Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Aghanejad, Iman
Doctor of Philosophy - PhD
University of British Columbia
It has been shown that a homogeneous and isotropic material with simultaneously negative values of permittivity and permeability exhibits counter-intuitive responses such as refraction to the same side of normal (negative refraction) and evanescent wave amplification. Consequently, a flat slab of this material can focus an object to the image plane and overcome the diffraction limit associated with conventional dielectric lenses. Such double-negative materials do not exist in nature, however, artificial structures comprised of arrays of resonant inclusions have been designed to mimic their responses and realize a flat superlens with sub-diffractive resolution. This thesis rigorously investigates the behaviour of these artificial heterogeneous structures known as negative-index metamaterials. In the first part of this work, we present spatial frequency maps of power flow in order to provide deeper insights into the flow of propagating waves in negative-index metamaterials. We show through k-space diagrams that the distribution of power can vary considerably across these structures. These observations allow us to categorize different metamaterials and identify the mechanism behind negative refraction at a given interface. We also discuss how k-space maps of power flow can be used to explain the high or low transmittance of the power into different superlenses made from negative-index metamaterials. In the second part of this work, we study evanescent wave amplification in negative-index metamaterial lenses and propose design strategies to improve their imaging resolution and robustness. We show how resonant modes arising in metamaterial superlenses cause imaging artifacts and reduce imaging fidelity. We choose a well-studied periodic structure consisting of an array of magnetodielectric cylinders for the lens medium under test. We demonstrate that the presence of these artifacts can lead to erroneous interpretation of the standard two-source resolution test for lenses. We show that artifacts can be mitigated by introducing point defects into the array, which move the resonant modes to higher spatial frequencies and in the case of finite lenses, suppress their amplitudes through radiation losses. This strategy enables more robust and reliable subwavelength imaging performance, improves the spatial resolution of the metamaterial lens, and reduces the deleterious effects of material losses.
Investigating Spatial-Frequency PowerFlow and Evanescent-WaveAmplification in MetamaterialSuperlensesbyIman AghanejadA 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 2019© Iman Aghanejad, 2019The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Investigating Spatial-Frequency Power Flow and Evanescent-Wave Amplification in Metamaterial Superlensessubmitted by Iman Aghanejad in partial fulfilment of the requirements ofthe degree of Doctor of PhilosophyDr. Lo¨ıc Markley, School of EngineeringSupervisorDr. Kenneth J. Chau, School of EngineeringSupervisorDr. Jonathan F. Holzman, School of EngineeringSupervisory Committee MemberDr. Murray Neuman, School of Arts and SciencesUniversity ExaminerDr. Ma´rio G. Silveirinha, Department of Electrical and Computer Engineering, University of LisbonExternal ExamineriiAbstractIt has been shown that a homogeneous and isotropic material with simul-taneously negative values of permittivity and permeability exhibits counter-intuitive responses such as refraction to the same side of normal (negativerefraction) and evanescent wave amplification. Consequently, a flat slabof this material can focus an object to the image plane and overcome thediffraction limit associated with conventional dielectric lenses. Such double-negative materials do not exist in nature, however, artificial structures com-prised of arrays of resonant inclusions have been designed to mimic theirresponses and realize a flat superlens with sub-diffractive resolution. Thisthesis rigorously investigates the behaviour of these artificial heterogeneousstructures known as negative-index metamaterials.In the first part of this work, we present spatial frequency maps of powerflow in order to provide deeper insights into the flow of propagating waves innegative-index metamaterials. We show through k-space diagrams that thedistribution of power can vary considerably across these structures. Theseobservations allow us to categorize different metamaterials and identify themechanism behind negative refraction at a given interface. We also discusshow k-space maps of power flow can be used to explain the high or lowtransmittance of the power into different superlenses made from negative-index metamaterials.In the second part of this work, we study evanescent wave amplifica-tion in negative-index metamaterial lenses and propose design strategies toimprove their imaging resolution and robustness. We show how resonantmodes arising in metamaterial superlenses cause imaging artifacts and re-duce imaging fidelity. We choose a well-studied periodic structure consistingof an array of magnetodielectric cylinders for the lens medium under test.We demonstrate that the presence of these artifacts can lead to erroneousinterpretation of the standard two-source resolution test for lenses. We showthat artifacts can be mitigated by introducing point defects into the array,which move the resonant modes to higher spatial frequencies and in the caseof finite lenses, suppress their amplitudes through radiation losses. Thisstrategy enables more robust and reliable subwavelength imaging perfor-iiiAbstractmance, improves the spatial resolution of the metamaterial lens, and reducesthe deleterious effects of material losses.ivLay SummaryAdvances in various research fields and technologies such as nanotech-nology and microbiology depend on improving conventional lenses and mi-croscopes. It has been suggested that artificial materials known as metama-terials can be designed to achieve superlenses with enhanced imaging resolu-tions. In this thesis, we rigorously analyze the behaviour of electromagneticwaves in metamaterial lenses and discuss their potential limitations. We alsopropose design strategies to improve their imaging resolution and robustnessand reduce the deleterious effects of material losses.vPrefaceThis work has been done under the guidance of Dr. Lo¨ıc Markley andDr. Kenneth Chau at the School of Engineering in The University of BritishColumbia.Portions of my thesis have been published in three journal articles:1. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley. “Avoidingimaging artifacts from resonant modes in metamaterial superlenses.”Phys. Rev. B., 100, 035137 (2019). I prepared the manuscript, whichwas further edited by Dr. Markley and Dr. Chau. The materials ofthis paper have been used in Chapters 1 and 4.2. Iman Aghanejad and Lo¨ıc Markley. “Spatial frequency maps of powerflow in metamaterials and photonic crystals: Investigating backward-wave modes across the electromagnetic spectrum.” Phys. Rev. B, 96,205157 (2017). I prepared the manuscript, which was further edited byDr. Markley. The materials of this paper have been used in Chapters2 and 3.3. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley. “Electromag-netic origins of negative refraction in coupled plasmonic waveguidemetamaterials.” Phys. Rev. B, 94, 165133 (2016). I prepared themanuscript, which was further edited by Dr. Markley and Dr. Chau.The materials of this paper have been used in Chapters 2 and 3.Portions of my thesis have also been presented at the following confer-ences:1. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley, “Avoidingimaging artifacts in metamaterial superlenses”, APS/IEEE Interna-tional Symposium, Atlanta, Georgia, USA (2019).2. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley, “K-space sig-natures of negative-index metamaterials”, APS/IEEE InternationalSymposium, San Diego, California, USA (2017).viPreface3. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley, “Enhancedtransmission into layered-plasmonic metamaterials through k-spaceharmonic coupling”, National Radio Science Meeting, Boulder, Col-orado, USA (2017).4. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley, “Rigorous anal-ysis of negative refraction in a layered plasmonic metamaterial”, AP-S/IEEE International Symposium, Fajardo, Puerto Rico (2016).5. Iman Aghanejad, Kenneth J. Chau, and Lo¨ıc Markley, “Real-k-spaceanalysis of electromagnetic waves in a plasmonic waveguide meta-material”, APS/IEEE International Symposium, Vancouver, Canada(2015).viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Microscopic Maxwell’s equations . . . . . . . . . . . . 31.1.2 Macroscopic Maxwell’s equations . . . . . . . . . . . . 31.2 Bulk eigenmodes of a homogeneous and isotropic medium . . 51.2.1 Bulk modes of the Veselago medium . . . . . . . . . . 101.3 Veselago flat lens . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Transmission over the propagating spectrum . . . . . 131.3.2 Transmission over the evanescent spectrum . . . . . . 161.3.3 Non-perfect homogeneous flat lens . . . . . . . . . . . 211.4 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.1 Artificial magnetization of current loops . . . . . . . . 241.4.2 Effective negative permeability for an array of mag-netic elements . . . . . . . . . . . . . . . . . . . . . . . 251.4.3 Effective negative permittivity for an array of thin wires 26viiiTABLE OF CONTENTS1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Chapter 2: Spatial frequency maps of power flow in metama-terials and photonic crystals . . . . . . . . . . . . . 302.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Split-ring resonator and thin-wire metamaterial . . . . . . . . 342.3 Photonic crystal . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Sub-wavelength magnetodielectric crystal . . . . . . . . . . . 402.5 Sub-wavelength plasmonic crystal . . . . . . . . . . . . . . . . 412.6 Coupled-plasmonic-waveguide metamaterial . . . . . . . . . . 422.7 Fishnet metamaterial . . . . . . . . . . . . . . . . . . . . . . . 462.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Chapter 3: k-space analysis of refraction and mode excitationefficiency across a metamaterial interface . . . . . 503.1 Refraction across air-metamaterial interfaces . . . . . . . . . 503.2 Mode excitation efficiency across a metamaterial interface . . 543.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Chapter 4: Evanescent-wave enhancement in negative-indexmetamaterials . . . . . . . . . . . . . . . . . . . . . . 614.1 Regular array magnetodielectric (MD) metamaterial . . . . . 624.1.1 Bulk modes in two-dimensional regular array MD meta-material . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Bound modes in infinite regular array metamaterialslabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1.3 Resonant modes in finite regular array metamateriallenses . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.4 Imaging performance of finite regular array metama-terial lens . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Defected array MD metamaterial . . . . . . . . . . . . . . . . 734.2.1 Bulk modes in two-dimensional defected array MDmetamaterial . . . . . . . . . . . . . . . . . . . . . . . 744.2.2 Defected array metamaterial slab . . . . . . . . . . . . 754.2.3 Defected array metamaterial lens . . . . . . . . . . . . 784.2.4 Imaging performance of finite defected metamateriallens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83ixTABLE OF CONTENTSChapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Appendices A:Clausius-Mossotti equation . . . . . . . . . . . . . . 95Appendices B:Effective permittivity in the presence of motion-freeelectrons . . . . . . . . . . . . . . . . . . . . . . . . . 97Appendices C:Light focusing with a flat Veselago slab . . . . . . . 99Appendices D:Surface bound modes of a Veselago half-sapce . . . . 101Appendices E: Slab bound modes of Veselago lens . . . . . . . . . . 103Appendices F: The response of Veselago lens to incident evanescentwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Appendices G:Bloch modes finite element simulation at a fixed realfrequency . . . . . . . . . . . . . . . . . . . . . . . . . 109xList of TablesTable 2.1 The categorization of periodic structures that supportbackward power across the electromagnetic spectrumbased on their distinguishing feature in the spatial fre-quency domain. . . . . . . . . . . . . . . . . . . . . . . 49xiList of FiguresFigure 1.1 The dispersion diagrams of air over the kxkz plane(ky = 0) for a fixed frequency ω. (a) The kx and kzcomponents can have real values. (b) The transversekx component have unlimited real values and theircorresponding longitudenal kz components are purelyreal for |kx|<k0 and purely imaginary for |kx|>k0.The wave vector k and the Poynting vector S of apropagating plane wave are shown by the black thinand the blue thick vectors, respectively. . . . . . . . . 6Figure 1.2 The frequency variation of relative constitutive pa-rameters of a double-negative medium modeled byconstitutive parameters in Eqs. 1.41 and 1.42. . . . . 10Figure 1.3 (a) The right-handed (RH) and (b) left-handed (LH)plane waves in which power and phase flow in thesame and opposite directions, respectively. . . . . . . 11Figure 1.4 The dispersion diagrams of Veselago medium over thekxkz plane (ky = 0) for the frequency ωo. (a) The kxand kz components can have real values. (b) Thetransverse kx component have unlimited real valuesand their corresponding longitudenal kz componentsare purely real for |kx|<ko and purely imaginary for|kx|>ko. The wave vector k and the Poynting vectorS of a a left-handed plane wave are shown by theblack thin and the red thick vectors, respectively. . . 12xiiLIST OF FIGURESFigure 1.5 Image field intensity of different object distributionsfor a conventional lens which only transfers the prop-agating components from source to image plane. (a)A point source is located at the origin. (b) A pairof symmetrically centered dipole sources separatedby λ0. (c) A pair of symmetrically centered dipolesources separated by 0.5λ0. (d) A pair of symmetri-cally centered dipole sources separated by 0.4λ0. Thelocation of the corresponding objects are indicated bythe gray lines. . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.6 Graphical description of plane-wave refraction fromthe right-handed free space (n = 1) into (a) the right-handed free space (n = 1) and (b) into the left-handedVeselago medium (n = −1) using equifrequency con-tour (EFC) analysis. Continuity of the tangentialwave vector components can be visualized throughthe black dashed lines. . . . . . . . . . . . . . . . . . 14Figure 1.7 An illustration of the focusing mechanism for a Vese-lago flat lens with thickness d embedded in air atωo. (a) The negative refraction of light rays at theinterfaces of the flat lens (solid arrows) in compari-son to their behaviour in free space (dashed arrows).(b) The field plot for radiation from a point sourceinfinitely extended along the y direction in air. (c)The field plot for imaging of propagating componentswithin the Veselago lens. . . . . . . . . . . . . . . . . 15Figure 1.8 An illustration of the evanescent enhancement withina Veselago flat lens embedded in air at frequency ωo.The behaviour of evanescent wave in air (without thepresence of lens) is shown by a dashed line for com-parison. . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 1.9 An illustration of the surface resonant modes at theinterface of a Veselago half-space (solid line) whichcan be excited by an incident evanescent wave fromair (dashed line) . . . . . . . . . . . . . . . . . . . . . 18Figure 1.10 Field distribution of (a) even bound mode and (b) oddbound mode guided along a slab of Veselago material 19xiiiLIST OF FIGURESFigure 1.11 Dispersion diagrams of even and odd bound modesguided along Veselago slabs of thickness a (dashedblack lines) and 5a (solid black lines). The geometryof the slabs and the magnetic field distributions (Hy)of even and odd bound modes are plotted in the insetdiagrams. The field plots correspond to a wave vectorkx = 1.7ko. The light line is indicated by the solidgray line. The dispersion diagram of an uncoupledsurface waves at a single interface is shown by theflat dotted black line at ωo. The flat solid red line in-dicates the operating frequency ωo in which the bothvalues of εr and µr are equal to −1. . . . . . . . . . . 20Figure 1.12 Field distribution of even solution (right plot) andodd solution (middle plot) and their superposition(left plot) for the response of a Veselago lens to anincident evanescent wave at frequency ωo . . . . . . . 20Figure 1.13 A Veselago lens dispersion diagram illustrating theappearance of resonant modes in a non-perfect flatslab. The flat dashed-dotted red line corresponds toa slight deviation in operating frequency from ωo andthe subsequent appearance of bound modes at theintersection points. These intersection points are in-dicated by red dots in the dispersion diagram. Theimaging performance would be further degraded fora thicker slab since the intersecting resonance pointappears at a lower spatial frequency. . . . . . . . . . . 21Figure 1.14 Illustration of the evanescent enhancement with in(a) a solid negative index slab and (b) a multi-layeredslab of alternating thin layers of positive and negativeindex media. . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.15 The geometry of different double-negative index meta-material structures (a) Arrays of split-ring resonatorsand thin wires. (b) The coupled plasmonic waveguidemade of metallic and dielectric layers. (c) Arrays ofplasmonic, dielectric or magnetodielectric inclusions. . 23Figure 1.16 A split ring resonator and its simplified LC circuitmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 1.17 A cubic array of SRRs . . . . . . . . . . . . . . . . . 25xivLIST OF FIGURESFigure 1.18 The dispersion diagram for the extracted effectivepermeability of a cubic array of split-ring resonators(SRRs) . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 1.19 (a) An array of thin wires. (b) The cubic unit cell ofthe structure. . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.1 The k-space diagrams for a backwards eigenmode prop-agating in-plane along the z axis of the split-ring res-onator and thin-wire (SRR-TW) metamaterial (θ =0). The Bloch harmonics are plotted on the ky = 0plane (a) and on the kx = 0 plane (b). Left-handedplane wave components in k space are plotted withred arrows, while right-handed components are plot-ted with blue arrows. The EFCs are given by thesolid and dotted curves inside and outside the firstBrillouin zone (BZ), respectively, where the first BZ isindicated by the light grey dashed rectangle. The unitcell is shown from the top (a, inset) and side (b, inset)with perfect electric conductor (PEC) boundary con-ditions (solid lines) and periodic boundary conditions(dashed lines) for the planes normal and parallel tothe y axis, respectively. . . . . . . . . . . . . . . . . . 35Figure 2.2 The k-space diagrams for a backwards eigenmode prop-agating in-plane along the diagonal axis of the SRR-TW metamaterial (θ = 45◦). The Bloch harmonicsare plotted on the ky = 0 plane (a) and on the kx = kzplane (b). Note that only the average power S mustbe normal to the EFC. Individual harmonic Poyntingvectors are free to point in any direction. . . . . . . 36xvLIST OF FIGURESFigure 2.3 The k-space diagrams for backward eigenmodes prop-agating along the (a) ΓK and (b) ΓM directions inthe hexagonal photonic crystal. All Bloch harmon-ics for this structure are transverse electric and mag-netic (TEM) right-handed plane waves plotted withblue arrows. The EFCs are given by the solid anddotted curves inside and outside the first BZ, respec-tively, where the first BZ is indicated by the lightgrey dashed hexagon. The photonic crystal geometrywith its rectangular unit cell is illustrated in the inset.Again, as seen above, individual harmonic Poyntingvectors are not restricted to being normal to the EFCs. 38Figure 2.4 The k-space illustration of Bloch harmonics for back-ward eigenmodes propagating in the magnetodielec-tric crystal (a) parallel to the z axis (θ = 0◦) and(b) diagonal to the x and z axes (θ = 45◦). Theschematic geometry of the magnetodielectric crystaland the square unit cell are shown in the inset diagram. 40Figure 2.5 The k-space illustration of Bloch harmonics for back-ward eigenmodes propagating in the plasmonic crys-tal (a) parallel to z axis (θ = 0◦) and (b) diago-nally to x and z axes (θ = 45◦). Left-handed planewave components are plotted with red arrows, whileright-handed components are plotted with blue ar-rows. The schematic geometry of the plasmonic crys-tal and the square unit cell are shown in the inset. . . 41Figure 2.6 The backward eigenmodes propagating parallel andperpendicular to the layers in the coupled-plasmonic-waveguide metamaterial. (a) The schematic diagramof a unit cell with periodicity a. (b) Mode magneticfield plots at quarter-period intervals over t1 < t2 <t3 < t4 for a wave propagating parallel to planes(left) and perpendicular to the planes (right). (c)The k-space illustration of harmonics for modes prop-agating parallel to planes (left) and perpendicular tothe planes (right) corresponding to the modes in (b).In (c) the attenuation contours corresponding to theimaginary parts of the calculated Floquet-Bloch wavevector are given by the green dashed curve. . . . . . . 43xviLIST OF FIGURESFigure 2.7 The backward eigenmodes propagating in a plane per-pendicular to the layers of coupled-plasmonic-waveguidemetamaterial. (a) The k-space illustration of Blochharmonics for out-of-plane backward modes propa-gating at angles θ = 20◦, θ = 45◦, and θ = 70◦. (b)The decomposition of total power flow along the refer-ence direction into left-handed and right-handed con-tributions versus propagation angle θ. As θ increases,dominant contributors to power flow shift from left-handed to right-handed plane-wave components. . . . 45Figure 2.8 The fishnet metamaterial. (a) The schematic diagramof the front view (left plot) and the side views (rightplot) in which the PEC screen and square hole arespecified by the blue and white color, respectively.(b) The k-space illustration of Bloch harmonics overthe kxkz plane (top view) and over the kzky plane(side view) for a backward eigenmode propagating inthe fishnet metamaterial along the kz axis (normalto the perforated screens). The cubic unit cell of thestructure is determined in (a) by the PEC boundaryconditions (solid lines) parallel to the xz plane andperiodic boundary conditions (dashed lines) along yaxis in order to investigate propagation parallel to thexz plane. . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.1 A diagram illustrating the coupled-plasmonic-waveguidemetamaterial and the planes of incidence for (a) end-fire in-plane refraction, (b) end-fire out-of-plane re-fraction, and (c) broadside out-of-plane refraction.The unit cell periodicity is a. . . . . . . . . . . . . . . 51xviiLIST OF FIGURESFigure 3.2 Description of wave refraction from free space intothe layered metamaterial based on the k-space com-ponents of the eigenmodes in the metamaterial andconservation of tangential wave vectors along the in-terface for (a) end-fire in-plane refraction, (b) end-fire out-of-plane refraction, and (c) broadside (out-of-plane) refraction. The real and imaginary parts of thewave vector are plotted with solid gray and dotted-dashed green contours, respectively. The repeatedEFCs of the layered metamaterial are also plotted bythe dotted lines outside of the first BZ. Continuity ofthe tangential wave vector components can be visu-alized through the dashed black lines. . . . . . . . . . 52Figure 3.3 Two different end-fire wave excitations of the nor-mal in-plane eigenmode propagating with kt = 0 inthe coupled-plasmonic-waveguide metamaterial. Incase A, a normally incident plane wave targets thefundamental m = 0 harmonic, while in case B, twooblique plane waves with incidence angles of ±53.2◦are superposed in order to simultaneously target thehigher-order harmonics m = ±1. The k-space powermaps of eigenmodes and the incident and transmittedmagnetic field at the interface are plotted at the leftand right side, respectively. The white arrows showthe direction of the time-and-space-averaged powerflow in both the incidence dielectric medium (n = 4)and the layered medium. It is shown that couplingto the higher order harmonics (the dominant k-spacecomponents) has increased the coupling efficiency bya factor of almost 40. The reflected power as a per-centage of the incident power is indicated by R. . . . 55xviiiLIST OF FIGURESFigure 3.4 Three different end-fire wave excitations of the out-of-plane eigenmode propagating with kt = k0 sin 20◦in the coupled-plasmonic-waveguide metamaterial. Aplane wave is incident from a dielectric medium of in-dex n = 4 with an angle −43.1◦ (case A), −4.9◦ (caseB), and 30.9◦ (case C) in order to target the m = +1,m = 0, and m = −1 component of the eigenmode,respectively. It is shown that coupling to the higherorder harmonics has increased the coupling efficiencymore than 220 times. The direction of power flowindicated by the white arrows demonstrates negativerefraction of power at the interface for cases A andB, and positive refraction of power for case C. . . . . 57Figure 3.5 Two different wave excitation of the eigenmode prop-agating with kt = 0 in the coupled-plasmonic-waveguidmetamaterial tilted by 45◦. In case A, a normally in-cident plane wave from a dielectric medium of indexn = 4 targets the fundamental harmonic, while incase B, an oblique plane wave with incidence anglesof 25.03◦ targets the dominant higher-order harmon-ics m = 1. It is shown that coupling to the higherorder harmonic (the dominant k-space components)has increased the coupling efficiency more than 8 times. 58Figure 3.6 Power transmission for two different interfaces of arectangular photonic crystal sample studied in Sec. 2.3.The transverse electric field is plotted for two Gaus-sian beams normally incident from free space ontothe vertical and horizontal interfaces. The low trans-mittance of the incident beam along the ΓK axis andits high transmittance along ΓM axis are consistentwith the k-space power plots of the photonic crystaleigenmodes propagating along these directions in (a)and (b), respectively. The photonic crytal geometryis partially shown in (c). Note that both reflectedbeams have been omitted for clarity. . . . . . . . . . . 59xixLIST OF FIGURESFigure 4.1 The equifrequency contour of the bulk eigenmodesat ωo in the regular array metamaterial (solid blacklines), alongside the EFC of an ideal homogeneousdouble-negative medium (solid gray lines). The com-plex longitudinal wave vector components kz+iαz areplotted as a function of the real transverse wave vec-tor component kx. The geometry of the regular arraymetamaterial with its square unit cell is illustrated inthe inset diagram. . . . . . . . . . . . . . . . . . . . . 