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Improving the excitation efficiency of Surface Plasmon Polaritons near small apertures in metallic films Mehfuz, Reyad 2013

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Improving the Excitation Efficiency ofSurface Plasmon Polaritons NearSmall Apertures in Metallic FilmsbyReyad MehfuzB.Sc., Bangladesh University of Engineering and Technology, 2004M.Sc., International Islamic University Malaysia, 2008A 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)November 2013c? Reyad Mehfuz, 2013AbstractLight incident onto a small aperture in a metal film can convert intolight waves bound to the surface of that film. At visible frequencies and be-yond, these surface-bound waves are commonly known as surface plasmonpolaritons (SPPs). In this work, we explore ways to enhance the excitationefficiency of SPPs in the vicinity of a slit aperture. We introduce a basicmethod to treat this problem in which the slit and adjacent metal surfaceare approximated as independent waveguides. By mapping out the elec-tromagnetic modes sustained by the waveguide components approximatingthe slit structure, we predict enhanced SPP excitation efficiency when wavevector matching is achieved between the waveguide modes. The concept ofwave vector matching is applied to investigate SPP coupling efficiencies forvarious slit geometries and material configurations. We consider slits withdimensions comparable to the incident wavelength, categorizing this workinto explorations of sub-wavelength and super-wavelength slits. We showthat SPP coupling from a sub-wavelength slit can be enhanced by placing adielectric layer onto the exit side of the metal surface. By varying the layerthickness, it is possible to tune the efficiency of SPP coupling, which can beenhanced significantly (about six times) relative to that without the layer.Broadband enhancement of SPP coupling from a sub-wavelength slit overmost of the visible spectrum is demonstrated using the same method. Wealso show that high-efficiency SPP coupling can be achieved using a super-wavelength slit. We hypothesize that higher-order modes in a large slit canassist wave vector matching and boost SPP coupling. Enhanced SPP ex-citation in a super-wavelength slit aperture is first shown using numericalsimulations, and later verified with experiments. Overall, the thesis demon-strates that simple wave vector matching conditions, similar to classical SPPcoupling methods based on prisms or gratings, can also be applied to de-scribe SPP coupling in small slit apertures. The thesis also provides insightsinto the role of different parameters, such as slit width, dielectric layer thick-ness and surrounding dielectric media, in realizing significant enhancementsin SPP coupling efficiencies.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 three journal articles:? R. Mehfuz, F. A. Chowdhury, and K. J. Chau, ?Imaging slit-coupledsurface plasmon polaritons using conventional optical microscopy?,Optics Express, vol. 20, pp. 10527-10537, 2012.? R. Mehfuz, M. W. Maqsood, and K. J. Chau, ?Enhancing the Effi-ciency of Slit-Coupling to Surface-Plasmon Polariton via DispersionEngineering,? Optics Express, vol. 18, pp. 18206-18216, 2010.? M. W. Maqsood, R. Mehfuz, and K. J. Chau, ?High-throughput diffrac-tion assisted surface-plasmon-polariton coupling by a super-wavelengthslit,? Optics Express, vol. 18, pp. 21669-21677, 2010.Portions of my thesis have also been presented at the following confer-ences:? R. Mehfuz and K. J. Chau, ?Surface plasmon polariton coupling froma slit: can bigger be better?? SPP6, Ottawa, Canada, 2013.? R. Mehfuz and K. J. Chau, ?Far-field detection and imaging of sur-face plasmon polaritons by engineering sub-wavelength slit-structure,?SPIE Optics+Photonics, San Diego, USA, 2012.? R. Mehfuz and K. J. Chau, ?Design of Metal-Dielectric Sub-WavelengthSlit Structure for High Efficiency Coupling of Surface Plasmon Polari-tons,? Photonics North, Ottawa, Canada, 2011.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Early work on light diffraction through apertures . . . . . . . 11.2 Maxwell?s equations . . . . . . . . . . . . . . . . . . . . . . . 41.3 Kirchhoff?s scalar diffraction theory . . . . . . . . . . . . . . . 61.4 Helmholtz-Kirchhoff integral equation for scalar diffraction . 71.5 Kirchhoff?s formulation of scalar diffraction by a plane screen 101.6 Fraunhofer diffraction and the Fourier transform . . . . . . . 121.7 Bethe?s vectorial diffraction theory . . . . . . . . . . . . . . . 151.8 Scalar wave equation for two-dimensional problems . . . . . . 161.9 Angular spectrum representation . . . . . . . . . . . . . . . . 171.10 Scalar diffraction through small slits . . . . . . . . . . . . . . 181.11 Computational solvers for Maxwell?s equations . . . . . . . . 191.11.1 Finite-difference time-domain method . . . . . . . . . 201.11.2 Finite-difference frequency-domain method . . . . . . 211.11.3 Application of FDTD and FDFD to model light trans-mission through a slit . . . . . . . . . . . . . . . . . . 22ivTABLE OF CONTENTS1.12 Recent research interest in light transmission through aper-ture in metallic films . . . . . . . . . . . . . . . . . . . . . . . 221.13 Surface plasmon polaritons . . . . . . . . . . . . . . . . . . . 241.13.1 Excitation of surface plasmon polaritons . . . . . . . . 291.14 Understanding aperture coupling to surface plasmon polaritons 321.15 Improving the excitation efficiency of surface plasmon polaritons 361.16 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Chapter 2: Improving the SPP Coupling Efficiency Near aSub-wavelength Slit . . . . . . . . . . . . . . . . . . 392.1 SPP coupling from a coated slit . . . . . . . . . . . . . . . . . 402.2 Numerical investigation of enhanced SPP coupling . . . . . . 442.2.1 Simulation design . . . . . . . . . . . . . . . . . . . . . 442.2.2 Control studies . . . . . . . . . . . . . . . . . . . . . . 462.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3 Experimental investigation of enhanced SPP coupling . . . . 542.3.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 552.3.2 Experimental design . . . . . . . . . . . . . . . . . . . 552.3.3 SPP measurement . . . . . . . . . . . . . . . . . . . . 572.3.4 SPP coupling efficiency measurements . . . . . . . . . 622.3.5 SPP propagation distance measurements . . . . . . . . 652.3.6 SPP lensing using an array of holes . . . . . . . . . . . 662.4 Broadband enhancement of SPP coupling . . . . . . . . . . . 672.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 672.4.2 Experimental design . . . . . . . . . . . . . . . . . . . 682.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 702.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Chapter 3: Improving the SPP Coupling Efficiency Near aSuper-wavelength Slit . . . . . . . . . . . . . . . . . 743.1 Hypothesis: enhanced SPP coupling due to higher-order modes 753.2 Simulation design . . . . . . . . . . . . . . . . . . . . . . . . . 813.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 813.4 Experimental design . . . . . . . . . . . . . . . . . . . . . . . 863.5 Experimental SPP coupling efficiency measurements . . . . . 923.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Chapter 4: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 97Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102vTABLE OF CONTENTSAppendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Appendices A:Drude Model and its Limitations . . . . . . . . . . . 112Appendices B: Fabrication Tools . . . . . . . . . . . . . . . . . . . . 117viList of TablesTable 2.1 Control simulation geometries and results. Fixed pa-rameters include w = 150 nm, d = 100 nm, t = 300 nm,and ?0 = 500 nm. . . . . . . . . . . . . . . . . . . . . . 47Table A.1 Plasma frequency and collision rate measured by Ze-man and Sachts for Ag, Au, and Al . . . . . . . . . . . 113Table B.1 Spin speed of the Laurell WS-400B-6NPP-LITE coaterversus the expected thickness and experimentally mea-sured PMMA layer thickness . . . . . . . . . . . . . . 119viiList of FiguresFigure 1.1 Huygens? secondary wavelet analysis of a wavefrontpassing through an opening in an opaque screen. . . 2Figure 1.2 Image of the Arago spot in the shadow of a circularobstacle placed in a coherent beam. This image wasused with permission. . . . . . . . . . . . . . . . . . . 3Figure 1.3 Theoretical construct consisting of a closed surface Sassociated with a volume V and surrounding an ob-servation point P0 used to illustrate Green?s theorem.Note that we take the normal direction to correspondto the outward normal. . . . . . . . . . . . . . . . . . 8Figure 1.4 Application of the Helmholtz-Kirchhoff integral theo-rem to solve for the scalar disturbance at an observa-tion point P0 when an opaque screen with an openingis placed between a source and the observation point. 11Figure 1.5 Configuration used to analyze scalar diffraction froman opening in an opaque screen. . . . . . . . . . . . . 13Figure 1.6 Configuration considered by Kowarz to analyze near-field diffraction from a slit. . . . . . . . . . . . . . . . 19Figure 1.7 Three-dimensional Yee Grid depicting the electric andmagnetic field components distributed over a discretespatial grid. . . . . . . . . . . . . . . . . . . . . . . . 21Figure 1.8 Application of the FDTD and FDFD method to solvethe two-dimensional problem of TM-polarized plane-wave incident onto a slit in a metallic film. Figure (a)shows the schematic of the simulation grid consistingof a metallic film with a small opening, immersed ina dielectric. The simulation space is surrounded by aperfectly matched layer (PML). Figures (b) and (c)show the calculated magnetic field distribution aboutthe aperture obtained using the FDFD and FDTDmethod, respectively. . . . . . . . . . . . . . . . . . . 23viiiLIST OF FIGURESFigure 1.9 A single interface between two disparate media can,under certain conditions, sustain surface waves thatobey Maxwell?s equations. . . . . . . . . . . . . . . . 25Figure 1.10 Schematic of the TM-polarized electromagnetic wavesand surface charges associated with SPPs at the in-terface of a metal and dielectric. . . . . . . . . . . . 27Figure 1.11 Dispersion relation of a SPP wave and a free-spaceplane wave. kspp and k0 are the wave vectors of a SPPwave and free-space plane wave, respectively. At anygiven frequency, kspp is larger than k0. . . . . . . . . . 28Figure 1.12 Schematic of light coupling to SPPs by using a prism.(a) Mechanism of wave vector matching by prism cou-pling, manifested in the (b) Kretschmann configura-tion and (c) Otto configuration. . . . . . . . . . . . . 30Figure 1.13 Schematic of free-space light coupling to SPPs usinga grating immersed in free-space. . . . . . . . . . . . 31Figure 1.14 Schematic of the coupling of free-space light to SPPsby an aperture. . . . . . . . . . . . . . . . . . . . . . 33ixLIST OF FIGURESFigure 1.15 (a) Three-dimensional plasmonic nano-focusing of lightwith asymmetrically patterned silver pyramids show-ing i) a scanning electron microscope (SEM) imageof the device, ii) a side-view of the scanning confo-cal Raman image at an incident wavelength of 710nm. (b) Splitting of surface plasmons into two chan-nels, showing i) a SEM image and ii) a near-field op-tical image at an incident wavelength of 1600 nm.(c) Sub-wavelength focusing of surface plasmon to a250-nm-wide Ag strip guide with a curved array ofholes, showing i) a SEM image (light gray, Ag anddark gray, Cr) and ii) a near-field optical image ofSPP focusing and guiding at a wavelength of 532 nm.(d) Surface plasmon propagation along a 18.6?m longsilver nano-wire at an incident wavelength of 785 nm,showing i) a sketch of optical excitation where I is theinput and D is the end of the wire, ii) a conventionaloptical microscopic image showing the bright spot tothe left is the focused exciting light, iii) a near-fieldoptical image corresponding to the white box in ii,and iv) a 2-?m-long cross-cut along the chain dottedline in iii. (e) Surface plasmon waveguiding, showingi) a SEM image and ii) photon scanning tunnelingmicroscopy image at an wavelength of 800 nm. . . . . 34Figure 1.16 (a) Basic two-dimensional configuration to be studiedconsisting of a slit in a metallic film. The slit struc-ture is conceptually divided into two uni-axial semi-infinite waveguides: (b) a vertical metal-dielectric-metal (MDM) waveguide, which sustains longitudi-nal and transverse wave vector components kz andkx, respectively, and (c) a horizontal metal-dielectric(MD) waveguide, which sustains a longitudinal wavevector component kspp along the metal-dielectric in-terface. Wave vector matching is achieved when thetransverse component of the wave vector in the MDMwaveguide matches with the longitudinal componentof the wave vector in the MD waveguide. . . . . . . . 37xLIST OF FIGURESFigure 2.1 (a) Real-space (left) and k-space (right) depictions ofthe modes scattering from the exit of a slit in a metalfilm when the metal film is completely immersed ina dielectric. (b) Real-space (left) and k-space (right)depictions of the modes scattering from the exit ofa slit in a metal film when the slit is filled with adielectric and the metal film is coated with a dielectriclayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.2 Simulation geometry used to study SPP coupling froman illuminated slit. The detectors D1 and D2 capturethe SPP modes, and the detector D3 captures the ra-diating modes. . . . . . . . . . . . . . . . . . . . . . 45Figure 2.3 Image of the FDTD-calculated instantaneous |Hy|2distribution for a slit of width w = 150 nm illumi-nated by a quasi-plane-wave of wavelength ?0 = 500 nm.The simulation geometry is depicted by the abovegraphic. Lines indicating the position of detectorsD1, D2, and D3 have been superimposed on the image. 48Figure 2.4 Basic principle used to enhance slit-coupling to SPPmodes near a sub-wavelength slit in a metallic film.(a) Dispersion curve of nspp as a function of ?0 forvarious d calculated from the roots of the eigenvalueequation for the SPP mode on an asymmetric, three-layer silver-glass-air waveguide. (b) nspp as a functionof d for ?0 = 500 nm. The dotted gray line corre-sponds to the refractive index of a plane-wave modein glass,?d. . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.5 Image of the FDTD-calculated instantaneous |Hy|2distribution for a slit of width w = 150 nm illumi-nated by a quasi-plane-wave of wavelength ?0 = 500 nmwhen a dielectric layer of thickness (a) 100 nm and (b)500 nm is placed on the metal film. The simulationgeometry is depicted by the above graphic. . . . . . 50Figure 2.6 (a) The time-averaged SPP intensity, ISPP (blue squares),and time-averaged radiated intensity, Ir (red circles),and (b) the corresponding SPP coupling efficiency, ?,as a function of d. The error bars describe the uncer-tainties in the measurement of ISPP and Ir due to,respectively, the finite amount of Ir captured by D1and D2 and the finite amount of ISPP captured by D3. 51xiLIST OF FIGURESFigure 2.7 SPP wavelength measured from the FDTD simula-tions (blue squares) and predicted from the modesolver (red line) as a function of dielectric layer thick-ness d. The dotted gray line indicates the value of?i = ?0/?d. The error bars describe the uncertaintyin the measurement of ?spp from the FDTD simula-tions due to variation in ?spp as a function of distancefrom the slit exit. . . . . . . . . . . . . . . . . . . . . 52Figure 2.8 Image of the FDTD-calculated instantaneous |Hy|2distribution for a slit illuminated by a quasi-plane-wave of wavelength ?0 = 500 nm when the slit widthis (a) 150 nm and (b) 300 nm. The dielectric layerthickness d = 100 nm is held constant. The simula-tion geometry is depicted by the above graphic. . . . 53Figure 2.9 a) The time-averaged SPP intensity, ISPP (blue squares),and time-averaged radiated intensity, Ir (red circles),and (b) the corresponding SPP coupling efficiency, ?,as a function of w. The error bars describe the uncer-tainties in the measurement of ISPP and Ir due to,respectively, the finite amount of Ir captured by D1and D2 and the finite amount of ISPP captured by D3. 54Figure 2.10 Simulation procedure to determine an upper boundfor the slit width, for an arbitrary set of parameters,when the slit is filled with a dielectric and loaded witha dielectric layer. Image of the FDTD-calculated in-stantaneous |Hy|2 distribution for a slit illuminatedby a quasi-plane-wave of wavelength ?0 = 500 nmwhen the slit width is (a) 150 nm, (b) 200 nm, (c)250 nm, (d) 300 nm, (e) 350 nm, and (f) 450 nm. Dispersion-less glass with the refractive index of 1.5 is used as thedielectric. The dielectric layer thickness d = 100 nmis held constant. Note the transition of the field pro-file in the slit as the slit width increases. . . . . . . . 56xiiLIST OF FIGURESFigure 2.11 (a) Re[kspp] corresponding to SPP modes propagat-ing along a silver metal surface coated with a dielec-tric layer of refractive index n = 1.5 and surroundedby air, for various layer thickness values, along withthe nk0 line. Complex kspp values are calculated bysolving the dispersion relation of a semi-infinite three-layer silver-glass-air waveguide, where the permittiv-ity of silver is fitted to experimental data. The circleshighlight, for a given dielectric layer thickness, thefrequency at which the momentum matching condi-tion Re[kspp] = nk0 is satisfied. It should be notedthat the momentum matching condition is an approx-imation and provides only a first-order procedure toestimate the optimal dielectric layer thickness. (b)Schematic of the experimental set-up. Polarized lightfrom a He-Ne laser (?0 = 632.8 nm) illuminates thesample and the far-field transmission image is cap-tured by an optical microscope (Zeiss Axio Imager)using a 100? objective lens with a numerical apertureof 0.90 in air ambient and recorded using a Si CCDcamera. . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 2.12 Representative microscope images of the slit and groovesfor (a) an uncoated sample, (b) a coated sample withPMMA layer thickness d = 80 nm, and (c) a coatedsample with PMMA layer thickness d = 120 nm. Thewidth of the slits is w = 150 nm. The left columnshows scanning electron microscope (SEM) images,and the middle and right columns show optical mi-croscope images under x-polarized and y-polarized il-lumination, respectively. The color of the optical mi-croscope images has been modified for clarity, but theimages are otherwise unprocessed. . . . . . . . . . . . 59xiiiLIST OF FIGURESFigure 2.13 Experiment to distinguish diffraction from a slit andSPP scattering from adjacent grooves. (a) SEM im-age of a representative sample consisting of two iden-tical slits of width w = 150 nm, one of which is flankedby grooves. The sample is coated with a PMMA layerof thickness d = 80 nm. Optical microscope image ofthe sample under (b) x-polarized illumination and (c)y-polarized illumination. We apply a subtraction pro-cedure to images (b) and (c) in which the region R2 issubtracted from R1. The resulting subtracted imagederived from (b) show bright spots at the groove loca-tion indicative of SPP scattering. These bright spotsare absent in the subtracted image derived from (c),suggesting the absence of SPPs. . . . . . . . . . . . . 61Figure 2.14 (a) SPP coupling efficiency, ?, as a function of thePMMA layer thickness, for a fixed slit width of w =150 nm. ? is measured using two methods. In thefirst method, labeled ?expt-1?, ? is calculated by us-ing Ing,L and Ing,R derived from the image of theslit and grooves under y-polarized illumination andthen using Eq. (2.13) (red circles). In the secondmethod, labeled ?expt-2?, ? is calculated by usingIng,L and Ing,R derived from the image of the slit withno grooves under x-polarized illumination and thenusing Eq. (2.13) (magenta diamond). We calculate ?from two-dimensional FDTD simulations modeling x-polarized plane-wave illumination of a coated slit forvarious d values (blue squares). We also calculate, forthe optimal case of d = 80 nm, the SPP coupling ef-ficiency when a 20- nm-deep dimple is present in thedielectric layer above the slit (cyan square), whichemulates possible non-planarity of the polymer layerdue to conforming to the slit walls. (b) shows the SPPcoupling efficiency measured using the method ?expt-1? (red circles) and calculated using FDTD simula-tion (blue squares) as a function of the slit width. Theerror bars in (a) and (b) correspond to the varianceof five independent measurements. . . . . . . . . . . 63xivLIST OF FIGURESFigure 2.15 (a) SEM image of an array of identical w = 150 nmslits where the groove spacing from the slits is var-ied from s = 1?m to s = 8?m in 1?m increments.The array is coated with a PMMA layer of thicknessd = 80 nm. (b) shows the corresponding microscopeimage of the array. (c) depicts profiles of the imageintensity along horizontal lines intersecting the slitand grooves for various slit-groove separation values.(d) The integrated SPP intensity normalized to theintegrated intensity of the slit as a function of the slit-groove separation (red circles), where the blue linecorresponds to an exponential fit. . . . . . . . . . . . 64Figure 2.16 Measurement of a focused SPP beam emitted froma curved array of sub-wavelength holes using opti-cal microscopy. (a) SEM image of three plasmoniclenses consists of a curved array of holes, with dif-ferent groove patterns milled adjacent to the lenses.The lenses consist of 17 holes, each of diameter '200 nm, milled into a semi-circle of radius 5?m. Op-tical microscope image of the three lenses under (b)x-polarized illumination and (c) y-polarized illumina-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 2.17 (a) Experimental setup to characterize broadband SPPcoupling using a slit-groove structure. Light from atungsten light source is directed through a polarizerand is incident onto the slit structure. An image ofthe top of the slit structure is collected by a 100 ? ob-jective lens with numerical aperture of 0.9. The spec-tral data are collected by a spectrum analyzer, whichis placed in the image plane of the microscope. (b,top) Spectrum of the light that is supplied from thetungsten source and (b, bottom) transmitted throughthe various slit structures. . . . . . . . . . . . . . . . 69xvLIST OF FIGURESFigure 2.18 (a) SEM image of a slit-groove structure with slit-to-groove separation distances of 3?m and 6?m. (b)Normalized intensity profile measured in the hori-zontal direction across the slit and grooves extractedfrom a transmission-mode microscope image of thestructure shown in (a). (c) and (d) show the spectracollected by sampling at slit-to-groove separation dis-tances of 3?m and 6?m. (e) and (f) plot the ratioof the groove spectra of the coated samples to thatof the uncoated samples for measurements taken atslit-to-groove separation distances of 3 and 6?m. . . 71Figure 2.19 (a) SEM image of a slit-groove structure with slit-to-groove separation distances of 8?m and 10?m.(b) Normalized intensity profile measured in the hor-izontal direction across the slit and grooves extractedfrom a transmission-mode microscope image of thestructure shown in (a). (c) and (d) show the spec-tra collected by sampling at slit-to-groove separationdistances of 8?m and 10?m, respectively. (e) and(f) plot the ratio of the groove spectra of the coatedsamples to that of the uncoated samples for measure-ments taken at slit-to-groove separation distances of8 and 10?m, respectively. . . . . . . . . . . . . . . . 72Figure 3.1 Magnetic field profiles of the (a) TM0 and (b) TM1modes at ?0 = 6.0 ? 1014 Hz in an infinite metal-dielectric-metal (MDM) waveguide having a dielectriccore thickness w = 150 nm. Silver is used as the metaland the core consists a dielectric of index 2.5. Thefield profiles are normalized so that the modes carryunit power in the upward direction. . . . . . . . . . . 76xviLIST OF FIGURESFigure 3.2 Formulation of a hypothesis for diffraction-assistedSPP coupling by a super-wavelength slit aperture. (a)Figure-of-merit and (b) the real transverse wave vec-tor component versus frequency and wavelength forTM0 and TM1 modes sustained in slits of differentwidths. (c) Diffraction spectrum corresponding tothe TM0 mode in a 200- nm-wide slit and the TM1modes in 350 nm-wide and 500 nm-wide slits. (d)Wave vector-space depiction of diffraction-assisted SPPcoupling from slits of width w = 200 nm, w = 350 nm,and w = 500 nm, immersed in a uniform dielectric ofrefractive index n = 1.75 . . . . . . . . . . . . . . . . 78Figure 3.3 Images of the FDTD-calculated instantaneous |Hy|2distribution (left) and the time-averaged |Hy|2 angu-lar distribution (right) for a slit of width values (a)w = 150 nm, (a) w = 350 nm, and (a) w = 500 nmimmersed in a dielectric (n=1.75) and illuminated bya quasi-plane-wave of wavelength ?0 = 500 nm. Acommon saturated color scale has been used to ac-centuate the fields on the exit side of the slit. . . . . . 82Figure 3.4 (a) SPP coupling efficiency as a function of optical slitwidth for dielectric refractive index values n = 1.0(squares), n = 1.5 (circles), n = 1.75 (upright tri-angles), n = 2.0 (inverted triangles), n = 2.5 (dia-monds). (b) The measured SPP intensity (circles),radiative intensity (squares), and total intensity (dia-monds). The shaded region indicates the sub-wavelength-slit-width regime. . . . . . . . . . . . . . . . . . . . . 84Figure 3.5 Wave vector mismatch as a function of refractive in-dex of the dielectric region for a fixed optical slitwidth nw = 600 nm and free-space wavelength ?0 =500 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 85xviiLIST OF FIGURESFigure 3.6 (a) SPP coupling from a slit in an opaque gold film.High-efficiency SPP coupling is achieved when thewave vector of light leaving the slit is wave vectormatched with the SPP mode. Frequency dependenceof the real parts of the (b) longitudinal and (c) lat-eral wave vector of the lowest-order modes in a Au-dielectric-Au waveguide, which approximates the slitstructure. The refractive index of the dielectric is as-sumed to be 1.4. Note the large increase in the lateralwave vector of the first-order mode relative to that ofthe zero-order mode, which provides a wave vectorboost needed for SPP coupling. . . . . . . . . . . . . 88Figure 3.7 Schematic of the experimental set-up. The slit-groovestructures etched into a gold film are immersed in ahigh-index fluid. TM-polarized light from a He-Nelaser (?0 = 632:8 nm) illuminates the sample and thefar-field transmission image is captured using a CCDcamera by oil immersion microscopy (100x objectivelens and numerical aperture of 1.3). We place a coverslide and silicone oil successively on top of the highindex fluid to avoid the possible damage of the eye-piece (as most of the high index fluid reacts). . . . . 89Figure 3.8 Scanning electron microscope of a typical slit struc-ture etched into a 300-nm-thick Au film by focusedion beam milling. The whitish boundary at the edgeof the slit represents the blunted slit edges resultingfrom bombardment of a spherical ion beam. . . . . . 