1(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2pi]e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2pi])TM−1∑m=0e−iκzmD(AmJ+jBmJ+j).(2.24)The spatial-frequency domain representation of the electromagnetic wavesolution in (2.24) provides little more physical insight over the spatial-domain representation of the wave solution in (2.18). However, we canfurther factorize and simplify the wave solution (2.24) knowing that the ma-282.3. Isolating Floquet-Bloch Modes by Fourier Transformationtrix relation between the wave coefficients in an arbitrary unit cell (AmJ+j ,BmJ+j), and the wave coefficients in the first unit cell (Aj , Bj) is given by(AmJ+jBmJ+j)= Umj(AjBj), (2.25)where Umj is the unit cell transfer matrix from layer j to mJ + j and isdetermined from the transmission and propagation matrices byUmj =mJ+j−1∏q=jT qP q. (2.26)Substituting (2.25) into (2.24), we get the wave solution in terms of Ajand Bj wave coefficientsH(κx, κz) = (2pi)2δ(κx − kx,0)J∑j=1dje−iκzzj−1(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2pi]e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2pi])T(M−1∑m=0(e−iκzDU j)m)( AjBj).(2.27)By matrix decomposition, the unit cell transfer matrix referenced fromlayer 1, U1, will beU1 = Q λ Q−1 (2.28)where Q is the eigenvector of U1 and λ is the eigenvalue of U1 whose diagonalelements are the corresponding eigenvalues λ1 and λ2. U1 can be related tothe referenced unit cell transfer matrix from layer j, U j , using the relationUmj = W j,1 Um1 W−1j,1= W j,1Q λ Q−1W−1j,1 .(2.29)Since the determinant of U1 is unity, the eigenvalues are inverses of eachother, λ2 = 1/λ1. These eigenvalues can be related to the Floquet-Blochmode, kFB, by292.3. Isolating Floquet-Bloch Modes by Fourier Transformationλ =(λ1 00 λ2)=(e+ikFBD 00 e−ikFBD). (2.30)Substituting (2.29) into (2.27) and relating the wave coefficients in layerj to the wave coefficients in the left half space, we arrive at the final formof the magnetic field solutionH(κx, κz) =(2pi)2δ(κx − kx,0)J∑j=1dje−iκzzj−1(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2pi]e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2pi])TW j,1Q︸ ︷︷ ︸LjM−1∑m=0(e−iκzDλ)m︸ ︷︷ ︸FBQ−1T0(1r)︸ ︷︷ ︸C,(2.31)where three distinctive matrix factors are highlighted - a layer matrix Ljdependent on the thickness and wave vector in the jth layer of the unit cell,a Floquet-Bloch matrix FB dependent on the eigenvalues of the unit cell,and a weighting matrix C dependent on the reflection coefficient.Deriving the factorized form of the magnetic field solution in equation(2.31) is the major contribution of this work. It shows how the Floquet-Bloch modes and the electromagnetic field in a lossy layered medium of finiteextent are inherently related. If the general magnetic field form (2.31), thethree factors can be simply re-written asLj =(L+j L−j), FB =(FB+ 00 FB−), C =(C+C−), (2.32)where the “forward” and “backward” elements of each matrix factor aredistinguished by the superscript “+” and “−”, respectively, the magnetic302.3. Isolating Floquet-Bloch Modes by Fourier Transformationfield solution can be written in a compact form asH(κx, κz) = (2pi)2δ(κx − kx,0)J∑j=1Lj FB C= (2pi)2δ(κx − kx,0)J∑j=1(L+jL−j)T (FB+C+FB−C−).(2.33)Compared to the spatial-domain solution in (2.18), which does not in-clude any clue to the Floquet-Bloch modes, the spatial-frequency domainsolution in (2.31) can distinctively describe the collective wave behavioracross repeated sets of unit cells through the elements of the Floquet-Blochmatrix. For a medium of finite extent (M < ∞), the FB matrix elementsare given byFB± =e−i(κz±kFB)M−12DM∆ 2piD[κz ±<(kFB)]∗ sinc(M [κz ± i=(kFB)]D/22pi),(2.34)where =(kFB) is the material loss, ±<(kFB) is the principal harmonic center,and the Dirac comb ∆ 2piD[κz ± <(kFB)] is describing an infinite comb ofdiscrete spatial-frequency harmonics spaced by 2pi/D and it is defined as∆ 2piD[κz ±<(kFB)] =∞∑N=−∞δ [κz − 2piN/D ±<(kFB)] . (2.35)In equation 2.34, the principal Floquet-Bloch harmonics of the forward el-ement FB+ and the backward element FB− are centered at <(kFB) and−<(kFB), respectively. Due to the convolution of the Dirac comb and thesinc function, the harmonic elements FB± are widened through the com-bined effects of finite extent (M <∞) and material loss (=(kFB) 6= 0).For a medium of infinite extent (M → ∞), the elements of the FBmatrix approachlimM→∞FB± =M∆ 2piD (κz ±<(kFB)) =(kFB) = 0∆ 2piD[κz ±<(kFB)] ∗ 2D√κ2z+=(kFB)2 =(kFB) 6= 0.(2.36)In the case of lossless materials (=(kFB) = 0), the FB± elements are discretespectra with peaks at harmonics of ±<(kFB). The resulting magnetic field312.4. Analysis of Electromagnetic Fields in Layered Metamaterialssolution will be equivalent to the classical Floquet-Bloch solution where thefield solution consists of discrete forward and backward Floquet-Bloch modeswith amplitudes that can be explicitly determined by (2pi)2∑Jj=1 L+j C+ and(2pi)2∑Jj=1 L−j C−, respectively. In the case of lossy materials (=(kFB) 6= 0),the FB± elements are continuous spectra with peaks centered about theprincipal harmonics of ±<(kFB) and broadened due to the effect of =(kFB).2.4 Analysis of Electromagnetic Fields inLayered MetamaterialsWe study the electromagnetic fields in a typical metal-dielectric layeredmetamaterial structure illuminated by a normal-incidence TM-polarized planewave (kx,0 = 0). For a bi-layer unit cell system, the factorized Fourier-domain magnetic field solution takes the formH(κz) = (2pi)2δ(κx − kx,0)(L+1 + L+2L−1 + L−2)T (FB+C+FB−C−), (2.37)where L+1 , L+2 and L−1 , L−2 are the forward and backward elements of layermatrices, respectively. Applying the Fourier-domain solution in (2.37) toanalyze a practical design of layered metamaterial, we assume the layeredsystem is bounded by free-space and the bi-layer unit cell consists of a 30-nm-thick Ag layer and a 30-nm-thick TiO2 layer. A layered combination of thesetwo materials has been modeled as a homogeneous left-handed medium [7].We analyzed two distinct Ag-TiO2 bi-layer systems, a system composed oftwo unit cells (M = 2) and another composed of ten unit cells (M = 10).Exciting each system by a plane wave with the free-space wavelength ofλ0 = 365 nm, Ag is assigned the complex refractive index of 0.076 + 1.605i(interpolated from experimental data [17]), and TiO2 is assigned the realrefractive index of 2.80 [7].For the unit cell cases M = 2 and M = 10, the matrix elements of themagnetic field solution and its modulus squared, besides the z-componentof the time-averaged spectral Poynting vector are highlighted in Figure 2.3.The concourse of broad spectral envelopes defined by L±1 (Ag layer) andL±2 (TiO2 layer), and finer spectral combs defined by FB+C+ and FB−C−results in the magnetic field spectrum. The Floquet-Bloch mode of the unitcell is kFB = 32.5 + 0.4i µm−1, corresponding to a Floquet-Bloch refractiveindex nFB = 1.89 + 0.02i. In Figure 2.3 c) and f), FB+C+ and FB−C−combs are offset; the former with a principal peak located at <(kFB) and322.4. Analysis of Electromagnetic Fields in Layered Metamaterialsthe latter with a principal peak located at −<(kFB). Increasing number ofunit cells M from 2 to 10 narrows the peaks of the combs and consequentlynarrows the peaks in the magnetic field spectrum. As a result, we will havebetter defined wavevectors for additional number of unit cells.Figure 2.3: Decomposition of the wave solution in a metal-dielectric bi-layersystem consisting of alternating layers of 30- nm-thick Ag and 30- nm-thickTiO2, assuming a normally incident TM-polarized wave with a free-spacewavelength of λ0 = 365 nm. a) The forward and backward components ofthe layer matrix |L1|2 corresponding to the 30- nm-thick Ag layer. b) Theforward and backward components of the layer matrix |L2|2 correspond-ing to the 30- nm-thick TiO2 layer. c), d), and e) depict the forward andbackward components of |FB C|2, the magnetic field spectrum |H|2, and thez-component of the time-averaged spectral Poynting vector, respectively, forthe case of 2 unit cells; f), g), and h) depict the same set of information forthe case of 10 unit cells. The horizontal gray lines in e) and h) correspondto zero values of the spectral Poynting vector . [5]Comparing the forward and backward magnitudes of FB for the M = 2332.4. Analysis of Electromagnetic Fields in Layered Metamaterialsand M = 10 cases, it can be noticed that the magnitude of FB+ C+ islarger than that of FB− C− in both cases. This indicates that the for-ward propagating components in the medium outweigh the backward prop-agating components. Carefully considering the z-component of the time-averaged spectral Poynting vector < Sz > shown in Figure 2.3 e) and h), itis remarkable that these negative spatial-frequency components are forward-propagating waves since they are characterized by a negative time-averagedspectral Poynting vector. Plotting the forward and backward componentsof the weighting matrix |C|2 versus the number of repetitions in Figure 2.4,we show that forward propagating waves |C+|2 generally exceeds backwardpropagating waves |C−|2 and as the number of repetitions increases the back-ward component gradually approaching zero. To achieve negative refraction,the system should support backward waves. Figure 2.4 shows that the exam-ined layered system is mainly supporting forward waves which means mostof the waves will be refracted positively.0 5 10 15 20 25 300.00.51.01.52.0|C|2Number of repetitions forward backwardFigure 2.4: Forward and backward components of the weighting matrix |C|2versus the number of repetitions. Here, we have assumed a metal-dielectricbi-layer system with a unit-cell made from a 30- nm-thick Ag layer and 30-nm-thick TiO2 layer, assuming a normally incident TM-polarized wave witha free-space wavelength of λ0 = 365 nm. [5]At frequencies near the bulk plasma frequency of metal, the real parts342.5. Elecromagnetic Field Band Diagrams in Layered Metamaterialsof the Floquet-Bloch refractive index in finely layered structures composedof metal turn to be negative, particularly for TM-polarized illumination [7,150]. Mimicking the refractive properties associated with negative-indexmedia, these finely layered media are shown to be capable of imaging witha planar slab (flat lensing) [4, 7]. As a continuation of this work, differentmetamaterial configurations composed of right-handed materials have beenanalyzed and the right-handedness of these systems has been consistentlyobserved, where the positive and negative spatial-frequency components ofthe wave are found to be both forward-propagating waves [151].2.5 Elecromagnetic Field Band Diagrams inLayered MetamaterialsBand diagrams are necessary for illustrating refractive properties. Theyare usually obtained by leveraging the Floquet-Bloch theorem [130, 131] toisolate discrete sets of real-valued wave vectors ~k and then use a linear eigen-value equation derived from the Helmholtz equation [143] to solve for thecorresponding frequencies ω(~k). To provide graphical information on phaseand group velocity in the medium, equi-frequency contours (EFCs) are thendisplayed as a function of real-valued wave-vector coordinates. This way ofsolving for band diagrams is usually applied to photonic crystals and it isbased on Floquet-Bloch theorem and Helmholtz equation. However, the ap-plication of Floquet-Bloch theorem is limited to lossless infinite systems, andin case of dispersion the Helmholtz equation turns into a non-linear eigen-value equation, which requires time-consuming iterative algorithms that aresensitive to initial guesses. As a result, EFC band diagrams are rarely usedto describe the electromagnetic properties of layered metamaterials [152].Recently, a finite-element method has been proposed to solve for complex-valued wave vectors ~k as a function of frequency ω(~k) in order to deriveband diagrams for layered metamaterials [138, 139, 152, 153].In this thesis, we present a new approach to obtain band diagrams forany layered metamaterial. Using the derived spatial Fourier magnetic fieldin Eq. 2.33, we solve for the corresponding x and z electric field components.The factorized Fourier-domain electric field components can be consequentlywritten asEx(κx, κz) = (2pi)2δ(κx − kx,0)J∑j=11j0ω(kz,j−kz,j)Lj FB C (2.38)352.5. Elecromagnetic Field Band Diagrams in Layered MetamaterialsandEz(κx, κz) = (2pi)2δ(κx − kx,0)J∑j=11j0ω(kx,0−kx,0)Lj FB C, (2.39)respectively. The spectral electric fields, as well as magnetic fields, aredefined as functions of kz, the spatial frequency along z, which exist fora fixed kx. Varying θ results in sweeping over different values of kx andconsequently having a two-dimensional k-space distributions of the electric~E(~k) and magnetic ~H(~k) fields as a function of the spatial-frequency variable~k = kxxˆ+ kz zˆ.The electric and magnetic field distributions define a plane-wave com-pletely described by the triad ~k-~E(~k)-~H(~k) [154–156]. For each plane wavecomponent, the time-averaged power flow is given by the spectral Poyntingvector〈~S(κx, κz)〉 = 12<[~E(κx, κz)× ~H∗(κx, κz)]. (2.40)The scalar potential of the spectral Poynting vector is defined byΦ(~k) =14pi∫ ∇~k′ · 〈~S(~k′)〉d~k′|~k − ~k′|. (2.41)Describing scalar potential as contour lines will form a diagram of bandsin k-space. In general, the spectral Poynting vector can be defined as〈~S(~k)〉 = ∇Φ +∇× ~R. (2.42)where ~R is the rotational component of the spectral Poynting vector. Basedon our observations working with layered systems, we found that solenoidalcomponents of 〈~S(~k)〉 are negligible. In this case the second term of Eq. 2.42will be almost zero (∇× ~R ≈ 0) and the direction of the spectral Poyntingvector will be the gradient of the scalar potential (contour lines).Based on the fundamental relationship of group velocity [157, 158]~vg =〈~S(~k)〉〈U(~k)〉, (2.43)the spectral Poynting vector have the same direction of the group velocity.Here, 〈U(~k)〉 is the time-averaged energy density which is the amount of362.5. Elecromagnetic Field Band Diagrams in Layered Metamaterialsenergy stored in the system per unit volume and it is generally defined by〈U(~k)〉 = 12(|E(~k)|2 + µ|H(~k)|2). (2.44)It can be noticed that the conventional EFC band diagrams defined byfrequency ω(~k) are similar to the EPC band diagrams defined by Φ(~k) sinceboth have gradients along the group velocity pointing to the direction ofthe time-averaged power flow. However, EPC band diagrams are unique forseveral reasons. First, like the optical density of states, the k-space energydensity distribution can be inferred from the density of EPCs, where a highEPC density indicates high energy density. Second, unlike the EFCs whichare limited to the Floquet-Bloch theorem assumptions of periodic, lossless,and infinite media, EPCs are applicable to any layered system since theycan be derived without invoking the Floquet-Bloch theorem.Third, as opposed to discrete modes with complex wave vectors for lossymaterials, the material loss effect using EPCs can be captured by a con-tinuum of plane-wave modes with real wave vectors endorsing EPCs to beplotted as a function of real wave-vector coordinates. Here we have to ac-knowledge that the presented EPC approach builds on the past studies ofthe anomalous refraction in infinite, lossless photonic crystals [154–156].These efforts are based on spatial-frequency decomposition of electric andmagnetic fields to examine phase and group velocity. In this work, we havegeneralized the theory by incorporating finite, lossy media and introducinga new method to extract EPCs.To demonstrate the refraction consistency of the derived EPCs with ge-ometric optics, we examine a positive and a negative index homogeneous,isotropic slabs, whose interface are aligned along the z plane. For homo-geneous slabs, the EPCs are semi-circular bands as shown in Figs. 2.5 (a)and (b). When the slabs are lossless and infinite, the EPC bands collapseto a semi-circular line emulating the plane-wave mode in a homogeneousmedium (ω = ck/n). The homogeneous, lossy and finite slab is illuminatedby an obliquely incident plane wave from free-space at an angle θ where theplane wave can be described in k-space by a wave-vector ~k0 = kx,0xˆ+ kz,0zˆ.The wave vector component parallel to the interface (kx,0), visualized by thevertical lines in Figs. 2.5 (a) and (b), is conserved based on the principal oftangential field continuity.372.5. Elecromagnetic Field Band Diagrams in Layered MetamaterialsFigure 2.5: Light refraction at an interface described by EPCs. EPCs forfinite, lossy slabs of thickness 2λ0 having either (a) a positive refractive indexn = 1.5 + 0.1i or (b) a negative refractive index n = −1 + 0.1i. The bottompanels depict the EPCs of an incident plane wave (λ0 = 400 nm) impingingon the slab. The resulting phase (blue arrow) and time-averaged Poyntingvector (red arrow) are graphically derived in k-space from the EPCs andthen depicted in real-space in the insets. [8]For the positive-index slab, the excited plane waves have parallel phaseand group velocities along a direction consistent with Snell’s law. For thenegative-index slab, the incident wave excites planes waves with anti-parallelphase and group velocities, where the phase velocity is directed towardsthe interface and the group velocity is directed away from the interface.The anti-parallel phase and group velocities are hallmarks of a left-handedelectromagnetic response and they impose negative refraction to the sameside of the normal, as first presented by Veselago [19].The spectral time-averaged Poynting vector in (2.40) is quite similar to382.6. Band Diagram Analysis of Layered Metamaterialsthe one proposed in Ref. [154, 155] to analyze energy propagation of discreteFloquet-Bloch modes in infinite, lossless dielectric photonic crystals. How-ever, now we have extended the work to accommodate a continuous range ofFourier field components in a finite, lossy periodic system. Moreover, the di-rection of the time-averaged power flow can be inferred based on the gradientof the equipotential contours. We applied this concept to derive band dia-grams which can be used to distinguish forward- and backward-propagatingcomponents of the wave in a medium and provide intuitive visualization ofphase and power flow [8].2.6 Band Diagram Analysis of LayeredMetamaterialsTo evaluate the functionality of our proposed method, we will deriveEPCs for a variety of layered metamaterials, including a thin silver layersustaining surface plasmon polariton (SPP) mode, Pendry’s silver slab lens,the Veselagos lens, and metal-dielectric multi-layered systems capable ofcanalization and far-field imaging. EPCs are calculated based on spatial-frequency representations for arbitrary lossy layered media of finite extentunder plane-wave illumination [5]. For accurate solutions, we check that thesolenoidal component of the spectral Poynting vector in each case is smallcompared to its ir-rotational component. Besides the numerical methods,such as the finite-element method, for analyzing the electric and magneticfield solutions in lossy finite layered media, our proposed analytical expres-sions yield an efficient analysis method and high-resolution solutions of theinternal field modes (up to 500 unique solutions