63Figure 4.2 Diagram of the simulation setup used to solve for thebound modes guided along infinite slabs of the regu-lar array MD metamaterial. Periodic boundary con-ditions are assigned to the top and bottom sides andperfectly matched layers are placed at the left andright sides of the domain. . . . . . . . . . . . . . . . . 64Figure 4.3 Dispersion diagrams of guided bound modes alonginfinite slabs of the regular array MD metamaterial.The results for a monolayer MD slab and a five-layerMD slab are shown by the dashed black lines and solidblack lines over the first BZ (up to |kx| = pi/a= 7.73ko),respectively. The even bound modes (correspond-ing to the upper lines at low spatial frequencies) in-tersect with ωo to produce a transmission peak atkx = 1.871ko for the five-layer and at kx = 3.183ko forthe monolayer slab. The slab geometries, unit cells,and bound mode field distributions corresponding tokx = 1.7ko are plotted in the inset diagrams. The dis-persion diagram of uncoupled single interface surfacewaves is shown by the dotted black line. The flat solidred line indicates the operating frequency ωo . . . . . 65Figure 4.4 A diagram of the transmission simulation setup of theinfinite regular array slab. . . . . . . . . . . . . . . . 66Figure 4.5 The transfer function of the slabs at ωo. The resultsfor a monolayer MD slab and a five-layer MD slabare shown by the dashed black lines and solid blacklines over the first BZ (up to |kx| = pi/a= 7.73ko),respectively. Transmission peaks correspond to thepoints where the dispersion curves in Fig. 4.3 intersectwith ωo. The ideal transfer function of Veselago slabsis shown in (b) by the flat gray line at 0 dB. . . . . . 67xxLIST OF FIGURESFigure 4.6 Magnetic field intensity plots of the dominant reso-nant modes for the regular array lenses of thickness5a. The lens has a width of 65a in (a) and 59a in(b) and (c). The resonance modes have an even lon-gitudinal symmetry (along the z axis). (a) The reso-nant mode with ωr = ωo and γ/ωo = 6.3437× 10−5has transverse even symmetry (along the x axis). (b)The resonant mode with ωr = 0.9998ωo and γ/ωo =4.3085× 10−5 has transverse odd symmetry (alongthe x axis). (c) The resonant mode with ωr = 1.0003ωoand γ/ωo = 1.1288× 10−4 has transverse even sym-metry. The inset diagrams plot the real part of thefield to show the transverse and longitudinal symme-tries of each mode. . . . . . . . . . . . . . . . . . . . . 68Figure 4.7 Imaging field plots for the regular array lenses withwidths 65a (left) and 59a (right). Each lens has thick-ness 5a and is illuminated by two x-polarized electricdipoles located at x= 0.25λo and −0.25λo. The mag-netic field intensity is plotted over the xz plane on theleft, while the normalized field intensity at the imageplane is plotted on the right in solid black. The imagefield without the lens is plotted in dotted black, alongwith the source plane field intensity of the object insolid gray. The source plane at z=−5a and the im-age plane at z= 5a are indicated by thin black linesin the field plots. . . . . . . . . . . . . . . . . . . . . . 70Figure 4.8 Field plots similar to the ones shown in Fig. 4.7 fortwo offset electric dipoles located at x= 0.13λo and0.63λo (left) and for two offset dipoles with widerspacing located at x= 0.25λo and −0.65λo (right). . . 71xxiLIST OF FIGURESFigure 4.9 The normalized field intensity at the image planefor imaging with lossless and lossy regular array MDlenses. The illuminating sources are four x-polarizedelectric dipoles located at x = 0.675λo, 0.225λo, −0.225λoand −0.675λo. The fields from MD lenses with loss-less and lossy inclusions (with electric and magneticloss tangents equal to 10−3) are plotted in solid blackline and solid red line with circle markers, respec-tively. The normalized source field intensity of theobject and image intensity without the lens are plot-ted in solid gray and dotted black, respectively. . . . 72Figure 4.10 The EFC of the bulk eigenmodes at ωo in the de-fected array metamaterial (solid and dashed blacklines), alongside the EFC of an ideal homogeneousdouble-negative medium (solid gray lines). The ge-ometry of the defected array metamaterial with itssquare unit cell is illustrated in the inset diagram. . . 74Figure 4.11 Diagrams of the simulation setups used to generatethe dispersion curves and transmission response ofthe infinite defected array slab are shown in (a) and(b), respectively. . . . . . . . . . . . . . . . . . . . . . 75Figure 4.12 The dispersion diagrams of guided bound modes alongthe defected array metamaterial slab over the firstBZ (up to |kx| = pi/2a= 3.98ko). The middle curverepresents an even bound mode while the upper andlower curves represent odd bound modes. The oddbound mode corresponding to the lowest line at lowspatial frequencies intersects with ωo to produce thefirst transmission peak at kx = 2.824ko. The MD slabgeometry and unit cell are plotted in the inset dia-grams alongside the magnetic field distribution of theguided odd mode at kx = 1.7ko. . . . . . . . . . . . . . 76Figure 4.13 The transfer function of the defected array metama-terial slab at ωo over the first BZ (up to |kx| = pi/2a =3.98ko). Transmission peaks correspond to the pointswhere the dispersion curves in Fig. 4.12 intersect withωo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77xxiiLIST OF FIGURESFigure 4.14 Magnetic field intensity plots of the dominant reso-nant modes for the defected array lens of width 59aand thickness 5a. The resonance modes have an oddlongitudinal symmetry (along the z axis). (a) Theresonant mode with ωr = 0.9998ωo and γ/ωo = 5.8441×10−5 has transverse even symmetry. (b) The resonantmode with ωr = ωo and γ/ωo = 4.1274× 10−5 hastransverse odd symmetry. The inset diagrams plotthe real part of the field to show the transverse andlongitudinal symmetries of each mode. . . . . . . . . 78Figure 4.15 Imaging field plots for the regular array lens (left)and the defected array lens (right). Each lens hasthickness 5a, width 59a, and is illuminated by twox-polarized electric dipoles located at x= 0.25λo andx=−0.25λo. . . . . . . . . . . . . . . . . . . . . . . . 80Figure 4.16 Imaging field plots for the defected array lens for twooffset electric dipoles located at x= 0.13λo and 0.63λo(left) and for two offset dipoles with wider spacinglocated at x= 0.25λo and −0.65λo (right). . . . . . . 80Figure 4.17 The normalized field intensity at the image plane forimaging with lossless and lossy defected array MDlenses. The illuminating sources are four x-polarizedelectric dipoles located at x= 0.675λo, 0.225λo, −0.225λoand −0.675λo. The fields from MD lenses with loss-less and lossy inclusions (with electric and magneticloss tangents equal to 10−3) are plotted in solid blackline and solid red line with circle markers, respec-tively. The normalized source field intensity of theobject and image intensity without the lens are plot-ted in solid gray and dotted black, respectively. . . . 81Figure 4.18 The minimum separation distance (∆) necessary toresolve a four symmetric dipoles in the image of theregular array lens (red circle markers) and the de-fected array lens (blue asterisk markers) as a functionof the electric and magnetic loss tangent of the MDinclusions (tanδ). The length of error bars is equal to0.005λo. . . . . . . . . . . . . . . . . . . . . . . . . . . 82xxiiiAcknowledgementsI would like to express my gratitude to my supervisors, Dr. Lo¨ıc Markleyand Dr. Kenneth Chau, for their precious support, and guidance. Theyhelped me to deal with tough situations, enhance my critical thinking skills,and improve the level of my research. Without their support conductingthis work would not have been possible. I would also thank Dr. JonathanHolzman, my committee member, for his feedback and suggestions. I wouldlike to thank all my friends and research colleagues, especially MohammedAl Shakhs, Asif Alnoor, Max Bethune-Waddell, Vincent Loi, Masoud Ah-madi, Connor Badowich, Nibirh Jawad, Ali Maleki, Cael Warner, and MarekSplawinski.Special thanks to my parents for their support and patience. I owe mydeepest gratitude to my wife, Ladan Ahmadi, for her unconditional love,support, patience, and sacrifices.xxivDedicationTo my parents and to my wifexxvChapter 1IntroductionHopes to advance important technologies such as imaging and microscopywill depend on the development of efficient ways to manipulate electro-magnetic fields at the finest spatial scales. Metamaterials hold promisefor achieving these capabilities and offer new perspectives to the electro-magnetic world [1]. Metamaterials are artificial structures in which theresponse of the media is defined by the complex interactions between theelectromagnetic fields and their sub-wavelength inclusions. In other words,metamaterials gain their electromagnetic properties from their structuralcomposition rather than atomic composition in natural materials. In theory,metamaterials can be designed to achieve desired responses beyond what canbe achieved using natural materials. Based on metamaterial technologies,anomalous phenomena such as negative refraction have been observed [2, 3]and novel devices such as flat superlenses with sub-diffraction resolutionshave been experimentally implemented [4, 5].In this thesis, we are particularly interested in materials with negative in-dex of refraction. Negative index is perhaps the most well known example ofan electromagnetic property not found in natural materials. The research to-wards negative index materials dates back to 1968 when V. Vesalago first de-scribed the behaviour of the electromagnetic fields in a hypothetical double-negative materials with simultaneously negative values of electric permittiv-ity ε and magnetic permeability µ [6]. He showed that such medium couldsustain anomalous left-handed propagating plane waves with anti-parallelphase and group velocities. As a result, a flat slab of this material, whichwe will refer to as a Veselago lens, can focus all the propagating componentsof an object by negatively refracting them at the two interfaces of the slab.The Veselago slab was later shown to amplify and restore the sub-wavelengthdetails of an object, smaller than half the wavelength of operation, whichare carried by evanescent waves from the object, and would become blurredwhen imaging with conventional dielectric lenses [7]. The physical origin ofevanescent amplification has been attributed to the coupling of evanescentwaves to the resonant bound modes guided along the Veselago slab [8, 9].These theoretical predictions attracted the attention of researchers to realize11.1. Maxwell’s equationsnegative index materials and implement a flat superlens which image beyondthe conventional diffraction limit. Since 2001, many different metamaterialstructures have been designed across the electromagnetic spectrum whichare able to mimic the Veselego medium and negatively refract electromag-netic beams [10–14]. The extraction of double-negative effective parametersfor these metamaterials has suggested that they might be viable candidatesfor Veselago superlensing [15–17]. Subwavelength-spaced symmetric pairsof point sources have also been resolved using flat lenses comprised of theseheterogeneous structures [4, 18, 19].These observations have led researchers to describe these structures ashomogeneous media with isotropic negative refractive index; however, a closeinspection of the fields throughout these heterogeneous structures as pro-vided in this thesis would suggest otherwise. We rigorously analyze therelationship between phase and power flow within a metamaterial structureby introducing a method to calculate the power flow of all Floquet-Blochharmonics, information which cannot be obtained from either conventionalanalysis of () or effective medium theory. Access to power flow of all har-monics enables us to demonstrate the origin of backward power, and indoing so verify the validity of the claimed isotropic left-handed responseand the validity of describing the medium by a simple negative effectiveindex of refraction n=−1. We also expand our analysis by studying theresponse of negative-index metamaterials to incident evanescent waves. Weshow that although a metamaterial flat lens can negatively refract and fo-cus electromagnetic beams its response can differ significantly from a perfectVeselago lens at high spatial frequency beyond the free-space cutoff. We in-vestigate the resonant modes in infinite and finite metamaterial lenses andtheir imaging artifacts. We show that if overlooked, these artifacts canlead to erroneous interpretation of the standard two-source resolution test.We propose design strategies to tailor the resonant modes and improve theresolution and robustness of imaging and reduce the deleterious effects ofmaterial losses.1.1 Maxwell’s equationsIn 1865, James Clerk Maxwell published a set of partial differential equa-tions that form the foundation of classical electromagnetism [20]. Theydemonstrate the behaviour of fluctuating electric and magnetic fields knownas electromagnetic waves in a material (either free space as a neutral mediumor a material which has a response to electric and magnetic fields). In this21.1. Maxwell’s equationsthesis, we consider time-harmonic fields and sources with an implicit e−iωtdependence. In the following subsections we explain microscopic and macro-scopic forms of Maxwell’s equations [21, 22].1.1.1 Microscopic Maxwell’s equationsThe microscopic Maxwell’s equations consider an electromagnetic mate-rial as free space, with an electric permittivity ε0 and magnetic permeabilityµ0, which can contain bound charges ρb and bound currents Jb at the atomicscales. These bound sources determine the electromagnetic response of thematerial. Besides them, free currents Jf and free charges ρf as primary elec-tromagnetic sources can also be present in space. The microscopic Maxwell’sequations relate the electric field E and magnetic field B to the total currentsource J = Jf + Jb and total charge source ρ= ρf + ρb in space by∇×E = iωB, (1.1)∇×B = µ0(J− iωε0E), (1.2)∇ ·E = ρε0, (1.3)∇ ·B = 0. (1.4)1.1.2 Macroscopic Maxwell’s equationsThe macroscopic Maxwell’s equations describe the large-scale behaviourof electromagnetic fields by averaging out atomic-scale field fluctuations.The large-scale behaviour for the bound sources (Jb and ρb) are given bythe averaged electric polarization P and averaged magnetic polarization Mof material particles asJb = −iωP +∇×M, (1.5)ρb = −∇ ·P. (1.6)Substituting the above expressions for the bound sources in 1.2 and 1.3 givesimportant relations between averaged electromagnetic fields and free sources(Jf and ρf )∇×E = iωB, (1.7)∇×H = Jf − iωD, (1.8)∇ ·D = ρf , (1.9)31.1. Maxwell’s equations∇ ·B = 0, (1.10)where the axillary fields D and H are defined byD = ε0E + P, (1.11)B = µ0(H + M), (1.12)which are known as constitutive relations. Using these equations and as-suming a linear and local relation between the averaged polarization P (M)and averaged field E (H) the relative permittivity εr and permeability µrhave been defined to model the large-scale electromagnetic response of amaterial in frequency domain ω:D = ε0εrE = εE, (1.13)B = µ0µrH = µH. (1.14)A material with invariable response with respect to spatial direction andlocation is known as isotropic and homogeneous medium, respectively. Fora homogeneous and isotropic material ε and µ are generally complex scalars(in which the imaginary part refers to the losses) and can be a functionof angular frequency ω for frequency dispersive materials. The constitu-tive parameters can be extracted by theoretical calculations or experimentalmeasurements for different materials. For instance, two theoretical methodsare briefly discussed in the Appendices A and B. It should be mentionedthat the above constitutive relations cannot model all complex material re-sponses. For instance, the polarization in a spatially dispersive material canbe induced by the electromagnetic fields in all other spatial locations andthus the material cannot be modeled by the above local constitutive param-eters [23]. In addition, the polarization P or M in a bi-anisotropic materialcan be induced by both electric and magnetic fields. These materials aremodeled by ε and µ together with the magnetoelectric coupling parameterswhich relate the electric polarization P to the magnetic field H and mag-netic polarization M to the electric field E in their constitutive equations[22].41.2. Bulk eigenmodes of a homogeneous and isotropic medium1.2 Bulk eigenmodes of a homogeneous andisotropic mediumHere, we discuss the behaviour of electromagnetic waves in a simple,lossless, homogeneous and isotropic material. We assume that the materialcan be modeled by scalar constitutive parameters ε and µ which are generallyfunctions of frequency ω. In the absence of primary sources (Jf = ρf = 0),the electromagnetic fields in this material are governed by the source-freeMaxwell’s equations∇×E = iωµH, (1.15)∇×H = −iωεE, (1.16)∇ ·E = 0, (1.17)∇ ·H = 0. (1.18)Taking another curl of electric fields in Eq. 1.15 and substituting Eq. 1.16results in the vector wave equation∇×∇×E− k2E = 0, k = ±ω√εµ (1.19)which governs electric fields in the absence of sources. The variable k is theeigenvalue of Eq. 1.19 and is equal to k0 = ω√ε0µ0 for air. The solutions ofEq. 1.19 for electric fields together with their corresponding magnetic fields(calculated from 1.15) determine the bulk eigenmodes (or normal modes)which can be sustained in the medium without the presence of any primarysources. They provide a complete set of basis functions and any general solu-tion for Maxwell’s equation can be expanded to these eigenmodes. The bulkmodes can provide a general description for the electromagnetic response ofthe medium.The eigenvalue equation 1.19 can be solved by transferring to the spatial-frequency space (k space). Here, we use a three-dimensional Fourier trans-formE(r) =∫∫∫ ∞−∞E(k)eik·r dk, (1.20)where r =xxˆ + yyˆ + zzˆ is the position vectors in the spatial space and k =kxxˆ + kyyˆ + kzzˆ is the position vectors in k space. According to this trans-formation, the curl and divergence operators in real space correspond tothe cross and scalar products in k space, respectively. Therefore, we canwrite the governing equations for the spatial-frequency components in a ho-mogeneous and isotropic material by replacing ∇× and ∇· in the above51.2. Bulk eigenmodes of a homogeneous and isotropic medium−1 0 1−3−2−10123kz/k0kx/k0−1 0 1−3−2−10123kz/k0kx/k00 1 2 3αz/k0a) b)kSkSFigure 1.1: The dispersion diagrams of air over the kxkz plane (ky = 0)for a fixed frequency ω. (a) The kx and kz components can have real val-ues. (b) The transverse kx component have unlimited real values and theircorresponding longitudenal kz components are purely real for |kx|<k0 andpurely imaginary for |kx|>k0. The wave vector k and the Poynting vectorS of a propagating plane wave are shown by the black thin and the bluethick vectors, respectively.source-free Maxwell’s equations with ik× and ik·, respectively. The electricand magnetic field components at spatial frequency k are governed byk×E(k) = ωµH(k), (1.21)k×H(k) = −ωεE(k), (1.22)k ·E(k) = 0, (1.23)k ·H(k) = 0, (1.24)and consequently the wave equation for the electric field component is givenby(|k|2 − k2)E(k) = 0. (1.25)Note that the orthogonality of electric field components with the vector k(Eq. 1.23) has been used in deriving the above wave equation.A nontrivial solution for this eigenvalue equation can be achieved whenthe vector k satisfies the equation|k|2 = kx2 + ky2 + kz2 = k2 = ω2εµ (1.26)61.2. Bulk eigenmodes of a homogeneous and isotropic mediumknown as the dispersion equation of the medium. The curve traced by thevectors k in k space for a fixed frequency ω is know as an equifrequencycontour of the medium. According to Eq. 1.26, the EFCs of homogeneousand isotropic media with εµ > 0 are spheres with radius k. Figure 1.1(a)plots the EFC of air over the kxkz plane where the vector k is identified bya black arrow. Now that we know the eigenvalues k, we can consider theircorresponding eigenmodes sustained in a bulk homogeneous and isotropicmedium with εµ > 0. The electric fieldE(k) = e0 (1.27)satisfies the wave equation for any complex vector e0 which is normal to thevector k. Using Eq. 1.21 the corresponding magnetic field of the eigenmodeis given byH(k) = h0 =k× e0ωµ. (1.28)Using the transform relation 1.20, the electric and magnetic fields of themode in real space can be given byE(r) = e0eik·r, (1.29)H(r) = h0eik·r. (1.30)As determined by the exponential terms in the above equations, the elec-tromagnetic fields are uniform over a plane normal to the vector k and thusit is known as a plane wave with the wave vector k = kkˆ. The amplitudeof electromagnetic fields are equal in each pair of normal planes and onlytheir phase are different by ka, where a is the distance between the planes.In other words, the plane wave propagates along the kˆ direction with thewavenumber k or with the phase velocity of vp =ω/k and moves the distanceλ= 2pi/k, known as wavelength, in one time period T = 2pi/ω. The planewaves propagate in air by the phase velocity of c=ω/k0 = 1/√ε0µ0 knownas the speed of light. The bulk properties of these modes can be used to de-termine the response of the medium. The dispersion equation (k = ω√εµ)together with wave impedance, defined as the ratio of the transverse com-ponents of the electric and magnetic fields (Z = e0/h0 =√µ/ε), are usedas secondary parameters (instead of constitutive parameters ε and µ) tocompletely describe a homogeneous and isotropic medium. Note that theelectric and magnetic fields are normal to the wave vector k (see Eqs. 1.27and 1.28) and these plane waves are defined as TEM polarized waves.The density and direction of time-averaged power flow for an electro-71.2. Bulk eigenmodes of a homogeneous and isotropic mediummagnetic wave can be given by the Poynting vectorS(r) =12Re(E(r)×H(r)∗), (1.31)where H∗ is the complex conjugate of magnetic fields [21]. By substitutingEqs. 1.29 and 1.30, the Poynting vector of a plane wave is given byS0 =12Re(e0 × h0∗). (1.32)As shown by the blue array in Fig. 1.1(a), the Poynting vector of a planewave propagating with wave vector k in air is in the same direction as thewave vector (i.e. it is in the same direction as the outward normal to theEFC).The group velocity vector vg of an electromagnetic wave can be definedbyvg = ∇kω, (1.33)where ∇k is the gradient in k space and angular frequency ω is a function ofk defined by the dispersion relation. The gradient of ω is normal to its andshows the direction of the maximum rate of change in ω. In the absence ofloss and high dispersion, the group velocity is equal to the energy velocityve =SUe + Um, (1.34)where Ue and Um are the electric and magnetic energy densities [21]. Thismeans that the gradient vectors or group velocities (which can be graphicallydetermined from the dispersion diagram of the medium) can be used todetermine the direction of power flow. For example, the frequency gradientof the circular EFC in air points in an outward direction and is aligned withthe corresponding wave vector and Poynting vector.So far, we explained that TEM polarized plane waves propagate in alldirections as bulk eignemodes of a homogeneous and isotropic medium. Wealso used their dispersion behaviour and wave impedance to describe themedium. In some cases, however, we are interested in describing the be-haviour of the medium with respect to a specific reference direction. Inthis way, the electromagnetic fields of the bulk eigenmodes are not generallynormal to that reference direction. We need to consider the transverse elec-tric (TE) polarization and transverse magnetic (TM) polarization in whichelectric field and magnetic field are the normal field component to the refer-81.2. Bulk eigenmodes of a homogeneous and isotropic mediumence direction, respectively. These bulk eigenmodes also provide a completeset of basis functions and any general solution for Maxwell’s equation canbe written as the superposition of the TE and TM polarized waves [24].The bulk behaviour of these polarized waves also completely describes theresponse of the homogeneous and isotropic medium. Let us consider thez axis as a reference direction and a TE polarized wave with a transverseelectric field component Ey and magnetic field components Hx and Hz. Sim-ilar to the previous procedure, we expand the electric fields to a completeset of basis components and find the solutions of the wave equation 1.19.However, instead of a three dimensional transformation, here we use thetwo-dimensional Fourier transformEy(r) =∫∫ ∞−∞Ey(kt, z)eikt·r dkt (1.35)in which kt = kxxˆ + kyyˆ is the transverse wave vector component and theintegral is taken over a transverse plane (normal to kz axis) instead of thewhole volume. This results in bulk eigenmodes given byE(kt, z) = e0eikzz yˆ, (1.36)H(kt, z) = (h0xxˆ+ h0z zˆ)eikzz =k×Eωµ, (1.37)and the dispersion relationk2z = k2 − kx2 − ky2 = k2 − kt2. (1.38)It can be inferred from this equation that spatial-frequency components witha transverse wave number |kt|<k have a real longitudinal wavenumber kzand thus their corresponding waves propagate with phase variation eikzzalong z axis. For higher transverse wavenumbers |kt|>k, however, kz = iαzis purely imaginary and the waves are evanescent, decaying exponentiallywith e−αzz along the z axis. The dispersion diagram of air in terms ofreal transverse wave vector component kx is plotted in Fig 1.1(b). It can beshown that the same dispersive behaviour can be achieved for a TM polarizedwave withHy, Ex and Ez field components. This dispersion relation togetherwith wave impedance for TE polarizationZt =e0yh0x= −ωµkz, (1.39)91.2. Bulk eigenmodes of a homogeneous and isotropic medium1 1.5 2 2.5−2−1012ωoωpω/ωo r,µrFigure 1.2: The frequency variation of relative constitutive parameters ofa double-negative medium modeled by constitutive parameters in Eqs. 1.41and 1.42.and for TM polarizationZt =e0xh0y=kzωε, (1.40)can be used to describe the behaviour of the medium along the z direction.1.2.1 Bulk modes of the Veselago mediumIt has been observed that natural materials have limited values of ε andµ across the electromagnetic spectrum. For example, nearly all materialshave unity permeability at optical frequencies. However, fundamental lawsof electromagnetics place no restrictions on allowable values of ε and µ. In1968, Veselago studied a hypothetical medium with simultaneously negativevalues of ε and µ [6]. Here, we disuse the bulk behaviour of a homogeneousand isotropic double-negative index medium. Let us consider a medium withfrequency-dispersive relative material parametersεr(ω) = 1− 2ωo2ω2(1.41)µr(ω) = 1− 2ωo2ω2(1.42)where both εr and µr are equal to −1 at the operating frequency ωo (seeFig. 