91Figure 3.9 (a) Representative SEM image of a sample of twoidentical slit structures (w = 450 nm) where the bot-tom slit is flanked by two grooves. (b) The far fieldimage of the sample when it is immersed in a highindex fluid (n = 1.7). The color map of the opti-cal microscope images has been modified for clarity,but the images are otherwise unprocessed. (c) Thenormalized intensity profile obtained from the areabounded by dotted boxes in (b). Note the enhancedlight intensity of the side lobes next to the slit due tothe presence of the grooves. . . . . . . . . . . . . . . . 92xviiiLIST OF FIGURESFigure 3.10 FDFD simulations of TM-polarized illumination (?0=632.8 nm) of a set of tapered slits (with parametersmatching slit pairs with a nominal width of w =350 nm used in the experiments) immersed in a di-electric medium where n = 1.6. (a) Energy densitydistribution for the case where the slit is (a) withoutgrooves and (b) with grooves. (c) Normalized energydensity profiles extracted from the simulations alongthe magenta line shown in (a) and (b), which approx-imates the object plane imaged by the microscope. . . 93Figure 3.11 SPP coupling efficiency (?) as a function of the exitslit width (w2) when immersed in index fluids of re-fractive index (a) 1.5, (b) 1.6, and (c) 1.7. . . . . . . 95Figure 3.12 FDFD simulation of the electromagnetic response of aslit structure with a nominal width of w = 350 nm at?0 = 632.8 nm, with a zoomed-in section highlightingthe fields at the slit exit. (b) Two-dimensional Fouriertransformation applied to the fields at the slit exit,revealing large spatial frequency components alongthe horizontal (x) axis. (c) Plot of Fourier amplitudeat the spatial frequency kspp as a function of the slitwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.1 SPP coupling efficiency as a function of dielectriclayer thickness calculated at representative wavelengthsin the infrared, visible, and UV. For the case of UVillumination, the metal is aluminum, the slit widthis 50 nm, and the incident wavelength is 250 nm. Forthe case of visible illumination, the metal is silver, theslit width is 100 nm, and the incident wavelength is500 nm. For the case of IR illumination, the metal isgold, the slit width is 150 nm, and the incident wave-length is 822 nm. . . . . . . . . . . . . . . . . . . . . . 100Figure A.1 Permittivity values of gold predicted by the Drudemodel and obtained from experimental measurements 114Figure A.2 Permittivity values of silver predicted by the Drudemodel and obtained from measurements . . . . . . . . 115Figure A.3 Permittivity values of aluminum predicted by the Drudemodel and obtained from measurements . . . . . . . . 116xixAcknowledgementsI have been fortunate to be surrounded by many wonderful people, andI want to take this opportunity to thank them all for their love, support andencouragement. First of all, I would like to express my sincere gratitude tomy PhD advisor, Dr. Kenneth Chau, for his relentless support and guidancethroughout my Doctoral studies. His confidence in my abilities was crucialfor my smooth transition to become a contributor of knowledge in the excit-ing area of nano-scale photonics. His scientific brilliance, boundless energyand exemplary vision will forever be an inspiration.I am thankful to Dr. Jonathan Holzman, Dr. Stephen O?Leary andDr. Murray Neumann for their valuable comments and suggestions thathelped enrich my work. I am grateful to Dr. Deborah Roberts for allowingme to use her microscope for the experimental portions of this work. Iwould like to thank Dr. Karen Kavanagh of Simon Fraser University (SFU)for supporting my research activities at the Nanoimaging facilities. I amspecially thankful to Dr. Li Yang and Chris Ballicki of 4D labs, SFU fortraining me on different nanofabrication tools.I have spent wonderful moments with my extraordinary research col-leagues. Waqas Maqsood, Faqrul Alam Chowdhury, Samuel Schaefer, Mo-hammed Al-Shakhs and Max Bethune-Waddell have been a consistent sourceof support, sharing their views on different research and non-research issues.During his short stay in Kelowna, Dr. Peter Ott from Heilbronn Universityhas tremendously motivated me through his excellent work ethic.I am grateful to Brad Armstrong for assisting me in many ways and es-pecially for getting me a quiet study room. Special thanks and appreciationto my friends in Kelowna, who made me feel at home despite being 8000miles away from home.Finally, it is my parents, in-laws and family members who have workedbehind the scene to always keep me motivated to go ahead. It will be futileto give simply thanks to them. Without the love, care and patience of myparents, wife and kids I would not have been able to do what I have done.xxDedicationTo four caring ladies: my mother, wife, daughter and sisterxxiChapter 1IntroductionThis thesis examines the interaction between electromagnetic waves anda small aperture in a metallic film. How small is small? We will focusour attention to a regime in which the aperture dimension is comparable tothe wavelength of the optical wave, parsing the work into the study of anaperture having sub-wavelength (smaller than the wavelength) and super-wavelength (greater than the wavelength) dimensions. We will also restrictour efforts to understanding and improving the process in which an opticalwave incident onto an aperture in a metallic film converts into a surface waveimmediately adjacent to the aperture exit. This phenomenon has been gain-ing greater research attention in recent years because it offers a means tocontrol light on very small size scales and may have potential application inminiaturized light-based devices. To treat this problem, we will develop andpropose a basic physical model to describe wave coupling in the vicinity ofan aperture in a metallic film, drawing upon approximate scalar diffractiontheories and analytical solutions to Maxwell?s equations. Predictions madeusing the physical model will be validated by two independent approaches:first, we will use computer simulations solving Maxwell?s equations to modeloptical wave interaction with apertures in metallic films in various config-urations and second, apertures in metallic films will be fabricated over alarge parameter range and their optical wave response will be experimen-tally measured. The result of this thesis work is a robust model of surfacewave excitation near small apertures and several experimentally and nu-merically validated approaches to optimize the efficiency of this excitationmechanism.1.1 Early work on light diffraction throughaperturesBefore proceeding further, it is appropriate to describe the rich historicalcontext of this topic and how it has evolved to become a field of modernresearch. The interaction of light with small apertures is a classical problem,11.1. Early work on light diffraction through aperturesFigure 1.1: Huygens? secondary wavelet analysis of a wavefront passingthrough an opening in an opaque screen.spanning several centuries, and has been instrumental to the formation ofour current understanding of light. In the 15th century, observations of theblurred edges of a shadow cast by a small pinhole in an opaque screen couldnot be explained by the predominant corpuscular theory of light champi-oned by Isaac Newton, which predicted that the shadow of a small pinholewould have sharp, well-defined borders due to the purely rectilinear prop-agation of light particles along well-defined rays. In 1670, Christiaan Huy-gens proposed an alternative picture viewed light as a collection of sphericalwaves forming a wavefront. Each point of the wavefront in turn becomesa source of secondary spherical waves which have tangents that collectivelyform an envelope defining the wavefront at a subsequent time and a fartherposition. Although this wave picture could explain the spreading shadowcast by a pinhole in terms of the natural spreading of spherical wavelets(refer to Fig. 1.1), it was not widely accepted over the corpuscular the-ory of light, due in large part to Newton?s enormous influence. It wasnot until 1803 that further evidence of the wave nature of light was pro-vided by Thomas Young, who experimentally showed that light transmittedthrough two closely spaced slits formed a diffraction pattern consisting ofbright and dark fringes. Later, in 1816, Augustin-Jean Fresnel modifiedHuygens? principle by adding the concept of wave interference. The brightand dark regions of a diffraction pattern due to light transmission throughslits could be explained by Huygens-Frensel theory in terms of the construc-tive and destructive interference of secondary wavelets emanating from the21.1. Early work on light diffraction through aperturesFigure 1.2: Image of the Arago spot in the shadow of a circular obstacleplaced in a coherent beam. This image was used with permission from Ref[1].slits. The theory was initially deemed to be incorrect because it made seem-ingly strange predictions - for example, it predicted that the center of theshadow of a small circular disk would exhibit a bright spot. However, thetheory was later vindicated by the experimental observation of this effect byDominique-Franois-Jean Arago (and hence named the ?Arago spot?). Al-though Fresnel?s wave theory of light was successful in describing diffractionthrough apertures, many at the time still supported the corpuscular the-ory of light. It was not until James Maxwell?s transcendental work on thedynamical equations of the electromagnetic field and the subsequent exper-imental verification by Heinrich Hertz of electromagnetic wave propagationpredicted by Maxwell?s equations that an electromagnetic wave picture oflight became fully entrenched in the scientific community.31.2. Maxwell?s equations1.2 Maxwell?s equationsMaxwell?s equations, published in 1873, are a set of differential equationsthat relate four vector fields - the displacement field, ~D, the electric field, ~E,the magnetic flux density, ~B, and the magnetic field, ~H - to the presence ofsources of fields - the charge density, ?, and the current density, ~J [2]. Theequations are given by~? ? ~D(~r, t) = ?, (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)where ~r is the position vector and t is the time variable. Since there aresix variables and only four equations, Maxwell?s equations as written aboveare not solvable. However, by imposing at least two constitutive relations- independent equations relating the vector fields and sources - it becomespossible to obtain analytical solutions. The most common constitutive re-lation is that the displacement field ~D and the magnetic flux density ~B arelinearly related to the electric field ~E and magnetic field ~H, respectively,giving~D(~r, t) = ro ~E(~r, t), (1.5)and~B(~r, t) = ?r?o ~H(~r, t), (1.6)where r is the relative electric permittivity, o is the free-space permit-tivity, ?r is the relative magnetic permeability, and ?o is the free-spacepermeability. Since most materials are non-magnetic at optical frequencies,we typically set ?r = 1. An additional assumption commonly made is thatthe current density ~J is linearly related to the electric field ~E, resulting ina relation widely known as Ohm?s law given by~J(~r, t) = ? ~E(~r, t), (1.7)where ? is the electric conductivity. Together, the constitutive relationsgiven by Eqs. 1.5, 1.6, and 1.7, correspond to a medium that is deemed tobe linear, isotropic, homogeneous, and instantaneously responsive. To revealthe wave nature of light implicit to Maxwell?s equations in such a medium,41.2. Maxwell?s equationslet us first make the assumption that a region is source-free, so that ? = 0and ~J = 0. Maxwell?s equations can then be succinctly re-written as~? ? ~E(~r, t) = 0, (1.8)~? ? ~H(~r, t) = 0, (1.9)~? ? ~E(~r, t) = ??r?o? ~H(~r, t)?t, (1.10)and~? ? ~H(~r, t) = ro? ~E(~r, t)?t. (1.11)The modified Maxwell?s equations can be manipulated into the form ofwave equations for the electric and magnetic fields. Taking the curl of bothsides of Eq. 1.10 and setting ?r = 1 results in~? ? ~? ? ~E(~r, t) = ~? ?(??o? ~H(~r, t)?t)= ??o??t(~? ? ~H(~r, t)). (1.12)Using Eqs. 1.5 and 1.11, Eq. 1.12 simplifies to~? ? ~? ? ~E(~r, t) = ??o?2 ~D(~r, t)?t2= ??oor?2 ~E(~r, t)?t2. (1.13)The left side of Eq. 1.13 can be written as~? ? ~? ? ~E(~r, t) = ~?(~? ? ~E(~r, t))? ~?2 ~E(~r, t), (1.14)which, in a source-free region, simplifies to~? ? ~? ? ~E(~r, t) = ?~?2 ~E(~r, t). (1.15)Combining Eq. 1.13 and Eq. 1.15, we get~?2 ~E(~r, t)? ?oor?2 ~E(~r, t)?t2= 0. (1.16)By defining the speed of light through the free-space permittivity and per-meability asc =1??oo, (1.17)51.3. Kirchhoff?s scalar diffraction theorywe arrive at a wave equation describing the electric field~?2 ~E(~r, t)?rc2?2 ~E(~r, t)?t2= 0, (1.18)which is also known as the Helmholtz wave equation. A similar treatmentcan be applied to yield a wave equation for the magnetic field, given by~?2 ~H(~r, t)?rc2?2 ~H(~r, t)?t2= 0. (1.19)Thus, Maxwell?s equations introduced the revolutionary concept that lightpropagates as a vector wave for both the electric and magnetic fields. Whilethis theory was initially met with resistance, the experimental verificationof the existence of electromagnetic waves at microwave frequencies by Hein-rich Hertz in 1888, and the countless experimental observations of electro-magnetic wave phenomena since have conclusively proved the validity andveracity of Maxwell?s theory.1.3 Kirchhoff?s scalar diffraction theoryWith the establishment of Maxwell?s equations, there was sufficient math-ematical theory to tackle the problem of light diffraction through an aper-ture. The first major contribution after the advent of Maxwell?s equationswas provided by Gustav Kirchhoff, who in 1882, introduced scalar diffrac-tion theory to accurately model the physical diffraction principles put forthby Huygens and Fresnel. Although scalar diffraction theory makes severalassumptions (which will be discussed here) that in fact make it an invalidsolution to Maxwell?s equations, it has been highly influential due to itspredictive powers and simplicity of use. To this day, introductory opticstextbooks use Kirchhoff?s scalar theory to derive the well-known Fraunhoferand Fresnel descriptions of light diffraction in the far-and near-field, respec-tively. Due to the importance of scalar diffraction theory in modern opticsand its use in this thesis, we will detail its foundation, its critical assump-tions, and various ways in which it has been used to describe light diffractionthrough an aperture. For interested readers, excellent discussions of scalardiffraction theory are given in the textbooks Introduction to Fourier Opticsby Joseph Goodman [3] and Principles of Optics by Max Born and EmilWolf [4].The starting point of scalar diffraction theory is the postulation of agoverning wave equation to solve. In contrast to the vector electric and61.4. Helmholtz-Kirchhoff integral equation for scalar diffractionmagnetic wave equations shown above to be derivable from Maxwell?s equa-tions, scalar diffraction theory assumes that the electric and magnetic fieldscan be treated as scalar quantities which obey a scalar wave equation?2 ?1c2?2u(~r, t)?t2= 0. (1.20)where u(~r, t) is a scalar function representing either the electric or magneticfield in free-space propagating at the speed of light. Equation 1.20 ignoresthe vectorial coupling between the electric and magnetic fields, as describedby Maxwell?s equations, which is a valid assumption in an isotropic andhomogeneous medium, but is not valid near boundaries. Nonetheless, let?sproceed with this line of thought and explore its implications.To analytically solve the scalar wave equation, let?s assume that thefunction u describes a monochromatic waveu(~r, t) = U(~r)e?i?t, (1.21)where U is the amplitude and ? is the frequency (the position and timedependences will be dropped in subsequent discussions for brevity). Sub-stitution of this candidate solution into the scalar wave equation yields thetime-independent Helmholtz equation expressed as(?2 + k2)U = 0, (1.22)where k is the wave number given by k = 2pi?/c = 2pi/?0 and ?0 is thefree-space wavelength.1.4 Helmholtz-Kirchhoff integral equation forscalar diffractionThe Huygens-Fresnel theory postulated that light diffraction can beviewed as the superposition of the secondary waves emanating from a sur-face. Kirchhoff made this picture mathematically rigorous, demonstratingthat the Huygens-Fresnel concept was an approximation to an integral solu-tion to the Helmholtz equation. This integral solution, derived from Green?stheorem, enabled the field at an arbitrary point to be expressed in termsof the value of the solution, and its first derivative, at points on a surfaceenclosing that point. Thus, Kirchhoff transformed the physical wave-basedpicture of diffraction into a mathematical solution of the wave equationsolvable by imposing appropriate boundary conditions.71.4. Helmholtz-Kirchhoff integral equation for scalar diffractionFigure 1.3: Theoretical construct consisting of a closed surface S associatedwith a volume V and surrounding an observation point P0 used to illustrateGreen?s theorem. Note that we take the normal direction to correspond tothe outward normal.Assume that we have a closed surface S surrounding a volume V andenclosing an observation point P0 (this is where we would like to calculatethe field U). Let U , which represents either the electric or magnetic scalarfield, be a complex-valued scalar function that has single-valued first andsecond partial derivatives continuous within and on S. If another complex-valued scalar function G satisfies the same continuity conditions imposed onU , then the scalar Green?s theorem provides a relationship between the twoscalar functions given by???V(G?2U ? U?2G)dV =??S(G?U?n? U?G?n)dS, (1.23)where the partial derivatives with respect to n are differentials in the di-rection of the outward surface normal (consistent with mathematical con-vention but opposite to the discussion in Principles of Optics by Max Bornand Emil Wolf [4]). If the function G is chosen so that it also satisfies thetime-independent scalar wave equation(?2 + k2)G = 0, (1.24)then the volume integrals on the left-hand side of Eq. 1.23 disappear and81.4. Helmholtz-Kirchhoff integral equation for scalar diffractionwe are left with ??S(G?U?n? U?G?n)ds = 0. (1.25)The question then becomes: what function should we select for G? Origi-nally, Kirchhoff was inspired by the spherical wave picture of Huygens andFresnel and chose the function G to represent a scalar spherical wave ema-nating from the observation point given byG =eikss, (1.26)where s represents the distance from the observation point to any arbitraryposition (it can be easily checked that this particular G satisfies the time-independent Helmholtz equation in spherical coordinates for s > 0). Aspherical wave seems to be a reasonable choice, except that it has a sin-gularity at s = 0 and Green?s theorem requires that G be continuous anddifferentiable both within and on S. Therefore, to exclude the discontinuityat P0, a small spherical surface Sa of radius a, is inserted about the pointP0. Green?s theorem is then applied over the surfaces S and Sa??S{eikss?U?n? U??n(eikss)}ds+??Sa{eikss?U?n? U??n(eikss)}ds = 0.(1.27)To further simplify this expression, we can develop the normal derivative ofG in the direction of s assuming a spherical coordinate system centered atP0,??s(eikss)=eikss(ik ?1s). (1.28)Substituting Eq. 1.28 into Eq. 1.27 and evaluation of the surface integral onSa in a spherical coordinate system yields??S{eikss?U?n? U??n(eikss)}ds= ???Sa{eikss?U?n? Ueikss(ik ?1s)}ds= ???Sa{eikaa?U?n? Ueikaa(ik ?1a)}a2d?, (1.29)where d? denotes the solid angle. If we allow the radius a to shrink to zero,all the terms on the right hand side of Eq. 1.29 go to zero except for the last91.5. Kirchhoff?s formulation of scalar diffraction by a plane screenone, which equals 4piU . We can then re-arrange the equation in terms of UasU =14pi??S{eikss?U?n? U??n(eikss)}ds, (1.30)which expresses the value of U at an observation point P0 in terms of thevalues of U , and its derivative on a surface bounding P0. This elegantsolution is known as the integral theorem of Helmholtz and Kirchhoff.1.5 Kirchhoff?s formulation of scalar diffractionby a plane screenWe are now in position to apply scalar diffraction theory to the prob-lem of light diffraction from an aperture in an opaque planar screen. Let?sconsider a monochromatic spherical wave disturbance from a point sourcelocated at the origin O which is situated on one side of an opening in anopaque screen. We will let P0 as before represent a point located on theother side of the opening where we want to determine the value of the scalardisturbance. With the help of the integral theorem, we can now find thevalue of the disturbance at P0 by carefully selecting an appropriate surfaceto surround that point. Let?s take a surface on the exit side of the screen, asshown in Fig. 1.4, which consists of a planar section S1 covering the opening,another planar section S2 covering the screen, and a third circular sectionS3 with a radius R that is centered on P0. The Helmholtz-Kirchhoff integraltheorem, applied over this closed surface, takes the formU =14pi[??S1+??S2+??S3]{eikss?U?n? U??n(eikss)}ds, (1.31)where s is the distance from P0 to an element on the surface. At this point,we need to make some assumptions to make the integral solvable. Let?sfirst assume that the field and its derivative over the section S2 are bothapproximately zero since every point on S2 lies in the geometric shadow ofthe opaque screen. It could also be reasoned that if we make the radius Rsufficiently large, the contributions to the surface integral over S3 diminishto zero. We are then left with just a surface integral term applied over theopening S1. At the opening, we assume that the field is everywhere thesame as if the screen were not present, which would then take the form of aspherical wave emanating from the originUS1 =Aeikrr, (1.32)101.5. Kirchhoff?s formulation of scalar diffraction by a plane screenFigure 1.4: Application of the Helmholtz-Kirchhoff integral theorem to solvefor the scalar disturbance at an observation point P0 when an opaque screenwith an opening is placed between a source and the observation point.111.6. Fraunhofer diffraction and the Fourier transformwhere A is a constant and r is the distance from the origin to an arbi-trary point on the surface S1. This assumption is perhaps approximatelytrue near the center of the opening but is questionable near the edges ofthe opening. We therefore restrict the size of the aperture to be greaterthan the wavelength in order to limit the relative contributions of fringingfields near the aperture edge. Altogether, these assumptions of the fieldand its derivative at the boundaries of the surface are known as Kirchhoff?sboundary conditions. If we insert Eq. 1.32 into Eq. 1.31 and develop thederivatives, dropping terms that are second order in r and s (this implicitlyassumes that the source and observation points are at distances from theaperture much larger than the wavelength), we arrive at the solutionU(P0) = ?iA2?0??S1eik(r+s)rs{cos(n, s)? cos(n, r)}ds, (1.33)where cos(n, s) and cos(n, r) denote the cosines of the angles between theoutward normal and the position vectors corresponding to s and r, respec-tively. This solution is known as the Fresnel-Kirchhoff diffraction formula,an approximate mathematical description of the transmission through anaperture in an opaque screen requiring that the dimension of the apertureexceed the wavelength (so that it does not strongly perturb the field exitingthe aperture) and that the source and observation points are spaced fromthe aperture at distances that are much greater than the wavelength.1.6 Fraunhofer diffraction and the FouriertransformThe Fresnel-Kirchhoff formula relates the scalar disturbance at a par-ticular point due to the presence of an opening in an opaque screen placedbetween that point and a source of the disturbance. We can further sim-plify the expression by making two assumptions: first, the distance betweenthe aperture to the source and observation points are much larger than thedimensions of the aperture (so far, they are only required to be much largerthan the wavelength) and second, the source and observation points arenearly aligned (paraxial) to a line that intersects the center of the apertureand is normal to the plane of the screen. For convenience, we will also nowshift the origin so that it coincides with the center of the aperture (labeledO?) and align the x?y? plane centered at O? to the plane of the screen andhave the z? axis pointing into the observation half space, as shown in Fig. 1.5.121.6. Fraunhofer diffraction and the Fourier transformFigure 1.5: Configuration used to analyze scalar diffraction from an openingin an opaque screen.131.6. Fraunhofer diffraction and the Fourier transformTo simplify the Fresnel-Kirchhoff formula in this configuration, let?s care-fully examine the terms in the integrand of Eq. 1.34 and see how each termvaries with respect to the opening of the aperture. Since the aperture is largecompared to the wavelength, the term eik(r+s) varies over many cycles as wescan over the area of the aperture and must be left inside the integrand. Onthe other hand, the source and observation points are far from the aperture,meaning that the term cos(n, s)?cos(n, r) does not vary appreciable over thearea of the aperture. We can therefore replace the term cos(n, s)? cos(n, r)by ?2 cos(?), where ? is the angle between the line connecting the sourceand observation points and the normal to the aperture, and take this factoroutside of the integrand (note that for normal incidence illumination andobservation, this factor becomes unity). If we also take the variables r ands - referenced with respect to an arbitrary position within the aperture -and replace them with r? and s? - referenced to the center of the aperturelocated at O? - we arrive at a simplified version of the Fresnel-Kirchhoffformula given byU(P0) 'iA?0cos(?)r?s???S1eik(r+s)ds. (1.34)If we let (x?0, y?0, z?0) and (x?1, y?1, z?1) represent the coordinates of the sourceand observation points, respectively, with respect to O? and (X,Y ) the co-ordinate at a point in the plane of the aperture, we have the geometricalrelationsr2 = (x?0 ?X)2 + (y?0 ? Y )2 + z?20s2 = (x?1 ?X)2 + (y?1 ? Y )2 + z?21r?2 = x?20 + y?20 + z?20s?2 = x?21 + y?21 + z?21 . (1.35)These relations can be combined to yieldr2 = r?2 ? 2(x0X + y0Y ) +X2 + Y 2s2 = s?2 ? 2(x1X + y1Y ) +X2 + Y 2. (1.36)Assuming that r? and s? are both much larger than the dimensions of theaperture, we can expand both r and s as power series in X/r?, Y/r?, X/s?,and Y/s? to yieldr ' r? ?x?0X + y?0Yr?+X2 + Y 22r?+ ...s ' s? ?x?1X + y?1Ys?+X2 + Y 22s?