versus kx).To support the derived EPC predictions, Maxwell’s equations are solvedand the full-wave solutions are simulated using the finite-difference frequency-domain (FDFD), a numerical technique to solve the frequency domain har-monic form of Maxwell’s curl equations [159, 160]. These simulations aresuitable for laterally asymmetric two-dimensional geometries and they canvisually provide the spatial distribution details of fields and energy densityin and about the layered medium. The simulations are conducted using theFDFD technique due to its superiority over other methods like the finite-difference time-domain (FDTD) method in realistic modeling of lossy mate-rials, where FDFD allows the direct use of tabulated complex permittivityvalues generally available in Refs. [17, 161]. The basic implementation ofthe FDFD simulation method starts with defining a spatial grid. Then, aplane wave source is placed in the simulation space, the boundary condi-392.6. Band Diagram Analysis of Layered Metamaterialstions are defined, and the steady-state distribution of the fields at a singlefrequency are calculated everywhere [162, 163]. The FDFD simulations inthis work are conducted over a discrete two-dimensional spatial grid withthe minimal spatial resolution of 1 nm. Simulating layered configurations,the finite simulation spaces are defined wide enough to have laterally longlayered media and to minimize boundary effects. The used FDFD MATLABcode is provided in the Appendix.To excite the SPP modes, we utilize the standard prism configuration byilluminating the layered system with a plane wave source from a high-indexdielectric medium to probe the surface plasmon modes of layered metama-terials at high lateral spatial frequencies (where kx >> k0). For the pur-pose of quantifying the imaging resolution of layered systems, an optically-thick chromium mask with two λ0/15-wide openings and distance T apartis placed between the light source and the layered system. The minimumdistance T between the two mask openings that can be clearly identified astwo high energy density spots in the image region can be considered as theminimum resolvable feature by the layered system.2.6.1 SPP Mode as a Window into the Negative-IndexWorldConsidering the simplest layered system that can sustain SPP modes, wederive the EPCs of a thin silver layer under the Kretschmann configurationto investigate the correlation between SPP modes and left-handed electro-magnetic behavior. The examined layered system consists of a 40-nm-thicksilver layer placed between a semi-infinite dielectric medium (n = 2) and asemi-infinite free-space. The light source is placed in the dielectric mediumwhere the thin silver layer is illuminated by a transverse-magnetic (TM)polarized plane wave with free-space wavelength of λ0 = 400 nm where thesilver permittivity is characterized by = −4.4 + 0.2i [17].Figure 2.6(a) shows the derived EPCs where the most prominent fea-ture is the high density vertical line localized just beyond the free-spacecutoff spatial frequency k0, and it corresponds to the SPP mode, a sub-wavelength guided mode propagating along the surface. we draw a verticalline on the EPC diagram at kx ' 1.1k0 and a wave vector ~k from the ori-gin to the point of highest line density along the vertical line to visualizethe SPP mode plane-wave excitation. While the phase velocity along ~kpoints away from the origin, the group velocity along the spatial frequencygradient points towards the origin. The anti-parallel phase and group veloc-ities relation is the distinct characteristic of a left-handed medium which is402.6. Band Diagram Analysis of Layered Metamaterialsachieved solely within the silver layer along the interface. The excited SPPmode is inherently dispersive due to its near-resonance operation, requiresTM polarization, and occurs at a spatial-frequency that is inaccessible tofree-space plane-wave illumination. To show the polarization effect, we usea transverse-electric (TE) polarized light source and we show the resultingEPCs in Figure 2.6(b) where the potential gradient has been inverted andthe group velocity is generally directed away from the origin.At surface plasmon resonance, an effective magnetic response is estab-lished in the 40-nm-thick silver layer by microscopic circulation of its elec-tric field as illustrated in the FDFD simulation 2.6(c), where the electricfields are the blue arrows and the magnetic fields are the contour lines.This phenomenon leads to a light behaviour analogous to the behavior ofa negative-index medium. The electric field circulation is coincided withthe background magnetic field. At the silver-air interface where the SPPmode resides, this synchronization is more pronounced. Using homogeniza-tion methods, an effective magnetic flux density can be determined and thecontributions from the in-plane electric field to the magnetic response canbe described by a negative permeability [122, 164]. In contrast to the shapedsplit-ring resonators that can achieve artificial magnetism at microwave fre-quencies, the silver layer with a smooth surface can enable electric circulationand attain artificial magnetism and left-handed behavior at visible and UVfrequencies over the smallest possible size scales.In multi-layer metal-dielectric systems, if the fill fraction of the dielectricis sufficiently low, the SPP mode across the metal dielectric interfaces canstill support the left-handed behavior. To show the effect of adding dielectriclayers to the system, we use the plane wave illumination (λ0 = 400 nm, TM-polarization) to excite a metal-dielectric five-layer waveguide made of three40-nm-thick silver layers separated by two 10-nm-thick dielectric (n=2) lay-ers where the dielectric fill fraction is 0.14. We then map out the derivedEPCs where the concentric elliptical contours are increasing close to the ori-gin as shown in Figure 2.6(d). It can be noticed that in this Figure phaseand group velocities are nearly anti-parallel at the predetermined surfaceplasmon resonance spatial frequeny in Figure 2.6(a) kx = 1.1k0. This ob-served left-handed behavior is consistent with the theoretical investigationsof backward-propagating SPP modes [122, 164], as well as the experimen-tal results of in-plane negative refraction of visible light in metal-dielectricwaveguides with fairly thin dielectric cores [165]. The FDFD simulation ofthe electric and magnetic fields in the multi-layer metal-dielectric system atthe surface plasmon resonance shows that the left-handed response arisesfrom microscopic electric field circulation similar to the case of the single412.6. Band Diagram Analysis of Layered Metamaterialssilver layer (Fig. 2.6(f)).Figure 2.6: Band diagrams reveal interesting propagation characteristics ofthe SPP mode. We first examine EPCs for a single 40-nm-thick silver layerilluminated by a plane wave (λ0 = 400 nm) incident from a dielectric (n = 2)prism for either (a) TM or (b) TE polarization. (c) depicts the electric (bluearrows) and magnetic (contour lines) fields in the 40-nm-thick silver layer atthe surface plasmon resonance. The x and z scales are the same. Under TM-polarized illumination at the same wavelength, we next examine EPCs for a5-layer stack of silver and silicon nitride (n = 2) layers, where the silver layerthickness is fixed at 40 nm and the thickness of the silicon nitride is either(d) 10 nm or (e) 40 nm. (f) depicts the electric (blue arrows) and magnetic(contour lines) fields in the multi-layer structure at the surface plasmonresonance for the case of dielectric thickness 10 nm. The insets in the EPCsdepict the geometrical configuration for each case. The blue and red arrowsdescribe the phase and time-averaged Poynting vector, respectively, of themost dominant mode excited at the spatial frequency kx ' 1.1k0. Thelocation of the dominant mode corresponds to the highest density of contourlines intersected by the vertical line describing the conserved wave-vectorcomponent of an incident plane wave.[8]Investigating the sensitivity of left-handedness response to the dielectric422.6. Band Diagram Analysis of Layered Metamaterialsfill fraction, we increase the thickness of the dielectric layers to 40 nm (di-electric fill fraction of 0.40) where we observe that the EPCs dramaticallychange and flatten out into a series of horizontal lines with an upward po-tential gradient as shown in Figure 2.6(e). This is an indication that thesystem is now supporting bulk mode propagation through the medium asopposed to guided mode propagation along the surface.2.6.2 Super-Resolving Silver Slab Lens and the VeselagoLensThe capability of a thin silver layer for imaging beyond the diffractionlimit, which was first proposed by Pendry, has been demonstrated theoreti-cally and experimentally in the literature. As in the Veselago lens, the per-fect resolution capability of Pendry’s silver slab is considered to emerge fromthe amplitude amplification of evanescent waves [27]. To excite evanescentwaves at special frequencies beyond the free-space cutoff, a thin planar airregion is adopted between the prism and the Pendry’s silver slab lens, as wellas the Veselago lens. Deriving the EPCs under evanescent wave illuminationconditions, we attempt to understand the origins of their resolving power.Beyond the considerable studies in this field, our presented EPC approachoffers two novelties. First, the evanescent wave coupling into the lenses isconducted using a realistic laboratory implementable prism-coupling con-figuration. Second, using the EPC diagrams an intuitive imaging analysisbased on phase and power flow in the lens, similar to the classical imaginganalysis, is provided.In Figures 2.7 and 2.8, we map out the EPCs for a Pendry’s silver lensand a Veselago lens with equivalent thicknesses of 40 nm. The object planeis placed close to the interface, 20 nm apart from the entrance of the lens,between a dielectric prism and air. For high-k mode plane-wave excita-tion with large lateral spatial frequencies, a prism with an unusually highrefractive index (n = 5) is used. Such prism configuration is a realistic em-ulation of the evanescent wave illumination conditions used in Ref. [27], thefirst propose of super-resolution imaging in these systems. The comparableEPCs of the two lenses justifies their analogy under near-field conditions.At large spatial frequencies, the behavior of both lenses is governed by thedense vertical lines outside the free-space cutoff (surface modes) and poten-tial gradients directed towards the origin (left-handed response).432.6. Band Diagram Analysis of Layered MetamaterialsFigure 2.7: Band diagram analysis of Pendry’s super-resolving layer. EPCsfor (a) Pendry’s 40-nm-thick silver slab lens for TM-polarized illuminationat the wavelength of λ0 = 357 nm. The inset describes the geometricalconfiguration. The red arrows trace out the time-averaged Poynting vectoralong a frequency contour. Simulations of a test object imaged by Pendry’ssilver slab lens when the two point-like features of the object are spaced by(b) λ0/2.5, (c) λ0/3.0 and (d) λ0/3.5. [8].Examining the mapped out EPCs, several observations can be made.First, more modes are supported by the single layers beyond the free-spacecutoff spatial frequency compared to the below cutoff. These modes arethe high-k modes which are the source of their super-resolution capabilities.Second, the time-averaged Poynting vectors of the high-k modes along apotential contour (group velocities) are mostly directed towards the normalline kx = 0. This emphasizes the capability of these lenses to collect andconcentrate light across their extent. Third, the density of modes graduallydiminishes as k increases. This is an explicit indication that there is a finite442.6. Band Diagram Analysis of Layered Metamaterialslimit to resolving capabilities of any layered system excited by the realisticprism configuration, even the Veselago lens.Figure 2.8: Band diagram analysis of Veselago super-resolving layer. EPCsfor (a) a Veselago lens ( = µ = −1) of equivalent thickness for TM-polarizedillumination at the wavelength of λ0 = 357 nm. The inset describes the geo-metrical configuration. The red arrows trace out the time-averaged Poyntingvector along a frequency contour. Simulations of a test object imaged bythe Veselago lens when the two point-like features of the object are spacedby (b) λ0/4, (c) λ0/5, and (d) λ0/6. [8].It is quite evident that the common EPC feature of Pendry’s silverlens and Veselago lens is the left-handed guided modes, the prime causeof sub-wavelength imaging. Here, we show how the resolution limits canbe quantified by estimating the spatial-frequency bandwidth over which theleft-handed modes exist. In Pendry’s silver slab, the EPC lines are densevertical lines just outside the free-space cutoff up to a spatial frequencybetween 1.5k0 to 2.0k0. This implies that the minimum resolvable feature452.6. Band Diagram Analysis of Layered Metamaterialsby Pendry’s silver lens can be just less than double the free-space diffrac-tion limit (λ/2). The FDFD simulations of the lens (Figs. 2.7(b)-(d)) showgradual disappearance of discrete lobes in the image region. This suggestthat the two objects become indiscernible and result in T = λ/3 as theminimum resolution, which is consistent with the EPC estimations. On theother hand, the left-handed modes in the Veselago lens stretch up to arounddouble the bandwidth of Pendry’s silver lens, between 3k0 to 4k0. This es-timation is supported by FDFD simulations (Figs. 2.8(b)-(d)) which showthat the Veselago lens’s minimum resolvable feature is about twice smallerthan that for Pendry’s silver lens (TV L = λ/6). The arguable results of fi-nite resolution for Veselago lens, which is theoretically shown to have perfectresolution, can be attributed to the use of a lossy, two-dimensional objectin the simulation. These structural parameters are adequate to violate therequired subtle conditions for perfect imaging.2.6.3 Canalization in Multi-Layered StructuresMetal-dielectric multi-layer systems can work as canal for light where thelight can pass across their extent with little to no diffraction as shown by thesimulations in the literature [6, 166]. Due to the complexity of multi-layersystems, which can be made of 40 alternating layers, their electromagneticproperties are commonly described by an effective permittivity tensor ofanisotropic form which is modeled based on the effective medium theory(EMT). We nominate the configuration proposed in Ref. [6] as a case study,where the layered system consists of 20 repetitions of a bi-layer unit cellcomposed of a 7.2-nm-thick silver layer ( = −15) and a 7.8-nm-thick siliconlayer ( = 14), at the wavelength λ0 = 600 nm. In Figure 2.9(a), we show theband diagram or the EPCs of this configuration derived from the internalelectric and magnetic fields besides the volumetric averaging of the localpermittivity using EMT which basically extracts an effective refractive indexfor the metamterial configuration and yields a single horizontal line locatedat kz ' 0.The band diagram consists of flat elliptical contours which are roughlysymmetric about kz ' 0, with two distinguished bands of horizontal lines,one centered at kz ' k0 (forward- propagating mode) and another centeredat kz ' −k0 (backward-propagating mode). Since the contours remain flatbeyond the free-space cutoff spatial frequency, an incident plane wave at anyangle of incidence can excite modes that pass energy across layers’ extentin either forward or backward direction. Considering the EMT method, theplotted EMT line show that EMT can precisely predict the dominant flat462.6. Band Diagram Analysis of Layered Metamaterialshorizontal contours of EPC, particularly those ones below the free-spacecutoff frequency, but it shows to be unsuccessful to predict their locationsin k-space.Figure 2.9: Band diagram for the metal-dielectric multi-layer systems stud-ied in Ref. [6] for the case of TM-polarized plane-wave (λ0 = 600 nm) illumi-nation from a high-index dielectric (n = 5) prism. The blue line depicts thesimplified EPC predicted from effective medium theory (namely, Maxwell-Garnett theory). FDFD simulations of the multi-layers imaging a test objectwhen the two point-like features of the object are spaced by (b) λ0/8, (c)λ0/9, and (d) λ0/10. [8].This reveals the limitations associated with EMT compared to the rigor-ously derived EPC values. The FDFD simulations shown in Figures 2.9(b)-(d) visually illustrate the ability of the examined multi-layer system to chan-472.6. Band Diagram Analysis of Layered Metamaterialsnel light straight through the medium even with light streams spaced λ/10apart from each other. The high resolving power observed in the simula-tions is due to employing the ideal lossless material parameters originallyused in Ref. [6]. In case of simulating light propagation through materialswith realistic losses, the resulting resolving power is expected to be reduced.2.6.4 Far-Field Flat Lens ImagingFor better understanding of the electromagnetic field interactions thatenable far-field imaging (flat lensing) in a recently presented layered sys-tem [7], we apply the introduced EPC as the fundamental field analysistechnique. The reported far-field flat lens is designed to work in free-spaceand it is made of three repetitions of a five-layer unit cell with the layersequence Ag (33 nm) - TiO2 (28 nm) - Ag (30 nm) - TiO2 (28 nm) - Ag(33 nm). To understand the external optical properties of this structure bysimple geometrical optics, it has been modeled as a homogeneous, isotropic,negative-index medium. This model works for describing the structure’s ex-ternal optical properties, but does not provide insight into the dynamic be-haviour of the internal fields. This insight can be gained by analyzing EPCsof the internal fields. The EPCs of the structure beside the derived band di-agrams using two homogenization methods (S-parameter method [167] andFloquet-Bloch modes [125]) are shown in Figure 2.10. The EPC has a promi-nent upward concavity band feature similar to the observed band curvaturein the EPC of the negative-index slab in Figure 2.5(b). Such upward EPCcurvature in the medium opposes the downward wavefront curvature in free-space and forces the exiting wavefront to re-form into a real image in thetransmitted region. Furthermore, the associated internal fields’ group veloc-ity is directed towards the normal kx = 0 line, a feature that is importantfor collecting and concentrating light to form a real image. Consistent withpast experiments, the imaging capability of this structure is fundamentallylimited to the free-space diffraction limit due to the absence of the EPCupward concavity feature beyond the free-space cutoff.S-parameter method and the Floquet-Bloch theorem are two of thewidely used homogenization methods for modeling metamaterials. Theyyield an infinite number of potential solutions in k space (Fig. 2.10(b)), butthe position and concavity of the dominant EPC band can be accuratelydescribed by only one of the solutions. It can be noticed that the bestmatching solution from the S-parameter method is at the m = 2 branch andthe best one from the Floquet-Bloch theorem is at the