1.2). Note that these equations must necessarily contain loss terms atlow frequencies, but we consider the limit at which losses are negligible at101.2. Bulk eigenmodes of a homogeneous and isotropic mediumthe frequencies of interest. We call the wave number and wavelength corre-sponding to the frequency ωo, ko = ωo/c and λo = 2pi/ko, respectively. Wecan also define ωp =√2ωo as the frequency where constitutive parametersare zero. We will refer to this medium as the Veselago medium and discusselectromagnetic wave solutions in this medium.Having simultaneously negative values of ε and µ results in εµ> 0 andk2> 0 similar to conventional double-positive index media. Plane waves canalso propagate in a double-negative medium (see the dispersion relation inEq. 1.26), however, from Maxwell’s equations (1.21, 1.22) we can deduce thatthe electric field, magnetic field, and wave vector in these plane waves form aleft-handed triad. This is in contrast to plane waves in conventional double-positive media such as air, in which the electric field, magnetic field andwave vector form a right-handed triad. According to Eq. 1.32, the Poyntingvector, however, forms a right-handed triad with the electric and magneticfields for both right-handed and left-handed plane waves. It can be con-cluded that the propagating bulk modes in a homogeneous double-negativeindex medium are backward waves in which energy and phase flow in theopposite directions. This is in contrast to conventional double-positive indexmedia in which energy and phase flow in the same direction (see Fig. 1.3).Therefore, a left-handed medium can be characterized by a negative re-kSEHzxyRHk SEHzxyLHa) b)Figure 1.3: (a) The right-handed (RH) and (b) left-handed (LH) plane wavesin which power and phase flow in the same and opposite directions, respec-tively.fractive index n= ck/ω in contrast to a conventional positive index for aright-handed medium.The dispersion diagram of bulk modes for a Veselego medium at fre-quency ωo (with εr =µr = − 1) is shown in Fig. 1.4(a). The EFC of themedium is a sphere with radius ko, however, the Poyinting vector is pointedinward everywhere instead of outward as for free space. The dispersion di-agram of the medium in terms of real transverse wave vector componentkx is also shown in 1.4(b). Assuming that the power flows in the positivez direction, the bulk modes over the propagating spectrum (|kx|<ko) are111.3. Veselago flat lens−1 0 1−3−2−10123kz/kokx/ko−1 0 1−3−2−10123kz/kokx/ko0 1 2 3αz/koa) b)kSkSFigure 1.4: The dispersion diagrams of Veselago medium over the kxkz plane(ky = 0) for the frequency ωo. (a) The kx and kz components can have realvalues. (b) The transverse kx component have unlimited real values andtheir corresponding longitudenal kz components are purely real for |kx|<koand purely imaginary for |kx|>ko. The wave vector k and the Poyntingvector S of a a left-handed plane wave are shown by the black thin and thered thick vectors, respectively.propagating left-handed plane waves with purely negative real longitudi-nal wave vector components kz =−√k2o − k2t . In other words, these bulkmodes have opposite longitudinal wave vectors with respect to their cor-responding modes in air which propagate with the same transverse wavevectors. For the evanescent spectrum (|kx|>ko), however, bulk eigenmodesare evanescent waves in the z direction with a purely imaginary longitudinalcomponent iαz = i√k2x − k2o , which is identical to free space.1.3 Veselago flat lensThe resolution of classic optical instruments is limited by Abbe’s diffrac-tion limit in which details of an object smaller than half the wavelength ofoperation become blurred at the image plane [25]. These sub-wavelengthdetails corresponding to high transverse spatial-frequency components inEq. 1.35, are carried by evanescent waves from the object, and are lost be-fore reaching the image plane. For further illustration Fig. 1.3(a) shows theimage of a point source located at the origin where only its propagating com-ponents restored in the image plane (i.e. the transfer function from sourceto image plane is a unit function which is one over the propagating spectrum121.3. Veselago flat lens(|kx|<k0) and is zero over the evanescent spectrum (|kx|>k0)). For a pairof centered point sources (as in Figs. 1.3(b)-(d)) we can use the Rayleighcriterion to define the image resolution as the minimum distance betweentwo point sources that results in a drop of 0.91 dB between their imagefield intensity peaks. According to this criterion a conventional lens whichonly restores propagating components of an object cannot image a pair ofpoint sources separated by less than half a wavelength (see Fig. 1.3(d)). Ithas been shown, however, that an infinitely long flat slab of homogeneousdouble-negative material with εr =µr =−1 embedded in air, called a Vese-lago flat lens, can overcome the conventional diffraction limit and restore allthe propagating and evanescent components of an object to the image plane[6, 7]. In the following sections, we discus the physical origins of perfectimaging.0 1−2−1.5−1−0.500.511.52Intensityx/λo0 1Intensity0 1Intensity0 1Intensitya) b) c) d)Figure 1.5: Image field intensity of different object distributions for a con-ventional lens which only transfers the propagating components from sourceto image plane. (a) A point source is located at the origin. (b) A pairof symmetrically centered dipole sources separated by λ0. (c) A pair ofsymmetrically centered dipole sources separated by 0.5λ0. (d) A pair ofsymmetrically centered dipole sources separated by 0.4λ0. The location ofthe corresponding objects are indicated by the gray lines.1.3.1 Transmission over the propagating spectrumIn this section, we explore how a flat slab of Veselago medium embeddedin air can restore all the propagating components of an object to the im-age plane through negative refraction at the slab interfaces. The refractive131.3. Veselago flat lensbehaviour of a plane wave crossing the interface between two homogeneousand isotropic media can be determined by imposing energy conservation andphase continuity at the boundary. These conditions are described by Snell’slawn1 sin θi = n2 sin θt, (1.43)in which n1 and n2 are the refractive indices of two media and θi and θt arethe angles of incident and refracted beams, respectively. We can also graphi-cally illustrate the refraction of electromagnetic beams by plotting the EFCsof the bulk modes sustain within two media at the frequency of operationand imposing the transverse phase continuity and energy conservation (seeFig. 1.6). The theoretical Snell’s law or graphical EFC analysis predicts that0 1−101n = 1kSSikz/kokx/ko0 1kSStn = 1kz/kokkSSn = 1n = 1θiθt0 1−101kSSin = 1kz/kokx/ko−1 0kSStn = −1kz/kokkSSn = −1n = 1θi θta)b)Figure 1.6: Graphical description of plane-wave refraction from the right-handed free space (n = 1) into (a) the right-handed free space (n = 1)and (b) into the left-handed Veselago medium (n = −1) using EFC analy-sis. Continuity of the tangential wave vector components can be visualizedthrough the black dashed lines.141.3. Veselago flat lens−0.5 0 0.5−1−0.500.51d2d d2z/λox/λo−1 −0.5 0 0.5 1z/λo−1 −0.5 0 0.5 1d2d d2z/λo0-11a) b) c)n = 1 −1 1 n = 1 n = 1 −1 1Figure 1.7: An illustration of the focusing mechanism for a Veselago flat lenswith thickness d embedded in air at ωo. (a) The negative refraction of lightrays at the interfaces of the flat lens (solid arrows) in comparison to theirbehaviour in free space (dashed arrows). (b) The field plot for radiationfrom a point source infinitely extended along the y direction in air. (c) Thefield plot for imaging of propagating components within the Veselago lens.the energy and phase velocities of a plane wave crossing the interface betweentwo isotropic and homogeneous media with similar handedness would refractto the other side of the interface normal (see Fig. 1.6(a)). This phenomenonis known as positive refraction. Media with different handedness, however,would refract the energy and phase velocities to the same side of the inter-face normal, an anomalous phenomenon known as negative refraction (seeFig. 1.6(b)). Consequently, the Veselago medium operating at frequency ωo(with n= εr =µr = −1) would negatively refract incident propagating wavesfrom air (with n= εr =µr = 1)) by angles equal in magnitude but oppositein sign to the incident angles. In addition, these two media have the samewave impedance over the propagating spectrum (see Eqs. 1.39 and 1.40)and thus the propagating waves would be fully transmitted at the interfacewithout any reflection. This means that all the propagating components ofa point source located in air would be negatively refracted at the interfaceand would be focused inside the Veselago medium at the same distance fromthe interface. For a flat slab of a Veselago medium embedded in air, negativerefraction will occur at both interfaces over the propagating spectrum. Thisresults in two focal points: one inside the slab and another outside the slabon the excident side. In other words, the Veselago lens has a unity transferfunction over the propagating spectrum at frequency ωo when the object and151.3. Veselago flat lensimage planes are separated by twice the lens thickness (for a more detaileddescription please see the Appendix C). Figure 1.7 plots the instantaneoustransverse electric field Ey at time t = 0 for a line current I = I0 cos(ωot)which is infinitely extended along the y axis. It can be seen that an infinitelylong flat Veselago slab can focus the radiated fields inside the lens at z = 0and over the image plane at z = d, where d is the thickness of the slab.Note that Fig. 1.7(c) does not illustrate the response of the Veselgao lensto the incident evanescent waves since they are 90 degree out of phase fromthe source current1 and therefore zero at t = 0.1.3.2 Transmission over the evanescent spectrumIn this section, we describe how a flat slab of the Veselago mediumembedded in air can restore all the evanescent components of an object tothe image plane at frequency ωo. As discussed in Section 1.2, a high spatial-frequency component with transverse wave vector component |kx|>ko iscarried by the evanescent waves e−αoz along the z direction in both Veselagoand air media where the attenuation coefficient αo =√kx2 − ko2. Now, letus consider a Veselago slab with thickness d centered at the origin in air andthe Fourier component of an object with spatial frequency |kx|>ko has anamplitude c1 over the source plane at z = −d. The incident evanescent wavefrom the object can result in multiple evanescent waves bouncing back andforth inside the slab due to the reflection at the two interfaces of the slab.We can model the superposition of all waves going along the z direction by asingle wave b+2 e−αoz and all waves going along −z direction by a single waveb−2 eαoz. It can be shown that by setting the amplitude b+2 = 0 and b−2 = c1,the magnetic fieldsHy(kx, z) = c1e−αo(z+d) z < −d2(1.44)= c1eαoz − d2< z <d2(1.45)= c1e−αo(z−d) z >d2(1.46)1Examining the Fourier spectrum of the cylindrical wave radiated by the line current[26]H20[ko(x2 + z2)1/2]=1pi∫ ∞−∞e−i(k2o−k2x)1/2|z|(k2o − k2x)1/2 eikxx dkxdemonstrates that each symmetric evanescent component pairing, i.e. with transversewave vector kx and −kx, has an instantaneous field 2e−αoz |z|(piαoz)−1 cos(kxx) sin(ωot).161.3. Veselago flat lensand corresponding electric fields both satisfy continuity of transverse elec-tric and magnetic fields (boundary conditions) at the interfaces and thuscomprise a unique solution to Maxwell’s equations. The above solution and−0.25 0 0.25d2 dd2n = 1 n = −1 n = 1z/λoc1 c1Figure 1.8: An illustration of the evanescent enhancement within a Veselagoflat lens embedded in air at frequency ωo. The behaviour of evanescent wavein air (without the presence of lens) is shown by a dashed line for comparison.its corresponding plot in Figure 1.8 demonstrates that the Veselago lensperfectly amplifies and restores all the incident evanescent components ofthe object to the image plane at z = d. Reconstructing all the propagatingand evanescent components of an object with unity transfer function to theimage plane overcomes the diffraction resolution limit in conventional lensesand results in a perfect imaging. It should be mentioned that the response ofa Veselago slab at the operating frequency ωo is independent of polarizationand a similar response can be achieved for the TE polarization.1.3.2.1 Bound modes of Veselago lensIn Sec. 1.3.1, we described negative refraction at the interfaces of a flatVeselago lens as the physical mechanism for imaging the propagating com-ponents. In this section, we discuss how the resonant bound modes guidedalong the Veselago lens are the physical origin of evanescent wave enhance-ment. The interface between Veselago half-space and air can guide TMpolarized resonant modes (eigenmodes) with a magnetic field given byHy(kx, z) = b1eαoz z < 0 (1.47)= b1e−αoz z > 0 (1.48)171.3. Veselago flat lensfor all values of transverse wave vector kx over the evanescent spectrumat frequency ωo (for a more detailed discussion please see Appendix D).Therefore, the dispersion diagram of the resonant modes is a flat line at thefrequency ωo. These guided modes have the amplitude b1 at the interface−0.25 0 0.25n = 1 n = −1z/λoFigure 1.9: An illustration of the surface resonant modes at the interfaceof a Veselago half-space (solid line) which can be excited by an incidentevanescent wave from air (dashed line)(z= 0) and decay exponentially with coefficient αo in both media (see solidlines in Fig. 1.9). These modes are confined to the interface and thus areknown as surface wave modes.Although a Veselego half space could negatively refract and focus all inci-dent propagating waves from air, a single interface cannot properly enhanceand restore the evanescent waves in the image plane. Consider that surfacemodes guided along a single interface are the solutions of the source-freeMaxwell’s equations at the operating frequency ωo. Applying an incidentevanescent wave (as an external source) increases the amplitude of the sur-face wave with the corresponding transverse wave vector, which eventuallyachieves an infinite value for the steady state solution. In other words,the excitation of surface modes exactly at their resonance frequency by theevanescent incident wave (shown by dashed line in Fig. 1.9) results in aninfinite reflection and transmission at the interface (shown by solid lines inFig. 1.9). Hence, these surface resonant modes correspond to poles of trans-mission and reflection functions at the interface between Veselago half-spaceand air. The over-amplification of incident evanescent waves destroys theinformation carried by the evanescent waves from the object (related to thesub-wavelength details of the object) and obscures the image of the object.The bound modes guided along a Veselago slab, however, can amplifyand restore all the evanescent component of an object [9]. The bound modes181.3. Veselago flat lens−0.25 0 0.25z/λo−0.25 0 0.25z/λoa) b)Figure 1.10: Field distribution of (a) even bound mode and (b) odd boundmode guided along a slab of Veselago materialof a Veselago lens arise from the symmetric and antisymmetric coupling ofsurface modes guided along the two interfaces of the slab (see Fig. 1.10)[27, 28]. The uncoupled surface modes of a single interface which resonatesat frequency ωo split into two slab bound modes with resonance frequenciesabove and below the frequency ωo. Using the analytic eigenmode solutionsfor a slab of dispersive material (see Appendix E), the dispersion diagramsof bound modes guided along Veselago slabs with thickness d= a= 0.0642λoand d= 5a= 0.321λo are shown in Fig. 1.11. As can be seen in this figure,the flat dispersion line of an uncoupled single interface (dotted line) has beensplit into two curves with resonance frequencies above ωo for even surfacebound modes and below ωo for odd surface bound modes. These dispersioncurves asymptotically approach the operating frequency (indicated by thesolid red line) as kx increases. This approach is faster for thicker slabs wherethe coupling between surface waves is weaker.The ideal Veselago lens always operates between the even and odd moderesonance frequencies, exciting each mode precisely in such a way as toprovide unity transmission across a source-to-image-plane distance of 2d [9].Figure 1.12 shows the response of the Veselago lens to an incident evanescentwave as a superposition of even and odd solutions. These field plots illus-trate how evanescent wave growth within a Veselago slab can be interpretedas the superposition of off-resonance excited even and odd bound modeswith destructive interference at the front face (z=−d/2) and constructiveinterference at the back face (z= d/2).191.3. Veselago flat lens0 1 2 3 4 5 6 7 8 9 10 11 120.70.80.911.11.2a xz0-11a axz5a 01-15a 5akx/koω/ωoFigure 1.11: Dispersion diagrams of even and odd bound modes guidedalong Veselago slabs of thickness a (dashed black lines) and 5a (solid blacklines). The geometry of the slabs and the magnetic field distributions (Hy)of even and odd bound modes are plotted in the inset diagrams. The fieldplots correspond to a wave vector kx = 1.7ko. The light line is indicated bythe solid gray line. The dispersion diagram of an uncoupled surface wavesat a single interface is shown by the flat dotted black line at ωo. The flatsolid red line indicates the operating frequency ωo in which the both valuesof εr and µr are equal to −1.−0.25 0 0.25n = 1 n = −1 n = 1z/λo−0.25 0 0.25n = 1 n = −1 n = 1z/λo−0.25 0 0.25n = 1 n = −1 n = 1z/λo+ =Figure 1.12: Field distribution of even solution (right plot) and odd solution(middle plot) and their superposition (left plot) for the response of a Veselagolens to an incident evanescent wave at frequency ωo201.3. Veselago flat lens0 1 2 3 4 5 6 7 8 9 10 11 120.70.80.911.11.2kx/koω/ωoFigure 1.13: A Veselago lens dispersion diagram illustrating the appearanceof resonant modes in a non-perfect flat slab. The flat dashed-dotted red linecorresponds to a slight deviation in operating frequency from ωo and thesubsequent appearance of bound modes at the intersection points. Theseintersection points are indicated by red dots in the dispersion diagram. Theimaging performance would be further degraded for a thicker slab since theintersecting resonance point appears at a lower spatial frequency.1.3.3 Non-perfect homogeneous flat lensAn ideal infinite Veselago slab with εr =µr =−1 always operates be-tween the slab bound modes and have unity transfer function for all trans-verse spatial frequencies when the object plane is separated from the imageplane by twice the lens thickness. As shown by the flat dash-dotted redline in Fig. 1.13, a slight change in the operating frequency (which slightlydistorts the ideal material parameters) causes the bound mode dispersioncurve to intersect with the operating frequency, introducing a resonancethat prohibits perfect imaging. These intersection points are indicated byred dots in the dispersion diagram. These resonances correspond to polesin the slab transfer function (similar to the single-interface resonances) [29]and degrade the imaging performance of the slab in two ways [30]. First,the transfer function starts to rapidly decay for spatial frequencies above thebound mode wavenumber, thereby establishing an upper spatial-frequencycutoff for unity transfer function. This fact would put an upper limit inimaging small details of an object and thus imaging resolution. Second,components of the object which have transverse spatial frequencies at or211.4. Metamaterialsnear the bound mode wavenumber are over-amplified throughout the slab,which prevents image formation. These bound modes must be suppressed byadding material losses to the slab, however, these losses reduce the evanes-cent amplification and further limit the resolution of the lens [30].To achieve a super-lens with sub-diffractive imaging resolution, the res-onant modes must be suppressed while still maintaining a spatial frequencycutoff above the free-space cutoff. It has been shown that a multi-layeredslab of alternating thin layers of positive and negative index media whichsupport coupled surface waves at layer interfaces (see Fig. 1.14) can shift theresonant modes to higher spatial frequencies and amplify evanescent wavesover a wider spatial frequency. Therefore, a layered structure can have im-proved resolution over a solid non-perfect negative-index slab of the samethickness [30–33].a) b)Figure 1.14: Illustration of the evanescent enhancement with in (a) a solidnegative index slab and (b) a multi-layered slab of alternating thin layers ofpositive and negative index media.1.4 MetamaterialsAs discussed above, superlensing can be theoretically achieved by an ho-mogeneous and isotropic double-negative medium. Although such a materialdoes not exist in nature, it can be realized by an artificial material known asa metamaterial [1]. As explained in Sec. 1.1.1, in a natural material the elec-tromagnetic response of the bulk medium is determined by the microscopicinteractions of the fields and the constituent particles. In metamaterials,however, artificial inclusions play the role of atomic particles and hence canbe tailored to achieve the desired responses. Since 2001, many differentstructures across the electromagnetic spectrum have been presented in the221.4. Metamaterialsa)ab)aac)Figure 1.15: The geometry of different double-negative index metamaterialstructures (a) Arrays of split-ring resonators and thin wires. (b) The coupledplasmonic waveguide made of metallic and dielectric layers. (c) Arrays ofplasmonic, dielectric or magnetodielectric inclusions.literature to mimic left-handed Veselago media (see Fig. 1.15). These struc-tures can support backward bulk eigenmodes in all direction of propagationand exhibit circular or spherical equifrequency contours that resemble thoseof left-handed media over the propagating spectrum. Experiments showingnegative refraction of electromagnetic beams have also been used as cor-roborating evidence of a left-handed response [12–14, 34, 35]. Like naturalmaterials in which large-scale responses can be defined through averagingout atomic-scale field fluctuations, homogenization theories have been ap-plied to average the complex field interactions and map the backward bulkmodes to a single left-handed plane wave propagating in an equivalent homo-geneous effective double-negative index material [17, 36–39]. These theoriesare generally assumed to be valid when the wavelength of operation andspatial field variations are much larger than the size of inclusions.The first experimental demonstration of negative refraction was per-formed at microwave frequencies using a composite metamaterial consistingof arrays of split-ring resonators (SRRs) and thin wires (TWs) [10, 34]. Inthese experiments, measured refraction angles were shown to be consistentwith Snell’s law when the structure was modeled using a double-negativehomogeneous and isotropic effective medium [2, 3]. Negative effective per-meability µ was attributed to artificial magnetic moments induced withinthe SRRs [40], while negative permittivity ε was attributed to the plasma-like behaviour of the thin-wire arrays [41] which will be described with moredetails in the following sections.231.4. MetamaterialsHlocL CFigure 1.16: A split ring resonator and its simplified LC circuit model.1.4.1 Artificial magnetization of current loopsHere, we show how artificial magnetism can be achieved by a split-ringresonator [42]. We simply model a split-ring resonator as a capacitive loadedconducting loop (see Fig. 1.16) and find the artificial magnetic moment m0induced in this magnetic element (see Eq. A.2). For simplicity we assumethat local magnetic field Hloc (which can be generally a superposition ofexternal magnetic fields and field scattered by other elements in an array) isuniform over the loop and is directed toward the loop normal. According tothe Faraday’s law in Eq. 1.15, a time varying magnetic field can induce anelectromotive force (emf) over a closed loop. Taking the integral of Eq. 1.15over surface of the loop results in the induced voltageVemf = iωµ0SHloc, (1.49)in which S is the area of the loop. Assuming negligible Ohmic and radiationlosses and modeling the loop by a LC resonator with impedanceZ =L(ω2 − ωLC2)iω, ωLC =1√LC, (1.50)the corresponding induced current in the loop can be calculated by I =Vemf/Z. This circulating current produces the magnetic momentm0 = SI = − µ0S2ω2HlocL(ω2 − ωLC2) , (1.51)and consequently the magnetic polarizability coefficientαm =m0µ0Hloc= − S2ω2L(ω2 − ωLC2) (1.52)241.4. Metamaterialscan be defined for this magnetic element, which is a function of frequencyand has a pole at resonance frequency ωLC .Having artificial magnetization is not restricted to circulating currentsin conducting loops. The induced circulating bound currents in dielectricinclusions (which are defined by J =−iωP) can also result in artificial mag-netization without the existence of magnetic materials [17].1.4.2 Effective negative permeability for an array ofmagnetic elementskEHFigure 1.17: A cubic array of SRRsAfter deriving the magnetic polarizability coefficient for an individualmagnetic element we can apply the Clausius-Mossotti equation (see Ap-pendix A) to extract the effective permeability of an array of coupled mag-netic elements [42]. For example, consider an array of SRRs shown inFig. 1.17. By substituting Eq. 1.52 in Eq. A.12 and under a quasi-staticcondition, the effective permeability of a cubic array of SRRs with latticeconstant a can be given byµeff = µ0Fω2 − ωpm2ω2 − ω02 , (1.53)in whichF =3− 2q3 + q, ω0 =ωLC√1 + q/3, ωpm =ωLC√1− 2q/3 , and q =S2a−3L. (1.54)As can be seen in Fig. 1.18 the extracted effective permeability has negativevalues for frequencies above the resonance frequency ω0 up to the magnetic251.4. Metamaterials0.9 1 1.1 1.2 1.3−2−1012ω0ωoωpmω/ωoµrFigure 1.18: The dispersion diagram for the extracted effective permeabilityof a cubic array of SRRsplasma frequency ωpm and can achieve the value −1 at frequency ωo givenbyωo = ωLC√66− q . (1.55)1.4.3 Effective negative permittivity for an array of thinwiresAs shown in Appendix B, a homogeneous conducting medium with motion-free electrons can be modeled by a negative permittivity because of the in-ductive response of electrons to an applied electric field. Similarly, an arrayof conducting wires aligned parallel to applied electric fields (see Fig. 1.19(a))can have an inductive response and a negative effective permittivity can beextracted for this medium. Here, we drive a simplified expression for theelectric plasma frequency of the effective medium [42]. A cubic unit cell withside a (see Fig. 1.19(b)) can be defined for an array of wires with lattice con-stant a. This unit cell contains a piece of thin wire with radius rw<<a andlength a. For a lossless conducting wire with impedance Z =−iωL, the ex-ternal electric field applies the voltage V =Ea across the wire, and conductthe currentI =EaZ=iEaωL. (1.56)261.5. Motivationa aakEHa) b)Figure 1.19: (a) An array of thin wires. (b) The cubic unit cell of thestructure.By defining the the averaged current density Jb = I/a2 for the unit cell andusing Eq. 1.5 the averaged electric polarization can be given byP =iJbω= − Eaω2L. (1.57)By substituting the averaged polarization in Eq. 1.11, we can define theeffective relative permittivityεr = 1 +Pε0E= 1− ωpeω2, (1.58)with the effective electric plasma frequencyωpe =1aε0L. (1.59)Using arrays of SRRs and thin wires (see Fig. 1.15(a)) and by adjustingtheir plasma frequencies both effective negative permeability and permittiv-ity and consequently double-negative index can be achieved over a desiredband of frequency.1.5 MotivationThe negative effective index associated with metamaterials are based onmodeling the backward bulk mode that propagates through a heterogeneousstructure with a single left-handed plane wave. In other words, negative in-dex models are “black-box” approximations which disguise the complexity of271.6. Thesis outlineinteractions within the media and conceptually replace metamaterials withan equivalent homogeneous medium. Such approximate models have beenuseful for the design and analysis of metamaterials without a need for com-plex full-wave solutions, but they are only valid under certain circumstancesand hide the physical mechanisms behind the metamaterial response. Forexample, a metamaterial composed of stacked plasmonic waveguides (seeFig. 