+ .... (1.37)141.7. Bethe?s vectorial diffraction theoryThe expressions in Eq. 1.37 can now be inserted into the exponential factor inEq. 1.34 and be evaluated up to an arbitrary order inX and Y . When carriedout to just the first order of X and Y , the integral describes Fraunhoferdiffraction that is valid when the source and observation points are locatedat distances that are much larger than the aperture dimension. When thehigher order terms are included, the integral describes Fresnel diffractionapplicable when the source and observation points are located at distancescomparable to or less than the aperture dimension.The integral expression describing Fraunhofer diffraction (that is, keep-ing only terms to the first order of X and Y in the exponential factor inEq. 1.34) can be cast into a form that is identical to the familiar Fouriertransform if we allow the source to be located infinitely far away from theaperture (so that the aperture is illuminated by essentially a plane wave).Lumping all the constant factors into a common constant C and denotingthe differential element as ds = dXdY , the disturbance at the observationpoint is proportional toU(P0) ' C??S1e?ik(X+Y )/s?dXdY, (1.38)which is, within a constant factor, the two-dimensional Fourier transformof the aperture function evaluated at spatial frequencies ?x = kX/s? and?y = kY/s?. Due to the imposition of Kirchhoff?s boundary conditions andthe assumption of plane-wave illumination, the aperture function in thiscase is a piecewise function that is unity over the aperture opening andzero everywhere else. In general, this aperture function can be any two-dimensional function.1.7 Bethe?s vectorial diffraction theoryKirchhoff?s scalar diffraction theory was limited because, fundamentally,it did not account for the vector nature of light as postulated in Maxwell?sequations and could only be applied to describe diffraction through an aper-ture that was larger than the wavelength. In 1944, Hans Bethe [5] intro-duced a new theoretical formalism to describe diffraction through small holes(smaller than the wavelength) by solving Maxwell?s equations in its full vec-tor form. Bethe considered a sub-wavelength circular hole of radius a ina perfectly conducting, opaque and negligibly thick metal film and deriveda complete solution that satisfied continuity conditions at the boundarieseverywhere. His analysis made only two assumptions: the hole was small so151.8. Scalar wave equation for two-dimensional problemsthat retardation can be neglected in the aperture and the incident field wasuniform over the opening of the hole. The famous result obtained by Bethewas that the transmitted power of light, Ptr, through a sub-wavelength holeis given byPtr =6427pik4a6Si, (1.39)where k is the wave number and Si is the incident flux per unit of area, whichcan be written in terms of the incident electric field Ei as Si = coE2i /2. Dueto the simplicity of the final result and its consistency with Maxwell?s equa-tions, Bethe?s solution has been widely accepted and used as a theoreticalbenchmark (see, for example, Refs. [6?8] in which Bethe?s formula is usedas a standard against which experimental and simulation data on aperturetransmission are compared).1.8 Scalar wave equation for two-dimensionalproblemsA common critique of Kirchhoff?s theory was that it solved the scalarwave equation, as opposed to the vector wave equation predicted by Maxwell?sequations. Although this restricts the generality of Kirchhoff?s theory, it isby no means a fatal flaw, since Maxwell?s equations can condense into thescalar wave equation for a large number of electromagnetic problems, suchas those which are fundamentally one- or two-dimensional in nature. Con-sider, for example, the case of a two-dimensional electromagnetic problemthat is independent of the y coordinate. Assuming monochromatic complexelectric and magnetic fields both proportional to e?i?t, two of Maxwell?sequations in free-space can be developed into the form~??H = ?ik ~E, (1.40)and~?? ~E = ik ~H, (1.41)where the relation k = ?/c has been used. If we develop the spatial deriva-tives and take all partial derivatives with respect to y to be zero, there aretwo independent sets of equations; one involving only Ey, Hx, and Hz, andanother involving only Hy, Ex, and Ez. Because we can break an arbitrarysolution into a linear combination of two solutions obtained by setting everycomponent of each set, in turn, to zero, we can introduce two types of fields161.9. Angular spectrum representationobeying eitherEx = Ez = Hy = 0??Hz?x+?Hx?z= ?ikEy??Ey?z= ikHx?Ey?x= ikHz (1.42)orHx = Hz = Ey = 0??Ez?x+?Ex?z= ikHy??Hy?z= ?ikEx?Hy?x= ?ikEz. (1.43)Alignment of the electric or magnetic field along the coordinate of indepen-dence can be used to classify the sets. We denote the set with non-zero Eyas describing transverse-electric (TE) polarization and the set with non-zeroHy as describing transverse-magnetic (TM) polarization. For transverse-electric polarization, combination of the equations in 1.42 yields a scalarwave equation completely in terms of Ey, given by?2Ey?x2+?2Ey?z2+ k2Ey = 0. (1.44)Similarly, combination of the equations in 1.43 yields a scalar wave equationcompletely in terms of Hy, given by?2Hy?x2+?2Hy?z2+ k2Hy = 0. (1.45)1.9 Angular spectrum representationGiven the scalar wave equations applicable to electromagnetic fieldspropagating in a two-dimensional plane, it becomes possible to introducethe powerful angular spectrum representation method to obtain and inter-pret solutions. Given the two-dimensional scalar wave equation?2u?x2+?2u?z2+ k2u = 0, (1.46)171.10. Scalar diffraction through small slitswhere u represents either the transverse electric or magnetic field component,a fundamental solution takes the formu ? ei(kxx+kzz), (1.47)where kx and kz are the spatial frequencies along the x and z directions,respectively, and constrained by the conditionk2 = k2x + k2z . (1.48)Let?s now consider the complex field along the line z = 0. We know thatany solution u(x, 0) can be represented as a linear combination of the fun-damental solutionsu(x, 0) =? +???f(kx)eikxxdkx, (1.49)where f(kx) is a function describing the spatial harmonic weighting in u(x, 0)and can be directly calculated through the one-dimensional Fourier trans-form of u(x, 0),f(kx) =? +???u(x, 0)e?ikxxdx. (1.50)Here, we have chosen the z = 0 line out of convenience. In general, foran electromagnetic field confined to a two-dimensional plane, the field dis-turbance (described by the field component orthogonal to the plane) alongany line in that plane can be decomposed into its spatial harmonics bythe Fourier transformation along that line, providing a powerful method toanalyze and interpret field distributions.1.10 Scalar diffraction through small slitsThe angular spectrum representation of two-dimensional fields was usedby Marek Kowarz, a student of Emil Wolf, to analyze near-field diffrac-tion from a small slit in an opaque screen [9]. As shown in Fig. 1.6, theslit geometry considered by Kowarz extends infinitely along one coordinate(the y coordinate), and is thus, a two-dimensional electromagnetic problem.Diffraction of a plane wave through the slit opening can be treated by firstpostulating the form of the fields at the exit of the slit located in the planez = 0. For simplicity, Kowarz assumed a top-hat profile of the fields at theslit exit given by a constant value over the slit opening and zero everywhere181.11. Computational solvers for Maxwell?s equationsFigure 1.6: Configuration considered by Kowarz to analyze near-field diffrac-tion from a slit.else. The spatial harmonic distribution at the exit of the slit can then be de-duced by taking the one-dimensional Fourier transform of the assumed fieldat the slit exit. Interestingly, Kowarz parsed the fields into two categories:a homogeneous contribution, in which kx is less than the free-space wavevector (describing a propagating plane wave), and an inhomogeneous con-tribution in which kx is greater than the free-space wave vector (describingan exponentially decaying evanescent wave). It was shown that when theslit width approached sub-wavelength dimensions, the near-field diffractionpattern at the exit of the slit is dominated by inhomogeneous wave compo-nents. We will refer to this simple and elegant depiction of the slit scatteringproblem in the subsequent sections of this thesis.1.11 Computational solvers for Maxwell?sequationsWith the availability of abundant computing power, numerical simula-tors based on Maxwell?s equations have become increasingly popular andwidely used. The basic idea is to define a region of space in terms of a grid,place a source disturbance in the simulation space, implement appropriateboundary conditions, and then solve for the electric and magnetic fields over191.11. Computational solvers for Maxwell?s equationsthe grid. This can be done in terms of real-valued fields that are solved inthe time-domain or complex-valued fields that are solved in the frequency-domain. The former can be achieved using the finite-difference time-domain(FDTD) method [10?14], while the latter can be achieved using the finite-difference frequency-domain (FDFD) method [15, 16]. Numerical solverslike FDTD and FDFD are popular due to their versatility and ability to ac-curately model the electromagnetic response of structures with complicatedgeometries. Here, we will briefly go over some of the salient features of thesesolvers, as they will both be used in the thesis work.1.11.1 Finite-difference time-domain methodThe finite-difference time-domain (FDTD) technique solves the time-domain, instantaneous form of Maxwell?s curl equations as written in Eqns. 1.3and 1.4 [17]. The temporal and spatial partial derivatives in Maxwell?sequations are discretized into difference equations using the central-differenceapproximation and implemented over a special type of spatial grid (that iseither one-, two-, or three-dimensional) known as the Yee grid [18] (Figure1.7). When solving for the fields over a finite geometry, it is common toplace a reflection-less absorber [19, 20], known as a perfectly-matched layer(PML), around the simulation space to prevent unphysical back reflectionsinto the simulation space. At a particular time step, the difference equa-tions are solved for each grid point to determine the local amplitude of thefield components at a particularly moment in time. The grid resolutionsmust be sufficiently fine to accurately portray the electromagnetic fields. Inpractice, the grid spacing is commonly ?min/10 to ?min/40, where ?min isthe wavelength inside the highest refractive index found in the simulationspace. Because the fields must be solved for all points in space and at everymoment in time, FDTD simulations - especially those that are two- andthree-dimensional - are known to be slow and time-consuming. It is impor-tant to note that since FDTD is a time-domain simulator, all constitutiveparameters must be real and positive. Therefore, to simulate the responseof materials known to have negative real parts of their permittivity (suchas metals), a Drude constitutive relation is used, which will be discussed inthe Appendix. Interested readers can find a comprehensive discussion onthe FDTD method in the textbook Computational Electrodynamics: TheFinite-Difference Time-Domain Method by Allen Taflove and Susan Hag-ness [21].201.11. Computational solvers for Maxwell?s equationsFigure 1.7: Three-dimensional Yee Grid depicting the electric and magneticfield components distributed over a discrete spatial grid.1.11.2 Finite-difference frequency-domain methodThe finite-difference frequency-domain (FDFD) technique solves the frequency-domain, harmonic form of Maxwell?s curl equations. Assuming harmonicdependence in the complex fields ~E, ~D, ~H, and ~B, Maxwell?s curl equationscan be re-written as~? ? ~E = i? ~B, (1.51)and~? ? ~H = i? ~D, (1.52)where, in a linear medium, ~D and ~E are related by a complex permittivity and ~B and ~H are related by a complex permeability ?. The advantageof FDFD over FDTD is that it can obtain solutions at a single frequencywith greater efficiency [22] and the constitutive parameters are permittedto be any complex value. The basic implementation is similar to FDTD -a spatial grid is defined, a source is placed in the simulation space, bound-ary conditions are defined, and the fields are solved everywhere. Unlikethe FDTD method, where the fields are obtained dynamically by iteratingthrough time, the FDFD method obtains the steady-state distribution ofthe fields.211.12. Recent research interest in light transmission through aperture in metallic films1.11.3 Application of FDTD and FDFD to model lighttransmission through a slitHere, we will briefly demonstrate the power of FDTD and FDFD solversfor modeling the scattering of light through apertures. We select the ge-ometry of a slit in a metallic film immersed in a dielectric (refractive indexn = 1.5, as shown in Figure 1.8 ), a simple two-dimensional configurationwhich will be the focus of this thesis. The metal is Ag and its constitutiveparameters have been fitted to values obtained in the literature. The slitis illuminated from below with a TM-polarized wave with a broad Gaus-sian profile. The width of the slit is 100 nm and the incident wavelength is632.8 nm. A perfectly-matched layer is placed around the perimeter of thesimulation space and the grid resolution is 1 nm. Figures 1.8 (b) and (c)depict the steady-state magnetic field distribution obtained from FDFD andFDTD, respectively, about the slit after illumination. Not surprisingly, thesolvers illustrate similar field features - a highly confined electromagneticmode within the slit, a quasi-cylindrical wave emanating from the slit exit,and a slight field enhancement on the film surface at the exit side of theslit (signatures of a surface electromagnetic wave that will be studied exten-sively here). Slight differences in the field distributions at the entrance sideof the metal film are due to the inherent difference between the steady-statefield profile visualized by FDFD, and the chosen instantaneous snap-shotof the field profile visualized by FDTD. Although the solvers are powerfulbecause they completely solve Maxwell?s equations without approximations(outside of linearity of the media), it should be warned that they providevery little physical insight and are time-consuming in that one simulationsolves for just a single data point (as opposed to an analytical descriptionwritten in terms of continuous variables). Nonetheless, they will provide auseful tool that will be used extensively in subsequent chapters.1.12 Recent research interest in lighttransmission through aperture in metallicfilmsIn 1998, Thomas Ebbesen and his colleagues [23, 24] demonstrated ex-perimentally that an array of small holes in a metallic film could exhibitnormal-incidence transmission several orders of magnitude greater than thatpredicted by Bethe?s theory applied to the hole array. This report was shock-ing at the time because it dramatically contradicted Bethe?s theory and221.12. Recent research interest in light transmission through aperture in metallic filmsFigure 1.8: Application of the FDTD and FDFD method to solve thetwo-dimensional problem of TM-polarized plane-wave incident onto a slitin a metallic film. Figure (a) shows the schematic of the simulation gridconsisting of a metallic film with a small opening, immersed in a dielectric.The simulation space is surrounded by a perfectly matched layer (PML).Figures (b) and (c) show the calculated magnetic field distribution aboutthe aperture obtained using the FDFD and FDTD method, respectively.231.13. Surface plasmon polaritonssuggested that light could ?squeeze? through sub-wavelength openings withgreat efficiency. At the time, the enhancement was loosely attributed to thepresence of surface plasmon polaritons (SPPs) excited on the metal film [25?29], a type of surface electromagnetic wave that was not considered in eitherKirchhoff?s scalar diffraction theory or Bethe?s electromagnetic vector-fieldmodel, but a detailed physical picture was absent.In subsequent years after the work of Ebbesen and colleagues, the roleof SPPs in the enhanced transmission was confirmed and a clearer pictureof the enhancement mechanism emerged. Using a simpler grating structureconsisting of an array of slits in an optically thick film, Porto et al. [25] usedthe transfer matrix method to solve Maxwell?s equations in the vicinityof the slits and show that the excitation of SPPs on both the entranceand exit surfaces of the film and presence of wave guide resonances withinthe slits were the key factors responsible for the anomalously high opticaltransmission. It should be noted that a critical assumption of this model isthat the slit sustains only its fundamental mode - a common approximationknown as the single-mode approximation - which is valid as long as thewavelength of incident light is much greater than the width of the opening.The vital role of SPPs in the transmission enhancement through hole arrays- in addition other aperture systems, such as slit arrays, single holes, andsingle slits - was later confirmed by other numerical and experimental works[11, 14, 16, 30?38]. From these works, an approximate physical picturecrystallized. When TM-polarized electromagnetic waves are incident ontoa sub-wavelength aperture in a metallic film, SPPs are excited near thevicinity of the aperture. Due to the confined nature of SPPs (which will bedescribed in the next section), they funnel electromagnetic energy throughthe aperture. When the SPPs arrive at the exit of the aperture, they scatterand radiate into the far field, resulting in transmission that is enhancedabove and beyond what would be expected by Bethe?s formula.1.13 Surface plasmon polaritonsIn this section, we will show that surface plasmon polaritons are a validsolution to Maxwell?s equations, highlight the conditions under which thesewaves exist for a simple geometry, and derive an expression for the wavevector of SPPs as a function of frequency. Let?s start with the most basicgeometry of a semi-infinite medium (?medium 1?) occupying the regionz > 0 and another semi-infinite medium (?medium 2?) occupying the regionz < 0, as shown in Fig. 1.9. Both media are non-magnetic. Medium 1 is241.13. Surface plasmon polaritonsFigure 1.9: A single interface between two disparate media can, undercertain conditions, sustain surface waves that obey Maxwell?s equations.251.13. Surface plasmon polaritonscharacterized by a local complex relative permittivity 1, and medium 2 ischaracterized by a local complex relative permittivity 2. As this geometryis two-dimensional (independent of the y coordinate), we can analyze thewave solutions for TE and TM polarization independently. Let?s begin withthe case of TM polarization. We assume a general harmonic form of theelectromagnetic fields about the interface consisting of a transverse-magnetic(TM) polarized wave propagating along the +x-direction and decaying alongthe z directions given byHy(z) ={H1eikxxekz1z z < 0H2eikxxe?kz2z z > 0,(1.53)Ex(z) ={?ikz1H1eikxxekz1z/?o1 z < 0ikz2H2eikxxe?kz2z/?o2 z > 0,(1.54)andEz(z) ={?kxH1eikxxekz1z/?o1 z < 0?kxH2eikxxe?kz2z/?o2 z > 0.(1.55)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 the xaxis, and kz1 and kz2 are the complex wave vector components along the zaxis in media 1 and 2, respectively. The allowable modes in the system canbe described in terms of either the x or z components of the wave vector,but it is customary to use kx because it uniquely defines the mode in bothmedia due to continuity of the fields across the interface. A map of allowablevalues of kx versus frequency is called the dispersion relation of the mode.To solve for values of kx, we impose continuity of the tangential compo-nent of the electric and magnetic fields at z = 0, yieldingkz1H11+kz2H22= 0, (1.56)andH1 ?H2 = 0, (1.57)which has a solution only if the determinant is zerokz1kz2= ?12. (1.58)If kz1 and kz2 are positive (to prevent runaway solutions far away from theinterface) and strictly real, the existence of a surface bound wave requires261.13. Surface plasmon polaritonsFigure 1.10: Schematic of the TM-polarized electromagnetic waves andsurface charges associated with SPPs at the interface of a metal and dielec-tric.that the real parts of the permittivities of media 1 and 2 have opposite signs,a condition that can be fulfilled if the interface is between a metal and adielectric. When this condition is fulfilled, the fields of the SPP wave decayalong both directions away from the interface. Typical electromagnetic fieldand surface charge density distributions associated with a SPP wave at aninterface between a metal and dielectric are shown in Fig. 1.13.A closed form expression for kx can be obtained if we develop the generalwave equation in the two media using the postulated forms of the fields givenabove, resulting ink2z1 = k2x ? k201, (1.59)andk2z2 = k2x ? k202, (1.60)where k0 is the free-space wave vector. Applying Eqs. 1.59 and 1.60 toEq. 1.58, we obtain the dispersion relation for a single metal-dielectric in-terfacekx = k0?121 + 2, (1.61)where kx represents the SPP wave vector, kspp and the Eq. 1.61 can bewritten askspp = k0?121 + 2. (1.62)271.13. Surface plasmon polaritonsFigure 1.11: Dispersion relation of a SPP wave and a free-space plane wave.kspp and k0 are the wave vectors of a SPP wave and free-space plane wave,respectively. At any given frequency, kspp is larger than k0.Figure 1.11 shows typical dispersion curves for a SPP wave the interfacebetween a metal and free-space and for a free-space plane wave. For agiven frequency, the wave vector of the SPP wave (kspp) is larger than thewave vector of a free-space plane wave (k0), which results in wave vectormismatch.The same exercise can be carried out for TE polarization. By postulatingthe existence of a TE-polarized, surface-bound electromagnetic wave andimposing continuity of the fields at the interface, the condition governingthe existence of such surface waves isE1 = E2, (1.63)andE1(kz1 + kz2) = 0, (1.64)where E1 and E2 are the complex amplitudes of the magnetic field in media1 and 2, respectively. Again, if kz1 and kz2 are positive (to prevent runawaysolutions far away from the interface) and strictly real, the only possiblesolution is that E1 = E2 = 0. In other words, surface electromagnetic wavescannot exist for TE-polarization at the interface of non-magnetic media.281.13. Surface plasmon polaritons1.13.1 Excitation of surface plasmon polaritonsA plane wave can excite SPPs when two conditions are satisfied: 1) thefrequency of the incident plane wave and excited SPP are identical, and2) the wave vector component of the plane wave along the metal-dielectricinterface matches the SPP wave vector. The first condition is required toconserve energy, and the second condition is required to conserve momen-tum. When both conditions are fulfilled, the incident plane wave couplesinto SPPs with a high degree of efficiency. Classical methods to achieve SPPcoupling involve discrete scattering elements, such as a prism or a grating,placed adjacent to a metal interface.Prism couplingThe concept of prism coupling to SPPs was developed by the ground-breaking works of Otto, Kretschmann and Raether [39?41]. A prism cou-pling system requires at least three different systems: two dielectric systemswith different refractive indices and a metallic system on which SPPs aresustained. Let us consider the case in which the higher-index dielectric isa prism and the lower-index dielectric is air. The goal in prism coupling isto boost the wave vector of an incident plane wave by having that planewave go through the prism. When the prism is placed in close proximity toa metal-air interface, under certain conditions it becomes possible to matcha wave vector component of the plane wave in the prism to that of theSPPs on the metal-air interface. When this matching is achieved, it is pos-sible to couple the incident plane wave to the SPP wave with high efficiency(Fig. 1.12(a)).In the Kretschmann-Raether configuration, a thin film of metal is de-posited on one side of a prism and the metal film is illuminated through theprism by a beam of TM-polarized light. For a metal film that is sufficientlythin (on the order of tens of nanometres), it is possible to couple the incidentbeam across the metal film and interact with SPPs on the other side of themetal film. For a prism index n > 1, there exists a critical angle, ?c, atwhich the following condition is satisfiedkspp = nk0 sin ?c, (1.65)where kspp is the SPP wave vector on the air-metal interface. At this criticalangle, incident light couples optimally to the SPP wave.In the Otto configuration, the prism is separated from the metal surfaceby a thin air gap. TM-polarized light is incident through the prism and291.13. Surface plasmon polaritonsFigure 1.12: Schematic of light coupling to SPPs by using a prism. (a)Mechanism of wave vector matching by prism coupling, manifested in the(b) Kretschmann configuration and (c) Otto configuration.directed at the air gap at sufficiently large angles so that light undergoestotal internal reflection (TIR). Evanescent waves generated by TIR tun-nel through the air gap and interact with SPPs at the air-metal interface.Similar to the Kretschmann-Raether configuration, there exists a certaincritical angle when the wave vector component of the incident light alongthe metal-air interface matches the SPP wave vector on the air metal in-terface, satisfying the condition Eq. 1.65. SPP coupling is optimal at thiscritical angle. In both the Otto and Kretschmann-Raether configurations,the excitation of SPPs is detected and measured by observing a drop in thereflected light intensity. At the current state-of-the art, prism-coupled SPPsare used commercially for biochemical sensing [42].301.13. Surface plasmon polaritonsFigure 1.13: Schematic of free-space light coupling to SPPs using a gratingimmersed in free-space.Grating couplingIt is also possible to couple incident light onto a metal surface by pat-terning the metal surface with a grating [41, 43]. Figure 1.13(b) showsa simple grating structure immersed in free-space with a grating constant(corresponding to the distance between two adjacent grooves) of a. TM-polarized light incident on the grating structure at a certain angle scattersinto discrete diffraction orders, which boosts the wave vector componentalong the metal surface by an amount equal to an integer multiple of thegrating spatial frequency g = 2pi/a. At a critical angle ?c, is it possible thatone of the diffracted components has a wave vector component that matchesthe SPP wave vector at the metal interface, fulfilling the following conditionkspp = k0 sin ?c ? ?g, (1.66)where ? is an integer corresponding to a particular diffracted component.At the critical angle, the diffracted component couples optimally to the SPPwave. Similar to prism coupling, SPP excitation in a grating structure isrealized by observing a drop in the reflected light intensity.Aperture couplingSince the pioneering work of Ebbesen and colleagues [23], it has beenrealized that patterning a metal surface with small apertures can also exciteSPPs. Figure 1.14 shows the schematic of aperture coupling. Similar tothe scattering mechanism employed in grating coupling, small apertures311.14. Understanding aperture coupling to surface plasmon polaritonsscatter incident light into a continuum of directions and it is believed thatcomponents of the scattered light can couple to SPPs when there is wavevector matching to the SPP wave vector. However, unlike grating coupling,the exact coupling mechanism in single apertures is not well establishedand there does not exist a well-defined succinct expression, such as thatused to describe optimal SPP coupling using a prism or a grating, thatcan be used to predict when high efficiency coupling to SPPs is achieved.A