m = 1 branch. Inthis section, we show that EPCs are useful for linking internal and external482.6. Band Diagram Analysis of Layered Metamaterialslight behaviors of layered structures as well as determining the best effec-tive medium model [168] through recognizing the one that most accuratelydescribes EPC features.Figure 2.10: (a) EPC for the flat lens structure shown in Ref. [7] to becapable of far-field imaging in the UV. The EPC is derived for TM-polarizedplane-wave (λ0 = 365 nm) illumination from a dielectric (n = 5) prism. Theinset depicts the geometrical configuration of the flat lens. (b) shows thesimplified EPCs derived using two common homogenization techniques: S-parameter method (blue solid) and Floquet-Bloch modes (red dashed). [8].2.6.5 Far-Field Flat Lens with Less MetalUsing the EPC concept and considering the absorption losses, we manageto engineer and design a metal-dielectric layered system that is capable offar-field imaging with minimal use of metal. We begin with a tri-layer unitcell template composed of a single TiO2 layer sandwiched by two silverlayers of identical thickness. While using the thinnest silver layers possible,we design a far-field imaging layered system with the essential phase and492.6. Band Diagram Analysis of Layered Metamaterialspower flow key features where the associated EPC should consist of a singledominant band with an upward concavity.Figure 2.11: (a) EPC of a new layered flat lens structure that is capableof far-field imaging, yet possesses about half the metallic fill fraction ofthe flat lens presented in Ref. [7]. The proposed structure consists of 8repetitions of a unit cell with the layer sequence Ag (7.5 nm) - TiO2 (25nm) - Ag (7.5 nm). The EPC is derived assuming TM-polarized plane-wave(λ0 = 330 nm) illumination from a dielectric (n = 2) prism. (b) shows anFDFD simulation of the proposed flat lens imaging a test object placed onits surface. Tapering of the magnetic energy density in the image region ata location spaced about a wavelength from the exit surface confirms thatthe structure is capable of forming real images in the far field. [8]Optimization of the layered system according to the specified designobjective and constraint yields an optimal design consisting of 8 repetitionsof the seed tri-layer unit cell with the sequence and thicknesses of Ag (7.5502.7. Summarynm) - TiO2 (25 nm) - Ag (7.5 nm). The proposed far-field flat lens designis operational at the UV wavelength of λ0 = 330 nm. Using less metals,we fulfill the set design constraint where the metal fill fraction of the newdesign (0.38) is about half the metal fill fraction of the structure discussedin the previous section (0.63). The new flat lens design achieves a prominentupward concavity EPC band feature as shown in Figure 2.11(a), the requiredEPC feature for far-field imaging. The FDFD simulations of this system inFigure 2.11(b) confirm its far-field imaging performance where a real imageof the apertures, though not distinct, is formed about a wavelength apartfrom the exit surface. The introduced less metal far-field flat lens possessesless internal losses due to lower metallic content and this yields a boostin light transmission. Therefore, this flat lens design can be consideredas a small step towards more practical flat lenses that work for real-worldapplications such as UV microscopy.2.7 SummaryConsidering a general layered system configuration with practical con-strains of loss and finite extent, we developed a new representation of theelectromagnetic field solutions in the Fourier domain. We then used the newfield representation to derive and plot the band diagrams. The developedFourier domain solution is a compact product of three terms where eachterm is dependent on a distinctive physical parameter of the layered system.There are two main contributions in this work. The first is the manifesta-tion of the Floquet-Bloch modes as one of the explicit terms of the Fourierdomain solution using only Maxwell’s equation and without invoking theFloquet-Bloch theorem. This work shows the functionality of representingthe electromagnetic field solutions of a layered system in Fourier domain toanalyze its complex electromagnetic properties. The promising conclusionsof this work suggest that the Fourier domain analysis method can be ap-plied to analyze the electromagnetic properties of more complex geometries.The second is the introduction of band diagrams as useful tool for designingpractical layered systems as well as validating conventional homogenizationmethods. We have used EPCs to design a layered system that mimic theleft-handed electromagnetic response, an anisotropic metamaterial that iscapable of canalization, and a far-field highly transmissive flat lens whichincorporates the minimum amount of metal.51Chapter 3Flat Lens Criterion bySmall-angle PhaseIn this chapter, we introduce a quantitative method based on the classicalray optics to analyze the imaging capability of any ultra-thin planar slabmade of homogeneous isotropic layers. Unlike the band diagram methodproposed in the previous chapter, this method use the small-angle phasebehavior as a criterion to predict the imaging capability of any planar slaband to estimate the image plane location.Here, we only consider the small-angle phase behavior and neglect thelarge angle and evanescent plane wave components to develop a simple,general and instant flat lens metric that can provide acceptable but not veryaccurate estimations before conducting any extensive analysis. To validatethis method, we applied the flat lens criterion to some of the already studiedstructures of near-field and far-field flat-lenses. Using this criterion, wederive approximate analytical expressions to show when a single layer or amulti-layer system can act as a flat lens. We ultimately demonstrate howthe introduced flat lens criterion is practical by effectively designing threedistinct flat lenses with novel functionalities.3.1 Flat Lens CriterionThe imaging response of a flat lens can be described by the classical rayoptics using the concept of point spread function (PSF) and paraxial ap-proximation. Flat lens analysis based on the PSF amplitude is a standardapproach [9, 28, 55, 58]. For a complete analysis, we also analyze the PSFphase and we underline its potential as a flat lens indicator. Considering aplanar medium with the same geometry of the layered system illustrated inchapter 2 (Fig. 2.2), we place a point source in the object region directlyon the entrance face of the medium at z = −d. We assume that the objectregion (z < −d) and the image region (z > 0) are composed of generallydissimilar, isotropic and homogeneous media. The transmitted light source523.1. Flat Lens Criterionin the image region can be decomposed in the xz plane as a uniform plane-wave spectrum parameterized by the wave vector ~kt = kt,xxˆ+ kt,z zˆ. Alongthe z axis, we only need to evaluate the phase of the plane-wave components( Φz) due to translational symmetry. The phase of the plane-wave compo-nents exiting the slab Φz(z = 0), referenced to a common initial value atthe source, can trace out the wavefront and map out the loci of equi-phasepoints in the image region.At an arbitrary point z0, the phase of the PSF can be written as Φz(z0) =Φz(z = 0) + kt,zz0. In case of ideal image formation at z0, the phase is in-variant for all plane-wave components at z0 based on Fermat’s principle,∂ Φz(z0)/∂ kt,z = 0. The image formation will suffer from spherical abber-ations and we will not have an ideal image since the monochromatic lightphase is not invariant for all plane-wave components at any point.Applying the concept of phase invariance on the small-angle plane-wavecomponents ( kt,x << kt,z ' kt), the paraxial image location in terms ofthe PSF phase can be defined bys = z0 ≡ −∂ Φz(z = 0)∂ kt,z∣∣∣kt,z= kt, (3.1)where image plane location s is the maximum working distance of the flatlens and it can be positive for a real image in the image region or negativefor a virtual image in the slab or object regions. Hence, a planar mediumcan be defined as flat lens if it is capable of producing a real paraxial image(s > 0) of a point source located on its entrance face. The flat lens criterionis defined for near zero numerical aperture (NA), the range of angles overwhich the system can accept light. Usually, this criterion has been used todescribe conventional optical systems. However, we show that this conditionis applicable to any flat lens since it depends only on the phase profile atthe slab exit.Considering the PSF interpretation as a transfer function relating theiso-planatic field quantities along object and image planes perpendicular tothe optical axis, the amplitude of the PSF is commonly presented versusthe normalized lateral wave vector kt,x/ kt [169]. On the other hand, thePSF phase is associated with spherical wavefront curvature and sphericalaberrations mapping onto high order terms which are difficult to distinguishon a graph and eventually makes the PSF phase inconvenient to be displayedversus kt,x/ kt. Knowing that Φz(z = 0) is dependant on kt,z in Eq. (3.1),here we suggest to plot the PSF phase as a function of qt = 1 − kt,z/ kt =1− cos θt, where θt is the angle of the transmitted light at the exit surface.533.1. Flat Lens CriterionOver the range 0 ≤ qt ≤ 1, qt describes propagating waves. Plotting thePSF phase versus qt, several observations can be evolved (Fig. 3.1). Positiveslope will be an indication of flat lens behavior, the slope at the y-intercept(∂ Φz(z = 0)/∂qt|qt=0) can be counted as an estimation for the paraxialimage location, and the departure from linearity will imply the occurrenceof spherical aberrations.A AqFz(z=0)A qq =1-cost tqFz(z=0)a) Divergent exiting wavefronts:virtual image formationb) Convergent exiting wavefronts:real image formation(z=0) q < 0F /z t(z=0) q 0F /z t >xz 0 1 0 10 0q =1-cost tqFigure 3.1: Curvature of the wavefronts exiting a planar slab for the cases of(a) virtual and (b) real image formation. By plotting the phase Φz(z = 0)versus qt = 1 − cos θt, the possibility of a flat lens can be determined byinspection from a positive slope. [4]In this work, we have applied the flat lens criterion to analyze the imagingof a point source located on the entrance face of the slab. Such analysis isextendable to volumetric objects which can be treated as collections of pointsources along the lateral and longitudinal directions. As we scan over thelateral direction along the translational symmetry of the slab, we will havea shift in the image point towards the same lateral direction with no effecton the location of the image plane. Contrary, the imaging capability of theslab will distinctly change as we scan over the longitudinal direction alongthe normal line to the slab. As long as the depth of an object is less than thecharacterized image plane location s of a flat lens, a real image of the objectplaced perfectly against the entrance face of the flat lens can be created.543.2. Methodology3.2 MethodologyValidating the proposed flat lens criterion as a general criterion to predictflat lens behavior, we first visit some past flat lens implementations operatingin both the near- and far-fields and calculate the paraxial image location.Then, we use this criterion as a metric to design new flat lens configurations.The suggested flat lens criterion predict the paraxial image plane locationusing the PSF derived from the transfer matrix method [170, 171].For past and new flat lens designs, the predictions of the image plane lo-cation have been compared with the full-wave electromagnetic simulations.These simulations display the solutions of Maxwell’s equations over a two-dimensional grid with pixel size of at most 1 nm using the finite-differencefrequency-domain (FDFD) technique. The utilized light source in simula-tions is a TM or TE polarized light source with normal incidence plane wavepropagating along the +z direction. Based on the use of TM or TE polar-ization, the simulation results are displayed in terms of the distribution ofmagnetic or electric energy density, respectively.To conduct near diffraction limit imaging simulations, an opaque chromiummasking layer with two λ0/10-wide openings spaced by λ0/2.5, which isslightly below the diffraction limit, is placed between the flat lens and theplane wave source. Such simulations reveal further information about theflat lens resolution capabilities near the diffraction limit which is one ofthe most appealing claimed features of metal-dielectric layered flat lenses.The used permittivity values in the transfer matrix method calculationsand FDFD simulations are from [17] for silver and gold and from [161] forchromium. The used permittivity values in the calculations of past publishedflat lens implementations are taken from the original works.3.3 Comparison with Past Flat Lens Results3.3.1 Pendry’s Silver Slab LensWe use the single silver layer flat lens configuration as the first testplatform for investigating the small-angle phase criterion. It has been shownthat a thin silver layer (Pendry’s silver slab) can work as a flat lens [27]mimicking the electromagnetic properties of a perfect negative refractiveindex slab (Veselago lens). Due to its evanescent wave amplification, thissingle layer flat lens was suggested as a feasible tool to image beyond thediffraction limit. Such super-resolution behavior is not directly predictableusing the small-angle phase which can only predict the existence and location553.3. Comparison with Past Flat Lens Resultsof the paraxial image. Considering Pendry’s setup, a 40-nm-thick silver layerwith complex permittivity of = −1.0+0.4i at wavelength of λ0 = 356.3 nmis placed 20 nm apart from an object. Applying the flat lens criterion, theimage plane location is predicted to be at s = 36 nm, which is comparableto the reported image location of 20 nm in [27]. This disparity could be dueto accounting for large angle plane-wave components and assuming idealnegative refraction in the reported image location by Pendry.AirCrAg 40nmAir36nmMagnetic energy densityMagnetic energy densitya) b)c) d)0nmCr Cr CrAir AirAirCr CrAirCrAir 60nmAirAirCr Cr20nmFigure 3.2: (a) PSF phase for Pendry’s silver slab lens consisting of a 40-nm-thick Ag layer with a permittivity of = −1.0 + 0.4i at a wavelengthof λ0 = 356.3 nm, where the object is 20 nm away from the entrance of theslab. The phase has been calculated at the exit of the slab (z = 0 nm) andthe paraxial image location has been predicted at z = 36 nm. (b) FDFD-simulated profile of the magnetic energy density at the image plane z = 36nm for the cases where the near diffraction limit spaced objects are imagedwithout (blue) and with (red) the silver slab lens. Simulated time-averagedmagnetic energy density distributions of the illuminated object are shown(c) without and (d) with the 40-nm-thick Ag layer. The yellow dashed linesin (d) show the positions where the PSF phase profiles have been calculatedin (a). [4]563.3. Comparison with Past Flat Lens ResultsThe phases of the transmitted light versus qt at the exit interface of thesilver layer, and at the paraxial image location are shown in Figure 3.2(a).The former phase profile at the exit interface is a clear indication of theexistence of a real paraxial image due to its positive slope. The secondphase profile at the image location of s = 36 nm shows to be consistent withthe Fermat’s principle of real image formation where the phase profile isflat at small angles. To investigate if the silver slab is capable of imagingslightly below the diffraction limit λ/2, a comparative simulation of themagnetic energy density at the image location (36 nm) is conducted usingan almost diffraction limit spaced object (λ/2.5) with and without the silverslab (Fig. 3.2(b)).For further highlight on imaging with resolution slightly below the diffrac-tion limit using a thin silver layer, we show the spatial energy density distri-butions for the two cases of with and without the silver slab in Figs. 3.2(c)and 3.2(d) respectively. Although it is difficult to firmly locate the imageplane location using the energy density simulations due to the decay of theenergy density near the exit interface, the energy density profile at 36 nm(Fig. 3.2(b)) is shown to be consistent with the small-angle phase prediction.3.3.2 Near-field Imaging with Silver LayersImaging with flat lenses is usually confined to near-field due to the dif-ficulties associated with satisfying the physical conditions for real imageformation at far-field using a flat slab. Here, we discuss the correlation be-tween the predicted real paraxial image and the reported near-field super-resolution image in some of the past UV imaging experiments on photoresistusing a silver layer.Using the proposed flat lens criterion, we reasonably explain some of theobserved near-field imaging behaviours of the past flat lens implementationsby analyzing their paraxial image locations (Fig. 3.3). Imaging beyond thediffraction limit or super-resolution imaging using very thin silver layersof thicknesses down to 36 nm has been shown in several experiments [9,10, 50, 54]. At UV wavelengths, the slope of the phase profiles of the pastimplemented flat lenses using a 36-nm-thick silver layer [9] and a 50-nm-thicksilver layer [10] are both positive (Fig. 3.3(a)). The corresponding paraxialimage locations have been predicted to be at s = 10 nm and s = 22 nm,respectively. Comparing the imaging capability of a metal layer versus adielectric layer of the same thickness, we examined the third [11, 53] andthe forth [12] flat lens configurations illustrated in Figure 3.3.573.3. Comparison with Past Flat Lens ResultsFigure 3.3: Flat lens criterion applied to past implementations. (a) PSFphase at the exit surface of lenses based on the 36-nm-thick silver layerstudied in Fang et al. [9], the 50-nm-thick silver layer studied in Melvilleet al. [10], and the 120-nm-thick silver layer studied by Melville et al. in[11, 12], along with the control used in [12] of a 120-nm-thick PMMA layer.