1.15(b)) which support backward propagation along the layers has beenshown to exhibit a nearly spherical EFC which mimics a homogeneous andisotropic left-handed medium. Experiments performed on this structurehave also shown nearly all-angle negative refraction of light beams and thelayered metamaterial has been characterized by an isotropic negative refrac-tive index [14]. The mechanism of negative refraction in a plane parallelto the layers can be attributed to the guidance of a backward guided modewith opposing phase and group velocities which can sustain along the layers.However, it is unclear whether in this structure with an anistropic geome-try, negative refraction in a plane perpendicular to the layers is also derivedfrom left-handed modes or if other mechanisms are present. This “black-box” approximation cannot also completely describe the response of themetamaterial structure. For example, matching the impedance of an inci-dent plane wave to the effective medium parameters may not be sufficient toprovide a high transmission at an metamaterial interface [43]. In addition,the effective parameters of propagating bulk modes may not be sufficientto determine the response of a metamaterial lens to evanescent waves. Ametamaterial structure may mimic a quasi-isotropic negative index responseover the propagating spectrum, however, the evanescent response can differsignificantly. The effect of finite lens dimensions, periodicity, and hetero-geneity on evanescent wave amplification and the impact of edge scatteringon imaging performance also needs further study.1.6 Thesis outlineThe thesis presents systematic and comprehensive analysis of negative-index metamaterials based on full-wave Maxwell’s solutions and with a min-imum approximations. The first chapter provides background informationabout the response of an ideal homogeneous and isotropic double-negativeindex medium and its imaging performance. These analytical results willbe used as a reference point in analyzing negative-index metamaterials andtheir imaging performance in the rest of this thesis. Chapter 2 analyzes thebehaviour of backward-wave eigenmodes propagating in bulk metamaterials281.6. Thesis outlinebased on the distribution of their power in the spatial frequency domain.The k-space power maps enables us to identify the spatial frequency originof backward power and distinguish between the electromagnetic responsesof structures with similar EFCs. Chapter 3 shows how k-space power mapscan be used to determine the source of negative refraction at different inter-faces of a metamaterial and to explain high or low transmittance of power.Chapter 4 analyzes the evanescent amplification within a metamaterial lens.We investigate resonant modes in finite metamaterial superlenses and theirimaging artifacts. We show that if overlooked, these artifacts can lead toerroneous interpretation of the standard two-source resolution test. Wepropose design strategies to tailor the resonant modes and improve the res-olution and robustness of imaging. The conclusions of this research and thepossible directions for future work are presented in Chapter 5.29Chapter 2Spatial frequency maps ofpower flow in metamaterialsand photonic crystalsThe objective of this chapter is to provide insight into the behaviourof backward-wave eigenmodes propagating in bulk periodic structures. Weanalyze several common metamaterial and photonic crystal structures thathave been described in the literature as negative index media. To addressthis objective from first principles, the electric and magnetic fields insidethe media are calculated from Maxwell’s equations, Fourier-transformed,and then paired to find a complete set of spatial-frequency harmonics, eachwith a well-defined wave vector and Poynting vector. Plotting the Poyntingvectors in k space provides a simple visual representation of power flowwithin a periodic heterogeneous structure. A similar expansion has beensuccessfully used to analyze photonic crystals [44], however, we proposea systematic and general procedure that can be applied to any periodicstructure, regardless of frequency, complexity of geometry, periodicity, orconstituent materials.Since heterogeneous structures can only mimic the electromagnetic be-haviour of left-handed media, spatial spectral power maps can be used toprovide a quantitative comparison with an ideal left-handed medium in orderto compare and contrast different metamaterials across the electromagneticspectrum. The power maps enable us to identify the spatial frequency originof backward power and distinguish between the electromagnetic responsesof structures with similar EFCs.2.1 MethodologyThe solutions of source-free Maxwell’s equations (Eqs. 1.15-1.18) for aninfinite medium with a periodic inhomogeneity are known as Bloch modes.The Floquet-Bloch theorem [45, 46] states that time-harmonic electric and302.1. Methodologymagnetic fields of Bloch modes have the form ofE(r) = ue(r)eikFB·r (2.1)H(r) = uh(r)eikFB·r (2.2)where ue(r) and uh(r) are periodic vector functions and kFB is the Floquet-Bloch wave vector. The periodic vector functions have the same periodicityas the medium expressed byu(r) = u(r + R), (2.3)where R is lattice vector of the structure. The Floquet-Bloch wave vectorcan generally be complex and expressed as kFB = k+ iα. The real part k isthe fundamental propagating wave vector and the imaginary part α is theattenuation vector. As the attenuation is embedded in the complex Floquet-Bloch wave vector, the vector functions ue(r) and uh(r) always representthe non-attenuating periodic component of E(r) and H(r), respectively. TheBloch modes supported by the periodic structure can be determined bysolving the wave equation∇× ( 1µr∇×E)− εrω2c2E = 0. (2.4)for the electric field or the wave equation∇× ( 1εr∇×H)− µrω2c2H = 0. (2.5)for the magnetic field. Here, µr and εr are the periodic relative perme-ability and permittivity of the medium, both of which can be complex anddispersive in general. Substitution of Eq. (2.1) and Eq. (2.2) into these waveequations yields∇× (1p∇× u) + ikFB × (1p∇× u) + i∇× (1pkFB × u)+kFB · kFBpu− kFBp(kFB · u)− qω2c2u = 0, (2.6)where u = ue, p = µr, and q = εr for the electric field and u = uh, p = εr,and q = µr for the magnetic field [47]. For a linear medium in which ma-terial parameters (ε, µ) are independent of electromagnetic fields and fora fixed frequency ω the above equation is a linear eigenvalue problem in312.1. Methodologywhich u and kFB are eigenfunction and eigenvalue, respectively. In order tosolve this equation constraints must be placed on the Floquet-Bloch wavevector kFB so that it can be expressed in terms of a single scalar eigenvalueλ. Different constraints can be used for different circumstances, with thecorresponding eigensolutions unique only to that constraint. In the caseof the infinite periodic structures studied in this chapter, we apply a con-straint by defining a direction of propagation and choose kFB = λkˆ. In thecase studied in the next chapter of refraction across an interface, however,continuity of phase at the interface imposes the constraint on kFB and thedirections of phase propagation and field decay are no longer necessarilyaligned. Equation (2.6) can be solved over a single unit cell for the complexeigenvalue λ and its corresponding eigenmode u(r) in order to compare theelectromagnetic response in different directions of propagation. The EFCassociated with a particular frequency is then given by tracing the Floquet-Bloch wave vector across all directions of propagation. Note that since thisformulation is evaluated at a single frequency, dispersive materials can betreated in the same manner as non-dispersive materials. In this work, wesolved the weak form of the eigenvalue equation (2.6) for ue or uh usingCOMSOL Multiphysics finite-element software package [47, 48] (for a moredetailed description please see the Appendix G.).It has often been assumed that when the periodicity is sub-wavelengththe Bloch mode can be approximated by a single plane wave with the wavevector kFB (see Eqns. 1.29 and 1.30). This has led to the modeling ofmetamaterials and photonic crystals by equivalent homogeneous media (us-ing effective parameters) [17, 36–39, 49] or graphically by EFC diagrams[14, 50]. Instead, a complete description of bulk modes that retains all spa-tial field information can be achieved by expanding the electric and magneticfields into a complete set of spatial frequency harmonics as follows,E(r) =∑l,m,nel,m,neiGl,m,n·reikFB·r (2.7)H(r) =∑l,m,nhl,m,neiGl,m,n·reikFB·r (2.8)where Gl,m,n is the reciprocal lattice vector in k space and l, m, n spanthe set of all integers. In this formulation, hl,m,n and el,m,n are the three-dimensional spatial Fourier transforms of the non-attenuating periodic com-ponents of the magnetic and electric field vectors, respectively. According tothis expression, the k space can be divided up into unit cells with reciprocallattice vector G. These unit cells are known as BZ. The reciprocal lattice322.1. Methodologyvector G in k space is related to the physical lattice vector R of the struc-ture in real space. Substituting the Fourier expansions of periodic functionsin Eq. 2.3 yields∑l,m,nel,m,neiGl,m,n·r =∑l,m,nel,m,neiGl,m,n·(r+R), (2.9)and results in the relationG ·R = 2piN. (2.10)where N is an integer number.The harmonic components of the electric and magnetic fields in Eqs. 2.7and 2.8, each have a real wave vector given by kl,m,n = k + Gl,m,n andan attenuation vector given by α = Im{kFB}. We can also see the fieldexpansion in Eq. 2.7 as the three dimensional Fourier transform in Eq. 1.20in which E(k) have only non-zero values at discrete points kl,m,n due to theperiodicity.By pairing corresponding electric and magnetic field harmonics, theBloch mode can be expressed as a spectrum of unique plane waves locatedat discrete real kl,m,n points at each BZ in k space. The time-averagedPoynting vector associated with each (non-attenuating) plane wave can becalculated from Eq. 1.32 and can be given bySl,m,n =12Re{el,m,n × h∗l,m,n}. (2.11)Each plane-wave harmonic therefore has a well-defined wave vector kl,m,nand Poynting vector Sl,m,n from which handedness can be defined. To cal-culate each harmonic’s handedness, the wave vector and Poynting vector areprojected onto a reference axis defined by the direction of propagation kˆ. Ifthe projected vectors are parallel (in the same direction), the harmonic isconsidered to be right handed, while if they are anti-parallel (in the oppositedirection), the harmonic is considered to be left handed. This generalizesthe definition of handedness for individual harmonics from definitions thatrely only on the direction of average power flow [51].Due to the orthogonality of the harmonics, the individual harmonicPoynting vectors can be related to the total time-and-space-averaged Poynt-ing vector through the following summationS =1V∫∫∫V12Re{ue(r)× u∗h(r)}dv =∑l,m,nSl,m,n (2.12)332.2. Split-ring resonator and thin-wire metamaterialwhere V is the volume of the unit cell. We can now map the total powerflow across all spatial frequency harmonics for a complete characterizationof the relationship between phase and power flow in periodic structures.This mapping can reveal cases where most of the power lies outside thefirst BZ, or reveal whether the power is carried primarily by right-handedor left-handed harmonics.We show that in a general heterogeneous medium, handedness can bedefined for individual harmonics but not for the medium itself. This is be-cause a consistent phase progression typically does not exist when a mediumis not homogeneous in the direction of propagation. We must therefore re-frain from associating any kind of handedness to the relative orientations ofkFB and S (or equivalently, to the complex sign relation of kFB in which thesign of imaginary and real parts are related to the direction of power flowand wave vector, respectively) and discuss only the handedness of harmonicswhich can be determined using the method described above.This method provides a systematic and generalized spatial frequencyanalysis that can be applied to arbitrary periodic structures (with dielectric,magnetic, and conductive modulations) under arbitrary polarization. In thefollowing sections, this method is applied to several common metamaterialsand photonic crystals to carefully investigate wave behaviour that has beendescribed in the literature as arising from negative refractive index.2.2 Split-ring resonator and thin-wiremetamaterialThe SRR-TW metamaterial and their double-negative effective param-eters have been discussed in the Introduction chapter. Here, we apply thek-space analysis to provide new insights into the response of a SRR-TWstructure. We consider a two-dimensional array of thin wires aligned withthe y axis inter-spaced between two orthogonal arrays of SRRs and considerpropagation parallel to the xz plane. The lattice has a periodicity a in thex and z directions and a periodicity d = 0.86a in the y direction. The SRRsand thin wires are modeled as PECs with radii of 0.016a and 0.02a, respec-tively. The SRRs embedded within crossed dielectric layers of permittivityεr = 7 and thickness 0.1a. Each unit cell contains two complete SRRs (ori-ented orthogonally to each other) and one complete thin wire, as illustratedin the inset diagrams of Fig. 2.1.We begin our study by solving the eigenvalue equation 2.6 to find theBloch mode periodic function uh(r) and the Floquet-Bloch wave vector kFB342.2. Split-ring resonator and thin-wire metamaterial−1 −0.5 0 0.5 1−1−0.500.511st BZky = 0S000 S100kSTWSRRzxkza/2pikxa/2pi−1 −0.5 0 0.5 1−1−0.500.511st BZkx = 0S001S00-1S000 S100kSTWSRRzykza/2pikyb/2pia) b)Figure 2.1: The k-space diagrams for a backwards eigenmode propagatingin-plane along the z axis of the SRR-TW metamaterial (θ = 0). The Blochharmonics are plotted on the ky = 0 plane (a) and on the kx = 0 plane (b).Left-handed plane wave components in k space are plotted with red arrows,while right-handed components are plotted with blue arrows. The EFCs aregiven by the solid and dotted curves inside and outside the first BZ, respec-tively, where the first BZ is indicated by the light grey dashed rectangle.The unit cell is shown from the top (a, inset) and side (b, inset) with PECboundary conditions (solid lines) and periodic boundary conditions (dashedlines) for the planes normal and parallel to the y axis, respectively.associated with each direction of propagation. Once ue(r) is calculated fromuh(r), both periodic functions are expanded to their Fourier harmonics andeach harmonic Poynting vector is then calculated. The EFCs associated witha particular frequency can also be traced by the fundamental and higher-order wave vectors across all direction of propagation inside and outsidethe first BZ, respectively. Figures 2.1 and 2.2 plot the harmonic Poyntingvectors alongside the repeated zone EFCs of the structure for Bloch wavespropagating parallel to and diagonally to the in-plane axes, respectively.The normalized frequency of operation is Ω = a/λ0 = 0.0647, where λ0 isthe wavelength in free space. The time-and-space-averaged Poynting vectorS of the global fields calculated by Eq. 2.12 is shown by the thick blackarrow with the fundamental wave vector shown by the thin black arrow.The location of the fundamental harmonic is also indicated by a black dot.352.2. Split-ring resonator and thin-wire metamaterial−1 −0.5 0 0.5 1−1−0.500.511st BZky = 0S000S0-10S100kSkza/2pikxa/2pi−0.5 0 0.5−1−0.500.511st BZkx = kzS001S00-1S000kS√k2x + k2za/2√2pi(b)kyb/2pia) b)Figure 2.2: The k-space diagrams for a backwards eigenmode propagatingin-plane along the diagonal axis of the SRR-TW metamaterial (θ = 45◦).The Bloch harmonics are plotted on the ky = 0 plane (a) and on the kx = kzplane (b). Note that only the average power S must be normal to the EFC.Individual harmonic Poynting vectors are free to point in any direction.Note that unlike the total power flow S, the individual harmonic Poyntingvectors Sl,m,n are not necessarily normal to their corresponding repeatedEFCs.In conventional EFC analysis, the Bloch mode is modeled as a singleplane wave located within the first BZ and no information about the har-monic distribution of power is available. Observing that the average powerflow points opposite to the fundamental wave vector would therefore seemto imply a left-handed response characterized by a single negative refractiveindex. By plotting the harmonic Poynting vectors in the spatial frequencydomain, however, information is provided about the Bloch mode field varia-tions across each unit cell that is not available from EFC diagrams nor fromeffective parameters. We can see from Figs. 2.1 and 2.2 that the electro-magnetic fields are a strong mixture of Bloch harmonics, containing bothleft-handed harmonics (in red) and right-handed harmonics (in blue), withonly a fraction of the total power residing within the first BZ (i.e. residingwithin the fundamental).The dominant harmonics in Fig. 2.1 can be separated into two distinctsets: the right-handed components are arranged longitudinally along the362.3. Photonic crystalaxis of propagation while the left-handed components are arranged in thetransverse direction along an axis perpendicular to the direction of propa-gation. We will show later that this distribution of right- and left-handedcomponents is not unique to this Bloch mode. The presence of one or bothof these harmonic sets can reveal the spatial frequency origin of backwardpower in metamaterials and photonic crystals. Let us now explore each ofthese harmonic sets further.The left-handed transverse harmonics S0,0,1 and S0,0,−1 shown in Fig. 2.1(b)can be interpreted as two plane waves bouncing back and forth within theparallel-plate waveguide formed by the unit cell PEC symmetry planes.These higher harmonics are symmetric with respect to the kxkz plane. Theyform standing waves in the direction transverse to the plane and propagatingwaves in the direction parallel to the plane. Together they form a backward“waveguide” mode in which power flows in the direction opposite to phase.As the direction of propagation changes in the xz plane, these left-handedharmonic pairs rotate about the ky axis to maintain their transverse posi-tion and continue to provide backward power flow to the mode (as shownin Fig. 2.2(b)). The SRR-TW structure can be therefore said to support abackward waveguide mode that is nearly isotropic in the xz plane.The right-handed longitudinal harmonics S0,0,0 and S1,0,0 provide an un-even power distribution of forward waves propagating in opposite directions(resulting in a partial cancellation of right-handed power). These waves aredistributed in k space at regular intervals with opposing power flow arisingdue to reflections off the lattice of inclusions. As the angle of propagation ischanged from parallel to the z axis (θ = 0◦) to diagonally between the x andz axes (θ = 45◦), the lattice periodicity re-distributes the higher-order har-monic power from S1,0,0 to S1,0,0 and S0,−1,0 as shown in Fig. 2.2(a). For allangles of propagation, however, the net contribution of these componentsto the total power flow ranges is minor (less than 4% for θ = 0◦). Fromthis analysis, we can conclude that the backward power in SRR-TW meta-material arises form pair of left-handed transverse harmonics (left-handedwaveguide modes) and the observed negative refraction is due to refractionof the right-handed free-space wave vector into the left-handed transverseharmonics.2.3 Photonic crystalPhotonic crystal (PC) structures with strong periodic modulations indielectric constant have also been shown to exhibit negative refraction of372.3. Photonic crystalpower at an interface [52, 53]. In contrast to SRR-TW structures, PCs relyon band-folding to achieve negative refraction rather than double-negativeeffective parameters. It has been shown that a two-dimensional square lat-tice of silicon cylinders in air can support backward Bloch modes with non-circular EFCs (i.e. the fundamental wave vector kFB and the averaged powerflow S are not always collinear for all direction of propagation). Althoughthis structure does indeed refract incident beams negatively, Lombardet etal. have shown through Fourier analysis that the refracted beam is a right-handed plane wave outside the first BZ, and that this behaviour should bedescribed through diffraction phenomenon rather than through electromag-netic left-handedness. [44].−1.5 −1 −0.5 0 0.5 1 1.5−2−10121st BZS1-1S11S02S0-2Γ KMMk SΓKΓMkza/2pikx√ 3a/2pi−1.5 −1 −0.5 0 0.5 1 1.5−2−10121st BZS11S-11S02kSkza/2pikx√ 3a/2pia) b)Figure 2.3: The k-space diagrams for backward eigenmodes propagatingalong the (a) ΓK and (b) ΓM directions in the hexagonal photonic crystal.All Bloch harmonics for this structure are TEM right-handed plane wavesplotted with blue arrows. The EFCs are given by the solid and dottedcurves inside and outside the first BZ, respectively, where the first BZ isindicated by the light grey dashed hexagon. The photonic crystal geometrywith its rectangular unit cell is illustrated in the inset. Again, as seen above,individual harmonic Poynting vectors are not restricted to being normal tothe EFCs.Here, we apply the k-space analysis to a two-dimensional photonic crys-tal composed of a dielectric containing a dense hexagonal (triangular) arrayof cylindrical holes. This crystal supports backward Bloch modes in all382.3. Photonic crystaldirections perpendicular to the holes and is an interesting case study forour analysis. Unlike the structure in [44], it has circular EFCs that couldbe associated with a two-dimensionally isotropic negative effective index[35, 52, 53]. In this photonic crystal, the holes have a diameter 2r = 0.8aand are arranged with a lattice constant a inside a dielectric medium ofdielectric constant εr = 12.96. The two-dimensional rectangular unit cell ofthe structure (identified by the solid black lines in the inset of the Fig. 2.3(a))has the width a and height√3a along the z and x directions, respectively.By using this unit cell and applying periodic boundary conditions the Blochmodes are solved for TE polarized waves (where the electric field lies per-pendicular to the xz plane) at a normalized frequency Ω = a/λ0 = 0.32and then expanded to its spatial-frequency harmonics. Figures 2.3(a) and2.3(b) plot the power maps alongside the EFC contours of the structure forBloch waves propagating along the symmetry axes ΓK and ΓM, respectively.Again, we can see that the Sm,n Poynting vectors are not necessarily normalto their corresponding EFCs. Although conventional EFC analysis wouldattribute backward power and observed negative refraction to an isotropicleft-handed response, these power maps provide a completely different per-spective. They reveal that backward modes are composed of right-handedTEM plane wave harmonics (where the Poynting vector and wave vectorare parallel) in which the fundamental harmonic contributes negligibly tothe total power. There are no left-handed components and backward powerflow arises from right-handed higher-order harmonics which propagate inthe backward direction with respect to the fundamental Bloch wave vec-tor. This insight reveals the right-handed origin of the observed negativelyrefracted beams in photonic crystal prisms [50, 53].These observations demonstrate that despite the presence of negative ef-fective refractive index and circular EFC diagrams, an isotropic left-handedVeselago medium cannot truly describe the behaviour of this photonic crys-tal. At this point, we must point out that the distribution of power acrosshigher-order right-handed harmonics occurs for negatively refracting pho-tonic crystals of all lattice spacings. In other words, the k-space distribu-tion does not change significantly when the unit cells are made extremelysub-wavelength and the medium enters what would typically be consideredto be an effective medium regime. We will see in the following section thathighly subwavelength crystals of magnetodielectric cylinders contain weakfundamental harmonics similar to those observed in photonic crystal struc-tures.392.4. Sub-wavelength magnetodielectric crystal2.4 Sub-wavelength magnetodielectric crystal−1 0 1−1011st BZkSzxS10S11S1-1kza/2pikxa/2pi−1 0 1−1011st BZkSS10S01 S11kza/2pikxa/2pia) b)Figure 2.4: The k-space illustration of Bloch harmonics for backward eigen-modes propagating in the magnetodielectric crystal (a) parallel to the z axis(θ = 0◦) and (b) diagonal to the x and z axes (θ = 45◦). The schematicgeometry of the magnetodielectric crystal and the square unit cell are shownin the inset diagram.It has been shown that dense arrays of MD sub-wavelength inclusionsembedded in a host dielectric can support electric and magnetic resonancesin the same frequency band and have been homogenized as double-negativemedia [12, 17, 23, 54]. Here, we apply k-space analysis to a two-dimensionalcrystal of nearly touching magnetodielectric cylinders in air. The cylindersare made of a magetodielectric medium with εr = µr = 20 and have adiameter 2r = 0.9a, where a is the lattice constant of the crystal. Thismetamaterial supports backward Bloch modes under TM polarization (i.e.the magnetic field lies perpendicular to the xz plane) at the normalized fre-quency Ω = a/λ0 = 0.0639 for all angles of propagation in the xz plane. Thecorresponding power maps for propagation along the z axis (θ = 0◦) andalong the diagonal (θ = 45◦) are illustrated in Figs. 2.4(a) and 2.4(b), respec-tively. Although backward power in this metamaterial structure has beenattributed to an isotropic left-handed response, our power maps suggest thatbackward power actually arises from higher-order right-handed harmonicssimilar to the photonic crystal (as shown in Fig. 2.3). A complete analysisof phase and power flow in this metamaterial therefore requires a harmonicexpansion and cannot be modeled solely through an effective negative re-402.5. Sub-wavelength plasmonic crystalfractive index or EFC diagrams.2.5 Sub-wavelength plasmonic crystal−2 −1 0 1 2−2−10121st BZkSzxkza/2pikxa/2pi−2 −1 0 1 2−2−10121st BZkSkza/2pi(b)kxa/2pia) b)Figure 2.5: The k-space illustration of Bloch harmonics for backward eigen-modes propagating in the plasmonic crystal (a) parallel to z axis (θ = 0◦)and (b) diagonally to x and z axes (θ = 45◦). Left-handed plane wave com-ponents are plotted with red arrows, while right-handed components areplotted with blue arrows. The schematic geometry of the plasmonic crystaland the square unit cell are shown in the inset.A dense two-dimensional array of metallic sub-wavelength cylinders hasbeen shown to support backward Bloch modes at optical frequencies in alldirections of propagation perpendicular to the metallic cylinders [13, 18].This structure, known as a plasmonic crystal, operates below the plasmafrequency of the metallic inclusions (with ω = 0.6ωpe) and has been shownto have a circular EFC with an inward pointing frequency gradient for TMpolarized waves. Here we study a plasmonic crystal with a square lattice ofmetal cylinders embedded in air. The cylinders have a relative permittivityεmetal = −1.78 and a diameter 2r = 0.9a. At the normalized frequencyΩ = a/λ0 = 0.0954, this metamaterial structure has been modeled as a two-dimensionally isotropic double-negative metamaterial.The spatial frequency power maps of the structure are plotted in Figs. 2.5(a)and 2.5(b) for backward Bloch modes propagating parallel and diagonallyto the lattice vectors, respectively. We can see from these diagrams that the412.6. Coupled-plasmonic-waveguide metamaterialconcentrated fields surrounding the plasmonic cylinders resulted in powerflow being distributed across many Bloch harmonics. Furthermore, we seethat backward power arises from a combination of both right-handed andleft-handed harmonics. The appearance of left-handed harmonic pairs (i.e.waveguide-mode pairs) with roughly isotropic behaviour resembles the left-handed response of the SRR-TW metamaterial. Although the physical ge-ometry of the plasmonic crystal strongly resembles that of a PC or an MDmetamaterial, as do its EFCs and its homogenized effective refractive index,its electromagnetic behaviour is very different. Instead of most of the powerbeing contained in higher order harmonics with large longitudinal wave vec-tor components, the plasmonic crystals also have a strong fundamental andstrong left-handed components adjacent to the first BZ.2.6 Coupled-plasmonic-waveguide metamaterialMetamaterials composed of stacked plasmonic waveguides can also sup-port backward Bloch modes in all directions with roughly spherical backward-power EFCs at optical frequencies [14, 55]. Here, we study one such meta-material composed of a one-dimensional periodic repetition of five alternat-ing thin layers of silver (Ag) and titanium dioxide (TiO2), as illustrated inFig. 