major limitation of aperture coupling is that only a small portion oflight near the aperture effectively couples to SPPs, making it much lessefficient than prism or grating coupling in terms of the overall power carriedby an incident beam. To the best of our knowledge, the maximum SPPcoupling efficiency that has been experimentally achieved using an apertureis on the order of 20-30% (this is measured relative to the amount of lightleaving the aperture). Prior to the work of this thesis, the highest SPPcoupling efficiency achieved with a slit aperture was 22% [44], realized usinga sub-wavelength slit in a gold film at a free-space wavelength of 785 nm.The highest SPP coupling efficiency achieved with a hole aperture is 28%[45], realized using a sub-wavelength hole in a gold film at a free-spacewavelength of 800 nm. Although aperture coupling is less efficient thanprism and grating coupling, their advantages include inherent insensitivityto the incident angle, compactness and ease of integration into complexphotonic structures. These features make aperture-based SPP coupling afront-runner in the miniaturization of photonic devices.With the advent and widespread availability of advanced nanofabricationtechniques, the excitation and manipulation of SPPs near nano-structuredmetallic media has, for the past few decades, become a thriving area ofmodern optics research known as nano-photonics [30, 46?48]. In Figure 1.15,we highlight some representative, recent research work in which SPPs havebeen excited and manipulated on various nano-patterned surfaces or devices.1.14 Understanding aperture coupling to surfaceplasmon polaritonsAlthough it is generally accepted that small apertures are efficient cou-plers to SPPs (relative to the amount of light incident onto the aperture),the precise microscopic mechanism underlying the excitation process re-mains a mystery and has been the subject of continuing research. Basedon Kowarz?s application of scalar diffraction theory to describe near-fielddiffraction patterns near small slits, Thio and Lezec [53] proposed that the321.14. Understanding aperture coupling to surface plasmon polaritonsFigure 1.14: Schematic of the coupling of free-space light to SPPs by anaperture.331.14. Understanding aperture coupling to surface plasmon polaritonsFigure 1.15: (a) Three-dimensional plasmonic nano-focusing of light withasymmetrically patterned silver pyramids [49] showing i) a scanning elec-tron microscope (SEM) image of the device, ii) a side-view of the scanningconfocal Raman image at an incident wavelength of 710 nm. (b) Splittingof surface plasmons into two channels [50], showing i) a SEM image and ii)a near-field optical image at an incident wavelength of 1600 nm. (c) Sub-wavelength focusing of surface plasmon to a 250-nm-wide Ag strip guidewith a curved array of holes [51], showing i) a SEM image (light gray, Agand dark gray, Cr) and ii) a near-field optical image of SPP focusing andguiding at a wavelength of 532 nm. (d) Surface plasmon propagation alonga 18.6?m long silver nano-wire at an incident wavelength of 785 nm [52],showing i) a sketch of optical excitation where I is the input and D is the endof the wire, ii) a conventional optical microscopic image showing the brightspot to the left is the focused exciting light, iii) a near-field optical imagecorresponding to the white box in ii, and iv) a 2-?m-long cross-cut alongthe chain dotted line in iii. (e) Surface plasmon waveguiding [30], showingi) a SEM image and ii) photon scanning tunneling microscopy image at anwavelength of 800 nm.341.14. Understanding aperture coupling to surface plasmon polaritonsexit of a small hole or slit consists of a spectrum of evanescent waves gener-ated by diffraction from the aperture edges. The composition of the evanes-cent wave spectrum is determined by the Fourier transformation of the fieldprofile along a plane at the exit of the aperture and it is the compositesum of these evanescent waves that defines the surface wave generated atthe exit of a small aperture. Later works, from the same research group,concluded that surface waves near an aperture are formed by combiningthe composite evanescent wave and surface plasmon polaritons available atthe metal surface [54, 55]. Although these works provided insights intothe role of diffraction in aperture coupling to SPPs, they did not providequantitative estimates of SPP coupling from apertures. Later, Lalanne andhis colleagues theoretically examined SPP generation at the exit of a sub-wavelength slit [56, 57] by proposing a semi-analytical model that was usedto quantitatively estimate SPP coupling efficiency. SPP excitation was de-scribed microscopically based on a two-stage scattering mechanism. Thefirst stage describes the diffraction of light at the slit exit, which, like in thepicture proposed by Thio and Lezec, forms a spectrum of rapidly decayingevanescent waves. The second stage describes the subsequent launching ofrelatively slowly decaying surface plasmon polaritons [37]. Like many otherstudies of light interaction with a sub-wavelength slit, Lalanne invoked thesingle-mode approximation to formulate the model. The model was used tocalculate SPP excitation efficiency as a function of slit width, revealing anoptimum ratio between the slit width and operating wavelength ratio whereSPP coupling efficiencies were as high as 40% (the percentage light leavingthe slit that converts to SPPs). The model was also validated against ex-perimental data of the transmission through a two-slit system [58]. WhileLalanne?s model was restricted to sub-wavelength slit widths, Renger andcolleagues were the first to conduct experimental and analytical researchinto SPP excitation efficiency as a function of slit width beyond the sub-wavelength regime [59]. This work revealed an oscillatory dependence ofthe SPP coupling efficiency on the slit width and that optimizing the slitwidth could result in SPP conversion efficiencies as high as 50%. Additionalexperimental evidence of the sensitivity of SPP coupling on the slit widthwas later investigated by several other researchers [44, 60], who used near-field scanning optical microscopy or far-field leakage radiation microscopyto detect the SPPs excited by slits of variable width.While previous research has provided new insights into SPP coupling andhinted at the possibility of further enhancements in SPP coupling efficiency,some basic questions remain which will be addressed in this thesis. In allresearch conducted so far, the simplifying single-mode approximation was351.15. Improving the excitation efficiency of surface plasmon polaritonsused in which the field distribution in the aperture is restricted to be eitherthat of the lowest-order mode or a simple top-hat profile. What is the role,if any, of higher order modes in the aperture on SPP coupling? Also, pastinvestigations have been restricted to apertures immersed in semi-infinitedielectric media. What is the effect of loading the apertures with dielectriclayers and what effect does this have on the SPP coupling efficiency? Canfurther SPP coupling efficiency gains be realized? The literature has alsoso far reported on SPP coupling measurements at single wavelengths - isit possible to excite SPPs over a broad range of wavelengths and how canthis be achieved? In the next section, we will introduce our basic physicalmodel of SPP coupling from a single slit which goes beyond the single-modeapproximation. This model will be used to study optimal configurations toachieve high-efficiency SPP coupling from slits having both sub-wavelengthwidth and super-wavelength width.1.15 Improving the excitation efficiency ofsurface plasmon polaritonsIn this work, we propose a basic physical model to analyze SPP excitationat the exit of a slit in a metallic film. Inspired by Lalanne?s earlier work [56]describing SPP excitation as a two-step scattering process, we similarly de-scribe the SPP coupling process by analyzing the electromagnetic responseof two distinct segments: the slit and the adjacent metallic surface. To an-alytically describe the electromagnetic response of the slit, we approximatethe slit structure to be an infinite metal-dielectric-metal (MDM) waveguide,where the dielectric region represents the slit opening and the semi-infinitemetallic claddings represent the metallic walls of the slit. To describe theelectromagnetic response of the adjacent metallic surface, we approximatethe surface to be an infinite metal-dielectric (MD) waveguide. By concep-tually dividing the slit geometry into two infinite waveguide segments, thecomplex two-dimensional problem of SPP coupling from a slit is simplifiedinto two classical electromagnetic boundary value problems. The solutionsof the boundary value problems yield the modes of the waveguide segments,described in terms of a two-dimensional wave vector. The component of thewave vector parallel to the axis of the waveguide describes the propagationconstant and the component perpendicular to the axis describes the confine-ment of the mode. By mapping out the allowable wave vectors in the twowaveguide segments, it is now possible to investigate the coupling of wavesfrom one segment to the other by invoking the phase matching concept. We361.15. Improving the excitation efficiency of surface plasmon polaritonsFigure 1.16: (a) Basic two-dimensional configuration to be studied consist-ing of a slit in a metallic film. The slit structure is conceptually divided intotwo uni-axial semi-infinite waveguides: (b) a vertical metal-dielectric-metal(MDM) waveguide, which sustains longitudinal and transverse wave vectorcomponents kz and kx, respectively, and (c) a horizontal metal-dielectric(MD) waveguide, which sustains a longitudinal wave vector component ksppalong the metal-dielectric interface. Wave vector matching is achieved whenthe transverse component of the wave vector in the MDM waveguide matcheswith the longitudinal component of the wave vector in the MD waveguide.hypothesize that maximum coupling efficiency to SPPs is achieved whenthe mode in the MDM waveguide segment (approximating the slit) has aperpendicular wave vector component that matches the SPP wave vectorcomponent along the metal surface. By mapping the wave vectors of themodes in the wave guide segments as a function of wavelength, geometry, andmaterial composition, conditions for optimal SPP coupling can be predicted.We will then use numerical simulations and experimental measurement toconfirm the existence of maximal coupling under the conditions predictedby this model.371.16. Thesis outline1.16 Thesis outlineThe thesis is structured to present a new method to improve the exci-tation efficiency of surface waves near slits in metallic films. In Chapter 1,we have provided a detailed account of the rich history of the study of lighttransmission through apertures, introducing foundational concepts includ-ing Huygens? principle, Maxwell?s equations, scalar diffraction theory, andFraunhofer diffraction, in addition to more advanced concepts to be usedlater in the thesis such as angular spectrum representation and numericalsolvers of Maxwell?s equations. We have highlighted some recent researchresults that have sparked tremendous research interest in understanding theexcitation of SPPs in the vicinity of apertures in metallic films. We havediscussed the basic theory of SPPs on a metal-dielectric interface and haveintroduced a basic physical model that we will use in this thesis to improveand optimize SPP coupling near slit apertures. The next two chapters willprovide detailed discussions of the application of our model to optimize SPPcoupling from a slit and give numerical and experimental results that verifythe predictions made by the model. Chapter 2 will consider the geometry ofa slit that has sub-wavelength dimensions and is coated with a thin dielec-tric layer. Portions of this chapter have been published in [61, 62]. Chapter3 will consider the geometry of a slit that has super-wavelength dimensionsand is immersed in a high-index dielectric medium. Portions of this chapterhave been published in [63]. For both sub-wavelength and super-wavelengthslit geometries, it is shown that the proposed physical model is capable ofaccurately predicting conditions for optimal SPP coupling. In Chapter 4, weconclude the thesis by summarizing the assumptions made throughout thework, highlighting the key findings, and providing future directions of thework. Appendix A describes commonly used plasmonic materials and theirelectromagnetic properties over a broad range of frequencies. Appendix Bprovides a brief description of the equipment used to fabricate the devicesused in the experiments of Chapter 2 and Chapter 3.38Chapter 2Improving the SPP CouplingEfficiency Near aSub-wavelength SlitWe begin our investigation of SPP coupling near an aperture by con-sidering the simplest case of a slit in a metallic film having sub-wavelengthdimension. The slit geometry has been chosen, as opposed to a more com-plex aperture shape such as a hole, because it provides the simplest config-uration whose electromagnetic response can be analyzed in two dimensions.We have specifically focused on slits with sub-wavelength width because weare permitted in this regime to invoke the single-mode approximation todescribe the distribution of light in the slit. It should be noted that SPPcoupling from sub-wavelength slits immersed in air has been studied exten-sively, with typical SPP coupling efficiencies (percentage of light leaving theslit that couples to an adjacent surface wave) reported in the literature tobe around 20% (experiment) [60], 22% (experiment) [44] and 40% (calcula-tion) [56] at visible frequencies. In this Chapter, we will show that coatinga sub-wavelength slit with an ultra thin layer of dielectric can result in SPPcoupling efficiencies around 70% at visible frequencies. We will first de-velop this idea by considering a simple physical picture of the SPP couplingprocess: light is scattered by the exit of the slit, generating a distributionof evanescent waves of which a portion couples to an adjacent SPP mode.We conceptually divide the slit structure into two regions ? the near-fieldregion at the slit exit and the dielectric-coated region above the metal sur-face ? and then match the wave vector of the scattered light and that ofthe SPP mode. In general, the wave vector of light in the near-field of aslit is indeterminate. Diffraction of light at the slit exit yields a near-fieldlight distribution describable by a distribution of wave vector values orientedalong a continuum of directions. For initial design considerations, however,we make the simplifying assumption that a large fraction of light at the slitexit possesses wave vector values equal to that of light in a dielectric, nk0,392.1. SPP coupling from a coated slitwhere n is the refractive index of the dielectric, with a component orientedalong the metal surface that is amenable to SPP coupling. Due to the pres-ence of the thin dielectric layer, the wave vector of the SPP mode can becontinuously tuned over a wide range of values. We hypothesize that thereis an optimal dielectric layer thickness that leads to wave vector matchingbetween the SPP mode and the light scattered from the slit exit. This hy-pothesis is initially tested by conducting a series of numerical simulationsusing the FDTD method in which the SPP coupling efficiency is quantifiedas a function of the dielectric layer thickness. The simulation results revealthe existence of an optimal dielectric layer thickness resulting in peak effi-ciencies about four times greater than the efficiency without the dielectriclayer. We next experimentally test the hypothesis by fabricating a series ofslit structures coated with dielectric layers of varying thickness and measurethe SPP coupling efficiencies from the slits. The experimental results alsoindicate a marked enhancement of the SPP coupling efficiency by coatingthe slit with thin dielectric layers. Based on the evidence, we conclude thatadding a thin dielectric layer to a slit provides a simple method to improvethe SPP coupling efficiency from slits. The added benefit of the addition ofthe dielectric layer is that the metallic film is passivated, naturally prolong-ing the lifetime of the metal as a SPP-sustaining surface.2.1 SPP coupling from a coated slitFigure 2.1(a) depicts the geometry of the SPP coupling structure to bestudied consisting of a slit in a semi-infinite metal film with relative permit-tivity m is immersed in a dielectric medium with relative permittivity d.The film extends infinitely in the x- and y-directions and occupies the region?t < z < 0. A slit of width w oriented parallel to the z-axis and centredat y = 0 is cut into the metal film. A TM-polarized electromagnetic planewave of wavelength ?0 and wave vector nk0, where n =?d is the refractiveindex of the dielectric, is normally incident onto the bottom surface of thefilm located at z = ?t. A fraction of the electromagnetic plane-wave modeincident onto the slit couples into a guided mode in the slit. According tothe scalar diffraction picture put forward by Kowarz [9], light exiting the slitin the region z > 0 consists of radiating and evanescent modes with wavevector magnitude ki ' nk0; coupling to the SPP mode on the metal surfaceis not perfect because ki is less than the wave vector of the SPP mode on402.1. SPP coupling from a coated slitFigure 2.1: (a) Real-space (left) and k-space (right) depictions of the modesscattering from the exit of a slit in a metal film when the metal film iscompletely immersed in a dielectric. (b) Real-space (left) and k-space (right)depictions of the modes scattering from the exit of a slit in a metal film whenthe slit is filled with a dielectric and the metal film is coated with a dielectriclayer.the metal surface, kspp, given bykspp = k0?mdm + d. (2.1)Diffraction at the slit exit disperses the modes at the slit exit into acontinuum of directions described by ~k = kxx?+kz z?. Assuming the Kirchhoffboundary conditions at the exit of the slit (which can be considered theoriginal single-mode approximation), field distribution in the plane z = 0has the formE(x, 0) ={Ei |x| < w/20 |x| > w/2(2.2)where Ei is a constant. As the geometry is two-dimensional, we can de-compose the field distribution at the plane of the slit exit into a diffraction412.1. SPP coupling from a coated slitspectrum given byf(kx) =Eipisin(kxw/2)kx. (2.3)Diffracted modes that assume a real wave vector x-component kx < ki pos-sess a real wave vector z-componentkz =(k2i ? k2x)1/2, (2.4)corresponding to radiative modes that propagate away from the slit exit.On the other hand, diffracted modes that assume a real wave vector x-component kx > ki possess an imaginary wave vector z-componentkz = i(k2x ? k2i)1/2, (2.5)corresponding to evanescent modes confined to the slit region. A fraction ofthe evanescent modes with values of kx matching the wave vector of the SPPmode on the air-silver interface couple from the slit exit to the ?x-directedSPP modes.The total intensity radiated from the slit, Ir, is obtained by a summation,weighted by the squared amplitude distribution of the diffraction spectrum,of the intensities of the radiated modes [9]Ir =4|Ei|2pikw2? ki0sin2(kw/2)k2dk. (2.6)Likewise, the total intensity confined to the slit-exit plane, Ie, is obtainedby a summation of the intensities of the evanescent modesIe =4|Ei|2pikw2? ?kisin2(kw/2)k2e?2?k2x?k2i zdk. (2.7)A fraction of Ie constitutes the SPP mode intensity ISPP and the remainderconstitutes the intensity of decaying modes Id. The observable couplingefficiency of the radiating light from the slit to the SPP mode on the metalsurface can be expressed as? =ISPPIr + ISPP? 100%, (2.8)which assumes a value in the range 0% ? ? ? 100%.In contrast to SPP coupling efficiency calculations based on the totalincident beam, which is used for SPP couplers based on prisms and grat-ings, we define the SPP coupling efficiency of a slit by considering the light422.1. SPP coupling from a coated slitleaving the slit. There are two reasons for this approach. First, it is a well-established method in the literature [44, 56, 60]. Second, this definition isindependent of the incident beam size. If the SPP coupling efficiency froma slit were to be taken with respect to the incident beam, then in the limitof an infinite plane wave, the SPP coupling efficiency would approach zero.Thus, normalization of the SPP coupling efficiency by the light leaving theslit ensures that the calculated efficiency depends only on physics at the slitexit. Note that we are using intensities to define the SPP coupling efficiency.It is also possible to define the coupling efficiency in terms of field quantities,which are accessible through simulations, but an intensity-based definitionof coupling efficiency enables more direct comparison with intensity-basedexperimental measurements to appear later in the chapter.To enhance the SPP coupling efficiency from a slit, the configurationdepicted in Figure 2.1(b) is proposed. A semi-infinite metal film with a slitof width w is immersed in free-space with permittivity o. The slit in thefilm is completely filled with a lossless dielectric of relative permittivity d.A semi-infinite dielectric layer, also of relative permittivity d, is placed onthe metal film such that it occupies the region 0 < z < d. In the limitswhere d?? and d? 0, the SPP wave vector approacheskspp = k0?mdm + d, (2.9)andkspp = k0?mm + 1, (2.10)respectively. Thus, the thickness of the dielectric layer on the metal filmnow enables the continuous tuning of the SPP wave vector over a range thatspans nk0. At this point, we assume that the magnitude of the wave vector ofthe radiating and evanescent modes at the slit exit in the region z > 0 is ki 'nk0; in reality, the light radiating from the slit consists of a distribution ofwave vector magnitudes, of which a large fraction corresponds to nk0 [9, 53].By varying the thickness of the dielectric layer, d, the effective wave vectorof the SPP mode can be tuned such that the wave vector matching conditionkspp ' ki is achieved, enabling efficient coupling of light from the slit intothe SPP mode on the adjacent metal surface. The dielectric layer has anadded practical benefit of passivation of the underlying metal surface; thisis especially important for the case of silver, which is an excellent plasmonicmaterial but tarnishes rapidly in atmosphere.432.2. Numerical investigation of enhanced SPP coupling2.2 Numerical investigation of enhanced SPPcouplingTo test the hypothesis of enhanced SPP coupling, we use the two-dimensionalFDTD method to study the SPP-coupling efficiency as a function of the slitwidth ranging from 50 nm to 300 nm, the thickness of the dielectric layerranging from 0 nm to 700 nm, and the incident wavelength ranging from400 nm to 700 nm and determine a set of optimal parameters that yieldmaximum coupling. We will show in this section that judicious selection ofthe dielectric layer thickness and the slit width can yield coupling efficiency? ' 77% extending over the visible wavelength range 400 nm ? ?0 ? 700 nm;the achieved efficiency is ' 4 times more efficient than that observed for a slitwithout the dielectric layer. Design guidelines established by FDTD in thissection could potentially be used to assist the experimentalist in realizingcoupling geometries that yield optimal SPP coupling.2.2.1 Simulation designWe select the two-dimensional FDTD method to map out the electro-magnetic response of the structure. The simulation grid has dimensionsof 4000 ? 