(b) PSF phase at the exit surface of lenses based on metal-dielectric multi-layers studied by Belov et al. [6]. The inset in (b) shows a magnified view ofthe data near normal incidence. (c) Paraxial image location as a function ofunit cell repetition for the periodic metal-dielectric layered system studiedby Kotynski et al. [13]. (d) PSF phase at the exit surface of the geometrystudied by Xu et al. [7] where the flat lens composed of metal and dielectriclayers. [4]583.3. Comparison with Past Flat Lens ResultsAt a wavelength of λ0 = 341 nm, the slope of the phase profile at the exitface of the 120-nm-thick silver layer sandwiched between two 60-nm-thickPMMA layers is negative, which is the sign of having a virtual paraxial image(Fig. 3.3(a)). As predicted by the slope of the phase, the flat lens criterioncalculations located the image at s = −40 nm inside the flat lens region. Onthe other hand, the single 120-nm-thick PMMA layer configuration formsa virtual paraxial image at s = −150 nm. This insists that silver layer hassignificantly improved the imaging performance bringing the image planelocation closer to the image region (s = −40 nm). The reported experimentalresults in the literature are consistent with our real and virtual paraxialimage predictions.The discussed comparative analysis results showed that the real paraxialimage formation using a thin film is fully dependant on the film thickness. Toform a real image, the film thickness should be sufficiently smaller than thewavelength as stated before by Pendry. It can be also concluded that havinga real paraxial image within the image-capturing region is a prerequisitefor near diffraction limit resolution imaging. However, understanding andaffirming this postulation requires further work. In the next sections, wewill analytically discuss the film thickness effect on flat lens imaging.3.3.3 Anisotropic Metamaterial LensesThe periodic metal-dielectric bi-layer implementations of flat lens havebeen proposed to mimic the properties of anisotropic metamaterial for thepurpose of imaging as theorized in previous works [23, 49]. A metal-dielectricbi-layer system can be characterized by a flat wave vector diagram anddescribed by a highly anisotropic permittivity tensor, if the permittivityand thickness values of the metal ( M , dM ) and the dielectric ( D, dD)layers in the bi-layer composition satisfies the relations M/ D = − dM/ dDand M + D = 1 [6, 57, 63]. The characterized flat wave vector relationin bi-layer metal-dielectric flat lenses enables the front to back direct imageprojection across the slab.Considering three different multi-layer systems designed as anisotropicmetamaterial lenses [6, 13], we examine the consistency of the reported imag-ing properties with the predicted paraxial image locations (Fig. 3.3(b)). Atthe wavelength of λ0 = 600 nm, the calculated PSF phase of two lossless20-unit-cell systems presented in [6] have been analyzed and yield the re-spective paraxial image locations of s = 0.2 nm and s = −3 nm, where thecharacterized unit cell parameters of the first layered system are M = −1,D = 2, and dM/ dD = 1/2 and the second one are M = −14, D = 15,593.4. Validation of Flat Lens Criterion by Full-Wave Simulationsand dM/ dD = 14/15. The predicted paraxial image locations (s = 0.2 nm,s = −3 nm) are quite close to the exit interface of the slab as reported in thereferred work [6]. At the wavelength of λ0 = 422 nm, a different study [13]provided the detailed simulation data for a lossy multi-layer system with unitcell parameters of M = −5.637 + 0.214i, D = 2.62 and dM/ dD = 10/12.As the number of unit cell repetitions N increases from 1 to 50, the size ofthe spot light expands and shrinks at a fixed plane near the output face.Plotting the spot size versus N results in a consistently oscillating profilewith 6 evenly spaced local minima. We obtain identical oscillatory depen-dence on N by calculating the paraxial image locations using the PSF phase(Fig. 3.3(c)). This implies that the observed spot size variation in [13] ispartially due to image focusing and de-focusing at a fixed plane.3.3.4 Negative-Index Metamaterial LensEmulating the perfect negative index flat lens proposed by Veselago [19],a recent pioneer work introduced a metal-dielectric multi-layer flat lens as anisotropic negative index metamaterial [7]. Operating at the wavelength ofλ0 = 364 nm, a far-field image at 360 nm apart from the flat lens exit inter-face has been reported [7] which is found to be comparable to the predictedparaxial image location of s = 370 based on the PSF phase calculations (Fig.3.3(d)). The deviations from unity for the phase profile at the paraxial imagelocation can be attributed to the wavefront aberrations. The extrapolationof these deviations demonstrates the occurrence of wavefront aberrationslower than λ0/4 up to about unity NA.3.4 Validation of Flat Lens Criterion byFull-Wave SimulationsWe use the electromagnetic full-wave FDFD simulations of past flat lensconfigurations to validate the introduced flat lens criterion. This comparisonanalysis shows moderate agreement between the simulated image locationsand the paraxial image locations across near-field and eminent consistencyacross far-field regimes. We revisit four of the considered flat lenses inthe previous sections: 1) the 120-nm-thick silver layer presented in [11,12], 2) the 50-nm-thick silver layer presented in [10], 3) the metal-dielectricanisotropic metamaterial lens reported in [6], and 4) the metal-dielectricnegative-index metamaterial lens reported in [7]. We then conduct the full-wave simulation for each flat lens system.603.4. Validation of Flat Lens Criterion by Full-Wave SimulationsFigure 3.4: Comparison of paraxial image locations predicted by PSFphase and numerical simulations. FDFD-calculated time-averaged energydensity distributions for flat lenses consisting of (a) a 120-nm-thick silverlayer studied in [11, 12], (b) a 50-nm-thick silver layer studied in [10], (c)metal-dielectric multi-layers studied in [6], and (d) metal-dielectric multi-layers studied in [7]. In all cases, we use near diffraction limit spaced objectsconsisting of two, λ0/10-wide openings spaced λ0/2.5 apart in an opaquemask that is illuminated by a TM-polarized plane wave. The yellow dashedline in each panel shows the corresponding paraxial image location calculatedfrom the slope of the output phase. [4]613.5. Flat Lens Condition for a Single LayerThe simulations and the paraxial image locations, depicted by dashedyellow lines, are shown in Figure 3.4. Except the first flat lens configurationwith the 120-nm-thick silver layer where the simulation of the energy densitydistribution shows divergent behaviour in the image region (Fig. 3.4(a)),the simulations of the other three flat lens systems show convergent energydensity distributions at certain spots in the image region (Fig. 3.4(b), (c),(d)). The narrowest plane in the simulation of the 120-nm-thick silver layerflat lens is located close to the predicted virtual paraxial image locations = −40 nm which implies that this system is not capable of forming animage of the object in the image region (Fig. 3.4(a)).On the other hand, imaging by lenses made of 50-nm-thick silver layerand anisotropic metamaterial is quite evident to provide imaging resolutionsslightly below the diffraction limit as it can be observed from the simula-tions. Two high energy density spots appear in the simulations near lenses’output faces at plane locations roughly match the estimated real paraxialimage locations calculated by the phase of the PSF [Figs. 3.4(b) and 3.4(c)].Even though the predicted paraxial image location is determined withoutaccounting for the evanescent plane-wave components which are believed tobe the main contributor in super-resolution imaging, surprisingly we foundreasonable agreement between our predictions and the exact full-wave sim-ulations.Finally, a single high energy density spot well-separated from the outputface is observable in the full-wave simulation of the negative-index metama-terial lens. This is a clear indication of having a diffraction-limited image ata position that is again quite consistent with the predicted real paraxial im-age location (Fig. 3.4(d)). Unlike the second and third considered lenses, thenegative-index metamaterial lens cannot resolve the defined near diffractionlimit spaced objects in the image region.3.5 Flat Lens Condition for a Single LayerTo precisely determine when a single homogeneous layer can generatea real paraxial image based on the expression (3.1), we develop compactanalytical expressions that can work as determinant conditions for imagingwith a flat slab. First, we define the most generic configuration where ahomogeneous, non-magnetic, isotropic layer of thickness d, characterized bya permittivity of = ′ + i ′′ and a permeability of µ = 1, is placed in free-space. For a point source with a plane-wave spectrum parameterized by thewave vector ~ko in the free-space object region where ~ko = ko,xxˆ + ko,z zˆ,623.5. Flat Lens Condition for a Single Layerthe transmitted plane waves through the single layer are modulated by atransmission coefficient t wheret =4 p(1 + p)2e−ikzd − (1− p)2eikzd . (3.2)The parameter p is defined as kz/( ko,z) for TM polarization and as kz/ ko,zfor TE polarization, kz is the propagation constant within the layer wherekz = k0√n2 − sin θ2, k0 is the magnitude of the free-space wave vector, nis the layer refractive index where n =√, and θ is the angle of incidence.The derived transmission coefficient t accounts for all included losses dueto multiple reflection and propagation within the slab. The phase of t isequivalent to the phase of the PSF at z = 0 ( Φz(z = 0)) from which theparaxial image location can be calculated.3.5.1 Flat Lens for TM PolarizationFor TM polarization and under the thin-film and paraxial approxima-tions, a lossy layer similar to Pendry’s silver slab yields a first order expres-sion of k0d as∂ Φz,TM (z = 0)∂qt∣∣∣qt=0.= −k0d(12( ′ − 1) + ′||2), (3.3)which must be positive for real image formation. To keep the slope of thephase positive, a purely permittivity dependant condition must be satisfied( ′ < ||2/(||2 + 2)). Such condition can be fulfilled if the single layer ismade of metals.The derived expression in Eq. (3.3) predicts a paraxial image location ofs = 1.86d for Pendry’s silver slab with = −1 + 0.4i [27]. This prediction isalmost double the theoretically expected image location for a Veselago lensof equivalent thickness s = d [19]. Since the proposed method for predictionof the paraxial image location using Eq. (3.3) is a function of the flat lensthickness d, we investigate the dependence of the paraxial image location son d by calculating s of a point source placed directly on the entrance ofan ideal Veselago lens and Pendry’s silver slab lens for different thicknesses.Figure 3.5 shows the thickness dependence of the paraxial image location.Although it has been assumed by Pendry that Veselago lens and thin silverslabs have identical paraxial image locations under electrostatic conditions,in contrary Figure 3.5 demonstrates the differences between the paraxialimage locations of Veselago lens and Pendry’s silver slab.633.5. Flat Lens Condition for a Single LayerFigure 3.5: Paraxial image location versus thickness for an illuminated ob-ject located at the entrance of the ideal Veselago lens (red) and Pendry’ssilver slab lens (blue) when illuminated at the wavelength of λ0 = 356.3 nm.For the ideal Veselago lens, the image location is equivalent to the slabthickness, s = d. [4]Ignoring the flat slab losses based on the thin-film and paraxial approx-imations, a more accurate flat lens criterion with higher order terms can bederived. For TM polarization, the flat lens criterion up to the third order ofk0d is given by∂ Φz,TM (z = 0)∂qt∣∣∣qt=0.=− k0d ′2 − ′ + 22 ′+ (k0d)3 ( ′ − 1)2(3 ′2 + 5 ′ + 6)24 ′,(3.4)which must be positive for real image formation. If we only considerthe first term in Eq. (3.4), negative permittivity of the slab ( ′ < 0) will bethe only prerequisite condition for real image formation, which is possibleusing metals such as silver at UV frequencies. However, considering thehigher order term with opposite sign in Eq. (3.4) denotes that the thicknessincrease of a negative permittivity layer generally oppose the necessary phasecondition for making a flat lens. This correlation between the layer thickness643.5. Flat Lens Condition for a Single Layerand the real paraxial image formation explains why the thick silver layer in[11, 53] did not achieve super-resolution imaging, whereas the thinner silverlayers in later attempts did [10, 50, 54].3.5.2 Flat Lens for TE PolarizationHaving the analytical expression of the transmission coefficient t for TEpolarization enables the derivation of another general flat lens condition fora TE-polarized wave. Under the thin-film and paraxial approximations, theflat lens criterion of a lossless layer for TE polarization is given by∂ Φz,TE(z = 0)∂qt∣∣∣qt=0.= −k0d3− ′2− (k0d)3 ( ′ − 1)2(3 ′ − 1)24. (3.5)If we only consider the first term of Eq. (3.5), imaging with a single flatlayer would be possible when ′ > 3, a condition that can be fulfilled bymany types of glasses and semiconductors. As TM polarization, the higherorder term for TE polarization (Eq. 3.5) plays a crucial role when the layeris not sufficiently thin.To investigate the new flat lens condition for TE polarization, we studya high-index dielectric (n = 4) layer with the nominal thickness of 50 nmplaced in air and illuminated by a TE-polarized wave of free-space wave-length λ0 = 365 nm. A real paraxial image location of s = 2 nm was pre-dicted based on the PSF phase analysis. Figure 3.6(a) shows the phase atthe image plane location.We then conduct two distinct FDFD simulations with and without thedielectric layer for imaging near diffraction limit spaced objects. An imageof these objects at the paraxial image location is formed by adding thedielectric layer as shown in Figure 3.6(b), where the electric energy densityprofile for both configurations at the image plane location are plotted. Thefull-wave FDFD simulations in Figs. 3.6(c) and (d) further demonstrate thecapability of a dielectric layer to form a near-field image of near diffractionlimit spaced objects in case of TE polarization. The yellow dashed linein Figure 3.6(d) shows the consistency between simulation and small-anglephase prediction.653.6. Flat Lens Condition for Multi-layersAirElectr ic energy densityc)s = 2nm50nmAirn = 4Electric energy densityd)a) b)Air50nmCrn = 1AirCr Cr AirCr AirCr CrCr Cr CrAir AirFigure 3.6: Flat lens for TE polarization based on a 50- nm-thick losslessdielectric (n = 4) layer immersed in air and illuminated at a wavelengthλ0 = 365 nm. (a) PSF phase at the paraxial image location s = 2 nm. (b)FDFD-simulated profile of the electric energy density at the paraxial imagelocation for the cases where the object is imaged without (blue) and with(red) the dielectric slab. Simulated time-averaged electric energy densitydistributions of the illuminated object are shown (c) without and (d) withthe 50-nm-thick dielectric layer. The yellow dashed line in each panel showsthe paraxial image location calculated by the PSF phase. [4]3.6 Flat Lens Condition for Multi-layersFor a more general case of multi-layer system, we derive the conditionfor flat lens imaging and we show that it is identical to the used conditionfor making bi-layer anisotropic metamaterial lenses. The flat lens criterionbased on Eq. (3.3) for a multi-layer system using a TM-polarized wave isapproximately∂ Φz,TM (z = 0)∂qt∣∣∣qt=0.= −k0∑idi(12( ′i − 1) +′i| i|2), (3.6)663.7. Broadband Flat Lens Designed by Small-angle Phasewhere di and i = ′i+ ′′i are respectively the thickness and the permittivityof the ith layer.We now consider the simplest multilayer system with a single bi-layerunit cell of metal and dielectric with dielectric parameters of dD and D,and metal parameters of dM and M . If the metal permittivity is complexwhere M = ′M + ′′M , the required dielectric permittivity condition forprojecting a real image across the unit cell is given byD =12− dM γMdD±√(12− dM γMdD)2− 2 , (3.7)where γM= ( ′M − 1)/2 + ′M/| M |2. Eq. (3.7) can be simplified toM/ D = − dM/ dD and M + D = 1 when the metal permittivity is realand large ( ′M >> 1) and the thicknesses of the metal and dielectric layersare comparable ( dM ≈ dD). These simplified conditions are identical to thedefined constraints based on the effective medium theory for designing a flatlens made of metamaterial with an anisotropic permittivity tensor [6].3.7 Broadband Flat Lens Designed bySmall-angle PhaseWe systematically design a bi-layer system that can consistently projectthe image at particular location using a TM-ploarized wave with broadbandwavelength spectrum of a large portion of the ultraviolet-visible. The designis based on using the paraxial image location in Eq. (3.1) as a merit functionfor an optimization routine.Silver-gold bi-layer system is found to be a practical result of this designprocess, where the thicknesses are 28 nm and 29 nm respectively. Over alarge spectral range (365 nm < λ0 < 455 nm), the paraxial image locationof the bi-layer flat lens stays comparatively stable between 35 nm and 37 nm(Fig. 3.7 (a)). The amplitude and phase of the PSF at the paraxial image lo-cation of s = 37 nm and for the wavelengths of λ0 = 365 nm and λ0 = 455 nmare shown in Figure 3.7(b). The response modeling of the bi-layer systemat the lower and upper bounds of the considered wavelength range revealsthe analogy between the energy density concentrations near the predictedparaxial image locations of 37 nm as shown by the full-wave electromagneticsimulations in Figure 3.7(c) and (d). Unlike other near-field flat lenses, theformed near-field image by the bimetallic broadband lens cannot resolve thetwo openings of the near diffraction limit spaced objects. This loss of reso-673.8. Far-field Immersion Flat Lenslution can be attributed to the compounded light interference and metalliclosses due to the addition of the gold layer.l =365nmol =455nmoc) d)a)AirAirAgAu28nm29nmb)l =0365nml =0455nmMagnetic energy densityMagnetic energy densityCr Air57nms = 37nmAirAg/AuAirCr CrCr Air57nms = 37nmAirAg/AuAirCr CrFigure 3.7: Engineering a broadband flat lens. (a) Paraxial image locationover the ultraviolet-blue spectrum for a bi-layer flat lens consisting of a 28-nm-thick silver layer and a 29- nm-thick gold layer immersed in air. (b) PSFphase (red) and amplitude (blue) at the image plane location (s = 37 nm)of the bi-layer flat lens at the wavelengths of λ0 = 365 nm (solid lines) andλ0 = 455 nm (dashed lines). Time-averaged energy density distributions forthe bi-layer system under plane-wave illumination at (c) λ0 = 365 nm and(d) λ0 = 455 nm. The yellow dashed line in each panel shows the paraxialimage location calculated by PSF phase. [4]3.8 Far-field Immersion Flat LensOne of the most challenging tasks in designing flat lenses is boostingthe paraxial image location and projecting the image at far-field, which isessential for imaging three-dimensional objects . Here, we apply a simpletechnique to further extend the image location of the multi-layer far-fieldflat lens studied in [7] by increasing the dielectric permittivity of the image683.8. Far-field Immersion Flat Lensregion material.a)d) n = 1.5525nme) n = 2.0679nmc) n = 1.3463nmb)Flat lens criterionFDFD simulationFigure 3.8: Enhancing the image plane location of the multi-layered flatlens system previously studied in [7] by immersion of the image region in adielectric. (a) PSF phase (red) and amplitude (blue) at the paraxial imagelocation for the cases where the dielectric medium has refractive index n =1.0, 1.3, 1.5, and 2.0. (b) Paraxial image location versus the refractive indexof the dielectric medium predicted by PSF phase (blue line) and FDFDsimulations (red circles). (c), (d), and (e) show FDFD-calculated magneticenergy density distributions of the immersed flat lens system for n = 1.3,1.5, and 2.0, respectively. The yellow dashed lines in panels (c)-(e) show theparaxial image location calculated by PSF phase. [4]As the dielectric permittivity or refractive index increases, the predictedparaxial image location of the far-field flat lens system linearly escalates asshown in Figure 