2.6(a). The individual layers have thickness 33 nm, 28 nm, 30 nm, 28nm, and 33 nm, producing a total periodicity of a = 152 nm. This structuresupports backwards Bloch waves in all directions under TM polarization(magnetic field is normal to x axis) at a wavelength of λ0 = 363.8 nm (cor-responding to a normalized frequency of Ω = a/λ0 = 0.4178) [14]. At thiswavelength, the metamaterial layers were modeled using the complex per-mittivites Ag = −2.522 + i0.251 and TiO2 = 7.835 + i0.392 (non-magneticlayers were considered). The wave vectors kFB have also complex valueswith different real and imaginary sign which has been used as a necessaryand sufficient condition for achieving negative indices in absorbing media,including plasmon-based geometries [56, 57]. The following results, however,show that it cannot be a sufficient condition.The snap-shots of the magnetic field in the layered structure at differ-ent points in time for energy propagation both parallel and perpendicularto the layers are shown in Fig. 2.6(b). Although the EFC of the unit cellsuggests that it behaves like an isotropic and left-handed Veselago medium,the actual field distributions reveal complex wave dynamics that underminethis approximation. For propagation along the layers (the field plot on theleft), the wave is a laterally confined mode that exhibits phase advancement422.6. Coupled-plasmonic-waveguide metamaterialzyxaTiO2Ag0-11t1t2t3t4k St1 t2 t3 t4Sk−101kSTop viewS1, S−1ky/k0,αy/k0 Top viewk−2 −1 0 1 2−5−4−3−2−1012345kS1stBZSide viewS1S−1kz/k0, αz/k0kx/k0,αx/k0−2 −1 0 1 2kS1st BZSide viewS1S−1kz/k0a)b)c)Figure 2.6: The backward eigenmodes propagating parallel and perpendicu-lar to the layers in the coupled-plasmonic-waveguide metamaterial. (a) Theschematic diagram of a unit cell with periodicity a. (b) Mode magnetic fieldplots at quarter-period intervals over t1 < t2 < t3 < t4 for a wave propagat-ing parallel to planes (left) and perpendicular to the planes (right). (c) Thek-space illustration of harmonics for modes propagating parallel to planes(left) and perpendicular to the planes (right) corresponding to the modes in(b). In (c) the attenuation contours corresponding to the imaginary partsof the calculated Floquet-Bloch wave vector are given by the green dashedcurve.opposing the direction of power flow. However, for propagation perpendic-ular to the layers (the field plot on the right), the wave exhibits an intricateand seemingly random field pattern from which a natural phase progressioncannot be visually extracted. Examination of the field patterns for thesetwo propagation directions suggests that the metamaterial may possess aleft-handed response, but this response is likely not isotropic.Here, we use k space power maps of backward-wave modes to achieve432.6. Coupled-plasmonic-waveguide metamateriala deeper insight to the electromagnetic behaviour of this layered structure.Figure 2.6(c) shows the power maps corresponding to propagation paralleland perpendicular to the plane of the layers. For both the parallel case (leftplot) and perpendicular case (right plot), the three-dimensional distribu-tions of the Poynting vectors are viewed from the top (projected onto thekzky-plane and shown in the plots on the top) and from the side (projectedonto the kxkz-plane and shown in the plots on the bottom). The EFCs(traced by the vector k) and attenuation contours (traced by the vector α)are also plotted by the gray and green curves, respectively. These contoursreveal an attenuation coefficient that is roughly isotropic, falling within therange of 0.22k0 to 0.26k0 as the direction of propagation changes. It can beobserved that the global fields consist of multiple plane wave componentswhose Poynting vectors are not necessarily normal to their correspondingEFCs and that very little power lies within the first BZ. This confirms thatno single plane-wave component (let alone the weak component lying inthe first BZ) can be assigned to the global field. For propagation parallelto the layers backward power arises primarily from the left-handed trans-verse harmonic pair similarly to the SRR-TW metamaterial. The presenceof this left-handed waveguide mode in which power and phase propagatein opposite directions is consistent with the capability of planar plasmonicwaveguides to support backward surface plasmon modes along their layersseen in the left plot of Fig. 2.6(b) [57]. For propagation perpendicular tothe layers, however, the harmonics are all right-handed with a dominanthigh-order longitudinal harmonic S1 providing the greatest contribution tobackward power. The combination of all harmonics gives rise to the highlyheterogeneous field pattern seen in the right plot of Fig. 2.6(b), whose ab-sence of discernible phase progression can now be explained by the multiplewave components propagating in different directions. The k-space maps forpropagation perpendicular to layers has strong similarities to those observedin photonic crystals and magnetodielectric media.Now we discuss the power maps of backward modes for all other anglesof propagation either in a plane parallel to the layers (in plane) or in a planeperpendicular to the layers (out of plane). Since this medium is uniformparallel to the layers, for any angle of in plane propagation the power mapsshown in the left plot of Fig. 2.6(c) rotate about the kx axis and maintain anisotropic left-handed response. For out-of-plane propagation, however, thek-space maps change dramatically due to the anisotropy of the structure.Figure 2.7(a) illustrates how wave propagation in the metamaterial smoothlyevolves from resembling a backward waveguide mode for propagation alongthe layers to scattering with a dominant right-handed higher-order harmonic442.6. Coupled-plasmonic-waveguide metamaterial−1 0 1−5−4−3−2−10123451st BZθ = 20S1S−1kSθkz/k0kx/k0−1 0 1kS1st BZθ = 45S1S−1θkz/k0−1 0 1kS1st BZθ = 70S1S−1θkz/k00 10 20 30 40 50 60 70 80 90−1−0.500.5Propagation Angle (deg)PowerFlow(normalized)LH componentsRH componentsTotal power flowa)b)Figure 2.7: The backward eigenmodes propagating in a plane perpendicularto the layers of coupled-plasmonic-waveguide metamaterial. (a) The k-spaceillustration of Bloch harmonics for out-of-plane backward modes propagat-ing at angles θ = 20◦, θ = 45◦, and θ = 70◦. (b) The decomposition of totalpower flow along the reference direction into left-handed and right-handedcontributions versus propagation angle θ. As θ increases, dominant con-tributors to power flow shift from left-handed to right-handed plane-wavecomponents.for propagation perpendicular to the layers. Power maps are depicted forpropagation angles of θ = 20◦, 45◦, and 70◦, where θ is defined as the an-gle between the reference axis (kˆ) and the plane of the layers. Again, weobserve that there is no significant power residing in the first BZ for allangles. The evolution of the harmonic Poynting vectors as a function ofpropagation angle is complicated. Whereas the Poynting vector S rotateswith the propagation angle about the origin, the harmonic Poynting vec-tors rotate by smaller amounts, each in differing directions. The locationsof the harmonic Poynting vectors are roughly anchored in k-space, shiftingby only small amounts versus propagation angle. Due to this property andthe rotation of the reference axis (kˆ), the plane wave components changefrom being left-handed to right-handed as the direction of propagation goesfrom in-plane to perpendicular out-of-plane. The changes in magnitude anddirection of the harmonic Poynting vectors are consistent with the evolution452.7. Fishnet metamaterialfrom waveguide modes with symmetric power maps (arising from transversestanding waves) for in-plane propagation to asymmetric right-handed scat-tering for out-of-plane propagation in which power flows in the negative xdirection.Figure 2.7(b) compares the power flow contributions along the referenceaxis from all left-handed components SLH and all right-handed componentsSRH as the propagation axis is tilted from in-plane (θ = 0◦) to perpendic-ular out-of-plane (θ = 90◦). The field amplitudes have been normalized toachieve a constant energy density in the unit cell with respect to propaga-tion direction. The summation of SRH and SLH yields the total power flowalong the reference axis, which, due to the normalization process, is also pro-portional to the group velocity along the reference axis (see Eq. 1.34) [23].The negative value of the total power flow for all angles indicates all-anglenegative group velocity, but here we see that it can arise from right-handedor left-handed plane-wave components. It can also be seen that the groupvelocity magnitude decreases for larger angles, which is consistent with theincreasing presence of standing waves made of counter-propagating planewave components. This provides further evidence of the general electromag-netic anisotropy of this structure. For a given propagation direction, therelative contributions of SLH and SRH can be used to determine the ori-gin of backward power (with respect to kFB) and consequently distinguishbetween backward power flow and left-handedness. The origin of back-ward power and negative refraction of power is predominantly left-handedfor small angles and right-handed for large angles. The sharp thresholdbetween the left-handed and right-handed regimes arises because the domi-nant plane wave components switch between left-handed and right-handedclassifications at similar angles. Overall, the coupled-plasmonic-waveguidemetamaterial demonstrates the usefulness of spatial frequency maps of powerflow to differentiate between different types of backward modes and iden-tify anisotropies that can be overlooked using traditional EFC analysis oreffective medium theory.2.7 Fishnet metamaterialThe final structure we analyze in this chapter is the “fishnet” structure,which is made of stacked metallic plates perforated by periodic hole arrays[58]. Here we study a simple structure in which perfectly conducting plateswith zero thickness are stacked in air along the z axis with a periodicityd = a/3. The schematic diagram of the structure and a single unit cell is462.7. Fishnet metamaterialyxabyzdFront view Side view −1011st BZS000 S100kSTop viewS00100-1kxa/2pi−1 −0.5 0 0.5 1−2−10121st BZkSSide viewS001S00-1kzd/2pi(b)kya/2pia)b)Figure 2.8: The fishnet metamaterial. (a) The schematic diagram of thefront view (left plot) and the side views (right plot) in which the PECscreen and square hole are specified by the blue and white color, respec-tively. (b) The k-space illustration of Bloch harmonics over the kxkz plane(top view) and over the kzky plane (side view) for a backward eigenmodepropagating in the fishnet metamaterial along the kz axis (normal to theperforated screens). The cubic unit cell of the structure is determined in(a) by the PEC boundary conditions (solid lines) parallel to the xz planeand periodic boundary conditions (dashed lines) along y axis in order toinvestigate propagation parallel to the xz plane.shown in Fig. 2.8(a). The PEC screens are perforated with square holesof side length b = a/3 arranged in a square lattice of periodicity a. Thebulk eigenmodes propagating in xz plane were determined by applying PECboundary conditions in planes of the cubic unit cell which are parallel to xzplane (solid lines) and periodic boundary conditions in planes along y axis(dashed lines). Bloch modes were investigated at a normalized frequency ofΩ = a/λ0 = 0.95 (d/λ0 = 0.317).In contrast to the periodic structures studied in previous sections, theEFC diagrams plotted in the top of Fig. 2.8(b) are not circular and the fun-damental wave vector kFB and the time-and-space-averaged Poynting vector472.8. SummaryS (the frequency gradient) are only collinear (anti-parallel) for propagationnormal to the plates (along the z axis). The spatial frequency power mapsof the backward Bloch mode propagating along z axis in Fig. 2.8(b), demon-strate that the fundamental harmonic is right handed and backward powerarises primarily from the left-handed transverse harmonic pair (that onceagain form a backward waveguide mode). This resembles the k-space powermaps of coupled-plasmonic-waveguide metamaterial for propagation alongthe layers and the left-handed response of SRR-TW and plasmonic crystalmetamaterials.2.8 SummaryWe have shown how decomposition of a backward Bloch mode into acomplete set of spatial frequency harmonics can provide new insights intothe behaviour of electromagnetic waves inside metamaterials and other pe-riodic structures that may otherwise be characterized by the same negativeeffective index of refraction. Mapping power flow across the harmonics in kspace reveals the phase and power relation (handedness) of each individualharmonic and provides a complete description of power and phase that is notavailable through effective medium theory or EFC analysis (repeated zoneor otherwise). We have re-examined metamaterial structures that have (a)circular or spherical EFCs resembling that of an isotropic left-handed Vese-lago medium, (b) wide-angle negative refraction of light beams, (c) negativeeffective parameters extracted through homogenization theory, and/or (d)subwavelength periodicity. We show through our analysis that these arenot sufficient criteria for confirming an isotropic left-handed response anddo not provide enough information to fully describe their electromagneticbehaviour. We have shown that the spatial frequency origin of backwardpower flow can change with respect to the direction of propagation or fromone periodic structure to another. The spatial frequency maps of power flowrevealed that backward power arises through one of two basic mechanisms:either from pairs of left-handed transverse harmonics or from right-handedhigher-order longitudinal harmonics. These spatial power maps provide anelectromagnetic signature that can be used to identify and categorize pe-riodic structures with similar dispersive behaviour. We have shown thatbackward power arises from higher-order right-handed harmonics in struc-tures such as photonic crystals, magnetodielectric crystals, and coupled-plasmonic-waveguide metamaterials (for propagation across their layers).Backward power is due to pairs of left-handed waveguide modes in SRR-482.8. SummaryBackward power through left-handedwaveguide harmonic pairsSRR/TWPlasmonic crystal−2 −1 0 1 2−2−1012kxa/2pi(a)kya/2pi−2 −1 0 1 2−2−10121st BZkSkxa/2pi(b)kya/2piPlasmonic waveguide(along the layers)FishnetBackward power through higher-orderright-handed harmonicsPhotonic crystal−1.5 −1 −0.5 0 0.5 1 1.5−2−1012kxa/2pi(a)ky√ 3a/2piMagnetodielectriccrystal−2 −1 0 1 2−2−1012kxa/2pi(a)kya/2piPlasmonic waveguide(normal to the layers)Table 2.1: The categorization of periodic structures that support backwardpower across the electromagnetic spectrum based on their distinguishingfeature in the spatial frequency domain.TW composites, plasmonic crystals, fishnet structures (for on-axis propaga-tion), and in coupled-plasmonic-waveguides metamaterials (for propagationalong their layers). These results are summarized in Table 2.1 with all sixstructures classified into two groups.49Chapter 3k-space analysis of refractionand mode excitationefficiency across ametamaterial interfaceIn this chapter, we use our knowledge regarding the distribution of powerflow across the spatial-frequency harmonics to determine the origin of re-fraction at different interfaces of a metamaterial structure, an insight whichcannot generally be achieved by conventional EFC analysis. We also corrob-orate the hypothesis that k-space maps of power flow can provide informa-tion about the coupling efficiency into various metamaterials and be usedto suggest an incident wave excitation that provides enhanced transmission.We choose the coupled-plasmonic-waveguide metamaterial for our analysis,since it can support the two mechanisms for backward power flow identifiedin the previous chapter, i.e. pairs of left-handed transverse harmonics andright-handed higher-order longitudinal harmonics.3.1 Refraction across air-metamaterial interfacesFor the coupled-plasmonic-waveguide studied in Sec. 2.6, negative re-fraction of polarized light beams has been observed along planes of inci-dence parallel to the layers (in plane) and perpendicular to the layers (outof plane) [14]. Note that end-fire illumination of a facet perpendicular tothe layers can access both planes of incidence (see Figs. 3.1(a) and (b)),whereas broad-side illumination of a facet parallel to the layers can onlyaccess out-of-plane incidence (see Fig. 3.1(c)). As shown in Fig. 2.6, theEFC of this layered structure at λ0 = 363.8 nm is roughly spherical witha radius equal to the wave number in free space and power flow oppositeto the fundamental Floquet-Bloch wave vector for all angles of propagation.Based on conventional EFC analysis, refraction into that medium from a503.1. Refraction across air-metamaterial interfaceszyxaTiO2Agazxazxa)b)c)Figure 3.1: A diagram illustrating the coupled-plasmonic-waveguide meta-material and the planes of incidence for (a) end-fire in-plane refraction, (b)end-fire out-of-plane refraction, and (c) broadside out-of-plane refraction.The unit cell periodicity is a.homogeneous medium would be graphically described by drawing the EFCsat the working frequency and invoking continuity of the tangential wavevector components across the interface [50]. Using the graphical procedure,or using Snell’s law with the effective index n = −1, we would predict thatwaves incident from air onto a metamaterial facet of any orientation wouldundergo negative refraction of both power and phase for all angles of inci-dence. In other words, the Poynting and wave vectors would refract to thesame side of the interface normal with an angle roughly equal to the angle ofthe incident wave. This refraction analysis is predicted on the assumptionthat the propagating Bloch mode in the layered structure can be describedby a single plane wave located at the first BZ. However, based on k-spacepower analysis this assumption is not generally valid.In order to achieve insights into the refractive behaviour at the interfaceof the coupled-plasmonic-waveguide medium, we expand the conventionalanalysis method by considering the power flow of all harmonic componentsof the bulk eigenmode excited by the incident wave. The bulk eignmodeexcited in this layered structure is found by solving eigenvalue equation 2.6while enforcing the continuity of the tangential wave vector at the inter-513.1. Refraction across air-metamaterial interfaces−1 0 1 2 3−101AirkiSikz/k0, αz/k0ky/k0−2 −1 0 1 2kSMetamaterialS1 , S−1kz/k0, αz/k0−1 0 1 2 3−4−3−2−101234kiSiAirkz/k0, αz/k0kx/k0−2 −1 0 1 2kSS1S−1Metamaterialkz/k0, αz/k0−3−2−101kiSiAirkx/k0,αx/k0−2 −1 0 1 2−4−3−2−101234kSMetamaterialS0S1S−1kz/k0kx/k0,αx/k0a)b)c)Figure 3.2: Description of wave refraction from free space into the layeredmetamaterial based on the k-space components of the eigenmodes in themetamaterial and conservation of tangential wave vectors along the interfacefor (a) end-fire in-plane refraction, (b) end-fire out-of-plane refraction, and(c) broadside (out-of-plane) refraction. The real and imaginary parts of thewave vector are plotted with solid gray and dotted-dashed green contours,respectively. The repeated EFCs of the layered metamaterial are also plottedby the dotted lines outside of the first BZ. Continuity of the tangential wavevector components can be visualized through the dashed black lines.face. Hence, the Floquet-Bloch wave vector in this equation is written interms of the eigenvalue λ as kFB = kt + λkˆn, where kt is the tangentialcomponent of the incident wave vector and kˆn is the unit normal to theinterface. According to this constraint the real and imaginary parts of the523.1. Refraction across air-metamaterial interfaceseigenvalue determine the normal component of the wave vector kn = Re{λ}and the attenuation coefficient α = Im{λ}, respectively, for each real wave-vector component kt. In this way, the direction of phase propagation isdetermined by the wave vector components kt and kn and the directionof attenuation is always normal to the interface. Figure 3.2 illustrates ex-amples of refraction in three unique configurations: in-plane refraction atend-fire (across a planar interface perpendicular to the layers, as shown inFig. 3.1(a)) with kt = kyyˆ and kˆn = zˆ, out-of-plane refraction at end-fire (asshown in Fig. 3.1(b)) with kt = kxxˆ and kˆn = zˆ, and out-of-plane refractionat broad-side (across a planar interface parallel to the layers, as shown inFig. 3.1(c)) with kt = kz zˆ and kˆn = xˆ. The real and imaginary parts of thefundamental wave vector kFB are plotted in Fig. 3.2 for various interfacesas a function of kt using solid gray and dash-dotted green contours, respec-tively. The repeated EFCs are also plotted by the dotted lines outside ofthe first BZ. Note that for incidence angles close to normal, there is verygood agreement between these contours and the quasi-spherical EFCs fromthe infinite medium plotted in Fig. 2.6(c).For each interface depicted in Fig. 3.2, the spatial frequency maps ofpower flow are displayed for the mode corresponding to a plane wave inci-dent from air at an angle of 20◦ with respect to the interface normal (i.e. themode with kt = k0 sin 20◦). The real part of the fundamental Floquet-Blochwave vector k = kt +kn is refracted at an angle of approximately 20◦ to thesame side of the normal as the incident wave vector. The average power flowsaway from the interface into the metamaterial with the averaged Poyntingvector S pointing directly opposite to k. Since conventional EFC analysisuses only k and S to model the power and phase behaviour of the bulk eigen-modes, it would predict negative refraction of both power and phase at theinterfaces. Including the Poynting vector of each individual Bloch harmonicprojected onto the plane of incidence, however, indicates that the nature ofrefraction at an air-metamaterial interface is highly dependent on the ori-entations of the facet and plane of incidence. Negative phase and powerrefraction is unambiguously achieved only under in-plane end-fire illumina-tion (Fig. 3.2(a)). For out-of-plane refraction either by end-fire illumination(Fig. 3.2(b)) or broadside illumination (Fig. 3.2(c)), the refracted wave con-sists of multiple plane-wave components in which the dominant componentsare outside the first BZ. Vector summation of the harmonic Poyting vectorsfor both out-of-plane cases is consistent with negative power refraction, butphase refraction is ambiguous due to the presence of multiple plane-wavecomponents. Figure 3.2(c) demonstrates that both the negative refraction533.2. Mode excitation efficiency across a metamaterial interfaceof power observed experimentally by Xu et al. and the double-negative pa-rameters extracted from reflection and transmission data at the broadsideinterface of an coupled-plasmonic-waveguide metamateral slab [14] origi-nates from refraction of higher-order right-handed components, not from anelectromagnetic left-handedness. This insight clearly demonstrates that aneffective index should not be assigned to a periodic structure based solelyon refraction observations.Similar analysis can be applied to other periodic structures studied inprevious chapter. For the SRR-TW and metamaterials in which the back-ward power arises form a pair of left-handed transverse harmonics the neg-ative refraction of electromagnetic beams can be attributed to refractionof the right-handed free-space wave vector into the left-handed waveguidemodes. However, for the photonic crystal and matamaterial structures thatthe backward power arises from longitudinal higher-order right-handed har-monics the observed negative refraction has a right-handed origin [50, 53].3.2 Mode excitation efficiency across ametamaterial interfaceIn this section, we corroborate the hypothesis that k-space maps of powerflow in a periodic structure can be used to suggest an incident wave exci-tation that provides enhanced transmission. Let us consider the end-fireillumination of the coupled-plasmonic-waveguide metamaterial. The Blochharmonics of the eigenmode in the metamaterial have tangential wave vectorcomponents given bykm,x = kx +2pima, (3.1)where a is the lattice constant of layered structure and 2pi/a is the reciprocallattice constant along the x axis (see Eq. 2.10). Conservation of the tan-gential wave vector would imply that the metamaterial could be excited byan incident plane wave through any combination of these harmonics. How-ever, the spatial frequency power flow maps demonstrate that each channelcorresponds to a particular harmonic with a specific contribution to the to-tal power flow of the excited mode. Channels can therefore have differentcoupling efficiencies which cannot be determined by the simple geometricalrepetition of EFCs in