1400 pixels with a resolution of 1 nm/pixel and is surroundedby a perfectly-matched layer to eliminate reflections from the edges of thesimulation space. We consider a structure consisting of an optically thick(300- nm-thick) silver film with a slit of width w; the dielectric in the slit andthe dielectric layer on the metal film consist of dispersion-less glass with re-fractive index n = 1.5. The permittivity of silver is modeled using the Drudemodel, which is in good agreement with the experimental data over the visi-ble frequency range, as will be discussed in the Appendix. The corners of theslit are chamfered with a chamfer radius of 3 nm. This was done to eliminateperfectly sharp edges of the slit which can result in highly localized electricdipoles that affect the slit throughput [10]. We use a TM-polarized incidentbeam, which is centered in the simulation space at x = 0 and propagates inthe +z-direction, with a full-width-at-half-maximum (FWHM) of 1200 nmand a waist located at z = 0. The beam width is selected many order largerthan the slit so that the electromagnetic wave incident on the slit can beapproximated as plane wave incidence. The incident electromagnetic wavehas a wavelength of 500 nm and is TM-polarized such that the magneticfield, Hy, is aligned along the y-direction.The control variables are the incident polarization (TM), the metal type(silver), the dielectric type (dispersion-less glass), and the thickness of the442.2. Numerical investigation of enhanced SPP couplingmetal layer (t = 300 nm). The independent variables are the slit width w,which is varied from 50 nm to 300 nm in increments of 50 nm, the dielectriclayer thickness d, which is varied from 0 nm to 700 nm in increments of 50 nm,and the free-space wavelength ?0, which is varied over the visible frequencyregime from 400 nm to 700 nm in increments of 100 nm. The dependentvariables are the intensity of the SPP modes coupled to the exit surface ofthe metal film, ISPP , and the intensity of the radiated modes leaving the slitregion, Ir, which will both be used to compute the SPP coupling efficiency,?.Figure 2.2: Simulation geometry used to study SPP coupling from anilluminated slit. The detectors D1 and D2 capture the SPP modes, and thedetector D3 captures the radiating modes.The dependent variables are quantified in the simulations by placingline detectors, D1, D2, and D3, in the simulation space to integrate theinstantaneous magnitude squared of the magnetic field crossing the planeof the detectors (as shown in Figure 2.2). The detectors, D1, D2, andD3, capture different components of the intensity pattern radiated from theexit of the slit. The line detectors D1 and D2 straddle the metal/dielectricinterface and are situated adjacent to the slit exit a length L1 = ?0 away fromthe edges of the slit. D1 and D2 have identical heights H1 = ?0/4+50 nm, ofwhich 50 nm extends into the metal and ?0/4 nm extends into the dielectricregion above the metal. The extension of the line detectors into the metalsurface has been selected to be larger than the skin depth, and the extensionabove the metal surface has been selected based on observed decay scalesof the SPP mode. D1 and D2 capture the intensity of the left- and right-452.2. Numerical investigation of enhanced SPP couplingpropagating SPP modes that are coupled from the slit and flow on the metalsurface. The line detector D3 is centered on the slit and extends over thedielectric region above the metal surface with a height of H2 = 3?0/4 anda length of L2 = 2?0 + w. D3 captures the intensity of light radiated awayfrom the slit that is not coupled to the surface of the metal.Because the detectors indiscriminately capture the intensity crossing thedetector plane, Ir captured by D1 and D2 and ISPP captured by D3 consti-tute sources of error. The fraction of Ir captured by D1 and D2 is estimatedby calculating the acceptance angle formed by the line detectors D1 and D2with respect to the exit of the slit and integrating Eqn. 2.6 over this angle.At ?0 = 500 nm, < 3% of Ir is captured by D1 and D2. The fraction ofISPP captured by D3 is estimated from the attenuation of the SPP fieldsin the z-direction. At ?0 = 500 nm, < 8% of ISPP is captured by D3. Itshould be noted that the placement of the detectors and our definition ofefficiency ignores the effects of reflection of the incident beam, propagationloss through the slit, and absorption loss of SPPs propagating on the metal-dielectric interface to the detectors. The detectors placement here has beenlargely influenced by later experimental configurations and measurements.It is also possible, for example, to measure Ir and ISPP by placing a hor-izontal detector across the exit of the slit; however, this detection schemesuffers from near-field resonance effects within the slit which yields efficiencyvalues inconsistent with far-field measurements.The time-averaged intensities of the SPP mode and the radiated modesare quantified byISPP = ??D1|Hy|2d`+?D2|Hy|2d`?, (2.11)andIr = ??D3|Hy|2d`?, (2.12)where the angled brackets indicate time-averaging of the quantity within thebrackets.2.2.2 Control studiesThree control simulations are performed to establish baseline SPP cou-pling efficiency values. As summarized in Table 2.1, the control structuresconsists of a silver film of thickness t = 300 nm having a slit width ofw = 150 nm, where 1) the film is surrounded by free-space and the slitis filled with free-space, 2) the film with one side is attached to glass, while462.2. Numerical investigation of enhanced SPP couplingthe other side is open to free-space and the slit is filled with glass, and 3)the film is surrounded by and the slit is filled with glass. The slits are illu-minated with a wide gaussian beam, which can be approximated as a planewave at the slit, with a wavelength ?0 = 500 nm, when the light is incident atthe bottom of the structures. Each geometry represents different situationsin which the magnitude of the wave vector (or equivalently, wavelength) ofthe light radiating in the immediate vicinity of the slit exit is mismatchedwith the SPP wave vector (or wavelength) on the metal surface. The wave-length of the emanating light, ?i, and of the SPP mode, ?spp, are given inTable 2.1.Simulation Geometry ?i ?spp ?500 nm 473 nm 20%500 nm 473 nm 21%333 nm 292 nm 25%Table 2.1: Control simulation geometries and results. Fixed parametersinclude w = 150 nm, d = 100 nm, t = 300 nm, and ?0 = 500 nm.Shown in Figure 2.3 is a snap-shot of the instantaneous |Hy|2 distributionfor illumination of the silver film immersed in free-space. The slit sustains afield-symmetric mode that carries electromagnetic energy across the extentof the slit. A significant portion of the fields radiating from the slit exitpropagates in free-space away from the metal surface (captured by the D3detector), and a lesser portion of the fields couple into confined left- andright-propagating modes on the metal surface (captured by the detectorsD1 and D2). The wavelength of the confined mode on the metal surface is?spp = 480?10 nm, where the error corresponds to the observed variation ofthe SPP mode wavelength as a function of distance from the slit. An SPPcoupling efficiency of ' 20% is measured for the slit immersed in free-space.SPP coupling efficiencies of ' 21% and ' 25% are measured, respectively,for the dielectric-filled slit immersed in free-space and the slit immersed in472.2. Numerical investigation of enhanced SPP couplingFigure 2.3: Image of the FDTD-calculated instantaneous |Hy|2 distribu-tion for a slit of width w = 150 nm illuminated by a quasi-plane-wave ofwavelength ?0 = 500 nm. The simulation geometry is depicted by the abovegraphic. Lines indicating the position of detectors D1, D2, and D3 havebeen superimposed on the image.dielectric (see Table 2.1).2.2.3 ResultsVariations in the thickness of the dielectric layer on the metal film enabletuning of the SPP wave vector, kspp, and hence, the effective refractiveindex of the SPP mode, nspp = kspp/k0. The dispersion curves for nspp asa function of ?0 and d are determined by iteratively solving the complexeigenvalue equation for the wave vector kspp of the mode sustained by anasymmetric, three-layer silver-glass-air waveguide, where the thickness of theglass layer is d. As shown in Figure 2.4(a), nspp increases asymptotically as?0 decreases from 1900 nm to 400 nm. For finite d, the nspp dispersion curvesare bound between the curves corresponding to d = 0 (air-silver interface)and d >> ?0 (glass-silver interface). As shown in Figure 2.4(b), at a fixed?0 = 500 nm, nspp as a function of d can assume a continuum of values inthe range 1.05 < nspp < 1.72. The phase matching condition nspp ' n = 1.5is predicted to occur for a dielectric layer thickness d ' 75 nm.Figure 2.5 displays the instantaneous |Hy|2 distributions for the illumi-482.2. Numerical investigation of enhanced SPP couplingFigure 2.4: Basic principle used to enhance slit-coupling to SPP modesnear a sub-wavelength slit in a metallic film. (a) Dispersion curve of nsppas a function of ?0 for various d calculated from the roots of the eigenvalueequation for the SPP mode on an asymmetric, three-layer silver-glass-airwaveguide. (b) nspp as a function of d for ?0 = 500 nm. The dotted grayline corresponds to the refractive index of a plane-wave mode in glass,?d.nation of silver films with dielectric layer thicknesses of d = 100 nm andd = 500 nm. The slit width w = 150 nm is held constant. High-efficiencySPP coupling is evident for d = 100 nm; the presence of the d = 100 nmdielectric layer on the metal film yields negligible radiated intensity and rel-atively high SPP intensity. SPP coupling efficiency drops as the dielectriclayer thickness increases to d = 500 nm; the |Hy|2 distribution reveals an in-crease in the radiated intensity and a reduction in the SPP intensity relativeto that for d = 100 nm.Figure 2.6 displays the time-averaged radiated intensity Ir and time-averaged SPP intensity ISPP as a function of d, along with the correspondingSPP coupling efficiency, ?. Appropriately selecting the dielectric layer thick-ness can yield both enhanced SPP coupling and reduced radiation from theslit. For values of d in the range 50 nm < d < 150 nm, ISPP is near-maximumand Ir is near-minimum, yielding an SPP-coupling efficiency ' 77%. Forvalues of d > 150 nm, the efficiency curve drops and flattens. In the limitswhere d ? 0 nm and d ? 700 nm, the SPP-coupling efficiencies approach? ? 20% and ? ? 31%, respectively.The layer thickness corresponding to high coupling efficiency approxi-mately coincides with the layer thickness where the SPP wavelength, ?spp,492.2. Numerical investigation of enhanced SPP couplingFigure 2.5: Image of the FDTD-calculated instantaneous |Hy|2 distribu-tion for a slit of width w = 150 nm illuminated by a quasi-plane-wave ofwavelength ?0 = 500 nm when a dielectric layer of thickness (a) 100 nm and(b) 500 nm is placed on the metal film. The simulation geometry is depictedby the above graphic.matches the wavelength of light at the slit exit ?i = ?0/n. Figure 2.7 plotsthe SPP wavelength as a function of the dielectric layer thickness d. The SPPwavelength has been obtained by two methods: direct measurement from theFDTD-calculated |H2y | distributions and calculation via the SPP modal so-lutions of the asymmetric silver-glass-air waveguide. There is a good matchbetween the SPP wavelength values obtained by the FDTD simulations andthe modal solutions. As d increases from 0 to 500 nm, the SPP wavelengthreduces from 480 to 300 nm. At the layer thickness d ' 100 nm, yieldingmaximum SPP coupling efficiency, kspp nearly matches ki, which has beenassumed to be nk0 (although this is not generally true, as the modes at theexit of the slit assume a distribution of wave vector magnitudes, it has been502.2. Numerical investigation of enhanced SPP couplingFigure 2.6: (a) The time-averaged SPP intensity, ISPP (blue squares), andtime-averaged radiated intensity, Ir (red circles), and (b) the correspondingSPP coupling efficiency, ?, as a function of d. The error bars describe theuncertainties in the measurement of ISPP and Ir due to, respectively, thefinite amount of Ir captured by D1 and D2 and the finite amount of ISPPcaptured by D3.so far a useful and accurate approximation). The correlation between thehigh coupling efficiency and wavelength similarity between the SPP modeand the plane-wave mode in the dielectric suggest that phase-matched cou-pling is the primary culprit in the efficiency enhancement.Changing the slit width affects the distribution of the modes at the exitof the slit, which in turn affects the SPP coupling efficiency. Figure 2.8displays the instantaneous |Hy|2 distributions for illumination of silver filmswith slit widths of w = 150 nm and w = 300 nm. The dielectric layerthickness d = 100 nm is held constant. The narrower slit shows weakeroverall transmission through the slit, with the majority of the transmittedfield coupled into the bounded SPP modes on the metal surface at the exitside of the slit. The wider slit exhibits greater overall transmission, with asignificant portion of the transmission radiating from the metal surface. Asw increases, a greater percentage of the modes at the slit exit are propagatingmodes that radiate away from the metal surface. In the limit where the slitwidth is very large (w >> ?0), a ray picture can be used where the majorityof the incident light rays propagate directly through the slit and away fromthe slit exit.Increasing the slit width generally reduces the efficiency of SPP couplingfrom the slit structure. The influence of the slit width on ISPP , Ir, and ?512.2. Numerical investigation of enhanced SPP couplingFigure 2.7: SPP wavelength measured from the FDTD simulations (bluesquares) and predicted from the mode solver (red line) as a function ofdielectric layer thickness d. The dotted gray line indicates the value of?i = ?0/?d. The error bars describe the uncertainty in the measurementof ?spp from the FDTD simulations due to variation in ?spp as a function ofdistance from the slit exit.is plotted in Figure 2.9. As w increases from 50 nm to 300 nm, Ir increasesmonotonically, while ISPP peaks at w = 250 nm and then decreases at w =300 nm. The corresponding SPP coupling efficiency monotonically decreasesfrom ' 83% to ' 18% as the slit width increases from 50 nm to 300 nm.To investigate the wavelength-sensitivity of the coupling structure, theelectromagnetic response of a coupling structure with slit width w = 200 nmand dielectric layer thickness d = 100 nm is studied over free-space wave-lengths ranging from 400 to 700 nm, in increments of 100 nm. SPP- couplingefficiencies of 66%, 67%, 66%, and 65% are observed at wavelengths of 400,500, 600, and 700 nm, respectively. The coupling efficiency is largely insensi-tive to wavelength because the condition kspp ' ki is achieved via near-fieldperturbation of the SPP mode using a d << ?0 layer. That is, the dielec-tric layer shifts the kspp wave vector commensurately throughout the visiblefrequency range and quasi-phase matching is obtained over a large spectralrange.To conclude this section, the FDTD results show that it is possible toenhance the efficiency of slit-coupling from a free-space plane-wave mode522.2. Numerical investigation of enhanced SPP couplingFigure 2.8: Image of the FDTD-calculated instantaneous |Hy|2 distributionfor a slit illuminated by a quasi-plane-wave of wavelength ?0 = 500 nm whenthe slit width is (a) 150 nm and (b) 300 nm. The dielectric layer thicknessd = 100 nm is held constant. The simulation geometry is depicted by theabove graphic.into a SPP mode on a metal film by filling the slit and placing an ultra-thindielectric layer on the exit side of the metal film. Varying the thickness ofthe dielectric layer enables tuning of the SPP wave vector. When the SPPwave vector is matched with the wave vector magnitude of the modes exitingthe slit, coupling efficiencies ' 80% can be achieved, ' 4-times enhancementrelative to the case without the dielectric layer. In the next section, we willdescribe the experimental implementation of this concept and validation ofthe SPP coupling enhancements predicted by simulations.532.3. Experimental investigation of enhanced SPP couplingFigure 2.9: a) The time-averaged SPP intensity, ISPP (blue squares), andtime-averaged radiated intensity, Ir (red circles), and (b) the correspondingSPP coupling efficiency, ?, as a function of w. The error bars describe theuncertainties in the measurement of ISPP and Ir due to, respectively, thefinite amount of Ir captured by D1 and D2 and the finite amount of ISPPcaptured by D3.2.3 Experimental investigation of enhanced SPPcouplingWe next conduct a series of experiments to verify SPP coupling en-hancement conferred by the addition of a dielectric layer. The experimentaldesign is based on slit structures fabricated in an opaque silver film. Theslits are flanked by two parallel grooves, placed at a distance s from theslit on the top surface of the silver film. The slit is illuminated from belowusing TM-polarized visible-light, and the top of the slit is viewed under anoptical microscope. SPPs excited by the slit are incident onto the groovesand scattered from the metal surface into the far field. The SPPs scatteredby the grooves are then captured and detected in the far field by an op-tical microscope. Other methods used in the literature to measure SPPsinclude leakage radiation microscopy and near-field optical microscopy [64?69]. Leakage radiation microscopy works by providing a pathway for SPPsto radiate from a metal surface, which is commonly achieved by surroundingone side of a thin metal film with a dielectric of higher refractive index thanon the other side. SPPs radiated in a characteristic emission cone away fromthe metal surface are then captured by an oil-immersion optical microscopeand identified by distinctive features in the Fourier plane of the microscope542.3. Experimental investigation of enhanced SPP couplingimage. Near-field optical microscopy, on the other hand, uses a nano-scaletip to locally sample the SPP fields and generates a SPP image by rasteringthe tip over a region near the defect site. We have chosen the slit-groovedetection method here because it is simpler than the other methods and isfeasible with the available equipment.2.3.1 FabricationWe build the devices by sequentially evaporating a 5- nm-thick chromiumadhesion layer and then a 300- nm-thick, optically-opaque silver layer ontoa glass substrate that is 200?m thick. We use a mechanical surface profiler(Tencor P10 profiler), with a vertical resolution of 1 A?, to verify the targetedthickness of the silver films and quantify surface roughness. The measuredthickness of the silver films is 300? 5 nm and the average surface roughnessis Ra ? 3 nm. We fabricate a series of slits having a fixed length of 3?mand widths ranging between 100 nm ? w ? 300 nm using a FEI Dual BeamStrata-235 focused ion beam (FIB) system. The FIB system is a high-vacuum particle microscope/mill that uses Ga+ ions with high kinetic energyto locally bombard the sample and ablate material. The lower bound of theslit width is set by the milling resolution of the FIB tool and the upperbound is set by the restriction that the dielectric-filled slit sustains only thelowest order mode (which was confirmed by performing FDTD simulations ofplane-wave, normal incidence illumination of the slit having a variable widthand then visually corroborating that the simulated field structure in theslit was consistent with that of the lowest order mode; see for example, thesimulations depicted in Fig. 2.10). Each slit is flanked by two parallel groovesplaced 1?m on both sides from the edges of the slit. The grooves have adepth of ' 100 nm, a width of ' 200 nm, and a length of 2?m. We create 6identical sets of slits and grooves, each on its own separate glass substrate.Five of the substrates are spin-coated with a layer of PMMA (n = 1.49), withlayer thicknesses varying from 60 nm to 140 nm in increments of 20 nm. Theremaining substrate is left uncoated and serves as an experimental control.Based on surface profiler (Tencor P10 profiler) measurements, we find thatthe achieved PMMA layer thicknesses are within ?4 nm of their nominalthickness.2.3.2 Experimental designThe optical response of the structure is characterized by illuminatingthe bottom of the structure through the glass slide with TM-polarized col-552.3. Experimental investigation of enhanced SPP couplingFigure 2.10: Simulation procedure to determine an upper bound for theslit width, for an arbitrary set of parameters, when the slit is filled with adielectric and loaded with a dielectric layer. Image of the FDTD-calculatedinstantaneous |Hy|2 distribution for a slit illuminated by a quasi-plane-waveof wavelength ?0 = 500 nm when the slit width is (a) 150 nm, (b) 200 nm,(c) 250 nm, (d) 300 nm, (e) 350 nm, and (f) 450 nm. Dispersion-less glasswith the refractive index of 1.5 is used as the dielectric. The dielectric layerthickness d = 100 nm is held constant. Note the transition of the field profilein the slit as the slit width increases.562.3. Experimental investigation of enhanced SPP couplinglimated He-Ne laser beam at a wavelength ?0 = 632.8 nm and viewing thetop of the structure under an optical microscope (Zeiss Axio Imager) witha 100? objective lens, as shown in Fig. 2.11(b). At the chosen visible wave-length, the experimental range of dielectric layer thickness values is expectedto yield a range of SPP momentum values that spans the momentum oflight in the dielectric (shown in Fig. 2.11(a)). The complex kspp values arecalculated by solving the dispersion relation of a semi-infinite three-layersilver-glass-air waveguide, where the permittivity of silver is fitted to ex-perimental data [70]. We examine the structure under illumination withlight that is TM- or x-polarized (electric field perpendicular to the slit axis)and TE- or y-polarized (electric field along the slit axis). Due to the nat-ural TM polarization of SPPs as discussed in Chapter 1, a slit illuminatedwith TE-polarized light does not sustain SPP coupling, whereas a slit il-luminated with TM-polarized light produces a SPP beam emanating fromthe slit perpendicular to the slit axis. The experiment is designed so thatSPPs scattered by the grooves produce a bright spot in the resulting micro-scope image at the groove location. Due to the placement of the grooves onthe transmission side of the optically-opaque metal film, the grooves shouldbe visible only when they are illuminated by SPPs propagating along themetal surface and are otherwise invisible. It is noted that the detectionof radiated SPPs from isolated grooves is fundamentally different than thedetection methodology used in leakage radiation microscopy, which allowsSPPs to freely radiate from the metal surface at any location.2.3.3 SPP measurementOne of the immediate challenges of measuring SPPs coupled from a slitusing transmission-mode optical microscopy is the isolation of weak SPP sig-natures from light diffracted through the slit. In the absence of the polymerlayer, the microscope image of the slit region [Fig. 2.12(a)] under x-polarizedillumination is dominated by a diffraction pattern from the slit consistingof a bright main lobe centered on the slit with subsidiary lobes spanningseveral microns to the side of the slit. Any light scattered from the groovelocations is disguised by the side lobes of the diffraction pattern, precludingunambiguous identification of SPPs by direct observation. The overwhelm-ing diffraction from the slit can be mitigated by adding a polymer layer,which is predicted to better match the momentum of SPPs with the mo-mentum of light and enhance SPP coupling efficiency. With the additionof a 80- nm-thick polymer layer, the microscope image of the slit region[Fig. 2.12(b)] now reveals distinctive SPP signatures as bright spots at the572.3. Experimental investigation of enhanced SPP couplingFigure 2.11: (a) Re[kspp] corresponding to SPP modes propagating alonga silver metal surface coated with a dielectric layer of refractive index n =1.5 and surrounded by air, for various layer thickness values, along withthe nk0 line. Complex kspp values are calculated by solving the dispersionrelation of a semi-infinite three-layer silver-glass-air waveguide, where thepermittivity of silver is fitted to experimental data. The circles highlight,for a given dielectric layer thickness, the frequency at which the momentummatching condition Re[kspp] = nk0 is satisfied. It should be noted that themomentum matching condition is an approximation and provides only afirst-order procedure to estimate the optimal dielectric layer thickness. (b)Schematic of the experimental set-up. Polarized light from a He-Ne laser(?0 = 632.8 nm) illuminates the sample and the far-field transmission imageis captured by an optical microscope (Zeiss Axio Imager) using a 100?objective lens with a numerical aperture of 0.90 in air ambient and recordedusing a Si CCD camera.582.3. Experimental investigation of enhanced SPP couplingFigure 2.12: Representative microscope images of the slit and grooves for(a) an uncoated sample, (b) a coated sample with PMMA layer thicknessd = 80 nm, and (c) a coated sample with PMMA layer thickness d = 120 nm.The width of the slits is w = 150 nm. The left column shows scanningelectron microscope (SEM) images, and the middle and right columns showoptical microscope images under x-polarized and y-polarized illumination,respectively. The color of the optical microscope images has been modifiedfor clarity, but the images are otherwise unprocessed.592.3. Experimental investigation of enhanced SPP couplinggroove locations. The bright spots have an intensity comparable to thatof the main lobe of the diffraction pattern from the slit, indicating that asignificant portion of the light in the slit couples into SPPs. The momen-tum matching conferred by the polymer layer is highly sensitive to the layerthickness. As the polymer layer thickness increases to 120 nm, the intensityof the bright spots at the groove locations reduces and it again is difficultto distinguish the diffraction pattern from SPP signatures at the groovelocations [Fig. 2.12(c)]. Comparative microscope images under y-polarizedillumination show similar diffraction patterns, albeit without any SPP sig-natures at the groove location, regardless of the presence of the polymerlayer or its thickness.We next quantitatively determine the SPP coupling efficiency. A com-parative experiment is designed consisting of two identical slits of widthw = 150 nm and length 3?m, aligned and off-set along the direction of theslit axis so that the transmission through the slits are independent. Groovesare created on both sides of slit 1 [top slit in Fig. 2.13(a)], spaced s = 1?mfrom the center of the slit, while the metal surface adjacent to slit 2 [bottomslit in Fig. 2.13(a)] is left pristine. Figure 2.13(b) shows a microscope imageof the slits under x-polarized illumination. By comparing the images of theslits, image artifacts due to diffraction from the slit and SPP scattering fromthe grooves can be separated. For example, subtraction of the region of themicroscope image encompassing the slit without the grooves (R2) from theregion encompassing the slit and the grooves (R1), yields a resulting im-age in which the diffraction from the slit is suppressed and SPP signaturesfrom the grooves are isolated. Under y-polarized illumination, the same sub-traction procedure yields an image without any observable SPP signatures[Fig. 2.13(c)].We use the brightness of different portions of the microscope image ofthe slit and grooves to measure the SPP coupling efficiency. The brightnessof the slit is proportional to the amount of light that diffracts and radiatesfrom the slit exit. The brightness of the grooves is proportional to theamount of light that has converted into SPPs at the slit exit and radiatesupon striking the grooves. We define the quantities Ig,L and Ig,R as theintegrated intensity over a region encompassing the left and right grooves,respectively, the quantities Ing,L and Ing,R as the integrated intensity overa region spaced s = 1?m to the left and right of slit 2, respectively, and thequantity Is as the integrated intensity over a region encompassing exit side602.3. Experimental investigation of enhanced SPP couplingFigure 2.13: Experiment to distinguish diffraction from a slit and SPPscattering from adjacent grooves. (a) SEM image of a representative sam-ple consisting of two identical slits of width w = 150 nm, one of which isflanked by grooves. The sample is coated with a PMMA layer of thicknessd = 80 nm. Optical microscope image of the sample under (b) x-polarizedillumination and (c) y-polarized illumination. We apply a subtraction pro-cedure to images (b) and (c) in which the region R2 is subtracted from R1.The resulting subtracted image derived from (b) show bright spots at thegroove location indicative of SPP scattering. These bright spots are absentin the subtracted image derived from (c), suggesting the absence of SPPs.of slit 2. The SPP coupling efficiency is now defined as? =Ig,L + Ig,R ? Ing,L ? Ing,RIs + Ig,L + Ig,R ? Ing,L ? Ing,R? 100%, (2.13)where the numerator describes the intensity contributions to the image dueto SPP scattering from the grooves and the denominator describes the in-tensity contributions to the image due to both SPP scattering from thegrooves and diffraction from the slit. The efficiency as defined in Eq. (2.13)describes, to a good approximation, the fraction of light at the slit exit thatcouples into SPPs. The intensities used in the Eq. (2.13) to experimentallymeasure the SPP coupling efficiency are consistent with the time averageintensities Eq. (2.11 and 2.12), which are used for the calculation of SPPcoupling efficiency in the simulation. For the representative case depictedin Fig. 2.13 where w = 150 nm and d = 80 nm, an efficiency of ? = 52? 