3.8(b). The full-wave simulations in Figures 3.8(c), (d) and693.9. Summary(e) visually demonstrate this effect where the separation distance betweenthe maximum energy density spot in the image region and the exit of the lensincreases as the dielectric permittivity of the image region is augmented. Thepositions of the maximum energy density spot obtained from simulations arequite consistent with the paraxial image locations predicted by the phaseof the PSF as shown in Figure 3.8(b). Although the dielectric immersionmethod provides a premium method to enhance the working distance of flatlenses, it increases abberations at the image plane location as shown by theplotted PSF phase and amplitude at the paraxial image location for variousrefractive index values (Fig. 3.8(a)).3.9 SummaryWe have promoted a general flat lens criterion based on the small-anglephase behaviour for flat lens structures that are composed of a single ormultiple layers of homogeneous isotropic media. This criterion showed to beconsistent with the far-field flat lens implementation studied in [7], and moreinterestingly the super resolution near-field flat lenses presented in [6, 10,11, 13, 50]. The Analytical expressions of the flat lens criterion for singleand multi-layer systems provide a single metric for predicting real imageformation. Designing a flat lens for TE polarization which is capable ofimaging slightly below the diffraction limit, a broadband flat lens that worksover part of the UV-visible spectrum, and an immersion flat lens with anadjustable far-field paraxial image location up to several wavelengths fromthe exit surface are the novel outcomes of the proposed flat lens criterion.70Chapter 4Transparency of Thin Layers:Light Transparency Boost byOpposite SusceptibilityCoatingIn this study, we examine optical light transmission through metal-dielectric bilayer systems. It has been shown before that dielectric coatingof metallic films can enhance light transmission through metals. However,here we show that in theory the transmission enhancement phenomenonin metal-dielectric systems is reciprocal, where the transparency of a di-electric layer can be increased by adding a very thin metallic layer. In ageneral sense, coating a thin base layer with another thin layer of oppositesusceptibility sign can make the base layer more transparent. To experi-mentally validate the proposed light transmission hypothesis, we measurethe transmitted light through dielectric-coated silver films and silver-coatedsilicon nitride membranes and we found that experimental measurementsare favorably comparable to the theoretical calculations. We particularlyshow that the optical transparency of a silicon nitride membrane can be en-hanced over a narrow-band of the visible spectrum by the addition of a thinsilver layer. This study can be considered as the first work that demonstratethe reciprocity concept with respect to light transmission enhancement inmetal-dielectric bilayer systems.4.1 TheoryConsidering the bi-layer configuration in Figure 4.1(a) and based on thestandard transfer matrix methods [42], the transmittance T of a normallyincident monochromatic plane wave from the left half-space onto the bilayerat frequency ω has the general form714.1. TheoryT = f(k, i, t, 1, 2, d1, d2), (4.1)where d1, 1, χ1, and n1 are respectively the thickness, the complex per-mittivity, the complex susceptibility, and the complex refractive index ofthe first layer or the base layer. In the case of oblique light incidence, lighttransmittance will be a function of the same parameters beside the angle oflight incidence and we will not get the maximum light transmission due tothe presence of surface waves.Similarly, the properties of the second layer or the coating layer aredenoted by d2, 2, χ2, and n2. The permittivity values of both layers areallowed to be complex where they have the general form of = ′+ i′′. Theleft and right half-spaces have the dissimilar real permittivity values of iand t, respectively, and the free-space wave number is defined by k = ω/cwhere ω is the angular wave frequency and c is the speed of light.In the limit where the thickness of the coating layer d2 goes to zero, asimplified condition for transmission enhancement can be derived from thetransmittance partial derivative with respect to d2. Assuming high figuresof merit for both base and coating layers where complex permittivities arepredominantly real such that∣∣∣∣ ′1′′1∣∣∣∣ 1 , and ∣∣∣∣ ′2′′2∣∣∣∣ 1 , the transmissionenhancement condition will be given bysgn(∂T∂d2|d2=0)≡ sgn(− (′1 − i)(′2 − t))> 0. (4.2)If the bi-layer is immersed in air (i = t = 1), the transmission enhance-ment condition can be simplified to a function of the real susceptibilities ofthe base and coating layers where χ1 χ2 < 0.This condition distinctly shows that independent of the layer orderingsufficiently thin bi-layers with opposite susceptibilities are more transparentthan the individual layers alone. To achieve the highest light transmission,the optimal coating layer thickness should satisfy the following equationtan(2ϕ2) =2n1n2 sin(2ϕ1)(′1 − 1)(′1 − ′2)(1 + ′1) + (1− ′1)(′1 + ′2) cos(2ϕ1), (4.3)where any permittivity that satisfies Eq. 4.2 is permissible to be used inEq. 4.3, and the angle parameters ϕ1 and ϕ2 are defined as ϕ1 = n1kd1 andϕ2 = n2kd2, respectively.724.1. TheoryFigure 4.1: (a) Ideal configuration of a bi-layer immersed in two half-spaces,illuminated at normal incidence from the left half-space. (b) Predictednormal-incidence transmittance at a wavelength of 650 nm through a bi-layer composed of a 50-nm-thick base layer of silicon nitride and a coatinglayer of silver of variable thickness. A positive derivative of the transmit-tance in the limit of zero coating layer thickness can be used as an indicatorof transmission enhancement. [14]From Eq. 4.2, the permittivities of the base and coating layers shouldsatisfy either of the following inequalities′1 < 1 < ′2 or ′1 > 1 > ′2 (4.4)The practical implementation of the first condition in (4.4) is the well-734.2. Methodologyexamined configuration of a metallic layer coated with a dielectric layer. Thesecond condition in (4.4) can be related to the configuration of a dielectriclayer coated with a metallic layer, which has not been investigated yet.Although the metal-coating dielectric layer configuration for the purposeof light transmission enhancement is counter-intuitive, here we show that itcan be described by a full solution of Maxwell’s equations. Assuming thatthe dielectric base layer is a 50-nm-thick silicon nitride and the metal coatinglayer is silver, we calculate and plot the percentage normal-incidence trans-mittance of the bi-layer system at a wavelength of 650 nm as the thicknessof the silver coating layer is increased from 0 nm to 30 nm (Fig. 4.1(b)).According to this illustration, a maximum transmission boost of 10% canoccur when the thickness of the silver layer is 6.1 nm.In these calculations, the silicon nitride’s optical constants are takenfrom [172]. Although the permittivity of very thin metals is affected by theelectron scattering at the surface, we used the complex optical constantsof bulk silver [17] in the illustrated calculations in Figure 4.1(b). For moreaccurate results, the optical constants of thin metal films should be measuredand used in future works. The optimal silver layer thickness of 6.1 nmobtained from Figure 4.1(b) is comparable to the optimal thickness of 6.4nm extracted from Eq. 4.3. The slight discrepancy is due to the neglect ofsilver layer losses in Eq.4.3.In general, when we have a very thin dielectric layer with the base layerthickness less than a quarter of a wavelength, the transmissivity of lightthrough the dielectric layer increases as we reduce the dielectric layer thick-ness. This work introduces another way for light transmission enhancementthrough thin positive susceptibility dielectric layers by addition of a negativesusceptibility (NS) layer. Pairing a thin dielectric layer with a NS layer canreduce the effective optical path length of the dielectric layer and resultingin increased transmission as if the layer thickness were reduced.4.2 MethodologyWe examine the optical transmission enhancements due to combinationsof metallic and dielectric layers by first fabricating thin film samples of dif-ferent materials based on their properties in the visible region. We preparedmetal, metal-oxide, and elemental semiconductors samples using the mag-netron sputter deposition (Angstrom Engineering Nexdep). The metal filmsare made of silver with target purity of 99.99%, the metal-oxide films aresputtered from a titanium dioxide target (99.9%), and the elemental semi-744.2. Methodologyconductors are deposited using the targets of undoped silicon, boron-dopedp-type silicon, and germanium, where all the semiconductor targets havethe similar high purity of 99.999%. Silver has been selected among commonmetals due to its highest figure of merit as well as its real negative suscepti-bility. As for other materials, TiO2 has a high figure of merit, a real, positivesusceptibility, and high chemical stability; Si and Ge have modest figures ofmerit, real, positive susceptibilities, and can be tuned by impurity doping.All these materials are compatible with physical vapor deposition.The films are made at room temperature when the base vacuum pressureof the deposition chamber reaches at least ∼ 5.0× 10−5 torr and the argongas pressure is about 3.0 × 10−3 torr. Depositing at moderate rates, thedielectric and silver films are deposited at 1.0 A˚/s and 2.0 A˚/s, respectively.The rotating platform that holds the substrates is spaced about 20 cm abovethe 5-cm-diameter targets.To test transmission enhancement, two families of samples are made.One for testing the transparency enhancement by dielectric coating and theother one based on metallic coating. For the first family of samples, allsamples are sputtered on borosilicate glass substrates. The sputtered baselayers are made of silver and the sputtered coating layers are made of eitherTiO2, Si, p-Si, or Ge. Five samples are made for each type of coating layerwhere the dielectric coating layer thickness varies from 18 nm to 60 nm whilethe silver base layer thickness is fixed. A well-established method for real-izing transparent conductors is coating silver films with TiO2 [69], which isconsidered as benchmark in this study. Although the effects of semiconduc-tor coatings like Si or Ge on the transparency enhancement of silver filmshave yet to be investigated, relevant studies have examined substrates ofoptically-thick metals coated with very thin layers of Si and Ge from whicha wide range of reflected colors is visible [173–176].For the other family of samples where a dielectric layer is coated withmetal, a free-standing silicon nitride membrane is used as the base layerbecause transmission enhancements in a metal-coated dielectric layer are es-timated to be most prominent when it is bounded by air. The free-standingsilicon nitride membrane is 50-nm-thick with the dimensions of 0.5 mm ×0.5 mm, manufactured by SPI Supplies. According to the manufacturersspecifications, the surface roughnesses for all membrane samples are betterthan 0.5 nm root mean squared, and the thickness variation between dif-ferent membrane samples is less than 5 nm. The silicon nitride membranebase layer is sputter-coated by a very thin silver layer.The metal film growth process in general starts by the formation of smallislands that overlap as deposition continues. For very thin metal films, when754.3. Results and Discussionthe islands are just interfered, the metal films will have the maximum surfaceroughness and the utmost surface contact area with atmosphere which cansignificantly increase the oxidation rate of metals. Therefore, to preventatmospheric corrosion, in some cases, the ultra-thin silver coating layer ispassivated by an additional sputtered layer of TiO2.The quartz crystal monitoring system in the sputtering deposition sta-tion is calibrated and used to measure the thicknesses of the deposited films.The calibration was conducted by using the stylus-based profilometer (KLATencor Alphastep) to measure the thicknesses of a series of thin-film samplesof different materials with variable deposition thicknesses.The visible and infrared light transmission measurements (400 nm to1800 nm) of the dielectric-coated silver films is conducted using a FilmetricsF20 analyzer system, while the visible spectrum transmission measurements(400 nm to 750 nm) of the silver-coated membranes is performed in a con-focal setups to suit the small aperture of the membranes using a Schott-Fostec DDL fiber optic non-plane wave light source connected to an OceanOptics USB4000 spectrometer. Under the coherent laser illumination ofvarious wavelengths (365 nm, 470 nm, 590 nm), the microscopically zoomedin transmission images of silver-coated membranes are collected by a ZeissAxioimager microscope and captured by a monochrome CCD camera.4.3 Results and DiscussionSilver coated with a film of high-index, low-loss dielectric, such as TiO2,have been shown to be the most pronounced recipe for light transmissionenhancement through silver. The photographs and the normalized trans-mittance spectra of the TiO2-coated silver films and the uncoated sampleare shown in Figure 4.2(a). The measured spectral intensities transmittedthrough the sample I(λ) are normalized to that of air Io(λ), where the nor-malized transmittance spectra can be expressed as I(λ)/Io(λ). While baresilver has the transmittance spectrum that consistently decays from blueto red, addition of a TiO2 coating produce a transmittance peak within aband known as transparency band at a wavelength dependent on the coatingthickness. Varying the coating thickness from 18 nm to 57 nm, the peakwavelengths of the transparency bands shift from ∼400 nm to ∼780 nm.The photographs of the fabricated samples visually exhibit group of colorsincluding light blue, greenish blue, yellow, and brown. This range of colors isthe consequence of the shift in the maximum light transmission peak alongthe visible wavelengths as we increase the coating layer thickness.764.3. Results and DiscussionFigure 4.2: Changing the optical properties of semi-transparent silver bysputtered TiO2 coatings. (a) Experimental and (b) calculated normal-incidence transmittance spectra for 23-nm-thick silver layers that are eitheruncoated or coated with a TiO2 layer ranging in thickness from 18 nm to 57nm. The experimental spectra are obtained from the average of 5 indepen-dent measurements, where each measurement is made from an average of40 traces. Photographs of the samples placed on the printed UBC logo areshown at the top of panel (a) to highlight the visible appearance changescaused by the thin TiO2 layer. The leftmost photograph is of uncoated silverand the adjacent images are of coated silver (in order of increasing coatinglayer thickness from left to right). [14]774.3. Results and DiscussionFigure 4.3: Changing the optical properties of semi-transparent silver byvarious sputtered elemental semiconductor coatings. Experimental normal-incidence transmittance spectra for (a) 23-nm-thick silver coated with sput-tered silicon, (b) 23-nm-thick silver coated with sputtered p-type silicon, and(c) 18-nmthick silver coated with sputtered germanium. The experimentalspectra are obtained from the average of 5 independent measurements, whereeach measurement is made from an average of 40 traces. Photographs of thesamples placed on the printed UBC logo are shown at the top of each cor-responding panel to highlight the visible appearance changes caused by thethin layers. [14]784.3. Results and DiscussionFigure 4.4: Experimental measurements of the transmission enhancement ofa 50-nm-thick silicon nitride membrane conferred by coating the membranewith 10-nm-thick silver layers in three different configurations: single-sidedcoating with silver, single-sided coating with silver followed by a 10-nm-thickTiO2 passivating layer, and double-sided coating with passivated silver. (a)shows tabulated the average normalized transmittance values for the baremembrane and the three silver-coated membranes at the wavelengths of400 nm, 420 nm, 440 nm, and 460 nm. Cells in the table correspondingto transmission enhancement (beyond experiment error) are shaded green.