conventional methods [50]. To test this hypothesis, weconsider the case of a dielectric-metamaterial interface under end-fire illumi-nation in which the refractive index of the dielectric medium is sufficientlyhigh (n = 4) to enable an incident plane wave to phase match to the first543.2. Mode excitation efficiency across a metamaterial interfaceorder harmonics of the eigenmode. Changing the angle of incidence enablesan incident plane wave to phase match to either the fundamental harmonicof the eigenmode or the first-order harmonics of the eigenmode. Using acombination of multiple incident plane waves, multiple harmonics can alsobe phase matched simultaneously.0 1 2 3 4 5−4−3−2−101234kAkBkBSASBSBn=4kz/k0kx/k0−3 −2 −1 0 1 2kSS1S−1Metamaterialkz/k0ABR = 79%R = 3%0−11Figure 3.3: Two different end-fire wave excitations of the normal in-planeeigenmode propagating with kt = 0 in the coupled-plasmonic-waveguidemetamaterial. In case A, a normally incident plane wave targets the fun-damental m = 0 harmonic, while in case B, two oblique plane waves withincidence angles of ±53.2◦ are superposed in order to simultaneously targetthe higher-order harmonics m = ±1. The k-space power maps of eigenmodesand the incident and transmitted magnetic field at the interface are plottedat the left and right side, respectively. The white arrows show the directionof the time-and-space-averaged power flow in both the incidence dielectricmedium (n = 4) and the layered medium. It is shown that coupling to thehigher order harmonics (the dominant k-space components) has increasedthe coupling efficiency by a factor of almost 40. The reflected power as apercentage of the incident power is indicated by R.Here, we study the excitation of an in-plane eigenmode propagating alongthe layers (corresponding to θ= 0◦, or equivalently, kt = kx = 0). The k-space map of this mode in Fig. 3.3 shows that the two first-order harmonics(m = ±1) carry 78% of the total power, whereas the fundamental harmonic(m = 0) carries less than 1%. A normally incident plane wave phase matchedto the fundamental (case A) generally should yield low coupling efficiencyinto the structure, an observation that has been noted by others and hasbeen attributed to impedance mismatch at the interface [55]. On the otherhand, an incident plane wave matched to the m = ±1 components (case553.2. Mode excitation efficiency across a metamaterial interfaceB) should yield significantly greater coupling efficiency. To quantify thecoupling efficiency of different excitation channels, we carry out full-wavesimulations of wave incidence from a dielectric half space onto the layeredstructure half space using COMSOL Multiphysics. The power flow of theexcited eigenmode in the direction normal to the interface for different exci-tation methods (albeit with equal total incident power flow in that direction)provides a quantitative comparison between their coupling efficiencies. Theplots on the right side of Figure 3.3 show the magnetic field distributionsfor end-fire normal illumination of the in-plane eignmode using two differ-ent excitation channels: one using a normally incident plane wave phasematched to the m = 0 component (case A) and another using two obliqueplane waves phase matched to the m = ±1 components (case B). The lefthalf of each panel plots the incident fields, and the right half of each panelplots the transmitted fields. The direction of the time-and-space-averagedpower flow is shown by the white arrows and the reflected fields have beenomitted for clarity. Both excitation methods couple to the same eigenmode,which carries power and phase in opposing directions. Note that the spatialfield distribution of the transmitted mode for both cases is consistent withthat of the in-plane eigenmode plotted in the left side of Fig. 2.6(b), exceptthat the mode is now launched from an interface and attenuates heavily intothe lossy medium. For normal incidence excitation (case A), 79% of powerwill be reflected, whereas the reflection for oblique excitation (case B), isreduced to 3%. The coupling to the higher order harmonics also results ina coupling efficiency over 39 times greater than that achieved by normal-incidence plane-wave excitation. The dramatic improvement in a couplingefficiency confirms that the relative power maps can be used as means toobtain highly efficient excitation channels across interfaces.The presence of multiple excitation channels means that it is possible fordifferent plane waves to excite the same bulk eigenmode. Figure 3.4 illus-trates three different incident plane waves that excite the same kt = k0 sin 20◦mode within the coupled-plasmonic-waveguide metamaterial, albeit with dif-ferent efficiencies. The angles of incidence are selected to be −43.1◦ (case A),−4.9◦ (case B), and 30.9◦ (case C) to achieve phase matching to the m= +1,m= 0, and m=−1 components, respectively, where a positive angle of inci-dence is measured in the clockwise direction from the interface normal (−zaxis). Illumination tailored to the m= 0 harmonic (case B) yields to 76%reflection of the incident power. Illumination tailored to the m= +1 har-monic (case A) and m=−1 harmonic (case C), on the other hand, reducethe refraction to 5% and 7%, respectively. The coupling efficiency in caseA and C is 298 and 220 times grater than case B, respectively. Again, this563.2. Mode excitation efficiency across a metamaterial interface0 1 2 3 4 5−4−3−2−101234kBkAkCSBSASCn=4kz/k0kx/k0−3 −2 −1 0 1 2k SS1S−1Metamaterialkz/k0ABCR = 5%R = 76%R = 7%0−11Figure 3.4: Three different end-fire wave excitations of the out-of-plane eigenmode propagating with kt = k0 sin 20◦ in the coupled-plasmonic-waveguide metamaterial. A plane wave is incident from a dielectric mediumof index n = 4 with an angle −43.1◦ (case A), −4.9◦ (case B), and 30.9◦(case C) in order to target the m = +1, m = 0, and m = −1 component ofthe eigenmode, respectively. It is shown that coupling to the higher orderharmonics has increased the coupling efficiency more than 220 times. Thedirection of power flow indicated by the white arrows demonstrates negativerefraction of power at the interface for cases A and B, and positive refractionof power for case C.result is consistent with the dominant power flow of the two first higherorder harmonics in the k-space power maps of the bulk eigenmode.Interestingly, the possibility of multiple excitation channels challengesthe notion that refraction angles across a metamaterial interface can beuniquely defined. As shown by the white refracted arrows in Fig. 3.4, anincident wave that is inclined at either positive or negative angles can excitethe same eigenmode. Illumination phase matched to the m= 0 harmonic(case B) and m= +1 harmonic (case A) results in negative refraction ofpower, but with different angles of incidence. On the other hand, illumi-nation phase matched to the m=−1 harmonic (case C) results in positiverefraction of power. Clearly, such refractive behaviour cannot be accuratelydescribed by assigning a single refractive index value to the metamaterialand invoking Snell’s law.We can also use a similar procedure to analyze refraction for a differentinterface cut. Figure 3.5 shows the results for a case in which the interface isinclined at 45◦ relative to the layers. The tilted layered structure indeed sup-ports the same backward-wave Bloch modes, however, their circular EFCs573.2. Mode excitation efficiency across a metamaterial interface0 1 2 3 4 5−4−3−2−101234kAkBSASBn=4kz/k0kx/k0−3 −2 −1 0 1 2kSS1S−1Metamaterialkz/k0ABR = 50%R = 14% 0−11Figure 3.5: Two different wave excitation of the eigenmode propagating withkt = 0 in the coupled-plasmonic-waveguid metamaterial tilted by 45◦. Incase A, a normally incident plane wave from a dielectric medium of index n =4 targets the fundamental harmonic, while in case B, an oblique plane wavewith incidence angles of 25.03◦ targets the dominant higher-order harmonicsm = 1. It is shown that coupling to the higher order harmonic (the dominantk-space components) has increased the coupling efficiency more than 8 times.repeat along the diagonal axis. In fact, the tilted layered structure has pe-riodicity a along the diagonal axis and according to Eq. 2.10 its reciprocallattice vector is given byG =2√2pia(xˆ + zˆ). (3.2)The spatial power maps for a bulk eigenmode propagate normal to the in-terface (with an angle θ = 45 with respect to layers) show that a smallportion of power is carried by the fundamental harmonic and most of thepower is carried by the right-handed higher-order harmonic S1. A normallyincident plane wave (case A), which phase matches to the fundamental har-monic, yields 50% reflection of the incident power. Illumination tailored tothe m = +1 harmonic (case B), on the other hand, reduce the reflection to14% and results in a coupling efficiency over 8 times garter than case A.Similarly, k-space maps of power flow can be used to explain the high orlow transmittance of power into different metamaterial or photonic crystalstructures. For example, Fig. 3.6 shows the coupling efficiency for the pho-tonic crystal studied in Sec. 2.3. Although the circular EFC of the photoniccrystal would imply an isotropic response, the full-wave simulation results583.2. Mode excitation efficiency across a metamaterial interfaceS1-1S11S02S0-2Γ K−1.5 −1 −0.5 0 0.5 1 1.5S11S-11S02ΓMkza/2piΓKΓM(a)(b)(c)Figure 3.6: Power transmission for two different interfaces of a rectangularphotonic crystal sample studied in Sec. 2.3. The transverse electric fieldis plotted for two Gaussian beams normally incident from free space ontothe vertical and horizontal interfaces. The low transmittance of the inci-dent beam along the ΓK axis and its high transmittance along ΓM axis areconsistent with the k-space power plots of the photonic crystal eigenmodespropagating along these directions in (a) and (b), respectively. The photoniccrytal geometry is partially shown in (c). Note that both reflected beamshave been omitted for clarity.show a different transmission behaviour at different interfaces which is con-sistent with our k-space power map results. Figure 3.6(c) illustrates that aGaussian beam normally incident from free space onto the vertical interfaceis almost completely reflected at the interface. This result is consistent withthe k-space power maps of the excited mode, once we recognize that thefundamental harmonic in Fig. 3.6(a) carries almost zero power (four ordersof magnitude less than the total power). A plane wave incident along theΓK axis cannot phase match to a harmonic carrying any significant power,leading to poor coupling efficiency and high reflectivity.By contrast, Figure 3.6(c) illustrates that a beam normally incident ontothe horizontal interface is highly transmitted through the photonic crystal593.3. Summarywhich is again consistent with the k-space power maps of the excited modeshown in Fig. 3.6(b). In fact, a plane wave incident along the ΓM axis canphase match to the dominant higher-order harmonic S0,2, thereby couplingstrongly to a mode inside the photonic crystal.3.3 SummaryWe have shown that knowledge regarding the distribution of power flowacross the harmonics within a metamaterial can be used to determine theorigin of negative refraction at different interface of a metamaterial. Forexample, we showed that a coupled-plasmonic-waveguide metamaterial witha spherical EFC can negatively refract both power and phase only for in-plane propagation along the layers. The fields for out-of-plane propagationcontain non-negligible harmonics whose phase velocity components in thedirection of propagation are not all identical, and thus the concept of phaserefraction is undefined for the total field (unlike the case of homogeneousmedia). The power maps also demonstrate that both the negative refractionof electromagnetic beams and the extracted double-negative parameters canoriginate from refraction of higher-order right-handed components, not onlyfrom electromagnetic left-handedness.Due to existence of higher-order harmonics, several excitation channelscan exist for a periodic structure. The efficiency of an excitation channelcan be roughly predicted by the relative power residing in the correspond-ing harmonic. For example, we have shown that plane-wave coupling tothe dominant first-order harmonics as opposed to the weak fundamentalharmonic significantly increases the excitation efficiency. The presence ofdifferent excitation channels also means that it is possible for multiple in-cident waves to excite the same eigenmode, casting doubt on the practiceof uniquely characterizing refraction angles across a metamaterial interfaceand then invoking Snell’s law approximations.60Chapter 4Evanescent-waveenhancement innegative-index metamaterialsAs discussed in Chapter 1, an ideal superlens is a flat infinite slab ofhomogeneous double-negative material with εr =µr =−1 embedded in air,which achieves perfect imaging by restoring all the spatial frequency compo-nents of an object to the image plane. The left-handed plane waves propa-gating in the bulk material and the bound modes guided along the materialslab are responsible for negatively refracting propagating components andperfectly amplifying evanescent components, respectively, to restore themin the image plane. When the constitutive parameters of an infinite ho-mogeneous slab deviate from −1, however, the bound modes appear at theoperating frequency. These resonances degrade the imaging performance ofthe slab in two ways. First, the transfer function starts to rapidly decayfor spatial frequencies above the bound mode wave number, thereby estab-lishing an upper spatial-frequency cutoff. Second, components of the objectwhich have transverse spatial frequencies at or near the bound mode wavenumber are overamplified throughout the slab, which prevents image for-mation. These bound modes can be suppressed by adding material lossesto the slab; however, these losses reduce the evanescent amplification andfurther limit the resolution of the lens. Alternating thin layers of positiveand negative index media (e.g. by introducing line vacancy defects) hasbeen suggested as a strategy to shift the resonant modes to higher spacialfrequencies and improve the imaging performance of a non-perfect lens.Negative-index metamaterial lenses have been designed which can resolvesubwavelength-spaced sources in the image plane. In previous chapters, westudied the backward-wave Bloch modes that propagate in a bulk negative-index metamaterial and can provide negative refraction to enable imagingwith a flat lens. In this chapter, we extend our study to evanescent waveamplification in negative-index metamaterials and their superlensing perfor-614.1. Regular array MD metamaterialmance. We show that although a metamaterial lens can mimic the responseof a perfect lens over the propagating spectrum and focus propagating com-ponents of an object, the evanescent response can differ significantly. Theexcitation of resonant modes within the lens limits the spatial frequencybandwidth and introduces imaging artifacts that degrade the imaging fi-delity and robustness. We choose a well-studied periodic structure consistingof an array of magnetodielectric cylinders for the lens medium under test.We demonstrate that the presence of these artifacts can lead to erroneousinterpretation of the standard two-source resolution test for lenses. We showthat artifacts can be mitigated by introducing point defects into the array,which move the resonant modes to higher spatial frequencies and in the caseof finite lenses, suppress their amplitudes through radiation losses. Thisstrategy enables more robust and reliable subwavelength imaging perfor-mance, improves the spatial resolution of the metamaterial lens, and reducesthe deleterious effects of material losses.4.1 Regular array MD metamaterialWe start our study by analyzing a regular two-dimensional magentodi-electric metamaterial, which will be referred as regular array MD metama-terial (see the inset diagram in Fig. 4.1). As discussed in Sec. 2.4, thismetamaterial has been shown to mimic the left-handed behaviour of a Vese-lago medium below the free-space cutoff and has been reported to be capableof super-resolution imaging [12, 16, 19, 54]. By choosing lossless materialswith matched electric and magnetic properties, the MD array offers arguablythe closest metamaterial approximation to a homogeneous Veselago super-lens. It features a high degree of spatial symmetry and exhibits identicalresponses for TE and TM polarization. Such material parameters might notbe practically implemented, however, the magnetodielectric arrays provide aplatform for studying the physics and potential limitations of metamateriallenses.4.1.1 Bulk modes in two-dimensional regular array MDmetamaterialHomogenization methods have been used to show that effective bulkparameters of propagating backward-wave Bloch modes within a metamate-rial may provide a quasi-isotropic negative-index response [2, 18, 19]. Herewe show that this bulk response is not sufficient to determine the evanes-cent response and that further investigation is required. The metamaterial624.1. Regular array MD metamaterial−4 −3 −2 −1 001234aazxkz/kokx/ko0 1 2 3 4αz/koFigure 4.1: The equifrequency contour of the bulk eigenmodes at ωo inthe regular array metamaterial (solid black lines), alongside the EFC of anideal homogeneous double-negative medium (solid gray lines). The complexlongitudinal wave vector components kz+iαz are plotted as a function of thereal transverse wave vector component kx. The geometry of the regular arraymetamaterial with its square unit cell is illustrated in the inset diagram.under test consists of magnetodielectric cylinders with εr =µr = 20 whichare embedded in air as the host medium. The cylinders extend infinitelyalong the y axis with diameter 2r= 0.893a, where a is the lattice constantof the array. The two-dimensional bulk modes of infinite regular array MDmetamaterial are extracted at a fixed frequency by solving the wave equa-tion over a unit cell for different transverse wavevector components kx. Asshown by the solid black line in Fig. 4.1, this periodic structure can sustainbackward-wave Bloch modes for all directions of propagation. The circularEFC (traced by the fundamental Floquet wavevectors) mimics the homoge-neous Veselago medium when the periodicity is equal to a= 0.0646λo, whereλo is the free-space operating wavelength. Similar to the Veselago medium,a slab of this metamaterial can focus all the incident propagating waves fromair through coupling with the backward-wave Bloch modes sustained withinthe slab.As can be seen in Fig. 4.1, however, the dispersive behaviour of the bulkmodes deviates from the ideal Veselago medium over the evanescent spec-trum, with the bulk modes even taking on complex wave vectors at highertransverse spatial frequencies. This would naturally lead us to suspect that634.1. Regular array MD metamaterialPBCPBCPML PMLkxz−d/2 d/20Figure 4.2: Diagram of the simulation setup used to solve for the boundmodes guided along infinite slabs of the regular array MD metamaterial.Periodic boundary conditions are assigned to the top and bottom sides andperfectly matched layers are placed at the left and right sides of the domain.the slab bound modes also differ from those of the Veselago lens, potentiallyimpacting the imaging performance of the MD metamaterial lens. In thefollowing section we further investigate the evanescent spectrum response bydirectly solving for the bound modes sustained in a slab of this metamaterialand analyze its response to an incident wave.4.1.2 Bound modes in infinite regular array metamaterialslabsIn this section, we study the resonant bound modes guided along aninfinitely long slab of the regular array MD metamaterial embedded in air.We start our investigation by choosing idealized lossless material parame-ters in order to isolate the effect of resonant modes on imaging and establisha baseline for its superlensing performance. We use numerical solutions ofthe wave equation for an infinite MD-cylinder metamaterial slab to find itsbound modes and estimate its transmission response over the evanescentspectrum. We solve for the eigenmodes of a single unit cell of the slabas a function of transverse wavevector kx. Figure 4.2 shows the schematicdiagram of the simulation setup for a five-layer slab in which periodic bound-ary conditions (PBCs) are assigned to the top and bottom sides and perfectmatch layers (PMLs) are placed at the left and right sides of the domain.The height of the domain is equal to the slab periodicity a and the width isset large enough for the evanescent tails of the slab bound modes to be neg-ligibly small at the PML interfaces (ensuring a real valued eigenfrequency).Figure 4.3 shows the frequency dispersion diagram of guided boundmodes along a monolayer slab and a five-layer slab plotted over the first644.1. Regular array MD metamaterial0 1 2 3 4 5 6 70.920.940.960.9811.021.041.06aaxza a-1015aaxz5a 5a-101kx/koω/ωoFigure 4.3: Dispersion diagrams of guided bound modes along infinite slabsof the regular array MD metamaterial. The results for a monolayer MD slaband a five-layer MD slab are shown by the dashed black lines and solid blacklines over the first BZ (up to |kx| = pi/a= 7.73ko), respectively. The evenbound modes (corresponding to the upper lines at low spatial frequencies)intersect with ωo to produce a transmission peak at kx = 1.871ko for the five-layer and at kx = 3.183ko for the monolayer slab. The slab geometries, unitcells, and bound mode field distributions corresponding to kx = 1.7ko areplotted in the inset diagrams. The dispersion diagram of uncoupled singleinterface surface waves is shown by the dotted black line. The flat solid redline indicates the operating frequency ωoBrillouin zone. An incident evanescent wave can be amplified by couplingto the slab mode with the corresponding transverse wavevector. For exam-ple, consider the magnetic field distribution of even and odd modes plottedas insets in Fig. 4.3. These modes resemble coupled surface modes withinthe Veslago slab (see Fig. 1.11) and can similarly amplify incident evanes-cent waves. As can be seen from the dispersion curves of the monolayerslab (dashed black line) and the five-layer slab (solid black line), however,they can mimic the ideal lens behaviour only until the dispersion curves ofthe even modes intersect with the flat line at ωo (the even modes providethe lowest spatial frequency intersections). These intersection points areindicated by red dots in the dispersion diagrams. As the thickness of the654.1. Regular array MD metamaterialPBCPBCPML PMLkx L1 L2Portz−d −d/2 d/2 d0Figure 4.4: A diagram of the transmission simulation setup of the infiniteregular array slab.slab increases, the curves appear closer to the dispersion line of uncoupledsurface modes guided along a single interface (shown by the dotted line andcalculated using a very thick slab where the coupling between the surfacemodes at each interface is negligible). This pushes the intersection pointwith ωo to lower spatial frequencies, thereby reducing the spatial bandwidthof the lens. For example, the first resonance of the five-layer slab at theoperating frequency occurs for the even bound mode at kx = 1.871ko.To quantify the response of the metamaterial slabs to propagating andevanescent waves at ωo, we use full-wave simulations to calculate the transferfunction for plane waves transmitted from the source plane (z=−d) to theimage plane (z= d) as a function of transverse spatial frequency kx. We usethe simulation setup shown in Fig. 4.4 and solve for the transmitted fieldgiven an incident plane wave with a transverse wavevector kx. In this setup,the dashed lines represent periodic boundary conditions, the incident waveis generated through a simulation port (shown in red), and PMLs are usedto terminate the scattered waves. The transmission is then calculated as theratio of the transmitted Fourier coefficient at the image plane (L2) to theincident Fourier coefficient at the source plane (L1) usingT =∫L2Ht(x, z)e−ikxxdx∫L1Hi(x, z)e−ikxxdx, (4.1)where Hi(x, z) is the incident magnetic field and Ht(x, z) is the transmittedmagnetic field.Figure 4.5 plots the amplitude of the total transmission function for amonolayer slab (dashed black line) and a five-layer slab (solid black line)alongside the ideal transmission of a Veselago slab (flat gray line). Theresponse of the metamaterial slabs to propagating waves are similar to theVeselago slab over the propagating spectrum, with nearly uniform 0 dB664.1. Regular array MD metamaterial0 1 2 3 4 5 6 7−80−60−40−2002040kx/koTransmission(dB)Figure 4.5: The transfer function of the slabs at ωo. The results for a mono-layer MD slab and a five-layer MD slab are shown by the dashed black linesand solid black lines over the first BZ (up to |kx| = pi/a= 7.73ko), respec-tively. Transmission peaks correspond to the points where the dispersioncurves in Fig. 4.3 intersect with ωo. The ideal transfer function of Veselagoslabs is shown in (b) by the flat gray line at 0 dB.transmission. However, the metamaterial slabs only mimic the ideal lensbehaviour over the evanescent spectrum below the transmission resonances(indicated by the peaks in the response). These peaks correspond to thepoints where the dispersion curves intersect with ωo, and cause the imagingperformance of the metamaterial slab to be restricted in the following twoways. Firstly, the amplification of evanescent waves is dramatically reducedafter the appearance of resonances, placing an upper spatial frequency limitto the evanescent components of an object that can be reconstructed in theimage plane. In other words, the resonance peaks limit the minimum focalspot diameter corresponding to the image of an illuminating point source.Secondly, the over-amplification of the object components at or near theresonances manifest as high-intensity lobes which obscure the image of theobject. In the following section we study the dominate resonant modes (ator very close to the operating frequency) within a finite metamaterial lens.674.1. Regular array MD metamaterial4.1.3 Resonant modes in finite regular array metamateriallensesIn this section, we use full-wave eigenmode simulations to study theresonant modes and imaging artifacts of a MD metamaterial lens with finitetransverse width. As discussed in previous sections, bound modes guidedalong an infinite slab at the operating frequency (identified by the red dotsat the dispersion diagram intersection points) appear as high-intensity lobeswhich obscure the image of the object. Here, we consider a finite width forthe lens which reduces the continuous dispersion curves of infinitely longslabs to discrete resonances in ω-k space. Due to scattering from the topand bottom edges of the lens, these modes are generally not completelybounded, resulting in a complex resonant frequency ωr + iγ, where γ is adamping term that accounts for radiation loss.−1−0.5 0 0.5 1−2.5−2−1.5−1−0.500.511.522.5z/λox/λo−1−0.5 0 0.5 1z/λo−1−0.5 0 0.5 1z/λo0-11(a) (b) (c)Figure 4.6: Magnetic field intensity plots of the dominant resonant modesfor the regular array lenses of thickness 5a. The lens has a width of 65a in(a) and 59a in (b) and (c). The resonance modes have an even longitudinalsymmetry (along the z axis). (a) The resonant mode with ωr = ωo andγ/ωo = 6.3437× 10−5 has transverse even symmetry (along the x axis).