6%is measured, considerably higher than the efficiency value ( 20%) previouslyreported for nano-slits [44, 60], which do not use the dielectric coating tech-nique described here.612.3. Experimental investigation of enhanced SPP coupling2.3.4 SPP coupling efficiency measurementsThe measured results of SPP coupling efficiency as a function of thepolymer layer thickness yields the efficiency curve shown in Fig. 2.14(a). Thecoupling efficiency has a sharp peak at a layer thickness of 80 nm, nearly sixtimes higher than the efficiency without the layer. The measurements arecompared with numerical two-dimensional FDTD simulations of x-polarizedillumination (at ?0 = 632.8 nm) of a slit in a metal film for discrete d valuesranging from 0 to 200 nm. For fair comparison, we conduct another set ofsimulations for a simulation structure that is identical to the experimentsconsisting of a slit flanked by two grooves.We extract ? from the simulations by directly measuring the SPP inten-sity at the metal surface and light intensity radiated from the slit. Similarto the experimentally-measured efficiency, the FDTD-calculated efficiencypeaks at a layer thickness of 80 nm. Two observations suggest that the en-hanced coupling is due to momentum matching. First, the experimentaland simulated efficiency plots both peak at a d value that agrees, withina factor of 2, to the d value predicted to satisfy Re[kspp] ' nk0 usingthe simplistic design procedure described in Fig. 2.11(a), which assumeda momentum of light in the slit describable by nk0. Second, the exper-imental and simulated efficiency plots both drop-off for d values slightlyoff the optimal value. The drop-off in the simulated ? for d > 80 nmis not as sharp as that observed in the experiments. We attribute thisdiscrepancy to surface roughness present in the experiment but absent inthe simulations. Due to the tighter confinement of SPPs near the surfacefor increasing d (which can be inferred from the dispersion diagrams inFig. 2.11(a)), losses due to surface roughness should be more pronouncedfor thicker polymer layers and is evident in the efficiency plots. Over therange from 150 nm ? w ? 300 nm, we observe good agreement betweenthe experimentally-measured and FDTD-calculated SPP coupling efficien-cies as a function of the slit width (Fig. 2.14(b)). Over this slit width range(which corresponds to a normalized range of 0.24 ? w/?0 ? 0.47), similarreductions in the SPP coupling efficiency as a function of slit width weretheoretically predicted in Ref. [56] using a semi-analytical model of geomet-rical diffraction from a slit in an infinitely thick Au medium followed by SPPlaunching on an adjacent flat Au surface. It should be noted that the errorin the experimentally-measured SPP coupling efficiency for the thinnest slitwidth w = 100 nm is larger due to weak transmission through the slit.622.3. Experimental investigation of enhanced SPP couplingFigure 2.14: (a) SPP coupling efficiency, ?, as a function of the PMMAlayer thickness, for a fixed slit width of w = 150 nm. ? is measured using twomethods. In the first method, labeled ?expt-1?, ? is calculated by using Ing,Land Ing,R derived from the image of the slit and grooves under y-polarizedillumination and then using Eq. (2.13) (red circles). In the second method,labeled ?expt-2?, ? is calculated by using Ing,L and Ing,R derived from theimage of the slit with no grooves under x-polarized illumination and thenusing Eq. (2.13) (magenta diamond). We calculate ? from two-dimensionalFDTD simulations modeling x-polarized plane-wave illumination of a coatedslit for various d values (blue squares). We also calculate, for the optimalcase of d = 80 nm, the SPP coupling efficiency when a 20- nm-deep dimpleis present in the dielectric layer above the slit (cyan square), which emulatespossible non-planarity of the polymer layer due to conforming to the slitwalls. (b) shows the SPP coupling efficiency measured using the method?expt-1? (red circles) and calculated using FDTD simulation (blue squares)as a function of the slit width. The error bars in (a) and (b) correspond tothe variance of five independent measurements.632.3. Experimental investigation of enhanced SPP couplingFigure 2.15: (a) SEM image of an array of identical w = 150 nm slitswhere the groove spacing from the slits is varied from s = 1?m to s = 8?min 1?m increments. The array is coated with a PMMA layer of thicknessd = 80 nm. (b) shows the corresponding microscope image of the array. (c)depicts profiles of the image intensity along horizontal lines intersecting theslit and grooves for various slit-groove separation values. (d) The integratedSPP intensity normalized to the integrated intensity of the slit as a functionof the slit-groove separation (red circles), where the blue line corresponds toan exponential fit.642.3. Experimental investigation of enhanced SPP coupling2.3.5 SPP propagation distance measurementsNext, we measure SPP propagation distance to reassure that the groovebrightness is resulted due to the scattering of the SPP modes. We use a singlesnapshot in the vicinity of the slit under the optical microscope to calculatethe SPP propagation distance. We demonstrate this by creating a lineararray of 8 identical slits [shown in Fig. 2.15(a)] of width w = 150 nm andlength 3?m, again aligned and off-set along the direction of the slit axis sothat the transmission through the slits are non-interfering. Pairs of groovesare then milled next to the slits, where the slit groove spacing, s, variesfrom 1 to 8?m in 1?m increments. A 80 nm-thick polymer layer is then ap-plied to the device. The entire array encompasses an area of approximately20?m?50?m, well within the field of view of laboratory-grade microscopesusing a 100? objective lens. The array is illuminated by x-polarized lightand a microscope image of the entire array is captured [Fig. 2.15(b)]. Inaddition to the expected diffraction pattern emerging from the slits, the ar-ray image contains bright spots at the groove locations, which diminish ass increases but are still visible for the largest (s = 8?m) slit-groove separa-tions. Analysis of the image yields line plots, shown in Fig. 2.15(c), of theintensity distribution along the x direction, from which SPP signatures canbe clearly identified as intensity spikes amidst a background diffraction pat-tern. An exponential fit to the intensity of the SPP signature as a functionof slit-groove separation [Fig. 2.15(d)] yields an intensity decay constant of9.35?m, which is in good agreement with the expected intensity decay con-stant ? = 1/(2Im[kspp]) = 9.26?m extracted from the imaginary part of theSPP wave vector for d = 80 nm, n = 1.5, and ?0 = 632.8 nm. This agree-ment reinforces that the groove brightness arises from groove illuminationby SPP modes, rather than by higher-order guided modes in the polymerlayer (which are expected to be cutoff at ?0 = 632.8 nm for d = 80 nm).Massively-parallel SPP measurements can potentially be performed byimaging large two-dimensional arrays, composed of hundreds or thousandsof slit-groove structures, under a conventional microscope. The maximumnumber of slit-groove structures that can be accommodated using this tech-nique is dependent on the field-of-view of the microscope, the length of theslits, and the minimum separation distance between slits, which must begreater than the SPP decay constant to ensure minimal cross-talk. For ex-ample, a microscope with a 100?m field-of-view could image approximately250 slits, assuming a slit length of 2?m and a separation distance betweenslits of 20?m, chosen to accommodate a SPP decay constant of 10?m.652.3. Experimental investigation of enhanced SPP couplingFigure 2.16: Measurement of a focused SPP beam emitted from a curvedarray of sub-wavelength holes using optical microscopy. (a) SEM image ofthree plasmonic lenses consists of a curved array of holes, with differentgroove patterns milled adjacent to the lenses. The lenses consist of 17 holes,each of diameter ' 200 nm, milled into a semi-circle of radius 5?m. Opticalmicroscope image of the three lenses under (b) x-polarized illumination and(c) y-polarized illumination.2.3.6 SPP lensing using an array of holesWe next use optical microscopy to revisit the SPP lensing experimentdescribed in Ref. [51], in which a focused SPP beam emitted from a curvedarray of subwavelength holes is measured using near-field scanning opticalmicroscopy. Assuming that the momentum of light in a dielectric-filledsub-wavelength hole is the same as in a dielectric-filled sub-wavelength slit(that is, describable by nk0), the aforementioned principles of momentummatching should also apply here. We fabricate three identical, off-set semi-circular lens arrays consisting of 17 holes, each with a diameter of 200 nm <?0, arranged into a semi-circle of radius 5?m. To enhance SPP couplingfrom the holes to SPPs, the entire array is coated with an 80- nm-thickpolymer layer. As shown in Fig. 2.16(a), we create different groove patternsnext to the arrays to probe different aspects of the SPP beam. The groovepattern next to the top lens probes the confinement of the SPP beam within662.4. Broadband enhancement of SPP couplingthe cone defined by the arc angle of the array; the groove pattern next tothe middle lens measures the SPP distribution at the focal plane; the groovepattern next to the bottom lens is expected to disrupt any SPP lensingeffects. The image of the top and middle lens arrays under x-polarizedillumination reveals a single sharp, bright spot at the groove located at thefocus (Fig. 2.16(b)). The absence of SPP signatures at any of the othergrooves means that the SPP beam from the lens array is confined withinthe cone and focused onto a single point in the focal plane, consistent withthe results from Ref. [51]. As expected, SPP focussing is absent from theimage of the bottom lens array. Under y-polarized illumination, the threelens arrays produce similar diffraction patterns, with no pronounced groovesignatures to indicate SPP focussing (Fig. 2.16(c)).2.4 Broadband enhancement of SPP couplingAlthough metallic surfaces can generally support SPPs at frequenciesbelow the plasma frequency, it is challenging to efficiently and compactlycouple to SPPs at the same time over a broad frequency range. This capa-bility, however, would be potentially useful for the development of integratedplasmonic devices, broadband photovoltaic devices, and optical informationprocessing devices that use wavelength multiplexing. In recent years, therehas been several research reports [71?79] of SPP excitation in devices de-signed specifically to enable coupling over a broad frequency range. This hasbeen achieved, for example, using periodic or aperiodic slit arrays [72, 73, 79]or a single dielectric coated asymmetric nano-slit [77]. Similar to the obser-vations by Chen et al. (this work was conducted in the near-infrared showinga 100 nm SPP coupling bandwidth ranging from 710 nm to 810 nm whileconsidering just a single dielectric layer thickness), we have observed in ourFDTD simulations previously shown that broadband SPP excitation in thevisible is possible by coating a sub-wavelength slit with a thin dielectriclayer. In this section, we will experimentally verify this prediction by mea-suring the broadband SPP coupling from slit-groove structures illuminatedwith incoherent white light spanning approximately a 200 nm bandwidthfrom 480 nm up to 680 nm.2.4.1 FabricationTest samples are fabricated by successive sputtering of a 5-nm-thick ad-hesion layer of Cr followed by an optically opaque 300-nm-thick layer ofAg at a deposition rate of 1 A?/s onto a glass microscope slide. For each672.4. Broadband enhancement of SPP couplingsample, two 20-?m-long slits of identical width (w = 100 nm) are milledinto the metallic film using focused ion beam (FIB) milling. The extra longslits are employed in these studies to facilitate spectral measurements usingthe spectrometer. In the region adjacent to the slits, we mill two grooves,each with a length of 22?m and depth of 100 nm, parallel to the slit edgesplacing them each side of the slit. The grooves are placed to scatter SPPsfrom the metal surface into the far-field, which provides a measurable signalto quantify SPPs. For one slit (labeled ?sample 1?), the groove spacingsfrom the edges of the slit are s = 3?m and s = 6?m, and for the otherslit (labeled ?samples 2?), the groove spacings are s = 8?m and s = 10?m.We create three identical sets of each sample: one of each sample is leftuncoated and serves as an experimental control, another of each sample isspin-coated with a 160-nm-thick layer of PMMA, and the last of each sam-ple is spin-coated with a 300-nm-thick layer of PMMA. Based on additionalsimulations, which are not shown, it is predicted that the slits coated witha 160 nm thick PMMA layer will exhibit broadband SPP coupling enhance-ment over the visible. This enhancement is predicted to be absent when theslits are coated with the thicker 300 nm PMMA layer.2.4.2 Experimental designBroadband SPP coupling from the slits is measured using the setupshown in Fig. 2.17 (a). We illuminate the structures at normal incidencewith incoherent white light provided by a tungsten lamp and polarized intothe TM configuration using a broadband linear polarizer. Note that theincoherence of the incident light is not modeled by our previous FDTDsimulations, which assume a perfectly coherent incident wave. The lightincident on the slit generates SPPs, which propagate along the sliver/PMMAinterface and are scattered by the grooves. The top of the slit structure isthen imaged using a far-field optical microscope (Zeiss Axioimager) with a100? objective lens with a numerical aperture of 0.9. Color images of theslit structures are captured by a camera in the image plane. To make spectrameasurements, the camera is removed and a fibre-optic collector of a visible-frequency spectrometer (sensitive to a wavelength range from 400 nm to 800nm) samples a portion of the image plane. Figure 2.17 (b, top) shows thespectrum of the beam sampled by the spectrometer when the slit structure isremoved, providing a good representation of the light spectrum incident ontothe slit. With a slit structure mounted on the microscope stage, positioningthe fibre collector at the location of the image of the slit yields the spectrashown in Figure 2.17 (b, bottom). Similar slit transmission spectra are682.4. Broadband enhancement of SPP couplingFigure 2.17: (a) Experimental setup to characterize broadband SPP cou-pling using a slit-groove structure. Light from a tungsten light source isdirected through a polarizer and is incident onto the slit structure. An im-age of the top of the slit structure is collected by a 100 ? objective lenswith numerical aperture of 0.9. The spectral data are collected by a spec-trum analyzer, which is placed in the image plane of the microscope. (b,top) Spectrum of the light that is supplied from the tungsten source and (b,bottom) transmitted through the various slit structures.692.5. Summarymeasured for the coated and uncoated slits.2.4.3 ResultsThe experimental results collected for coated and uncoated versions ofsample 1 and sample 2 are summarized in Figures 2.18 and 2.19, respectively.We first extract the intensity profiles across the slit-groove structures fromcolor images taken with the microscope, normalizing the peak intensities tounity (shown in panel (b) of Figures 2.18 and 2.19). The profiles reveal aconsistent variation of the groove brightness between the various coated anduncoated samples; for any given groove location on either of the samples, thegroove brightness is highest for samples coated with a 160-nm-thick PMMAlayer, followed by uncoated samples, and then samples coated with a 300-nm-thick layer. We next analyze the spectral distribution of the SPP signalcollected from the grooves (shown in panels (c) and (d) of Figures 2.18 and2.19). For spectral measurement, we subtract the baseline noise signal fromthe raw spectral data. The measured SPP spectra span from approximately420 nm to 680 nm, encompassing nearly the entire visible range. Adding a160-nm-thick PMMA layer provides a uniform, broadband enhancement ofthe SPP signal compared to control measurements for the uncoated samples.This enhancement is observed for both samples at all sampled distancesfrom the slit. Increasing the PMMA layer thickness to 300 nm significantlyreduces the SPP signal compared to the control measurements, confirmingthat the thickness of the PMMA layer must be carefully tailored to achieveSPP enhancement over a large spectral range. To quantify the effect of thePMMA coating on the SPP intensity coupled from the slit, we take theratio of the SPP spectra with and without the coating and plot the spectralratio for the various groove locations (shown in panels (e) and (f) of Figures2.18 and 2.19). We consistently observe that the presence of the 160 nmPMMA layer leads to an increase in the SPP intensity by a factor rangingapproximately from 1.2-2.0 over most of the visible spectrum. On the otherhand, adding the 300 nm PMMA layer results in a general reduction of theSPP intensity.2.5 SummaryWe have proposed and demonstrated a simple method for enhancingthe efficiency of slit-coupling from a free-space plane-wave into SPPs on ametal film. The key element of the coupling scheme involves an ultra-thin702.5. SummaryFigure 2.18: (a) SEM image of a slit-groove structure with slit-to-grooveseparation distances of 3?m and 6?m. (b) Normalized intensity profile mea-sured in the horizontal direction across the slit and grooves extracted froma transmission-mode microscope image of the structure shown in (a). (c)and (d) show the spectra collected by sampling at slit-to-groove separationdistances of 3?m and 6?m. (e) and (f) plot the ratio of the groove spectraof the coated samples to that of the uncoated samples for measurementstaken at slit-to-groove separation distances of 3 and 6?m.712.5. SummaryFigure 2.19: (a) SEM image of a slit-groove structure with slit-to-grooveseparation distances of 8?m and 10?m. (b) Normalized intensity profilemeasured in the horizontal direction across the slit and grooves extractedfrom a transmission-mode microscope image of the structure shown in (a).(c) and (d) show the spectra collected by sampling at slit-to-groove separa-tion distances of 8?m and 10?m, respectively. (e) and (f) plot the ratio ofthe groove spectra of the coated samples to that of the uncoated samples formeasurements taken at slit-to-groove separation distances of 8 and 10?m,respectively.722.5. Summarydielectric layer placed on the exit side of the metal film. Varying the thick-ness of the dielectric layer enables tuning of the SPP wave vector. When athickness is selected which yields wave vector matching, the SPP couplingefficiency is enhanced by several times relative to that without the layer.This has been demonstrated both through numerical simulations using theFDTD method and experimental measurements. The experiments were im-plemented by fabricating slit-groove structures and visualizing their opticalresponse under a microscope. The enhanced coupling efficiency conferredby the addition of the layer results in SPP signatures that are visible undernaked-eye inspection through an optical microscope and appear as distinc-tive bright spots located at the groove scatterers. This method can be usedto perform parallel SPP measurement, which we have demonstrated using alinear array of slits and grooves, and can be extended for massively-parallelSPP measurement using larger two-dimensional arrays accommodating theentire microscope field-of-view. A SPP lensing experiment previously per-formed using near-field optical microscopy has also been re-visited here usingoptical microscopy. We have also experimentally shown that broadband en-hancement of SPP coupling from a sub-wavelength slit can be achieved overmost of the visible spectrum with the same technique. In addition to en-hancing SPP coupling efficiency, the method of using a thin dielectric layerhas the added benefit of passivation and protection of the SPP-sustainingmetal surface. The enhanced SPP excitation through a sub-wavelength slitcan be useful to realize nanoscale integration of photonic structures. Theresults also show the potential of miniaturization of SPP bio-chemical sensorin nano-scale.73Chapter 3Improving the SPP CouplingEfficiency Near aSuper-wavelength SlitNearly all studies of surface plasmon polariton (SPP) coupling from aper-tures have restricted the aperture size to less than the wavelength of lightin the surrounding dielectric. Why is this so? One reason may be thatsmaller apertures offer greater potential for miniaturization, which supportsthe long-term goal of making compact information devices based on SPPs.Another, more practical reason is that smaller apertures are easier to ana-lyze. Since the original diffraction theories of Kirchhoff and Bethe [5, 80], ithas been widely assumed that the fields in a small aperture can be describedas a constant - an assumption now known as the single-mode approxima-tion [13, 31, 32, 56, 81]. With the majority of efforts thus far focused onthe study of sub-wavelength apertures, it may be worth questioning if smallaperture dimensions are a necessary condition for efficient SPP generationand whether or not bigger apertures, for the purpose of SPP coupling, canbe better.Previous studies on SPP excitation from large apertures suggest thatefficient SPP coupling can be achieved even when the aperture size exceedsthe wavelength [59]. Renger et al. theoretically calculated the SPP scatter-ing cross section in the vicinity of isolated grooves and slits as a function ofaperture size spanning sub-wavelength and super-wavelength regimes andfound an oscillatory dependence. For aperture sizes exceeding the wave-length, this behavior was correlated to the presence of higher order modesin the aperture, although no causal mechanism linking higher order modesto the SPP scattering cross section was offered in this work. Later, Kihm etal. [60] experimentally measured the SPP coupling efficiency from small andlarge slits in air using a near-field scanning optical microscope and also foundan oscillatory slit-width dependence. The authors applied a simple scalardiffraction model using the single-mode approximation ? which precludes743.1. Hypothesis: enhanced SPP coupling due to higher-order modesthe possibility of higher order modes ? and found qualitative agreementwith their experimental data, despite the questionable validity of the singlemode approximation for super-wavelength slit values.The goal of this chapter is to investigate SPP coupling from large aper-tures to reveal how higher order modes in the aperture interact with SPPs,by using the simple model of SPP coupling in which the slit and surroundingmetallic surface are treated like independent, semi-infinite waveguides. Weadopt the standard configuration consisting of a single slit in a metal filmimmersed in a dielectric with refractive index n, illuminated with transverse-magnetic polarized light. As in Chapter 2, we adopt a slit configuration, asopposed to a hole, because it can be analyzed in two dimensions and pro-vides the most basic system to understand the physics of SPP coupling.Our hypothesis is that the onset of higher order modes in a slit - broughton by increasing the slit width w beyond the wavelength - boosts the lateralspatial frequency components in the slit, so that a larger portion of light di-rectly couples to SPPs by wave vector matching [63]. Intuitively, the boostin the spatial frequency components for larger slit widths is associated withthe transition of the least-attenuated mode in the slit from the lowest orderTM0 mode, which has no nodes across its field profile, to the higher orderTM1 mode, which has two nodes, as shown in Fig. 3.1. It is interesting tonote that the TM0 mode in a metal-dielectric-metal waveguide convergesto the transverse electromagnetic (TEM) mode of a parallel plate perfectelectric conductor in the limit that the conductivity of the metal approachesinfinity [82]. In this Chapter, we will show that by judicious selection of thesurrounding refractive index, a super-wavelength slit can couple to SPPswith efficiency comparable to or even greater than that of a sub-wavelengthslit.3.1 Hypothesis: enhanced SPP coupling due tohigher-order modesConsider a semi-infinite layer of metal (silver) with relative permittiv-ity m that extends infinitely in the x- and y-directions and occupies theregion ?t < z < 0. A slit of width w oriented parallel to the z-axis andcentred at x = 0 is cut into the metal film. The metal film is immersed ina homogeneous dielectric medium with relative permittivity d and refrac-tive index n =?d. The slit is illuminated from the region below it witha TM-polarized electromagnetic plane wave of wavelength ? = ?0/n andwave vector ~kp = kz?, where k = 2pi/?. The +z-axis defines the longitudinal753.1. Hypothesis: enhanced SPP coupling due to higher-order modesFigure 3.1: Magnetic field profiles of the (a) TM0 and (b) TM1 modes at?0 = 6.0 ? 1014 Hz in an infinite metal-dielectric-metal (MDM) waveguidehaving a dielectric core thickness w = 150 nm. Silver is used as the metal andthe core consists a dielectric of index 2.5. The field profiles are normalizedso that the modes carry unit power in the upward direction.direction, and the x-axis defines the transverse axis. The electromagneticwave couples into a guided mode in the slit having complex wave vectorkx = kz z? + kxx?, where kz and kx are the longitudinal and transverse com-ponents of the complex wave vector, respectively. The attenuation of theguided mode in the slit can be characterized by a figure of merit (FOM)defined asFOM =|Re[kz]||Im[kz]|, (3.1)where FOM >> 1 describes a propagating mode. When the guided modeexits the slit, electromagnetic energy is coupled into plane-wave modes and?x-propagating SPP modes. The SPP modes have complex wave vector?ksppx?, where Re[kspp] and Im[kspp] describe the spatial periodicity andattenuation, respectively, of the SPP field along the transverse direction.The slit structure can be conceptually divided into two regions - (a) theregion before the slit exit can be approximated as an infinite metal-dielectric-metal (MDM) waveguide along the x-direction, and (b) the dielectric regionabove the metal surface. A SPP coupling scheme based on the slit structureis designed by first mapping kz and kx of the TM0 and TM modes sustainedin the slit for varying slit width. The longitudinal wave vector componentskz of the TM0 and TM1 modes in the slit are calculated by solving theexponential and oscillatory forms of the complex eigenvalue equation [60],respectively, for an infinite MDM waveguide using the Davidenko methodwith an iterative solving scheme [83]. Details of this procedure as used inthis work can be found in the Master?s thesis of Waqas Maqsood, a colleaguewith whom I collaborated [84]. The parameter m is modeled by fitting to763.1. Hypothesis: enhanced SPP coupling due to higher-order modesexperimental data of the real and imaginary parts of the permittivity ofsilver [70], and d is assumed to be real and dispersion-less. Figure 3.2(a)shows FOM curves for TM0 and TM1 modes in slits of varying width forthe representative case where the slit is immersed in a dielectric with arefractive index n = 1.75. The FOM values for the TM0 modes are largelyinsensitive to variations in the slit width and gradually decrease as a functionof increasing frequency. FOM curves for the TM1 modes are characterized bya lower-frequency region of low figure of merit and a higher-frequency regionof high figure of merit, separated by a kneel located at a cutoff frequency.The cutoff slit width wc for the TM1 mode at a given frequency ? is thethreshold slit width value below which the TM1 mode is attenuating. Ata fixed visible frequency ? = 6.0 ? 1014 Hz (? = 285 nm), wc ? 