(b) shows the average transmittance spectra for the bare membranes (reddashed), the membrane that is coated on a single side by silver (green line),the membrane that is coated on a single side by passivated silver (orangeline) and the membrane that is coated on both sides by passivated silver(blue line). The error has a magnitude comparable to the line widths andhas not been explicitly plotted for clarity of presentation. [14]794.3. Results and DiscussionAs shown in Figure 4.2(b), the experimental data are quite consistentwith our theoretical calculations which are based on solving the transferfunctions and calculating the transmitted light through ideally planar lay-ered media with optical constants taken from [17] and [177] for silver andTiO2, respectively. The nominal variations between the peaks of the mea-sured and calculated transparency bands can be due to the discrepanciesbetween the actual and assumed optical constants of silver and TiO2, aswell as the surface roughness of the films which is neglected in the model.We show that coating silver with semiconductors can also boost trans-mission and form transparency bands. The photographs and the normalizedtransmittance spectra of bare silver films and coated ones with sputtered Si,p-Si, and Ge are shown in Figure 4.3. The induced visible-frequency trans-parency bands of the sputtered Si samples are interestingly analogous inmagnitude and spectral position to those caused by sputtered TiO2. Hence,in applications such as transparent conductors or metal-based heat-reflectingwindows, sputtered Si coating can be proposed as an alternative to sput-tered TiO2. The transparency bands made by the sputtered p-Si samples lieover the visible range between the free-space wavelengths of ∼600 nm and∼1100 nm, while those made of sputtered Ge spread completely outside thevisible spectrum between ∼900 nm and ∼1800 nm. The distinct spectralvariations between Si-coated and Ge-coated samples are due to the largeindex differences between bulk Si and Ge.On the other hand, the evident spectral variations between Si-coatedand p-Si-coated samples can be surprisingly attributed to the slight indexdifferences between bulk Si and p-Si. This observation suggests furtherinvestigation on the authenticity of using the optical constants of bulk p-Sifor the sputtered p-Si. The illustrated spectral transmittance in Figure 4.3show the capability of tailoring the induced transparency bands of silver-coated films across the entire visible and near-infrared using three differenttypes of semiconductors. Although this semiconductor coating method canbe useful for fabricating absorbers, optical filters, or solar cell coatings, theaccuracy of theoretically modeling such sputtered simiconductors (Si, p-Si,Ge) is limited by the high dependence of their optical properties on growthconditions [178, 179] which is not well-characterized.In the next experimental part of this work, we examine the 50-nm-thicksilicon nitride membrane samples to show the capability of transmission en-hancement through membranes by sputtered silver coating. The table ofaverage normalized transmittance of bare and coated silicon nitride mem-brane samples in the blue and green parts of the visible spectrum is shownin Figure 4.4. It includes the numerical transmittance of a bare membrane804.3. Results and Discussionas control spectra, a membrane with single-sided silver coating, a membranewith single-sided silver coating passivated by TiO2, and a membrane withdouble-sided passivated silver. The thickness of the silver coating layer ischosen based on the deduced information from the scanning electron micro-scope images which characterizes the minimum thickness at which sputteredsilver can form a continuous film around 10 nm.Figure 4.5: Experimental transmittance change over the entire visible spec-trum for a 50-nm-thick silicon nitride membrane coated with a 10-nm-thicksilver layer that is passivated by a 10-nm-thick TiO2 layer (red line witherror bars). Also shown are calculations of the transmittance change for thethree-layer system assuming various silver layer thicknesses. The error barsin the experimental measurement represent one standard deviation. [14]To mitigate sources of uncertainty, the spectra are averaged across manymeasurements. To mitigate thickness variations across membrane samples,the control transmittance spectra are averaged over measurements of threedistinct bare membranes. To mitigate local thickness variations of a givenmembrane, the transmittance spectra are averaged over measurements takenat 10 different locations on the membrane. To account for random noise fromthe light source and spectrometer, the transmittance spectrum is averagedover 150 measurements for all measurements at each location. As demon-strated in Figure 4.4, the spectral measurements of all three silver-coated814.3. Results and Discussionmembranes show moderate transmission enhancements in the blue part ofthe visible spectrum where all transmission boost peaks at 400 nm, thelower wavelength bound of the measurement. The maximum measured en-hancement of 6±1% is achieved by single-side silver coating. Silver coatedmembranes passivated with TiO2 yields similar transmission enhancementsas the non-passivated one, except over a larger wavelength range and withslightly smaller transmission enhancements around the peak values.Comparing the experimental measurements and theoretical calculations,Figure 4.5 shows the transmittance change measurements of the passivatedsingle-side silver-coated membrane over the full visible spectrum besides thecalculated transmittance change for various silver layer thicknesses. Thetransmittance change describes the measured variations between the trans-mitted spectral intensities of bare membrane (Ib(λ)) and silver-coated mem-brane (Ic(λ)), where it is defined by [Ic(λ)−Ib(λ)]/Ib(λ). While similar pos-itive transmittance changes can be observed at blue wavelengths betweenmeasurements and calculations, they start to diverge at larger wavelengths.This divergence can be attributed to the index differences between bulk andsputtered materials, the consideration of perfectly smooth layers in theoret-ical calculations, and the possibility of having sputtered layers with nano-thickness variations. The theoretical model, which is based on the idealconditions of perfect layer planarity and sharp boundaries, predicts greaterthan 10% transmittance change over the entire visible spectrum using a verythin 7-nm-thick silver coating layer.According to the model calculations of the silver coated membranes, theenhancement effect is quite dependant on the silver layer thickness whereincreasing the thickness from 7 nm to 16 nm is sufficient to completelyeradicate this effect. The most robust transmission enhancement predictionsas a function of the silver thickness are located around the blue part ofthe spectrum. This observation explains why the measured transmissionenhancement is limited to the blue frequency range. Fabricating smoothersilver layers will most probably improve the experimental enhancement.A strategy that can be explored in future for developing smoother silverlayers is the addition of a seed layer of germanium or nickel [180, 181] thatmay also change the optical constants of silver. To show if the transmis-sion enhancement conferred by thin silver coatings is visually observable,comparative microscope images of a bare membrane and an identical mem-brane coated with 10 nm of Ag and 10 nm of TiO2 are shown in Figure 4.6,where the membrane samples are illuminated by a normal-incidence laser atwavelengths of 365 nm, 470 nm, and 590 nm.824.3. Results and DiscussionFigure 4.6: Microscope images of (left column) an uncoated 50-nm-thickSi3N4 membrane and (right column) an identical membrane coated with 10-nm-thick Ag and 10-nm-thick TiO2 under laser illumination at wavelengthsof (a) 365 nm, (b) 470 nm, and (c) 590 nm. The images were collectedusing a monochrome camera and have been false-colored to reflect the colorof laser illumination. The percentages on the images in the right columnindicate the percent change in the average image brightness relative to theadjacent images in the left column. [14]834.4. SummaryAt the wavelength of 365 nm below the spectral measurement wavelengthlimit (400 nm), a modest image brightness increase of about 2% is measur-able. In a good agreement with the spectral measurements in Figure 4.5,the images turn dimmer by 9% and 28% at wavelengths of 470 nm and 590nm.4.4 SummaryWe develop a theory based on Maxwell’s equations to generally describethe well-known coating method, which is used specifically to enhance thetransmittance of metal films by dielectric coating. The theory include twogeneral conditions, one to determine the possibility of transmission enhance-ment, and the other one to calculate the optimal thickness of the coatinglayer. The proposed theory emphasizes that the transmission enhancementeffect is not limited to metal films coated with dielectric, but rather anycoated material can be made more transparent if the base layer and thecoating layer have opposite sign susceptibilities and the appropriate thick-nesses.We conducted two distinct series of experiments that confirm the validityof the introduced theory. In the first set of experiments, we examined sil-ver films coated with a well-explored dielectric (TiO2) and the less-exploredsemiconductors (Si, p-Si, Ge). We show up to ∼70% light transmission en-hancement in the coated silver films where the transmittance can be spec-trally tailored depending on the type and thickness of the dielectric coatingover the visible and near-infrared part of the spectrum. In the second set ofexperiments, we investigated the possibility of boosting light transmissionthrough a dielectric layer by metal coating. The transmission measurementsof silver-coated silicon nitride membranes show a modest light transmissionenhancement of 6±1% in the blue part of the spectrum. Similar transmissionenhancements are observed for the passivated single-side and double-sidesilver-coated membranes.84Chapter 5Surface Plasmon Resonance:Copper as Good as GoldSurface plasmon resonance sensing is a well-established method in diversesensing applications. To enhance the sensitivity of SPR sensors, the SPRcoupling efficiency should be improved. For this purpose, various meth-ods like metal thickness optimization [182], dielectric coating [95], nano-structural modification [96, 98–100] and deposition process modulation [102,103] have been implemented. It has been shown that metals with high fig-ures of merit (low loss) are the best metals for making SPR sensors. Hence,metals like silver, gold, and copper with fairly low loss have been widelyused for fabricating SPR sensors. Although copper has the modest priceamong the three and gold is the most expensive (Cu≈ $2.7/lb, Ag≈ $237/lb,Au≈ $17000/lb), gold continued to be the most preferable metal due to itsacceptable SPR performance and chemical stability.In this work, we introduce copper as a cheaper alternative metal forsurface plasmon applications by enhancing the fairly poor SPR couplingefficiency of copper thin films. We examine the influence of different deposi-tion parameters on the SPR couplings of copper, silver and gold thin films,and we find a new DC sputtering deposition recipe for making copper nano-films with enhanced surface plasmon coupling. Using this deposition recipe,we fabricated 40-nm-thick copper films that have optical resonances compa-rable to gold. Our method significantly improves the quality factor of thesurface plasmon resonance of copper thin films (up to 200% improvement),but shows minimal effect on silver and gold thin films.5.1 MethodologyWe fabricate silver, gold, and copper thin films using the magnetronsputter deposition station (Angstrom Engineering Nexdep) at different slewrates and deposition rates. SPR is measured by recording the intensity ofreflected light from a metal film at different angles of incidence. Over a nar-855.1. Methodologyrow range of angles, the reflected intensity shows a sharp and pronounceddip that is an indicative of SPR. We measure the SPR of the fabricatedthin films using our home built SPR station which was made based on theKretschmann’s total internal reflection (TIR) coupling method [183], as il-lustrated in Figure 5.1. The SPR measurements are then qualified basedon the SPR parameters: the sharpness and the depth of the SPR dip. Thesharper and deeper the dip, the more sensitive the SPR phenomenon be-comes.Figure 5.1: Schematic of light coupling to surface plasmon waves usingKretschmann configuration.The thin film fabrication process starts with a standard glass cleaningmethod using RO water and acetone to clean the bare glass substrates [184].The glass substrates are then mounted on a rotating platform spaced about20 cm above the 5-cm-diameter material source (target). Thin-films of metalare deposited on the cleaned glass substrates from 99.99% metal targets bybombarding the target with Argon ions, as illustrated in Figure 5.2. Themagnets on the back of the target are used to discharge the ejected targetatoms and start film growth on the glass substrates. All thin films aresputtered at room temperature, where the deposition base vacuum pressurekept to be at least ∼ 5.0× 10−5 torr, and the argon gas pressure 3.0× 10−3torr. For optimal SPR responses, we set the thicknesses of all depositionsto ∼40 nm for copper [95], and ∼50 nm for gold and silver [182].865.1. MethodologyFigure 5.2: Schematic of the sputtering process in the vacuum depositionchamber.Examining the influence of modifying different deposition parameters onthe SPR couplings of copper, silver and gold thin films, we mainly variedtwo nano-deposition parameters in this work: the deposition rate which isthe growth rate of thin films, and the slew rate which is the maximum powerchange allowed per second controlling the rate of power change on the tar-get. To increase the rate of deposition of our targets, the applied voltageacross the targets has to be increased. This will consequently increase theapplied power and heat up the target. The required power for reaching cer-tain deposition rate varies from one target to another based on the targetsthermal conductivity, thermal coefficient of expansion, mechanical strengthcharacteristics, and melting point. We usually keep an eye on the powerapplied to the target, but the critical quantity is really the power density,which is the applied power divided by the target’s surface area. Since ar-bitrarily increasing power can cause many adverse effects to the target, wegradually increased the deposition rate of our 2 inch diameter targets tofind the maximum safe deposition rates of copper, silver and gold targets attheir maximum secure power density. Knowing the maximum safe deposi-tion rates, we sputtered copper and gold at deposition rates as high as 10.0A˚/s, and silver at higher deposition rates upto 20.0 A˚/s. All depositionsare conducted at both low (4.0 %/s) and high (99.9 %/s) slew rates.875.2. Results and DiscussionsThe SPR station consists of a BK7 right-angle prism (n = 1.517) mountedon a rotary platform, two HeNe laser sources with yellow (λo = 594.0 nm)and red (λo = 638.2 nm) operational wavelengths, and a mobile silicon pho-todetector (DET36A). These two independent laser sources with two differ-ent wavelengths are used to confirm the consistency of the proposed methodfor SPR coupling enhancement in copper films. The glass substrate withmetal coating is placed on the back of the prism. The light beam intensityreflected by the installed thin film is detected at various incident angles.A distinct dip in the reflected beam intensity at certain angles is a clearindication of surface plasmon coupling. To account for the non-planarityacross metal films, we conduct six reflected intensity measurements at threedifferent sites on each sample.To compare different SPR spectra measured from different samples, weuse the standard quality factor metric of the average reflected beam intensity.The standard quality factor is defined by the ratio of the absorbed peakenergy to the peak linewidth, typically the full width at half-maximum [185].Since we are measuring reflected light rather than absorbed light, the qualityfactor (QF) will be given byQF =IR,minFWHM, (5.1)where IR,min is the minimum reflected light intensity and FWHM is the fullwidth at half-maximum of the reflected beam intensity dip. To investigatethe morphological cause of the observed SPR enhancement measurements,we take visual photographs, scanning electron microscope (SEM) images,and atomic force microscopy (AFM) images of the silver, gold, and copperfilms sputtered at different deposition parameters. The photographs arecaptured in a dark room by a high resolution SLR camera from the sameviewing angle and under similar light conditions. The SEM images aretaken in high vacuum by a Field Emission Scanning Electron Microscope at100000x magnification. The AFM images are taken by the Bruker DimensionIcon AFM using the peak force tapping mode over the scan area of 5µm x5µm.5.2 Results and DiscussionsThe intensity measurements of the reflected red laser beam from thefabricated 50-nm-thick silver and gold films at low and high deposition/slewrates show that sputtering rates can play a small but not insignificant role885.2. Results and Discussionsin changing the SPR quality (Fig. 5.3). Conducting the same measurementsfor the 40-nm-thick copper films fabricated at low and high deposition/slewrates, we noticed fairly significant improvements in the quality of the SPRdips (depth and acuteness). To clearly demonstrate this prominent SPRresponse enhancement, we plot copper SPR measurements alongside gold’sSPR responses (Fig. 5.4). The measurements are conducted using yellowand red coherent laser sources. Comparing the SPR measurement curves ofcopper and gold, it can be noticed that SPR responses of copper thin filmssputtered at high deposition rates are comparable to gold’s SPR responses.Figure 5.3: Show the SPR measurements of the 50-nm-thick substrates de-posited at low and high, deposition/slew rates, for a) silver, and b) gold,using a coherent red He-Ne laser with free-space wavelength λo = 632.8 nm.To quantitatively compare the SPR responses of silver, copper and goldfilms made at different deposition/slew rates, we calculate the SPR dipquality factors. We illustrate the calculated quality factors through side byside bars in Figure 5.5. Silver thin films showed to sustain the sharpest dipsand the highest SPR quality factors at all deposition/slew rates. The SPRquality factors of gold thin films demonstrate almost constant levels in allsituations, whereas the SPR quality factors of copper thin films interestinglyexhibit quite large improvements as deposition rate increases.895.2. Results and DiscussionsFigure 5.4: Show how the SPR dips of the 40-nm-thick copper (blue lines)become sharper and deeper comparable to the 50-nm-thick gold SPR dips(red lines), when the deposition rate of copper is increased from low depo-sition rate (1.0 A˚/s, blue) to high deposition rate (10.0 A˚/s, green). SPRmeasurements using a coherent yellow λo = 594.0 nm a) b), and red λo =632.8 nm c) d) He-Ne laser.In general, it can be observed that the SPR quality factors of copperbecome comparable to that of gold as we increase deposition rate. For theparticular case in which the copper films are made at low slew rate and veryhigh deposition rate and excited using a red laser beam, the SPR qualityfactors improved by around 200% and reach the highest level (∼0.5), whichis better than the best quality factors achieved by gold. The detailed tablesof SPR parameters and quality factors are provided in the Appendix.905.2. Results and DiscussionsFigure 5.5: The quality factor bar charts with error bars for silver (blue),copper (green), and gold (yellow) at different deposition/slew rates usingyellow a) b), and red c) d) lasers. The deposition rates I, II, III, and IVrespectively correspond to 1.0 A˚/s, 7.0 A˚/s, 14.0 A˚/s, and 20.0 A˚/s forsilver, and 1.0 A˚/s, 3.0 A˚/s, 7.0 A˚/s, and 10.0 A˚/s for copper and gold.To probe the cause of this phenomenon, visual photographs (Fig. 5.6),SEM images (Fig. 5.7), and AFM images (Fig. 5.8) of silver, gold and coppersubstrates sputtered at different deposition rates and fixed low slew rate areinvestigated. Looking at the photographs, the two silver substrates (Fig. 5.6(a), (b)), the two gold substrates (Fig. 5.6 (c), (d)) and the two coppersubstrates (Fig. 5.6 (e), (f)) look visually identical. As for SEM, we wouldexpect to have different SEM images for copper films rather than silver orgold films due to the differences in copper SPR responses. However, theSEM images show featureless smooth surface for copper thin films (Fig. 5.7(e), (f)), minimal surface roughness variation for gold thin films (Fig. 5.7(c), (d)), and diverse surface roughness for silver thin films (Fig. 5.7 (a),(b)) at low and high deposition rates.915.2. Results and DiscussionsFigure 5.6: Photograph images at low and high deposition rates for the50-nm-thick silver a) b), the 50-nm-thick gold c) d), and the 40-nm-thickcopper e) f) thin films deposited on glass substrates.Figure 5.7: Scanning electron microscope (SEM) images at low and highdeposition rates for the 50-nm-thick silver a) b), the 50-nm-thick gold c) d),and the 40-nm-thick copper e) f) thin films deposited on glass substrates.925.3. SummaryFigure 5.8: The atomic force microscopy (AFM) images at low and highdeposition rates for the 50-nm-thick silver a) b), the 50-nm-thick gold c) d),and the 40-nm-thick copper e) f) thin films deposited on glass substrates.The AFM images show fairly small surface roughness differences betweencopper films deposited at low and high deposition rates (Fig. 5.8 (e), (f)),and similar surface roughness features for silver (Fig. 5.8 (a), (b)) and gold(Fig. 5.8 (c), (d)). The average surface roughness differences for silver, goldand copper are 5.0 nm, 1.16 nm, and 0.12 nm, respectively. These