(b) The resonant mode with ωr = 0.9998ωo and γ/ωo = 4.3085× 10−5 hastransverse odd symmetry (along the x axis). (c) The resonant mode withωr = 1.0003ωo and γ/ωo = 1.1288× 10−4 has transverse even symmetry.The inset diagrams plot the real part of the field to show the transverse andlongitudinal symmetries of each mode.684.1. Regular array MD metamaterialAs discussed in previous section, the bound mode guided by an infinitefive-layer regular array slab at ωo has longitudinal even symmetry and alow transverse wavevector kx = 1.871ko. Not surprisingly, the dominantresonant modes within a finite regular lens also have even symmetry alongthe z axis, resulting in non-directive scattering from the edges of the lens.These resonant modes can have symmetric or anti-symmetric field distribu-tions in the transverse directions (across the x= 0 plane). As can be seen inFig. 4.6, the resonant fields are weakly confined to the lateral side of the lensand appear as side lobes across the image plane. The low spatial frequencycomponents of the resonant modes make them easy to excite, resulting instrong imaging artifacts. For example, a regular array five-layer lens withwidth 65a has a resonant mode at the operating frequency (see Fig. 4.6(a)). When this lens is used for imaging, the mode is strongly excited at itsresonance and completely obscures the object image.By carefully choosing the lens width, the resonance frequency of thedominant modes can be detuned from the operating frequency to reducethe mode amplitudes; however this mitigation is limited. For example, aregular five-layer lens with width 59a does not have a resonant mode at ωo(see Figs. 4.6(b) and (c)), however, the off-resonance mode fields are notwell confined to the lens and can lead to strong imaging artifacts. Therobustness of this lens in imaging different objects will be studied further inin the following section.Taking the Fourier transform of the magnetic field over the surface of thelens at z= 2.5a shows that resonant mode fields in Figs. 4.6(b) and (c) havedominant transverse spatial variations equal to kx = 1.93ko and kx = 1.82ko,respectively. These have similar spatial frequencies to the even bound modeof the infinite slab at the operating frequency (i.e. they are located near thefirst red dot in Fig. 4.3 at kx = 1.871ko), which demonstrates consistencybetween the dispersion of the infinite and finite slab modes. Furthermore,far from the lenses, the field patterns exhibit low spatial variations withtransverse wavenumbers near ko. This can be attributed to the fact that theslabs can also support an even bound mode near the light line just above ωo(see the solid line in Fig. 4.3).4.1.4 Imaging performance of finite regular arraymetamaterial lensIn this section, we study the imaging performance of the regular arraylenses using full-wave simulations. The object to be imaged consists of mul-tiple two-dimensional electric dipoles with moments aligned parallel to the694.1. Regular array MD metamaterial−0.5 0 0.5 1−2−1.5−1−0.500.511.52z/λox/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)−0.5 0 0.5 1z/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)01Figure 4.7: Imaging field plots for the regular array lenses with widths 65a(left) and 59a (right). Each lens has thickness 5a and is illuminated bytwo x-polarized electric dipoles located at x= 0.25λo and −0.25λo. Themagnetic field intensity is plotted over the xz plane on the left, while thenormalized field intensity at the image plane is plotted on the right in solidblack. The image field without the lens is plotted in dotted black, alongwith the source plane field intensity of the object in solid gray. The sourceplane at z=−5a and the image plane at z= 5a are indicated by thin blacklines in the field plots.x axis. Each dipole emits a transverse magnetic field Hy with a uniformsource-plane Fourier spectrum across both propagating and evanescent do-mains. This enables the response of each lens to be studied across a broadspatial bandwidth. All lenses have a thickness of 5a and are centered at theorigin. The source and image planes are separated by a distance equal totwice the lens thickness and are located at z=−5a and z= 5a, respectively.Figure 4.7 shows the imaging performance of lenses with width of 65a(left) and 59a (right) for a pair of symmetrically centered dipole sourcesseparated by 0.5λo. In this figure, the field plots on the left show the mag-netic field intensity distribution over the xz plane, while the plots on theright show the normalized transverse field intensity in dB (20 log |Hy|) at theimage plane. The field intensity at the source and image planes for emptyspace without the lenses are also plotted for reference. The intensity plotsin this figure and all other following figures are normalized with respect tothe maximum value of the image intensity for the regular array lens withwidth 59a. This provides a quantitative comparison of the performance of704.1. Regular array MD metamaterial−0.5 0 0.5 1−2−1.5−1−0.500.511.52z/λox/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)−0.5 0 0.5 1z/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)01Figure 4.8: Field plots similar to the ones shown in Fig. 4.7 for two offsetelectric dipoles located at x= 0.13λo and 0.63λo (left) and for two offsetdipoles with wider spacing located at x= 0.25λo and −0.65λo (right).lenses.The simulation results for the regular lens with width 65a illustrateshow excitation of unconfined modes at their resonances completely obscureimaging. Figure 4.7, however, shows that the lens with width 59a anddetuned resonance frequencies can produce two distinct peaks at the imageplane for a pair of symmetrically centered dipole sources separated by 0.5λo.These imaging discrepancies reflect that the regular array lens is not robustto variation in the width of the lens. The image from a lens of width 59a hashigh side lobe levels and image peaks laterally shifted from the true objectpositions. This demonstrates that the resonant modes can still producestrong imaging artifacts and can reduce imaging fidelity.Figure 4.8 illustrates the robustness of the regular array lens with width59a to object configurations which are not symmetrically centered on thelens. The plot on the left side shows the image fields for the same twodipoles offset by 0.28λo along the x axis. As shown in the left figure, thesuperposition of the object image with the resonant mode excited withinthe regular array lens produces an image in which the location of bothsources are no longer identifiable. In the right plot of Fig. 4.8, the sourcesare separated further to a distance of 0.9λo and offset by 0.2λo. Despitethe larger separation distance, the regular array lens still cannot locate thebottom source at x=−0.65λo in the image of the lens. This implies thatthe resolution of the objects observed in Fig. 4.7 may simply have been a714.1. Regular array MD metamaterial−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−40−30−20−100x/λoMagneticfieldintensity(dB)Regular lens w/o lossRegular lens w lossFigure 4.9: The normalized field intensity at the image plane for imagingwith lossless and lossy regular array MD lenses. The illuminating sources arefour x-polarized electric dipoles located at x = 0.675λo, 0.225λo, −0.225λoand −0.675λo. The fields from MD lenses with lossless and lossy inclusions(with electric and magnetic loss tangents equal to 10−3) are plotted in solidblack line and solid red line with circle markers, respectively. The normalizedsource field intensity of the object and image intensity without the lens areplotted in solid gray and dotted black, respectively.consequence of the object spectrum matching the spectrum of a resonancemode within the regular lens. When the spectra are mismatched, however,the objects are no longer resolved.Here, we briefly discuss the minimum separation distance necessary toresolve symmetric dipoles with a lens with width 59a. We first considera pair of centered dipoles (as in Fig. 4.7) and use the Rayleigh criterionto define the image resolution as the minimum distance between two pointsources that results in a 0.91 dB drop between their image peaks. Simula-tion results demonstrate that a pair of dipoles is still resolvable when theseparation between them is reduced to 0.3192λo. We then consider an ob-ject composed of four evenly spaced dipoles symmetrically centered over thesource plane. The four dipoles provide a more complex source to test theimaging resolution under more robust conditions. Our simulation resultsdemonstrate that in order to resolve the object the separation between thedipoles should be increased to 0.4286λo. These results further confirm thatthe effect of resonant modes should not be ignored in analyzing the imaging724.2. Defected array MD metamaterialperformance and that resolving a pair of symmetric point sources does notprovide a sufficiently rigorous test for super-resolution.We now investigate the effect of material loss on the imaging perfor-mance of the regular array lens. Figure 4.9 plots the image field intensitycorresponding to an object composed of four symmetric dipoles separatedby 0.45λo. Two cases are considered: in the first, the lenses are composed oflossless MD cylinders (εr =µr = 20), while in the second, low loss cylindersare used in which the electric and magnetic loss tangents are equal to 10−3(εr =µr = 20 + 0.02i). In the absence of loss, the regular array lens gener-ates an image that can resolve the point sources, although it has prominentside lobes and peaks that are laterally shifted from the true object posi-tions (due to interference with the resonant mode fields). In the presenceof absorption, the resonance modes are suppressed in the regular array lens,however, the spatial resolution is also reduced and the lens can no longerresolve the inner point objects.4.2 Defected array MD metamaterialIt has been shown that a multi-layered slab of alternating thin layers ofpositive and negative index media which support coupled surface waves atlayer interfaces can amplify evanescent waves over a wider spatial frequencyand thus have improved resolution over a solid negative-index slab of thesame thickness [30–33]. Considering that a single MD monolayer guidesbound modes along the layer [see Fig. 4.3] we can expect that an anisotropicmulti-layered stack of MD monolayers, which corresponds to introducing airgaps (i.e. line vacancy defects) parallel to the slab interfaces between eachlayer of MD cylinders, has improved imaging resolution with respect to aregular array metamaterial. Introducing uniformly spaced vacancy point de-fects in the regular array MD metamaterial instead of line vacancy defects(see the inset diagram in Fig. 4.10) can also provide geometric symmetry andresult in supporting backward-wave Bloch modes in all directions of propa-gation. Therefore, we can expect that a slab of the isotropic defected arrayprovides amplification over a wide range of the evanescent spectrum, whilemimicking the focusing mechanism of a perfect lens over the propagatingspectrum. In the following sections we compare the result for the proposeddefected array and the regular array studied in previous section and investi-gate our hypothesis of improved imaging through the introduction of pointdefects.734.2. Defected array MD metamaterial−4 −3 −2 −1 001234zx 2a2akz/kokx/ko0 1 2 3 4αz/koFigure 4.10: The EFC of the bulk eigenmodes at ωo in the defected arraymetamaterial (solid and dashed black lines), alongside the EFC of an idealhomogeneous double-negative medium (solid gray lines). The geometry ofthe defected array metamaterial with its square unit cell is illustrated in theinset diagram.4.2.1 Bulk modes in two-dimensional defected array MDmetamaterialHere, we analyze bulk modes within our proposed defected array meta-material designed by introducing uniformly spaced vacancy point defects inthe regular array MD metamaterial studied in previous section. We can alsoview the defected array as a cross-hatched grid of MD cylinders made by ver-tical and horizontal MD cylinder monolayers which are spaced by distance2a along the longitudinal (z) and transverse (x) directions, respectively. Asshown by the solid black line in Fig. 4.10, the propagating backward-waveBloch modes in the defected array can also mimic the circular EFC of theVeselago medium by adjusting the lattice constant to 2a= 0.1284λo. Giventhe shape of the EFC, we expect that a slab of this structure focuses allincident propagating components through negative refraction (just like theregular MD array).Although the dispersive behaviour of dominant bulk modes in the regu-lar and defected array metamaterials are very similar over the propagatingspectrum the behaviour over the evanescent spectrum is very different. Fur-thermore, the defected array supports a secondary mode that propagates at744.2. Defected array MD metamaterialhigher spatial frequencies and is not present in the regular array response(indicated by the dashed lines in Fig. 4.10). This mode can be attributedto the appearance of additional coupled resonances introduced by the va-cancy defect points inside the structure (similar to how alternating thinlayers of positive and negative index media can support coupled surfacewaves at layer interfaces [59]). As a result, we would expect that the boundmodes within a defective array slab would not resemble those of the regu-lar array lens and may not even be evanescent inside the slab. It has beenshown elsewhere, however, that similar bound modes in a multi-layered slabcan amplify evanescent waves over a wider spatial frequency and thus haveimproved resolution over a solid negative-index slab of the same thickness[30–33]. In the following section, we directly solve for the bound modes ofa slab of the defected array metamaterial and analyze the slab response toincident waves.4.2.2 Defected array metamaterial slabPBCPBCPML PMLkxz0−d/2 d/2PBCPBCPML PMLkx L1 L2Portz−d d0−d/2 d/2(a) (b)Figure 4.11: Diagrams of the simulation setups used to generate the disper-sion curves and transmission response of the infinite defected array slab areshown in (a) and (b), respectively.In this section, we solve for the bound modes guided along an infiniteslab of the defected array metamaterial and analyze the slab response toincident waves using the methods described in the Sec. 4.1.2 (see Fig. 4.11for diagrams of the simulation setups). We study a defected array slab ofthickness 5a and compare it to the response of the five-layer regular arraymetamaterial slab. The dispersion diagram of bound modes over the first BZof the defected array slab (up to |kx| = pi/2a= 3.89ko) and the geometry ofthe defected array metamaterial are shown in Fig. 4.12. Note that unlike thehomogeneous double-negative medium and the regular array metamaterial,higher-spatial frequency propagating modes are present in the defected array(shown by the dashed lines in Fig. 4.10) that prevent surface modes fromforming at a single interface. This fundamentally changes how the boundmodes are formed since they do not arise from simple coupling between754.2. Defected array MD metamaterial0 1 2 30.9911.011.025a2axz5a-101kx/koω/ωoFigure 4.12: The dispersion diagrams of guided bound modes along the de-fected array metamaterial slab over the first BZ (up to |kx| = pi/2a= 3.98ko).The middle curve represents an even bound mode while the upper and lowercurves represent odd bound modes. The odd bound mode corresponding tothe lowest line at low spatial frequencies intersects with ωo to produce thefirst transmission peak at kx = 2.824ko. The MD slab geometry and unit cellare plotted in the inset diagrams alongside the magnetic field distributionof the guided odd mode at kx = 1.7ko.the surface modes of the slab. As can be seen in Figure 4.12, the defectedarray slab can guide odd bound modes with a flat dispersion profile nearωo and the first intersection point pushed up to kx = 2.824ko. The near-resonant coupling of incident evanescent waves at ωo to these odd boundmodes provides amplification over a wide range of the evanescent spectrumthat results in higher spatial resolution imaging [60]. The transfer function ofthe defected array slab at ωo is shown in Fig. 4.13 in order to quantitativelydemonstrate the response of the slab to an incident wave. The defected arrayslab has a nearly unity transfer function over the propagating spectrum andprovides amplification for higher spatial frequencies up to the first resonancepeak (coinciding with the first intersection point in Fig. 4.12).We can gain insights into the operation of the defected array slab byconsidering the bound modes of multi-layered MD slabs composed of eitherlongitudinally stacked transverse monolayers spaced with air (wherein each764.2. Defected array MD metamaterial0 1 2 3−80−60−40−2002040kx/koTransmission(dB)Figure 4.13: The transfer function of the defected array metamaterial slabat ωo over the first BZ (up to |kx| = pi/2a = 3.98ko). Transmission peakscorrespond to the points where the dispersion curves in Fig. 4.12 intersectwith ωo.layer is parallel to the x axis) or transversely stacked longitudinal monolay-ers spaced with air (wherein each layer is parallel to the z axis). Note that asingle MD monolayer guides bound modes along the layer (see Fig. 4.3) and amulti-layered stack of MD monolayers forms a coupled-waveguide structurewith similar properties to the anisotropic multi-layered stack of alternatingthin layers of positive and negative index media [59]. First, we consider theslab composed of transverse monolayers. This corresponds to introducingair gaps (i.e. line vacancy defects) parallel to the slab interfaces betweeneach layer of MD cylinders. The dispersion diagram of bound modes in thisstructure approach the diagram of a single monolayer MD slab in Fig. 4.3with the resonance pushed to higher spatial frequencies. Below the reso-nance point, however, these modes are far from ωo and cannot provide theamplification of the regular array MD slab. This is not unexpected, giventhe evanescent decay across each of the air gaps between layers. Secondly,we consider the slab composed of longitudinal monolayers. This correspondsto introducing air gaps (line vacancy defects) that are perpendicular to theslab interfaces. These monolayers act as transmission lines, allowing the slabto sustain bound modes with relatively flat dispersion near ωo. The hori-774.2. Defected array MD metamaterial−1−0.5 0 0.5 1−2.5−2−1.5−1−0.500.511.522.5z/λox/λo−1−0.5 0 0.5 1z/λo0-11(a) (b)Figure 4.14: Magnetic field intensity plots of the dominant resonant modesfor the defected array lens of width 59a and thickness 5a. The resonancemodes have an odd longitudinal symmetry (along the z axis). (a) The reso-nant mode with ωr = 0.9998ωo and γ/ωo = 5.8441×10−5 has transverse evensymmetry. (b) The resonant mode with ωr = ωo and γ/ωo = 4.1274× 10−5has transverse odd symmetry. The inset diagrams plot the real part of thefield to show the transverse and longitudinal symmetries of each mode.zontal monolayers couple poorly to incident waves, however, and the overallamplification of the slab is minimal. Although both structures shift theresonant slab modes to higher spatial frequencies, evanescent waves are notstrongly amplified and propagating waves can only be focused if the sourceand image planes are nearly adjacent to the slab interfaces. The proposeddefected array slab, however, combines both layered structures, providinghigh coupling and amplification over a wide range of the evanescent spec-trum, while mimicking the focusing mechanism of a perfect lens over thepropagating spectrum.4.2.3 Defected array metamaterial lensAs discussed in previous section, the first resonant mode guided by thedefected array slab at ωo (identified by the red dot in Fig. 4.12) is an oddmode with a transverse wavevector kx = 2.824ko (this is higher than thatof the regular array slab). The dominant resonant modes within a finitedefected array lens have the same odd symmetry along the z axis as the784.2. Defected array MD metamaterialinfinite slab. As can be seen in Figs. 4.14 for a lens of width 59a, the edgescattering of the lens is directed away from the image plane. The resonantmode fields are therefore confined closer to the lens surface with amplitudesthat are suppressed by this radiation loss. The higher spatial frequencyof the resonances also results in weaker excitation by the highly decayingevanescent components of the object field. Compared to the regular arraylens, we see a significant decrease in imaging artifacts in the defected arraylens. Simulation results show that although the defected array lens hasa resonant mode at the operating frequency (see Fig. 4.14(b)), the modefields do not extend into the image plane. This means that compared tothe regular array lens, the performance of the defected array lens is lesssensitive to the location of resonances and is more robust to the transversewidth. The resonant mode fields shown in Figs. 4.14(a) and 4.14(b) havedominant transverse spatial variations equal to kx = 2.68ko and kx = 2.82ko,respectively, which occur near the spatial frequency of the odd bound modeof the infinite slab at the operating frequency (indicated by the red dot atkx = 2.824ko in Fig. 4.12). Unlike the regular array lens, the field patternsof the defected array lens do not exhibit the same low spatial variations farfrom the lens because the lens does not support odd bound modes near thelight line close to ωo (see Fig. 4.12). In the next section, we demonstrateand quantify the improvements in imaging performance of the defected arraylens over the regular array lens.4.2.4 Imaging performance of finite defected metamateriallensIn this section, we study the imaging performance of the defected arraylens with thickness of 5a and width of 59a using full-wave simulations. Fig-ures 4.15 and 4.16 shows the imaging performance of the defected array lensfor similar object configurations used in Figs. 4.7 and 4.8 for the regulararray lens. This provides a quantitative comparison of the performance ofeach lens across different object distributions.Figure 4.15 illustrates that both the regular array lens and the defectedarray lens produce two distinct peaks at the image plane for a pair of sym-metrically centered dipole sources separated by 0.5λo. The image from thedefected array lens (right plot), however, is much clearer with lower side lobelevels and more distinct peaks with respect to the regular array lens (leftplot). The image peaks of the defected array lens are also at the true objectpositions. These imaging discrepancies reflect the differences between theexcited resonant modes within the regular and defected array lenses and the794.2. Defected array MD metamaterial−0.5 0 0.5 1−2−1.5−1−0.500.511.52z/λox/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)−0.5 0 0.5 1z/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)01Figure 4.15: Imaging field plots for the regular array lens (left) and thedefected array lens (right). Each lens has thickness 5a, width 59a, and isilluminated by two x-polarized electric dipoles located at x= 0.25λo andx=−0.25λo.