300 nm.The least attenuating mode in the slit can be identified at a particularfrequency and slit width by the mode with the largest FOM. The TM0mode is dominant for w < wc, and the TM1 mode is dominant for w > wc.The real part of the transverse wave vector component, Re[kx], of theguided mode in the slit describes the component of electromagnetic momen-tum in the transverse plane parallel to the plane of the metal surface andexists in the dielectric core of the slit. Values of kx are obtained from therelationkx =?k2 ? k2z, (3.2)where k = nk0 is the magnitude of the wave vector in the dielectric core ofthe slit. Note that kx assumes a different value in the metallic portions of theslit. Figure 3.2(b) shows Re[kx] values over the visible-frequency range forthe TM0 mode in a slit of width w = 200 nm and for the TM1 mode in slitsof widths w = 350 nm and w = 500 nm. At the frequency ? = 6.0?1014 Hz,Re[kx] for the TM0 mode in the w = 200 nm slit is nearly two orders ofmagnitude smaller than Re[kx] for the TM1 mode in the w = 350 nm andw = 500 nm slits. Values of Re[kx] for the TM1 mode generally increase fordecreasing slit width. Given the parameters in Fig. 3.2(b) and for a fixed? = 6.0 ? 1014 Hz, Re[kx] for the TM1 mode increases from 8.5 ? 106 m?1to 1.3? 107 m?1 as the slit width decreases from 500 nm to 350 nm.At this point, we propose a simple physical picture of SPP excitation bya slit aperture in terms of a three-step process (similar, in principle, to thetheoretical work proposed in Ref. [56, 57]): 1) guided modes carry electro-magnetic energy through the slit, 2) when the guided modes encounter theslit edges, they diffract into a continuum of spatial frequencies (extrapolatedto the far-field, the propagating parts of this spatial frequency continuumproduce a characteristic far-field diffraction pattern) , and 3) some of these773.1. Hypothesis: enhanced SPP coupling due to higher-order modesFigure 3.2: Formulation of a hypothesis for diffraction-assisted SPP cou-pling by a super-wavelength slit aperture. (a) Figure-of-merit and (b) thereal transverse wave vector component versus frequency and wavelength forTM0 and TM1 modes sustained in slits of different widths. (c) Diffractionspectrum corresponding to the TM0 mode in a 200- nm-wide slit and theTM1 modes in 350 nm-wide and 500 nm-wide slits. (d) Wave vector-space de-piction of diffraction-assisted SPP coupling from slits of width w = 200 nm,w = 350 nm, and w = 500 nm, immersed in a uniform dielectric of refractiveindex n = 1.75783.1. Hypothesis: enhanced SPP coupling due to higher-order modesdiffracted modes are phase-matched to the SPP wave vector and excite SPPsat the surface adjacent to the slit exit. In the previous paragraphs, we havealready discussed the guided modes and their wave vector components Re[kx]and Re[kz]. Next, we examine the diffraction process at the slit exit, whichgenerates a distribution of transverse spatial frequency components, ?, com-monly called the diffraction spectrum. Relying on scalar diffraction theoryand the work of Kowarz [9], we can approximate the diffraction spectrumby Fourier transformation of the transverse field profiles of the guided moderight at the slit exit. Figure 3.2(c) shows the normalized diffraction spectrumfor slit widths w = 200 nm, w = 350 nm, and w = 500 nm, at a fixed fre-quency ? = 6.0?1014 Hz. The peak transverse spatial frequency component,?p, is the spatial frequency at which the diffraction spectrum peaks, whichis close to, but does not exactly match Re[kx] of the guided mode. For theparameters in Fig. 3.2(c), ?p shifts from 1.6?107 m?1 to ?p = 8.3?106 m?1as the slit width increases from w = 200 nm to w = 500 nm, indicating aspatial frequency boost due to diffraction as the slit width increases. It isnoteworthy that ?p < Re[kspp] for all slit width values.We propose a simple picture of diffraction-assisted SPP coupling inwhich SPP coupling at the slit exit is mediated by diffraction of the guidedmode, yielding a net real transverse wave vector component Re[kx] + ?p.Coupling from the diffracted mode at the slit exit to the SPP mode ad-jacent to the slit exit is optimized when the wave vector-matched con-dition Re[kx] + ?p = Re[kspp] is satisfied. This proposed phase match-ing condition is similar to the more-established (and experimentally veri-fied) equation governing SPP coupling from a grating, which is given byk0n sin(?) ?mg = Re[kspp], where n is the index of the dielectric, ? is theangle of incidence, m is an integer corresponding to the diffraction order,and g is the grating spatial frequency. The proposed equation for couplingthrough a large slit is similar to the equation for grating coupling becauseboth take into account contributions from a wave vector component of anincident wave and a spatial frequency contribution due to scattering. Thereare, however, some subtle differences. Whereas the grating coupling equa-tion considers an incident plane wave, we now have an incident guided mode.Moreover, whereas the grating coupling equation considers scattering froma periodic grating, which produces a discrete spectrum of spatial frequen-cies spaced by g, we now have mode scattering from the slit exit, whichproduces a continuous spectrum that can be described, by good approxima-tion, through scalar diffraction theory applied to an aperture.Let?s examine why this picture is particularly suitable to describe SPPcoupling from a larger slit. Because Re[kspp] is generally larger than both793.1. Hypothesis: enhanced SPP coupling due to higher-order modesRe[kx] and ?p, large and commensurate contributions from both Re[kx] and?p are required to fulfill wave vector matching. In a sub-wavelength slit,the TM0 mode has Re[kx] << ?p and SPP coupling at the slit exit requiresa sufficiently small slit width to generate large diffracted spatial frequencycomponents to match with Re[kspp]. On the other hand, a super-wavelengthslit sustains a TM1 mode with Re[kx] ' ?p. The large contributions ofRe[kx] to the net real transverse wave vector component reduces the requiredcontributions from ?p needed for wave vector matching. As a result, wavevector matching with the SPP mode adjacent to the slit exit can be achievedwith a relatively large slit aperture. This proposed SPP coupling process ispresented in Fig. 3.2(d) based on the data in Figs. 3.2(a)-(c) for w = 200 nm,w = 350 nm and w = 500 nm at a fixed ? = 6.0? 1014 Hz.It is important to note that the proposed hypothesis of SPP couplingfrom a slit provides a plausible physical interpretation that is easy to un-derstand but highly approximate in nature. Here, SPP excitation is theresult of two contributions, one from the transverse wave vector componentof the incident guided modes and another from the transverse spatial fre-quency component, ?p, associated with the diffracted guided mode after itencounters the slit edges. At first glance, it may appear that there is doublecounting of the contribution of Re[kx] - it is relied upon to describe both theintrinsic lateral momentum of the guided mode and also contributes in thecalculation of the diffraction spectrum of the diffracted mode. However, theintrinsic lateral momentum of the guided mode and momentum due to scat-tering are physically independent effects - one arises from the nature of theguided mode and the other from the interaction of that mode with a struc-tural discontinuity. In this case, the structural discontinuity is an abrupttermination of the waveguide which allows the guided mode profile itself tobe used to approximate the diffraction process. In the case of a grating,for example, the structural discontinuity is a periodic spatial function whichgives rise to a discrete diffraction spectrum. It is also possible to estimate thespatial frequencies of the diffracted guided mode by numerically calculatingthe field profile after the guided mode has scattered from the slit edges, butthis relies on numerical simulation and provides less physical insight. In theabsence of any work to provide a meaningful physical interpretation of slitcoupling to SPPs in the super-wavelength regime, we should keep in mindthat our goal here is to formulate a plausible, physically insightful hypothe-sis that can be analytically calculated and tested. Going forward, it is nowpossible to evaluate the coupling condition and then measure or simulatethe SPP coupling efficiency to determine if the coupling condition succeedsin predicting SPP coupling maxima.803.2. Simulation design3.2 Simulation designWe postulate that enhanced SPP coupling efficiency provides evidenceof wave vector matching between the SPP wave vector and the wave vec-tor of the incident guided mode, boosted by diffraction from the slit exit..SPP coupling efficiency of a slit immersed in a dielectric is modeled us-ing finite-difference-time-domain (FDTD) simulations. The simulation gridhas dimensions of 4000? 1400 pixels with a resolution of 1 nm/pixel and issurrounded by a perfectly-matched layer to eliminate reflections from theedges of the simulation space. The incident beam is centered in the simula-tion space at x = 0 and propagates in the +z-direction, with a full-width-at-half-maximum of 1200 nm and a waist located at z = 0. The incidentelectromagnetic wave has a free-space wavelength ?0 = 500 nm and is TM-polarized such that the magnetic field, Hy, is aligned along the y-direction.Control variables of this study include the type of metal (chosen as sil-ver), the thickness of the metal layer (set at t = 300 nm), the polarizationof the incident electromagnetic wave (TM), the angle of incidence of theincident electromagnetic wave (normal), and the wavelength of the incidentelectromagnetic wave (?0 = 500 nm). The independent variables include thewidth of the slit, w, which varies from 100 to 800 nm, and the refractiveindex of the surrounding dielectric n, which varies from 1.0 to 2.5. The de-pendent variables are the time-averaged intensity of the SPP modes coupledto the metal surface at the slit exit, ISPP , the time-averaged intensity of theradiated modes leaving the slit region, Ir, and the SPP coupling efficiency,?. The dependent variables are quantified by placing line detectors in thesimulation space to capture different components of the intensity patternradiated from the exit of the slit, similar to the method employed in Chap-ter 2. The ISPP detectors straddle the metal/dielectric interface, extending50 nm into the metal and ?0/4 nm into the dielectric region above the metal,and are situated adjacent to the slit exit a length ?0 away from the edges ofthe slit. The Ir detector captures the intensity radiated away from the slitthat is not coupled to the surface of the metal. The coupling efficiency isthen calculated by the equation? =ISPPISPP + Ir? 100%. (3.3)3.3 Simulation resultsThe numerical simulations provide evidence of high-throughput and high-efficiency SPP coupling from a slit of super-wavelength width. Figure 3.3813.3. Simulation resultsFigure 3.3: Images of the FDTD-calculated instantaneous |Hy|2 distribu-tion (left) and the time-averaged |Hy|2 angular distribution (right) for a slitof width values (a) w = 150 nm, (a) w = 350 nm, and (a) w = 500 nmimmersed in a dielectric (n=1.75) and illuminated by a quasi-plane-wave ofwavelength ?0 = 500 nm. A common saturated color scale has been used toaccentuate the fields on the exit side of the slit.823.3. Simulation resultsdisplays representative snap-shots of the instantaneous |Hy|2 intensity andtime-averaged |Hy|2 angular distribution calculated from FDTD simula-tions for plane-wave, TM-polarized, normal-incidence illumination of a slitimmersed in a dielectric (n = 1.75) for slit width values w = 200 nm,w = 350 nm, and w = 500 nm. Radiative components of the field in thedielectric region above the slit propagate away from the metal-dielectricinterface, and plasmonic components propagate along the metal-dielectricinterface. For w = 200 nm [Fig. 3.3(a)], the incident plane wave couples intoa propagative TM0 mode in the slit, which is characterized by intensity max-ima at the dielectric-metal sidewalls. Diffraction of the TM0 mode at theexit of the slit yields a relatively strong radiative component with an angularintensity distribution composed of a primary lobe centred about the longi-tudinal axis and a relatively weak plasmonic component. For w = 350 nm[Fig. 3.3(b)] and w = 500 nm [Fig. 3.3(c)], the incident plane wave couplesprimarily into the TM1 mode in the slit, which is characterized by an inten-sity maximum in the dielectric core of the slit. The high-throughput SPPcoupling is evident by the large SPP intensities observed for w = 350 nm.Diffraction of the TM1 mode at the w = 350 nm slit exit yields a relativelyweak radiative component with an angular intensity distribution skewed athighly oblique angles and a relatively strong plasmonic component. Fur-ther increasing the slit width to w = 500 nm increases the total throughputthrough the slit, but reduces the efficiency of SPP coupling. Diffraction ofthe TM1 mode at the w = 500 nm slit exit yields a strong radiative com-ponent with an angular intensity distribution composed of two distinct sidelobes and a relatively weak plasmonic component.Trends in the SPP coupling efficiencies calculated from the FDTD simu-lations are compared to qualitative predictions from the model of diffraction-assisted SPP coupling described in Fig. 3.2. Figure 3.4(a) plots the FDTD-calculated SPP coupling efficiencies as a function of the optical slit width nwfor dielectric refractive index values ranging from n = 1.0 to n = 2.5. Forsub-wavelength slit width values nw < ?0, highest SPP coupling efficiencyis observed for the smallest optical slit width. This trend is consistent withdiffraction-dominated SPP coupling predicted to occur for sub-wavelengthslit widths, in which small slit width is required to yield large diffracted spa-tial frequencies to achieve wave vector matching. For super-wavelength slitwidth values nw > ?0, the SPP coupling efficiencies exhibit periodic mod-ulations as a function of optical slit width, qualitatively agreeing with thegeneral trends observed in experimental data measured for a slit in air [60]and theoretical predictions based on an approximate model for SPP cou-pling from a slit [57]. The data in Fig. 3.4 reveals that the magnitude of833.3. Simulation resultsFigure 3.4: (a) SPP coupling efficiency as a function of optical slit widthfor dielectric refractive index values n = 1.0 (squares), n = 1.5 (circles), n =1.75 (upright triangles), n = 2.0 (inverted triangles), n = 2.5 (diamonds).(b) The measured SPP intensity (circles), radiative intensity (squares), andtotal intensity (diamonds). The shaded region indicates the sub-wavelength-slit-width regime.843.3. Simulation resultsFigure 3.5: Wave vector mismatch as a function of refractive index of thedielectric region for a fixed optical slit width nw = 600 nm and free-spacewavelength ?0 = 500 nm.the fluctuations in the SPP coupling efficiencies are highly sensitive to thedielectric refractive index. For refractive index values n = 1.0, 1.5, 1.75, and2.0, the SPP coupling efficiency rises as nw increases above ?0 and reacheslocal maxima of ? = 14%, 44%, 68%, and 48% at a super-wavelength op-tical slit width nw ' 600 nm, respectively. The rapid increase ? as the slitwidth increases from sub-wavelength slit width values to super-wavelengthslit width values is attributed to the disappearance of the TM0 mode in theslit and the emergence of the TM1 mode in the slit, which boosts the netreal transverse wave vector component at the slit exit to enable wave vectormatching. It is interesting to note that the SPP coupling efficiency peak atnw = 600 nm observed for lower refractive index values is absent for n = 2.5.Figure 3.4(b) displays the time-averaged radiative intensity Ir, SPP in-tensity ISPP , and total intensity It = ISPP + Ir, as a function of the opticalslit width for n = 1.75. Although the smallest optical slit width gener-ally yields high SPP coupling efficiency, the total throughput and the SPPthroughput is low. As the optical slit width increases to w ' ?0 from sub-wavelength values, an increase in Ir and a decrease in ISPP yield low SPPcoupling efficiency. In the super-wavelength range of optical slit width val-ues, 520 nm < nw < 700 nm, concurrently high SPP throughput and high853.4. Experimental designSPP coupling efficiency (? > 50%) are observed. For the optical slit widthvalue nw ' 600 nm, ISPP is about an order of magnitude larger than ISPPfor the smallest slit width value nw = 175 nm. As the optical slit widthis further increased nw > 700 nm, Ir is significantly greater than ISPP ,resulting again in low SPP coupling efficiencies.Variations in the peak SPP coupling efficiency at a fixed optical slit widthnw = 600 nm for varying n can be qualitatively explained by the mismatchbetween the net real transverse wave vector component Re[kx] +?p and thereal SPP wave vector Re[kspp]. Figure 3.5 plots the transverse wave vectormismatch Re[kspp] ? (Re[kx] + ?p) as a function of the dielectric refractiveindex at a constant optical slit width value nw = 600 nm. The wave vectormismatch increases monotonically from ?0.4 ? 107 m?1 to 2.4 ? 107 m?1as the refractive index increases from n = 1.0 to n = 2.5, crossing zero atn = 1.75. Coincidence between the n value that yields peak SPP couplingefficiency at nw = 600 nm and that which yields zero wave vector mismatchsupports the hypothesis that optimal SPP coupling efficiency occurs whenRe[kspp] = (Re[kx] + ?p), and that this condition can be achieved using asuper-wavelength slit aperture immersed in a dielectric. The relatively largewave vector mismatch for n = 2.5 is also consistent with the noted absenceof a SPP coupling efficiency peak at nw = 600 nm.3.4 Experimental designIn the previous section, it was shown through numerical simulations thata super-wavelength slit immersed in a dielectric can achieve SPP coupling ef-ficiencies on the order of 60% by immersing the slit in a high-index dielectricmedium n ' 1.7?1.8. The SPP coupling enhancement is highly sensitive tothe width of the slit and the wavelength, and is believed to occur due to wavevector matching between the higher order modes in the slit and the adjacentSPP mode. Here, we pursue further validation of this concept by design-ing experiments, performing measurements, and running addition numericalsimulations to quantify SPP coupling efficiency from large slit apertures andto understand the physical mechanism underlying this enhancement.In contrast to the previous section which employed silver as the metal-lic surface, the experiment and simulations performed in this section willemploy gold as the SPP-sustaining surface. It was discovered in early ex-perimentation that silver reacts vigorously with the high-refractive indexfluids to be used and severely tarnishes. Gold, on the other hand, remainedinert. We expect that a super-wavelength slit in a gold film will show similar863.4. Experimental designSPP coupling enhancement as was previously shown for silver. Figure 3.6highlights the frequency dependence of the wave vector components of thetwo lowest-order TM modes in a semi-infinite MDM waveguide (where themetal regions are gold and the dielectric is glass with n = 1.5). Note thatthe first order mode, which becomes dominant when the slit width is com-parable to and larger than the wavelength, has a significantly larger lateralwave vector component compared to that of the lowest order mode. Thislateral wave vector boost in the slit conferred by the first order mode is hy-pothesized to enable wave vector matching to the adjacent SPP mode andthus, high SPP coupling efficiency.To experimentally verify this hypothesis, we fabricate test samples byfirst evaporating a 5- nm-thick adhesion layer of Cr followed by an opticallyopaque 300- nm-thick layer of Au onto a glass microscope slide. Pairs of3-?m-long slits of identical width, aligned and off-set along the direction ofthe slit axis, are milled into the metallic film using focused ion beam (FIB)milling (using a FEI Strata Dual Beam 235). In the region adjacent to oneof the slits, parallel, 2-?m-long, 200- nm-wide and 100- nm-deep grooves aremilled 1?m from the edges of the slit. The region adjacent to the other slitis kept pristine and serves as a local control. The width of the slit pairs isdesigned to range from 100 nm to 600 nm in steps of 50 nm. We vary therefractive index of the medium surrounding the slits over the range from1.5 to 1.7, in steps of 0.1, by placing one of four transparent index fluids(Cargille Liquids) between the patterned Au film and a cover slip.To measure the SPP coupling efficiency, we use an experimental con-figuration shown in Figure 3.7 consisting of a He-Ne laser, a polarizer andan optical microscope. The sample is illuminated by a TM-polarized (po-larized with the electric field oriented perpendicular to the slit axis) beam(?0 = 632.8 nm) from the transparent substrate side of the structure. Weemploy immersion microscopy in which a cover slip is placed on top of thehigh index liquid coating the slit structure and the gap between the coverslip and the objective lens is filled with silicone oil (n = 1.4). The slit-groovestructures are then imaged with a 100? oil immersion lens with a numericalaperture of 1.3. SPPs excited near the slit apertures propagate along the Au-fluid interface and are scattered by the grooves, producing a tell-tale opticalsignal that is imaged in the far-field. Due to placement of the grooves onthe transmission side of the optically-opaque metal film, the grooves shouldbe visible only when they are illuminated by SPPs propagating along themetal surface and are otherwise invisible. The SPP coupling efficiency canbe quantified, to good approximation, by analyzing the relative brightnessof the region adjacent to the slits with and without grooves.873.4. Experimental designFigure 3.6: (a) SPP coupling from a slit in an opaque gold film. High-efficiency SPP coupling is achieved when the wave vector of light leavingthe slit is wave vector matched with the SPP mode. Frequency dependenceof the real parts of the (b) longitudinal and (c) lateral wave vector of thelowest-order modes in a Au-dielectric-Au waveguide, which approximatesthe slit structure. The refractive index of the dielectric is assumed to be 1.4.Note the large increase in the lateral wave vector of the first-order moderelative to that of the zero-order mode, which provides a wave vector boostneeded for SPP coupling.883.4. Experimental designFigure 3.7: Schematic of the experimental set-up. The slit-groove struc-tures etched into a gold film are immersed in a high-index fluid. TM-polarized light from a He-Ne laser (?0 = 632:8 nm) illuminates the sampleand the far-field transmission image is captured using a CCD camera by oilimmersion microscopy (100x objective lens and numerical aperture of 1.3).We place a cover slide and silicone oil successively on top of the high indexfluid to avoid the possible damage of the eye-piece (as most of the high indexfluid reacts).893.4. Experimental designDue to the spherical beam profile of the FIB, the fabricated slits ac-quire a tapered profile, as illustrated in a representative scanning electronmicroscope image of a slit in Figure 3.8. The width of the tapered regioncan be estimated from the high-brightness region of the SEM image of theslit edge [31]. For each of the fabricated slit pairs, we quantify the actualslit widths w1 and w2 corresponding to the width at the bottom and topof the metal film, respectively. As we are considering SPP coupling at thetop surface of the metal film, we use w2 as the true measurement of the slitwidth.Figure 3.9 highlights the experimental analysis used to quantify the SPPcoupling efficiency. Slits of identical width are fabricated on top of eachother, where only one of the slits possesses adjacent grooves. The slits arethen illuminated by TM-polarized laser light and an image of the slits is cap-tured. We use the brightness of different portions of the microscope image ofthe slit and grooves to measure the SPP coupling efficiency. The brightnessof the slit image is proportional to the amount of light that diffracts andradiates from the slit exit. The brightness of the image of the grooves isproportional to the amount of light that has converted into SPPs at the slitexit and radiates upon striking the grooves. We define the quantity Ig as theintegrated intensity over a region encompassing the left and right grooves,respectively, the quantity Ing as the integrated intensity over a region spaceds = 1?m to the left and right of slit 2, respectively, and the quantity Is asthe integrated intensity over a region encompassing exit side of slit 2. TheSPP coupling efficiency is now defined as? =Ig ? IngIs + Ig ? Ing? 100%, (3.4)where the numerator describes the intensity contributions to the image dueto SPP scattering from the grooves and the denominator describes the inten-sity contributions to the image due to both SPP scattering from the groovesand diffraction from the slit. The efficiency as defined in Eqn. 3.4 providesa reasonable approximation about the fraction of light at the slit exit thatcouples into SPPs.Figure 3.10 highlights the simulation analysis used to quantify the SPPcoupling efficiency. Here, we employ the FDFD method to model the elec-tromagnetic response of the slits in a gold film. The FDFD method is used,as opposed to the FDTD method, because the Drude model used in FDTDdoes not accurate predict the imaginary part of the permittivity of gold atthe operating wavelength of ?0 = 632.8 nm (see Appendix for a compari-son between the Drude model of the permittivity of gold and experimental903.4. Experimental designFigure 3.8: Scanning electron microscope of a typical slit structure etchedinto a 300-nm-thick Au film by focused ion beam milling. The whitishboundary at the edge of the slit represents the blunted slit edges resultingfrom bombardment of a spherical ion beam.913.5. Experimental SPP coupling efficiency measurementsFigure 3.9: (a) Representative SEM image of a sample of two identical slitstructures (w = 450 nm) where the bottom slit is flanked by two grooves. (b)The far field image of the sample when it is immersed in a high index fluid(n = 1.7). The color map of the optical microscope images has been modifiedfor clarity, but the images are otherwise unprocessed. (c) The normalizedintensity profile obtained from the area bounded by dotted boxes in (b).Note the enhanced light intensity of the side lobes next to the slit due tothe presence of the grooves.measurements). To measure the SPP coupling efficiency, we simulate theelectromagnetic response of the slits (having identical tapered geometries asmeasured for the fabricated slits) under two conditions: one in which themetal region adjacent to the slit has grooves and another in which the metalregion does not have grooves. To mimic the experimental measurement, weextract the time-averaged energy density across a plane that is 300 nm abovethe metal surface and approximates the object plane imaged by the micro-scope. A plot of the energy density profile is shown in Figure 3.10(c), whichreveals increased intensity of the side lobe features adjacent to a primarypeak due to the presence of the grooves. The observed side-lobe enhance-ment is similar to the experimental results shown in Figure 3.9(c). The SPPcoupling efficiency is quantified from the simulated energy density profilesin a manner analogous to Eqn. 3.4.3.5 Experimental SPP coupling efficiencymeasurementsThe evolution of the SPP