quitesimilar SEM and AFM results for copper films deposited at low and highdeposition rates suggest that the SPR enhancement effect can be attributedto finer surface features or chemical composition variation. To analyze thechemical composition of the sputtered thin metal films, we used the inte-grated elemental analysis tool with SEM system (Energy Dispersive Spec-troscopy). However, this chemical analysis method cannot provide accurateresults since it is based on analyzing the SEM back scattered electrons withquite long penetration effects (around 1 µm), while the maximum thicknessof the thin metal films is only 50 nm. For accurate chemical analysis of thesethin films, we can try other methods like x-ray photolectron spectroscopy.5.3 SummaryWe have proposed a simple nano-film fabrication procedure to producecopper films with surface plasmon quality factors comparable to gold films.935.3. SummaryThe considered parameters in this work are the film thickness, the deposi-tion rate and the slew rate. The film thicknesses are fixed to the optimalreported values while the deposition rate and the slew rate are the variableparameters. Experimental results showed significant improvement in surfaceplasmon coupling performances for copper films deposited with higher depo-sition rates, while minor enhancement is noticed for silver and gold using thesame fabrication process. As the deposition rate increases, the improvementin the SPR quality factors of copper thin films are found to be more consis-tent and prominent when the depositions are conducted at low slew rates.The maximum measured quality factor improvement of 200% is achieved forcopper thin films deposited at low slew rate and the high deposition rate of10 A˚/s. We believe that replacing of gold films with our modified copperfilms can significantly reduce the manufacturing cost of commercial SPRsensors.94Chapter 6ConclusionThis thesis has investigated the interaction of light with sub-wavelengthlayered systems made of metals and/or dielectrics. Ray optics and the elec-tromagnetic wave theory of light in layered media has been deeply studiedbefore. However, reported observations of the abnormal electromagneticproperties in sub-wavelength layered systems (layered metamaterials), suchas negative phase velocity, super-resolution, canalization, and far-field imag-ing, highlight the necessity of further exploring the intrinsic electromagneticbehaviours of layered metamaterials.In this thesis, we have discussed the electromagnetic wave theory of lay-ered systems with the practical constraints of loss and finite extent. Wehave developed the theory and derived a new representation for the elec-tromagnetic field solutions. The developed theory yields new criteria andconditions that facilitate the design of new layered metamaterials. In Chap-ter 2, we followed a bottom-up approach starting with Maxwell’s equationsto derive a new expression of the electromagnetic fields in Fourier domain.The new representation is a compact product of three terms where eachterm is dependent on a particular physical parameter of the layered system.The Floquet-Bloch modes, which have been used before for modeling infi-nite lossless systems, surprisingly show up in one of the three terms, withoutinvoking the Floquet-Bloch theorem.This Fourier domain representation of the fields shows its capability ofdecomposing the wave function to analyze complex electromagnetic prop-erties. We extracted the corresponding band diagrams of the Fourier do-main representation and used them to graphically describe a wide rangeof refractive properties of layered metamaterials and to validate the con-ventional homogenization methods. Although the derived Fourier domainelectromagnetic field solutions can be only used for planar flat systems, thefollowed strategy in this electromagnetic study is not limited to layered meta-materials and it is applicable to any metamaterial structure. Using banddiagrams, we analyzed and distinguished the abnormal intrinsic electromag-netic behaviours that cause the external convergence of light. Analyzingthe external behaviour of light just after exiting the planar media, we have95Chapter 6. Conclusionintroduced a quantitative method in Chapter 3 based on the small-anglephase behaviour to predict the image plane location and consequently theimaging capability of single and multiple layer slabs made of homogeneousisotropic media. Using the proposed flat lens criterion, we designed three flatlenses with novel functionalities. Although the proposed flat lens criterionprovides new insights into the field of flat lens imaging and enables a newmethod for flat lens design over large parameter spaces, it cannot supplantthe existing flat lens analysis methods due to several limitations.A recognized limitation in the flat lens criterion derivations is the dis-regard of the interactions between the object and lens, where the reflectedwaves from the source or object have been neglected. However, the interac-tion effect on calculations become more prominent when the object-to-lensseparation is small [61]. Accounting for these interactions would proba-bly provide more accurate predictions but it will lead to object-dependentpredictions where the flat lens criterion is not general. Since the flat lens cri-terion is based on the small-angle phase alone, the other limitation that canbe noticed is the lack of information provided by the flat lens criterion fordescribing the resolution, contrast, or fidelity of the image. To study large-angle plane-wave components with NA near unity or evanescent plane-wavecomponents with NA greater than one, alternative criteria can be developedin future studies by calculating and analyzing the phase of large-angle andevanescent plane-wave components. Another noteworthy limitation of flatlens design is its associated restriction on configurations where the thicknessof the layered stack should be much smaller than the wavelength of the lightsource. This limitation is due to the dependence of flat-lens imaging on theinterference of multiple reflected waves within the stack, which is best re-alized using coherent laser light. However, imaging with partially coherentlight source from narrow-band light-emitting diodes, which are low-cost andamenable to fluorescence imaging, should be possible using sufficiently thinstacks. To investigate the compatibility of flat lenses with light sources ofdifferent level of coherence, the impact of light coherence on imaging qualityshould be further examined.In future works, the correlation between the small-angle flat lens criterion(NA near zero) and super-resolution imaging (NA greater than unity), whichhas been shown to be accurately predicted by the small-angle criterion, hasto be fully understood and established by examining more case studies andrigorously investigating the small-angle flat lens criterion limitations in pre-dicting super-resolution imaging. Consequently, a flat lens aberration theorythat accommodates both propagating and evanescent components should bedeveloped. Future flat lens engineering will predominantly focus on practi-96Chapter 6. Conclusioncal challenging features such as imaging over the entire visible spectrum orsimultaneously imaging with TM and TE polarizations.Since light transmission through thin films with complex permittivityis challenging due to the associated loss, we have discussed the theory ofoptical light transmission through a single bi-layer unit cell made of metaland dielectric in Chapter 4. For the first time, we have theoretically andexperimentally shown that transmission enhancement phenomenon in metal-dielectric systems is reciprocal. We show that a thin coating layer on top ofa thin base layer can make the base layer more transparent when the twolayers have opposite sign susceptibilities. The developed theory can predictthe possibility of transmission enhancement and it can estimate the optimalcoating layer thickness for the most transparent bi-layer system. Althoughwe experimentally demonstrated that a thin silver coating layer can enhancelight transmission through a silicon nitride membrane, the achieved opticaltransparency enhancements were limited to a narrow-band of the visiblespectrum with the maximum transmittance enhancement of 6±1%. Theo-retically, enhancement factors greater than 10% is attainable using thinnersilver coatings. However, the experimental implementations of silver-coatedmembranes are restricted by challenges in making perfectly planar and con-tinuous silver films below 10 nm in thickness. In future, the use of seedinglayers to improve the smoothness of the silver layers and the consequencesof having bi-layers with opposing magnetic susceptibility can be explored.Exploring the plasmonic properties of single layers of metals, we haveproposed an easy nano-film fabrication procedure for making copper nano-films with enhanced surface plasmon coupling in Chapter 5. The depositionmethod is based on the modification of two deposition parameters: thedeposition rates and the slew rates. We have been able to produce copperfilms that have optical resonances comparable to gold films. Implementingthe introduced nano-film fabrication method, copper nano-films can workas a cheaper substitute for gold nano-films in surface plasmon applications.In future work, copper films could be passivated for making more sensitive,robust and chemically stable SPR sensors.Overall, this thesis provides a fundamental theoretical study of electro-magnetic fields in layered metamaterials with a discussion of three commonapplications. This work introduced a general method for characterizing andcommunicating the electromagnetic properties of lossy finite layered meta-materials. Moreover, the thesis proposed three engineering methods forthree practical implementations of layered metamaterials to achieve design-ing new systems with new features. 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Zhang, “Experimental real-ization of three-dimensional indefinite cavities at the nanoscale withanomalous scaling laws,” Nature Photonics, vol. 6, no. 7, pp. 450–454,2012.114Appendices115Appendices ASurface Plasmon PolaritonsSurface plasmon polariton waves are a valid solution of Maxwell’s equa-tions under certain conditions. Having two semi-infinite non-magnetic mediawhere the interface is at z = 0, the first and the second media are charac-terized by the local complex relative permittivities 1 and 2, respectively.Starting with a TM polarized wave, the electromagnetic fields propagatingabout the interface along the +x-direction and decaying along the z direc-tions are given byHy(z) ={H1eikxxekz1z z < 0H2eikxxe−kz2z z > 0, (A.1)Ex(z) ={ −ikz1H1eikxxekz1z/ω01 z < 0ikz2H2eikxxe−kz2z/ω02 z > 0,(A.2)andEz(z) ={ −kxH1eikxxekz1z/ω01 z < 0−kxH2eikxxe−kz2z/ω02 z > 0. (A.3)where H1 and H2 are the complex amplitudes of the magnetic field in media1 and 2, respectively, kx is the complex wave vector component along thex axis, and kz1 and kz2 are the complex wave vector components along thez axis in media 1 and 2, respectively. Due to continuity of the fields acrossthe interface, kx is the uniquely defined mode in both media.Imposing the continuity of the tangential components of the electric fieldsat z = 0, we getkz1H11+kz2H22= 0, (A.4)andH1 −H2 = 0, (A.5)116Appendices A. Surface Plasmon Polaritonswhich has a solution only ifkz1kz2= −12. (A.6)For positive real kz1 and kz2, surface waves can exist if the real parts of thepermittivities of the two media have opposite signs. Such condition can besatisfied at the interface between a metal and a dielectric media.Conducting the same exercise for TE polarization, the complex ampli-tudes of the electric fields across the interface should satisfyE1 = E2, (A.7)andE1(kz1 + kz2) = 0. (A.8)As in the TM polarization case, kz1 and kz2 are positive and real. Therefore,the only possible solution will be E1 = E2 = 0, which indicates that surfaceelectromagnetic waves cannot exist at the interface of non-magnetic mediafor TE-polarization.117Appendices BEvanescent WaveAmplification with a ThinSilver LayerFrom Maxwell’s equations, the general form of the wavevector kz forpropagating waves in free-space [27] is given bykz = +√ω2c−2 − k2x − k2y, (B.1)where ω2c−2 > k2x + k2y. For larger values of the transverse wavevector, kzwill be given bykz = +i√k2x + k2y − ω2c−2, (B.2)where ω2c−2 < k2x+k2y. Electromagnetic waves defined by the latter wavevec-tor are known as evanescent waves where they decay exponentially with z. Ina medium defined by the electromagnetic properties and µ, the wavevectorof the evanescent waves is given bykz = +i√k2x + k2y − µω2c−2, (B.3)where µω2c−2 < k2x + k2y.Working with a sub-wavelength thin layer of silver, the electrostatic limitcan be applied to decouple electrostatic and magnetostatic fields. For TM-polarized fields, the transmission coefficient becomes independence of µ anda function of , which is negative at optical frequencies. Based on the elec-trostatic limitω c√k2x + k2y. (B.4)118Appendices B. Evanescent Wave Amplification with a Thin Silver LayerThe transmission coefficient T of the thin layer assuming the electrostaticlimit becomeslimk2x+k2y→∞T = limk2x+k2y→∞2 kz kz + k′z2k′zk′z + kz× exp(ik′zd)1− (k′z−kzk′z+kz)2 exp(2ik′zd)=4 exp(ikzd)(+ 1)2 − (− 1)2 exp(2ikzd) .(B.5)Assuming that is becoming negative, the transmission coefficient becomeslim→−1limk2x+k2y→∞T = lim→−14 exp(ikzd)(+ 1)2 − (− 1)2 exp(2ikzd)= exp(−ikzd)= exp(+d√k2x + k2y),(B.6)which means the thin silver layer does amplify evanescent waves.119Appendices CS-parameter MethodFor a normally incident planewave, the effective wavevector kz(eff) withinthe layered system along the z direction based on the S-parameter method [167],is given bykz(eff)d = ± arccos(1− S211 + S2212S21) + 2mpi, (C.1)where S11 is the reflection coefficient, S21 is the transmission coefficient, dis the total thickness of the layered system, and m is an integer.To derive the above expression, we first substitute the layered system witha homogeneous medium that has identical scattering parameters. Applyingthe boundary conditions at the front (z1 = 0) and back (z2 = d) interfaces,we getE+l eiklzzl+1 + E−l e−iklzzl+1 = E+l+1eik(l+1)zzl+1 + E−l+1e−ik(l+1)zzl+1 , (C.2)H+l eiklzzl+1 +H−l e−iklzzl+1 = H+l+1eik(l+1)zzl+1 +H−l+1e−ik(l+1)zzl+1 , (C.3)where E+ and H+ represents all wave components propagating towardspositive zˆ direction, and E− and H− represents those propagating towardsnegative zˆ direction. The subscripts l and l+1 are used to denote the regionson the left and right side of the interfaces, respectively.Knowing that H = ±kµωE, Eq. (C.3) can be rewritten in terms of electric fieldE+l eiklzzl+1 − E−l e−iklzzl+1= pl(l+1)[E+l+1eik(l+1)zzl+1 − E−l+1e−ik(l+1)zzl+1 ],(C.4)wherepl(l+1) =µlk(l+1)zµl+1klz. (C.5)The semi-infinite half spaces on both sides of the layered system areassumed to be free-space and denoted by the subscript 0 from the frontside and t from the back side. Knowing the electric wave components, the120Appendices C. S-parameter Methodreflection coefficient R and the transmission coefficient T are defined asR =E−0E+0, (C.6)T =E+tE+0. (C.7)After few lines of manipulations, we can re-write Eq. (C.2) and Eq. (C.4)asE+l eiklzzl+1 =12(1 + pl(l+1))[E+l+1eik(l+1)zzl+1 +Ql(l+1)E−l+1e−ik(l+1)zzl+1 ],(C.8)E−l e−iklzzl+1 =12(1 + pl(l+1))[Ql(l+1)E+l+1eik(l+1)zzl+1 + E−l+1e−ik(l+1)zzl+1 ],(C.9)whereQ(l+1)l =1− p(l+1)l1 + p(l+1)l= −Ql(l+1). (C.10)Taking the ratio of Eq. (C.8) and Eq. (C.9), after few manupulations weobtaineR =E−0E+0=ei2k0zz1Q01+[1− (1/Q201)]ei2(k1z+k0z)z1(1/Q01)ei2k1zz1 +Q12ei2k1zz2=Q01 +Q12ei2k1z(z2−z1)1 +Q01Q12ei2k1z(z2−z1)ei2k0zz1 .(C.11)Knowing that Q12 = Q10 = −Q01, k1z = nk0, z1 = 0 and z2 − z1 = d,we getR =Q01(1− ei2nk0d)1−Q201ei2nk0d= S11. (C.12)121Appendices C. S-parameter MethodRecalling that,E+0 eik0zdt =12(1 + p01)[E+t eik1zz0 +Q01E−t e−ik1zz0 ],(C.13)andE+1 eik1zdt =12(1 + p1t)[E+t eiktzd +Q1tE−t e−iktzd],(C.14)we can solve for E+tE+t =2(1 + p1t)[E+1 ei(k1z−ktz)d], (C.15)and the transmission coefficient will beT =(1−Q201)eink0d1−Q201ei2nk0d= S21. (C.16)Using Eq. C.12 and Eq. C.16, we solve for Q201Q201 =eink0d − S21eink0d − S21ei2nk0d , (C.17)andQ201 =S2111− 2S21eink0d + S221ei2nk0d. (C.18)Equating Eq. (C.18) and Eq. (C.17), we geteink0d − S21eink0d − S21ei2nk0d =S2111− 2S21eink0d + S221ei2nk0d. (C.19)After few lines of simplification, we obtain this expressioneink0d + e−ink0d2=1− S211 + S2212S21. (C.20)Since eink0d+e−ink0d2 = cos(nk0d), we get the final expression for effectivewave vector as Eq. C.1.122Appendices DMaxwell-Garnett MethodAssume that a plane wave is incident on a metal-dielectric layered struc-ture where the thicknesses tm and td are fairly small compared to the wave-length and m and d are the dielectric constants of the metal and thedielectric layers, respectively. For TE polarization where electric field isperpindicular to the plane of incidence, the electric displacement D in thelayered medium can be considered to be uniform. The corresponding metaland dielectric electric fields are Em and Ed, and related to the uniform elec-tric displacement byEm =Dm, (D.1)Ed =Dd, (D.2)The averaged mean field over the total volume isE =tmDm+ tdDdtm + td, (D.3)The effective dielectric constant ⊥ is:⊥ =DE=(tm + td)mdtmd + tdm=mdfmd + fdm, (D.4)where fm = tm/(tm + td) and fd = td/(tm + td) = 1− fm are the metal anddielectric filling factors, respectively.For TM polarization where electric field is parallel to the plane of incidence,the electric field E now is uniform. The corresponding metal and dielectricelectric displacements areDm = mE, (D.5)123Appendices D. Maxwell-Garnett MethodandDd = dE. (D.6)The averaged mean field over the total volume isD =tmmE + tddEtm + td, (D.7)Therefore, the effective dielectric constant ‖ will be:‖ =DE=tmm + tddtm + td= fmm + fdd, (D.8)The expressions of introduced ‖ and ⊥ can be re-written as [186]⊥ =dm(1− p)m + pd , (D.9)‖ = (1− p)d + pm, (D.10)where f1 = p and f2 = (1− p), or as [81]‖ =d + ηm1 + η, (D.11)⊥ =dm(η + 1)m + ηd, (D.12)where η = td/tm = p/(1− p).124Appendices EFinite Difference FrequencyDomain MATLAB Code1 % This MATLAB code s imu la t e s two−dimens iona l o p t i c a ls t r u c t u r e s us ing the2 % f i n i t e −d i f f e r e n c e frequency−domain method .34 f unc t i on [R,T,m, F , Dr1 , Dr2 , Dspp ] = fdfd2d ( lam0 ,UR2,ER2,RES2 ,NPML, kinc , pol , nx1 , nx2 , ny4 , ny5 )56 % INPUT ARGUMENTS7 % lam0 i s the f r e e space wavelength8 % UR2 conta in s the r e l a t i v e pe rmeab i l i t y on a 2X gr id9 % ER2 conta in s the r e l a t i v e p e r m i t t i v i t y on a 2X gr id10 % NPML i s the s i z e o f the PML on the 1X gr id11 % RES2 = [ dx2 dy2 ]12 % kinc i s the i n d i c e n t wave vec to r13 % pol i s the p o l a r i z a t i o n ( ’E’ or ’H’ )1415 % OUTPUT ARGUMENTS16 % R conta in s d i f f r a c t i o n e f f i c i e n c i e s o f r e f l e c t e dwaves17 % T conta in s d i f f r a c t i o n e f f i c i e n c i e s o f t ransmit tedwaves18 % m conta in s the i n d i c e s o f the harmonics in R and T19 % F i s the computed f i e l d2021 %% HANDLE INPUT AND OUTPUT ARGUMENTS22 c = 3e8 ;23 % DETERMINE SIZE OF GRID24 [ Nx2 , Ny2 ] = s i z e (ER2) ;25 dx2 = RES2(1) ;125Appendices E. Finite Difference Frequency Domain MATLAB Code26 dy2 = RES2(2) ;27 % 1X GRID PARAMETERS28 Nx = Nx2/2 ; dx = 2∗dx2 ;29 Ny = Ny2/2 ; dy = 2∗dy2 ;30 % COMPUTE MATRIX SIZE31 M = Nx∗Ny;32 % COMPUTE REFRACTIVE INDEX IN REFLECTION REGION33 e r r e f = ER2( : , 1 ) ; e r r e f = mean( e r r e f ( : ) ) ;34 u r r e f = UR2( : , 1 ) ; u r r e f = mean( u r r e f ( : ) ) ;35 n r e f = s q r t ( e r r e f ∗ u r r e f ) ;36 i f e r r e f <0 && urre f <037 n r e f = − n r e f ;38 end39 % COMPUTE REFRACTIVE INDEX IN TRANSMISSION REGION40 e r t rn = ER2 ( : , Ny2) ; e r t rn = mean( e r t rn ( : ) ) ;41 urtrn = UR2( : , Ny2) ; urtrn = mean( urtrn ( : ) ) ;42 ntrn = s q r t ( e r t rn ∗ urtrn ) ;43 i f e r t rn<0 && urtrn<044 ntrn = − ntrn ;45 end4647 %% INCORPORATE PERFECTLY MATCHED LAYER BOUNDARYCONDITION48 % PML PARAMETERS49 N0 = 376 .73032165 ; %f r e e spaceimpedance50 amax = 3 ;51 cmax = 1 ;52 p = 3 ;53 % INITIALIZE PML TO PROBLEM SPACE54 sx = ones (Nx2 , Ny2) ;55 sy = ones (Nx2 , Ny2) ;56 % COMPUTE FREE SPACE WAVE NUMBERS57 k0 = 2∗ pi /lam0 ;58 % Y PML59 N = 2∗NPML;60 f o r n = 1 : N61 % compute Y−PML value62 ay = 1 + amax∗(n/N) ˆp ;63 cy = cmax∗ s i n ( 0 . 