−0.5 0 0.5 1−2−1.5−1−0.500.511.52z/λox/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)01−0.5 0 0.5 1z/λo−50 −40 −30 −20 −10 0Magnetic field intensity (dB)01Figure 4.16: Imaging field plots for the defected array lens for two offsetelectric dipoles located at x= 0.13λo and 0.63λo (left) and for two offsetdipoles with wider spacing located at x= 0.25λo and −0.65λo (right).differences between the resulting imaging artifacts.We now investigate the robustness of the defected array lens to objectconfigurations which are not symmetrically centered. Figure 4.16 shows thatthe object configurations which could not be resolved by the regular array804.2. Defected array MD metamaterial−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−40−30−20−100x/λoMagneticfieldintensity(dB)Defecetd lens w/o lossDefected lens w lossFigure 4.17: The normalized field intensity at the image plane for imag-ing with lossless and lossy defected array MD lenses. The illuminatingsources are four x-polarized electric dipoles located at x= 0.675λo, 0.225λo,−0.225λo and −0.675λo. The fields from MD lenses with lossless and lossyinclusions (with electric and magnetic loss tangents equal to 10−3) are plot-ted in solid black line and solid red line with circle markers, respectively. Thenormalized source field intensity of the object and image intensity withoutthe lens are plotted in solid gray and dotted black, respectively.lens (see Fig. 4.8) can be resolved and located in the image by the defectedarray lens without difficulty.Here, we briefly discuss the minimum separation distance necessary toresolve symmetric dipoles with the defected array lens. We first consider apair of centered dipoles (see Fig. 4.15). Simulation results demonstrate thata pair of dipoles is still resolvable when the separation between the electricdipoles is reduced to 0.2951λo (for the regular array lens the minimum dis-tance was 0.3192λo). We then consider an object composed of four evenlyspaced dipoles symmetrically centered over the source plane. Our simula-tion results demonstrate that in order to resolve the object the separationbetween the dipoles should be increased to 0.3239λo (for the regular arraylens the minimum distance was 0.4286λo). These results further confirmthat the imaging resolution has been enhanced in the defected array lens.We now investigate the effect of material loss on the imaging performanceof the defected array lens. We apply similar conditions used for imaging withthe regular array lens in Fig. 4.9 for a better comparison. Figure 4.17 plots814.2. Defected array MD metamaterial0 0.005 0.01 0.0150.320.340.360.380.400.420.440.460.480.500.520.54Regular array MD lensDefected array MD lens∗tanδ∆/λoFigure 4.18: The minimum separation distance (∆) necessary to resolve afour symmetric dipoles in the image of the regular array lens (red circlemarkers) and the defected array lens (blue asterisk markers) as a functionof the electric and magnetic loss tangent of the MD inclusions (tanδ). Thelength of error bars is equal to 0.005λo.the image field intensity corresponding to a similar object composed of foursymmetric dipoles separated by 0.45λo. Again two cases are considered:in the first, the lenses are composed of lossless MD cylinders (ε=µ= 20),while in the second, low loss cylinders are used in which the electric andmagnetic loss tangents are equal to 10−3 (ε=µ= 20 + 0.02i). The defectedarray lens has higher resolution and (unlike the regular lens) can resolve allfour objects even at the presence of the material loss.We systematically investigated the imaging resolution of the regular ar-ray lens and the defected array lens by gradually varying levels of materialloss. Figure 4.18 plots the minimum separation distance (∆) necessary toresolve four symmetric dipoles as the electric and magnetic loss tangent(tanδ) of the MD inclusions is varied. The plot shows that a lossy defectedarray lens with tanδ < 0.006 has a higher resolution than the lossless regularlens. While the regular array lens can resolve sources separated by a dis-tance ∆ = 0.5λo for tanδ < 0.003, the same resolution can be achieved by thedefected array lens for much higher loss tanδ < 0.014 (which is within therange reported for low-loss magnetodielectric materials [61]). This demon-strates the improved imaging robustness of the defected array lens and the824.3. Summaryimportance of designing superlenses with resonance modes in mind.4.3 SummaryIn this chapter, we have investigated the impact of resonant modes onthe imaging performance of a negative-index metamaterial lens. We showedthat although a metamaterial lens can mimic the response of a perfect lensover the propagating spectrum, the evanescent response can differ signifi-cantly. The excitation of resonant modes within the lens limits the spa-tial frequency bandwidth and introduces imaging artifacts that degrade theimaging fidelity and robustness. Furthermore, we show that the standardtwo-source resolution test can be unreliable when applied to superlenses. Inparticular, some demonstrations of super-resolution arise simply as a con-sequence of the object spectrum matching the spectrum of the dominantresonant mode. For example, we show that a regular array of MD cylinderscan image a pair of illuminating point sources when symmetrically centeredin front of the lens but not when that symmetry is broken. Achieving robustimaging performance therefore requires an understanding of how resonantmodes produce artifacts and requires that all super-resolution claims aresupported by comprehensive resolution testing.We also showed that the resolution and robustness of a metamateriallens can be improved by tailoring its resonant modes to avoid artifacts.Using the MD cylindrical metamaterial as an example, we introduced defectsinto the array to push the resonant modes to high spatial frequencies anddirect the edge-scattered fields away from the image plane. We show thatthese defects also mitigate the effects of material losses on subwavelengthresolution. This work provides deeper insight into the behaviour of resonantmodes and their effects on imaging and provides a comprehensive methodto study super-resolution imaging in metamaterial lenses.83Chapter 5ConclusionThis thesis has investigated the power flow of propagating backwardwaves and amplification of evanescent waves in metamaterial superlenses.Homogenization methods have been deeply studied before and used to modelmetamaterial structures with effective parameters. However, this researchfill a knowledge gap in the field of metamaterials by studying these hetero-geneous structures from other perspectives which provide new insights totheir response and reveals their limitations in mimicking an ideal negative-index homogeneous medium. This knowledge has enabled us to propose newdesign strategies and improve their superlensing performance.In this thesis, we investigated the behaviour of backward-wave eigen-modes propagating in bulk metamaterial structures which have been de-scribed in the literature as negative index media. To address this objectivefrom first principles, the electric and magnetic fields inside the media are cal-culated from Maxwell’s equations, Fourier-transformed, and then paired tofind a complete set of spatial-frequency harmonics, each with a well-definedwave vector and Poynting vector. Plotting the Poynting vectors in k spaceprovides a simple visual representation of power flow within a periodic het-erogeneous structure. This is a systematic and general procedure that canbe applied to any periodic structure, regardless of frequency, complexity ofgeometry, periodicity, or constituent materials. Access to power flow of allharmonics enables us to demonstrate the origin of backward power and indoing so, verify the validity of the claimed isotropic left-handed responseand the validity of describing media by a simple negative effective indexof refraction n = −1. We have shown that the spatial frequency originof backward power flow can change with respect to the direction of prop-agation or from one periodic structure to another. The knowledge aboutharmonic power flows has also been used to determine the origin of neg-ative refraction at different interface of a metamaterial. The power mapsdemonstrate that both the negative refraction of electromagnetic beams andthe extracted double-negative parameters can originate from refraction ofhigher-order right-handed components, not only from an electromagneticleft-handedness. It has been shown that the existence of higher-order har-84Chapter 5. Conclusionmonics provides several excitation channels for a periodic structure whichtheir coupling efficiency can be roughly predicted by the relative power re-siding in the corresponding harmonic.In this thesis, We have introduced a comprehensive method to studythe resonant modes within a metamaterial lens and analyze its superlesningperformance. The metamaterial under test is a two-dimensional array ofMD cylinders with optimized matched parameters which provides a usefulplatform for clearly demonstrating the relevant physics. Numerical solutionsof the wave equation have been used for metamaterial slabs to find theirresonant modes and estimate their response over the evanescent spectrum.We have shown that a metamaterial flat lens can negatively refract and focuselectromagnetic beams, however, its response can differ significantly from anideal homogeneous negative-index lens at high spatial frequencies beyond thefree-space cutoff. The excitation of resonant modes within the lens limits thespatial frequency bandwidth and introduces imaging artifacts that degradethe imaging fidelity and robustness. We showed that the standard two-source resolution test can be unreliable when applied to metamaterial lenses.We also showed that the resolution and robustness of a metamaterial lenscan be improved by tailoring its resonant modes to avoid artifacts. Using theMD cylindrical metamaterial as an example, we introduced defects into thearray to push the resonant modes to high spatial frequencies and direct theedge-scattered fields away from the image plane. These defects also mitigatethe effects of material losses on subwavelength resolution.Although k-space power maps provide a new insight to the couplingat a metamaterial interface they cannot quantitatively analyze the couplingefficiency. This method also disregards the excitation of other possible eigen-modes at the interface. The dispersion behaviour of resonant mode can beused to determine and improve imaging limitations for a metamaterial lens,however, it cannot solely provide a quantitative description for the meta-material response. The coupling efficiency of incident evanescent waves tothe resonant modes is an important factor which is not considered in thismethod.In future studies an experimental setup can be designed to test the cou-pling efficiency at a metamaterial interface for tailored incident waves andverify the predicted result. Our method should be extended to analyze theresonant modes within three-dimensional metamaterial lenses. This methodcan also be extend to study the effect of absorption in metamaterial inclu-sions from the first step by analyzing resonant modes with complex wavevectors. The superlensing performance of other metamaterial structurescan be re-examined by investigating their resonant modes and testing their85Chapter 5. 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Clausius, Die mechanische Behandlung der Electricitat.Kessinger Publishing, 1879. → pages 95[63] C. Caloz and T. Itoh, Electromagnetic Metamaterials: TransmissionLine Theory and Microwave Applications. New York, USA: Wiley-IEEE Press, 2005. → pages 9992Chapter 5. Bibliography[64] C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of3d metamaterial/photonic crystals.,” Opt. Express, vol. 19, pp. 19027–19041, Sep 2011. → pages 10993Appendices94Appendices AClausius-Mossotti equationHere, we explain a simplified form of Clausius-Mossotti equation, a ho-mogenization method which relates the constitutive parameters to the polar-izability coefficients of inclusions [62]. Assume that a material is constructedof a cubic array of the same particles in which the dominant scattering re-sponse of each particle to the applied electromagnetic fields can be modeledby an electric dipole moment and a magnetic dipole moment. By suchassumption the response of each particle in this crystal can be simply de-scribed by polarizability coefficients (αe,αm) which relate the induced dipolemoments (p0,m0) to the local fields at the center of particle (Eloc,Hloc). Fora single particle (located at the origin of the coordinate system) the inducedelectric and magnetic dipole moments can be given byp0 = ε0αeEloc, (A.1)m0 = µ0αmHloc, (A.2)where the electric polarizability coefficient αe and the magnetic polarizabil-ity coefficient αm are generally a function of frequency. Note that it hasbeen assumed that the particle does not have a bi-anisotropic response andthus magnetodielectic coupling terms, between p0 (m0) and Hloc (Eloc),have not been considered in the above equations. The local fields in absenceof the inclusion can be written asEloc = Earray + Eext = Cp0ε0+ Eext, (A.3)Hloc = Harray + Hext = Cm0µ0+ Hext, (A.4)where Earray and Harray are the fields produced by other dipoles and theEext and Hext are the applied external fields. In the above equations, Earrayand Harray are related to dipole moments p0 and m0 through the interactionconstant C, respectively. Note that the possible electric field produced bythe array of magnetic dipoles and magnetic field produced by the array ofelectric dipoles are ignored. Using Eqs. A.1-A.4 the relation between the95Appendices A. Clausius-Mossotti equationinduced dipole moments and the applied fields can be given byp0 =ε0Eextαe−1 − C , (A.5)m0 =µ0Hextαm−1 − C . (A.6)For a crystal with cubic lattice constant a, the averaged electric and mag-netic polarizations can be given byP = a−3p0 =a−3ε0Eextαe−1 − C , (A.7)M = a−3m0 =a−3µ0Hextαm−1 − C , (A.8)where a3 is the volume of each unit cell and thus a−3 is the number ofparticles per volume. Assuming that the space-averaged field E (H) is equalto the applied field Eext (Hext), the constitutive relations are given byD = ε0εrE = ε0E + P = ε0E(1 +a−3αe−1 − C ), (A.9)B = µ0µrH = µ0(H + M) = µ0H(1 +a−3αm−1 − C ), (A.10)and the relative permittivity and permeability of the medium can be givenbyεr = 1 +a−3αe−1 − C , (A.11)µr = 1 +a−3αm−1 − C . (A.12)The above assumptions are assumed to be valid under quasi-static conditionswhen fields have non-significant spatial variations inside a unit cell. Forfurther simplification it has also been assumed that all the induced dipoleshave the same phase and the produced fields by the dipoles have negligibletime delays. It has been shown that under these conditions the interactionconstant C is equal to 1/3a3 and thus constitutive parameters of the mediumcan be find by only knowing the polarizibility αe and αm for an individualconstitutive particle.96Appendices BEffective permittivity in thepresence of motion-freeelectronsHere, we describe the effective permittivity of a medium in the pres-ence of motion-free electrons [21]. Assume a conductive material which hasmotion-free electrons of mass m and charge e subject to an applied time-harmonic electric field E. Using the Newton’s equationeE = −m(ω2x¯+ iωγ), (B.1)the electron displacement x¯ is given byx¯ =−eEmω(ω + iγ), (B.2)where γ is the damping constant. By defining the dipole moment p0 = ex¯,the averaged electric polarization can also be given byP = Np0 =−Ne2Em(ω2 + γ2)+ iNγe2Emω(ω2 + γ2), (B.3)where N is the number of electrons per volume.The effective permittivity of the medium can be derived from the av-eraged electric polarization at this state; however, we first derive the cor-responding bound current of the medium in order to provide a more clearinsight to the response of electrons. Using Eq. 1.5 the bound currents of themedium can be given byJb = −iωP = Nγe2Em(ω2 + γ2)+ iNωe2Em(ω2 + γ2). (B.4)The first term of the induced current is in phase with the applied electricfield which demonstrates an Ohmic response (J = σE) originated from the97Appendices B. Effective permittivity in the presence of motion-free electronsdamping term of electron motions. The second term, however, has a 90degree phase deference which demonstrates that the induced current lag be-hind the applied electric field and thus electrons have an inductive response.This kinetic inductance originates from the time delay which is needed toaccelerate an electron with mass m subject to an applied electric force eE.Using the achieved electric polarization in Eq. B.3 and utilizing Eqs. 1.11and 1.13, the corresponding effective relative permittivity of the medium canalso be given byεr = 1 +Pε0E= (1− Ne2mε0(ω2 + γ2)) + iNγe2mε0ω(ω2 + γ2). (B.5)This permittivity has generally a complex value and is a function of fre-quency. For a loss less medium with vanishing damping factor (γ → 0) thepermittivity is purely real and given byεr = 1− ωpe2ω2, ωpe =Ne2mε0(B.6)At plasma frequency ωpe the permittivity of the medium is zero. Themedium has a negative permittivity up to that frequency and a positivepermittivity for higher frequencies (see Fig. 1.2). We can also define theoperating frequency ωo = ωpe/√2 as the frequency where εr = −1.98Appendices CLight focusing with a flatVeselago slabWe start this section by describing the refraction behaviour of a propa-gating plane wave at the single interface between air with ε1 =µ1 = 1 (firstmedium over z < 0) and the Veselago medium with ε2 =µ2 = − 1 (the sec-ond medium over z > 0) [63]. The response of these homogeneous mediaare independent of TM and TE polarization, however, we chose TM polar-ized waves with non-zero magnetic component Hy and electric componentsEx and Ez propagate in xz plane to illustrate the results. As shown inSec. 1.2, the bulk modes of each medium can be expressed by the magneticfield distributionH(kx, z) = b1eik1zz yˆ z < 0 (C.1)= b2eik2zz yˆ z > 0 (C.2)and the corresponding electric field distributionE(kx, z) =k1zb1ωε1eik1zzxˆ− kxb1ωε1eik1zz zˆ z < 0 (C.3)=k2zb2ωε2eik2zzxˆ− kxb2ωε2eik2z zˆ z > 0 (C.4)in whichk1z = −k2z =√ko2 − kx2. (C.5)The continuity of transverse magnetic field components at the interface im-pose that b1 should be equal to b2. The fact that k1z =−k2z and ε1 =−ε2 = 1demonstrate that the continuity of transverse electric field component Ex isalso satisfied at the interface and thus these bulk modes comprise the uniquesolution of the Maxwell’s equations.Let us consider propagating Fourier components of an object (with |kx|<ko)which are distributed over a source plane at z=−d/2 (see Eq. 1.35). Ac-cording to the above equations, each component with transverse wave vector99Appendices C. Light focusing with a flat Veselago slabkx would be carried in air (−d/2<z < 0) by the modeH(kx, z) = b1eik1zz yˆ, (C.6)and in the Veselago medium (z > 0) byH(kx, z) = b1e−ik1zz yˆ. (C.7)This fact demonstrates that the longitudinal phase change k1zd/2 in airwould be compensated after traveling a longitudinal distance d/2 in theVeselago medium. Thus, all propagating components of the object restorein the image plane at z = d/2 with a unity transfer function. This is cor-responding to negatively refracting all incident waves from air without anyreflections and imaging them at a distance d from the object. Accordingly, aflat slab of Veselago lens with thickness d can compensate the phase changesof bulk modes traveling longitudinal distance d in air and restore the objectover the image plane separated by distance 2d from the source plane.100Appendices DSurface bound modes of aVeselago half-sapceHere, we solve for the TM polarized resonant modes guided along thesingle interface between air (first medium over z < 0) and the Veselagomedium (the second medium over z > 0) [27]. As shown in Sec. 1.2, the bulkmodes of each medium can be expressed by the magnetic field distributionH(kx, z) = b1eα1zz yˆ, z < 0 (D.1)= b2e−α2zz yˆ, z > 0 (D.2)and the corresponding electric field distributionE(kx, z) =−iα1zb1ωε1eα1zzxˆ− kxb1ωε1eα1zz zˆ, z < 0 (D.3)=iα2zb2ωε2e−α2zzxˆ− kxb2ωε2e−α2zz zˆ, z > 0 (D.4)in whichα1z =√kx2 − k102, (D.5)α2z =√kx2 − k202. (D.6)The continuity of transverse magnetic field components at the interface im-pose that b1 should be equal to b2. In addition, by applying the continuity oftransverse electric field component Ex the resonance condition can be givenbyα1zα2z= −ε1ε2. (D.7)For air with εr = µr = 1 and Veselago medium which has εr = µr = −1at the operating frequency ωo the above condition would be satisfied forall values of kx over the evanescent spectrum (|kx|>ko)). The dispersiondiagram of guided modes which determines the relation of resonance fre-101Appendices D. Surface bound modes of a Veselago half-sapcequencies versus the wave vector kx will be a flat line at frequency ωo. Thegroup velocity of guided modes (∂ω/∂kx) is equal to zero which means thatthe forward traverse power flow in air has the same value as the backwardpower flow in the Veselago medium. The same results can also be derivedfor TE polarization by replacing ε and µ.It might be interesting to also show the resonant condition based on thewave impedance of two media. At the interface the impedance seen towardair and the Veselago medium (see Eqs. 1.39 and 1.40) are purely imaginaryand have equal but opposite sign values at frequency of ωo. This conditionis analogous to the resonance condition in a LC resonator.102Appendices ESlab bound modes ofVeselago lensHere, we solve for the TM polarized resonant modes guided along aninfinite Veselago slab with thickness d (second medium over−d/2<z <d/2))embedded in air (first medium over z <−d/2 and z >d/2) [9, 28]. Due tolongitudinal symmetry of the problem there are two solutions with evenand odd longitudinal symmetries for each transverse wave vector kx. Asshown in Sec. 1.2, the even solution can be expressed by the magnetic fielddistributionH(kx, z) = b1eα1z(z+d2)yˆ z < −d2(E.1)= b2 cosh(α2zz)yˆ −d2< z <d2(E.2)= b1e−α1z(z− d2 )yˆ z >d2(E.3)and corresponding electric field distributionE(kx, z) =−iα1zb1ωε1eα1z(z+d2)xˆ− kxb1ωε1eα1z(z+d2)zˆ z < −d2(E.4)=−iα2zb2ωε2sinh(α2zz)xˆ− kxb2ωε2cosh(α2zz)zˆ −d2< z <d2(E.5)=iα1zb1ωε1e−α1z(z−d2)xˆ− kxb1ωε1e−α1z(z−d2)zˆ z >d2(E.6)in whichα1z =√kx2 − k102, (E.7)α2z =√kx2 − k202. (E.8)103Appendices E. Slab bound modes of Veselago lensApplying the continuity of transverse electromagnetic fields at the interfacesof the slab results in the amplitude relationb2 =b1cosh(α2zd2 ), (E.9)and the resonant condition (dispersion relation) for the even solution givenbycoth(α2zd2) = −α2zε1α1zε2, (E.10)ore−α2zd =α2zε1 + α1zε2α2zε1 − α1zε2 . (E.11)We now consider an odd solution which can be expressed by the magneticfield distributionH(kx, z) = −b1eα1z(z+ d2 )yˆ z < −d2(E.12)= b2 sinh(α2zz)yˆ −d2< z <d2(E.13)= b1e−α1z(z− d2 )yˆ z >d2(E.14)and corresponding electric field distributionE(kx, z) =iα1zb1ωε1eα1z(z+d2)xˆ+kxb1ωε1eα1z(z+d2)zˆ z < −d2(E.15)=−iα2zb2ωε2cosh(α2zz)xˆ− kxb2ωε2sinh(α2zz)zˆ −d2< z <d2(E.16)=iα1zb1ωε1e−α1z(z−d2)xˆ− kxb1ωε1e−α1z(z−d2)zˆ z >d2(E.17)Applying the boundary conditions results in the amplitude relationb2 =b1sinh(α2zd2 ), (E.18)and resonant condition for odd solutions given bytanh(α2zd2) = −α2zε1α1zε2, (E.19)104Appendices E. Slab bound modes of Veselago lensore−α2zd = −α2zε1 + α1zε2α2zε1 − α1zε2 . (E.20)It might be interesting to find the above even and odd resonant condi-tions based on the wave impedance of media. At the interface z=−d/2 theadmittance toward the right direction YR can be achieved for even and oddsolutions by considering a PEC and a perfect magnetic conductor (PMC)at the center of the lens (z = 0), respectively. This will results in the evenwave admittance YRo and odd wave admittance YRe given byYRe =ωε2iα2zcothα2zd2, (E.21)YRo =ωε2iα2ztanhα2zd2. (E.22)There is air at the left side of the interface at z=−d/2 and thus accordingto Eq. 1.40 the wave admittance of this bulk medium can be given byYL =ωε1iα1z. (E.23)Applying resonant conditions YRe = −YL and YRo = −YL results in thesimilar dispersion equations of E.10 and E.19 for even and odd solutions,respectively.It may also be interesting to mention that an evanescent wave withinthe slab after passing the longitudinal distance 2d (decaying by e−2α2zd)and reflecting two times from the interfaces by reflection coefficientRTM =ε1k2z − ε2k1zε1k2z + ε2k1z=ε1α2z − ε2α1zε1α2z + ε2α1z(E.24)to return to its original location has an unchanged amplitude and phase ateven and odd resonances. This means the dispersion equations of E.10 andE.19 satisfies the condition to have guided modes along the slab.105Appendices FThe response of Veselagolens to incident evanescentwavesHere, we solve for the response of an infinite Veselago slab with thick-ness d (second medium over |z|<d/2) embedded in air (first medium over|z|>d/2) to an incident TM polarized evanescent wave. Due to the longitu-dinal symmetry, this problem can be solved as a superposition of even andodd solution for each transverse wave vector kx [9]. For the even solutionwe can solve the problem for two symmetric evanescent waves incident fromthe right and left sides of the lens (see Fig. 1.12). As shown in Sec. 1.2, aneven solution can be expressed by the magnetic field distributionHey =c12e−α1z(z+d) + be1eα1z(z+d2), z < −d2(F.1)= be2 cosh(α2zz), −d2< z <d2(F.2)=c12eα1z(z−d) + be1e−α1z(z− d2 ), z >d2(F.3)and its corresponding transverse electric field distribution−iωEex =α1zc12ε1e−α1z(z+d) − α1zbe1ε1eα1z(z+d2), z < −d2(F.4)=−α2zbe2ε2sinh(α2zz), −d2< z <d2(F.5)= −α1zc12ε1eα1z(z−d) +α1zbe1ε1e−α1z(z−d2). z >d2(F.6)106Appendices F. The response of Veselago lens to incident evanescent wavesApplying the continuity of transverse fields at the interfaces of the slabresults in the amplitude relations given bybe1 =c12α1zε2 cosh(α2zd/2)− α2zε1 sinh(α2zd/2)α1zε2 cosh(α2zd/2) + α2zε1 sinh(α2zd/2)e−α1zd/2, (F.7)be2 =c122α1zε2e−α1zd/2α1zε2 cosh(α2zd/2) + α2zε1 sinh(α2zd/2). (F.8)The odd solution can also be achieved for two anti-symmetric evanescentwaves incident from the right and left side of the lens (see Fig. 1.12). Ina similar procedure, the odd solution can be given by the magnetic fielddistributionHoy =c12e−α1z(z+d) + bo1eα1z(z+d2) z < −d2(F.9)= −bo2 sinh(α2zz) −d2< z <d2(F.10)= −c12eα1z(z−d) − bo1e−α1z(z−d2) z >d2(F.11)and the electric field distribution−iωEox =α1zc12ε1e−α1z(z+d) − α1zbo1ε1eα1z(z+d2) z < −d2(F.12)=α2zbo2ε2cosh(α2zz) −d2< z <d2(F.13)=α1zc12ε1eα1z(z−d) − α1zbo1ε1e−α1z(z−d2) z >d2(F.14)in whichbo1 =c12α1zε2 sinh(α2zd/2)− α2zε1 cosh(α2zd/2)α1zε2 sinh(α2zd/2) + α2zε1 cosh(α2zd/2)e−α1zd/2 (F.15)bo2 =c122α1zε2e−α1zd/2α1zε2 sinh(α2zd/2) + α2zε1 cosh(α2zd/2)(F.16)Note that the solutions for the dispersion equations of even modes (Eq. E.10)and odd modes (Eq. E.19) corresponding to the poles of even solution(Eq. F.7) and odd solution (Eq. F.15), respectively. This means that excitingthe lens at its resonances results in an infinite transmission and reflection.It can be shown that for air with εr = µr = 1 and Veselago medium which107Appendices F. The response of Veselago lens to incident evanescent waveshas εr = µr = −1 at the operating frequency ωo the amplitude relationsbo1 = −be1, (F.17)bo2 = −be2 = −c1. (F.18)exist for all values of kx and the superposition of even and odd solutionsresults in the evanescent wave growth within the lens (with the magneticfield distribution expressed in Eq. 1.44).108Appendices GBloch modes finite elementsimulation at a fixed realfrequencyHere we describe how to solve the Bloch modes of a periodic structureat a fixed real frequency using a finite element method (FEM) solver [64].This method is particularly useful for calculating and for analyzing periodstructures constructed of frequency dispersive materials.Let us solve the wave equation for magnetic field which is given by∇× ( 1εr∇×H)− µrω2c2H = 0. (G.1)The Floquet-Bloch theorem states that time-harmonic magnetic field ofBloch modes in a periodic structure has the form ofH(r) = uh(r)eikFB·r, (G.2)where uh(r) is periodic vector function and kFB is the Floquet-Bloch wavevector. Substitution of Eq. G.2 in Eq. G.1 and using the identities∇× (QA) = ∇Q×A +Q∇×A, (G.3)A× (B×C) = B(A ·C)−C(A ·B), (G.4)results in the eigenvalue equation∇× ( 1εr∇× uh) + ikFB × ( 1εr∇× uh) + i∇× ( 1εrkFB × uh)+kFB · kFBεruh − kFBεr(kFB · uh)− µrω2c2uh = 0. (G.5)The electric field of Bloch modes can also be calculated from magnetic field109Appendices G. Bloch modes finite element simulation at a fixed real frequencybyE(r) =1−iωε∇×H =1−iωε(∇× uh + ikFB × uh)eikFB·r. (G.6)Now we derive the weak form expression of equation G.5 in order tosolve this eigenvalue equation by the finite element method. We start bymultiplying the leftside of Eq. G.5 with a test function u˜h and use theidentity∇ · (A×B) = B · (∇×A)−A · (∇×B) (G.7)which results ink2FBεru˜h · uh − 1εr(kFB · u˜h)(kFB · u) + i 1εru˜h ·[kFB × (∇× uh)]− i(∇× u˜h) · 1εr(kFB × uh)− (∇× u˜h) · 1εr(∇× uh)− µrω2c2u˜h · uh+ i∇ · [u˜h × 1εr(kFB × uh)]+∇ · [u˜h × 1εr(∇× uh)]. (G.8)By excluding the last two divergence terms from the above expression, thecorresponding functional for the Bloch modes or the weak form expressioncan be given byFH(u˜h,uh) =k2FBεru˜h · uh − 1εr(kFB · u˜h)(kFB · u) + i 1εru˜h ·[kFB × (∇× uh)]− i(∇× u˜h) · 1εr(kFB × uh)− (∇× u˜h) · 1εr(∇× uh)− µrω2c2u˜h · uh. (G.9)The Bloch modes can be solved by enforcing the integral of the weak formequation over the unit cell to zero. In fact, this integral is equal to0 =∫∫∫VdV FH(u˜h,uh)=∫∫∫VdV u˜h ·[∇× ( 1εr∇× uh) + ikFB × ( 1εr∇× uh)+ i∇× ( 1εrkFB × uh) + kFB · kFBεruh − kFBεr(kFB · uh)− µrω2c2uh]110Appendices G. Bloch modes finite element simulation at a fixed real frequency+∮SdS u˜h ·[nˆ× 1εr(∇× uh − ikFB × uh)], (G.10)in which V is the volume of unit cell, S is the surface of the unit cell, and nˆis the unit normal vector of the surface. Note that the divergence theoremand the vector identityA · (B×C) = B · (C×A) (G.11)have been applied in deriving the above equation. Equation G.10 demon-strates that weak form solutions satisfy the wave equation G.5 over the unitcell and a natural boundary condition over the surface of the unit cell. Ac-cording to Eq. G.6, the surface integral in Eq. G.10 enforces the tangentialelectric field (nˆ×E) to be zero over the surface of the unit cell which is knownas perfect electric boundary condition. This will be the default boundarycondition in the absence of any other constraints over the boundary.The weak form equation can be solved by implementing equation G.9in COMSOL Multiphysics as a FEM solver. Note that uh is approximatedwith Lagrange interpolation elements for a two-dimensional problem andwith curl interpolation elements for a three-dimensional problem. The weakform for electric fields can also be expressed from equation G.9 by replacinguh with ue and swapping εr and µr. In this case perfect magnetic boundarycondition will be the default boundary condition in the absence of any otherconstraints over the boundary.111
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Investigating spatial-frequency power flow and evanescent-wave amplification in metamaterial superlenses
Text