coupling efficiency as a function of slit widthis shown in Figure 3.11, for three different values of the surrounding refrac-923.5. Experimental SPP coupling efficiency measurementsenergy densityglassgoldfluidTM-polarized lightmeasurement plane1 mma no groove b groove cFigure 3.10: FDFD simulations of TM-polarized illumination (?0= 632.8nm) of a set of tapered slits (with parameters matching slit pairs with anominal width of w = 350 nm used in the experiments) immersed in a di-electric medium where n = 1.6. (a) Energy density distribution for the casewhere the slit is (a) without grooves and (b) with grooves. (c) Normalizedenergy density profiles extracted from the simulations along the magentaline shown in (a) and (b), which approximates the object plane imaged bythe microscope.tive index. For convenience, the plots have been divided into shaded andunshaded areas denoting the sub-wavelength and super-wavelength regimes,respectively. In general, there is good quantitative agreement between theSPP coupling efficiency values measured through experiment and predictedthrough simulation. The SPP coupling efficiency in the sub-wavelengthregime decreases as a function of increasing slit width, consistent with manyprevious observations reported in the literature [56, 60]. As the slit widthtransitions into the super-wavelength regime, the SPP coupling efficiencyrises sharply, reaches a tell-tale peak just above the wavelength threshold,and then drops. This feature in the SPP coupling efficiency is observed forall values of the surrounding refractive index, but is most pronounced forthe highest refractive index value of n = 1.7. It should be noted that thepeak SPP coupling efficiencies are lower than that predicted for silver in theprevious section due to the intrinsically higher losses in gold and the factthat groove scattering into the far-field under-estimates the SPPs presenton the surface. In reality, the grooves do not perfectly scatter SPPs intothe far-field and a portion of SPPs remain on the surface after interactingwith the grooves. Nonetheless, the observation of a local peak in the SPPcoupling efficiency as the slit width increases to super-wavelength values933.6. Summarysupports the hypothesis that there is increased wave vector matching to theadjacent SPP mode due to the onset of higher order modes.Can we confirm that the enhanced SPP coupling efficiency for super-wavelength slit values is due to a boost in the lateral wave vector componentin the slit? Experimentally, it is not possible to locally access the near-fieldsassociated with the interior of the slit without also perturbing the fields.Our best approach, perhaps, is to use the local field distribution in the slitcalculated through simulation. As discussed in Chapter 1, the field distribu-tion in a two-dimensional electromagnetic problem can be decomposed intoits spatial frequency components by Fourier transformation. We apply thistechnique here, by using the FDFD method to calculate the field distribu-tion in the vicinity of the slit and then applying a fast Fourier transformto the magnetic field in a region encompassing the slit opening, as shownin Figure 3.12. As shown in Figure 3.12(c), the Fourier amplitude of thefields along the x-axis at a spatial frequency of kspp is boosted at the samesuper-wavelength slit width values which undergo an enhancement in cou-pling efficiency. This indicates a correlation between the coupling efficiencyand the spatial Fourier amplitude at kspp and supports the original hypoth-esis that higher SPP coupling efficiency in super-wavelength slits is due tothe boost in the lateral wave vector component.3.6 SummaryIn this Chapter, we have proposed and validated a method to achievehigh-efficiency SPP coupling method from a super-wavelength slit aperture.The crux of this method is to exploit the onset of higher-order modes in theslit when the aperture is larger than the wavelength. This work is the first toexplore SPP coupling from super-wavelength slits by explicitly treating theinteraction between the higher-order modes (which become non-evanescentwhen the slit width is increased) and SPP modes. The conclusions willassist in the continued development of SPP devices by providing a newhigh-efficiency and high-throughput methods for coupling to SPP modes.943.6. Summarysubwavelengthabcfluid index = 1.5fluid index = 1.6fluid index = 1.7Figure 3.11: SPP coupling efficiency (?) as a function of the exit slit width(w2) when immersed in index fluids of refractive index (a) 1.5, (b) 1.6, and(c) 1.7.953.6. SummaryfluidgoldzxHy030-3030-30|H| y20k=0 ksppk mx ( m )-1kmz(m)-1abcFigure 3.12: FDFD simulation of the electromagnetic response of a slitstructure with a nominal width of w = 350 nm at ?0 = 632.8 nm, witha zoomed-in section highlighting the fields at the slit exit. (b) Two-dimensional Fourier transformation applied to the fields at the slit exit,revealing large spatial frequency components along the horizontal (x) axis.(c) Plot of Fourier amplitude at the spatial frequency kspp as a function ofthe slit width.96Chapter 4ConclusionThis thesis has examined the interaction of light with small slit aper-tures in metallic films, focussing on the problem of light coupling from theexit of a slit aperture to surface waves sustained on the adjacent metal sur-face. Light interaction with small apertures is a centuries-old problem whichhas been critical to the development of basic concepts in optics such as theHuygens-Fresnel theory, Kirchhoff?s scalar diffraction theory, and Fresneland Fraunhofer diffraction. There has been renewed interest in the prob-lem of light transmission through small apertures since the recent discoverythat apertures in metallic films can efficiently excite surface electromagneticwaves, known as surface plasmon polaritons (SPPs). Although there havebeen significant research efforts to apply SPPs in miniature light-based de-vices, the basic physical mechanism of SPP coupling from an illuminatedaperture have not been fully elucidated and remains an active area of re-search.In this thesis, we have proposed to study SPP coupling from a slit aper-ture using a simple model in which the slit and adjacent metal surface aretreated as independent, semi-infinite waveguides. This approach enables theelectromagnetic response of a slit structure to be treated as two waveguideboundary value problems. By mapping the electromagnetic solutions (ormodes) of the waveguide sub-components, it is possible to describe lightcoupling from the slit to the SPP mode adjacent to the slit in terms of con-ventional wave vector matching. Based on this approximation, we have in-vestigate different methods to improve SPP coupling efficiencies from variousslit apertures by determining conditions that lead to wave vector matchingto the SPP mode. In Chapter 2, we have introduced the possibility of usinga dielectric layer to facilitate wave vector matching between the light at theexit of a slit and the SPP mode. Experiments and simulations have beenconducted to show that a sub-wavelength slit coated with a thin dielectriclayer can exhibit SPP coupling efficiencies that are many times greater thanthe coupling efficiencies without the dielectric layer. In Chapter 3, we haveshown high efficiency SPP coupling can even be achieved from a slit with di-mensions larger than the wavelength by exploiting the onset of the first-order97Chapter 4. Conclusionmode in the slit. This hypothesis was first developed through simulationsshowing large SPP coupling enhancements from a slit with the width wasincreased above the wavelength and then further validated by experimentalmeasurements. A SPP coupling condition for large slits, analogous to thewell-established SPP coupling condition for gratings, was proposed and ver-ified over a limited range of parameters. Overall, the results presented inthis thesis illustrate that SPP coupling from a slit can be understood andoptimized through simple waveguide models and that judicious tuning ofparameters such as the slit width, dielectric layer coating thickness, or therefractive index of the dielectric surrounding the slit can yield significantenhancements in SPP coupling efficiencies.It is important to recall the key assumptions that have been used to gen-erate the conclusions of our this work and to discuss the limitations of theseassumptions. We have in general restricted our work to the study of the slitgeometry, a simple geometry that has been assumed to be independent ofone coordinate and can be analyzed in two dimensions. Although the analy-sis approach should also be valid for more complex aperture geometries suchas holes, specific conditions for optimal SPP coupling from slits obtained inour work should not a priori be expected to apply to other geometries. Inour analysis in Chapter 2, we have invoked a variant of the single-mode ap-proximation in which the field in the slit is assumed to be constant, which isgenerally accepted to be valid only for apertures with deep sub-wavelengthdimensions. In the electromagnetic simulations performed throughout thethesis, we have consistently assumed that the metal surfaces are perfectlysmooth. In sections where these simulation results have been compared toexperiment, it should be noted that fabricated metal surfaces are not per-fectly smooth and have average surface roughness on the order of 1.3 nm.Simulations performed using the FDTD method have employed the Drudemodel to describe the electromagnetic response of metals. As discussed inthe Appendix, the Drude model is not a completely accurate model for thepermittivity of common plasmonic materials such as Ag, Au, and Al, andis particularly poor at predicting the imaginary part of the permittivity ofthese metals. In our experiments to measure the SPP coupling efficiency,we have made the experimental assumption that a single groove can com-pletely scatter SPPs into the far-field. Of course, the scattering efficiency ofa single groove is not perfect and the SPP coupling efficiency inferred fromthe far-field brightness of single grooves likely underestimates the true SPPcoupling efficiency. In our proposed coupling condition given in Chapter3, we have assumed that a guided mode in a slit incident onto the exit ofthe slit can be have a lateral wave vector component boosted by diffraction98Chapter 4. Conclusionfrom the slit exit. Although this proposal draws upon the well-establishedscattering concept used to describe grating coupling, the implementation ofthis condition for a guided mode incident onto a slit exit leads to the appear-ance of double counting of the lateral wave vector component of the guidedmode. As discussed in Chapter 3, the appearance of double counting arisesfrom the application of approximate scalar diffraction theory to describethe scattering process at the slit exit and can be alleviated by taking, forexample, the numerically calculated field profile after the guided mode hasscattered from the slit exit. Certainly, there is room for future refinementand improvement of the model to describe coupling to SPPs from largerapertures.Although our experimental and numerical studies have been restrictedto enhancing SPP coupling at visible frequencies, it should be noted that thestrategies we have proposed to achieve SPP coupling enhancement can be ex-ploited at other frequencies. To briefly demonstrate this, we have conductedFDFD simulations in which a sub-wavelength slit in a opaque metallic filmcoated with a thin, lossless dielectric layer (of variable thickness and fixedrefractive index of n = 1.5) is excited with ultraviolet (UV), visible, andinfrared (IR) light. For visible illumination, we have assumed a metallicfilm composed of silver, and for UV and IR illumination, we have assumedmetallic films composed of aluminum and gold, respectively. As shown inFigure 4.1, the SPP coupling efficiency from a sub-wavelength slit subjectedto either UV, visible, or IR illumination can be dramatically enhanced bysimply tuning the dielectric layer thickness, based on the principles intro-duced and discussed in this work.This thesis represents a first step towards better optimization of SPPcoupling from apertures by using simple physical models of the coupling pro-cess to predict conditions that achieve wave vector matching. Future workin this research area should be focused on improving the physical model ofSPP coupling, extending its capability to treat more complex geometries,and finding suitable applications for the SPP coupling enhancement meth-ods proposed and demonstrated in this work. Although our physical modelfor SPP coupling can predict conditions that lead to wave vector matchingand therefore, optimal SPP coupling, it still does not provide quantitativeestimates of the amount of SPP coupling. Improving the model to enablequantitative estimates of coupling would need to go beyond simple wave vec-tor matching concepts and may incorporate field matching algorithms [85].To extend our treatment to more complex geometries, three-dimensionalanalysis techniques and simulations would have to be developed. Ultimately,the goal of such efforts would be to create design tools that would enable a99Chapter 4. ConclusionFigure 4.1: SPP coupling efficiency as a function of dielectric layer thicknesscalculated at representative wavelengths in the infrared, visible, and UV. Forthe case of UV illumination, the metal is aluminum, the slit width is 50 nm,and the incident wavelength is 250 nm. For the case of visible illumination,the metal is silver, the slit width is 100 nm, and the incident wavelength is500 nm. For the case of IR illumination, the metal is gold, the slit width is150 nm, and the incident wavelength is 822 nm.100Chapter 4. Conclusionpractitioner to specify desired SPP wave properties (such as beam amplitudeand phase profile) and retrieve an optimized aperture geometry.In this thesis, we have explored SPP excitation using sub-wavelengthand super-wavelength slits. Of the two, SPP excitation by sub-wavelengthslits is a more well-established technique, and our contribution has been tointroduce a coating method to refine this method to achieve very high effi-ciencies and therefore, high signal-to-noise ratios. It is important to keep inmind, however, that the overall SPP throughput for a sub-wavelength slitis still small and limited by its surface area. On the other hand, SPP exci-tation by super-wavelength slits has received very little research attention,and our contribution has been to make an initial attempt to develop a phys-ical model for SPP excitation in this slit-width regime. We have observedthat it is possible to boost the SPP coupling efficiency of a super-wavelengthslit, but this generally requires the slit to be immersed in a high-index di-electric fluid. A super-wavelength slit inherently allows more light to passthrough than a sub-wavelength one and thus has greater SPP throughput,but cannot achieve the same efficiency values. 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Zhou, ?Broad-band surface plasmon resonance spectroscopy for determination ofrefractive-index dispersion of dielectric thin films,? Applied Physics Let-ters, vol. 90, p. 181112, 2007.[73] B. L. Feber, J. Cesario, H. Zeijlemaker, N. Rotenberg, and L. Kuipers,?Exploiting long-ranged order in quasiperiodic structures for broadbandplasmonic excitation,? Applied Physics Letters, vol. 98, p. 201108, 2011.[74] R. M. Gelfand, L. Bruderer, and H. Mohseni, ?Nanocavity plasmonicdevice for ultrabroadband single molecule sensing,? Optics Letters,vol. 34, pp. 1087?1089, 2009.108Bibliography[75] B. Roberts and P. C. Ku, ?Broadband characteristics of surface plasmonenhanced solar cells,? 35th IEEE Photovoltaic Specialists Conference(PVSC), pp. 2952?2954, 2010.[76] F. Djidjeli, E. Jaberansary, H. M. H. Chong, and D. M. Bagnall,?Broadband plasmonic couplers for light trapping and waveguiding,?Proceedings of SPIE, vol. 7712, pp. 77122R?8, 2010.[77] J. Chen, Z. Li, M. Lei, S. Yue, J. Xiao, and Q. Gong, ?Broadbandunidirectional generation of surface plasmon polaritons with dielectric?film?coated asymmetric single?slit,? Optics Express, vol. 19, pp. 26463?26469, 2011.[78] R. Verre, K. Fleischer, O. Ualibek, and I. V. Shvets, ?Self-assembledbroadband plasmonic nanoparticle arrays for sensing applications,? Ap-plied Physics Letters, vol. 100, p. 031102, 2012.[79] J.-S. Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats,?Broadband and broadangle SPP antennas based on plasmonic crystalswith linear chirp,? Scientific Reports, p. 829, 2012.[80] G. Kirchhoff, ?Zur theorie der lichtstrahlen,? Annalen Der Physik,vol. 254, pp. 663?695, 1883.[81] A. G. Brolo, R. Gordon, B. Leathem, and K. L. Kavanagh, ?Surfaceplasmon sensor based on the enhanced light transmission through arraysof nanoholes in gold films,? Langmuir, vol. 20, pp. 4813?4815, 2004.[82] R. Buckley and P. Berini, ?Radiation suppressing metallo-dielectric op-tical waveguides,? Journal of Lightwave Technology, vol. 27, pp. 2800?2808, 2009.[83] S. H. Talisa, ?Application of davidenkos method to the solution of dis-persion relations in lossy waveguiding systems,? IEEE Transactions onMicrowave Theory and Techniques, vol. 33, pp. 967 ? 971, 1985.[84] M. W. Maqsood, Metal Waveguides for Multi-Axial Light Guidingat Nanometer Scales. Master?s Thesis, The University of BritishColumbia, Nov. 2011.[85] A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Con-trol of Laser Radiation (Wiley Series in Pure and Applied Optics).Wiley-Interscience, 2002.109Bibliography[86] B. Wang and P. Lalanne, ?Surface plasmon polaritons locally excited onthe ridges of metallic gratings,? Journal of Optical Society of America.A, vol. 27, pp. 1432?1441, 2010.[87] J. Feng, V. S. Siu, A. Roelke, V. Mehta, S. Y. Rhieu, G. T. R. Palmore,and D. Pacifici, ?Nanoscale plasmonic interferometers for multispectral,high-throughput biochemical sensing,? Nano Letters, vol. 12, pp. 602?609, 2012.[88] P. Drude, ?Zur elektronentheorie der metalle,? Annalen Der Physik,vol. 306, p. 506, 1900.[89] P. Drude, ?Zur elektronentheorie der metalle; ii. teil. galvanomagnetis-che und thermomagnetische effecte,? Annalen Der Physik, vol. 308,p. 369, 1900.[90] E. J. Zeman and G. C. Schatz, ?An accurate electromagnetic theorystudy of surface enhancement factors for silver, gold, copper, lithium,sodium, aluminum, gallium, indium, zinc, and cadmium,? Journal ofPhysical Chemistry, vol. 91, pp. 634 ? 643, 1987.[91] D. R. Lide, CRC Handbook of Chemistry and Physics. CRC Press,88th ed., 2007.[92] M. J. Madou, Fundamentals of Microfabrication: the Science of Minia-turization. CRC Press, 2nd ed., 2002.110Appendices111Appendices ADrude Model and itsLimitationsMetals are characterized by a real permittivity that is negative. Thefrequency-dependent permittivity of metals can be modeled to good ap-proximation by a simple kinetic model of an electron gas proposed by PaulDrude in 1900 [88, 89]. The model begins by considering a density of un-bounded free electrons. From the equation of motion for an electron in thegas under the influence of a time-varying electric field, the permittivity ofthe electron gas can be shown to be(?) = 1??2p?(? + i/?). (A.1)where 1/? is the collision frequency and ?p is the plasma frequency. Re-writing Eq. A.1 into distinctive real and imaginary parts yields(?) = 1??2p(?2 + 1/?2)+ i?2p??(1 + ?2?2). (A.2)Assuming negligible damping 1/? ' 0, the permittivity can be written sim-ply as(?) = 1??2p?2, (A.3)which is negative when the frequency is less than the plasma frequency.Thus, for a low-loss electron gas, the plasma frequency sets an upper fre-quency bound below which the gas behaves like a metal.Three metals that have been commonly used to sustain SPPs are gold(Au), silver (Ag), and aluminum (Al). As seen in Table A.1, a commonfeature of these metals is that their collision rates are several orders of mag-nitude less than the plasma frequency, meaning that they all have relativelylow loss. Of the three metals, Ag has the lowest losses, followed by Au andthen Al. Although Al has the largest losses, its high plasma frequency means112Appendices A. Drude Model and its LimitationsTable A.1: Plasma frequency and collision rate measured by Zeman andSachts [90] for Ag, Au, and AlMetal Plasma frequency ?p (Hz) Collision rate ??1 (s?1)Silver 2.186? 1015 5.139? 1012Gold 2.15? 1015 17.14? 1012Aluminium 2.911? 1015 31.12? 1012it retains its metallic-like behavior at higher frequencies than Au and Ag,making it attractive for application in the deep UV. Ag is perhaps the mostideal metal in terms of its low loss, but it is reactive to sulphur in the atmo-sphere and tarnishes easily. Au has moderate losses and is only metallic forfrequencies below the red, but boasts the advantages of chemical inertnessand ease of deposition. In this thesis, we will use both Ag and Au to formSPP-sustaining surfaces.Because we have used the Drude model in the FDTD simulators to de-scribe the permittivity of metals, it will be useful to briefly compare predic-tions of the permittivity made by the Drude model (based on the parametersin Table A.1) against experimentally-measured values of the permittivityavailable in the literature. For Au and Ag, we use experimental data fromJohnson and Christy [70]. For Al, we use experimental data from the CRCHandbook of Chemistry and Physics [91]. As seen in Figures A.1, A.2, andA.3, the Drude model is fairly accurate in modeling the real part of the per-mittivity for all three metals over a wide frequency range. Discrepancies inthe Drude model predictions of the imaginary part of the permittivity for allthree metals versus the experimental data are due to interband transitions,which are not included in Eqn. A.1 but can be implemented, for instance, byrestricting the free-electron approximation and modeling electrons as bound,resonant entities.113Appendices A. Drude Model and its LimitationsFigure A.1: Permittivity values of gold predicted by the Drude model andobtained from experimental measurements [70].114Appendices A. Drude Model and its LimitationsFigure A.2: Permittivity values of silver predicted by the Drude model andobtained from measurements [70].115Appendices A. Drude Model and its LimitationsFigure A.3: Permittivity values of aluminum predicted by the Drude modeland obtained from measurements [91].116Appendices BFabrication ToolsWe have used a number of nanofabrication tools to build various slitdevices used in the experiments of this thesis. The general methodologyemployed was to deposit a thin layer of metal (either gold or silver) on amicroscope glass slide, etch features into the metal film using focused ionbeam milling, and apply various dielectric coatings. This appendix willbriefly discuss the salient features of the various tools used.Physical vapor depositionWe have used three different types of deposition systems based either onthermal evaporation, electron-beam evaporation, or sputtering.Thermal evaporationThermal evaporation is based on evaporating a source material in vac-uum using resistance-heating. In vacuum, the vaporized particles traveldirectly from the source to the substrate, where they condense back to asolid state. Thermal evaporation is one of the simplest ways to depositmaterial onto a substrate, but is limited to low-melting-point metals. Onemajor disadvantage is that the process is wasteful of the source materialand evaporated films generally exhibit poor adhesion. One way to increaseadhesion is to first lay down an ultra-thin (thickness smaller than 10 nm)layer on the substrate before depositing the source material.For the experiments in Chapter 2, we used a NRC 3115 thermal evapo-rator deposition system, which is a custom built system designed and manu-factured by the Simon Fraser University Technical Centre. This system wasused to deposit a 300- nm-thick silver film onto a glass substrate. A 5- nm-thick chromium layer was used to increase adhesion between the substrateand silver. The measured surface roughness of the film was approximately1.1 - 1.4 nm.117Appendices B. Fabrication ToolsElectron-beam evaporationElectron beam (e-beam) evaporation is based on bombarding a sourcematerial with an electron beam given off by a charged tungsten filamentunder high vacuum. The electron beam applies immense heat to the targetmaterial and vaporizes it. The vaporized particles then precipitate into solidform, coating everything in the vacuum chamber including the substrate.E-beam deposition provides high material deposition rates with relativelylow waste of the source material. It is economically suitable for depositingexpensive metals such as gold and platinum.For the experiments in Chapter 3, we used a Lesker PVD 75 DepositionSystem to deposit a 300- nm-thick gold film onto microscope glass substrates.A 5- nm-thick chromium layer was used to increase adhesion between theglass substrate and gold film. The measured surface roughness of the filmwas on the order of 1.1 - 1.3 nm.Sputter depositionSputter deposition is based upon ion bombardment of a source material,the target. Ion bombardment results in a vapor consisting of atoms fromthe target material. The most common approach for growing thin filmsby sputter deposition is to use a magnetron source in which positive ionspresent in the plasma of a magnetically enhanced glow discharge bombardthe target.For the broadband experiments in Chapter 2, we used a Lesker PVD75 Deposition System to deposit a 300- nm-thick silver film onto microscopeglass substrates. A 5- nm-thick chromium layer was used to increase theadhesion. The measured surface roughness of the film was on the order of1.3 - 1.5 nm.Focused ion beam millingFocused ion beam milling is a direct etching process that does not requirethe use of masking and process chemicals and is capable of sub-micrometerfeature resolution. The ion source, usually liquid gallium, is heated up andthen a high acceleration voltage is applied to extract Ga+ ions. Electrostaticlenses and other control mechanisms are used to guide and accelerate theions towards the substrate. The incident ions bombard the substrate andmaterial from the substrate is etched. To etch a desired feature, the focusedion beam is rastered digitally across the area to be milled. FIB is simple118Appendices B. Fabrication Toolsand versatile, and it has gained widespread use for fabricating miniaturizedstructures such as high-aspect-ratio surgical burbs, vertical field emitters,and probe tips [92]. In our experiments, we have used a FEI Strata DualBeam 235 system, with a focused ion beam resolution of 7 nm and a focusedelectron beam resolution of 3 nm.Spin coatingWe used Laurell WS-400B-6NPP-LITE precision spin coater to coatPMMA on metal substrates. We use the standard spinning speeds pro-vided in the table below to target the desired thickness. After spinning,substrates are baked at a temperature of 180 ? for 1 minute. The thicknessis re-checked by a high-sensitivity surface profiler (Tencor P10) with 1 A?resolution.Table B.1: Spin speed of the Laurell WS-400B-6NPP-LITE coater versusthe expected thickness and experimentally measured PMMA layer thicknessSpeed (rpm) Expected thickness ( nm) Measured thickness ( nm)3100 60 58.02500 80 78.61400 100 101.31200 120 122.51050 140 140.0980 160 160.3119

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