5∗ pi ∗n/N) ˆ2 ;126Appendices E. Finite Difference Frequency Domain MATLAB Code64 s = ay∗(1− i ∗cy∗N0/k0 ) ;65 % inco rpo ra t e va lue in to PML66 sy ( : ,N−n+1) = s ;67 sy ( : , Ny2−N+n) = s ;68 end69 % X PML70 f o r n = 1 : N71 % compute X−PML value72 ax = 1 + amax∗(n/N) ˆp ;73 cx = cmax∗ s i n ( 0 . 5∗ pi ∗n/N) ˆ2 ;74 s = ax∗(1− i ∗cx∗N0/k0 ) ;75 % inco rpo ra t e va lue in to PML76 sx (N−n+1 , :) = s ;77 sx (Nx2−N+n , : ) = s ;78 end79 % COMPUTE TENSOR COMPONENTS WITH PML80 ER2xx = ER2 . / sx .∗ sy ;81 ER2yy = ER2 .∗ sx . / sy ;82 ER2zz = ER2 .∗ sx .∗ sy ;83 UR2xx = UR2 . / sx .∗ sy ;84 UR2yy = UR2 .∗ sx . / sy ;85 UR2zz = UR2 .∗ sx .∗ sy ;86 % OVERLAY MATERIALS ONTO 1X GRID87 ERxx = ER2xx ( 2 : 2 : Nx2 , 1 : 2 : Ny2) ;88 ERyy = ER2yy ( 1 : 2 : Nx2 , 2 : 2 : Ny2) ;89 ERzz = ER2zz ( 1 : 2 : Nx2 , 1 : 2 : Ny2) ;90 URxx = UR2xx ( 1 : 2 : Nx2 , 2 : 2 : Ny2) ;91 URyy = UR2yy ( 2 : 2 : Nx2 , 1 : 2 : Ny2) ;92 URzz = UR2zz ( 2 : 2 : Nx2 , 2 : 2 : Ny2) ;93 % CLEAR TEMPORARY VARIABLES94 c l e a r N0 amax cmax p sx sy n N ay cy s ;95 c l e a r UR2 ER2 ER2xx ER2yy ER2zz UR2xx UR2yy UR2zz ;9697 %% PERFORM FINITE−DIFFERENCE FREQUENCY−DOMAIN ANALYSIS98 % FORM DIAGONAL MATERIAL MATRICES99 ERxx = diag ( spar s e (ERxx ( : ) ) ) ;100 ERyy = diag ( spar s e (ERyy ( : ) ) ) ;101 ERzz = diag ( spar s e (ERzz ( : ) ) ) ;102 URxx = diag ( spar s e (URxx ( : ) ) ) ;103 URyy = diag ( spar s e (URyy ( : ) ) ) ;127Appendices E. Finite Difference Frequency Domain MATLAB Code104 URzz = diag ( spar s e (URzz ( : ) ) ) ;105 % COMPUTE DERIVATIVE OPERATORS106 NS = [Nx Ny ] ;107 RES = [ dx dy ] ;108 BC = [−2 −2 0 0 ] ;109 [DEX,DEY,DHX,DHY] = yeeder2d (NS, k0∗RES,BC, k inc /k0 ) ;110 % At the end o f t h i s code , the func t i on code o f theYee g r id de r iva tove111 % opera to r s on a 2D gr id i s inc luded112113 % COMPUTE FIELD MATRIX114 switch pol115 case ’E ’ ,116 A = DHX/URyy∗DEX + DHY/URxx∗DEY + ERzz ;117 case ’H ’ ,118 A = DEX/ERyy∗DHX + DEY/ERxx∗DHY + URzz ;119 otherwise ,120 e r r o r ( ’ Unrecognized p o l a r i z a t i o n . ’ ) ;121 end122 % COMPUTE SOURCE FIELD123 micrometers =1;124 nanometers = micrometers /1000 ;125 w0=1200 ∗ nanometers ;126 xa = [ 0 : Nx−1]∗dx ;127 ya = [ 0 : Ny−1]∗dy ;128 [Y,X] = meshgrid ( ya , xa ) ;129 f s r c = exp(− i ∗( k inc (1 ) ∗X+kinc (2 ) ∗Y) ) ; % plane wavesource130 f s r c = f s r c ( : ) ;131 % COMPUTE SCATTERED−FIELD MASKING MATRIX132 Q = ze ro s (Nx,Ny) ;133 Q( : , 1 :NPML+2) = 1 ;134 Q = diag ( spar s e (Q( : ) ) ) ;135 % COMPUTE SOURCE VECTOR136 f = (Q∗A−A∗Q) ∗ f s r c ;137 % PREPARE MEMORY138 c l e a r NS RES BC DEX DEZ DHX DHZ;139 c l e a r ya X Y f s r c ;140 c l e a r ERxx ERyy ERzz URxx URyy URzz ;141 % COMPUTE FIELD128Appendices E. Finite Difference Frequency Domain MATLAB Code142 F = A\ f ; %backward d i v i s i o n i s usedhere ! !143 F = f u l l (F) ;144 F = reshape (F , Nx,Ny) ;145146 %% COMPUTE DIFFRACTION EFFICIENCIES147 % EXTRACT REFLECTED AND TRANSMITTED WAVES148 Fre f = F ( : ,NPML+1) ;149 Ftrn = F ( : , Ny−NPML) ;150 % REMOVE PHASE TILT151 p = exp(+ i ∗ kinc (1 ) ∗xa ’ ) ;152 Fre f = Fre f .∗ p ;153 Ftrn = Ftrn .∗ p ;154 % COMPUTE SPATIAL HARMONICS155 Fre f = f f t s h i f t ( f f t ( Fre f ) ) /Nx ;156 Ftrn = f f t s h i f t ( f f t ( Ftrn ) ) /Nx ;157 % COMPUTE WAVE VECTOR COMPONENTS OF THE SPATIALHARMONICS158 m = [− f l o o r (Nx/2) : f l o o r (Nx/2) ] ’ ;159 kx = kinc (1 ) − 2∗ pi ∗m/(Nx∗dx ) ;160 kzR = conj ( s q r t ( ( k0∗ n r e f ) ˆ2 − kx . ˆ 2 ) ) ;161 kzT = conj ( s q r t ( ( k0∗ntrn ) ˆ2 − kx . ˆ 2 ) ) ;162 x l e f t = round ( nx1 /2) −round ((1000∗ nanometers ) /dx ) ;163 x r i g h t = round ( nx2 /2) + round ((1000∗ nanometers ) /dx ) ;164 Dr1 yb = round ( ny5 /2) ;165 Dr1 ya = round ( ny5 /2) + round (300∗ nanometers ) /dy ;166 Dr1 y=Dr1 yb : Dr1 ya ;167 Dr2 y=Dr1 ya ;168 Dspp yb=round ( ny4 /2) − round ((50∗ nanometers ) /dy ) ;169 Dspp ya= round ( ny5 /2) ;170 Dspp y=Dspp yb : Dspp ya ;171 F rad1 = F( x l e f t , Dr1 y ) .∗ conj ( F( x l e f t , Dr1 y ) ) ; %f i e l d source172 Dr1 = sum( F rad1 ) ;% f r e e space de t e c t o r l e f tv e r t i c a l |173 F rad2 = F( x l e f t : x r i gh t , Dr2 y ) .∗ conj (F( x l e f t :x r i gh t , Dr2 y ) ) ; % f i e l d source H∗ conj (H)174 Dr2 = sum ( F rad2 ) ;% f r e e space de t e c t o r toph o r i z o n t a l175 F spp = F( x l e f t , Dspp y ) .∗ conj ( F( x l e f t , Dspp y ) ) ;129Appendices E. Finite Difference Frequency Domain MATLAB Code176 Dspp = sum( F spp ) ; % SPP177 % COMPUTE DIFFRACTION EFFICIENCY178 switch pol179 case ’E ’ ,180 R = abs ( Fre f ) . ˆ2 .∗ r e a l (kzR/ kinc (2 ) ) ;181 T = abs ( Ftrn ) . ˆ2 .∗ r e a l (kzT∗ u r r e f / k inc (2 ) /urtrn ) ;182 case ’H ’ ,183 R = abs ( Fre f ) . ˆ2 .∗ r e a l (kzR/ kinc (2 ) ) ;184 T = abs ( Ftrn ) . ˆ2 .∗ r e a l (kzT∗ e r r e f / k inc (2 ) /e r t rn ) ;185 end1 f unc t i on [DEX,DEY,DHX,DHY] = yeeder2d (NS,RES,BC, k inc )2 % YEEDER2D Yee Grid Der ivatove Operators on a 2DGrid3 %4 % [DEX,DEY,DHX,DHY] = yeeder2d (NS,RES,BC, k inc ) ;5 %6 % Input Arguments7 % =================8 % NS [Nx Ny ] 1X gr id s i z e9 % RES [ dx dy ] 1X gr id r e s o l u t i o n10 % BC [ x lo xhi y lo yhi ] boundary c o n d i t i o n s11 % −2: pseudo−p e r i o d i c ( r e q u i r e s k inc )12 % −1: p e r i o d i c13 % 0 : D i r i c h l e t14 % kinc [ kx ky ] i n c i d e n t wave vec to r15 % This argument i s only needed f o r pseudo−p e r i o d i c boundar ies .16 %17 % Note : For normal ized gr ids , use dx=k0∗dx and kinc=kinc /k018 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%19 % VERIFY INPUT/OUTPUT ARGUMENTS20 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%21 % VERIFY NUMBER OF INPUT ARGUMENTS22 e r r o r ( nargchk (3 , 4 , narg in ) ) ;23 % VERIFY NUMBER OF OUTPUT ARGUMENTS24 e r r o r ( nargchk (1 , 4 , nargout ) ) ;130Appendices E. Finite Difference Frequency Domain MATLAB Code25 % EXTRACT GRID PARAMETERS26 Nx = NS(1) ; dx = RES(1) ;27 Ny = NS(2) ; dy = RES(2) ;28 % DETERMINE MATRIX SIZE29 M = Nx∗Ny;30 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%31 % DEX32 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%33 % INITIALIZE MATRIX34 DEX = spar s e (M,M) ;35 % PLACE MAIN DIAGONALS36 DEX = spd iags (−ones (M, 1 ) ,0 ,DEX) ;37 DEX = spd iags (+ones (M, 1 ) ,+1 ,DEX) ;38 % CORRECT BOUNDARY TERMS (DEFAULT TO DIRICHLET)39 f o r ny = 1 : Ny−140 neq = Nx∗(ny−1) + Nx ;41 DEX( neq , neq+1) = 0 ;42 end43 % HANDLE BOUNDARY CONDITIONS ON XHI SIDE44 switch BC(2)45 case −2,46 dpx = exp(− i ∗ kinc (1 ) ∗Nx∗dx ) ;47 f o r ny = 1 : Ny48 neq = Nx∗(ny−1) + Nx ;49 nv = Nx∗(ny−1) + 1 ;50 DEX( neq , nv ) = +dpx ;51 end52 case −1,53 f o r ny = 1 : Ny54 neq = Nx∗(ny−1) + Nx ;55 nv = Nx∗(ny−1) + 1 ;56 DEX( neq , nv ) = +1;57 end58 case 0 , %D i r i c h l e t59 otherwise ,60 e r r o r ( ’ Unrecognized x−high boundary cond i t i on . ’ );61 end62 % FINISH COMPUTATION63 DEX = DEX / dx ;131Appendices E. Finite Difference Frequency Domain MATLAB Code64 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%65 % DEY66 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%67 % INITIALIZE MATRIX68 DEY = spar s e (M,M) ;69 % PLACE MAIN DIAGONALS70 DEY = spd iags (−ones (M, 1 ) ,0 ,DEY) ;71 DEY = spd iags (+ones (M, 1 ) ,+Nx,DEY) ;72 % HANDLE BOUNDARY CONDITIONS ON YHI SIDE73 switch BC(4)74 case −2,75 dpy = exp(− i ∗ kinc (2 ) ∗Ny∗dy ) ;76 f o r nx = 1 : Nx77 neq = Nx∗(Ny−1) + nx ;78 nv = nx ;79 DEY( neq , nv ) = +dpy ;80 end81 case −1,82 f o r nx = 1 : Nx83 neq = Nx∗(Ny−1) + nx ;84 nv = nx ;85 DEY( neq , nv ) = +1;86 end87 case 0 ,88 otherwise ,89 e r r o r ( ’ Unrecognized y−high boundary cond i t i on . ’ );90 end91 % FINISH COMPUTATION92 DEY = DEY / dy ;93 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%94 % DHX95 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%96 % INITIALIZE MATRIX97 DHX = spar s e (M,M) ;98 % PLACE MAIN DIAGONALS99 DHX = spd iags (+ones (M, 1 ) ,0 ,DHX) ;100 DHX = spd iags (−ones (M, 1 ) ,−1 ,DHX) ;101 % CORRECT BOUNDARY TERMS (DEFAULT TO DIRICHLET)102 f o r ny = 2 : Ny132Appendices E. Finite Difference Frequency Domain MATLAB Code103 neq = Nx∗(ny−1) + 1 ;104 DHX( neq , neq−1) = 0 ;105 end106 % HANDLE BOUNDARY CONDITIONS ON XLOW SIDE107 switch BC(1)108 case −2,109 dpx = exp(+ i ∗ kinc (1 ) ∗Nx∗dx ) ;110 f o r ny = 1 : Ny111 neq = Nx∗(ny−1) + 1 ;112 nv = Nx∗(ny−1) + Nx ;113 DHX( neq , nv ) = −dpx ;114 end115 case −1,116 f o r ny = 1 : Ny117 neq = Nx∗(ny−1) + 1 ;118 nv = Nx∗(ny−1) + Nx ;119 DHX( neq , nv ) = −1;120 end121 case 0 ,122 otherwise ,123 e r r o r ( ’ Unrecognized x−low boundary cond i t i on . ’ ) ;124 end125 % FINISH COMPUTATION126 DHX = DHX / dx ;127 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%128 % DHY129 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%130 % INITIALIZE MATRIX131 DHY = spar s e (M,M) ;132 % PLACE MAIN DIAGONALS133 DHY = spd iags (+ones (M, 1 ) ,0 ,DHY) ;134 DHY = spd iags (−ones (M, 1 ) ,−Nx,DHY) ;135 % HANDLE BOUNDARY CONDITIONS ON YLOW SIDE136 switch BC(3)137 case −2,138 dpy = exp(+ i ∗ kinc (2 ) ∗Ny∗dy ) ;139 f o r nx = 1 : Nx140 neq = nx ;141 nv = Nx∗(Ny−1) + nx ;142 DHY( neq , nv ) = −dpy ;133Appendices E. Finite Difference Frequency Domain MATLAB Code143 end144 case −1,145 f o r nx = 1 : Nx146 neq = nx ;147 nv = Nx∗(Ny−1) + nx ;148 DHY( neq , nv ) = −1;149 end150 case 0 ,151 otherwise ,152 e r r o r ( ’ Unrecognized y−low boundary cond i t i on . ’ ) ;153 end154 % FINISH COMPUTATION155 DHY = DHY / dy ;134Appendices FQuality Factor TablesTable F.1: The dip properties of the reflected light intensities from 50 nmof silver sputtered using different deposition parameters at two differentwavelengthsAg (50 nm) λo = 594 nm λo = 632.8 nmSlew rate(%/s)Dep. rate(A˚/s)FWHM (o) Dip QF FWHM (o) Dip QF4.01.0 0.65 ± 0.09 0.48 ± 0.01 0.75 ± 0.10 0.62 ± 0.09 0.44 ± 0.01 0.72 ± 0.1020.0 0.71 ± 0.04 0.49 ± 0.02 0.70 ± 0.06 0.57± 0.06 0.39 ± 0.01 0.70 ± 0.0799.91.0 1.01 ± 0.12 0.60 ± 0.02 0.60 ± 0.08 0.84 ± 0.09 0.57 ± 0.02 0.68 ± 0.0820.0 0.94 ± 0.11 0.53 ± 0.02 0.57 ± 0.07 0.79 ± 0.08 0.37 ± 0.01 0.47 ± 0.05Table F.2: The dip properties of the reflected light intensities from 50 nm ofgold sputtered using different deposition parameters at two different wave-lengthsAu (50 nm) λo = 594 nm λo = 632.8 nmSlew rate(%/s)Dep. rate(A˚/s)FWHM (o) Dip QF FWHM (o) Dip QF4.01.0 4.34 ± 0.08 0.93 ± 0.01 0.21 ± 0.003 2.44 ± 0.09 0.76 ± 0.01 0.31 ± 0.0210.0 4.40 ± 0.44 0.85 ± 0.01 0.20 ± 0.02 2.51± 0.11 0.66 ± 0.01 0.26 ± 0.0199.91.0 4.42 ± 0.43 0.84 ± 0.01 0.19 ± 0.02 2.41 ± 0.07 0.66 ± 0.01 0.27 ± 0.0110.0 4.29 ± 0.32 0.80 ± 0.01 0.19 ± 0.01 2.60 ± 0.09 0.57 ± 0.01 0.22 ± 0.01Table F.3: The dip properties of the reflected light intensities from 40 nmof copper sputtered using different deposition parameters at two differentwavelengthsCu (40 nm) λo = 594 nm λo = 632.8 nmSlew rate(%/s)Dep. rate(A˚/s)FWHM (o) Dip QF FWHM (o) Dip QF4.01.0 4.60 ± 0.14 0.62 ± 0.01 0.134 ± 0.003 2.76 ± 0.11 0.46 ± 0.01 0.167 ± 0.00510.0 3.06 ± 0.04 0.90 ± 0.01 0.29 ± 0.01 1.57 ± 0.08 0.82 ± 0.02 0.53 ± 0.0499.91.0 4.08 ± 0.53 0.48 ± 0.02 0.12 ± 0.02 2.98 ± 0.09 0.61 ± 0.03 0.20 ± 0.0110.0 4.87 ± 0.33 0.78 ± 0.02 0.16 ± 0.01 3.27 ± 0.13 0.81 ± 0.01 0.25 ± 0.01135Appendices GExperimental ToolsIn the experiments of this thesis, we have used various experimentaltools to fabricate nano-layers, and conduct light transmission and reflectionmeasurements, thickness measurements, and morphological studies. Thisappendix will briefly discuss the prominent features of the used experimentaltools.Sputter Deposition SystemSputter deposition is a physical vapor deposition method based uponvaporizing a source material (target) by bombarding the target with Argonions. To discharge the ejected atoms from the target and start film growthon a substrate, the most common approach is to use a magnetron source onthe back of the target.For the experiments in Chapter 4 and 5, we used an Angstrom Engineer-ing Nexdep Deposition System to deposit different metals and dielectricswith various film thicknesses onto microscope glass substrates and siliconnitride membranes.Atomic Force MicroscopeAtomic-force microscopy (AFM) is a very-high-resolution type of scan-ning probe microscopy to measure local properties, such as height, with aprobe. The demonstrated resolution of AFM is 1000 times better than theoptical diffraction limit, in the order of fractions of a nanometer.The AFM images in Chapter 5 are taken by the Bruker Dimension IconAFM using the peak force tipping mode over the scan range of 5µm x 5µm.The measured average surface roughness of silver, gold and copper filmsfabricated using different deposition parameters are shown in the followingtable.136Appendices G. Experimental ToolsTable G.1: The measured average surface roughness of silver, gold and cop-per films fabricated at different deposition and slew rates.Slew rate(%/s)Dep. rate(A˚/s)Average surface roughness (nm)Ag (50 nm) Au (50 nm) Cu (40 nm)4.01.0 7.210 0.702 0.57510.0 / 20.0 2.210 0.727 0.45899.91.0 7.050 0.525 0.47810.0 / 20.0 2.990 1.680 0.545Scanning Electron MicroscopeScanning electron microscopy (SEM) is a type of nano-imaging that usesa focused beam of high energy electrons to produce high-resolution imagesof solid specimens. The interaction of electrons with atoms in the sampleproduces various signals that contain information about the sample’s exter-nal morphology (texture), chemical composition, and crystalline structure.A resolution better than 1 nanometer is achievable by SEM.The SEM images in Chapter 5 are taken by the Tescan Mira3 XMU FieldEmission SEM using the high vacuum mode over various magnification levelsand at different sites of the samples. The presented SEM images in thisthesis are the best taken images at the highest magnification that providesan acceptable resolution.ProfilometerProfilometer is a measuring instrument used to measure the thicknessand the surface roughness of thin films. In Chapter 4, we used a mechanicalstylus-based high resolution profilometer (KLA Tencor Alphastep) to mea-sure the thicknesses of a series of thin-film samples in order to calibrate thequartz crystal monitoring system in the sputtering deposition station.The stylus-based profilometers use a physically moving probe along thesurface to acquire the surface height as a function of position. This processis performed by continuously monitoring the sample force pushing up theprobe as it scans along the surface. The acquired sample forces are thenused to reconstruct the surface. The probe’s shape and tip size can affectmeasurements and restrict resolution. Typical profilometers are capable ofmeasuring small vertical features ranging from 10 nanometres to 1 millime-tre.137Appendices G. Experimental ToolsSpectrometerSpectrometer is an instrument that measures the intensity of light as afunction of wavelength. In this work, we used the high-performance OceanOptics USB4000 spectrometer with a wavelength range, from 200 to 1100nm for conducting the visible light transmission measurements of the silver-coated membranes.Filmetrics F20 AnalyzerFilmetrics are usually used to measure the thickness and optical con-stants of dielectric and semiconductor thin films, where the measured filmsmust be optically smooth and within the thickness range set by the system.Selecting the appropriate mode, filmetrics can also be used for reflection andtransmission measurements. In this work, we used the Filmetrics F20 ana-lyzer with a wide wavelength range from 380 to 1700 nm for conducting thevisible and infrared light transmission measurements of the dielectric-coatedsilver films.138
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Engineering optical properties using layered metamaterials Al Shakhs, Mohammed Hashim 2017
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Title | Engineering optical properties using layered metamaterials |
Creator |
Al Shakhs, Mohammed Hashim |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | This thesis explores the concept of metamaterials; a fairly recent concept in the literature which has attracted the attention of researchers due to their abnormal electromagnetic properties. We will particularly consider one dimensional version of a metamaterial made of layers. At the first glance, layered metamaterials are simply multi-layer thin films. The distinguishing feature of layered metamaterials is that they usually incorporate metals whereas most thin film structures in the past have only incorporated dielectrics. The immense interest in certain layered configurations of metals and dielectrics, particularly when the thicknesses are really thin compared to the wavelength, is due to their exhibition of seemingly counter-intuitive or impossible properties such as refraction to the same side of normal (negative refraction), evanescent wave amplification, or light focusing with a flat interface (flat lensing). The simple configuration of layered metamaterials and their interesting properties are the prime motivations of this work. In this thesis, we first start with a very generic electromagnetic description of the optical properties of layered structures. This general description appears to be novel due to presenting theory in new form. We use this understanding to explain how and why certain layered structures can exhibit negative refraction or flat lensing. This investigation has also led to several new predictions of new optical properties of layered metamaterial structures. We conclude this work by various experimental studies which validate the predictions of the work and also explore fabrication challenges in the making of layered metamaterials. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-08-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0351982 |
URI | http://hdl.handle.net/2429/62525 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical Engineering |
Affiliation |
Applied Science, Faculty of Engineering, School of (Okanagan) |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-09 |
Campus |
UBCO |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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